Introduction to Econometrics 1

Multiple regression

In the previous section, we have covered a regression with one explanatory variable. In this section, we will cover the regression model with more than one explanatory variables. We will also cover the analysis of variance in terms of the Chow test. We will also cover the adjusted R2 and how it affects the inclusion or exclusion of independent variables. The R2 is affected by adding independent variables. We will also cover the ANOVA table single factor or analysis of variance. It is used to show the decomposition of the variance explained by the independent variables and the error term in the regression equation.

Let’s take as an example a financial model related to the returns of a share price in relation to market returns and price earnings ratio expressed in percentages.

The dependent variable is share price returns expressed in percentage and the independent variables are market price returns and price earnings ratio expressed in percentages.

The mathematical equation will be as follows:

yi = Where: yi = share price returns. It is the dependent variable.

1

A numerical example will be very helpful to understand the calculations that are involved in a multiple regression analysis with two independent variables. In this example, we use the difference of the numerical values from their means displayed in the following columns . Please consider the example that we have covered in simple regression section by adding an additional independent variable. The dependent variable is annual profits and the independent variables are expenditures for R&D and marketing expenses.

Year Annual profits(£000) (y)

Expenditures for R&D(£000) (X1)

Marketing expenses(£000) (X2)

2003 31 5 42002 40 11 102001 30 4 82000 34 5 71999 25 3 51998 20 2 3

A convenient way to solve the multiple regression problem is to construct the following Table. Please sum each variable. Then, find the mean of the dependent and independent variables. Then, subtract from each value from its mean.

Year(n=6)

Annual profits(£000) (y)

Expenditures for R&D(£000) (X1)

MarketingExpenses (£000) (X2)

y x1 x2

2003 31 5 4 0 0 -2.1666672002 40 11 10 10 6 3.8333332001 30 4 8 -1 -1 1.8333332000 34 5 7 3 0 0.8333331999 25 3 5 -6 -2 -1.1666671998 20 2 3 -11 -3 -3.166667

∑y = 180 ∑x1 = 30

x1y x2y x1x2

2

0 -2.16667 0 0 4.6960 38.33333 23 36 14.69

0 0 -1.83333 1 3.360 3.333333 0 0 0.69

10 5.833333 2.333333 4 1.3630 31.66667 9.5 9 10.03

∑x1y =100 77 33 ∑ = 50

Please calculate , 1 and 2 . I have included the required equations.

First of all, we calculate b1 and b2.

443.1652941ˆ

1

1089174133003850ˆ

2

3

844.0652550ˆ

2

Then, we substitute the numerical values of 1 and b2 in the following equation to find the intercept.

= 30 – ( x 5) – ( x 6.166667) = 30 – 7.22 – 5.20 = 17.58.

The regression equation may now be written as:

I have included the Excel output that you will get by running a multiple regression. Please use the Data analysis pack in Tools. Then, select regression. In the Y range, input the numerical values of the dependent variable. In the X range, input the numerical values of both independent variables. Select labels and confidence levels. Then, select the cell that your output will be displayed. Select by clicking on the residuals and residuals plots. This will show the error term of your multiple regressions as the difference between actual and predicted values.

4

SUMMARY OUTPUT

Regression StatisticsMultiple R 0.929919713R Square 0.864750673Adjusted R Square 0.774584455Standard Error 3.303045922Observations 6

ANOVA

df SS MS FSignificance

FRegression 2 209.2697 104.6348 9.590628 0.049739643Residual 3 32.73034 10.91011Total 5 242

Confidence Intervals

CoefficientsStandard

Error t Stat P-value Lower 95% Upper 95%Lower 95.0%

Upper 95.0%

Intercept 17.58426966 3.760573 4.675954 0.018476 5.616435321 29.552104 5.616435 29.5521

x1 1.443820225 0.763074 1.892111 0.154831-

0.984622726 3.8722632 -0.98462 3.872263

x2 0.842696629 0.914227 0.921759 0.424636-

2.066783462 3.7521767 -2.06678 3.752177

RESIDUAL OUTPUT

Observation Predicted Y Residuals1 28.1741573 2.8258432 41.89325843 -1.893263 30.1011236 -0.101124 30.70224719 3.2977535 26.12921348 -1.129216 23 -3

5

We will solve the above example by using equations related to the sum of squares of deviations of the variables from their means.The dependent variable is annual profits and the independent variables are expenditures for R&D and marketing expenses. It is required to calculate the regression equation, the residual sum of squares, the coefficient of multiple determination and the correlation coefficient. In addition, calculate the standard errors and the 95% confidence interval of the intercept and the coefficients . Finally, calculate the t-statistics for the intercept and the coefficients.

Year Annual profits(£000) (y)

Expenditures for R&D(£000) (x1)

Marketing expenses(£000) (x2)

2003 31 5 42002 40 11 102001 30 4 82000 34 5 71999 25 3 51998 20 2 3

The equations related to the sum of squares of deviations of the variables from their means are as follows:

The equations for , 1 and 2 are as follows:

6

The first step is to construct the Table wit the relevant calculations to facilitate you to calculate the coefficients , and the intercept .

Year(n=6)



MarketingExpenses (£000) (x2)

2003 31 5 4 0 0 -2.1666672002 40 11 10 10 6 3.8333332001 30 4 8 -1 -1 1.8333332000 34 5 7 3 0 0.8333331999 25 3 5 -6 -2 -1.1666671998 20 2 3 -11 -3 -3.166667

∑y = 180 ∑x1 = 30

y2 x12 x2

2 x1x2 x1y x2y961 25 16 20 155 124

1600 121 100 110 440 400900 16 64 32 120 240

1156 25 49 35 170 238625 9 25 15 75 125400 4 9 6 40 60

5642 200 26321

8100

0118

7

We calculate first 1 and 2 .

The equations related to the sum of squares of deviations of the variables from their means are as follows:

7


Finally, the intercept will be calculated as follows:

Thus, the regression equation is:

The residual sum of squares is calculated from the following equation:

The coefficient of multiple determination is calculated from the following equation:

The correlation coefficient is as follows:

(to 4.d.p.).

8

The mathematical formulas of the estimated variance of the error term as estimation of , the variances, standard errors and the covariance of the coeficients and the intercept are as follows:

Once we know the coeficients and the standard errors, then, we can calculate the confidence intervals. The sample size is 6 and the degrees of freedom are n-3 or 6-3 = 3 egrees of freedom. From the t-distribution and with 5% significance level the value is 3.182.

The confidence intervals with 95% confidence level for the intercept and the coefficients are as follows:

9

The t-statistics of the intercept and the coefficients are calculated as follows:

t-statistic(

t-statistic

Please complete the following calculation to become familiar with the t-statistic.

t-statistic =

I have attached the Excel output of the multiple regression equation. Please check the calculations of the R square, the residual, the intercept and the coefficients.

SUMMARY OUTPUT

10


ANOVA






Upper 95.0%

Intercept 17.58426966 3.760573 4.675954 0.018476 5.616435321 29.552104 5.616435 29.5521

x1 1.443820225 0.763074 1.892111 0.154831-

0.984622726 3.8722632 -0.98462 3.872263

x2 0.842696629 0.914227 0.921759 0.424636-

2.066783462 3.7521767 -2.06678 3.752177

I have attached another example with the following dataset :

Share Market PE3.526787 8.73209 0.499819922-4.34533 -5.19815 -4.21807425.222709 6.21865 0.877518989-4.99619 -5.5393 -0.745571156

11

-3.04336 7.69808 -0.317399673-2.375422 -4.99735 2.0611907912.651303 5.42777 2.629939417-0.68924 -1.5424 -0.8809008470.205664 1.4639 0.2530117572.4783 3.6528 -0.920953642

0.237407 -0.1494 -1.1487126780.329728 0.16688 0.372533199-0.26869 -0.1444 -1.7140759240.064769 0.097873 4.630805383-0.5873 -0.09911 -2.530835538

0.329225 -0.08344 -1.038471775-0.11849 0.122767 0.5309048290.011541 -0.45767 -1.857415164-0.18757 -0.53046 1.227869802-0.38752 -0.11118 0.642577585-0.26835 -0.28947 1.8390492010.262798 -0.17676 -0.954329890.355054 -1.15686 -2.503369578-1.34302 -0.5771 0.877111656-0.77964 0.578182 -1.55602547-0.04649 -0.05331 0.0405596920.098381 -0.23054 -0.522472998-0.09585 -0.66625 2.93017109-0.0059 -0.50071 -0.497160825-0.05415 -0.53128 -1.885502172

The stochastic equation in Econometrics will incorporate the error term and it will be as follows:

yi= +

I will include the steps on how to perform a multiple regression in Excel.

1. Plot or insert the data in Excel.2. Press tools…..then select data analysis……then select regression…..3. Then in input Y range select the label and the data of the dependent variable.

In our case, it is the share price returns. 4. Then, in input X range select the labels and the data of the two independent

variables. In our case, it is the market price returns and the price earnings ratio.

5. Select the box labels and confidence levels. Select the required confidence levels. For example, 90%, 95% or 99%.

6. Press output range box and select the cell that your output will be displayed.7. Finally, select the box residuals in order that the residuals table and data will

be displayed.

12

The regression Excel output will be as follows:

SUMMARY OUTPUT


13

ANOVA


FRegression 2 55.88963329 27.94481665 12.70789 0.000129025Residual 27 59.37335129 2.199013011Total 29 115.2629846

Coefficients Standard Error t Stat P-value Lower 95%Upper 95%

Intercept -0.260766997 0.273942189 -0.951905212 0.349589 -0.82284956 0.3013156Market returns 0.405779907 0.087380092 4.643848503 7.93E-05 0.226490889 0.5850689P/E ratio 0.131419379 0.154479732 0.850722473 0.402405 -0.18554664 0.4483854

The intercept and the coefficients of the variables in relation to the t - statistics are as follows:

(-0.95) (4.64) (0.85)

The residual will be as follows:

RESIDUAL OUTPUT

ObservationPredicted Share

returns Residuals1 3.348225697 0.1785613032 -2.924408515 -1.4209214853 2.377959224 2.8447497764 -2.606486136 -2.3897038645 2.821246723 -5.864606723

14

6 -2.017710802 -0.3577111987 2.287338016 0.3639649848 -1.002409369 0.3131693699 0.366504857 -0.160840857

10 1.100434692 1.37786530811 -0.472353622 0.70976062212 -0.144092364 0.47382036413 -0.54462441 0.2759344114 0.387525469 -0.32275646915 -0.633584679 0.04628467916 -0.431100589 0.76032558917 -0.141179432 0.02268943218 -0.690580635 0.70212163519 -0.314651119 0.12708111920 -0.22143446 -0.1660855421 -0.136541402 -0.13180859822 -0.457910095 0.72070809523 -1.059188817 1.41424281724 -0.379673112 -0.96334688825 -0.23064426 -0.5489957426 -0.277068794 0.23057879427 -0.422978574 0.52135957428 -0.146036594 0.05018659429 -0.529281622 0.52338162230 -0.724141271 0.669991271

The multiple regression output in EViews 6

Before loading and transferring your data in EViews 6, you have to insert the numerical values or the data in Excel. Do not use long titles for each time – series. Use abbreviations. For example, share for the first time – series, market for the second time – series and PE, price earnings ratio for the third time - series. Name the sheet of the Excel file and delete the other sheets. For example, the name of the sheet is reg2. Once the Excel file is ready, you close it and you open the statistical package EViews 6. You press file and then you select new worksheet. Then, in workfile structure type, select unstructured / undated. Insert the number of observations. In our case, the number is 30. You will get and untitled worksheet. Then press file, then, import, then, read text - lotus – excel. Select the Excel file and press OK. Excel spreadsheet import

15

screen will open. In the box, upper left data cell, select A2. This is the cell that your first observation starts. In Excel 5 + sheet name write reg2. This is the name of your filename. In the box name of series, please write share then press space bar and then market and then PE. These are the names of the variables.

Once you done these steps, you will be able to see the time series transferred from Excel to EViews 6. Press file, then, save as. Then, write the filename reg2 and save it with the extension wf.

You click on the time series for example share to do the following tests. You press view and then you select descriptive statistics and tests……… histogram and stats or correlogram or unit root test. The same thing you do for the file market and PE. You open it and then you press view.

To run a regression, you press quick, and then, estimate equation. In the specification box, you write:

share c market PE and you press OK.

Then, your output will be displayed. From the top menu you have options to do forecast and check the residuals or error term for further tests.

The dependent variable is share price returns and the independent variables are market price returns and

Dependent Variable: SHAREMethod: Least SquaresDate: 02/08/15 Time: 11:12Sample: 1 30Included observations: 30

Variable Coefficient Std. Error t-Statistic Prob.

C -0.260767 0.273942 -0.951905 0.3496MARKET 0.405780 0.087380 4.643849 0.0001

PE 0.131419 0.154480 0.850722 0.4024

R-squared 0.484888 Mean dependent var -0.127295Adjusted R-squared 0.446732 S.D. dependent var 1.993636

16

S.E. of regression 1.482907 Akaike info criterion 3.720525Sum squared resid 59.37335 Schwarz criterion 3.860645Log likelihood -52.80788 Hannan-Quinn criter. 3.765351F-statistic 12.70789 Durbin-Watson stat 1.765635Prob(F-statistic) 0.000129

The intercept and the coefficients of the variables in relation to the t - statistics are as follows:

(-0.95) (4.64) (0.85)

There are five basic assumptions of the linear and multiple regression models to be valid. The coefficients have to be BLUE. They must have the smallest possible variance and be consistent to achieve best fit regression analysis. The multiple regression model that we will cover is as follows:

Our aim is to show that the coefficients are unbiased and have minimum variance in the multiple regression equation. In addition, the explanatory variables are not collinear to satisfy the fifth assumption, namely, that the independent variables are uncorrelated with the error term, the dependent variable and between them. For example, if advertising and revenues are higly correlated, it will be difficult to separate the effects of the variables under study. The coefficients have to be BLUE.

The acronyms of BLUE stands for the following words:

B: BestL: LinearU: UnbiasedE: Estimator

We would like that the coefficients are BLUE by comparing the hypothesis testing of the F-statistic. Please check the regression example that we covered to understand the relation of the F-statistic with the coefficients.

The first assumption is that the error term is normally distributed.

17

The second assumption is that the mean of the error term is zero.

The third assumption is that the variance of the error term is the same in each time period for all values of the independent variables. This is a homoskedastic situation.

The fourth assumption is that the numerical value of the error term is uncorrelated with its value in other time period.

The fifth assumption is that the independent variables are uncorrelated with the error term, the dependent variable and between them.

If the above assumptions are violated, then, you should consider diagnosing the problems related to multicollinearity, heteroskedasticity, autocorrelation, and errors in variables.

Thus, the assumptions related to the error term as follows:

E( )=0, cov(x1, , cov(x2,

When expressing the residual or error term in terms of the multiple regression equation, we have the following equation:

Multicollinearity

Multicollinearity is a violation of the fifth assumption. It is a case when the independent variables are highly correlated with the dependent variables. The estimated coefficients may be statistically insignificant although R2 is very significant. In addition, the standard errors could be very high or the t-ratios very low. The confidence intervals for the parameters of interest are thus very wide. When the explanatory variables are highly intercorrelated, it becomes difficult to disentangle the separate effects of each of the explanatory variables on the explained variable. It can be overcome or reduced by collecting more data or dropping one of the highly collinear variables.

Heteroskedasticity

Heteroskedasticity is a violation of the third assumption. The assumption is that the variance of the error term is the same in each time period for all values of the independent variables. Once that we have heteroskedasticity, then, the coefficients are not BLUE. We have biased estimates and larger variance in our data set. The problem is detected by observing the high values of the residuals in relation to the estimated data. A possible solution to this problem is to use a log function for both the dependent and independent variables. In the Econometrics book that I will prepare in the future, I will explain in detail through different examples how to perform the tests.Other heteroskedasticity tests are the ARCH test, Glejser’s test, Breusch and Pagan’s test, White’s test and detection of the residuals in the likelihood ratio test.

18

Autocorrelation

One of the assumptions of multiple regressions is that the value which the error term assumes in one period is uncorrelated to its value in any other time period. This ensures that the average value of the dependent variable depends only on the independent variable and not on the error term. The Durbin-Watson statistic tests for first-order autocorrelation. If there is autocorrelation, it leads to biased standard errors and thus to incorrect statistical tests. Autocorrelation is tested by using the test of the Durbin-Watson statistic.

Errors in variables

It refers to measurement errors related to the independent variables. For example, the data set is not consistent and there are gaps in the measurement history. A solution to this problem is to replace the independent variable with another one that is not correlated with the error term.

Exercise

You are given the following data for a sample size of n = 28.

It is required to calculate the following:

19

a) Calculate the intercept

In addition, please calculate their standard errors. b) Estimate the regression equation.

Solution


Thus, the regression equation is as follows:

Multiple regression with more than 2 independent variables

I have attached an example of 30 observations. It is related to a multiple regression with 3 independent variables. The dependent variable is the discount and the independent variables are the the market effect, the size effect, and the book to market effect. The data were gathered from datastream. It is required to calculate the multiple regression equation.

The multiple regression has the following format:

where: yi is the discount. is the intercept. is the market effect. is the size effect.

20

is the book-to-market effect.

dis markt s bm-2.36697 -9.12939 -1.70722 1.927814-0.51194 0.577933 0.669331 2.8669240.254743 2.162936 0.020789 0.225917-2.48828 -4.14315 -2.2856 -4.685693.491507 12.83122 -2.13122 3.176416-0.63164 -1.75343 0.887992 -0.751224.183572 2.886911 -1.57409 -0.188150.316764 -8.61904 -1.03613 2.5486413.375268 1.049722 -4.61812 2.9391050.938469 -1.5605 -1.5808 -2.33380.071534 5.700149 1.721926 4.103659.627071 10.81373 1.635702 0.7578495.921342 10.37399 1.888181 2.964885-0.85242 6.860824 0.805653 -6.209395.53793 5.806177 3.93546 -0.37606

2.480958 6.140132 -2.02608 -0.26034-0.54454 1.165931 1.246596 -0.059190.565246 -2.2651 -1.97628 -0.169530.608492 2.957873 2.345294 1.576884.826277 6.006215 1.126188 1.1268593.465707 2.439138 1.841884 -1.545847.663349 7.959142 2.43412 -0.045667.751392 5.265374 -1.00022 2.38889112.2131 21.38698 0.325796 -1.36015

2.025977 -1.55935 5.17537 1.816525-4.81799 -4.10934 0.590867 -0.737670.505699 -1.50866 -1.73348 0.08419.726398 11.7856 -4.2814 1.041869-3.51059 -2.34779 -0.2577 0.463281

Solution

By plotting the data in Excel and using the data analysis toolbox, you will get the following output.

SUMMARY OUTPUT

Regression StatisticsMultiple R 0.273516R Square 0.074811

Standard Error 0.498851Observations 29

ANOVA


F

21

Regression 3 0.503055 0.167685 0.673834 0.576168Residual 25 6.221306 0.248852Total 28 6.724362


Error t Stat P-value Lower 95%Upper 95%

Intercept -0.18839 0.103061 -1.82797 0.079513 -0.40065 0.023866markt 0.018152 0.01424 1.274672 0.214146 -0.01118 0.04748s 0.020127 0.042147 0.477546 0.637122 -0.06668 0.106931bm 0.000836 0.041498 0.020146 0.984087 -0.08463 0.086304

t-statistics (-1.83) (1.27) (0.48) (0.02)

Exercise

You are required to calculate the confidence interval of the prediction with 95% confidence level of the multiple regression equation. The sample size is 6. Please consider the following regression:

You are given the following data:

The predicted

22

The standard error of the prediction is 0.245 and the degrees of freedom are n-3 or 6-3 = 3. From the table of the t-distribution the value is 3.182 for the 5% significance level.

The 5% significance level of the prediction is as follows:

Please complete the calculations for the upper and lower confidence level.

F-statistic calculation based on the multiple regression

It is very important to understand the analysis of variance based on the coefficients of the multiple regression. It is required to calculate the F-statistic based on the following sample size which consists of 6 observations.

Year(n=6)



MarketingExpenses (£000) (x2)

2003 31 5 4 0 0 -2.1666672002 40 11 10 10 6 3.8333332001 30 4 8 -1 -1 1.8333332000 34 5 7 3 0 0.833333

23

1999 25 3 5 -6 -2 -1.1666671998 20 2 3 -11 -3 -3.166667

∑y = 180 ∑x1 = 30

y2 x12 x2

2 x1x2 x1y x2y961 25 16 20 155 124

1600 121 100 110 440 400900 16 64 32 120 240

1156 25 49 35 170 238625 9 25 15 75 125400 4 9 6 40 60

5642 200 26321

8100

0118

7

The mathematical formula for the F-statistic is as follows:

I have attached the Excel output with the F-statistic calculation.

SUMMARY OUTPUT


ANOVA


FRegression 2 209.2697 104.6348 9.590628 0.049739643Residual 3 32.73034 10.91011

24

Total 5 242 Confidence Intervals



Upper 95.0%

Intercept 17.58426966 3.760573 4.675954 0.018476 5.616435321 29.552104 5.616435 29.5521

x1 1.443820225 0.763074 1.892111 0.154831-

0.984622726 3.8722632 -0.98462 3.872263

x2 0.842696629 0.914227 0.921759 0.424636-

2.066783462 3.7521767 -2.06678 3.752177

Analysis of variance or Chow test

The predictive test for stability is based on the fact that the prediction errors have zero mean. You nedd two datasets to estimate this test. The first sample size n1 is used to estimate the regreesion equation and then used to get the predictions for the second sample size n2. The F-test mathematical equation that is used in this test is as follows:

Where: SRSS is the total of the sum residual of squares from both regressions. RSS1 is the residual sum of squares of the first sample. RSS2 is the residual sum of squares of the second sample. n1 is the sample size of the first sample. n2 is the sample size of the second sample. The degrees of freedom are n2 and (n1-k-1).

25

Example

You are given the following data for different time periods. The sample sizes are n1= 20 and n2 = (20-2-1) = 17. Pease calculate the F-test and compare it with the F-value from the F-tables. Based on the result accept ot reject the hypothesis of stability.

From 2000 – to 2005: RSS1 = 0.1141From 2007 - to 2014: RSS2 = 0.0432 RSS from both regressions = 0.3761

Solution

The hypotheses are as follows:

H0: There is stability and the variances are constant.H1: There is no stability and the variances are not constant.

Plug the numbers in the above F-test equation.

Then, please look at the end of the book for the F-tables to find the F-value for 5% significance level. The degrees of freedom are 20 and 17. 20 referes to the vertical column and 17 refers to the horizontal column. The F-value is 2.23. Therefore, at the 5% significance level we cannot reject the hypothesis H0 of stability.

I have attached the stability Chow Forecast test in EViews 6 for different datasets. Let me explain how you do it. You run first the regressions for the different samples. You can check at the bottom of the test the RSS1 and RSS2 or the sum squared residuals for different regressions. Then, you click on view and you select stability tests. Please focus on the first two lines of the test that are bold. Check the F-statistic with the related p-values. Check if it is significant or not and acccordingly accept or reject the stability effect. Please remember that the test is applicable for diffierent time periods to show the stability of the variances in different time span. In this case the F-statistic is not significant at the 5 % significance value and the sample evidence suggests to reject the null hypothesis of stability.

Chow Forecast Test: Forecast from 30 to 30

F-statistic 0.095234 Prob. F(1,27) 0.7600Log likelihood ratio 0.105632 Prob. Chi-Square(1) 0.7452

Test Equation:Dependent Variable: SHAREMethod: Least Squares

26

Date: 04/24/16 Time: 11:56Sample: 1 29Included observations: 29


C -0.300281 0.280689 -1.069799 0.2942MARKET 0.424144 0.086175 4.921896 0.0000

R-squared 0.472914 Mean dependent var -0.129817Adjusted R-squared 0.453393 S.D. dependent var 2.028876S.E. of regression 1.500007 Akaike info criterion 3.715289Sum squared resid 60.75056 Schwarz criterion 3.809585Log likelihood -51.87168 Hannan-Quinn criter. 3.744821F-statistic 24.22506 Durbin-Watson stat 1.874055Prob(F-statistic) 0.000038

Exercise of Chow breakpoint dates

Please consider the following dataset. The dependent variable is fund returns and the independent variables are size, mrket returns, and book to market returns. We want to apply Chow breakpoint dates to see the stability effect in different time periods or observations.

Fund return Size Market returnBook to market return

8.826227396 9.960064 6.770160683 -0.9656252914.3672068 5.355746 2.101215141 0.414592015

-3.744606009 -0.83212 -2.256078026 -0.4427797463.653337534 0.455159 3.415008092 0.489551486

-2.953694894 2.587989 -0.948914314 -1.8014539-2.422063841 -5.26719 -1.376494263 -0.724588286-2.739912699 -2.81245 -2.109216247 0.033230007-9.315254996 0.08267 -7.732750354 -0.682711797-14.94682585 1.892262 -15.8749113 -1.064875722-6.339207193 -5.17364 -5.148485371 -1.253981349

27

0.642977938 1.008631 0.24888198 0.937226066-3.606672395 -1.95412 -3.445206217 0.227324759-10.95272614 0.025582 -11.00011018 0.5205531780.400824294 -1.79843 0.244664806 0.6193559031.869571463 -2.02135 3.38517113 0.241543165

-3.897889112 0.929765 -0.924758847 0.07628458-11.29111047 1.537612 -8.775426604 -1.60504611-6.661286528 -1.18146 -7.103029456 2.763327169-4.448910028 0.796062 2.350319644 -0.904642691-6.364596029 0.908112 2.623642306 0.922935303

0.67082299 0.109951 -1.092979559 0.969071134-0.306823529 -1.66881 -0.566832617 1.797172793-4.701484083 -3.97912 -5.525479768 -2.189439024-7.227822193 -0.54654 -6.09870943 0.572655503-6.819388756 1.794597 -7.128801365 0.5947627363.798680926 2.22648 1.900347699 -0.241326231

-6.407574151 2.90064 -4.765574598 -0.100919561-6.800821645 4.582849 2.256769855 1.151969143

0.78018847 1.950273 -3.014224796 0.1502333773.281922722 -1.35779 -5.225298329 1.348764503

-5.902120198 -5.19986 -4.372702903 -0.9663620032.735700817 -0.28909 -3.009988946 -0.1402873421.127754174 -2.51951 -5.27213103 1.354648138

-6.979359505 1.359845 -9.017072966 -0.4995759880.132010282 -2.71113 2.694147467 -0.502174863

-1.783030396 -0.52075 -4.799965131 -0.896890172-6.070046292 2.410862 -7.305311366 -1.3709710933.932558537 0.709116 2.718674687 1.149926139

-5.498280124 -1.54017 -6.271749703 1.148277-1.026654953 -4.29409 2.547264128 0.063203037

1.29505961 -3.60993 3.746687032 -1.431831064-5.210391155 -7.10815 -1.127785203 -2.82488946-4.169418282 -6.16405 0.629884119 3.649972499-2.417722014 4.466414 -7.612368855 0.904322589-5.836022668 -5.06333 -6.631707976 0.327308887-4.34147211 8.042842 -10.15385517 -0.557336145

-6.013067376 -2.31934 -6.779492991 0.4645981982.351678237 -2.94247 5.807435413 0.362367951

-6.051033528 -6.47283 -5.191925454 5.5202052545.507247639 -0.0779 4.307946209 -4.1095313645.568041295 3.477585 2.141515594 -1.4359075664.050537123 0.95435 3.458736009 0.127594979

2.61824883 7.26415 -2.328445821 1.925220354-2.663025875 -7.75868 0.405165127 0.37703302-2.58735089 -4.16073 -3.001386449 -0.460596869

-18.49933968 -5.2051 -13.01239192 1.756621858-1.023099967 0.476781 -2.350246928 -0.998376504-6.992559532 -10.2425 -17.09413351 3.549554357-7.949112283 -0.11357 -9.429005018 0.456761373-10.99642206 -4.59335 -14.96744007 3.643166785-12.55287795 1.871977 -8.705122947 3.860200829-8.469066505 5.660192 5.107106143 -4.0571251568.947897286 1.23615 11.5829977 -1.2650910291.108790499 3.339655 3.200897507 -4.584911994

28

-2.478781079 3.986295 -4.792938648 -0.70736197413.26262082 0.448869 12.2606341 -0.140015364

-1.943407791 5.521951 -7.683346632 -0.8938766086.884125763 4.200894 6.296174259 -4.774779412-9.07257895 -0.40545 -14.81734321 1.2947414234.508494188 -7.34258 7.062232477 6.6638861413.41288662 2.428614 -0.922061916 3.83377619126.59266267 3.27669 15.66395515 -0.218511978

-9.770788482 12.44936 -19.02650795 3.44646497222.02580384 5.658625 -9.789377382 3.025288068

-13.12702576 -2.89127 17.84848758 -8.354330581-13.94608983 -2.56212 -2.867582535 -1.480902962-5.411771556 -1.62255 4.237393994 -5.2776033585.387498221 5.286432 2.683161051 -0.6522882951.306410511 2.040671 -4.275937147 -0.7888977615.44616019 -0.13068 6.778060323 0.880962704

-9.068237655 1.879668 -6.645082878 -6.81811859-4.03622483 -5.39266 2.160831737 -2.368982491

-16.29317457 1.90084 -5.304864768 -6.458821572.001570939 -2.21752 -1.30755068 0.9191446884.294160506 3.951971 3.005900556 -1.65017838

-4.825510415 1.283857 -5.165205523 0.046750836-10.30923811 -3.81126 -4.275694375 -3.43372624310.33376783 0.268692 13.31939629 -0.7390212745.352167827 3.495569 2.454867193 2.780105608

-9.732250156 -4.4211 -17.02418316 -0.967065568-8.973101425 -2.86788 -9.983114132 1.6771428151.301330571 2.530577 -0.088339962 -1.633333621

-14.93474615 -10.2433 -7.379219909 -0.114463999-20.25512925 1.787003 -16.96223063 1.31924314-11.48327474 10.58094 -10.26966727 -4.053618976-1.983759531 -1.11783 -4.146557016 0.0952221350.042296285 -1.77422 2.555558476 0.948573591

-3.732611732 -2.45459 0.106627672 3.3044426254.83643739 1.177549 1.868378125 -0.300813688

0.166514103 3.920495 -2.201628006 0.163661169-5.167810897 0.998025 -5.176270858 -3.21846229-10.32433827 -2.3444 -9.114406255 -1.525630134-10.17605333 2.571829 -10.6757764 -38.07453099-4.421339244 -4.67418 0.468311538 -2.324806591-9.258995579 -2.60953 -7.92730825 3.342569585-1.051830185 -5.61631 5.552752453 -2.6005188913.761018401 4.65114 2.817569926 -2.871721699

Solution

First of all, run the regression equation in EViews 6 as follows:

fr c s mr bm

Where: fr is fund returns. s is size and is is the first independent variable. mr is market return and it is the second independent variable.

29

bm is book to market return and it is the third independent variable.

Click view and select the stability tests and then the first one Chow breakpoint test.The total number of observations are 107.

The hypotheses are as follows:

H0: There is stability and the variances are constant.H1: There is no stability and the variances are not constant.

I have included the output of the Chow breakpoint test for 20, 40 60, 80 and 100 observations.

