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Financial Economics Assignment
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BUS5ATE Advanced Time Series Analysis
Part 1: Implications of Unit Root
Asset Prices: the efficient market hypothesis shows that returns follow a stochastic, rather
unpredictable pattern and at such asset prices tend to exhibit a stochastic process or
presence of unit root. Evidently, asset prices could be risk-free in the case of fixed income
securities and mortgage-backed securities and risky in the case of equities, foreign exchange
and commodities. A typical case of unpredictability and stochastic movements can be
aligned to equity progression or trend analysis that follows a random walk both with drift
and without drift. In terms of time horizon, the prices of an asset is more likely to change in
the long-term than in the short-terms with much greater risk expected to affect the value
and return of the asset in the long-term. Previous empirical findings exemplify this axiom to
mean that indeed asset prices have a unit root - in other words they behave sporadically. A
typical instance would be the actions of the Central bank in deciding what path the
economy will take by altering the policy lending rate to either contract or expand the
monetary system, their consequential effect will transcend to volatility in asset prices
following a shift in investor’s perception about the expected rate of inflation and return on
investment (ROI).
Purchasing Power Parity Hypothesis: exchange rate is a core determinant in income
determination, for every good produced, same price is expected irrespective of the country
where the good is sold or purchased. The implication of the hypothesis is rather flawed
because trade policies differ by country which will affect cost of imports and exports both
ways, in addition currency value especially in the domestic market differs relatively,
empirically studies have rejected the null hypothesis of a unit root for real exchange rate
even though nominal exchange rate is supposed to match the national price levels.
Growth and Convergence: the periods between economic fluctuation and growth are
sensitive ones. Empirical evidence have increasingly shown that capital converge at same
level so if the zero mean means that the difference in per capita income between two
countries is transitory and fluctuates around zero. This implies that two countries per capita
output converges and becomes same in the absence of unit root.
Convergence of Real Interest Rate: the implication of having two nations with same interest
rate is predicated on international macroeconomics. Interest rate has been found to have
unit root which essentially means that capital markets of the two nations are not fully
integrated. In essence, a myriad of factors would definitely cause the interest rate to
dissipate overtime namely the dynamics of inflation rate, the likelihood for interest to
change overtime due to the actions of the respective countries Central bank, the intensity of
the financial market especially currencies and equities where offshore investments flow and
non-economic factors.
Part 2: Unit Root in Real Interest Rate
The consumption-based asset pricing model explains the rudiments of real interest rate
which is simply nominal interest rate less of inflation rate or simply put interest rate
adjusted for expected inflation rate. This defines real interest rate as core variable in the
determination and evaluation of policies at the Central bank (Taylor, 1993) and the neo-
classical growth model (Cass, 1965; Koopmans, 1965). The postulation was further
expressed with empirical results using U.S. data which summarises the persistence and
exploration of the cause of real interest rate.
The relation exhibits a persistent and substantial shock following the advanced periods of
post-war. A combination of unit root and cointegration test was found to be central in
analysing the effects of shocks on real interest rate in other words does real interest rate
behaves like a random walk. Further research also shows the non-linearity in variable
behaviour and the concept of fractional integration, both of which are linked to persistent
shocks and structural breaks reaction in between periods. For instance, pre-war and post-
war effects of shocks on interest rate are to be mentioned. However, a major limitation
emanates from the inability of the study to address the root causes of real interest rate.
Hence, the need for a theoretical model as opined by Lucas (1978), and Breeden (1979)
termed the consumption-based asset pricing model which discusses the components of real
interest rate at different time periods, taking into consideration the risk factor and the per
capita consumption. The U.S. data was used by Rose (1988) to empirically test the
theoretical model using the 3-month real treasury bills rate and annualised growth. It was
found that a huge divergence exist in the 1970s, 1980s and 2001-2005.
More elaborate theoretical model was employed to examine the changes in macroeconomic
policies majorly monetary and fiscal policy – with intent of altering the steady-state real
interest rate using the Euler’s equation. In summary most researchers were open to the idea
of allowing changes in preferences over the extended period (Clark, 2007).
The concept of forecasting the behaviour of inflation was also brought to light to ascertain
the level of real interest rate for all the maturity spectrum of U.S securities. The theoretical
postulation was also empirically tested using the concepts of ex-ante real interest rate
(EARR) and ex-post real interest rate (EPRR); since business agents and investors alike make
futuristic decision on the basis of expected inflation rate - in other words EARR is a measure
for examining decisions; the study also starts by drawing an analogy and distinction
between both concepts via time horizon and persistence properties (Sun and Philip, 2004).
The structural vector autoregression (SVAR) became the method of estimation after
evaluating unit root test in real interest rate notably the result yielded a non-rejection of the
null hypothesis of unit root in a study that also includes the U.S. 3month treasury bills,
inflation and EPRR. In summary, an extension was made including the updated results on
unit root and cointegration for U.S. data both of which presented mixed outcomes and
confidence interval for the sum of the AR coefficients.
