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Chapter 1 : Introduction and ReviewChapter 1 : Introduction and Review

1.1 What is econometrics?

1.2 Review of linear regression

1.3 Review of univariate non-seasonal ARIMA models

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1.11.1 What is econometrics?What is econometrics?

• Econometrics is the main technique used in this course

• It is a branch in economics that brings together economic theory and statistics to study economic phenomena.

• Econometric theory is concerned with the development and extension of statistical techniques appropriate for economic data (e.g. VAR models, Cointegration analysis, etc.)

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• In this course, we are primarily concerned with applied econometrics.

• In recent years, a branch in econometrics, has emerged with emphasis on financial applications.

• Examples of econometric tools in terms of financial applicability :

1. Test whether financial markets are weak-form efficient.2. Test whether the Capital Asset Pricing Model (CAPM) and the

Abitrage Pricing Theory (APT) are appropriate models for the determination of returns on risky assets.

3. Measuring and forecasting the volatility of bond or stock returns (e.g. GARCH models).

4. Modelling long-term relationships between prices and exchange rates (cointegration analysis).

5. Forecasting the correlation between the stock indices of two counties.

and so on.

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Economic Data vs. Financial DataEconomic Data vs. Financial Data• Financial data and Economic data often differ in

terms of their frequency, accuracy, seasonality and other properties.

• Lack of data is a serious problem for economic data analysis. For example, it might be that the data required on government budget deficits, on population figures, which are measured only on an annual basis.

• Measurement errors and data revisions are the two other major problems with economic data. Economic data are often estimated based on sample information.

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• These issues are rarely of concern in finance.

• Generally the prices and other entities recorded are those at which trades actually took place.

• Also, financial data are observed at much higher frequencies than economic data. Asset prices are often available at minute-by-minute frequency!!

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• On the other hand, financial data are often very “noisy” (i.e. it is more difficult to separate the trends and other patterns from the random behaviour).

• Financial data are almost always not normally distributed, but most techniques in econometrics assume they are.

• High frequency data often contain additional patterns which are the result of the way the market works (e.g. clustering). These features need to be considered in the model-building process.

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• Regression analysis is certainly the single most important technique in econometrics.

• A statistical technique that attempts to explain movements in one variable (dependent) as a function of movements in a set of other variables (explanatory).

1.21.2 Review of Linear RegressionReview of Linear Regression

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Yt = 0 + 1Xt + t ; t ~ N(0, 2)• The ’s are the coefficients0 is the constant (intercept) term in the

regression equation.1 is the slope coefficient. It measures the

amount Y will change when X changes by 1 unit.

2 is the variance of t , the random disturbance term.

Simple Linear Regression ModelSimple Linear Regression Model

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• Examples of regressions used in Financial analysis:• How asset returns vary with their level of market

risk.• Measuring the relationship between spot prices and

market risk.• Constructing an optimal hedge ratio.

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Slope CoefficientsSlope Coefficients

• Recall that we are trying to explain or predict changes in Y using X.

1 defines the relationship for us.

• For linear models, the slope is constant.

2 11

2 1

Y YY

X X X

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Linear RegressionLinear Regression

• i.e., the dependent variable has a linear relationship with the explanatory variable

• cannot use linear regression methods if the equation is not linear in the coefficients

210

10

tt

tt

XY

XY

10

t

XYt

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Expected Value of Expected Value of YYtt

0 + 1Xt is the deterministic part of the regression equation.

• i.e. it is the value of Yt determined by a given value of Xt , which we assume to be non-stochastic.

• can also think of it as the expected value of Yt given Xt

E(Yt | Xt) = 0 + 1Xt

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Stochastic Error Term (Stochastic Error Term (tt))

• Almost always the case that Xt alone cannot explain all the variations in Yt .

• Can add more variables.

• But there will still be some variation in Yt that cannot be explained by the model.

• Sources of error : random shocks, measurement errors etc.

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Example : Capital Asset Pricing Model (CAPM)Example : Capital Asset Pricing Model (CAPM)

• A fundamental idea of modern finance is that an investor needs a financial incentive to take a risk

• Said differently, the expected return on a risky asset, R, must exceed the return on a safe, or a risk free investment, Rf .

• Thus the expected excess return, R – Rf , should be positive.

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• According to the CAPM, the expected excess return on an asset is proportional to the expected excess return on a portfolio of all available assets (the “market portfolio”)

i.e. R – Rf = (Rm – Rf )

• So, a stock with < 1 (> )1 has less (more) risk than the market portfolio and therefore has a lower (higher) expected excess return them the market portfolio.

• The ’s are usually estimated by least squares regression.