Chow Breakpoint Test: 20 40 60 80 100 Varying regressors: All equation variablesEquation Sample: 3 107

F-statistic 2.500424 Prob. F(20,83) 0.0020Log likelihood ratio 50.45822 Prob. Chi-Square(20) 0.0002Wald Statistic 101.7974 Prob. Chi-Square(20) 0.0000

The F-statistic, the log likelihood ratio and the Wald statistic p – values are all significant at the 5% significance level. The sample evidence suggests not to reject the null hypothesis of stability. In other words, the variances for different time periods are constant.

I have also attached a residual plot to facilitate your learning.

-30

-20

-10

0

10

20

30

-30

-20

-10

0

10

20

30

10 20 30 40 50 60 70 80 90 100

Residual Actual Fitted

30

Exercise

You are given the following two datasets.

Dataset 1 Dataset 2

Sx1 = 48 Sx1 = 47Sx2 = 35 Sx2 = 34Sx1x2 = 32 Sx1x2 = 33Sx1y =100 Sx1y =102Sx2y =78 Sx2y =80Syy = 240 Syy = 244n =10 n =15

It is required to calculate the regression equation, the residual sum of squares, the coefficient of multiple determination, and the correlation coefficient. In addition, calculate the standard errors and the 95% confidence interval of the intercept and the

31

coefficients . Finally, calculate the t-statistics for the intercept and the coefficients for each dataset.

We compute the first dataset.






32



33



t-statistic(

t-statistic


t-statistic =

34

Please compute the second dataset.








35



t-statistic(

36

t-statistic


t-statistic =

Exercise

You are given the following multiple regression equation from a sample of n = 20 observations. It was estimated based on the ordinary least squares method.

t-statistics (1.2) (2.1) (3.2) (2.4)

In parentheses, I have included the t-statistics. The R2 = 0.892.

You are required to comment what will happen to R2 if we drop or add an explanatory variable?

Solution

37

If an explanatory variable falls, then the R2 will fall. It is important to check the t-statistic if it is significant at the 5% or 1% significance level to determine if R 2 will rise.

Exercise

Based on the following summary output, please, calculate the R square of the multiple regression.

SUMMARY OUTPUT


ANOVA Df SS MS F Significance

38





Upper 95.0%

Intercept 17.58426966 3.760573 4.675954 0.018476 5.616435321 29.552104 5.616435 29.5521

x1 1.443820225 0.763074 1.892111 0.154831-

0.984622726 3.8722632 -0.98462 3.872263

x2 0.842696629 0.914227 0.921759 0.424636-

2.066783462 3.7521767 -2.06678 3.752177

Solution

You are given the following mathematical equation:

R2 =

Another formula to use is as follows:

R2 =

Exercise

It is given the following multiple regression equation. The sample size is 20. The dependent variable is share prices and the independent variables are stock index and the numbers of buyers.

SE (123.43) (0.768) (7.342)

t-statistics (0.7976) (19.98) (4.66)

p (0.0000) (0.0000) (0.0000)

R2 = 0.885 F = 110.324

39

It is required the following:

(a) Please interpret the regression equation output.(b) Test the hypotheses that the coefficient = 0 under the null hypothesis.

Compute the t value and then decide whether to accept or reject the null hypothesis.

Solution

(a) The share prices are positively related to the stock index and the number of buyers. If the stock index increase, the average price of the share price will increase by about 15.345 dollars. Following the same way of thinking, by keeping the stock index constant, the average share price increase by 34.23 dollars justified by the increase of the number of buyers. The R2 number of 0.768 means that the two explanatory variables account for 76.8% of the variation in the share prices. All the p-values are significant.

(b) Please state the hypotheses as follows:

Under the null hypothesis, the stock index has no effect on the share price. Under the alternative hypothesis it affects the share prices.

The t-value is as follows:

The degrees of freedom are n-3. The sample size is 20. Therfore, 20 – 3 = 17

Do we reject or accep the null hypothesis with 5% significance level?

Solution

The degrees of freedom are n – 3 or 20 – 3 = 17. With 95% confidence level, the t- critical value of a two-tail t-test is 2.110. The calculated t-statistic is 19.98 and is higher than the critical value of 2.110. Therefore, the suggested evidence suggests that we can reject the null hypothesis with 95% confidence level. The econometrician would assume that the true value of the parameter is not equal to 0.

You should compare the value of the test statistic from the sample with the critical value and reject the null hypothesis if the test statistic exceeds the critical value. The value of the t-statistic 19.98 exceeds the critical value ± 2.110. Therefore, the sample evidence suggests to reject the null hypothesis.

40

Exercise

Please consider the following dataset and ANOVA table single factor. Compute te F - ratio. ANOVA is the acronyms of analysis of variance. It is used to show the decomposition of the variance explained by the independent variables and the error term in the regression equation. The mathematical formula for the F-statistic is as follows:

Where: d.f is the degrees of freedom.

The dependent variable is share return and the independent variables are risk free rate and size. The null hypothesis is that and = 0. The degrees of freedom are k-1 in the numerator and n – k in the denominator. Thus, we have 3 – 1 = 2 degrees of freedom in the numerator and 72 – 3 = 69 in the denominator.

ObservationsShare return

Risk free rate Size

1 3.526787 2.755836 0.823441

41

2 -4.34533 2.725726 4.458782

3 5.222709 3.092531 4.232887

4 -4.99619 3.089416 2.137064

5 -3.04336 2.802173 1.658494

6 -2.375422 2.409178 0.20628

7 2.651303 2.220845 0.028879

8 -0.68924 2.522834 0.774512

9 0.205664 2.669247 0.259913

10 2.4783 2.969892 0.325758

11 0.237407 1.994892 0.008251

12 0.329728 1.861559 0.008251

13 -0.26869 2.469416 0.929169

14 0.064769 2.503892 0.633557

15 -0.5873 2.265607 0.711829

16 0.329225 2.606003 0.008251

17 -0.11849 2.639092 2.19223

18 0.011541 2.572749 1.91165

19 -0.18757 2.540823 0.008251

20 -0.38752 2.259086 0.008251

21 -0.26835 2.608419 0.148522

22 0.262798 2.558915 0.191428

23 0.355054 3.042134 0.145221

24 -1.34302 2.168168 0.167979

25 -0.77964 3.415726 0.609765

26 -0.04649 2.244892 0.864844

27 0.098381 3.257392 0.072066

28 -0.09585 2.592576 0.960441

29 -0.0059 2.533267 0.00825130 -0.05415 2.678596 0.00825131 -0.00384 2.595078 0.24341132 -0.00799 3.029892 0.11551733 -0.0538 2.502834 0.40678534 -0.00541 2.801892 0.18714635 -0.0178 2.985979 0.55309536 -0.00335 2.873892 0.45271137 -0.03195 2.738983 0.25866138 -0.05187 2.188128 0.74260939 -0.00569 2.984753 0.14239940 -0.0009 2.774178 7.44718541 -0.15083 2.867511 0.88980242 -0.00748 3.143092 0.12376843 -0.78198 3.162392 0.20139644 -0.01982 2.054598 0.79945545 -0.04005 3.108503 0.4456446 -0.002 2.633226 0.272711

42

47 -0.00483 2.47726 0.15064248 -0.05627 2.887051 1.91840749 -0.01541 2.370292 0.7508650 -0.00233 2.802692 0.7508651 -0.00399 2.830845 0.7508652 -0.00422 1.625607 0.06806953 -0.00349 1.647749 0.16961754 -0.0038 2.855726 0.49380155 -7.53E-05 3.073577 0.7508656 -0.00448 2.516488 1.52307557 -0.01442 2.503017 0.33665858 -0.04197 2.515186 0.88538359 -0.05222 2.581392 0.14122860 -0.01357 2.804629 0.08251261 -0.01134 2.831559 0.08251262 -0.00305 2.935229 0.08251263 -0.0007 2.551392 0.08251264 -0.01616 2.944589 0.08251265 -0.00178 3.145347 0.08251266 -0.0375 4.196281 1.68386767 -0.04155 4.112035 0.66009768 -0.23705 4.090387 0.42163769 -9.67E-17 2.403577 2.98867270 -0.0583 2.703371 0.38615771 -0.04783 2.461197 2.88058172 -0.04177 2.928787 1.489131

Please find attached the ANOVA table single factor.

Anova: Single Factor

SUMMARY

Groups Count Sum Average Variance

Share return 72 -5.72171 -0.07947 1.633991

Risk free 72 195.3111 2.712654 0.212762

Size 72 57.48029 0.798337 1.469171

ANOVASource of Variation SS df MS F P-value F crit

Between Groups 293.5463 2 146.7731 132.78943.61E-

38 3.038267Within Groups 235.4306 213 1.105308

43

Total 528.9769 215

F-ratio =

As the variance explained by the independent variables becomes larger relative to the unexplained variance, then, we will get a large number of the F-ratio. It indicates to us that we can reject the null hypothesis of no effect. The two independent variables affect the dependent variable. The F - ratio value exceeds the F critical value and therefore, we reject the null hypothesis that the explanatory or independent variables are equal to zero. The F ratio test is a measure of the overall significance of the estimated regression line. It is also related to R2 through the following equation:

Exercise

Please consider the following regression equation:

Where: y is the dependent variable and represents the hedge fund price. x1 is the independent variable and represents the hedge fund index. x2 is the second independent variable and represents the hurdle rate. It is required the following:

(a) Interpret the regression results(b) At the 5% significance level, test that the hurdle rate has no effect on the

hedge fund price.(c) Test the joint hypotheses through a t-test statistic that the two explanatory

variables affect substantially the hedge fund price. State the null and alternative hypotheses. The sample size is 25.

44

Exercise

Please consider the following dataset. Construct an ANOVA table and calculate the F-ratio statistic. Then, compare the F-ratio with the critical value and test the joint hypotheses of the independent variables related to significance at the 5 % significance value. The samle size is n = 20.

ObservationsShare return

Risk free rate Size

1 3.526787 2.755836 0.823441

2 -4.34533 2.725726 4.458782

3 5.222709 3.092531 4.232887

4 -4.99619 3.089416 2.137064

5 -3.04336 2.802173 1.658494

6 -2.375422 2.409178 0.20628

7 2.651303 2.220845 0.028879

8 -0.68924 2.522834 0.774512

9 0.205664 2.669247 0.259913

10 2.4783 2.969892 0.325758

11 0.237407 1.994892 0.008251

12 0.329728 1.861559 0.008251

13 -0.26869 2.469416 0.929169

14 0.064769 2.503892 0.633557

45

15 -0.5873 2.265607 0.711829

16 0.329225 2.606003 0.008251

17 -0.11849 2.639092 2.19223

18 0.011541 2.572749 1.91165

19 -0.18757 2.540823 0.008251

20 -0.38752 2.259086 0.008251

Exercise

Please consider the following dataset.

y x1 x2

1 2 33 4 45 7 57 8 79 10 810 2 913 11 1014 12 1616 14 18

(a) Estimate the regression equation.(b) From the summary output outline the t-statistics and the standard errors.(c) Estimate the 95% confidence intervals for .(d) Check the R2 and the F-statistic and compare it with F critical region.(e) Check the value of the TSS, ESS and RSS and comment on the variability.

46

Exercise

You are given the following data and mathematical equations based on a sample of 30 observations.

Please calculate , 1 and 2 . I have included the required equations.

47

(a) It is required to calculate the multiple regression coefficients.(b) Estimate their standard errors.(c) Estimate their t – statistics.(d) Estimate 95% confidence intervals for .(e) Show in Excel the ANOVA table and interpret the F ratio by comparing it with the F- critical region. Compare the variability of the explained sum of squares versus the residual sum of squares.

Exercise

Please find attached the regression results based on a sample of 25 open-ended mutual funds that are traded in the United Kingdom.

1 + 12.45x2 +

SE (0.32) (5.32) (6.21)

R2 = 0.48 F = 110.23 p-value = (0.0000)

Where: y is the dependent variable and is related to the price of the funds. x1 is the independent variable and it is related to the mutual funds index. x2 is the management fees that are charged by the funds.

(a) Calculate the t-statistics.(b) Interepret the regression results.(c) At the 95% confidence level, thest the hypothesis that the management fees

has no effect on the price of the funds. What test are going to use?(d) Are the coefficients of the two variables significant? What the R2 is telling us.

48

Exercise

Please consider the personal consumption in the UK in relation to disposable after-tax income and mortgage rates charged by banks.

Where: y is the personal consumption. x1 is the disposable after-tax income. x2 is the mortgage rate.

(a) Calculate the marginal propensity to consume, (MPC). Please check my book introduction to mathematical economics for clarification concerning the marginal propensity to consume.

(b) Please check and explain the signs of the coefficients.Would you expect a negative sign for mortgage rates.

(c) Calculate the standard error of each coefficient.(d) Is the coefficient significantly different from zero?

49

Exercise

You are given the following ANOVA table with missing data of a three variable regression results.

Source of variation Sum of squares (SS)

d.f. Mean sum of squares (MSS)

Explained sum of squares due to regression, (ESS)

20.456 2

Residual sum of squares, (RSS)Total variation, (TSS)

25.123 20


(a) Please complete the table.(b) Compute the value of RSS.(c) What are the d.f of the ESS and RSS. The RSS has n-3 degrees of freedom.(d) Calculte the F-ratio.

50

Exercise

Please consider the following table that represents the fees of a closed-end mutual fund in relation to the net asset value nominated in billions pounds.

Fees nominated in percentages Net asset value in billions pounds0.98 200.88 300.76 400.65 500.62 600.58 700.59 800.42 900.48 1000.47 1100.37 1200.34 1400.23 1500.22 160

(a) Draw a scatterplot to show the relationship between fees and net asset value.(b) Estimate the regression equation and interpret the results.(c) Compute the t-statistics.(d) Are the coefficients of the fees related to the net asset value? Which test are

you going to use?

51

(e) Transform your variables to natural logarithmic and compare the R2 of the linear and log linear regressions.Draw your conclusion based on the findings?

(f) Use the p - values from the regression output to conclude if the t-statistics are significant. Check your results based on the 5 % significance level. p-values below the 5% are significant.

(g) Check the adjusted R2 and the standard errors of the logarithmic regression.

Exercise

Please consider the Cobb-Douglas production function where the dependent variable is output and the independent variables are labour input and capital input. I have attached the table with the relevant data. The output is nominated in millions. The labour input is nominated in thousands of persons and the capital is nominated in millions USD.

Year Output Labour , (x1) Capital, (x2)2000 1056780 2678 20455402001 1105780 3456 22456202002 1294780 4567 23689302003 1378940 5623 24678402004 1489620 6278 30878502005 1589230 6589 31567602006 1683490 7654 33678702007 1734890 7894 38907802008 1892340 8765 40357902009 1923450 8124 45789202010 2098340 9534 50987402011 2209870 10345 54678602012 2381340 11567 55678702013 2487650 12567 6087790

It is required to calculate the following:

52

(a) Estimate the regression equation. The dependent variable is output and the independent variables are labour and capital. Interpret the slope coefficients of labour and capital.

(b) Draw a scattergraph to see the relationship between the variables.(c) Test the hypotheses that and = 0.(d) Calculate the t-statistics.(e) Is the coeeficient of labour statistically different from zero?(f) Did you expect negative or positive signs of the coefficients of the

independent variables?

Example

I have attached a detailed example of multiple regressions that was performed during different time periods based on different sectors of investment trusts.There is also references based on the literature review of my PhD thesis. You can see the intercept and the coefficients values and their related t-statistics. In addition, the table include the adjusted R2 and the Durbin Watson statistic and we will cover it in the subsequent chapter related to autocorrelation. The aesterix denote the level of significance at the 5% or 1% significance level. The example is related to Fama and French’s three factor model of UK excess NAV return. The dependent variable is UK excess NAV return and the independent variables are market, size, and book – to – market.

You are required to consider carefully the following table and spot the level of significance and how it affects the adjusted R2 of the whole multipl regression model. Please write the regression equation for each period. I have included interpretation of the following table.

Table 1: Fama and French’s three factor model of UK excess NAV return

We use 12, 36, 60 and 109 observations by applying a rolling methodology. The sample includes 16 sectors of UK investment trusts with total number of 120 funds. We use two style indices to measure the size effect, and the book-to-market effect. Specifically, the UK size effect is measured as the difference between the return on the FTSE Smaller Companies index and the return on the FTSE 100 index. The FTSE 100 is used as a proxy for the return on large companies. The UK book to maket effect is measured as the difference between the return on the FTSE 350 Growth index and the return on the FTSE 350 Value index.

53

AITC Category Coefficients 1Y 3Y 5Y 9YGlobal Growth Adj R2 0.89 0.71 0.67 0.48

3.87( 3.30)**

0.72 ( 1.27)

1.91(3.01)**

-0.56(-0.89)

Market 1.68(9.70)**

0.81(9.74)**

0.65(4.45)**

0.69(5.85)**

Size 1.29(2.99)**

0.73(1.91)*

0.33(3.25)**

0.49(3.30)**

Book-to-market -1.85(-1.99)*

0.71(2.11)*

-0.05(-0.22)

0.39(1.58)

D/W 2.10 2.11 1.96 1.95Global Growth & Income

Adj R2 0.87 0.71 0.83 0.55

1.60(2.95)**

1.18(2.02)*

0.37(0.97)

-0.46(-1.15)

Market 0.95(8.60)**

0.82(10.89)**

0.86(17.78)**

0.66(6.92)**

Size 0.66(4.22)**

-0.00(-0.02)

-0.05(-1.18)

-0.02(-0.17)

Book-to-market 0.72(1.55)

-0.28(-0.52)

-0.11(-0.48)

-0.18(-0.77)

D/W 2.10 2.13 1.98 2.04Global Smaller Companies

Adj R2 0.89 0.74 0.67 0.49

3.54(3.24)**

-0.42(-0.33)

1.91 (3.01)**

-0.45(-0.70)

Market 1.64(10.34)**

1.13(10.06)**

0.65(4.45)**

0.71(5.77)**

Size 1.25(2.99)**

0.20(0.99)

0.33(3.25)**

0.49(3.30)**


-0.89(-1.78)*

-0.05(-0.22)

0.39(1.57)

D/W 2.03 1.95 1.96 1.97UK Growth Adj R2 0.90 0.84 0.80 0.78

2.27(1.60)

0.81 (1.50)

0.53(1.29)

0.12(0.34)

Market 0.82(6.64)**

0.69(13.15)**

0.82(15.56)**

0.82(13.54)**

Size 0.20(2.36)**

0.68(5.96)**

0.02(2.18)*

0.83(9.65)**


-0.63(-1.73)*

-0.40(-1.26)

0.06(0.23)

D/W 2.00 1.98 2.05 2.01UK Growth & Income Adj R2 0.84 0.81 0.77 0.76

1.88(1.08)

1.23 (2.02)*

0.22(0.46)

0.06(0.16)

Market 0.74(4.29)**

0.68(11.00)**

0.86(13.55)**

0.77 (9.24)**

Size 0.23(2.25)*

0.70(5.70)**

-0.11(-1.03)

0.82 (9.48)**

Book-to-market -0.13(-1.09)

-0.67(-2.43)**

-0.63(-1.83)*

0.06(0.22)

D/W 2.07 1.95 2.10 1.93

54

UK Smaller Companies

Adj R2 0.89 0.84 0.69 0.79

2.43(1.65)*

0.85(1.58)

1.97(2.43)**

0.13(0.38)

Market 0.84(5.92)**

0.68(13.44)**

0.54(4.76)**

0.83(13.94)**

Size 0.20(2.32)*

0.70(6.42)**

0.60(5.21)**

0.84(9.54)**


-0.67(-1.86)*

-0.10(-0.84)

0.06(0.23)

D/W 2.04 1.97 1.99 2.06North America Adj R2 0.53 0.76 0.79 0.69

-0.93(-0.89)

0.99(1.53)

0.48 (1.05)

0.64(1.77)*

Market 0.47(3.52)**

0.92(10.11)**

0.95 (14.82)**

0.77(9.53)**

Size -0.61(-1.86)*

0.67(2.33)*

0.62(2.98)**

0.43(4.37)**

Book-to-market 1.51(-1.80)*

-0.21(-1.30)

-0.09(-0.98)

0.19(0.97)

D/W 2.05 2.09 1.99 2.06North America Smaller Companies

Adj R2 0.79 0.84 0.61 0.69

1.98(1.37)

0.82(1.44)

0.28(0.25)

0.63(1.73)*

Market 0.58(3.16)**

0.86(2.97)**

0.69(6.23)**

0.71(9.56)**

Size 0.93(3.34)**

0.49(3.46)**

0.40(1.73)*

0.43(4.40)**


0.25(0.79)

-0.43(-1.76)*

0.18(0.94)

D/W 1.95 1.92 1.97 2.07Far East (Including Japan)

Adj R2 0.68 0.78 0.75 0.53

-1.45(-0.66)

0.93(1.38)

0.87(1.60)

-0.17(-0.32)

Market 0.63(2.07)**

0.96(10.05)**

0.91(12.96)**

0.82(11.06)**

Size 0.12(0.34)

0.24(1.92)*

0.14(1.08)

0.27(2.03)*


-0.16(-1.03)

-0.37(-1.21)

0.24(1.09)

D/W 1.96 2.15 2.08 1.97Far East (Excluding Japan)

Adj R2 0.63 0.44 0.24 0.25

-1.75(-0.49)

0.85(0.68)

0.85(0.81)

-0.09(-0.11)

Market 0.29(1.35)

0.79(4.28)**

0.57(2.54)**

0.39(4.01)**

Size -0.13(-0.17)

-0.13(-0.42)

-0.26(-0.74)

0.08(0.63)

Book-to-market 2.10(1.86)*

0.84(1.19)

-0.12(-0.20)

0.49(2.31)*

D/W 2.04 2.02 1.92 1.92Japan Adj R2 0.91 0.74 0.77 0.53

55

4.40(5.00)**

-0.11(-0.08)

-0.14(-0.24)

-0.18(-0.33)

Market 1.70(10.85)**

1.16(9.78)**

1.13(16.43)**

0.81

(19.99)**Size 0.83

(2.04)*0.24

(0.86)0.51

(3.05)**0.28

(2.07)*Book-to-market -0.76

(-2.82)**-0.84

(-1.72)*-0.42

(-1.21)0.53

(1.55)

D/W 1.93 2.03 1.96 1.98Japanese Smaller Companies

Adj R2 0.80 0.80 0.65 0.53

3.54 (1.99)*

0.35(0.32)

0.40(0.48)

-0.18(-0.33)

Market 9.90(9.50)**

1.15(12.57)**

0.98(8.01)**

0.81(19.99)**

Size 5.40(6.00)**

0.84(3.96)**

0.59(2.82)**

0.28(2.07)*


-0.51(-0.82)

-0.37(-0.77)

0.26(1.78)*

D/W 1.95 1.92 1.96 1.98Europe Adj R2 0.37 0.78 0.75 0.41

0.84(1.36)

0.68(1.19)

0.67(1.33)

-0.03(-0.07)

Market 0.60(2.85)**

0.73(10.05)**

0.79(11.05)**

0.67(5.23)**

Size 0.32(0.75)

0.16(1.52)

0.04(0.50)

0.16(0.93)


-0.03(-0.12)

-0.02(-0.08)

-0.02(-0.08)

D/W 1.94 1.99 1.97 2.08European Smaller Companies

Adj R2 0.89 0.90 0.88 0.71

0.69(0.88)

-0.11(-0.28)

0.48(1.52)

-0.48(-1.28)

Market 0.90(12.45)**

0.86(21.13)**

0.84(16.27)**

0.84(10.13)**

Size 0.44(2.30)*

0.32(3.19)**

0.23(2.66)**

0.49(3.69)**


0.40(1.92)*

0.02(0.11)

0.30(-1.36)

D/W 2.02 2.05 2.02 2.06Country Specialists Far- East

Adj R2 0.83 0.87 0.80 0.53

1.65(2.45)**

0.32(0.52)

1.16(1.82)*

-0.17(-0.30)

Market 0.93(7.91)**

0.99(13.50)**

0.94(13.62)**

0.82(6.62)**

Size 0.64(3.58)**

0.32(3.45)**

-0.02(2.31)*

0.27(2.11)*


0.25(-4.18)**

-0.07(-0.91)

0.24(1.04)

56

D/W 2.01 2.09 2.06 1.97Sector Specialists Property

Adj R2 0.67 0.58 0.30 0.69

3.17(1.50)

0.22(0.20)

0.99(0.94)

0.60(1.64)*

Market 0.90(4.39)**

0.70(6.03)**

0.51(3.23)**

0.70(9.53)**

Size 1.03(1.53)

0.46(2.00)*

0.35(2.04)*

0.42(4.28)**

Book-to-market -1.95(-3.40)**

-0.40(-1.58)

-0.23(-1.09)

0.19(0.99)

D/W 2.05 2.01 2.09 2.04Total observations 12 36 60 109Source: author calculation* represents t –value that is statistically significant at 5% significance level** represents t-value that is statistically significant at 1% significance level

According to table 1 , the adjusted R squared for most AITC categories is higher than 0.50. In addition, there is no evidence of first order autocorrelation in the above regressions as on average the Durbin/Watson statistic (D/W) for each sector is close to 2.00. For example, the European Smaller Companies category has an adjusted R squared value of 0.89 for the first 12 months and 0.88 for the first 60 months. Similarly, UK Growth has an adjusted R squared value of 0.90 for the first 12 months and 0.84 for the first 36 months.

As indicated above, is used to measure the ability of managers to outperform the base index. A positive and statistically significant indicates a skilled fund manager whose decisions add value to the fund. Rhodes (2000) argued, that “persistent performance shows that some fund managers are able to outperform their peers. This implies that the fund managers must either have access to information that is insider or not widespread or make use of information in a speedier way than other managers. As markets become more efficient it will be more difficult for any fund manager to outperform the market continuously”(Rhodes,2000,p.7). On the other hand, negative

values or statistically insignificant values represent inferior or neutral performance of the manager. In other words, a negative indicates a poorly performing manager whose decisions affect negatively the value of the fund. According to Table 44, the results are mixed. Ten out of the sixteen sectors display an that is positive and statistically significant at the 1% and 5% level (in one-tailed tests). In more detail, Global Growth and Income shows a positive and statistically significant for the first year at the 1% level and at the 5% significance level for the first three years. Japan displays a significant for the first year at the 1% significance level. Global Smaller Companies show a positive and statistically significant at the 1% level the first five years. Finally, Japanese Smaller Companies and Country Specialist Far East display a positive and statistically significant for the first 12 months at the 1% and 5% levels respectively. On the other hand, the rest of the sectors show a mixed picture of positive and negative alphas that are not statistically significant at the 5% or 1% levels. Furthermore, only two alphas are significant for the full nine years at the 5% level in a one-tailed test. So while there is some evidence for managerial performance persistence in the short-term, there is little evidence for persistence in the long-run.

Most of the sectors show significant t-statistics for both the market return and size. The t-statistic for the UK size effect measured as the difference between the return on

57

the FTSE Smaller Companies index and the return on the FTSE 100 index is statistically significant at the 1% and 5% level. On the other hand, the UK book-to-market effect measured as the difference between the return on the FTSE 350 Growth index and the return on the FTSE 350 Value index is negatively and positively significant for some of the sectors and statistically insignificant for most of the sectors. Thus, consistent with Pontiff (1997), we find that the book-to-market factor does not seem to have any significant explanatory power. Through his cross-sectional regression, Pontiff found that the book-to-market effect only influenced funds with low premiums and discounts.

Example from my PhD thesis entitled as a multifactor model of investment trust discounts. A comparative study of UK investment trusts and US closed-end funds.

Econometrics refers to the application of economic theory to the economic variables or financial theory to financial data through statistical techniques in order to test hypotheses and estimate their coefficients. I will refer to the literature review of factor selection of a multifactor risk model and expectations concerning their coefficients. It is an excellent example to help you understand how research methods are used in postgraduate and research degrees.

The most common approach for choosing factors that affect stock returns are firm specific attributes. A well known model is the three factor model of Fama and French (1993). Fama and French group common stocks according to their size and their ratios of book to market value.

From the standpoint of both the academic researcher and practitioner, it is crucial to be able to identify which factors best capture stock return variation and specifically the discount variation. As a result, there has been a proliferation of research that attempts to identify the various factors. The importance of market risk, book-to-market risk and small firm risk in explaining cross-sectional differences in stock returns is discussed in Fama and French (1993). On the other hand, Carhart (1997) adds a fourth factor to these three risk measures, which is based on the results of Jegadeesh and Titman (1993), that mutual funds are affected by momentum, namely that past winners will continue to be next winners the following year and past losers will continue to be losers. In other words, a firm’s past return helps to predict

58

future returns (Chan, Jegadeesh and Lakonishok,1996; Chopra, Lakonishok and Ritter, 1992; DeBondt and Thaler,1985; Rosenberg, Reid and Lanstein, 1984). We found some evidence for this managerial performance persistence in our empirical studies. Momentum creates the suspicion that management performance might influence the discount return. Momentum is defined as the difference between funds NAV return and the sector’s average NAV returns. Past performance is measured over the full history of the funds. We test the momentum anomaly documented in Carhart’s four-factor model.

We add two more factors to these four risk measures. The first one is based on the results of Lee, Shleifer and Thaler (1991) that closed-end funds are subject to systematic “investor sentiment risk”. In more detail, one possible explanation for variations in the discount is based on the existence of two categories of investors. The first category includes rational investors, namely arbitrageurs, who make rational decisions in accordance with their preferences. The second category includes the so-called noise traders. These investors do not act fully rationally, and their investment decisions are considered as unpredictable. In some periods they overestimate expected returns relative to the rational expectations and in other periods they underestimate them. Therefore, prices of securities are a function of both categories.

The interaction of these two categories of investors may help to explain the variation of the discount. For arbitrageurs to buy funds that are characterized by a constant discount would be costly and ,therefore, not always profitable. In more detail, if investment trusts trade at a discount and then at a premium to NAV, an apparent arbitrage profit can be realised by selling the fund’s share above the risk free rate. However, if the discount stays relatively constant over the investment horizon, the arbitrageurs make no profit. According to the descriptive statistics presented in chapter 5, there is a wide range between the lower and upper bounds of the discount in each AITC and CEFA category, which suggests that there is active interaction between both parties. This active interaction can be measured quantitatively by using as proxies the retail flows of each sector or by using an investor sentiment index.

The retail flow measure was first used by Malkiel (1977) to measure general investor sentiment regarding investment companies. According to him “when individuals were showing little enthusiasm for open-end funds, it is probable that they showed a similar lack of enthusiasm for closed- end funds”(Malkiel, 1977, p.856). The relationship between changes in fund discounts and net redemptions of open-end funds can be seen in Figure 1

Figure 1 : Relationship between changes in fund discounts and net redemptions

P SC P

PA PA So

Change in D0 D0

59

Discount D1 D1

Q Sales Flow Q

Closed-end funds Open-end funds

Source: Malkiel (1977), the valuation of closed-end investment company shares

In Figure 1, according to Malkiel (1977), “ S0 is the supply curve of open-end funds, which is infinitely elastic because open-end funds will sell as many new shares or redeem outstanding ones at net asset value (PA). The supply curve for the closed-end funds (Sc) is perfectly inelastic as there is a fixed supply of shares. Shifts in the demand curve for open-end shares will result in changes in the quantity of shares outstanding or net sales” (Malkiel, 1977, p.856). Shifts in the demand curve for closed-end funds will result in changes in fund discounts or premiums. In a perfect market, demand shifts will not change the level of the discount but due to the interactions of noise and rational traders, we expect net sales of open-end funds to be associated with changes in the discount. Malkiel (1977) found that an increase in net sales of open-end funds was associated with an increase in the average discount, but the coefficients were not significant. The problem with his study was that he only used 24 US closed-end funds and not all companies could be included in his cross-sectional regression because of missing data. So the explanatory power of his model was weak.

On the other hand, Gemmill and Thomas (2002) conducted a more detailed analysis to find out whether investor sentiment can explain fluctuations in the discount. They found that monthly flows of retail investment into particular sectors were closely related to changes in sector discounts. According to their findings investor sentiment not only affects the discount in the short-term but may also influence its level over periods of several years. They investigated 11 UK sectors which consisted of 158 closed-end equity funds.

The second additional factor is management expenses. Malkiel (1977) defines management expenses as a percentage of fund assets. The problem is that the ratio of expenses to cash flow is not constant over the life of a fund. As the long-term expense to cash flow ratio is not observable, it can be proxied by current monthly expenses as a percentage of fund assets. In this chapter, we define expenses as the monthly management charge (or staff costs), auditors and custody fees, directors fees and marketing costs as a percentage of fund assets. We test two contradictory hypotheses. The first one is Barclay, Holderness and Pontiff’s (1995) view that a possible explanation of the discount is that as managers try to protect their private benefits, expenses increase over the long term and have an effect on the discount. The second one is Malkiel (1977) who claims that an expense ratio will not be significant due to the difficulty of measuring it accurately. The ratio of expenses to cash flow is not constant over the life of a fund.

The market factor is the difference between the return on the relevant index and one-month risk-free rate. The UK size effect is measured as the difference between the return on the FTSE Smaller Companies index and the return on the FTSE 100 index. The FTSE 100 is used as a proxy for the return on large companies. The size effect is

60

used to test the anomaly documented in Fama and French’s three-factor model. The UK book-to-market effect is measured as the difference between the return on the FTSE 350 Growth index and the return on the FTSE 350 Value index. The US size effect is measured as the difference between the return on S&P Small Cap and the return on the S&P 500 Composite as it is used as a proxy for the return on large companies. The US book-to-market effect is measured as the difference between the return on the S&P 500/Barra Growth index and the return on the S&P 500/Barra Value index.

From the theory, In terms of signs of the independent variables, we expect to find positive values for the coefficients of size, market and sentiment and negative values are expected for the book-to-market effect and momentum. In more detail, size effect was documented in Fama and French’s model in terms that small firms outperform big companies and therefore we expect a positive effect. Market effect has been significant in return explanation based on the CAPM. Sentiment is expected to have a positive value results based on the hypothesis of Lee, Shleifer and Thaler (1991) that closed-end funds are subject to systematic “investor sentiment risk”. In more detail, one possible explanation for variations in the discount is based on the existence of two categories of investors. The first category includes rational investors, namely arbitrageurs, who make rational decisions in accordance with their preferences. The second category includes the so-called noise traders. Book-to-market effect is expected to have a negative value based on the hypothesis of Pontiff (1997) who found that the book-to market effect is negative and insignificant and affects funds with low premiums and dicounts. Momentum is expected to have a positive sign during the short-term according to the hypothesis of Jegadeesh and Titman (1993). In more detail, Jegadeesh and Titman (1993) show that buying past winners and selling past losers generates significant profits when returns are measured over three to twelve-month periods. This hypothesis is tested later by using deciles and trying to test the significance between rankings. In contrast, momentumt is expected to have a negative sign over the long term as we assume that information both private and public were incorporated in the prices of funds. Expenses are expected to have positive or negative values as we test two contradictory hypotheses. The first one is Barclay, Holderness and Pontiff’s (1995) view that a possible explanation of the discount is that as managers try to protect their private benefits, expenses increase over the long term and have an effect on the discount. The second one is Malkiel (1977) who claims that an expense ratio will not be significant due to the difficulty of measuring it accurately. The ratio of expenses to cash flow is not constant over the life of a fund.

Fama and French (1993) show the importance of market risk, small firm and book-to-market risk in explaining cross-sectional differences in stock returns. The famous model of Fama and French (1993) was constructed and implemented on various portfolios of shares to explain various anomalies in financial markets in terms of size, book/market ratio etc. This leads to the three factor model. Equation (1) enables us to test the significance of these three factors in relation to the excess discount return. We use the excess discount return because changes in the discount can be interpreted as returns, and the discount was shown to be non-stationary.

(1)

61

where: is the excess discount return for each sector at time t

is the excess market return

is the size factor (small minus big), i.e the difference

between the return on a portfolio of small stocks and the return on a

portfolio of large stocks.

is the book to market factor (high minus low), i.e the

difference between the return on a portfolio of high-book-to-market

stocks and the return on a portfolio of low-book-to- market stocks

is the distrurbance term

We test the significance of the estimated coefficients using t-statistics and F-tests. We also use the Durbin/Watson statistic to check for autocorrelation. The estimation results are shown in Table 2 for the UK and Table 3 for the US.

Table 2 Fama and French’s three-factor model of the UK excess discount return

The regressions by sector are estimated over the period January 1990 to January 2003 by using monthly observations. The sample covers 90 funds for UK investment trusts. The t-statistic for each factor is displayed in parentheses. In addition, we provide tests of autocorrelation.

Factor 1 Factor 2 Factor 3Fund Name by AITC category

D/W F-Ratio AdjustedR2

Market(

Size Book to market

Global Growth 2.07 30.43 0.36 0.26 (7.22)**

0.16 (4.41)**

-0.36 (-2.65)**

Global Growth & Income

2.01 37.52 0.41 0.34 (3.19)**

0.11 (2.30)*

-0.36 (-2.52)**

Global Smaller Companies

1.94 47.17 0.47 0.41 (7.24)**

0.07(1.78)*

-0.40 (-3.23)**

UK Growth 2.05 40.40 0.43 0.28 (3.01)**

0.15 (3.52)**

-0.41 (-3.05)**

UK Growth & Income

1.97 62.85 0.54 0.44 (9.92)**

0.11 (3.03)**

-0.33 (-2.64)**


1.97 49.72 0.49 0.41 (5.54)**

0.14 (3.32)**

-0.25 (-1.77)*

North America 2.06 35.43 0.40 0.46 (8.08)**

0.10 (1.79)*

-0.31 (-1.88)*

North America Smaller Companies

2.07 25.75 0.32 0.26 (2.24)*

0.13 (2.54)**

-0.64 (-3.43)**

Far East (Including 2.06 45.77 0.46 0.39 0.14 -0.21

62

Japan) (4.05)** (3.18)** (-1.51)Far East (Excluding

Japan)2.02 24.33 0.31 0.27

(7.31)**0.14

(3.07)**-0.24

(-1.41)Japanese Smaller

Companies2.03 34.57 0.39 0.30

(3.23)**0.11

(2.19)*-0.43

(-2.78)*Japan 1.95 46.12 0.47 0.40

(5.58)**0.15

(3.52)**-0.27

(-1.92)*Europe 2.03 49.67 0.49 0.36

(3.47)**0.12

(2.55)*-0.40

(-2.84)**Country Specialists

– Far East2.02 38.17 0.42 0.39

(5.89)**0.15

(3.37)**-0.45

(-2.41)**Sector Specialists –

Property2.06 30.78 0.37 0.25

(7.28)**0.17

(4.70)**-0.36

(-2.65)**European Smaller

Companies2.05 34.67 0.39 0.30

(3.23)**0.11

(2.19)*-0.42

(-2.73)**Average 2.02 0.42

Source : author calculation

* Statistically significant at 5% level

** Statistically significant at 1 % level

From Table 2, we see that on average, three-factor model can explain 42% of the variations in the UK excess discount return by taking into consideration market, size, and the book-to-market effect. As expected the coefficients of the market and size effect are positive and statistically significant for all sectors, while the coefficients of the book-to-market effect are negative and significant for most of the 16 sectors. Thus we find that the book-to-market factor does have an explanatory power. It has a significant negative influence on excess discount return. Our results contradict Pontiff (1997), who found that the book-to-market effect is insignificant and affects funds with low premiums and discounts.

On the other hand, there is no evidence of autocorrelation in the times series of 158 monthly observations from January 1990 to January 2003. On average, the D/W statistic is 2.02. The F-ratio is used to test the overall significance of the regression test. Overall, all the sectors show a significant F-ratio which implies that the model as a whole has a significant degree of explanatory power. As an example, Europe has an F-ratio of 49.67 which exceeds the tabulated value of 3.04 at the 5% significance level with degrees of freedom of 2 and 155.

To summarise the UK results, the UK book-to-market effect, measured as the difference between the return on the FTSE 350 Growth index and the return on the FTSE 350 Value index has a negative and statistically significant influence for most of the AITC sectors. As an example, UK Growth and North America Smaller Companies display significant t-statistics of -3.05 and -3.43. The coefficient for the UK size effect, measured as the difference between the return on the FTSE Smaller Companies index and the return on the FTSE 100 index, is statistically significant at the 1% and 5% level for most of the sectors. For example, Global Growth and UK Growth and Income show significant t-statistics of 4.41 and 3.03. Finally, the market effect is positive and statistically significant for all sectors. Global Growth and UK

63

Growth and Income display significant t-statistics of 7.22 and 9.92. Far East (excluding Japan) and Japan show t-statistics of 7.31 and 5.58.

Table 3 Fama and French’s three factor model of the US excess discount return

The regressions by sector are estimated over the period January 1990 to January 2003 by using monthly observations. The sample covers 30 funds for US closed-end funds. The t-statistic of each factor is displayed in parentheses. In addition, we provide tests of autocorrelation.

Factor 1 Factor 2 Factor 3Fund Name by CEFA category


Market(

Size Book to market

Equity Income 1.99 84.25 0.62 0.62 (11.73)**

0.02(0.55)

0.00(0.01)

Global Equity 1.96 67.45 0.56 0.58 (13.20)**

0.14 (2.50)**

0.01(0.92)

Growth and Income 1.96 67.00 0.48 0.55 (7.53)**

0.09(0.82)

0.06(0.29)

Growth Domestic 1.90 117.14 0.69 0.67 (13.2)**

0.22(2.31)*

0.60 (3.86)**

Average 1.95 0.59 Source : author calculation

* Statistically significant at 5% level

** Statistically significant at 1 % level

From Table 3, we see that the three-factor model can explain 59 % of the variation of the US excess discount returns by taking into consideration market, size, and the book-to-market effect. As expected, coefficients are statistically significant for the market and size effect for most of the sectors. There is no evidence of autocorrelation in the time series of 158 observations from January 1990 to January 2003. On average

64

the Durbin/Watson statistic is 1.95. Overall, all the sectors show a significant F-ratio, implying that the model as a whole has explanatory power. As an example, the Growth and Income sector has an F-ratio of 67.00 which exceeds the tabulated value of 3.04 at the 5% significance level with degrees of freedom of 2 and 155.

To summarise the US results, the US book-to-market effect is statistically insignificant for most of the CEFA sectors except the Growth Domestic sector which has an unexpected positive sign. One possible explanation is that this sector is sensitive to the book to-market index as it is dominated from growth funds and behaves in an irregular way. The t-statistics of the US size effect are statistically significant at the 1% significance level for Global Equity 2.50 and the Growth Domestic sector 2.31. On the other hand, the market effect is highly statistically significant for all sectors. Equity Income, Global Equity, and Growth and Income display significant t-statistics of 11.73, 13.20, and 7.53.

Carhart’s(1997) four-factor model

Carhart (1997) constructed his four-factor model using Fama and French’s (1993) three-factor model plus an additional factor capturing Jegadeesh and Titman’s (1993) one-year momentum anomaly. This was motivated by the three-factor model’s inability to explain cross-sectional variation in portfolio returns (Fama and French (1993)). Chan, Jegadeesh and Lakonishok (1996) suggest that the momentum anomaly is market inefficiency due to slow reaction to information. However, the effect is robust to time periods (Jegadeesh and Titman (1993), Asness, Liew, and Stevens (1996)).

Equation (2) enables us to test the significance of the four factors for each sector in relation to the excess discount return.



is the size factor (small minus big), i.e the difference between

the return on a portfolio of small stocks and the return on a portfolio of

large stocks.

is the book to market factor (high minus low), i.e the

the difference between the return on a portfolio of high book-to-market

stocks and the return on a portfolio of low book-to- market stocks

65

(RNAVfd,t – RNAVse,t) is the momentum factor measured as the

difference between the fund and sector return


We test the significance of the estimated coefficients using t-statistics and F-tests. We also use the Durbin/Watson statistic to check for autocorrelation. The estimation results are shown in Table 4 for the UK and Table 5 for the US.

Table 4 Carhart’s four-factor model of UK excess discount return

The regressions by sector are estimated over the period January 1990 to January 2003 by using monthly observations. The sample covers 90 funds for UK investment trusts. The t-statistic of each factor is displayed in parentheses. In addition, we provide tests of autocorrelation.

Factor 1 Factor 2 Factor 3 Factor 4Fund Name by AITC category


Market(

Size Book to market

Momentum(RNAVfd,t

– RNAVse,t)

Global Growth 2.05 24.23 0.37 0.28 (3.03)**

0.15 (3.22)**

-0.35 (-2.50)**

-0.56(-1.18)

Global Growth & Income

2.10 47.02 0.54 0.48 (9.88)**

0.07 (2.03)*

-0.32 (-2.42)**

0.79(1.23)

Global Smaller Companies

1.98 39.61 0.50 0.45 (9.56)**

0.05(1.51)

-0.38 (-2.85)**

0.12(0.39)

UK Growth 2.08 32.70 0.45 0.33 (2.58)**

0.14 (2.75)**

-0.39 (-2.99)**

-1.24(-1.71)*

UK Growth & Income

2.00 50.80 0.56 0.44 (10.58)**

0.09 (2.84)**

-0.24 (-2.01)*

-3.01 (-2.64)**


2.05 30.77 0.43 0.47 (8.29)**

0.11 (2.03)*

-0.31 (-1.87)*

-0.57 (-1.93)*

North America 1.96 27.92 0.41 0.47 (8.25)**

0.09(1.64)

-0.27 (-1.58)

-0.56(-1.84)*

North America Smaller Companies

2.02 32.11 0.45 0.32 (9.23)**

0.12 (3.35)**

-0.41 (-2.89)**

-0.08(-0.39)

Far East (Including Japan)

2.07 32.15 0.44 0.37 (9.71)**

0.15 (3.74)**

-0.22(-1.50)

-1.13(-0.91)

Far East (Excluding Japan)

2.11 29.68 0.43 0.39 (3.12)**

0.09 (1.75)*

-0.22 (-1.77)*

-1.21 (-2.72)**

Japanese Smaller Companies

2.10 19.93 0.33 0.32 (6.23)**

0.18 (3.74)**

-0.33 (-1.88)*

-0.71(-1.83)*

Japan 2.05 24.23 0.37 0.28 (7.54)**

0.15 (4.17)**

-0.33 (-2.42)**

-0.51(-1.51)

Europe 2.00 32.65 0.45 0.33 (2.77)**

0.12 (2.62)**

-0.41 (-2.76)**

-0.08(-0.42)

Country Specialists 2.04 21.17 0.34 0.33 0.16 -0.38 -0.68

66

- Far East (2.60)** (2.93)** (-2.09)* (-1.82)*Sector Specialists –

Property1.95 22.09 0.35 0.41

(3.44)**0.04

(0.55)-0.26

(-1.51)-0.88

(-1.77)*European Smaller

Companies2.03 31.57 0.37 0.28

(2.63)**0.11

(2.20)*-0.42

(-2.67)**-0.31(1.20)

Average 2.04 0.42Source : author calculation

* Statistically significant at 5 % level

** Statistically significant at 1% level

Table 4 shows that on average, the four-factor model can explain 42% of the variation in the UK excess discount return by taking into consideration market, size, the book-to-market factor and momentum. As expected the coefficients for the UK are positive and statistically significant for the market and the size effect. In addition, the coefficients are negative and statistically significant for the book-to-market effect for some of the sectors. The explanatory power is no better than the three-factor model. However, momentum is only negative and statistically significant (at the 5% level in one-tailed tests) in five out of the sixteen sectors. Thus, we have very limited evidence that managerial performance has any influence on the excess discount return in the UK in the long-run. There is no evidence of autocorrelation. On average the Durbin/Watson statistic is 2.04. Overall, all the sectors show a significant F-ratio. As an example, the North America Smaller Companies sector has an F-ratio of 32.11 which exceeds the tabulated value of 2.65 at the 5% significance level with degrees of freedom of 3 and 154.

To summarise the UK results, the book-to-market effect is negative and statistically significant for some of the AITC sectors and insignificant for most of the sectors. As an example, UK Growth and Global Growth and Income display t-statistics of -2.99 and -2.42. The UK size effect is positive and statistically significant at the 1% and 5% level for most of the sectors. For example, Global Growth and Far East including Japan show significant t-statistics of 3.22 and 3.74. Also, the market effect is positive and highly statistically significant for all sectors at the 1% level. For example, UK Growth and Income and UK Smaller Companies display t-statistics of 10.58 and 8.29. Japan and Sector Specialist Property shows t-statistics for the market effect of 7.54 and 3.44.

Table 5 Carhart’s four -factor model of the US excess discount return

The regressions by sector are estimated over the period January 1990 to January 2003 by using monthly observations. The sample covers 30 funds for US closed-end funds. The t-statistic of each factor is displayed in parentheses. In addition, we provide tests of autocorrelation .

Factor 1 Factor 2 Factor 3 Factor 4Fund Name by CEFA category

D/W F-ratio AdjustedR2

Market(

Size Book to market

Momentum(RNAVfd,t – RNAVse,t)

Equity Income

1.93 83.32 0.68 0.65 (12.41)**

0.44 (2.94)**

0.08(1.05)

0.09(0.24)

67

Global Equity 2.02 52.00 0.57 0.61 (8.01)**

0.46 (2.81)**

0.10(0.93)

0.06(0.13)

Growth and Income

1.92 77.23 0.66 0.55 (12.20)**

0.48 (2.92)**

0.16(-2.01)*

0.03(0.11)

Growth Domestic

1.97 89.41 0.70 0.68 (14.29)**

0.53 (3.32)**

0.12(1.40)

-0.00(-0.04)

Average 1.96 0.65

Source : author calculation

* Statistically significant at 5% significance level

** Statistically significant at 1 % significance level

Table 5 shows that, on average, the four-factor model can explain 65% of the variation in the US excess discount return by taking into consideration the market effect, size, the book-to-market effect and momentum. Thus, there is a small improvement compared with the three-factor model. The coefficients for the US are statistically significant for the market and the size effect for all sectors. There is no autocorrelation. On average the Durbin/Watson statistic is 1.96. Overall, all the sectors show a significant F-ratio. As an example, Growth Domestic sector has an F-ratio of 89.41 which exceeds the tabulated value of 2.65 at the 5% significance level with degrees of freedom of 3 and 154.

However, the US book-to-market effect is statistically insignificant for most of the CEFA sectors. As an example, Global Equity has an insignificant t-statistic of 0.93. The t-statistic for the US size effect is statistically significant at the 1% significance level for Global Equity 2.81 and Growth Domestic sector 3.32. Finally, the market effect is highly statistically significant for all sectors. Equity Income, Global Equity and Growth and Income display significant t-statistics of 12.41, 8.01, and 12.20. However, the momentum variable is insignificant in all four sectors.

Guirguis six-factor model (I)

In this section we extend the four-factor model of Carhart (1997) by adding sentiment and expenses, as defined earlier. The purpose is to strengthen the four factor model and finally verify which factors are statistically significant and able to explain the fluctuations and persistence of the excess discount return over the longer term.

Equation (3) enables us to test the significance of the six factors in relation to excess discount return.



68



large stocks.

is the book to market factor (high minus low), ie the

difference between the return on a portfolio of high book-to-market

stocks and the return on a portfolio of low book-to- market stocks.

(RNAVfd,t – RNAVse,t) is the momentum factor

(Flowt – Flowt -1) is the difference of retail flows by sector used as a

proxy for investor sentiment

(expt) is the expense factor.


We test the significance of the estimated coefficients using t-statistics and F-tests. We also use the Durbin/Watson statistic to check for autocorrelation. The estimated results are shown in Table 6 for the UK and Table 7 for the US.

Table 6 Guirguis six -factor model of the UK excess discount return.

Equation (3) measures the sensitivity of the excess discount related to the market, size, book- to-market factor, momentum, sentiment and expenses. The regressions by sector are estimated over the period January 1990 to January 2003 by using monthly observations. The sample covers 90 funds for UK investment trusts. The coefficient and t-statistic of each factor is displayed in parenthesis. In addition, we provide tests of autocorrelation.

Factor 1 Factor 2 Factor 3 Factor 4 Factor 5 Factor 6Fund

Name byAITC

category

D/W F/ratio Adj R2 Market Size Book to market

Momentum

(RNAVfd,t

– RNAVse,t)

Sentiment(Flowt - Flowt -1)

Expense(expt)

Global Growth

2.12 17.25 0.39 0.28 (3.98) **

0.13 (2.48) **

-0.01(-0.57)

-0.26(-1.25)

1.49 (3.20)**

0.59(1.38)

Global Growth &

Income

1.93 50.24 0.66 0.43 (10.49) **

0.12 (3.50) **

-0.14(-1.23)

0.33(0.66)

0.87 (3.54)**

-0.25(-1.06)

Global Smaller

Companies

1.96 42.83 0.62 0.40 (7.91)**

0.11 (3.16)**

-0.09(-0.68)

0.60(-1.96)*

0.82 (7.50)**

-0.04(-0.14)

UK Growth

2.04 60.65 0.70 0.58 (12.90)**

0.06 (2.11) *

-0.15(-1.41)

-2.76 (-6.89)**

0.88 (3.18)**

0.08(0.46)

69

UK Growth &

Income

1.97 57.98 0.69 0.45 (11.31)**

0.10 (3.22)**

-0.14(-1.31)

-2.43 (-2.54)**

0.89 (6.60)**

0.68(1.84)*

UK Smaller

Companies

1.99 45.06 0.63 0.40 (9.90)**

0.09 (2.83)**

-0.11(-0.90)

-0.22(-0.49)

1.02 (5.81)**

-0.04(-0.09)

North America

2.02 72.44 0.73 0.42 (9.42)**

0.13 (3.52)**

-0.14(-1.07)

0.02(0.16)

0.98 (13.41)**

0.03(0.06)

North America Smaller

Companies

1.97 70.10 0.73 0.43 (10.61)**

0.11 (3.82)**

-0.17(-1.42)

-1.02(-1.47)

0.97 (11.60)**

-0.11(-0.55)

Far East (Including

Japan)

2.05 65.31 0.71 0.45 (11.27)**

0.13 (4.18)**

-0.18(-1.42)

-1.44(-1.53)

0.68 (6.52)**

-0.49(-2.45)**

Far East (Excluding

Japan)

2.06 62.34 0.70 0.37 (4.78)**

0.12 (3.11)**

-0.15(-1.30)

-1.20 (-3.57)**

0.92 (7.97)**

-0.55(-1.73)*

Japanese Smaller

Companies

1.97 87.79 0.77 0.44 (10.13)**

0.10 (3.07)**

-0.15(-1.33)

0.15(1.05)

0.93(12.98)**

1.13 (2.29)**

Japan 1.94 92.07 0.78 0.46 (11.41)**

0.12 (3.98)**

-0.19 (-1.82)*

0.17(1.42)

0.94(14.58)**

-0.59(-2.71)*

Europe 2.02 49.04 0.65 0.49 (11.26)**

0.10 (3.12)**

-0.34 (-2.81)**

0.15(0.87)

0.32 (3.89)**

-0.34(-0.98)

Country Specialists – Far East

1.97 41.03 0.61 0.39(9.67)**

0.12 (3.33)**

-0.17(-1.23)

-0.27(-0.91)

0.78(8.89)**

-0.21(-0.64)

Sector Specialists – Property

1.97 95.97 0.79 0.43 (9.82)**

0.11 (3.66)**

-0.20 (-1.80)*

-0.15(-0.58)

1.02 (14.78)**

0.14(0.87)

European Smaller

Companies

1.96 53.41 0.61 0.39 (9.67)**

0.12(3.33)**

-0.17(-1.23)

-0.27(-0.91)

0.78 (8.89)**

-0.21(-0.64)


* Statistically significant at 5 % significance level

** Statistically significant at 1% significance level

Table 6 shows that, on average, the six-factor model can explain 67% of the variation in the excess discount return by taking into consideration market, size, book-to- market effect, momentum, sentiment and expenses. The coefficients for the UK are statistically significant for the market, the size effect and sentiment. Book-to-market effect, momentum and expenses show t-statistics that are insignificant or negatively significant. In more detail, momentum is negative and statistically significant, (at the 1% and 5% level in one tailed tests), in four out of the sixteen sectors. Thus, we have limited evidence of managerial performance persistence over the long-run. In addition, our results contradict the hypothesis proposed by Barclay, Holderness and Pontiff’s (1995) view that a possible explanation of the discount is that as managers try to protect their private benefits, expenses increase over the long term and have an effect on the discount. There is no autocorrelation in the times series of our 158 monthly observations from January 1990 to January 2003. On average the Durbin/Watson statistic is 2.00. Overall, all the sectors show a significant F-ratio. As

70

an example, Japan sector has an F-ratio of 92.07 which exceeds the tabular value of F =2.27 at the 5% significance level with degrees of freedom of 5 and 152.

To summarise the UK results, the book-to-market effect and expenses are statistically insignificant for most of the AITC sectors. For example, Global smaller companies, UK smaller companies and North America smaller companies have insignificant t-statistics for the book-to-market effect of -0.68, -0.90, and -1.42. In addition, the same sectors have insignificant t-statistics for expenses of -0.14, -0.09, and -0.55. The t-statistics for the UK size effect and sentiment are statistically significant at the 1% level for most of the sectors. For example, North America Smaller companies, Japanese Smaller companies, Japan and Country specialist Far East have significant t-statistics for the size effect of 3.82, 3.07, 3.98 and 3.33. The same sectors display a statistically significant sentiment effect of 11.60, 12.98, 14.58, and 8.89.

As in the other models, the market effect is positive and highly statistically significant for all sectors. UK Growth, Far East Including Japan, and European Smaller Companies display significant t-statistics of 12.90, 11.27 and 9.67.

Table 7 Guirguis six-factor model (I) of the US excess discount return.

The regressions by sector are estimated over the period January 1990 to January 2003 by using monthly observations. The sample covers 30 funds for US closed-end funds. The t-statistic of each factor is displayed in parentheses. In addition, we provide tests of autocorrelation.

Factor 1 Factor 2 Factor 3 Factor 4 Factor 5 Factor 6Fund Name

byCEFA

category

D/W F/Ratio Adj R2

Market Size Book to market


Sentiment(Flowt - Flowt -1)

Expense(expt)

Equity Income

1.98 43.14 0.62 0.40 (7.95)**

0.11 (3.26)**

-0.09(-0.69)

0.60(1.98)*

0.83 (7.53)**

-0.05(-0.20)

Global Equity

2.00 77.15 0.75 0.43 (9.70)**

0.11 (3.32)**

-0.17(-1.44)

-0.17(-0.61)

0.92 (8.80)**

0.18(0.91)

Growth and Income

2.00 56.53 0.68 0.44 (11.19)**

0.10 (3.38)**

-0.20 (-2.48)**

-2.60 (-3.04)**

0.81 (6.01)**

0.61(1.60)

Growth Domestic

2.02 46.28 0.64 0.40 (9.86)**

0.09 (2.77)**

-0.09(-0.72)

-0.22(-0.50)

1.02 (6.06)**

0.05(0.11)


* Statistically significant at 5% significance level

** Statistically significant at 1 % significance level

On average, for the US the six-factor model can explain 66 % of the variation in the excess discount return by taking into consideration market, size, book-to- market effect, momentum, sentiment and expenses. The coefficients for US are statistically

71

significant for the market and the size effect for all sectors. There is no autocorrelation as on average the Durbin/Watson statistic is 2.00. Overall, all the sectors show a significant F-ratio. As an example, Global Equity sector has an F-ratio of 77.15 which exceeds the tabulated value of 2.27 at the 5% significance level with degrees of freedom of 5 and 152. The momentum variable is significant in just two of the four sectors and the expenses variable is insignificant in all sectors. The sentiment factor is highly significant in all four sectors. In more detail, Equity Income, Global Equity, Growth and Income and Growth Domestic category have t-statistics of 7.53, 8.80, 6.01 and 6.06.

The US book-to-market effect is statistically insignificant for most of the CEFA sectors. As an example, Global Equity display an insignificant t-statistic of -1.44. The t-statistic for the US size effect is statistically significant at the 1% significance level for all sectors. Global Equity, Equity Income and Growth and Income have t-statistic of 3.32, 3.26 and 3.28. Similarily, the market effect is highly statistically significant for all sectors. Equity Income, Global Equity and Growth and Income have t-statistics of 7.95, 9.70, and 11.19.

Guirguis six-factor model (II)

In this section we extend the four-factor model of Carhart (1997) by changing the sentiment factor from retail flow to investor sentiment index. The only behavioural data available was for the US market. Investors are said to be confident when the news about the future is good and stock prices are relatively high. According to Yale Management School, the sample of US individual investors from 1989 to 1998 was purchased from W.S.Ponton Inc, a list of “High-Grade Multi-Investors.” Starting in 1999, the sample was a random sample of high-income Americans purchased from Survey Sampling, Inc. We use only US investor sentiment index constructed from Yale School of Management as it was the most suitable one to represent investor sentiment on the market. It is measured in percentage of respondents who support a particular view. It is derived from the responses to a single question that has been asked consistently through time since 1989 to a consistent sample of respondents. The wordings of the question focused on how pessimistic or optimistic are investor’s towards the stock market.

Equation (4) enables us to test the significance of these six factors in relation to excess discount return.

where: is the excess discount return of each sector at time t



72


large stocks.

(Rg,t – Rv,t) is the book to market factor (high minus low), i.e the

difference between the return on a portfolio of high-book-to-market

stocks and the return on a portfolio of low-book-to- market stocks

(RNAVfd,t – RNAVse,t) is the momentum factor

(ISIt ) is the difference in percentage of respondents as a proxy of

sentiment factor.

(expt) is the expense factor.


We test the significance of the estimated coefficient using t-statistics and F-tests. We also use Durbin/Watson statistic to check for autocorrelation.

Table 8 Guirguis six factor model (II) of US excess discount return

The regressions by sector are estimated over the period October 1996 to January 2003 by using monthly observations. The sample covers 30 funds for US Closed-end funds. The t-statistic of each factor is displayed in parentheses. In addition, we provide tests of autocorrelation.