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• The table below gives the estimated ’s for six U.S. stocks

Company Estimated

Kellogg (breakfast cercal) 0.24

Waste Management (waste disposal)

0.38

Sprint (long distance telephone) 0.59

Walmat (discount retailer) 0.89

Barnes and Noble (book retailer) 1.03

Best Buy (electronic equipment) 1.80

Microsoft (software) 1.83

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Finding the “Best Fit”Finding the “Best Fit”

• Recall that we are trying to estimate a linear relationship such that the line passing through the data best represents the underlying true relationship.

• Difference between the actual values and the estimated values is the residual.

• Best fit minimize the residuals.

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Ordinary Least SquaresOrdinary Least Squares

• A regression estimation techniques that calculates estimates of the slope parameters in our linear regression model so as to minimize the squared residuals :

N

iie

1

2

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• Goodness of Fit measure :

R2 , Adjusted R2

• Hypothesis testing :• Significance of individual coefficients

– t test

• Overall significance of model– F test

• Significance of linear restrictions– partial F test

• Serial correlations– Durbin-Watson test (1st order)

– Lagrange Multipler test (higher order)

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1.31.3 Review of Univariate Non-seasonal Review of Univariate Non-seasonal ARIMA ModelsARIMA Models

• Univariate forecasts based on a statistical analysis of the past data. Differs from conventional regression methods in that the mutual dependence of the observations is of primary interest.

• Forecasts are linear functions of the sample observations.

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Box-Jenkins MethodologyBox-Jenkins Methodology

• Identification of the type of model to be used is a critical step. Differs from other univariate techniques in that a thorough study of the properties of a time series is carried out before applying a forecasting technique.

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Principle of ParsimonyPrinciple of Parsimony

• Find efficiently parameterized models. A model with a large number of parameters will achieve a good historical fit, but post-sample forecasts are likely to be poor.

• By building a model based on past realizations of a time series we are implicitly assuming that there is some regularity in the process generating the series.

• One way to view such regularity is through the concept of stationarity.

• The use of Box-Jenkins modeling techniques requires a stationary process.

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• A stochastic process is a collection {Xt : t = 1, 2, …, T} of random variables ordered in time. Example : the error term in a linear regression model is assumed to be a stochastic process.

• A stochastic process is weakly stationary if for all t values

E[Xt] = var(Xt) = 2

cov(XtXt-k) = k ti.e. its statistical properties do not change over time.

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• More precisely, a stationary series is one for which the mean and variance are constant across time and the covariance between current and lagged values of the series (autocovariances) depends only on the distance between the time points.

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AutocorrelationsAutocorrelations

• Covariances are often difficult to interpret because they depend on the units of measurement of the data.

• Correlations, on the other hand, are scale-free. Thus, we can obtain the same information about the time series by computing the autocorrelations of a time series.

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Autocorrelations CoefficientAutocorrelations Coefficient

• The autocorrelation coefficient between Xt and Xt-k is

• A graph of the autocorrelations is called a correlogram.

• Knowledge of the correlogram implies knowledge of the process which generated the series and vice versa.

2var

cov

k

t

kttk X

XX

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Partial AutocorrelationsPartial Autocorrelations

• Another important function in the Box-Jenkins methodology is the partial autocorrelation function.

• It measures the strength of the relationship between observations in a series controlling for the effect of the intervening time periods.

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Box-Jenkins MethodologyBox-Jenkins Methodology

• Identification. Determine, given a sample of time series observations, what the model of the [stationary] data is.

• Estimation. Estimate the parameters of the chosen model

2...

2211

11

2...

2211

2...

2211

,~;...

...:),(

,~;...:)(

,~;...:)(

NXXXqpARMA

NXqMA

NXXXXpAR

dii

tqtqtt

tptptt

dii

tqtqtttt

dii

ttptpttt

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• Diagnosting checking and Model Selection1) t-test for the significance of individual coefficients

2) Ljung Box-Pierce (Q) test for the correlations of the residuals

3) Minimize the Akaike Information Criteria and the Schwarz Bayesian Criteria

• Forecasting with fitted model.