Factor 1 Factor 2 Factor 3 Factor 4 Factor 5 Factor 6Fund Name

byCEFA

category

D/W F/ratio Adj R2 Market Size Book to market


Sentiment(ISIt )

Expense(expt)

Equity Income

1.97 2.83 0.32 0.06(0.66)

-0.11(-0.98)

-0.22(-0.56)

-0.49 (-8.35)**

0.63 (2.37)**

3.75(1.73)*

Global Equity 2.07 2.38 0.26 0.25 (2.28)*

-0.13(-0.99)

0.12(0.55)

-0.20(-2.08)*

0.03(0.39)

2.37(1.44)

Growth and 2.04 3.18 0.36 0.35 0.13 -0.21 -3.10 0.29 -1.19

73

Income (2.65)** (1.46) (-1.09) (-1.36) (1.93)* (-0.87)Growth Domestic

2.03 3.50 0.39 0.33 (3.52)**

-0.04(-0.43)

-0.19(-1.05)

-1.26(-1.63)

0.21(1.34)

1.11(0.52)


* Statistically significant at 5 % significance level

** Statistically significant at 1% significance level

Table 8 shows that, on average, the six-factor model can explain 0.33 % of the variation in the excess discount return by taking into consideration market effect, size, book-to- market, momentum, sentiment and expenses. The coefficients for US market are statistically significant for 3 out of four sectors at the 1% and 5% for one-tailed tests. On the other hand, there is no autocorrelation in the times series of our 26 monthly observations from October 1996 to January 2003. On average the Durbin/Watson statistic is 2.03. Overall, all the sectors show a significant F-ratio. As an example, Equity Income sector has an F-ratio of 2.83 which exceeds the tabulated value of F 2.27 at the 5% significance level with degrees of freedom of 5 and 152. The US size, book-to-market effect, momentum and expenses are statistically insignificant for most of the sectors of CEFA. On the other hand, the results for the sentiment factor are mixed. In more detail, Equity Income, Growth and Income show a statistically significant sentiment effect of 2.37 and 1.93. A possible explanation of this mixed picture is that the sample is small and surveys were initially conducted at six-month intervals from October 1996 to April 2001. Then surveys were conducted at a monthly basis. Our sample includes only 25 observations which reduce the explanatory power of the model. Thus, a possible area of future research is to include a larger sample measured on a daily basis and try to isolate the interaction of arbitrageurs and noise traders to identify shifts in the discounts of closed end funds.

In this section we attempted to extend the three-factor of Fama and French’s (1993) model in order to explain the existence and persistence of the excess discount return. We added two more factors to these four risk measures namely market, size, book-to-market, and momentum. The first one is based on the results of Lee, Shleifer and Thaler (1991) that closed-end funds are subject to systematic “investor sentiment risk”. We measure this sentiment in two ways as the changes in retail flows and investor sentiment index. This sentiment measure excludes the funds during their first six month of trading. The reason is to eliminate the influence of premiums that correspond to underwriting fees and start-up costs.

The expense ratio can be proxied monthly as percentage of total assets. These include the annual management charge (or staff costs), auditors and custody fees, directors fees and marketing costs.

On average the six factor model can explain 67% of the variation in the excess discount return in the UK market by taking into consideration market, size, book-to-market effect, momentum, sentiment and expenses. In contrast, Fama and French (1993) three factor and four factor model can explain 42% of the variation of the

74

excess discount return. The coefficients of the six-factor model for UK are statistically significant for market, size effect and sentiment and statistically insignificant for book-to-market, momentum, and expenses. On the other hand, there is no autocorrelation in the times series. On average the Durbin/Watson statistic is 1.98. The six-factor model is very robust with emphasis on sentiment namely the interaction of arbitrageurs and noise traders.

On the other hand, on average the six-factor model can explain 66 % of the variation of the excess discount return in the US market by taking into consideration market effect, size, book-to-market, momentum, sentiment and expenses. In contrast, Fama and French (1993), three-factor model can explain 59% of the excess discount return variation and Carhart’s (1997) four factor model can explain 65% of the variation.

The other version of my six factor model applied in the US market can explain 33 % of the variation of the excess discount returns by changing the sentiment factor from retail flow to investor sentiment index. The only behavioural data available was for the US market from the Yale management school. The index was measured in percentage of respondents who support a particular view. Our sample includes only 25 observations which decrease the explanatory power of the model. Thus, a possible area of future research is to include a larger sample measured on a daily basis and try to isolate the interaction of arbitrageurs and noise traders to identify shifts in the discounts of closed end funds.

75

Heteroskedasticity

One of the assumptions of the simple and the multiple regression is that the variance of the error term, , is constant and common. This is known as the homoskedasticity assumption.

E( )2 = σ2

This assumption ensures that each observation is equally reliable, so that estimates of the regression coefficients and tests of hypotheses are not biased. If the errors do not have a common variance, we say that they are heteroskedastic. This leads to biased estimates of the standard errors, higher variance of the estimators and thus incorrect statistical tests and confidence intervals. To overcome this problem we use EViews 6 heteroskedastic tests. We will analyze various tests based mainly on the ordinary least squares, (OLS) residuals. We will also discuss the application of heteroskedasticity on the residuals of the likelihood ratio test. The presence of heteroskedasticity shows that the standard errors and the t-statistics are biased. To overcome the problem we transform the variables to natural logarithmic linear form, ln. Another alternative to eliminate the heteroskedasticity problem and get efficient estimates of the coefficients is to deflate the variables by using the reciprocal of the square root or the original value of one of the independent variable. If you have more than one independent variables that show heteroskedastic problem, then, use the square root of the mean value of the dependent variable as a transformed value. You could also use the ratio from a new variable. The problem that could arise is spurious correlation between the new variable related to the dependent and independent variables. The above methods are applied when the variance of the model is unknown. When the error variance is known, then, the transformed regression is obtained by dividing each dependent and independent variable by its own standard deviation. In EViews 6 you select the weighted least squares option because each variable is weighted or divided by the relevant dataset standard deviation.

I will give you an example of using square root reciprocal as deflators. This method is based on the square root transformation. Please consider the following ordinary least square,OLS, equation. After conducting a scatterplot, we have found that the independent variable x1 shows heteroskedastic patterns with the error term. We divide each part of the regression with the square root of x1, namely, .

76

Then, run the regression in Excel and obtain the estimated ordinary least squares, OLS residuals . Regress the estimated squared residuals on the independent variable x1 and the squared independent variable each time separately and choose the regression model with the highest R2.

I have included a graph to show the homoskedastic effect. There is constant or equal variance along the line between the dependent and independent variable. There are no substantial deviations.

y

* ********* * * * * * * * * ****** * * * * * 0 x

I have also included a graph to show the heteroskedastic effect. The variance is not constant and is usually revealed in cross – sectional data. It decreases and then increases along the line of the explanatory variable x.Thus, we check the squared residual values in relation to the independent or explanatory variables. To overcome the problem, we use ln linear form to regress the dependent variable in relation to the independent and check again the residual or error term.

y * * * * * * * * * * * * * * * * * * * * * * * * 0

77

x

To check for heteroskedasticity plot the squared residuals across each observations and against each independent variable.

For example, let’s consider the following dataset and the squared residuals plot. Then, plot in a line graph the squared residuals along the independent variable. In our case, it is income and it is expressed in thousands.

ObservationResiduals squared Incomes

1 44602.429 10312 42083.497 12143 36.062242 14144 5379.947 16485 23000.459 18216 66332.716 20147 84.647914 21458 4059.1177 22349 1745.0951 2245

10 16.634201 229811 103264.69 234512 25512.72 241113 4.1959599 246714 103464.45 251815 23159.766 283916 9754.9813 311517 6983.4371 323118 62439.153 335119 42570.8 356120 135202.23 380021 108.91533 412522 6519.9155 433123 56651.487 437824 4525.1809 624525 71088.105 8100

78

Squared residuals plot

020000400006000080000

100000120000140000160000

Observations

Squa

red

resi

dual

s

I have plotted the squared residuals against the independent variable incomes. Please check the variability and determine if it is constant or unequal.

Squared residuals plot

0

20000

40000

60000

80000

100000

120000

140000

160000

1031

1414

1821

2145

2245

2345

2467

2839

3231

3561

4125

4378

8100

Incomes

Squa

red

resi

dual

s

Series1Incomes

79

The most common heteroskedastic tests that are available in EViews 6 are the Breusch - Pagan – Godfrey test, the Harvey test, the Glejser test, the ARCH and White test. The Breusch - Pagan – Godfrey test regress the squared residuals on the original regressors. The Harvey test regresses the ln of the squared residuals on the original regressors. The Glejser test regresses the absolute residuals on the original regressors. The ARCH test regresses the squared residuals on lagged squared residuals and a constant. The White test regresses the squared residuals on the cross product of the original regressors and a constant.

First of all, you need to collect your data and define the dependent and independent variable, then transfer your data from Excel to EViews 6. Then, estimate the regression equation by selecting the method of least squares. The regression equation in EViews 6 will be as follows:

share c market

Press OK and you will get the output of the regression equation. Then go to the upper left and press view, then, select residual tests and then heteroskedasticity tests. Then, you have the option to select the following heteroskedasticy tests:

The Breusch - Pagan – Godfrey test The Harvey test The Glejser test The ARCH The White test

To sum up, the heteroskedasticity tests are related to test the residual or error term that has a constant variance. In addition to the test outptut, EViews 6 provide a residual graph. The most common heteroskedasticity tests are the ARCH and the White tests.

Let’s try together an example to facilitate your learning experience. I have attached the data that I will use in this example.

Share returns (y) Market returns (x)

3.526787 8.73209-4.34533 -5.198155.222709 6.21865-4.99619 -5.5393-3.04336 7.69808-2.375422 -4.997352.651303 5.42777-0.68924 -1.54240.205664 1.46392.4783 3.6528

0.237407 -0.1494

80

0.329728 0.16688-0.26869 -0.14440.064769 0.097873-0.5873 -0.09911

0.329225 -0.08344-0.11849 0.1227670.011541 -0.45767-0.18757 -0.53046-0.38752 -0.11118-0.26835 -0.289470.262798 -0.176760.355054 -1.15686-1.34302 -0.5771-0.77964 0.578182-0.04649 -0.053310.098381 -0.23054-0.09585 -0.66625-0.0059 -0.50071-0.05415 -0.53128

This is the regression output that you will get in EViews 6.

Dependent Variable: SHAREMethod: Least SquaresDate: 04/16/16 Time: 10:16Sample: 1 30Included observations: 30


C -0.284046 0.271224 -1.047275 0.3039MARKET 0.422744 0.084654 4.993807 0.0000


Then, press view. Select the residual tests and then the heteroskedasticity tests. In this case, I choose the Breusch - Pagan – Godfrey test. It regress the squared residuals on

81

the original regressors. At the bottom of the test, you can see the regressors. In our case, we have one regressor and it is the market. Thus, you estimate the error term. You square it and finally you regress it in relation to the explanatory variable, which in our case is the market. I have attached the relevant dataset to facilitate your learning experience. You will find a datset series in EViews 6 denoted as resid and it means the residual or the error term. This time series is very important to test heteroskedasticity and regresses it with the explanatory variable and a constant according to the relevant test.

Error termError term squared (

) Market returns (x)0.119395394 0.01425526 8.73209

-1.863797548 3.473741301 -5.198152.877858835 8.282071477 6.21865

-2.370438465 5.618978517 -5.5393-6.01363018 36.16374794 7.698080.021223476 0.000450436 -4.997350.640792535 0.410615073 5.427770.24684642 0.060933155 -1.5424

-0.129144582 0.016678323 1.46391.218147284 1.483882807 3.65280.584611161 0.341770209 -0.14940.543226718 0.295095267 0.166880.076400441 0.005837027 -0.14440.307440007 0.094519358 0.097873-0.26135563 0.068306766 -0.099110.648544973 0.420610581 -0.083440.11365722 0.012917964 0.122767

0.489064424 0.239184011 -0.457670.320724953 0.102864496 -0.53046

-0.056473111 0.003189212 -0.111180.138067899 0.019062745 -0.289470.621568434 0.386347318 -0.176761.128155735 1.272735363 -1.15686

-0.815008271 0.664238482 -0.5771-0.740016695 0.547624709 0.5781820.260092699 0.067648212 -0.053310.479886601 0.23029115 -0.230540.469849348 0.22075841 -0.666250.489818322 0.239921988 -0.500710.454491603 0.206562617 -0.53128

82

If you regress the squared error term as the dependent variable in relation to the market return as the independent variable, then, you will get the following output.

Heteroskedasticity Test: Breusch-Pagan-Godfrey

F-statistic 5.965804 Prob. F(1,28) 0.0212Obs*R-squared 5.269244 Prob. Chi-Square(1) 0.0217Scaled explained SS 24.11401 Prob. Chi-Square(1) 0.0000

Test Equation:Dependent Variable: RESID^2Method: Least SquaresDate: 04/16/16 Time: 10:17Sample: 1 30Included observations: 30


C 1.710507 1.137900 1.503213 0.1440MARKET 0.867473 0.355158 2.442499 0.0212

R-squared 0.175641 Mean dependent var 2.032161Adjusted R-squared 0.146200 S.D. dependent var 6.699751S.E. of regression 6.190656 Akaike info criterion 6.548300Sum squared resid 1073.078 Schwarz criterion 6.641713Log likelihood -96.22450 Hannan-Quinn criter. 6.578183F-statistic 5.965804 Durbin-Watson stat 1.347827Prob(F-statistic) 0.021154

Please state the hypotheses:

H0: The error term is homosedasticH1: The error term is heteroskedastic

Please focus on the output of the first two sentences of the table that are bold. According to the above table, both the F-statistic and the Chi-squared statistics are significant. We reject the null hypothesis of homoskedasticity. Heteroskedasticity effect is present in the price change returns expressed as percentages. Specifically, the F-statistic is 5.9658 with a significant p-value of 0.02 and the probability Chi-square value is also significant with a numerical value of 0.02 and a Langrange LM statistic of 5.269. Please note the number 1 close to Prob. Chi-square(1), which mens that we have one explanatory variable.

Please feel free to e-mail me if you have more questions. Thanks for your participation and I wish you all the best in your future career in the Financial Services sector.

83

The Harvey test regresses the natural logarithmic ln of the squared residuals on the original regressors.I have attached the relevant dataset based on the above example. Thus, you regress the ln squared error as the dependent variable in relation to the market returns as the independent variable.

Error term Error term squared ln squared error ( ) Market returns (x)0.119395394 0.01425526 -4.250629302 8.73209

-1.863797548 3.473741301 1.245232198 -5.198152.877858835 8.282071477 2.114093115 6.21865

-2.370438465 5.618978517 1.726149889 -5.5393-6.01363018 36.16374794 3.588057179 7.698080.021223476 0.000450436 -7.705294746 -4.997350.640792535 0.410615073 -0.890099065 5.427770.24684642 0.060933155 -2.797977831 -1.5424

-0.129144582 0.016678323 -4.093645429 1.46391.218147284 1.483882807 0.39466217 3.65280.584611161 0.341770209 -1.073616671 -0.14940.543226718 0.295095267 -1.220457034 0.166880.076400441 0.005837027 -5.14353362 -0.14440.307440007 0.094519358 -2.35895062 0.097873-0.26135563 0.068306766 -2.683746461 -0.099110.648544973 0.420610581 -0.866047858 -0.083440.11365722 0.012917964 -4.349136402 0.122767

0.489064424 0.239184011 -1.430522103 -0.457670.320724953 0.102864496 -2.274342734 -0.53046

-0.056473111 0.003189212 -5.747981315 -0.111180.138067899 0.019062745 -3.960019381 -0.289470.621568434 0.386347318 -0.951018527 -0.176761.128155735 1.272735363 0.241168413 -1.15686

-0.815008271 0.664238482 -0.409114034 -0.5771-0.740016695 0.547624709 -0.602165063 0.5781820.260092699 0.067648212 -2.693434356 -0.053310.479886601 0.23029115 -1.468410902 -0.230540.469849348 0.22075841 -1.510686344 -0.666250.489818322 0.239921988 -1.427441457 -0.500710.454491603 0.206562617 -1.577151681 -0.53128

I have attached the output that you will get in EViews 6.

84

Heteroskedasticity Test: Harvey


Test Equation:Dependent Variable: LRESID2Method: Least SquaresDate: 04/17/16 Time: 10:41Sample: 1 30Included observations: 30


C -1.796732 0.442085 -4.064223 0.0004MARKET 0.155153 0.137982 1.124444 0.2704




Please focus on the output of the first two sentences of the table that are bold. According to the above table, both the F-statistic and the Chi-squared statistics are not significant. We can not reject the null hypothesis of homoskedasticity. Heteroskedasticity effect is not present in the price change returns expressed as percentages. Specifically, the F-statistic is 1.26 with a unsignificant p-value of 0.27 and the probability Chi-square value is also not significant with a numerical value of 0.25 and a Langrange LM statistic of 1.296.

Please feel free to e-mail me if you have more questions. Thanks for your participation.

The Glejser test regresses the absolute residuals on the original regressors. I have attached the relevant dataset based on the above example. Thus, you regress the absolute error term as the dependent variable in relation to the market returns as the independent variable. In Excel, you use the function = abs() to get the absolute value.

Error termAbsolute error term (

) Market returns (x)0.119395394 0.119395394 8.73209

-1.863797548 1.863797548 -5.19815

85

2.877858835 2.877858835 6.21865-2.370438465 2.370438465 -5.5393

-6.01363018 6.01363018 7.698080.021223476 0.021223476 -4.997350.640792535 0.640792535 5.427770.24684642 0.24684642 -1.5424

-0.129144582 0.129144582 1.46391.218147284 1.218147284 3.65280.584611161 0.584611161 -0.14940.543226718 0.543226718 0.166880.076400441 0.076400441 -0.14440.307440007 0.307440007 0.097873-0.26135563 0.26135563 -0.099110.648544973 0.648544973 -0.083440.11365722 0.11365722 0.122767

0.489064424 0.489064424 -0.457670.320724953 0.320724953 -0.53046

-0.056473111 0.056473111 -0.111180.138067899 0.138067899 -0.289470.621568434 0.621568434 -0.176761.128155735 1.128155735 -1.15686

-0.815008271 0.815008271 -0.5771-0.740016695 0.740016695 0.5781820.260092699 0.260092699 -0.053310.479886601 0.479886601 -0.230540.469849348 0.469849348 -0.666250.489818322 0.489818322 -0.500710.454491603 0.454491603 -0.53128

I have attached the output that you will get in EViews 6.

Heteroskedasticity Test: Glejser


Test Equation:Dependent Variable: ARESIDMethod: Least Squares

86

Date: 04/17/16 Time: 10:42Sample: 1 30Included observations: 30


C 0.770168 0.208943 3.686019 0.0010MARKET 0.125378 0.065215 1.922539 0.0648




Please focus on the output of the first two sentences of the table that are bold. According to the above table, both the F-statistic and the Chi-squared statistics are not significant. We can not reject the null hypothesis of homoskedasticity. Specifically, the F-statistic is 3.696 with a unsignificant p-value of 0.06 and the probability Chi-square value is also not significant with a numerical value of 0.06.


The ARCH test regresses the squared residuals on lagged squared residuals and a constant. I have attached the output that you will get in Eviews 6.

Heteroskedasticity Test: ARCH

F-statistic 0.383134 Prob. F(1,27) 0.5411Obs*R-squared 0.405756 Prob. Chi-Square(1) 0.5241

Test Equation:Dependent Variable: RESID^2Method: Least SquaresDate: 04/17/16 Time: 10:43Sample (adjusted): 2 30Included observations: 29 after adjustments


87

C 1.853995 1.339443 1.384153 0.1776RESID^2(-1) 0.118251 0.191043 0.618978 0.5411

R-squared 0.013992 Mean dependent var 2.101744Adjusted R-squared -0.022527 S.D. dependent var 6.807298S.E. of regression 6.883546 Akaike info criterion 6.762617Sum squared resid 1279.347 Schwarz criterion 6.856913Log likelihood -96.05795 Hannan-Quinn criter. 6.792149F-statistic 0.383134 Durbin-Watson stat 2.026327Prob(F-statistic) 0.541119



Please focus on the output of the first two sentences of the table that are bold. According to the above table, both the F-statistic and the Chi-squared statistics are not significant. We can not reject the null hypothesis of homoskedasticity. Heteroskedasticity effect is not present in the price change returns expressed as percentages .

Please feel free to e-mail me if you have more questions. Thanks for the participation.

The White test regresses the squared residuals on the cross product of the original regressors and a constant. First of all, I have attached the simple regression equation

share c market by selecting the following option box:

The White test is very common as it provides the researcher with corrected standard errors and t-statistics. It corrects the heteroskedasticity problem and provide us with efficient and unbiased estimators.

White Heteroskedasticity-Consistent Standard Errors & Covariance

Dependent Variable: SHAREMethod: Least SquaresDate: 04/17/16 Time: 10:43Sample: 1 30Included observations: 30White Heteroskedasticity-Consistent Standard Errors &

88

Covariance


C -0.284046 0.231710 -1.225872 0.2305MARKET 0.422744 0.172422 2.451802 0.0207


I have attached the relevant dataset based on the above example. Thus, you regress the squared residual or squared error term as the dependent variable in relation to the market returns and market returns squared as the independent variables. If you have two independent variables, then, you run the auxiliary regression of the squared residuals on the original values, their squared values, and their cross – products. For example, consider the following multiple regression with two independent variables.

The regression of the squared residuals is as follows:

In our case, we are dealing with a simple regression equation with one independent variable:

Where : the dependent variable is the error term squared.

Error term squared () Market returns (x1)

Market retunrs squared ( )

0.01425526 8.73209 76.249395773.473741301 -5.19815 27.020763428.282071477 6.21865 38.671607825.618978517 -5.5393 30.6838444936.16374794 7.69808 59.260435690.000450436 -4.99735 24.973507020.410615073 5.42777 29.460687170.060933155 -1.5424 2.378997760.016678323 1.4639 2.143003211.483882807 3.6528 13.342947840.341770209 -0.1494 0.022320360.295095267 0.16688 0.027848934

89

0.005837027 -0.1444 0.020851360.094519358 0.097873 0.0095791240.068306766 -0.09911 0.0098227920.420610581 -0.08344 0.0069622340.012917964 0.122767 0.0150717360.239184011 -0.45767 0.2094618290.102864496 -0.53046 0.2813878120.003189212 -0.11118 0.0123609920.019062745 -0.28947 0.0837928810.386347318 -0.17676 0.0312440981.272735363 -1.15686 1.338325060.664238482 -0.5771 0.333044410.547624709 0.578182 0.3342944250.067648212 -0.05331 0.0028419560.23029115 -0.23054 0.0531486920.22075841 -0.66625 0.443889063

0.239921988 -0.50071 0.2507105040.206562617 -0.53128 0.282258438

I have attached the output that you will get in Eviews 6.

Heteroskedasticity Test: White


Test Equation:Dependent Variable: RESID^2Method: Least SquaresDate: 04/17/16 Time: 10:44Sample: 1 30Included observations: 30White Heteroskedasticity-Consistent Standard Errors & Covariance


C 0.219509 0.428405 0.512386 0.6125MARKET 0.313314 0.412730 0.759124 0.4544

MARKET^2 0.165266 0.116086 1.423648 0.1660


90



The R2 value times the number of observations follows the x2 distribution with degrees of freedom equal to the number of the independent variables excluding the intercept. If the p-value of the calculated chi-square is below the 1% or 5% significance level, then, reject the null hypothesis of homoskedasticity or the existence of no heteroskedasticity. In contrast, if the p – value of the chi-square value is above the 5% significance level, then, do not reject the null hypothesis.

Please focus on the output of the first two sentences of the table that are bold. According to the above table, both the F-statistic and the Chi-squared statistics are significant. We reject the null hypothesis of homoskedasticity. Heteroskedasticity effect is present in the price change returns expressed as percentages. Specifically, the F-statistic is 6.80 with a significant p-value of 0.004 and the probability Chi-square value is also significant with a numerical p-value of 0.0066 .


I have also attached the graph that you will get in terms of the residual, actual and fitted once you perform the White test. You can see from the graph the variations and deviations between the actual and fitted numerical values.

91

-8

-6

-4

-2

0

2

4-6

-4

-2

0

2

4

6

5 10 15 20 25 30

Residual Actual Fitted

Exercise

You are given a datset that is related to household expenditures in relation to their disposable income. The dependent variable is household expenditures and the independent variable is disposable income. The dataset comprises of 25 U.K households. Please compute the regression equation and select from Excel to include the residual table. Then, check the residual values in relation to the explanatory variable for heteroskedasticity effect. If there are such effect, then, transform the values by using the natural logarithmic ln linear form and check again if heteroskedasticity persists. The effected will be revealed It as increases and decreases in the residual values are associated with increases in the independent numerical values.

Please find attached the monthly dataset expressed in thousands pounds:

Observations Expenditures (y)

Incomes (x)

1 1980 22342 3130 32313 3172 35614 1223 12145 4072 43316 6145 6245

92

7 3962 43788 2651 25189 1045 1031

10 3956 412511 8210 810012 3417 335113 3251 380014 1213 141415 1381 164816 2062 241117 2831 311518 2500 283919 1780 182120 2080 201421 1945 214522 2013 224523 1834 234524 2104 229825 2276 2467

Solution

Please insert your data in Excel and run the linear regression by using the data analysis toolbox and selecting residual output.

The output in Excel will be as follows:

SUMMARY OUTPUT


ANOVA df SS MS F Significance F

Regression 1 60782933 60782933 1667.092 5.7279E-23Residual 23 838590.6 36460.46Total 24 61621523

93



Intercept -203.1098322 83.0775 -2.44482 0.02257 -374.9684948 -31.25116954 -374.96849Incomes 1.00573905 0.024632 40.83003 5.73E-23 0.954783253 1.056694847 0.9547833

The regression equation is as follows:

R2 = 0.986

t-stat (-2.44) (40.83) RSS = 838590.6

Once you get the residual output, copy and past the numerical values of the independent variable incomes adjacent to the residuals. Then, use the chart wizard in Excel to plot your data in a line or scatter format. There is another way to do it. You can select the residual plot in the regression function in addition to residual output. This will help you to see the pattern of variation and deviations from the regression line.

RESIDUAL OUTPUT

ObservationPredicted

Expenditures ResidualsIncomes (x)

9 833.8071284 211.1929 10314 1017.857375 205.1426 1214

14 1219.005185 -6.00518 141415 1454.348122 -73.3481 164819 1628.340978 151.659 182120 1822.448615 257.5514 201421 1954.20043 -9.20043 2145

1 2043.711206 -63.7112 223422 2054.774335 -41.7743 224524 2108.078505 -4.0785 229823 2155.34824 -321.348 234516 2221.727017 -159.727 2411

94

25 2278.048404 -2.0484 24678 2329.341096 321.6589 2518

18 2652.183331 -152.183 283917 2929.767309 -98.7673 3115

2 3046.433038 83.56696 323112 3167.121724 249.8783 3351

3 3378.326925 -206.327 356113 3618.698558 -367.699 380010 3945.563749 10.43625 4125

5 4152.745993 -80.746 43317 4200.015729 -238.016 43786 6077.730535 67.26946 6245

11 7943.376473 266.6235 8100

I have attached the layout of the chart that you will get in Excel. In EViews 6 , you get automatically the chart of the residual output with the variations.

Residual plot

-500-400-300-200-100

0100200300400

0 5 10 15 20 25 30

Incomes

Resi

dual

s

As you can see from the chart there is strong evidence of heteroscedasticity. The residual or error term is not constant. To solve this problem, we transform the original data in ln linear format as follows:

ln Expenditures ln Incomes7.590852124 7.7115498.048788284 8.0805478.062117583 8.1777977.109062136 7.1016768.311889558 8.3735548.723394022 8.739536

95

8.284504227 8.3843477.882692206 7.831226.951772164 6.9382848.282988693 8.3248219.013108202 8.9996198.136518252 8.117014

8.08671792 8.2427567.100851909 7.2541787.230563153 7.4073187.631431665 7.7877977.948385285 8.0439847.824046011 7.9512077.484368643 7.5071417.640123173 7.6078787.573017256 7.6708957.607381426 7.7164617.514254653 7.7600417.651595574 7.7397947.730174795 7.810758

Then, we do the same thing as we did above. We run again the regression equation by using ln numbers and the residual output and plot.

SUMMARY OUTPUT


ANOVA


FRegression 1 5.672052 5.672052459 965.1566 2.76E-20Residual 23 0.135167 0.005876821Total 24 5.807219



Lower 95.0%

Intercept -0.19467 0.258333-

0.753551944 0.458763 -0.72907 0.339735 -0.72907ln Incomes 1.015243 0.032679 31.06697001 2.76E-20 0.947641 1.082845 0.947641

The regression equation is as follows:

R2 = 0.9767

96

t-stat (0.75) (31.07) RSS = 0.135

Could you please see the differences with the previous regression and the following chart. Everything is different in terms of R2, residual sum of squares, coefficient, intercept, standard errors and t-statistics. So, please be very careful with the heteroskedastic problem as you will get biased estimators that will distrort the interpretation of the results.

RESIDUAL OUTPUT

ObservationPredicted ln

Expenditures Residuals ln Incomes1 7.634427 -0.04358 6.9382844842 8.00905 0.039738 7.1016759723 8.107782 -0.04566 7.2541778464 7.015258 0.093804 7.407317715 8.306523 0.005366 7.507141086 8.678084 0.04531 7.6078780737 8.317481 -0.03298 7.6708948318 7.755923 0.126769 7.711548989 6.849376 0.102396 7.7164608

10 8.257048 0.025941 7.73979445811 8.942132 0.070976 7.76004068112 8.046073 0.090445 7.78779687813 8.173732 -0.08701 7.81075811714 7.170085 -0.06923 7.83122021515 7.325559 -0.095 7.95120715616 7.711838 -0.08041 8.04398443117 7.97193 -0.02354 8.08054696618 7.877739 -0.05369 8.11701408819 7.426904 0.057465 8.17779668320 7.529176 0.110947 8.24275634621 7.593154 -0.02014 8.32482129922 7.639414 -0.03203 8.373553741

97

23 7.683658 -0.1694 8.38434727824 7.663103 -0.01151 8.73953642325 7.735149 -0.00497 8.999619341

According to the following chart, you can see that the residual or error term in relation to the independent variable becomes homoskedastic. Thus, transforming the data to ln has helped us to solve the problem of heteroskedasticity.

Residual plot

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0 2 4 6 8 10

Ln Incomes

Resi

dual

s

98

The likelihood ratio test and analysis of heteroskedasticity on the residuals

Please find attached the following dataset of n = 30. The dependent variable is share returns and the independent variables are the risk free rate, the size, the market returns, the age, the hurdle rate and the management performance. Run the logit regression equation in EViews 6 and then check for heteroskedasticity by running the regression equation of the squared residuals in relation to the independent variable, the squared independent variable and their cross products. The likelihood ratio test is applicable for large sample test.

Solution

Dependent Variable: share returnsMethod: ML - Binary Logit Date: 04/22/16 Time: 17:05Sample: 2 73Included observations: 72Convergence achieved after 7 iterationsCovariance matrix computed using second derivatives

Variable Coefficient Std. Error z-Statistic Prob.

C -0.397262 3.719469 -0.106806 0.9149Rf 0.607784 1.340246 0.453487 0.6502

Size -0.195838 0.633367 -0.309202 0.7572Mr -2.947545 1.216692 -2.422590 0.0154

Age -0.038684 0.133300 -0.290200 0.7717Hr 0.089281 0.161129 0.554100 0.5795Mp -0.048030 0.046253 -1.038402 0.2991

McFadden R-squared 0.424735 Mean dependent var 0.069444S.D. dependent var 0.255992 S.E. of regression 0.218670

99

Akaike info criterion 0.484607 Sum squared resid 3.108085Schwarz criterion 0.705950 Log likelihood -10.44587Hannan-Quinn criter. 0.572725 Restr. log likelihood -18.15837LR statistic 15.42500 Avg. log likelihood -0.145081Prob(LR statistic) 0.017197

Obs with Dep=0 67 Total obs 72Obs with Dep=1 5

I have attached the table with the relevant data. Based on the following data check for White tests heteroskedasticity. You will regress the squared residuals on all the explanatory variables, their squares and their cross products. Therefore, you will regress the following.

will be regressed on x1 , x2, x3, x4, x5, x6, and so on

To sum up, the dependent variable is the squared residuals and the independent variables are x1, x2, x3, x4, x5, x6, their squared values and their cross products.

If you have difficulty please e-mail me.

Residuals Risk free (x1)

Size (x2) Market return (x3)

Age (x4) Hurdle rate (x5)

Mgt performance (x6)

-0.02516 2.755836 0.823441 5.397865001 40 10.34 1-0.00662 2.725726 4.458782 6.857003219 25 5.677 1.5-0.02722 3.092531 4.232887 7.989673776 86 10.35 0.67-0.03513 3.089416 2.137064 3.379482658 36 2.345 1-0.00133 2.802173 1.658494 -0.45807375 50 3.13 1.75-0.01652 2.409178 0.20628 7.497715958 13 4.31 1.75-0.00979 2.220845 0.028879 -0.92491506 14 10.45 1.5-0.00098 2.522834 0.774512 -8.40360195 57 0 1.5-0.00015 2.669247 0.259913 1.42834889 72 2.144 2.25-0.00063 2.969892 0.325758 14.91365445 71 5.678 2.45-0.00521 1.994892 0.008251 7.948571731 3 8.99 2.31

100

-0.00434 1.861559 0.008251 6.81028783 3 2.76 2.31-0.03533 2.469416 0.929169 5.783098584 10 15 1.67-0.02719 2.503892 0.633557 -0.67634733 2 1.23 1.23-0.02316 2.265607 0.711829 6.876247514 10 1.34 1.6-0.19276 2.606003 0.008251 6.928020847 6 2.34 1.3-0.00763 2.639092 2.19223 -2.23667513 48 3.456 1.670.914445 2.572749 1.91165 8.918700935 17 4.311 1.32-0.36925 2.540823 0.008251 -1.6115003 32 2.67 0.32

-0.1796 2.259086 0.008251 2.446540443 27 3.21 0.480.529751 2.608419 0.148522 -1.40503816 24 7.89 0.130.318986 2.558915 0.191428 7.779562137 22 2.134 0.450.319484 3.042134 0.145221 5.438933357 24 1.34 0.67-0.01753 2.168168 0.167979 6.333734893 18 5 1.50.919461 3.415726 0.609765 -0.93701863 12 10 1.23-0.00194 2.244892 0.864844 0.084759516 24 1.33 2.34-0.00017 3.257392 0.072066 11.41224438 100 15 2.22-0.00235 2.592576 0.960441 -0.28480562 46 11 1.5-0.00275 2.533267 0.008251 0.89586715 48 14 1.5-0.00384 2.678596 0.008251 3.650721789 48 12 1.5-0.00799 2.595078 0.243411 0.080447353 24 13 1.5

-0.0538 3.029892 0.115517 7.968449474 4 13 1.5-0.00541 2.502834 0.406785 -3.33913367 24 10 1.5

-0.0178 2.801892 0.187146 1.174795116 12 15 1.5-0.00335 2.985979 0.553095 -5.23778486 36 10 1.5-0.03195 2.873892 0.452711 2.603855611 2 12 1.5-0.05187 2.738983 0.258661 6.248583233 24 15 1-0.00569 2.188128 0.742609 -7.40827572 10 10 1.5

-0.0009 2.984753 0.142399 -5.46280949 84 12 1.25-0.15083 2.774178 7.447185 12.74202837 36 20 0.23-0.00748 2.867511 0.889802 1.764284504 56 15 1-0.78198 3.143092 0.123768 0.320108901 5 11 0.33-0.01982 3.162392 0.201396 0.538832773 56 21 0.4-0.04005 2.054598 0.799455 -4.55270147 15 12 0.5

-0.002 3.108503 0.44564 -6.16328308 90 13 0.1-0.00483 2.633226 0.272711 6.60581503 90 14 0.5-0.05627 2.47726 0.150642 6.444594868 36 15 0.8-0.01541 2.887051 1.918407 3.828884296 14 16 1.5-0.00233 2.370292 0.75086 -0.03430596 14 8 2-0.00399 2.802692 0.75086 3.07357262 14 6 2-0.00422 2.830845 0.75086 3.52403273 14 5 2-0.00349 1.625607 0.068069 -3.02022138 24 4 1.5

-0.0038 1.647749 0.169617 -1.99379246 24 3 1.5-7.53E-05 2.855726 0.493801 -4.70154363 84 10 2-0.00448 3.073577 0.75086 -6.45093819 36 2 1.5-0.01442 2.516488 1.523075 5.770408932 24 3 1.5-0.04197 2.503017 0.336658 -5.22058709 4 15 1-0.05222 2.515186 0.885383 8.476452598 4 4 1.5-0.01357 2.581392 0.141228 5.945917922 24 10 1.5-0.01134 2.804629 0.082512 -3.63438734 7 17.5 1.5-0.00305 2.831559 0.082512 -3.01410765 36 11.34 1.5

-0.0007 2.935229 0.082512 5.530374864 84 11 1.5-0.01616 2.551392 0.082512 -0.04525887 3 12 1.5-0.00178 2.944589 0.082512 9.559315003 72 14 1.5

-0.0375 3.145347 0.082512 10.42954088 12 15 1.5

101

-0.04155 4.196281 1.683867 9.71768063 60 11 1-0.23705 4.112035 0.660097 10.7369913 24 12 1

-9.67E-17 4.090387 0.421637 -0.59326216 72 13 12-0.0583 2.403577 2.988672 10.65900268 6 9 1.2

-0.04783 2.703371 0.386157 2.721150814 10 8 1.2-0.04177 2.461197 2.880581 8.251326888 10 7 1.2-0.10652 2.928787 1.489131 3.99353016 5 6 1

Exercise

You are given two datasets of 10 observations each. You are required to calculate the Goldfeld – Quandt test to find out if the error term or residuals are homoskedastic or heteroskedastic.

The data are related to the monthly salary and expenditures of 10 employees at JP morgan. The dependent variable is annual salary and the independent variable is expenditures. The data are expressed as thousands and nominated in Pounds.

Statistician 1y x

8000 50007000 40456000 30005000 24674000 10003000 27892000 17895000 45679000 80007010 3145

Statistician 2y x

9076 90008367 8100

102

6456 62564000 3567

3134 27892346 17785789 37282145 20006678 57898100 7567

a) State the hypotheses of the residuals or error term.b) Calculate the regression equations from the data gathered from the two statisticians by stating the R2 , the standard errors and the mean square variance of the residuals.c) Calculate the F-ratio for both datasets for 5% significance level.d) Accept or reject the hypothesis of homoskedasticity.e) Transform the variables by using the natural logarithmic ln linear format.f) Run two regressions for each dataset.g) Calculate the F-ratio for both datasets.h) Accept or reject the hypothesis of homoskedasticity.

Solution

a) H0: The residuals are constant and therefore homoskedastic.

H1: The residuals are not constant and therefore heteroskedastic.

b) I have attached the Excel regression equation for the first dataset.

R2 = 0.616

SE (997.088) (0.2468)

SUMMARY OUTPUT


103

ANOVA



Coefficients Standard Error t Stat P-value Lower 95%Upper 95%

Intercept 2434.897 997.0883717 2.442007212 0.040438 135.6056 4734.188X 0.884337 0.24683455 3.58271121 0.00716 0.315135 1.453539

c ) I have attached the Excel regression equation for the second dataset.

R2 = 0.95

SE (429.2888) (0.076117)

SUMMARY OUTPUT

Regression Statistics

Multiple R 0.975255

R Square 0.951122Adjusted R Square 0.945012

Standard Error 600.8309

Observations 10

ANOVA


FRegression 1 56197292 56197292.31 155.6721085 1.59E-06Residual 8 2887983 360997.823Total 9 59085275

104



Intercept 806.0896 429.2888 1.877732704 0.097244025 -183.853 1796.032x 0.9497 0.076117 12.47686293 1.59218E-06 0.774174 1.125225

d) The F-ratio test for the Goldfeld – Quandt test is as follows:

From the F-distribution with 5 % significance level and n-2 degrres of freedom for each dataset or 10 - 2 = 8, the F-value from the tables is 3.44 and the F-ratio test is 0.1693. The F-ratio test is significant and lies inside the critical region. The sample evidence suggests that we can not reject the homoskedasticity hypothesis.

I have also attached the White’s test for the first datatset.

Heteroskedasticity Test: White




C 1245431. 2016142. 0.617730 0.5563X 487.0253 1007.909 0.483204 0.6437

X^2 -0.078642 0.107989 -0.728242 0.4901

R-squared 0.154142 Mean dependent var 1705835.Adjusted R-squared -0.087532 S.D. dependent var 1462453.S.E. of regression 1525116. Akaike info criterion 31.55636Sum squared resid 1.63E+13 Schwarz criterion 31.64714Log likelihood -154.7818 Hannan-Quinn criter. 31.45678F-statistic 0.637810 Durbin-Watson stat 0.863562

105

Prob(F-statistic) 0.556597

Please focus on the output of the first two sentences of the table that are bold. According to the above table, both the F-statistic and the Chi-squared statistics are not significant. We cannot reject the null hypothesis of homoskedasticity.

Please make yourself comfortable with the different tests that measure heteroskedasticity.

Please complete (e), (f,) (g), (h) questions……….. please feel free to e-amil me for different levels of difficulties.

Good luck……………

Exercise of Ramsey’s test

You are given one dataset of 10 observations each. You are required to calculate the Ramsey’s test to find out if the error term or residuals are homoskedastic or heteroskedastic.

The data are related to the monthly salary and expenditures of 10 employees at JP morgan. The dependent variable is annual salary and the independent variable is expenditures. The data are expressed as thousands and nominated in Pounds.

Statistician y x

8000 50007000 40456000 30005000 24674000 10003000 27892000 17895000 45679000 80007010 3145

106

To estimate the Ramsey’s test you regress the estimated error term or estimated residuals from the regression equation on the estimated square and cube dependent variable namely . The dependent variable in our case is annual salary.

The regression equation and the hypotheses testing will be as follows:

The hypotheses testing are as follows:

H0: = 0 and the error term is homoskedastic.

H1: and the error term is heteroskedastic.

Solution

The first step is to calculate the estimated residuals based on the above dataset. I have included the output in Excel in terms of the estimated dependent variable and the estimated error term .

RESIDUAL OUTPUT

Observation Predicted y Residuals1 6856.58154 1143.4182 6012.0398 987.96023 5087.90772 912.09234 4616.55615 383.44385 3319.23391 680.76616 4901.31264 -1901.317 4016.97573 -2016.988 6473.66366 -1473.669 9509.59227 -509.592

10 5216.13658 1793.863

107

I have included the dataset that will be used to solve the above regression, which is as follows:

RESIDUAL OUTPUT Observation Predicted y Residuals y squared y cube

1 6856.58154 1143.418 47012710 3.22346E+112 6012.0398 987.9602 36144623 2.17303E+113 5087.90772 912.0923 25886805 1.3171E+114 4616.55615 383.4438 21312591 983907717265 3319.23391 680.7661 11017314 365690412316 4901.31264 -1901.31 24022866 1.17744E+117 4016.97573 -2016.98 16136094 648182978918 6473.66366 -1473.66 41908321 2.713E+119 9509.59227 -509.592 90432345 8.59975E+11

10 5216.13658 1793.863 27208081 1.41921E+11

I have attached the output of the above equation.

SUMMARY OUTPUT

Regression StatisticsMultiple R 0.209095R Square 0.043721Adjusted R Square -0.2295Standard Error 1526.553Observations 10

ANOVA


FRegression 2 745805 372902.5 0.160019 0.855158Residual 7 16312546 2330364Total 9 17058351


Error T Stat P-value Lower 95%Upper 95%

Intercept -1094.04 2255.252 -0.48511 0.642413 -6426.86 4238.78y squared 9.05E-05 0.000166 0.543666 0.603546 -0.0003 0.000484y cube -8.8E-09 1.57E-08 -0.55977 0.593087 -4.6E-08 2.84E-08

108

Conclusion

The coefficients are not significant different from zero. All the p-values are aboe the 5% significance level. The coefficients to be significance they must be below the 5% significance level. We accept H0 and the error term is homoskedastic.

Exercise

You are given the following information related to the intercept, the independent variable, the number of observations, the coefficients, the standard errors, the t-statistics, and the R2.

Variable Coefficients Standard errors, (SE) t-statisticsIntercept 0.234 0.045 5.2Independent variable

8.221 8.214 0.9996

R2 = 0.654 n = 30 2 = 17.34

After plotiing the residuals against the independent variable, it was found that the variance of the residuals increased with x.

In contrast, by transforming the variables by ln function, it was found that there was no heteroskedasticity effect. I have attached all the information related to the intercept, the independent variable, the number of observations, the coefficients, the standard errors, the t-statistics, and the R2.

Variable Coefficients Standard errors, (SE) t-statisticsIntercept 0.122 0.031 3.935Independent variable

2.221 3.214 0.691

R2 = 0.721 n = 30 = 10.56


(a) Please compare both tables and comment on the changes related to the intercept, the independent variable, the number of observations, the coefficients, the standard errors, the t-statistics, and the R2.

(b) Please explain the difference between heteroskedastic and homoskedastic effect.

(c) Please comment on the changes of the variance and R2 in the regression equation by using natural logarithmic ln. Please comment on the changes of the coefficients and the standard errors. In which table are they biased and unefficient?

109

Exercise

I will give you an example of using square root reciprocal as deflators. This method is based on the square root transformation. Please consider the following ordinary least square,OLS, equation. After conducting a scatterplot, we have found that the independent variable x1 shows heteroskedastic patterns with the error term. We divide each part of the regression with the square root of x1, namely, .

Then, run the regression in Excel and obtain the estimated ordinary least squares regression which will have different coefficients after the transformation from the original regression equation. Check the residuals . Regress the estimated squared residuals on the independent variable x1 and the squared independent variable each time separately and choose the regression model with the highest R2. Try different heteroskedastic tests in order to accept or reject the null hypothesis of homoskedasticity. For example, use the Breusch – Pagan, the White and the Glejser test.

Let’s take an example to fully understand the application of the square root transformation. Divide each variable with the square root of the first independent variable .

110

y x1 x2

9076 9000 92348367 8100 87896456 6256 67894000 3567 38903134 2789 32452346 1778 25675789 3728 46782145 2000 23456678 5789 57438100 7567 8231

Run the regression. I have attached the output in EViews 6.

……………………

Dependent Variable: YMethod: Least SquaresDate: 09/02/16 Time: 16:42Sample: 1 10Included observations: 10


C 345.4160 577.5074 0.598115 0.5686X1 0.177116 0.668650 0.264886 0.7987X2 0.786860 0.676766 1.162676 0.2831

R-squared 0.959033 Mean dependent var 5609.100Adjusted R-squared 0.947328 S.D. dependent var 2562.232S.E. of regression 588.0400 Akaike info criterion 15.83479Sum squared resid 2420537. Schwarz criterion 15.92557Log likelihood -76.17396 Hannan-Quinn criter. 15.73521F-statistic 81.93496 Durbin-Watson stat 2.743481Prob(F-statistic) 0.000014

Construct the tables with the relevant ratios.

95.67 0.01 94.87 97.33

111

92.97 0.01 90 97.6681.62 0.01 79.09 85.8366.97 0.02 59.72 65.1359.34 0.02 52.81 61.4555.64 0.02 42.17 60.8894.81 0.02 61.06 76.6247.96 0.02 44.72 52.4487.77 0.01 76.09 75.4893.12 0.01 86.99 94.62

Run the regression. I have attached the output in EViews 6.

……………..

Dependent Variable: y / sqrt(x1)Method: Least SquaresDate: 09/02/16 Time: 19:17Sample: 1 10Included observations: 10


1/sqrt(x1) 1083.989 1522.429 0.712013 0.5032x1 / sqrt(x1) 0.422835 0.782749 0.540193 0.6085x2 / sqrt(x1) 0.819956 0.697460 1.175631 0.2843


Could you please see the difference in the coefficients, t-statistics and standard errors between the two regressions.

Test the residuals for heteroskedasticity. Compare the Breusch – Pagan- Godfrey, the White and the Glejser test. Test the null hypothesis for homoskedasticity.

To facilitate you, I have attached the Breusch – Pagan- Godfrey test. Once you have run the regression equation, then, press view---- then---- residual tests ----- heteroskedasticity ---- select the Breusch – Pagan – Godfrey test.

112

Heteroskedasticity Test: Breusch-Pagan-Godfrey




C -57.40305 525.3184 -0.109273 0.9165X1 0.907957 8.338245 0.108891 0.9168X2 -0.110792 7.429710 -0.014912 0.9886

XSQURE 3761.196 16217.71 0.231919 0.8243


Exercise

It is given the following dataset:

Share returns (y) Market returns (x)

3.526787 8.73209-4.34533 -5.198155.222709 6.21865-4.99619 -5.5393-3.04336 7.69808-2.375422 -4.997352.651303 5.42777-0.68924 -1.5424

113

0.205664 1.46392.4783 3.6528

0.237407 -0.14940.329728 0.16688-0.26869 -0.14440.064769 0.097873-0.5873 -0.09911

0.329225 -0.08344-0.11849 0.1227670.011541 -0.45767-0.18757 -0.53046-0.38752 -0.11118-0.26835 -0.289470.262798 -0.176760.355054 -1.15686-1.34302 -0.5771-0.77964 0.578182-0.04649 -0.053310.098381 -0.23054-0.09585 -0.66625-0.0059 -0.50071-0.05415 -0.53128

Perform and compare the following heteroskedasticy tests:

The Breusch - Pagan – Godfrey test The Harvey test The Glejser test The ARCH The White test

Which tests show heteroskedastic effect and which tests show homoskedastic effect?

To facilitate your learning, calculate the absolute and the squared values of the residuals and plot them against the independent variable which is market returns. What conclusions you draw by performing the various tests?Would you use the weighted least squares, WLS, transformation to eliminate heteroskedasticity? Please compare the R2 of the original regression with the WLS if it is applicable.

Develop a suitable model with more independent variables that could capture the deviations of share returns. Test the model with the above mentioned heteroskedastic tests. Would you transform your data to eliminate the heteroskedastic problem? Will you use a dummy variable? Please interpret and explain the investment reasoning of the intercept and the slope. Based on the White test check the t-statistics and the p – values of the error term. If there is heteroskedastic problem, click the heteroskedastic corrected standard errors and t-statistics in EViews 6 and compare your results with the original regression.

114

I have attached a recent article that is testing the ARCH and GARCH heteroskedasticity effect on the total returns of Jeffries Commodity Performance Index. I am trying to support you with research skills that are needed for postgrasuate studies.

An application of ARCH and GARCH models to test total returns of Jeffries Commodity Performance Index.

Abstract

Autoregressive Conditional Heteroskedastic models (ARCH), and Generalized Autoregressive Conditional Heteroskedastic models, (GARCH) take into account the non-linearity that arises in the financial time series. Well known anomalies such as the calendar effects, January effect and seasonality’s have been studied indicating the weakness of the Efficient Market Hypothesis, (EMH) and showing the direction of a non random path for price changes. In this paper, we are going to test the volatility of the Thomson Reuters Jeffries Commodity Performance Index returns,(TR/JCPI), for changing conditional variances to spot any anomaly that contradict the validity of the Efficient Market Hypothesis. The data that we will use are daily returns starting from 07/07/2008 to 10/27/2010, which total to 585 observations.

Keywords: Autoregressive Conditional Heteroskedastic models, Generalized Autoregressive Conditional Heteroskedastic models, Jeffries Commodity Performance Index total returns, Efficient Market Hypothesis.

115

Introduction

The study will focus on modeling the total returns of the Thomson Reuters Jeffries Commodity Performance Index by using Autoregressive Conditional Heteroskedastic models and Generalized Conditional Heteroskedastic models. Using EViews 6 the models will be tested to validate the hypotheses that will be formulated. The existence of seasonality in terms of January effect’s will shed light to determine if luck or skills allow investors to outperform the market. The analysis will be based on the Thomson Reuters Jeffries Commodity Performance Index,(TR/JCPI). According to Wikipedia, it consists of 19 commodities, as quoted on the New York Mercantile Exchange, (NYMEX), Chicago Board of Trade, (CBOT), London Metal Exchange, (LME), Chicago Mercantile Exchange, (CME) and Commodity Exchange, (COMEX) exchanges. They are divided into 4 groups, which are petroleum, liquidity assets, highly liquid assets, diverse commodities. The commodities are: Aluminum, cocoa, coffee, copper, corn, cotton, crude oil, gold, heating oil, lean hogs, live cattle, natural gas, nickel, orange juice, silver, soybeans, sugar, unleaded gas and wheat.

Models which include a changing conditional variance are called autoregressive conditional heteroskedastic models. Engle (1982), suggested this type of model for estimating time series with conditional heteroskedasticity. Since then, different types of ARCH models appeared in the literature with different specifications such as mean reversion, mean dependency and variance dependency. For example, GARCH models capture all the history of shocks in a series, TARCH and EGARCH models allow for asymmetric shock to volatility and ARCH – M models are used in financial markets where the expected return on an asset is related to expected risk.

The models will be used to check the validity of the Efficient Market Hypothesis. Fama (1970) has defined the EMH into three types of efficiency. Strong, semi – strong and weak forms. The weak forms suggest that current prices of stocks and commodities fully reflect the information implied by all historical movements. Semi – strong efficient market is when prices reflect not only all historical data, but also relevant information that is related to the investment vehicle. A market is strong form efficient if stock and commodity prices fully reflect not only publicly available information but also private information. No investor could make abnormal profits by using any information. Most of the studies that have done show that markets are efficient in the weak and semi – strong forms. Fama (1970), classified the empirical approaches to test market efficiency based on predictability of returns, event studies and private information. The weak form of efficiency is based on the random walk hypothesis.

The rest of the paper is organized as follows. Section 1 describes the methodology and the data. Section 2 is an analysis of statistical and econometric tests and Section 3 summarizes and concludes.

116

1.Methodology and data description

The methodologies that will be used are autoregressive volatility models and specifically autoregressive and generalized autoregressive conditionally heteroscedastic models. Heteroskedasticity is a violation of the OLS assumption that the variance error term is constant for all observations. This leads to biased estimates of the standard errors, inefficient estimates of the coefficients and incorrect confidence intervals and statistical tests. In other words, we are checking if the variance of the errors is not constant and is creating volatility clusters. In addition, we are trying to spot if the January effect has affected the time series of the commodity index by showing higher positive returns in the month of January that could explain variance persistence.

The mean equation of the ARCH (1,5) is as follows:

(1)

Where:

rt: is the expected return and in our case are logarithmic and percentage daily returns of Jeffries Commodity Index.

The conditional variance equation is estimated by regressing the squared residuals on a constant and p lags:

(2)

Where:

The null hypothesis that there is no ARCH effect of order p is H0: The alternative hypothesis is that H1:

The mean equation of the GARCH(1,1) that we will use is as follows:

(3)

Where:

117

yt is the dependent variable and in our case is logarithmic and percentage daily returns of Jeffries Commodity Index.

is a constant. is the error term.

The variance equation is as follows:

(4)

Where:

According to Brooks, (2002), the log –likelihood function, (LIF) of the model that will be maximised is calculated as follows:

(5)

The hypotheses that we are going to formulate and test are as follows:

The null hypothesis, H0, states that by using autoregressive volatility models, returns expressed as percentages and logarithmic of the Thomson Reuters Jeffries Commodity Performance Index returns,(TR/JCPI) are constant and similarly distributed over time.

The alternative hypothesis, H1, states that by using autoregressive volatility models, returns expressed as percentages and logarithmic of the Thomson Reuters Jeffries Commodity Performance Index returns,(TR/JCPI) are not constant and similarly distributed over time.

To test for non-normality the Jarque – Bera statistic is analysed in relation to a histogram and descriptive statistics. Then, we will run an Augmented Dickey – Fuller’s, (ADF), statistic to test for stationarity and calculate and compare the critical values. Finally, we will test for seasonality by using a t-test paired two sample of means. The sample will be divided into sub-sample of two periods 2009-2010 to test for the persistence of the January effect. If seasonality exists, we would expect to find significantly positive mean returns in January and this positive return is higher then the returns in April. We are testing the hypothesis of Arsad and Coutts, (1997), that returns in April are significantly positive than in January. Their methodology was based on a dummy variable regression model for the period 1935-1994 using FT-30 index. They analysed sub-samples of the data and found that in all of the sub-samples April displays positive mean daily returns.

The data that we will use are total daily returns starting from 07/07/2008 to 10/27/2010 of the Thomson Reuters Jeffries Commodity Performance Index returns,

118

(TR/JCPI), which total to 585 observations. The return financial series are calculated by taking the log difference of the index and by using the daily price change returns expressed as percentages. The reason of using these two different methods is to show logarithms and percentages in spotting volatility changes.

The logarithmic formula that we have used is:

(6)

Where: Rt is the daily return for month t, Pt is the closing daily price for month t, and Pt-1 is the previous daily closing price for month t-1.

The daily price change returns expressed as percentages is given by the formula

x 100 (7)

Where: Rt is the daily return for month t, P1 is the price of a financial asset at the end of period 1, and P0 is the price of a financial asset at the beginning of the period.

Figure 1 shows the fluctuations of the prices of the Thomson Reuters Jeffries Commodity Performance Index, (TR/JCPI) starting from 2000.

Source: Thomson Reuters Jeffries Commodity Performance Index, (TR/JCPI) website.

According to Figure 1, there was a tremendous increase of 586,845 dated 07/03/08 and then a continuous decrease that has reached a high low of 246,266. It is obvious that there was an intense negative volatility cluster that resulted that the various commodities that constitute the index to fall sharply. In contrast, during the years 2003 to 2008, there was a continuous increase, which showed an upward positive trend. Unfortunately, we have a limited dataset that restrict our analysis for the January effects.

2. Statistical, econometric tests and analysis

Figure 2 and Table 1 shows the normality tests of the total returns calculated by taking the log difference of the index.

119

Table 1 and Figure 2 display Jarque - Bera normality test, which is calculated by taking the log difference of the Thomson Reuters Jeffries Commodity Performance Index total returns,(TR/JCPI).

0

20

40

60

80

100

-0.06 -0.04 -0.02 0.00 0.02 0.04 0.06

Series: LRSample 2 585Observations 584

Mean -0.000974Median -0.000774Maximum 0.064717Minimum -0.074723Std. Dev. 0.017810Skewness -0.160579Kurtosis 4.624725

Jarque-Bera 66.74330Probability 0.000000

Source: Author’s calculation based on E-views software.

We state the hypotheses as follows:

H0: The log daily difference returns of the Thomson Reuters Jeffries Commodity Performance Index total returns, (TR/JCPI) are normally distributed.

H1: The log daily difference returns of the Thomson Reuters Jeffries Commodity Performance Index total returns, (TR/JCPI) are not normally distributed.

According to Table 1 and Figure 2, the Jarque – Bera statistic is 66.74, which is statistically significant at the 5% significance value. The joint test of the null hypothesis that sample skewness equals 0 and sample kurtosis equals 3 is rejected. The probability of the p- value 0.0000 is very significant, so we can reject H0 of normality. The distribution is leptokurtic and slightly negatively skewed. The mean is -0.000974 and the standard deviation is 0.017810. There is a high dispersion from the actual data.

Table 2 and Figure 3 display Jarque Bera normality test, which is calculated by taking the daily price change returns expressed as percentages of Thomson Reuters Jeffries Commodity Performance Index total returns,(TR/JCPI).

120

0

20

40

60

80

100

-6 -4 -2 0 2 4 6

Series: RSample 2 585Observations 584

Mean -0.081590Median -0.077364Maximum 6.685728Minimum -7.199912Std. Dev. 1.777365Skewness -0.064443Kurtosis 4.613409




H0: The daily price change returns expressed as percentages of the Thomson Reuters Jeffries Commodity Performance Index total returns,(TR/JCPI) are normally distributed.

H1:. The daily price change returns expressed as percentages of the Thomson Reuters Jeffries Commodity Performance Index total returns,(TR/JCPI) are not normally distributed.

According to Table 2 and Figure 3, the Jarque – Bera statistic is 63.75, which is above the critical value at 5% significance value. The probability of the p- value 0.0000 is very significant, so we can reject H0 of normality. The distribution is leptokurtic, as the value 4.61 is bigger than 3 and slightly negatively skewed. The mean is -0.081590 and the standard deviation is 1.777365. There is significant dispersion from the mean of the actual data.

Tables 3 and 4 show the ADF tests of the log difference and the daily price change returns expressed as percentages of the Thomson Reuters Jeffries Commodity Performance Index total returns,(TR/JCPI).

Table 3 shows the ADF tests of the log difference of the Thomson Reuters Jeffries Commodity Performance Index total returns,(TR/JCPI).

121

ADF Test Statistic -10.54738 1% Critical Value* -3.4440 5% Critical Value -2.8669 10% Critical Value -2.5696

*MacKinnon critical values for rejection of hypothesis of a unit root.

Augmented Dickey-Fuller Test EquationDependent Variable: D(LR)Method: Least SquaresDate: 08/28/13 Time: 18:38Sample(adjusted): 7 585Included observations: 579 after adjusting endpoints

Variable Coefficient Std. Error t-Statistic Prob. LR(-1) -0.980186 0.092932 -10.54738 0.0000

D(LR(-1)) -0.056470 0.084169 -0.670911 0.5025D(LR(-2)) -0.040042 0.073412 -0.545442 0.5857D(LR(-3)) -0.003290 0.059801 -0.055022 0.9561D(LR(-4)) 0.038236 0.041583 0.919517 0.3582

C -0.000936 0.000746 -1.255455 0.2098R-squared 0.520784 Mean dependent var -2.57E-05Adjusted R-squared 0.516602 S.D. dependent var 0.025641S.E. of regression 0.017827 Akaike info criterion -5.205851Sum squared resid 0.182109 Schwarz criterion -5.160656Log likelihood 1513.094 F-statistic 124.5404Durbin-Watson stat 1.999997 Prob(F-statistic) 0.000000


For a level of significance of one per cent, the critical value of the t-statistic from Dickey-Fuller’s table is -3.4440. According to Table 3 and to the sample evidence, we can reject the null hypothesis namely the existence of a unit root with one, five and ten per cent significance level. The ADF test statistic is -10.54738, which is smaller than the critical values, (-3.4440, -2.8669, -2.5696). In other words, the log difference of the Thomson Reuters Jeffries Commodity Performance Index total returns, (TR/JCPI) are stationary series. In addition, there is no autocorrelation problem as shown from the Durbin – Watson statistic.

122

Table 4 shows the ADF tests of the daily price change returns expressed as percentages of the Thomson Reuters Jeffries Commodity Performance Index total returns,(TR/JCPI).