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Guidelines for Box-Jenkins IdentificationGuidelines for Box-Jenkins Identification

Model CorrelogramPartial

Correlogram

AR(p) Dies offTruncates after

Lag p

MA(q)Truncates after

Lag qDies off

ARMA(p,q) Dies off Dies off

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Applied Example (Box-Jenkins Modelling Applied Example (Box-Jenkins Modelling of Dow-Jones Industrial Index)of Dow-Jones Industrial Index)

• DJI index is an index of 30 industrial firms’ stock prices

• dataset : dji.txt• The values in this data set represent monthly

averages of the end-of-day values for the index over the period January 1984 to February 1994

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data djm;data djm;infile 'd:\teaching\ms4221\dji.txt';infile 'd:\teaching\ms4221\dji.txt';input date:monyy5. djiam @@;input date:monyy5. djiam @@;format date monyy5.;format date monyy5.;title 'Dow Jones Index Data';title 'Dow Jones Index Data';title2 'Monthly Average';title2 'Monthly Average';proc print;proc print;run;run;symbol1 i=join;symbol1 i=join;proc gplot data=djm;proc gplot data=djm;format date year4.;format date year4.;plot djiam*date/vminor=1;plot djiam*date/vminor=1;run; run; proc arima data=djm;proc arima data=djm;identify var=djiam;identify var=djiam;identify var=djiam(1);identify var=djiam(1);estimate q=1 method=ml;estimate q=1 method=ml;estimate p=1 method=ml;estimate p=1 method=ml;estimate p=1 q=1 method=ml;estimate p=1 q=1 method=ml;run;run;

SAS Program: Example 1.2

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The ARIMA Procedure Name of Variable = djiam Mean of Working Series 2381.074 Standard Deviation 772.4888 Number of Observations 122 Autocorrelations Lag Covariance Correlation -1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1 Std Error 0 596739 1.00000 | |********************| 0 1 578547 0.96951 | . |******************* | 0.090536 2 557449 0.93416 | . |******************* | 0.153642 3 537611 0.90092 | . |****************** | 0.194709 4 518674 0.86918 | . |***************** | 0.226313 5 500626 0.83894 | . |***************** | 0.252195 6 483171 0.80969 | . |**************** | 0.274117 7 465384 0.77988 | . |**************** | 0.293066 8 448773 0.75204 | . |*************** | 0.309610 9 432649 0.72502 | . |*************** | 0.324237 10 417192 0.69912 | . |************** | 0.337264 11 401879 0.67346 | . |*************. | 0.348941 12 384770 0.64479 | . |*************. | 0.359437 13 369178 0.61866 | . |************ . | 0.368796 14 354548 0.59414 | . |************ . | 0.377207 15 338543 0.56732 | . |*********** . | 0.384801 16 323456 0.54204 | . |*********** . | 0.391597 17 309078 0.51794 | . |********** . | 0.397699 18 294007 0.49269 | . |********** . | 0.403190 19 278698 0.46703 | . |********* . | 0.408095 20 263224 0.44110 | . |********* . | 0.412453 21 248311 0.41611 | . |******** . | 0.416302 22 233269 0.39091 | . |******** . | 0.419697 23 219661 0.36810 | . |******* . | 0.422671 24 207382 0.34753 | . |******* . | 0.425291 "." marks two standard errors

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Name of Variable = djiam Period(s) of Differencing 1 Mean of Working Series 21.87372 Standard Deviation 81.18672 Number of Observations 121 Observation(s) eliminated by differencing 1 Autocorrelations Lag Covariance Correlation -1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1 Std Error 0 6591.284 1.00000 | |********************| 0 1 2030.074 0.30799 | . |****** | 0.090909 2 -196.681 -.02984 | . *| . | 0.099158 3 -592.709 -.08992 | . **| . | 0.099233 4 -831.381 -.12613 | .***| . | 0.099904 5 -848.580 -.12874 | .***| . | 0.101211 6 -270.315 -.04101 | . *| . | 0.102556 7 141.561 0.02148 | . | . | 0.102691 8 -734.798 -.11148 | . **| . | 0.102728 9 -801.911 -.12166 | . **| . | 0.103723 10 212.541 0.03225 | . |* . | 0.104896 11 938.667 0.14241 | . |***. | 0.104978 12 -580.823 -.08812 | . **| . | 0.106563 13 164.689 0.02499 | . | . | 0.107163 14 509.608 0.07732 | . |** . | 0.107211 15 -549.975 -.08344 | . **| . | 0.107671 16 -502.865 -.07629 | . **| . | 0.108204 17 -605.287 -.09183 | . **| . | 0.108648 18 -86.271660 -.01309 | . | . | 0.109287 19 -704.905 -.10695 | . **| . | 0.109300 20 -1192.666 -.18095 | ****| . | 0.110162 21 -496.859 -.07538 | . **| . | 0.112591 22 -74.696413 -.01133 | . | . | 0.113008 23 -374.542 -.05682 | . *| . | 0.113017 24 947.819 0.14380 | . |*** . | 0.113253 "." marks two standard errors