ADF Test Statistic -10.61150 1% Critical Value* -3.4440 5% Critical Value -2.8669 10% Critical Value -2.5696

*MacKinnon critical values for rejection of hypothesis of a unit root.

Augmented Dickey-Fuller Test EquationDependent Variable: D(R)Method: Least SquaresDate: 08/28/13 Time: 18:40Sample(adjusted): 7 585Included observations: 579 after adjusting endpoints

Variable Coefficient Std. Error t-Statistic Prob. R(-1) -0.988133 0.093119 -10.61150 0.0000

D(R(-1)) -0.050056 0.084265 -0.594032 0.5527D(R(-2)) -0.032258 0.073457 -0.439141 0.6607D(R(-3)) 0.002205 0.059846 0.036849 0.9706D(R(-4)) 0.040629 0.041580 0.977130 0.3289

C -0.078746 0.074288 -1.060009 0.2896R-squared 0.521579 Mean dependent var -0.002572Adjusted R-squared 0.517404 S.D. dependent var 2.561163S.E. of regression 1.779218 Akaike info criterion 4.000533Sum squared resid 1813.898 Schwarz criterion 4.045728Log likelihood -1152.154 F-statistic 124.9378Durbin-Watson stat 2.000263 Prob(F-statistic) 0.000000


According to Table 4, for a level of significance of one, five and ten per cent, the critical value of the t-statistic from Dickey-Fuller’s table is -3.4440, -2.8669 and -2.5696. According to Table 4, we can reject the null hypothesis namely the existence of a unit root with one, five and ten per cent significance level. The ADF test statistic is -10.61150, which is smaller than the critical values, (-3.4440, -2.8669, -2.5696). In other words, the log difference of the Thomson Reuters Jeffries Commodity Performance Index total returns,(TR/JCPI) are stationary series. The Durbin – Watson statistic is 2.00, so there is no autocorrelation.

123

Table 5 shows the Autoregressive Conditional Heteroskedastic test, (ARCH 1,5) of the daily price change returns expressed as percentages of the Thomson Reuters Jeffries Commodity Performance Index total returns,(TR/JCPI).

ARCH Test:F-statistic 14.29835 Probability 0.000000Obs*R-squared 64.17012 Probability 0.000000

Test Equation:Dependent Variable: RESID^2Method: Least SquaresDate: 08/24/13 Time: 14:10Sample(adjusted): 12 585Included observations: 574 after adjusting endpoints

Variable Coefficient Std. Error t-Statistic Prob. C 1.419641 0.331044 4.288379 0.0000

RESID^2(-1) -0.030329 0.041412 -0.732374 0.4642RESID^2(-2) 0.071553 0.040282 1.776300 0.0762RESID^2(-3) 0.109845 0.040139 2.736618 0.0064RESID^2(-4) 0.229664 0.040287 5.700630 0.0000RESID^2(-5) 0.161264 0.041393 3.895883 0.0001

R-squared 0.111795 Mean dependent var 3.128923Adjusted R-squared 0.103976 S.D. dependent var 6.013453S.E. of regression 5.692247 Akaike info criterion 6.326485Sum squared resid 18404.15 Schwarz criterion 6.371983Log likelihood -1809.701 F-statistic 14.29835Durbin-Watson stat 2.021232 Prob(F-statistic) 0.000000


The presence of ARCH(1,5) in the residuals is calculated by regressing the squared residuals on a constant and p lags. In this case, we use the daily price change returns expressed as percentages of the Thomson Reuters Jeffries Commodity Performance Index total returns,(TR/JCPI).

According to Table 5, both the F-statistic and the Lagrange Multiplier, LM- statistics are significant. ARCH effect is present in the daily price change returns expressed as percentages of the Thomson Reuters Jeffries Commodity Performance Index total returns,(TR/JCPI). Specifically, the F-statistic is 14.29835 with a significant p-value of 0.000 and the LM – statistic is 64.17012 with a significant p-value of 0.000.

124

Figure 4 shows the residuals graph of the daily price change returns expressed as percentages of the Thomson Reuters Jeffries Commodity Performance Index total returns,(TR/JCPI).

-8

-6

-4

-2

0

2

4

6

8

100 200 300 400 500

R Residuals


According to Figure 4, we can see that there is a significant positive and negative volatility clusters depicted by the illustrated arrows, which confirms the ARCH effect on the time series of Thomson Reuters Jeffries Commodity Performance Index total returns,(TR/JCPI) based on the Lagrange Multiplier, (LM). This figure shows that the mean and variance are not constant or non stationary.

Table 6 shows the Autoregressive Conditional Heteroskedastic test, (ARCH 1,5) of the log difference of the Thomson Reuters Jeffries Commodity Performance Index total returns,(TR/JCPI).

ARCH Test:F-statistic 14.54688 Probability 0.000000Obs*R-squared 65.15890 Probability 0.000000

Test Equation:Dependent Variable: RESID^2Method: Least SquaresDate: 08/24/13 Time: 14:16Sample(adjusted): 12 585Included observations: 574 after adjusting endpoints

Variable Coefficient Std. Error t-Statistic Prob. C 0.000142 3.32E-05 4.262611 0.0000

RESID^2(-1) -0.034103 0.041408 -0.823595 0.4105RESID^2(-2) 0.068032 0.040261 1.689790 0.0916RESID^2(-3) 0.117220 0.040071 2.925303 0.0036

125

RESID^2(-4) 0.231715 0.040267 5.754497 0.0000RESID^2(-5) 0.161789 0.041387 3.909143 0.0001

R-squared 0.113517 Mean dependent var 0.000314Adjusted R-squared 0.105714 S.D. dependent var 0.000605S.E. of regression 0.000572 Akaike info criterion -12.08552Sum squared resid 0.000186 Schwarz criterion -12.04002Log likelihood 3474.544 F-statistic 14.54688Durbin-Watson stat 2.022482 Prob(F-statistic) 0.000000

Source: Author’s calculation based on E-views software.The presence of ARCH(1,5) in the residuals is calculated by regressing the squared residuals on a constant and p lags. In this case, we use the log difference of the Thomson Reuters Jeffries Commodity Performance Index total returns,(TR/JCPI). According to Table 6, both the F-statistic and the Lagrange Multiplier, LM- statistics are significant. LM statistic is defined as the multiplication of the observation with R-squared. ARCH effect is present in the log difference of the Thomson Reuters Jeffries Commodity Performance Index total returns,(TR/JCPI). Specifically, the F-statistic is 14.54688 with a significant p-value of 0.000 and the LM – statistic is 65.15890 with a significant p-value of 0.000.

Figure 5 shows the residuals graph of the log difference of the Thomson Reuters Jeffries Commodity Performance Index total returns,(TR/JCPI).

-0.08

-0.06

-0.04

-0.02

0.00

0.02

0.04

0.06

0.08

100 200 300 400 500

LR Residuals


According to Figure 5, we can see that there is a significant positive and negative volatility clusters depicted by the arrows, which confirms the ARCH effect on the time series of Thomson Reuters Jeffries Commodity Performance Index total returns,(TR/JCPI) based on the Lagrange Multiplier, (LM).

Table 7 displays the GARCH(1,1) model of the log difference of the Thomson Reuters Jeffries Commodity Performance Index total returns,(TR/JCPI).

Dependent Variable: LR

126

Method: ML - ARCHDate: 08/31/13 Time: 16:03Sample: 2 585Included observations: 584Convergence achieved after 21 iterationsBollerslev-Wooldrige robust standard errors & covariance

Coefficient Std. Error z-Statistic Prob. C 7.15E-05 0.000559 0.127973 0.8982

Variance EquationC 7.61E-07 1.12E-06 0.680912 0.4959

ARCH(1) 0.042498 0.017616 2.412435 0.0158GARCH(1) 0.953173 0.018179 52.43310 0.0000

R-squared -0.003455 Mean dependent var -0.000974Adjusted R-squared -0.008645 S.D. dependent var 0.017810S.E. of regression 0.017886 Akaike info criterion -5.456046Sum squared resid 0.185555 Schwarz criterion -5.426115Log likelihood 1597.166 Durbin-Watson stat 2.055019


According to Table 7, the z-statistic, the coefficients and the p- values of the variance equation of the log difference are very statistically significant. The log likelihood is maximised at the value of 1597.166. This means that the conditional variance is highly persistent. The ARCH has a coefficient of 0.042 and a significant z – statistic of 2.41.The GARCH effect has a coefficient of 0.95 and a very significant p-value of 0.0000. There is strong evidence of variance persistence and high volatility clustering.

Table 8 shows the GARCH(1,1) model of the daily price change returns expressed as percentages of the Thomson Reuters Jeffries Commodity Performance Index total returns,(TR/JCPI).

Dependent Variable: RMethod: ML - ARCHDate: 08/31/13 Time: 16:05Sample: 2 585Included observations: 584Convergence achieved after 15 iterationsBollerslev-Wooldrige robust standard errors & covariance

Coefficient Std. Error z-Statistic Prob. C 0.017548 0.055908 0.313861 0.7536

Variance EquationC 0.007689 0.011153 0.689394 0.4906

ARCH(1) 0.042456 0.017272 2.458056 0.0140GARCH(1) 0.953188 0.017865 53.35472 0.0000

R-squared -0.003116 Mean dependent var -0.081590Adjusted R-squared -0.008305 S.D. dependent var 1.777365S.E. of regression 1.784731 Akaike info criterion 3.752371Sum squared resid 1847.453 Schwarz criterion 3.782302Log likelihood -1091.692 Durbin-Watson stat 2.059260


According to Table 8, the z-statistic, the coefficients and the p- values of the variance equation of the daily price change returns expressed as percentages are very statistically significant. The log likelihood is maximised at the value of -1091.692. The coefficients of the lagged squared error and lagged conditional variance are very close to unity. This means that the conditional variance is highly persistent. The

127

ARCH effect after applying the Bollerslev-Wooldrige robust standard errors & covariance heteroskedasticity test shows a coefficient of 0.042 and a z – statistic of 2.46. The GARCH effect has a coefficient of 0.95 and a significant p-value of 0.0000 at the 5% significance level.

Tables 9 and 10 are used to test the hypothesis of Arsad and Coutts, (1997), that returns in April are significantly positive than in January. In other words, we are trying to prove that the market follows a random walk under the weak form hypothesis. If, we found evidence of the January effect, then, luck in contrast to skills allow investors to outperform the market.

Table 9 displays a t-Test: Paired Two Sample for Means for daily percentage changes in total returns for the years 2009 – 2010.

January AprilMean -0,34407 0,033532Variance 3,584054 2,010858Observations 39 39Pearson Correlation -0,09435 Hypothesized Mean Difference 0 df 38 t Stat -0,95465 P(T<=t) one-tail 0,172894 t Critical one-tail 1,685954 P(T<=t) two-tail 0,345788 t Critical two-tail 2,024394

Source: Author’s calculation based on Excel software.


H0: There is no significant difference of daily price change returns expressed as percentages of the Thomson Reuters Jeffries Commodity Performance Index total returns,(TR/JCPI) between the sample means of January and April.

H1: There is significant difference of daily price change returns expressed as percentages of the Thomson Reuters Jeffries Commodity Performance Index total returns,(TR/JCPI) between the sample means of January and April.

According to Table 9, the mean return of April is 0,034, but not significantly positive from January. In January, we have a negative mean return of -0.34. The t-statistic is -0,95 and for a two-tailed test is within the critical area of ±2,02. Thus, we accept the null hypothesis in terms that there is no significant difference of the means of daily price change returns for the months of January and April. Thus, skills and not luck allow investors to outperform the market. The limitation of this test is that the sample period was restricted for one year.Table 10 shows a t-Test: Paired Two Sample for Means of logarithmic difference total returns for the years 2009 – 2010.

January AprilMean -0,00362 0,000303Variance 0,000366 0,000202Observations 39 39

128

Pearson Correlation -0,07692 Hypothesized Mean Difference 0 df 38 t Stat -0,99292 P(T<=t) one-tail 0,163515 t Critical one-tail 1,685954 P(T<=t) two-tail 0,32703 t Critical two-tail 2,024394

Source: Author’s calculation based on Excel software.


H0: There is no significant difference of logarithmic difference total returns expressed as percentages of the Thomson Reuters Jeffries Commodity Performance Index total returns,(TR/JCPI) between the sample means of January and April.

H1: There is significant difference of logarithmic difference total returns expressed as percentages of the Thomson Reuters Jeffries Commodity Performance Index total returns,(TR/JCPI) between the sample means of January and April.

According to Table 10, the mean return of April is 0,0003, but not significantly positive from January. In January, we have a negative mean return of -0.004. The t-statistic is -0,99 and for a two-tailed test is within the critical area of ±2,02. Thus, we cannot reject the null hypothesis in terms that there is no significant difference of the means of logarithmic daily price change returns for the months of January and April.

According to the results of the above tables, our evidence contradicts Arsad and Coutts, (1997), that returns in April are significantly positive than in January. Thus, volatility clusters are not the result of the January effect and investors use skills instead of luck to outperform the market. The limitation of this approach is that, we have used a limited dataset that did not show evidence of January effects. In addition, we did not use a dummy variable regression model.

3. Summarizes and Concludes

The author’s attempted to explain, illustrate, test and analyse the total returns of the Thomson Reuters Jeffries Commodity Performance Index by using Autoregressive Conditional Heteroskedastic models, (ARCH) and Generalized Autoregressive Conditional Heteroskedastic models, (GARCH).

We have concluded that the conditional variance is highly persistent. There is a significant positive and negative volatility clusters, which confirms the ARCH and GARCH effect’s on the time series of Thomson Reuters Jeffries Commodity Performance Index total returns,(TR/JCPI) based on the F-statistic and the Lagrange Multiplier, (LM). Thus, we reject the null hypothesis that the returns expressed as percentages and logarithmic of the commodity index are constant and similarly distributed over time.

Our evidence contradicts Arsad and Coutts, (1997), that returns in April are significantly positive than in January. Thus, volatility clusters and variance

129

persistence are not the result of the January effect and investors use skills instead of luck to outperform the market. If we take into consideration the US recession in addition to the autoregressive models, we conclude that the market is semi – strong efficient, as prices reflected not only all historical data, but also relevant macroeconomic information such as the US recession in 2008. It has created a large negative volatility of the commodity prices. Future research should focus on regressive relationships between the returns of the Jeffries Commodity Index and other macroeconomic indicators.

References

Alexander, C., (2003), Market Models. A Guide to Financial Data Analysis. John Wiley and Sons Ltd. ISMA Centre. The Business School For Financial Markets.

Arsad, M., and Coutts, A., (1997), Some Anomalous Evidence Regarding Market Efficiency. Journal of Financial Economics, Vol.5.

Brooks,C.,(2002), Introductory econometrics for Finance. Cambridge University Press.

Fama, E.F.,(1970), Efficient Capital Markets: A Review of Theory and Empirical Work. Journal of Finance, Vol.25, pp 383 – 417.

Wikipedia internet source. www. wikepedia.org.

I have attached a second recent article that is testing the TGARCH, EGARCH, and PARCH heteroskedasticity effect on the spot AUD/USD exchange rate volatility. I am trying to support you with research skills that are needed for postgrasuate studies. I

130

would like that you get the best from your studies and get recruited in the best places of the Financial Services sector around the U.K.

Application of a TGARCH, EGARCH, and PARCH models to test the spot AUD/USD exchange rate volatility.

Abstract

In this article, we have tested the volatility of the returns of the spot exchange rate of AUD/USD for changing conditional variances. Threshold generalized autoregressive conditional heteroskedastic models, (TGARCH), exponential generalized autoregressive conditional heteroskedastic models, (EGARCH) models and power autoregressive conditional heteroskedastic models, (PARCH), take into account the non-linearity that arises in financial time series. We have checked the volatility clusters for a long period of time that arises in the financial times series of returns or the fact that large and small values occur persistently in clusters. We have found that the AUD/USD spot exchange rate time series is non-normal. The natural logarithmic monthly returns of the AUD/USD are a stationary series. By applying a TGARCH (1,1) model, we have found that there is no leverage effect. By applying an EGARCH (1,1) model, we have found that the asymmetry term is not statistically significant at the 5% significance level. Negative shocks do not imply a higher next period conditional variance than positive shocks of the same sign. By applying a PARCH (1,1) model, we have found that the asymmetric term is statistically significant at the 5% significance level. Positive shocks related to good news implied a higher next period conditional variance than negative shocks related to bad news of the same extent. By comparing the three models in terms of Akaike information criterion and Schwarz criterion, we have concluded the PARCH (1,1) is the best – fit model, as it has the lowest values. There is no serial correlation in the residuals of the TGARCH (1,1), EGARCH (1,1), and PARCH (1,1) models. The residuals of the TGARCH (1,1), EGARCH (1,1) and PARCH (1,1) are homoskedastic and there is no additional ARCH effect. Finally, there are negative and positive shocks that are driving the AUD/USD exchange spot rate away from normality. The data that we have used are monthly returns starting from 01/01/1990 to 01/01/2013, which total to 276 observations. The total dataset includes 277 observations. The data was obtained from the Federal Reserve Statistical Release Department and the symbol of the series is H.10.

Keywords: Threshold generalized autoregressive conditional heteroskedastic models, (TGARCH), exponential generalized autoregressive conditional heteroskedastic models, (EGARCH) models and power autoregressive conditional heteroskedastic models, (PARCH), AUD/USD exchange spot rate.

Introduction

131

This article has focused on modeling the natural logarithmic returns of the AUD/USD exchange spot rate, by using TGARCH, EGARCH and PARCH models. The aim will be to spot the volatility clusters of the time series. By using E-Views 6, the models will be tested to validate the hypotheses that will be formulated.

Models, which include a changing conditional variance, are called autoregressive conditional heteroskedastic models. Engle (1982) suggested this type of model for estimating time series with conditional heteroskedasticity. Since then, different types of ARCH models appeared in the literature with different specifications such as mean reversion, mean dependency and variance dependency of TGARCH, and EGARCH. In contrast, the PARCH model measures the standard deviation rather than the conditional variance. According to EViews 6 User’s Guide II, (p.199), “the power parameter of the standard deviation can be estimated and the optional parameters are added to capture asymmetry of up to order r”. GARCH models capture all the history of shocks in a series in a changing variance environment, TGARCH and EGARCH models allow for asymmetric shock to volatility. EGARCH model was calibrated by Nelson in (1991). The advantage of the EGARCH model is that the variance will be positive even if the parameters are negative.

Diagnostic checking of the residuals of the volatility models will be done by checking the correlogram squared residuals, the ARCH LM test and the quantile – quantile, (QQ), theoretical distribution plots for normality. Specifically, the correlogram is used to check for serial correlation in the residuals. The ARCH LM test is used to check whether the standardized residuals have additional ARCH effect. Finally, the QQ plot is used to check whether the data are normally distributed. The quantile of normality and the quantile of residuals should lie on a straight line. Otherwise, there is deviation from linearity.

The rest of the paper is organized as follows. Section 1 describes the methodological issues and data descriptions. Section 2 is an analysis of statistical and econometrical tests and Section 3 summarizes and concludes.

1.Methodological issues and data descriptions

The methodologies that we have used are autoregressive volatility models. Heteroskedasticity is a violation of the OLS assumption that the variance error term is constant for all observations. This leads to biased estimates of the standard errors, inefficient estimates of the coefficients and incorrect confidence intervals and statistical tests. In other words, we are checking if the variance of the errors is not

132

constant and is creating volatility clusters. Autoregressive models have been studied by various researchers such as: Alexander,(2003), Brooks, (2002), Nelson, (1991), Bollerslev, (1986), Bollerslev, Chou, and Kroner, (1992), Bollerslev, Engle and Nelson, (1994), Bollerslev, Jeffrey, and Wooldridge, (1992), Ding, Granger and Engle, (1993), Engle, (1982), Engle and Bollerslev, (1986), Engle, Lilien and Robins, (1987), Glosten, Jaganathan and Runkle, (1993), Schwert, (1989), Taylor, (1986) and Zakoian, (1994).

The mean equation of the TGARCH(1,1), EGARCH(1,1), and PARCH(1,1) is as follows:

(1)

Where:yt is the dependent variable and in our case is the logarithmic monthly returns of the spot exchange rate AUD/USD.

is a constant.

The variance equation of the TGARCH (1,1) is as follows:

(2)

Where:

According to EViews 6 User’s Guide II, (p.199), the conditional variance equation of the EGARCH model is as follows:

(3)

Where:

133

We assume according to Nelson, (1991), that the follows a generalized error distribution, (GED). The leverage effect represented by is exponential.

The hypotheses that have been formulated and tested are as follows:

According to EViews 6 User’s Guide II, (p.199, 200), the standard deviation equation of the PARCH model is as follows:

(4)

Where:


The return financial series are calculated by taking the natural logarithmic monthly returns of the AUD/USD spot exchange rate. The logarithmic formula that we have used is:

(5)

Where: Rt is the monthly return for month t, Pt is the closing price for month t, and Pt-1 is the previous closing price for month t-1.

Descriptive statistics will be displayed and to test for non-normality the Jarque – Bera statistic is analysed. We check for stationary of the series by applying the Augmented

134

Dickey – Fuller’s stationary test, (ADF). Finally, we apply different types of Autoregressive Heteroskedastic models to test for persistence of volatility clusters.

The data that we have used are monthly logarithmic returns starting from 01/01/1990 to 01/01/2013 of the AUD/USD spot exchange rate, which total to 276 observations. The data was obtained from the Federal Reserve Statistical Release Department and it is denoted by the symbol, H.10.

Figure 1 shows the fluctuations of the percentage returns of the spot rate AUD/USD.

AUD/USD

-20

-15

-10

-5

0

5

10

1990

-02

1991

-07

1992

-12

1994

-05

1995

-10

1997

-03

1998

-08

2000

-01

2001

-06

2002

-11

2004

-04

2005

-09

2007

-02

2008

-07

2009

-12

2011

-05

2012

-10

Date

Perc

enta

ge re

turn

Source: Author’s calculation based on monthly data obtained from the US Federal Reserve Statistical Release Department.

According to Figure 1, there was a continuous discount /premium fluctuations of the AUD against the USD. The greatest discount of the spot rate was recorded in October 2008 and the value was -15.89%. There is a persistent negative and positive volatility clusters that resulted to this fluctuations. In May, 2009, the spot rate value has recorded a premium of 6.85 percent. In May, 2010, the value of the spot rate has recorded a discount of -5.93 percent. As, we can see from Figure 1, large and small values occur persistently in clusters. 2. Statistical and econometrical tests

The following Figure and Table display descriptive statistics, the normality tests of the natural logarithmic returns of the AUD/USD spot rate.

Table 1 and Figure 2 display Jarque - Bera normality test and descriptive statistics, which are calculated by taking the log difference of the AUD/USD spot rate. The dataset has covered the period starting from 01/01/1990 to 01/01/2013.

135

0

10

20

30

40

50

-0.15 -0.10 -0.05 0.00 0.05

Series: LRSample 1990M02 2013M01Observations 276

Mean 0.001072Median 0.002359Maximum 0.071211Minimum -0.173060Std. Dev. 0.026888Skewness -1.127986Kurtosis 8.927606


Source: Author’s calculation based on EViews 6 software.Significant at the 5% significance level.


H0: The natural logarithmic monthly returns of the AUD/USD spot exchange rate are normally distributed.

H1: The natural logarithmic monthly returns of the AUD/USD spot exchange rate are not normally distributed.

According to Table 1 and Figure 2, the Jarque – Bera statistic is 462.60, which is very significant at the 5% significance level. The joint test of the null hypothesis that sample skewness equals 0 and sample kurtosis equals 3 is rejected. The probability of the p- value 0.0000 is very significant, so we can reject H0 of normality. The distribution is leptokurtic and slightly negatively skewed. The AUD/USD spot rate time series is non-normal. The mean is 0.001072 and the dispersion around the mean is 0.026888.

The following Table show the ADF test of the monthly natural logarithmic returns of the AUD/USD spot exchange rate for the period 01/01/1990 to 01/01/2013.

Table 2 shows the ADF test of the natural logarithmic monthly returns of the AUD/USD spot exchange rate for the period starting from 01/01/1990 to 01/01/2013.

Null Hypothesis: LR has a unit root

136

Exogenous: ConstantLag Length: 0 (Automatic based on SIC, MAXLAG=15)

t-Statistic Prob.*

Augmented Dickey-Fuller test statistic -11.64366 0.0000Test critical values: 1% level -3.453997

5% level -2.87184510% level -2.572334

*MacKinnon (1996) one-sided p-values.

Augmented Dickey-Fuller Test EquationDependent Variable: D(LR)Method: Least SquaresDate: 10/21/13 Time: 20:19Sample (adjusted): 1990M03 2013M01Included observations: 275 after adjustments


LR(-1) -0.661475 0.056810 -11.64366 0.0000C 0.000819 0.001529 0.535537 0.5927

R-squared 0.331824 Mean dependent var 0.000115Adjusted R-squared 0.329376 S.D. dependent var 0.030931S.E. of regression 0.025330 Akaike info criterion -4.506408Sum squared resid 0.175159 Schwarz criterion -4.480104Log likelihood 621.6310 Hannan-Quinn criter. -4.495851F-statistic 135.5748 Durbin-Watson stat 1.911204Prob(F-statistic) 0.000000


For a level of significance of one per cent, the critical value of the t-statistic from Dickey-Fuller’s table is -3.45. According to Table 2 and to the sample evidence, we can reject the null hypothesis namely the existence of a unit root with one, five and ten per cent significance level. The ADF test statistic is -11.64, which is smaller than the critical values, (-3.4749, -2.87, -2.57). In other words, the monthly natural logarithmic returns of the AUD/USD are a stationary series.

Table 3 displays the Threshold TGARCH (1,1) model of the natural logarithmic monthly returns of the AUD/USD spot rate. We have used the normal distribution. The dataset has covered the period starting from 01/01/1990 to 01/01/2013.

Dependent Variable: LRMethod: ML - ARCH (Marquardt) - Normal distributionDate: 10/21/13 Time: 20:45Sample: 1990M02 2013M01Included observations: 276

137

Convergence achieved after 26 iterationsPresample variance: backcast (parameter = 0.7)GARCH = C(2) + C(3)*RESID(-1)^2 + C(4)*RESID(-1)^2*(RESID(-1)<0) + C(5)*GARCH(-1)


C 0.000238 0.001450 0.164156 0.8696

Variance Equation

ω 4.35E-05 3.11E-05 1.401445 0.1611RESID(-1)^2 0.119815 0.086485 1.385384 0.1659

RESID(-1)^2*(RESID(-1)<0) 0.057151 0.081022 0.705375 0.4806

GARCH(-1) 0.790205 0.074457 10.61290 0.0000

R-squared -0.000965 Mean dependent var 0.001072Adjusted R-squared -0.000965 S.D. dependent var 0.026888S.E. of regression 0.026901 Akaike info criterion -4.548434Sum squared resid 0.199000 Schwarz criterion -4.482847Log likelihood 632.6838 Hannan-Quinn criter. -4.522115Durbin-Watson stat 1.317331


According to Table 3, the asymmetric term, RESID(-1)^2*(RESID(-1)<0) has a coefficient of 0.057, a z-statistic of 0.71 and a p-probability of 0.48, which is not statistically significant at the 5 % significance level. Positive shocks related to good news have not implied a higher next period conditional variance than negative shocks related to bad news of the same extent. The ARCH effect is not significant. Only, the GARCH effect is statistically significant as the p-probability is 0.0000.

The mean and variance equation (1) and (2) will be used in combination with Table 3 to show the coefficients .

Table 4 shows the estimation of the coefficients

Coefficients Coefficients Estimated

z-statistic Probability

c 0.0002 0.16 0.874.35E-05 1.40 0.16

0.12 1.39 0.17

138

0.057 0.71 0.480.79 10.61* 0.00

Source: Author’s calculation based on EViews 6 software.* Significant at the 5% significance level.

According to Table 4, all the coefficients are not significant, except the coefficient that captures the GARCH effect. Specifically, the constant c of the mean equation has a coefficient of 0.0002 and it is not significantly positive at 5% significance level. The coefficient of the constant of the variance equation is not statistically significant with a z- statistic of 1.40. The , which explains the ARCH effect is not significant. The leverage effect is not significant with a z-statistic of 0.71 and a p-value of 0.48, which implies that there is no leverage effect and persistent asymmetry in the time series. Finally, the coefficient of the variance equation, which measures the GARCH effect, is positively significant at the 5% significance level. The persistent asymmetry is a strong evidence of the interactions of noise traders and arbitrageurs. The noise traders by buying in the wrong price they have created persistent fluctuations or volatility clusters of the price of the AUD/USD spot exchange rate.

Table 5 shows the residuals test in terms of the correlogram of squared residuals of the TGARCH(1,1) model of the natural logarithmic monthly returns of the AUD/USD spot rate for the period 01/01/1990 to 01/01/2013.

AC PAC Q-Stat Prob1 -0.021 0.011 26.152 0.0722 0.045 0.006 26.746 0.0843 -0.021 -0.012 26.878 0.1084 -0.051 -0.068 27.651 0.1185 0.068 0.115 29.044 0.113

139

6 -0.081 -0.105 31.020 0.0967 -0.053 -0.092 31.864 0.1038 -0.016 0.010 31.942 0.1289 -0.039 0.012 32.408 0.147

10 0.045 0.030 33.017 0.16211 0.005 0.001 33.024 0.19612 -0.028 -0.072 33.267 0.226



H0: There is no serial correlation in the residuals.

H1: There is serial correlation in the residuals.

According to Table 5, the Q-statistic associated with the p-values is not significant, which is a sign that there is no serial correlation in the residuals of the TGARCH(1,1) model.

Table 6 displays the residuals test in terms of ARCH LM test of the TGARCH(1,1) model. The aim is to check whether the standardized residuals of the natural logarithmic monthly returns of the AUD/USD spot exchange rate exhibit additional ARCH effect for the period 01/01/1990 to 01/01/2013.


F-statistic 0.546217 Prob. F(1,273) 0.4605Obs*R-squared 0.549120 Prob. Chi-Square(1) 0.4587

Test Equation:

140

Dependent Variable: WGT_RESID^2Method: Least SquaresDate: 10/21/13 Time: 21:08Sample (adjusted): 1990M03 2013M01Included observations: 275 after adjustments


C 0.950693 0.118063 8.052435 0.0000WGT_RESID^2(-1) 0.044678 0.060452 0.739065 0.4605




H0: There is no ARCH effect or the residuals are homoskedastic.

H1: There is ARCH effect and the residuals are heteroskedastic.

According to Table 6, the probability of the Chi – square statistic is 0.46, which is greater than the 5% significance level. The observations multiplied by the R-squared or the LM test statistic has a value of 0.55. Therefore, we reject H1 and accept H0. In other words, the residuals of the TGARCH(1,1) are homoskedastic and there is no additional ARCH effect.

Figure 3 displays the QQ – plot of the residuals of the TGARCH(1,1) model of the natural logarithmic monthly returns of the AUD/USD spot exchange rate for the period 01/01/1990 to 01/01/2013. The purpose of this graph is to show that the residuals are normally distributed.

141

-.08

-.04

.00

.04

.08

.12

-.20 -.15 -.10 -.05 .00 .05 .10

Quantiles of RESID01

Qua

ntile

s of

Nor

mal

Source: Author’s calculation based on EViews 6 software.

According to Figure 3, there are negative and positive shocks that are driving the residuals of the AUD/USD exchange rate away from normality. The prerequisite that the series is normal is that the points in the QQ – plots should lie alongside the straight line, which is not the case in this Figure.

Table 7 shows the EGARCH(1,1) model of the natural logarithmic monthly returns of the AUD/USD spot exchange rate. We have used the generalized error distribution. The dataset has covered the period starting from 01/01/1990 to 01/01/2013.

142

Dependent Variable: LRMethod: ML - ARCH (Marquardt) - Generalized error distribution (GED)Date: 10/21/13 Time: 20:48Sample: 1990M02 2013M01Included observations: 276Convergence achieved after 42 iterationsPresample variance: backcast (parameter = 0.7)LOG(GARCH) = C(2) + C(3)*ABS(RESID(-1)/@SQRT(GARCH(-1))) + C(4) *RESID(-1)/@SQRT(GARCH(-1)) + C(5)*LOG(GARCH(-1))


C 0.000667 0.001455 0.458548 0.6466

Variance Equation

C(2) -0.858893 0.494951 -1.735311 0.0827C(3) 0.326125 0.095315 3.421562 0.0006C(4) -0.039778 0.056685 -0.701734 0.4828C(5) 0.918693 0.062897 14.60642 0.0000

GED PARAMETER 1.706175 0.174460 9.779723 0.0000



According to Table 7 and equation (3), the asymmetric term, RESID(-1)/@SQRT(GARCH(-1), has a coefficient of -0.040, a z-statistic of -0.70 and a p-value of 0.48, which is not statistically significant at the 5 % significance level. EGARCH represented by C(5), has a z – statistic of 14.61 and it is statistically significant at the 5% significance level. There is no evidence that positive shocks due to good news implied a higher next period conditional variance than negative shocks due to bad news of the same extent. The generalized error distribution, (GED) parameter has a coefficient of 1.71, a z-statistic of 9.78 and a p-value of 0.0000.

Table 8 shows the residuals test in terms of the correlogram of squared residuals of the EGARCH(1,1) model of the natural logarithmic monthly returns of the AUD/USD spot rate for the period 01/01/1990 to 01/01/2013.

AC PAC Q-Stat Prob1 -0.039 0.006 34.948 0.0892 0.033 0.009 35.284 0.106

143

3 -0.008 -0.011 35.305 0.1314 -0.037 -0.073 35.738 0.1495 -0.000 0.018 35.738 0.1816 0.004 0.036 35.742 0.2177 -0.033 -0.066 36.091 0.2438 0.030 0.047 36.378 0.2729 -0.022 -0.015 36.537 0.308

10 -0.012 -0.022 36.583 0.35011 0.029 0.066 36.853 0.38312 0.014 0.035 36.916 0.426





According to Table 8, the Q-statistic associated with the p-values is not significant, which is a sign that there is no serial correlation in the residuals of the EGARCH(1,1) model. All the p-values are above the 5% significance level.

Table 9 displays the residuals test in terms of ARCH LM test of the EGARCH(1,1) model. The aim is to check whether the standardized residuals of the natural logarithmic monthly returns of the AUD/USD spot exchange rate exhibit additional ARCH effect for the period 01/01/1990 to 01/01/2013.


F-statistic 0.791503 Prob. F(1,273) 0.3744

144

Obs*R-squared 0.794997 Prob. Chi-Square(1) 0.3726

Test Equation:Dependent Variable: WGT_RESID^2Method: Least SquaresDate: 10/21/13 Time: 21:09Sample (adjusted): 1990M03 2013M01Included observations: 275 after adjustments


C 0.942907 0.121704 7.747522 0.0000WGT_RESID^2(-1) 0.053758 0.060425 0.889665 0.3744






According to Table 9, the probability of the Chi – square statistic is 0.37, which is greater than the 5% significance level. The observations multiplied by the R-squared or the LM test statistic has a value of 0.79. Therefore, we reject H1 and accept H0. In other words, the residuals of the EGARCH(1,1) are homoskedastic and there is no additional ARCH effect.

Figure 4 displays the QQ – plot of the residuals of the EGARCH(1,1) model of the natural logarithmic monthly returns of the AUD/USD spot exchange rate for the period 01/01/1990 to 01/01/2013. The purpose of this graph is to show that the residuals are normally distributed.

145

-.08

-.06

-.04

-.02

.00

.02

.04

.06

.08

-.20 -.15 -.10 -.05 .00 .05 .10


Qua

ntile

s of

Nor

mal



Table 10 shows the PARCH (1,1) model of the natural logarithmic monthly returns of the AUD/USD spot exchange rate. We have used the student’s t distribution. The dataset has covered the period starting from 01/01/1990 to 01/01/2013.

Dependent Variable: LRMethod: ML - ARCH (Marquardt) - Student's t distribution

146

Date: 10/21/13 Time: 20:55Sample: 1990M02 2013M01Included observations: 276Convergence achieved after 89 iterationsPresample variance: backcast (parameter = 0.7)@SQRT(GARCH)^C(6) = C(2) + C(3)*(ABS(RESID(-1)) - C(4)*RESID( -1))^C(6) + C(5)*@SQRT(GARCH(-1))^C(6)


C 0.001040 0.001426 0.729200 0.4659

Variance Equation

C(2) 1.30E-06 1.00E-05 0.129446 0.8970C(3) 0.108492 0.094904 1.143167 0.2530C(4) 0.087149 0.178676 0.487748 0.6257C(5) 0.757152 0.107071 7.071528 0.0000C(6) 2.990119 2.111413 1.416169 0.1567

T-DIST. DOF 15.21706 10.48813 1.450885 0.1468



According to Table 10 and equation (4), the constant, the ARCH and GARCH effect are not significant. The coefficient C(5) *@SQRT(GARCH(-1))^C(6), which measures the asymmetric term is statistically significant at the 5% significance level. C(5) has a coefficient of 0.76, a z-statistic of 7.07 and a p-value of 0.0000. Positive shocks related to good news implied a higher next period conditional variance than negative shocks related to bad news of the same extent. The student’s t distribution has a coefficient of 15.22, a z-statistic of 1.42 and a p-value of 0.15.

Table 11 shows the residuals test in terms of the correlogram of squared residuals of the PARCH (1,1) model of the natural logarithmic monthly returns of the AUD/USD spot rate for the period 01/01/1990 to 01/01/2013.

AC PAC Q-Stat Prob1 -0.039 0.015 31.201 0.1822 0.060 0.049 32.296 0.1843 0.011 0.011 32.332 0.220

147

4 -0.027 -0.076 32.564 0.2525 0.011 0.025 32.604 0.2946 -0.018 0.014 32.707 0.3357 -0.025 -0.060 32.905 0.3748 0.026 0.050 33.115 0.4139 -0.009 -0.015 33.139 0.460

10 -0.009 -0.005 33.164 0.50811 0.022 0.055 33.316 0.55012 0.017 0.032 33.405 0.593





According to Table 11, the Q-statistic associated with the p-values is not significant, which is a sign that there is no serial correlation in the residuals of the PARCH (1,1) model. All the values are above the 5 % significance level.

Table 12 displays the residuals test in terms of ARCH LM test of the PARCH (1,1) model. The aim is to check whether the standardized residuals of the natural logarithmic monthly returns of the AUD/USD spot exchange rate exhibit additional ARCH effect for the period 01/01/1990 to 01/01/2013.


F-statistic 0.109173 Prob. F(1,273) 0.7413

148

Obs*R-squared 0.109929 Prob. Chi-Square(1) 0.7402

Test Equation:Dependent Variable: WGT_RESID^2Method: Least SquaresDate: 10/21/13 Time: 21:12Sample (adjusted): 1990M03 2013M01Included observations: 275 after adjustments


C 0.979695 0.118508 8.266904 0.0000WGT_RESID^2(-1) 0.019991 0.060504 0.330414 0.7413






According to Table 12, the probability of the Chi – square statistic is 0.74, which is greater than the 5% significance level. The observations multiplied by the R-squared or the LM test statistic has a value of 0.11. Therefore, we reject H1 and accept H0. In other words, the residuals of the PARCH(1,1) are homoskedastic and there is no additional ARCH effect.

Figure 5 displays the QQ – plot of the residuals of the PARCH (1,1) model of the natural logarithmic monthly returns of the AUD/USD spot exchange rate for the period 01/01/1990 to 01/01/2013. The purpose of this graph is to show that the residuals are normally distributed.

149

-.08

-.06

-.04

-.02

.00

.02

.04

.06

.08

-.20 -.15 -.10 -.05 .00 .05 .10


Qua

ntile

s of

Nor

mal




150

In this article, we have tested volatility clustering by applying threshold generalized autoregressive conditional heteroskedastic models, (TGARCH), exponential generalized autoregressive conditional heteroskedastic models, (EGARCH) and Power Generalized Autoregressive Conditional Heteroskedastic models, (PGARCH). We have tested the volatility of the logarithmic monthly returns of the spot exchange rate of AUD/USD. The data that we have used are monthly returns starting from 01/01/1990 to 01/01/2013, which total to 276 observations. The total dataset includes 277 observations. The data was obtained from the Federal Reserve Statistical Release Department and the symbol of the series is H.10.

We have found that the AUD/USD spot exchange rate time series is non-normal. The natural logarithmic monthly returns of the AUD/USD are a stationary series. By applying a TGARCH (1,1) model, we have found that there is no leverage effect. the asymmetry term, RESID(-1)^2*(RESID(-1)<0) has a coefficient of 0.057, a z-statistic of 0.71 and a p-probability of 0.48, which is not statistically significant at the 5 % significance level. Positive shocks related to good news have not implied a higher next period conditional variance than negative shocks related to bad news of the same extent. The ARCH effect is not significant. Only, the GARCH effect is statistically significant as the p-probability is 0.0000. There is no serial correlation in the residuals of the TGARCH (1,1) model. The residuals of the TGARCH (1,1) are homoskedastic and there is no additional ARCH effect. Finally, there are negative and positive shocks that are driving the AUD/USD exchange spot rate away from normality.

By applying an EGARCH (1,1) model, the asymmetric term RESID(-1)/@SQRT(GARCH(-1), has a coefficient of -0.040, a z-statistic of -0.70 and a p-value of 0.48, which is not statistically significant at the 5 % significance level. Negative shocks do not imply a higher next period conditional variance than positive shocks of the same sign. EGARCH represented by C(5), has a z – statistic of 14.61 and it is statistically significant at the 5% significance level. The generalized error distribution, (GED) parameter has a coefficient of 1.71, a z-statistic of 9.78 and a p-value of 0.0000. There is no serial correlation in the residuals of the EGARCH (1,1) model. The residuals of the EGARCH (1,1) are homoskedastic and there is no additional ARCH effect. Finally, there are negative and positive shocks that are driving the AUD/USD exchange spot rate away from normality.

By applying a PARCH (1,1) model, the coefficient C(5) *@SQRT(GARCH(-1))^C(6), which measures the asymmetric term is statistically significant at the 5% significance level. C(5) has a coefficient of 0.76, a z-statistic of 7.07 and a p-value of 0.0000. Positive shocks related to good news implied a higher next period conditional variance than negative shocks related to bad news of the same extent. The student’s t distribution has a coefficient of 15.22, a z-statistic of 1.42 and a p-value of 0.15. The constant, the ARCH and GARCH effect are not significant.

By comparing the three models in terms of Akaike information criterion and Schwarz criterion, we have concluded the PARCH (1,1) is the best – fit model, as it has the lowest values.

References

151

Alexander, C., (2003), “Market Models. A Guide to Financial Data Analysis”. John Wiley and Sons Ltd. ISMA Centre. The Business School For Financial Markets.

Brooks,C..(2002), Introductory econometrics for Finance. Cambridge University Press. pp.456, 470.

Bollerslev, T.,(1986), “Generalized Autoregressive Conditional Heteroskedasticity”. Journal of Econometrics, 31, pp.307 – 327.

Bollerslev, T., Chou, R,Y., and Kroner, K.F., (1992), “ARCH Modelling in Finance: A Review of the Theory and Empirical Evidence”. Journal of Econometrics, 52, pp.5 -59.

Bollerslev, T, Engle, R.F, and Nelson, D.B., (1994), “ARCH Models”. Chapter 49 in Robert F. Engle and Daniel L.McFadden (eds.). Handbook of Econometrics, Volume 4, Amsterdam: Elsvier Science B.V.

Bollerslev, T, and Wooldridge, J,M., (1992), “Quasi – Maximum Likelihood Estimation and Inference in Dynamic Models with Time Varying Covariances”. Econometric Reviews, 11, pp.143 – 172.

Dickey, D.A and Fuller, W.A.,(1979), “Distribution of Estimators for Time Series Regressions with a Unit Root”. Journal of the American Statistical Association 74, 427-31.

Ding, Z, Granger, C.W.J and Engle, R.F., (1993), “A Long Memory Property of Stock Market Returns and a New Model”. Journal of Empirical Finance, 1, pp.83 -106.

Engle, R.F., (1982), “Autoregressive Conditional Heteroskedasticity with Estimates of the Variance of U.K. Inflation”. Econometrica, 50, pp.987 – 1008.

Engle, R.F, and Bollerslev, T., (1986), “Modelling the Persistence of Conditional Variances”. Econometric Reviews, 5, pp.1 -50.

Engle, R.F, Lilien, D.M and Robins,R.P., (1987), “Estimating Time Varying Risk Premia in the Term Structure: The ARCH-M Model”. Econometrica, 55, pp.391 - 407.

EViews 6, (2007), “ User’s Guide II ”. Quantitative micro software. pp.199-200.

Glosten, L.R.R., Jaganathan and Runkle, D., (1993), “On the Relation between the Expected Value and the Volatility of the Normal Excess Return on Stocks”. Journal of Finance, 48, pp.1779 – 1801.

Loretan,M., Mazda,A. and Subramanian,S.(2005), Indexes of the Foreign Exchange value of the Dollar. It is published in the site of the Federal Reserve Bulletin, Winter 2005.

152

Nelson, D.B.,(1991), “Conditional Heteroskedasticity in Asset Returns: A New Approach”. Econometrica 59(2), pp.347-70.

Schwert, W., (1989), “Stock Volatility and Crash of ‘87”. Review of Financial Studies, 3, pp.77 – 102.

Taylor, S., (1986), “Modelling Financial Time Series, New York: John Wiley and Sons.

Zakoian, J.M., (1994), “Threshold Heteroskedastic Models”. Journal of Economic Dynamics and Control, 18, pp.931 -944.

I have added a research paper to help you understand the how the Wald test restrictions are used in modeling exchange rates by applying an ARCH model.

153

Application of an ARCH system equations of 4 spot exchange rates in terms of CAD/USD, DKK/USD, CHF/USD and JPY/USD to check the conditional variance, covariance and correlation.

Abstract

In this article, we have tested the volatility of the natural logarithmic monthly returns of the CAD/USD, DKK/USD, CHF/USD and JPY/USD spot exchange rates. We have applied an ARCH – conditional heteroskedasticity method to model the variance, the covariance of the error terms and the correlation of the spot exchange rates. We have used three multivariate ARCH specifications in terms of the conditional constant correlation, the diagonal VECH and the diagonal BEKK. By applying the system equations 1, which represent a diagonal VECH of a covariance specification multivariate ARCH model, we have found that the log likelihood value of the system equations model is 2935.896. The average log likelihood value is 2.66. The Akaike information criterion is -21.03. The Schwarz criterion is -20.58 and the Hannan – Quinn criterion is -20.85. By applying the system equations 2, which represent a constant conditional correlation of a covariance specification multivariate ARCH model, we have found that the log likelihood value of the system equations model is 2903.890. The average log likelihood value is 2.63. The Akaike information criterion is -20.88. The Schwarz criterion is -20.59 and the Hannan – Quinn criterion is -20.77. By applying the system equations 3, which represent a diagonal BEKK of a covariance specification multivariate ARCH model, we have found that the log likelihood value of the system equations model is 2891.647. The average log likelihood value is 2.62. The Akaike information criterion is -20.84. The Schwarz criterion is -20.63 and the Hannan – Quinn criterion is -20.75. The best fit model to help the arbitrageurs to craft their investment strategy in terms of buying, selling or holding large portfolios of spot exchange rates is the system ARCH equations 1. It has the largest value of the log likelihood and average log likelihood value and the lowest error term value of the Akaike and Hannan - Quinn information criterion. The Schwarz criterion has the highest value. By applying the Wald test of the symmetry restrictions of the natural logarithmic monthly returns of the spot exchange rate returns namely the CAD/USD, DKK/USD, CHF/USD and JPY/USD, we have found that the sample evidence suggests accepting the null hypothesis. In other words, the test has rejected the symmetry restrictions of the mean coefficients. The data that we have used started from 01/01/1990 to 01/01/2013 and represent the CAD/USD, DKK/USD, CHF/USD and JPY/USD spot exchange rates, which total to 277 observations. The natural logarithmic monthly returns total to 276 observations. The data was obtained from the Federal Reserve Statistical Release Department and it is denoted by the symbol, H.10.

Keywords: System equations ARCH and GARCH terms, CAD/USD, DKK/USD, CHF/USD and JPY/USD spot exchange rate, conditional variance, covariance and correlation.

Introduction

154

This article has focused on modeling the volatility of the natural logarithmic monthly returns of the CAD/USD, DKK/USD, CHF/USD and JPY/USD spot exchange rates by using an ARCH system equation to check the conditional variance, covariance and correlation. We have used EViews 6 to test and validate the hypotheses that will be formulated.

System ARCH is a convenient technique to show the variance and the covariance of the error term in an autoregressive form. There are three choices of multivariate specifications such as the diagonal VECH, the diagonal BEKK and the conditional constant correlation.

The purpose of using the ARCH system equations is to help the arbitrageurs to craft their investment strategy in terms of buying, selling or holding large portfolios of spot exchange rates. This will be done by selecting the best fit model in term of the maximum value of log likelihood, and average log likelihood. In addition, we are seeking the lowest error value of the Akaike information criterion, the Schwarz criterion and the Hanna – Quinn criterion. Specifically, Arbitrageurs or rational investors would open a short position to the system of expensive currencies portfolio and simultaneously purchase the same or very similar spot exchange rate portfolio at a spot rate in another market at a lower price to hedge their risks. The effect of this selling by arbitrageurs is to bring the price of the overpriced currencies portfolio down to its fundamental value. The same principle applies for undervalued or depreciated currencies. To earn profit arbitrageurs would buy underpriced currencies and open a short position to an overpriced to hedge their risk. They prevent with this technique underpricing to be substantial or long lasting.

The rest of the paper is organized as follows. Section 1 describes the methodology and the data. Section 2 is an analysis of statistical and econometrical tests and Section 3 summarizes and concludes.

1.Methodology and data description

The methodologies that we have used are system equations of multivariate ARCH models. Autoregressive models have been studied by various researchers such as: Alexander,(2003), Brooks, (2002), Nelson, (1991), Bollerslev, (1990), Bollerslev, (1986), Bollerslev, Chou, and Kroner, (1992), Bollerslev, Engle and Nelson, (1994), Bollerslev, Jeffrey, and Wooldridge, (1992), Donald, (1991), Donald and Monahan, (1992), Engle and Kroner, (1995), Ding, Granger and Engle, (1993), Engle, (1982), Engle and Bollerslev, (1986), Engle, Lilien and Robins, (1987), Glosten, Jaganathan and Runkle, (1993), Newey and West, (1994), Schwert, (1989), Taylor, (1986) and Zakoian, (1994).

The system equations 1 represent a multivariate ARCH model. The model is a diagonal VECH of a covariance specification. The ARCH maximum likelihood is calculated through the Marquardt algorithm. The estimation command, the estimated equations, the mean equations, the variance and covariance equations of the ARCH (1) and GARCH(1) multivariate system are as follows:

Estimation Command:=====================

155

ARCH(DERIV=AA) @DIAGVECH C(INDEF) ARCH(1,INDEF) GARCH(1,INDEF)

Estimated Equations:=====================LNRETCAD = C(1) (1)

Where: LNRETCAD is the natural logarithmic monthly returns of the spot exchange rate CAD/USD. C(1) is the constant of the equation.

LNRETCHF = C(2) (2)

Where: LNRETCHF is the natural logarithmic monthly returns of the spot exchange rate CHF/USD. C(2) is the constant of the equation.

LNRETDKK = C(3) (3)

Where: LNRETDKK is the natural logarithmic monthly returns of the spot exchange rate DKK/USD. C(3) is the constant of the equation.

LNRETJPY = C(4) (4)

Where: LNRETJPY is the natural logarithmic monthly returns of the spot exchange rate JPY/USD. C(4) is the constant of the equation.

Substituted Coefficients (mean equations):================================LNRETCAD = -0.000198594205929 (5) LNRETCHF = -0.00403814506366 (6)

LNRETDKK = -0.00367527995757 (7)

LNRETJPY = -0.00190342907729 (8)

Variance-Covariance Representation:=====================GARCH = M + A1.*RESID(-1)*RESID(-1)' + B1.*GARCH(-1)

Variance and Covariance Equations:=====================GARCH1 = M(1,1) + A1(1,1)*RESID1(-1)^2 + B1(1,1)*GARCH1(-1) (9)

GARCH2 = M(2,2) + A1(2,2)*RESID2(-1)^2 + B1(2,2)*GARCH2(-1) (10)



Where: GARCH1 is the conditional variance of the CAD/USD spot exchange rate. GARCH 2 is the conditional variance of the CHF/USD spot exchange rate. GARCH 3 is the conditional variance of the DKK/USD spot exchange rate. GARCH 4 is the conditional variance of the JPY/USD spot exchange rate.

COV1_2 = M(1,2) + A1(1,2)*RESID1(-1)*RESID2(-1) + B1(1,2)*COV1_2(-1) (13)





156


Where: COV 1_2 is the conditional covariance between the CAD and the CHF. COV 1_3 is the conditional covariance between the CAD and the DKK. COV 1_4 is the conditional covariance between the CAD and the JPY. COV 2_3 is the conditional covariance between the CHF and the DKK. COV 2_4 is the conditional covariance between the CHF and the JPY. COV 3_4 is the conditional covariance between the DKK and the JPY.

Substituted Coefficients:=====================GARCH1 = 1.53541081268e-05 + 0.195240705537*RESID1(-1)^2 + 0.76258984777*GARCH1(-1)

(19)

GARCH2 = 0.000186085932276 + 0.151025328609*RESID2(-1)^2 + 0.631849700201*GARCH2(-1)

(20)


(21)


(22)

COV1_2 = 2.8356351288e-05 + 0.0245260769584*RESID1(-1)*RESID2(-1) + 0.59715460081*COV1_2(-1)

(23)

COV1_3 = 3.2232504728e-05 + 0.0655861369301*RESID1(-1)*RESID3(-1) + 0.625219635364*COV1_3(-1)

(24)

COV1_4 = 3.48443071278e-05 + 0.14237717967*RESID1(-1)*RESID4(-1) -0.329718314306*COV1_4(-1)

(25)

COV2_3 = 0.000152706362993 + 0.161043984121*RESID2(-1)*RESID3(-1) + 0.638818634615*COV2_3(-1)(26)

COV2_4 = 9.40602772686e-05 + 0.0519183722581*RESID2(-1)*RESID4(-1) + 0.6856730049*COV2_4(-1)(27)

COV3_4 = 4.81238848873e-05 + 0.0523633483445*RESID3(-1)*RESID4(-1) + 0.781179497991*COV3_4(-1)(28)

The system equations 2 represent a multivariate ARCH model. The model is a constant conditional correlation of a covariance specification. The ARCH Maximum Likelihood is calculated through the Marquardt algorithm. The estimation command, the estimated equations, the mean equations, the variance and covariance equations of the ARCH (1) and GARCH(1) multivariate system are as follows:

Estimation Command:=====================ARCH(DERIV=AA) @CCC C ARCH(1) GARCH(1)


Where: LNRETCAD is the natural logarithmic monthly returns of the spot exchange rate CAD/USD. C(1) is the constant of the equation.

157

LNRETCHF = C(2) (30)

Where: LNRETCHF is the natural logarithmic monthly returns of the spot exchange rate CHF/USD. C(2) is the constant of the equation.

LNRETDKK = C(3) (31)

Where: LNRETDKK is the natural logarithmic monthly returns of the spot exchange rate DKK/USD. C(3) is the constant of the equation.

LNRETJPY = C(4) (32) Where: LNRETJPY is the natural logarithmic monthly returns of the spot exchange rate JPY/USD. C(4) is the constant of the equation.

Substituted Coefficients: (mean equations):=================================LNRETCAD = -7.39889179944e-05 (33)

LNRETCHF = -0.00224734526905 (34)

LNRETDKK = -0.00142461374593 (35)

LNRETJPY = -0.00151368914384 (36)

Variance and Covariance Representations:=====================GARCH(i) = M(i) + A1(i)*RESID(i)(-1)^2 + B1(i)*GARCH(i)(-1)

COV(i,j) = R(i,j)*@SQRT(GARCH(i)*GARCH(j))

Variance and Covariance Equations:=====================GARCH1 = C(5) + C(6)*RESID1(-1)^2 + C(7)*GARCH1(-1) (37)

GARCH2 = C(8) + C(9)*RESID2(-1)^2 + C(10)*GARCH2(-1) (38)




COV1_2 = C(17)*@SQRT(GARCH1*GARCH2) (41)







158

Substituted Coefficients:=====================GARCH1 = 1.32739414729e-05 + 0.202995283141*RESID1(-1)^2 + 0.766911037193*GARCH1(-1)

(47)


(48)


(49)


(50)

COV1_2 = 0.241402906757*@SQRT(GARCH1*GARCH2)

(51)


(52)


(53)


(54)


(55)


(56)

The system equations 3 represent a multivariate ARCH model. The model is a diagonal BEKK of a covariance specification. The ARCH Maximum Likelihood is calculated through the Marquardt algorithm. The estimation command, the estimated equations, the mean equations, the variance and covariance equations of the ARCH (1) and GARCH(1) multivariate system are as follows:

Estimation Command:=====================ARCH(DERIV=AA) @DIAGBEKK C(DIAG) ARCH(1,DIAG) GARCH(1,DIAG)


LNRETCHF = C(2) (58)

LNRETDKK = C(3) (59)

LNRETJPY = C(4) (60)

159

Substituted Coefficients:=====================LNRETCAD = 0.000598156398879 (61)

LNRETCHF = -0.00109184700772 (62)

LNRETDKK = -0.000926360627345 (63)

LNRETJPY = -0.00182102510883 (64)

Variance-Covariance Representation:=====================GARCH = M + A1*RESID(-1)*RESID(-1)'*A1 + B1*GARCH(-1)*B1

Variance and Covariance Equations:=====================GARCH1 = M(1,1) + A1(1,1)^2*RESID1(-1)^2 + B1(1,1)^2*GARCH1(-1) (65)

GARCH2 = M(2,2) + A1(2,2)^2*RESID2(-1)^2 + B1(2,2)^2*GARCH2(-1) (66)




COV1_2 = A1(1,1)*A1(2,2)*RESID1(-1)*RESID2(-1) + B1(1,1)*B1(2,2)*COV1_2(-1) (69)



COV2_3 = A1(2,2)*A1(3,3)*RESID2(-1)*RESID3(-1) + B1(2,2)*B1(3,3)*COV2_3(-1) (72) COV2_4 = A1(2,2)*A1(4,4)*RESID2(-1)*RESID4(-1) + B1(2,2)*B1(4,4)*COV2_4(-1) (73)



Substituted Coefficients:=====================GARCH1 = 7.48545113012e-06+0.0887305025778*RESID1(-1)^2+0.892558809051*GARCH1(-1)

(75)

GARCH2 = -3.04030767555e-06+0.0830928567535*RESID2(-1)^2+0.927099590292*GARCH2(-1)

(76)

GARCH3 = 3.92100107163e-06+0.101678776401*RESID3(-1)^2+0.901287145764*GARCH3(-1)

(77)

GARCH4 = 1.74128289923e-05+0.0501735508656*RESID4(-1)^2+0.926063342157*GARCH4(-1)

(78)

160

COV1_2 = 0.0858654234274*RESID1(-1)*RESID2(-1) + 0.909665271505*COV1_2(-1)

(79)


(80)


(81)


(82)


(83)


(84)

The data that we have used are logarithmic monthly returns starting from 01/02/1990 to 01/01/2013 of the CAD/USD, DKK/USD, CHF/USD and JPY/USD spot rate, which total to 276 observations. The data was obtained from the Federal Reserve Statistical Release Department and it is denoted by the symbol, H.10.

The logarithmic formula that we have used is:

(85)

Where: Rt is the monthly return for month t, Pt is the closing price for month t, and Pt-1 is the previous closing price for month t-1.

2. Statistical and econometrical tests

Table 1 shows the system equations 1, which represent a multivariate ARCH model. The model is a diagonal VECH of a covariance specification. The ARCH Maximum Likelihood is calculated through the Marquardt algorithm. The data that we have used are monthly returns starting from 01/01/1990 to 01/01/2013, which total to 277 observations. The natural logarithmic monthly returns total to 276 observations. The data was obtained from the Federal Reserve Statistical Release Department and the symbol of the series is H.10.

System: SYS01Estimation Method: ARCH Maximum Likelihood (Marquardt)Covariance specification: Diagonal VECHDate: 11/26/13 Time: 13:43Sample: 1990M02 2013M01Included observations: 276Total system (balanced) observations 1104Presample covariance: backcast (parameter =0.7)Convergence achieved after 56 iterations

161

Coefficient Std. Error z-Statistic Prob.