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Partial Autocorrelations Lag Correlation -1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1 1 0.30799 | . |****** | 2 -0.13777 | .***| . | 3 -0.04171 | . *| . | 4 -0.09698 | . **| . | 5 -0.07779 | . **| . | 6 0.00563 | . | . | 7 0.00572 | . | . | 8 -0.16507 | .***| . | 9 -0.06192 | . *| . | 10 0.07096 | . |* . | 11 0.09529 | . |** . | 12 -0.22195 | ****| . | 13 0.12195 | . |** . | 14 0.02227 | . | . | 15 -0.11899 | . **| . | 16 -0.01424 | . | . | 17 -0.12878 | .***| . | 18 0.05260 | . |* . | 19 -0.11561 | . **| . | 20 -0.22261 | ****| . | 21 -0.03616 | . *| . | 22 -0.00977 | . | . | 23 -0.10907 | . **| . | 24 0.06551 | . |* . |

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Maximum Likelihood Estimation Standard Approx Parameter Estimate Error t Value Pr > |t| Lag MU 21.50811 9.46053 2.27 0.0230 0 MA1,1 -0.35161 0.08613 -4.08 <.0001 1 Constant Estimate 21.50811 Variance Estimate 5959.094 Std Error Estimate 77.19517 AIC 1397.312 SBC 1402.903 Number of Residuals 121 Autocorrelation Check of Residuals To Chi- Pr > Lag Square DF ChiSq --------------------Autocorrelations-------------------- 6 2.56 5 0.7667 -0.001 -0.011 -0.063 -0.077 -0.096 -0.030 12 14.29 11 0.2173 0.063 -0.104 -0.089 -0.001 0.194 -0.163 18 19.39 17 0.3064 0.044 0.099 -0.112 -0.010 -0.101 0.042 24 30.14 23 0.1455 -0.075 -0.144 -0.042 0.040 -0.121 0.164

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Forecasts for variable djiam Obs Forecast Std Error 95% Confidence Limits 123 3921.6795 77.1952 3770.3798 4072.9793 124 3943.1876 129.7898 3688.8044 4197.5709 125 3964.6958 166.5283 3638.3063 4291.0853 126 3986.2039 196.5146 3601.0424 4371.3653 127 4007.7120 222.4955 3571.6287 4443.7952 128 4029.2201 245.7449 3547.5689 4510.8712 129 4050.7282 266.9772 3527.4624 4573.9940 130 4072.2363 286.6411 3510.4300 4634.0426 131 4093.7444 305.0401 3495.8769 4691.6119 132 4115.2525 322.3906 3483.3785 4747.1266 133 4136.7606 338.8540 3472.6190 4800.9022 134 4158.2687 354.5537 3463.3563 4853.1812

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Maximum Likelihood Estimation Standard Approx Parameter Estimate Error t Value Pr > |t| Lag MU 21.50030 10.22228 2.10 0.0354 0 AR1,1 0.31082 0.08703 3.57 0.0004 1 Constant Estimate 14.81748 Variance Estimate 6055.115 Std Error Estimate 77.81462 AIC 1399.216 SBC 1404.807 Number of Residuals 121 Autocorrelation Check of Residuals To Chi- Pr > Lag Square DF ChiSq --------------------Autocorrelations-------------------- 6 4.17 5 0.5257 0.040 -0.111 -0.055 -0.078 -0.099 -0.012 12 17.05 11 0.1063 0.078 -0.101 -0.120 0.032 0.192 -0.164 18 21.93 17 0.1874 0.034 0.113 -0.101 -0.032 -0.081 0.052 24 32.11 23 0.0978 -0.063 -0.157 -0.025 0.032 -0.114 0.156

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Maximum Likelihood Estimation Standard Approx Parameter Estimate Error t Value Pr > |t| Lag MU 21.50386 9.54223 2.25 0.0242 0 MA1,1 -0.34100 0.24812 -1.37 0.1693 1 AR1,1 0.01232 0.26329 0.05 0.9627 1 Constant Estimate 21.23899 Variance Estimate 6009.459 Std Error Estimate 77.5207 AIC 1399.309 SBC 1407.697 Number of Residuals 121 Autocorrelation Check of Residuals To Chi- Pr > Lag Square DF ChiSq --------------------Autocorrelations-------------------- 6 2.54 4 0.6378 -0.003 -0.014 -0.061 -0.077 -0.096 -0.029 12 14.36 10 0.1572 0.064 -0.103 -0.090 -0.000 0.195 -0.164 18 19.48 16 0.2446 0.044 0.099 -0.112 -0.010 -0.100 0.043 24 30.23 22 0.1130 -0.074 -0.145 -0.041 0.040 -0.122 0.164