C(1) -0.000199 0.000932 -0.213189 0.8312C(2) -0.004038 0.001682 -2.400459 0.0164C(3) -0.003675 0.001527 -2.406897 0.0161C(4) -0.001903 0.001797 -1.059111 0.2895

Variance Equation Coefficients

C(5) 1.54E-05 9.36E-06 1.641131 0.1008C(6) 2.84E-05 4.65E-05 0.610380 0.5416C(7) 3.22E-05 2.91E-05 1.107402 0.2681C(8) 3.48E-05 3.33E-05 1.045578 0.2958C(9) 0.000186 5.61E-05 3.316210 0.0009

C(10) 0.000153 3.91E-05 3.905382 0.0001C(11) 9.41E-05 7.99E-05 1.176990 0.2392C(12) 0.000157 3.87E-05 4.066967 0.0000C(13) 4.81E-05 3.83E-05 1.254928 0.2095C(14) 0.000305 0.000123 2.472924 0.0134C(15) 0.195241 0.063509 3.074200 0.0021C(16) 0.024526 0.051215 0.478884 0.6320C(17) 0.065586 0.051115 1.283121 0.1994C(18) 0.142377 0.088549 1.607897 0.1079C(19) 0.151025 0.038722 3.900238 0.0001C(20) 0.161044 0.035964 4.477900 0.0000C(21) 0.051918 0.044951 1.154997 0.2481C(22) 0.201548 0.038235 5.271357 0.0000C(23) 0.052363 0.039564 1.323505 0.1857C(24) 0.213685 0.073793 2.895740 0.0038C(25) 0.762590 0.076663 9.947267 0.0000C(26) 0.597155 0.645110 0.925663 0.3546C(27) 0.625220 0.298011 2.097972 0.0359C(28) -0.329718 0.366833 -0.898824 0.3687C(29) 0.631850 0.079720 7.925883 0.0000C(30) 0.638819 0.061119 10.45212 0.0000C(31) 0.685673 0.242232 2.830641 0.0046C(32) 0.600747 0.058177 10.32617 0.0000C(33) 0.781179 0.152202 5.132520 0.0000C(34) 0.358389 0.195785 1.830520 0.0672

Log likelihood 2935.896Schwarz criterion -20.58224Avg. log likelihood 2.659327Hannan-Quinn criter. -20.84927Akaike info criterion -21.02824

Covariance specification: Diagonal VECHGARCH = M + A1.*RESID(-1)*RESID(-1)' + B1.*GARCH(-1)M is an indefinite matrixA1 is an indefinite matrixB1 is an indefinite matrix

Transformed Variance Coefficients


M(1,1) 1.54E-05 9.36E-06 1.641131 0.1008M(1,2) 2.84E-05 4.65E-05 0.610380 0.5416M(1,3) 3.22E-05 2.91E-05 1.107402 0.2681M(1,4) 3.48E-05 3.33E-05 1.045578 0.2958M(2,2) 0.000186 5.61E-05 3.316210 0.0009M(2,3) 0.000153 3.91E-05 3.905382 0.0001M(2,4) 9.41E-05 7.99E-05 1.176990 0.2392

162

M(3,3) 0.000157 3.87E-05 4.066967 0.0000M(3,4) 4.81E-05 3.83E-05 1.254928 0.2095M(4,4) 0.000305 0.000123 2.472924 0.0134A1(1,1) 0.195241 0.063509 3.074200 0.0021A1(1,2) 0.024526 0.051215 0.478884 0.6320A1(1,3) 0.065586 0.051115 1.283121 0.1994A1(1,4) 0.142377 0.088549 1.607897 0.1079A1(2,2) 0.151025 0.038722 3.900238 0.0001A1(2,3) 0.161044 0.035964 4.477900 0.0000A1(2,4) 0.051918 0.044951 1.154997 0.2481A1(3,3) 0.201548 0.038235 5.271357 0.0000A1(3,4) 0.052363 0.039564 1.323505 0.1857A1(4,4) 0.213685 0.073793 2.895740 0.0038B1(1,1) 0.762590 0.076663 9.947267 0.0000B1(1,2) 0.597155 0.645110 0.925663 0.3546B1(1,3) 0.625220 0.298011 2.097972 0.0359B1(1,4) -0.329718 0.366833 -0.898824 0.3687B1(2,2) 0.631850 0.079720 7.925883 0.0000B1(2,3) 0.638819 0.061119 10.45212 0.0000B1(2,4) 0.685673 0.242232 2.830641 0.0046B1(3,3) 0.600747 0.058177 10.32617 0.0000B1(3,4) 0.781179 0.152202 5.132520 0.0000B1(4,4) 0.358389 0.195785 1.830520 0.0672


According to Table 1, the first four coefficients listed in the upper section, namely, C(1), C(2), C(3) and C(4) represent the mean equation. Specifically, the coefficient C(1) represents the CAD/USD spot exchange rate. It has a coefficient of -0.000199 and a p-value of 0.83. It is not significant as it is above the 5% significance level. The coefficient C(2) and C(3), which shows the CHF/USD and DKK/USD respectively are statistically significant at the 5% significance level. Finally, the C(4), which displays the JPY/USD is not statistically significant. Then, the table displays the variance coefficients. C(5) to C(14) are the coefficients for the constant matrix. The coefficients C(9), C(10), C(12) and C(14) show significant p-values. C(15) to C(24) are the coefficients for the ARCH term. C(15), C(19), C(20), C(22) and C(24) show a significant p-values of 0.0021, 0.0001, 0.0000, 0.0000, and 0.0038. C(25) to C(34) are the coefficients for the GARCH term. The coefficients C(25), C(27), C(29), C(30), C(31), C(32), C(33) display significant p-values at the 5% significance level. The log likelihood value of the system equations model is 2935.896. The average log likelihood value is 2.66. The Akaike information criterion is -21.03. The Schwarz criterion is -20.58 and the Hannan – Quinn criterion is -20.85.

The next part of the table 1 shows the diagonal VECH covariance model. GARCH represents the conditional variance matrix. M is the constant matrix coefficient. A1 is the coefficient matrix for the ARCH term and B1 is the coefficient matrix for the GARCH term. Equations (19) to (22) in the methodological section show the GARCH equations, which represent the conditional variance equations. For example, equation (19) is as follows:

GARCH1 = 1.53541081268e-05 + 0.195240705537*RESID1(-1)^2 + 0.76258984777*GARCH1(-1) (1.64) (3.07) (9.95) (0.10) (0.00) (0.00)Where: GARCH1 is the conditional variance equation of the CAD/USD spot exchange rate.

163

1.5354 represents M(1,1), which is the constant matrix coefficient. 0.1952 represents A1(1,1) which is the coefficient matrix for the ARCH term. 0.7626 represents B1(1,1) which is the coefficient matrix for the GARCH term.

The first parenthesis represents the z-statistic and the second parenthesis displays the p-value. The constant matrix coefficient is not significant. Both the coefficient matrix for the ARCH and GARCH term are significant as the p-values are below the 5% significance level. The conditional variance equations for GARCH 2, 3 and 4 have significant values for the coefficient matrices.

Equations (23) to (28) in the methodological section show the covariance equations, which represent the conditional covariance equations. The first parenthesis represents the z-statistic and the second parenthesis displays the p-value. The constant matrix coefficient, the coefficient matrix for the ARCH and GARCH term are not significant as the p-values are above the 5% significance level. For example, equation (23) is as follows:

COV1_2 = 2.8356351288e-05 + 0.0245260769584*RESID1(-1)*RESID2(-1) + 0.59715460081*COV1_2(-1) (0.61) (0.48) (0.93) (0.54) (0.63) (0.35)

Where: COV1_2 is the conditional covariance equation of the CAD/CHF spot exchange rate. 2.8356 represents M(1,2), which is the constant matrix coefficient. 0.0245 represents A1(1,2) which is the coefficient matrix for the ARCH term. 0.5972 represents B1(1,2) which is the coefficient matrix for the GARCH term.

The conditional covariance equation COV2_3, which represents the conditional covariance between the CHF and the DKK spot exchange rate displays significant values of the coefficient matrices. The COV1_4, which is the covariance of CAD/JPY is not significant. COV1_3, COV2_4, COV3_4, which represent CAD/DKK, CHF/JPY, and DKK/JPY display only a significant GARCH term of 0.04, 0.005 and 0.0000 respectively.

Table 2 shows the Wald test of the symmetry restrictions of the logarithmic monthly returns starting from 01/02/1990 to 01/01/2013. The data covers the spot exchange rate returns namely the CAD/USD, DKK/USD, CHF/USD and JPY/USD, which total to 276 observations. The data was obtained from the Federal Reserve Statistical Release Department and it is denoted by the symbol, H.10.

Wald Test:System: SYS01

Test Statistic Value df Probability

Chi-square 6.205460 4 0.1843

Null Hypothesis Summary:

Normalized Restriction (= 0) Value Std. Err.

C(1) -0.000199 0.000932C(2) -0.004038 0.001682C(3) -0.003675 0.001527

164

C(4) -0.001903 0.001797

Restrictions are linear in coefficients.Source: Author’s calculation based on EViews 6 software.Significant at the 5% significance level.


H0: The mean coefficients C(1)=C(2)=C(3)=C(4)= 0.

H1: The mean coefficients C(1)=C(2)=C(3)=C(4) ≠ 0.

According to Table 2, the probability of the Chi-square statistic is 6.21 and the p-value is 0.18. The sample evidence suggests to accept the null hypothesis. In other words, the test has rejected the symmetry restrictions.

Table 3 shows the system equations 2, which represent a multivariate ARCH model. The model is a constant conditional correlation of a covariance specification. The ARCH Maximum Likelihood is calculated through the Marquardt algorithm. The data that we have used are monthly returns starting from 01/01/1990 to 01/01/2013, which total to 277 observations. The natural logarithmic monthly returns total to 276 observations. The data was obtained from the Federal Reserve Statistical Release Department and the symbol of the series is H.10.

System: SYS02Estimation Method: ARCH Maximum Likelihood (Marquardt)Covariance specification: Constant Conditional CorrelationDate: 11/26/13 Time: 17:58Sample: 1990M02 2013M01Included observations: 276Total system (balanced) observations 1104Presample covariance: backcast (parameter =0.7)Convergence achieved after 62 iterations


C(1) -7.40E-05 0.000929 -0.079675 0.9365C(2) -0.002247 0.001559 -1.441825 0.1494C(3) -0.001425 0.001365 -1.043417 0.2968C(4) -0.001514 0.001687 -0.897015 0.3697

165


C(5) 1.33E-05 8.06E-06 1.647688 0.0994C(6) 0.202995 0.066791 3.039282 0.0024C(7) 0.766911 0.072270 10.61175 0.0000C(8) 0.000277 0.000105 2.631233 0.0085C(9) 0.224476 0.045657 4.916603 0.0000

C(10) 0.457997 0.144120 3.177881 0.0015C(11) 0.000129 4.49E-05 2.867505 0.0041C(12) 0.216475 0.050956 4.248249 0.0000C(13) 0.610380 0.080099 7.620348 0.0000C(14) 0.000527 0.000193 2.737024 0.0062C(15) 0.257042 0.088192 2.914578 0.0036C(16) 0.000274 0.261945 0.001046 0.9992C(17) 0.241403 0.062748 3.847151 0.0001C(18) 0.349273 0.058583 5.962018 0.0000C(19) 0.036081 0.059243 0.609045 0.5425C(20) 0.908953 0.009870 92.09174 0.0000C(21) 0.474856 0.043462 10.92576 0.0000C(22) 0.397918 0.049386 8.057360 0.0000


Covariance specification: Constant Conditional CorrelationGARCH(i) = M(i) + A1(i)*RESID(i)(-1)^2 + B1(i)*GARCH(i)(-1)COV(i,j) = R(i,j)*@SQRT(GARCH(i)*GARCH(j))

Transformed Variance Coefficients


M(1) 1.33E-05 8.06E-06 1.647688 0.0994A1(1) 0.202995 0.066791 3.039282 0.0024B1(1) 0.766911 0.072270 10.61175 0.0000M(2) 0.000277 0.000105 2.631233 0.0085A1(2) 0.224476 0.045657 4.916603 0.0000B1(2) 0.457997 0.144120 3.177881 0.0015M(3) 0.000129 4.49E-05 2.867505 0.0041A1(3) 0.216475 0.050956 4.248249 0.0000B1(3) 0.610380 0.080099 7.620348 0.0000M(4) 0.000527 0.000193 2.737024 0.0062A1(4) 0.257042 0.088192 2.914578 0.0036B1(4) 0.000274 0.261945 0.001046 0.9992R(1,2) 0.241403 0.062748 3.847151 0.0001R(1,3) 0.349273 0.058583 5.962018 0.0000R(1,4) 0.036081 0.059243 0.609045 0.5425R(2,3) 0.908953 0.009870 92.09174 0.0000R(2,4) 0.474856 0.043462 10.92576 0.0000R(3,4) 0.397918 0.049386 8.057360 0.0000


166

Significant at the 5% significance level.

According to Table 3, the first four coefficients listed in the upper section, namely, C(1), C(2), C(3) and C(4) represent the mean equation. Specifically, the coefficient C(1) represents the CAD/USD spot exchange rate. It has a coefficient of -7.40E-05 and a p-value of 0.94. It is not significant as it is above the 5% significance level. The coefficient C(2) and C(3), which shows the CHF/USD and DKK/USD respectively are not statistically significant at the 5% significance level as the p-value is 0.15 and 0.30 respectively. Finally the C(4), which displays the JPY/USD is not statistically significant. Then, the table displays the variance coefficients. C(5) to C(10) are the coefficients for the constant matrix. The coefficients C(6), C(7), C(8), C(9) and C(10) show significant p-values of 0.00, 0.00, 0.01, 0.00, 0.00 respectively. C(11) to C(16) are the coefficients for the ARCH term. C(11), C(12), C(13), C(14) and C(15) show a statistically significant p-values. C(17) to C(22) are the coefficients for the GARCH term. The coefficients C(17), C(18), C(20), C(21), and C(22), display significant p-values at the 5% significance level. The log likelihood value of the system equations model is 2903.890. The average log likelihood value is 2.63. The Akaike information criterion is -20.88. The Schwarz criterion is -20.59 and the Hannan – Quinn criterion is -20.77.

The next part of the table 1 shows the constant conditional correlation of the covariance model. GARCH represents the conditional variance matrix. M is the constant matrix coefficient. A1 is the coefficient matrix for the ARCH term and B1 is the coefficient matrix for the GARCH term. Equations (47) to (50) in the methodological section show the GARCH equations, which represent the conditional variance equations. For example, equation (47) is as follows:

GARCH1 = 1.32739414729e-05 + 0.202995283141*RESID1(-1)^2 + 0.766911037193*GARCH1(-1) (1.65) (3.04) (10.61) (0.099) (0.002) (0.000)Where: GARCH1 is the conditional variance equation of the CAD/USD spot exchange rate. 1.32739 represents M(1), which is the constant matrix coefficient. 0.20299 represents A1(1) which is the coefficient matrix for the ARCH term. 0.76691 represents B1(1) which is the coefficient matrix for the GARCH term.

The first parenthesis represents the z-statistic and the second parenthesis displays the p-value. The constant matrix coefficient and the coefficient matrix for the ARCH and GARCH term are significant as the p-values are below the 5% significance level. The conditional variance equations for GARCH 2, 3 and 4 have significant values for the coefficient matrices except the coefficient matrix of the GARCH term B1(4).

Equations (51) to (56) in the methodological section show the covariance equations, which represent the conditional covariance equations. The first parenthesis represents the z-statistic and the second parenthesis displays the p-value. The output displays that the correlation R(1,2) is 0.24. It is significant as the z-statistic is 3.85 and the p-value is 0.00 and below the 5% significance level. For example, equation (51) is as follows:


(3.85) (0.00)

Where: COV1_2 is the conditional covariance equation of the CAD/CHF spot exchange rate.

167

R(1,2) is 0.24 and shows the output of the constant conditional correlation between the two currencies.

The conditional covariance equation COV1_3, which represents the conditional covariance between the CAD and the DKK spot exchange rate, displays significant values of the correlation coefficient. The correlation R(1,3) is 0.35. It has a z-statistic of 5.96 and a p-value of 0.00. The COV1_4, which is the covariance of CAD/JPY is not significant. The correlation R(1,4) is 0.04. The z-statistic is 0.61 and the p-value is 0.54. The COV2_3, COV2_4, and COV3_4, which represent CHF/DKK, CHF/JPY, and DKK/JPY all statistically significant. The correlation R(2,3) is 0.91. It has a z-statistic of 92.09 and a p-value of 0.00. The correlation R(2,4) is 0.47. It has a z-statistic of 10.93 and a p-value of 0.00. Finally, the correlation R(3,4) is 0.40. It has a z-statistic of 8.06 and a p-value of 0.00.




Chi-square 2.293821 4 0.6819



C(1) -7.40E-05 0.000929C(2) -0.002247 0.001559C(3) -0.001425 0.001365C(4) -0.001514 0.001687

168



H0: The mean coefficients C(1)= C(2) =C(3)=C(4)= 0.

H1: The mean coefficients C(1)= C(2)=C(3)=C(4) ≠ 0.

According to Table 4, the probability of the Chi-square statistic is 2.29 and the p-value is 0.68. The sample evidence suggests to accept the null hypothesis. In other words, the test failed to reject the symmetry restrictions.

Table 5 shows the system equations 3, which represent a multivariate ARCH model. The model is a diagonal BEKK of a covariance specification. The ARCH Maximum Likelihood is calculated through the Marquardt algorithm. The data that we have used are monthly returns starting from 01/01/1990 to 01/01/2013, which total to 277 observations. The natural logarithmic monthly returns total to 276 observations. The data was obtained from the Federal Reserve Statistical Release Department and the symbol of the series is H.10.

System: SYS03Estimation Method: ARCH Maximum Likelihood (Marquardt)Covariance specification: Diagonal BEKKDate: 11/26/13 Time: 18:02Sample: 1990M02 2013M01Included observations: 276Total system (balanced) observations 1104Presample covariance: backcast (parameter =0.7)Convergence achieved after 93 iterations


C(1) 0.000598 0.000958 0.624331 0.5324C(2) -0.001092 0.001120 -0.974649 0.3297C(3) -0.000926 0.000957 -0.967527 0.3333C(4) -0.001821 0.001452 -1.254016 0.2098


C(5) 7.49E-06 4.36E-06 1.717914 0.0858C(6) -3.04E-06 2.23E-06 -1.365042 0.1722C(7) 3.92E-06 2.23E-06 1.755561 0.0792C(8) 1.74E-05 7.70E-06 2.262118 0.0237

169

C(9) 0.297877 0.045174 6.593932 0.0000C(10) 0.288258 0.026328 10.94880 0.0000C(11) 0.318871 0.026856 11.87347 0.0000C(12) 0.223995 0.038227 5.859642 0.0000C(13) 0.944753 0.018936 49.89229 0.0000C(14) 0.962860 0.006985 137.8454 0.0000C(15) 0.949361 0.007883 120.4268 0.0000C(16) 0.962322 0.011963 80.43952 0.0000



According to Table 5, the first four coefficients listed in the upper section, namely, C(1), C(2), C(3) and C(4) represent the mean equation. Specifically, the coefficient C(1) represents the CAD/USD spot exchange rate. It has a coefficient of 0.001 and a p-value of 0.53. It is not significant as the p-value is above the 5% significance level. The coefficient C(2), C(3), and C(4) which shows the CHF/USD, DKK/USD, and JPY/USD respectively are not statistically significant at the 5% significance level. Then, the table displays the variance coefficients. C(5) to C(8) are the coefficients for the constant matrix. The coefficients C(5), C(6), and C(7) are not statistically significant. C(9) to C(12) are the coefficients for the ARCH term. They are all significant. C(13) to C(16) are the coefficients for the GARCH term. The coefficients C(13), C(14), C(15), and C(16) are statistically significant at the 5% significance level. The log likelihood value of the system equations model is 2891.647. The average log likelihood value is 2.62. The Akaike information criterion is -20.84. The Schwarz criterion is -20.63 and the Hannan – Quinn criterion is -20.75.

GARCH represents the conditional variance matrix of the diagonal BEKK covariance model. M is the constant matrix coefficient. A1 is the coefficient matrix for the ARCH term and B1 is the coefficient matrix for the GARCH term. Equations (75) to (78) in the methodological section show the GARCH equations, which represent the conditional variance equations. For example, equation (75) is as follows:

GARCH1 = 7.48545113012e-06+0.0887305025778*RESID1(-1)^2+0.892558809051*GARCH1(-1) (1.72) (6.59) (49.89) (0.09) (0.00) (0.00)Where: GARCH1 is the conditional variance equation of the CAD/USD spot exchange rate. 7.4855 represents M(1,1), which is the constant matrix coefficient. 0.0887 represents A1(1,1) which is the coefficient matrix for the ARCH term. 0.8926 represents B1(1,1) which is the coefficient matrix for the GARCH term.

The first parenthesis represents the z-statistic and the second parenthesis displays the p-value. The constant matrix coefficient is not significant. Both the coefficient matrix for the ARCH and GARCH term are significant as the p-values are below the 5% significance level. Similar results show the conditional variance equations for GARCH 2, 3 and 4.

Equations (79) to (84) in the methodological section show the covariance equations, which represent the conditional covariance equations. For example, equation (79 -84) are as follows:

170







The conditional covariance equation COV2_3, which represents the conditional covariance between the CHF and the DKK spot exchange rate displays significant values of coefficient matrices. The COV1_4, which is the covariance of CAD/JPY is not significant. COV1_3, COV2_4, COV3_4, which represent CAD/DKK, CHF/JPY, and DKK/JPY display significant ARCH and GARCH terms.




Chi-square 2.559000 4 0.6341



C(1) 0.000598 0.000958C(2) -0.001092 0.001120C(3) -0.000926 0.000957C(4) -0.001821 0.001452



H0: The mean coefficients C(1) = C(2) =C(3)=C(4)= 0

H1: The mean coefficients C(1) = C(2)=C(3)=C(4) ≠ 0

171

According to Table 6, the probability of the Chi-square statistic is 2.56 and the p-value is 0.63. The sample evidence suggests accepting the null hypothesis. In other words, the test has rejected the symmetry restrictions.


In this article, we have tested the volatility of the natural logarithmic monthly returns of the CAD/USD, DKK/USD, CHF/USD and JPY/USD spot exchange rates. We have applied an ARCH – conditional heteroskedasticity method to model the variance, the covariance of the error terms and the correlation of the spot exchange rates. We have used three multivariate ARCH specifications in terms of the conditional constant correlation, the diagonal VECH and the diagonal BEKK. The data that we have used started from 01/01/1990 to 01/01/2013 and represent the CAD/USD, DKK/USD, CHF/USD and JPY/USD spot exchange rates, which total to 277 observations. The natural logarithmic monthly returns total to 276 observations. The data was obtained from the Federal Reserve Statistical Release Department and it is denoted by the symbol, H.10.

Fist of all, we have applied the system equations 1, which represent a diagonal VECH of a covariance specification multivariate ARCH model. We have found that the coefficient C(1), which represents the mean equations of the CAD/USD spot exchange rate has a coefficient of -0.000199 and a p-value of 0.83. It is not significant as it is above the 5% significance level. The mean coefficients C(2) and C(3), which shows the CHF/USD and DKK/USD respectively are statistically significant at the 5% significance level. The mean coefficient C(4), which displays the JPY/USD is not statistically significant. The variance coefficients C(5) to C(14) are the coefficients for the constant matrix. The coefficients C(9), C(10), C(12) and C(14) show significant p-values. C(15) to C(24) are the coefficients for the ARCH term. C(15), C(19), C(20), C(22) and C(24) show a significant p-values of 0.0021, 0.0001, 0.0000, 0.0000, and 0.0038. C(25) to C(34) are the coefficients for the GARCH term. The coefficients C(25), C(27), C(29), C(30), C(31), C(32), C(33) display significant p-values at the 5% significance level. The log likelihood value of the system equations model is 2935.896. The average log likelihood value is 2.66. The Akaike information criterion is -21.03. The Schwarz criterion is -20.58 and the Hannan – Quinn criterion is -20.85.

In addition, we have found that GARCH represents the conditional variance matrix. The conditional variance equations for GARCH 2, 3 and 4 have significant values for the coefficient matrices. The conditional covariance equation COV2_3, which represents the conditional covariance between the CHF and the DKK spot exchange rate displays significant values of the coefficient matrices. The COV1_4, which is the covariance of CAD/JPY is not significant. COV1_3, COV2_4, COV3_4, which

172

represent CAD/DKK, CHF/JPY, and DKK/JPY display only a significant GARCH term of 0.04, 0.005 and 0.0000 respectively.

By applying the system equations 2, which represent a constant conditional correlation of a covariance specification multivariate ARCH model, we have found different results. The coefficient C(1) represents the mean equation of the CAD/USD spot exchange rate. It has a coefficient of -7.40E-05 and a p-value of 0.94. It is not significant as it is above the 5% significance level. The mean coefficients C(2) and C(3), which shows the CHF/USD and DKK/USD respectively are not statistically significant at the 5% significance level as the p-value is 0.15 and 0.30 respectively. Finally the mean coefficient C(4), which displays the JPY/USD is not statistically significant. The variance coefficients for the constant matrix C(6), C(7), C(8), C(9) and C(10) show significant p-values of 0.00, 0.00, 0.01, 0.00, 0.00 respectively. C(11) to C(16) are the coefficients for the ARCH term. C(11), C(12), C(13), C(14) and C(15) show a statistically significant p-values. C(17) to C(22) are the coefficients for the GARCH term. The coefficients C(17), C(18), C(20), C(21), and C(22), display significant p-values at the 5% significance level. The log likelihood value of the system equations model is 2903.890. The average log likelihood value is 2.63. The Akaike information criterion is -20.88. The Schwarz criterion is -20.59 and the Hannan – Quinn criterion is -20.77.

Furthermore, we have found that GARCH represents the conditional variance matrix. The conditional variance equations for GARCH 2, 3 and 4 have significant values for the coefficient matrices except the coefficient matrix of the GARCH term B1(4). The conditional covariance equation COV1_3, which represents the conditional covariance between the CAD and the DKK spot exchange rate, displays significant values of the correlation coefficient. The correlation R(1,3) is 0.35. It has a z-statistic of 5.96 and a p-value of 0.00. The COV1_4, which is the covariance of CAD/JPY is not significant. The correlation R(1,4) is 0.04. The z-statistic is 0.61 and the p-value is 0.54. The COV2_3, COV2_4, and COV3_4, which represent CHF/DKK, CHF/JPY, and DKK/JPY all statistically significant. The correlation R(2,3) is 0.91. It has a z-statistic of 92.09 and a p-value of 0.00. The correlation R(2,4) is 0.47. It has a z-statistic of 10.93 and a p-value of 0.00. Finally, the correlation R(3,4) is 0.40. It has a z-statistic of 8.06 and a p-value of 0.00.

By applying the system equations 3, which represent a diagonal BEKK of a covariance specification multivariate ARCH model, we have found different results for the mean, variance and covariance coefficients. Specifically, the mean coefficient C(1) represents the CAD/USD spot exchange rate. It has a coefficient of 0.001 and a p-value of 0.53. It is not significant as the p-value is above the 5% significance level. The mean coefficients C(2), C(3), and C(4) which shows the CHF/USD, DKK/USD, and JPY/USD respectively are not statistically significant at the 5% significance level. C(5) to C(8) are the variance coefficients for the constant matrix. The variance coefficients C(5), C(6), and C(7) are not statistically significant. C(9) to C(12) are the coefficients for the ARCH term. They are all significant. C(13) to C(16) are the coefficients for the GARCH term. The coefficients C(13), C(14), C(15), and C(16) are statistically significant at the 5% significance level. The log likelihood value of the system equations model is 2891.647. The average log likelihood value is 2.62. The Akaike information criterion is -20.84. The Schwarz criterion is -20.63 and the Hannan – Quinn criterion is -20.75.

173

GARCH represents the conditional variance matrix of the diagonal BEKK covariance model. Both the coefficient matrix for the ARCH and GARCH term are significant as the p-values are below the 5% significance level. Similar results show the conditional variance equations for GARCH 2, 3 and 4. The conditional covariance equation COV2_3, which represents the conditional covariance between the CHF and the DKK spot exchange rate displays significant values of coefficient matrices. The COV1_4, which is the covariance of CAD/JPY is not significant. COV1_3, COV2_4, COV3_4, which represent CAD/DKK, CHF/JPY, and DKK/JPY display significant ARCH and GARCH terms.

By applying the Wald test of the symmetry restrictions of the natural logarithmic monthly returns of the spot exchange rate returns namely the CAD/USD, DKK/USD, CHF/USD and JPY/USD, we have found that the sample evidence suggests to accept the null hypothesis. In other words, the test has rejected the symmetry restrictions of the mean coefficients.

The best fit model to help the arbitrageurs to craft their investment strategy in terms of buying, selling or holding large portfolios of spot exchange rates is the system ARCH equations 1. It has the largest value of the log likelihood and average log likelihood value and the lowest error term value of the Akaike and Hannan - Quinn information criterion. The Schwarz criterion has the highest value.

References

Alexander, C., (2003), “Market Models. A Guide to Financial Data Analysis”. John Wiley and Sons Ltd. ISMA Centre. The Business School For Financial Markets.

Brooks,C..(2002), Introductory Econometrics for Finance. Cambridge University Press. pp.456, 470.

Bollerslev, T.,(1990), “ Modelling the Coherence in Short-run Nominal Exchange rates: A Multivariate Generalized ARCH Model”. The Review of Economics and Statistics, 72, pp.498-505.

Bollerslev, T.,(1986), “Generalized Autoregressive Conditional Heteroskedasticity”. Journal of Econometrics, 31, pp.307 – 327.

174

Bollerslev, T., Chou, R,Y., and Kroner, K.F., (1992), “ARCH Modelling in Finance: A Review of the Theory and Empirical Evidence”. Journal of Econometrics, 52, pp.5 -59.

Bollerslev, T, Engle, R.F, and Nelson, D.B., (1994), “ARCH Models”. Chapter 49 in Robert F. Engle and Daniel L.McFadden (eds.). Handbook of Econometrics, Volume 4, Amsterdam: Elsvier Science B.V.

Bollerslev, T, and Wooldridge, J,M., (1992), “Quasi – Maximum Likelihood Estimation and Inference in Dynamic Models with Time Varying Covariances”. Econometric Reviews, 11, pp.143 – 172.

Donald, A,W,K.,(1991), “Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimation”. Econometrica, 59, pp.817 - 858.

Donald, A,W,K., and Monahan, J,C., (1992), “An Improved Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimator”. Econometrica, 60, pp.953 -966.

Dickey, D.A and Fuller, W.A.,(1979), “Distribution of Estimators for Time Series Regressions with a Unit Root”. Journal of the American Statistical Association 74, pp.427-31.

Ding, Z, Granger, C.W.J and Engle, R.F., (1993), “A Long Memory Property of Stock Market Returns and a New Model”. Journal of Empirical Finance, 1, pp.83 -106.

Engle, R.F and Kroner, K.F, (1995), “Multivariate Simultaneous Generalized ARCH”. Econometric Theory, 11, pp.122 – 150.

Engle, R.F., (1982), “Autoregressive Conditional Heteroskedasticity with Estimates of the Variance of U.K. Inflation”. Econometrica, 50, pp.987 – 1008.

Engle, R.F, and Bollerslev, T., (1986), “Modelling the Persistence of Conditional Variances”. Econometric Reviews, 5, pp.1 -50.

Engle, R.F, Lilien, D.M and Robins,R.P., (1987), “Estimating Time Varying Risk Premia in the Term Structure: The ARCH-M Model”. Econometrica, 55, pp.391 - 407.

EViews 6, (2007), “ User’s Guide II ”. Quantitative micro software.

Glosten, L.R.R., Jaganathan and Runkle, D., (1993), “On the Relation between the Expected Value and the Volatility of the Normal Excess Return on Stocks”. Journal of Finance, 48, pp.1779 – 1801.

Loretan,M., Mazda,A. and Subramanian,S.(2005), Indexes of the Foreign Exchange value of the Dollar. It is published in the site of the Federal Reserve Bulletin, Winter 2005.

175

Nelson, D.B.,(1991), “Conditional Heteroskedasticity in Asset Returns: A New Approach”. Econometrica 59(2), pp.347-70.

Newey, W, and West, K.,(1994), “ Automatic Lag Selection in Covariance Matrix Estimation”. Review of Economic Studies, 61, pp.631 -653.

Schwert, W., (1989), “Stock Volatility and Crash of ‘87”. Review of Financial Studies, 3, pp.77 – 102.

Taylor, S., (1986), “Modelling Financial Time Series, New York: John Wiley and Sons.

Zakoian, J.M., (1994), “Threshold Heteroskedastic Models”. Journal of Economic Dynamics and Control, 18, pp.931 -944.

176

Documents

Introduction to Econometrics 1