Part 3: Basic Data Analysis
Here, we estimate the time plots and SACF of the variables 3-Month Treasury Constant
Maturity Rate, PCE deflator inflation rate, Ex post real interest rate (EPRR) and Consumption
growth rate.
The presence of serial correlation will mean that the SACF at all lags should be near zero
(significantly different from zero or not zero) and the Ljung-box Q-stat is insignificant.
However, the results presented below shows persistent and consistent residuals. In terms of
degree of persistence, ex post real interest rate (EPRR) and Consumption growth rate (CGR)
are likely to be forecastable given the persistent pattern of their time plots shown below
while 3-Month Treasury rate and PCE deflator inflation rate are likely to be forecastable
given the non-persistent or stochastic or volatile pattern of their time plots.
Fractional Integration (ARFIMA) as a test for degree of dependence: stationary processes
are more likely to have long memory in the presence of autocorrelation, which tends to
decay more slowly (Granger and Joyeux, 1980; Hosking, 1981).
Dependent Variable: DLOG(CGR) Method: Least Squares Date: 06/09/17 Time: 00:14 Sample (adjusted): 3 219 Included observations: 217 after adjustments Convergence achieved after 13 iterations MA Backcast: 2 Variable Coefficient Std. Error t-Statistic Prob. C 0.006381 0.008713 0.732341 0.4648 PCE 9.36E-05 0.002800 0.033427 0.9734 EPRR 0.000894 2.54E-05 35.20569 0.0000 _3TCM -0.001816 0.002714 -0.669117 0.5042 AR(1) 0.198698 0.070268 2.827727 0.0051 MA(1) -0.999822 0.028257 -35.38283 0.0000 R-squared 0.581020 Mean dependent var 0.038775 Adjusted R-squared 0.571091 S.D. dependent var 1.245687 S.E. of regression 0.815815 Akaike info criterion 2.458001 Sum squared resid 140.4318 Schwarz criterion 2.551455 Log likelihood -260.6932 Hannan-Quinn criter. 2.495753 F-statistic 58.52071 Durbin-Watson stat 2.063847 Prob(F-statistic) 0.000000 Inverted AR Roots .20 Inverted MA Roots 1.00
Correlogram of CGR
Date: 06/08/17 Time: 23:53
Sample: 1 219
Included observations: 219 Autocorrelation Partial Correlation AC PAC Q-Stat Prob .|. | .|. | 1 0.000 0.000 5.E-06 0.998
.|. | .|. | 2 -0.000 -0.000 5.E-06 1.000
.|. | .|. | 3 -0.000 -0.000 9.E-06 1.000
.|. | .|. | 4 0.000 0.000 9.E-06 1.000
.|. | .|. | 5 -0.000 -0.000 9.E-06 1.000
.|. | .|. | 6 -0.000 -0.000 1.E-05 1.000
.|. | .|. | 7 -0.000 -0.000 2.E-05 1.000
.|. | .|. | 8 -0.000 -0.000 3.E-05 1.000
Correlogram of EARR
Date: 06/08/17 Time: 23:55
Sample: 1 219
Included observations: 219 Autocorrelation Partial Correlation AC PAC Q-Stat Prob .|. | .|. | 1 -0.000 -0.000 4.E-06 0.998
.|. | .|. | 2 0.000 0.000 5.E-06 1.000
.|. | .|. | 3 -0.000 -0.000 5.E-06 1.000
.|. | .|. | 4 -0.000 -0.000 6.E-05 1.000
.|. | .|. | 5 -0.000 -0.000 6.E-05 1.000
.|. | .|. | 6 0.000 0.000 6.E-05 1.000
.|. | .|. | 7 -0.000 -0.000 7.E-05 1.000
.|. | .|. | 8 -0.000 -0.000 7.E-05 1.000 Correlogram of PCE
Date: 06/08/17 Time: 23:55
Sample: 1 219
Included observations: 219 Autocorrelation Partial Correlation AC PAC Q-Stat Prob .|******| .|******| 1 0.846 0.846 158.75 0.000
.|******| .|** | 2 0.792 0.271 298.73 0.000
.|******| .|** | 3 0.779 0.234 434.76 0.000
.|***** | .|. | 4 0.727 -0.009 553.68 0.000
.|***** | *|. | 5 0.660 -0.099 652.21 0.000
.|***** | .|* | 6 0.644 0.080 746.53 0.000
.|**** | *|. | 7 0.584 -0.100 824.31 0.000
.|**** | *|. | 8 0.511 -0.112 884.22 0.000 Correlogram of _3TCM
Date: 06/08/17 Time: 23:56
Sample: 1 219
Included observations: 219 Autocorrelation Partial Correlation AC PAC Q-Stat Prob .|******* .|******* 1 0.929 0.929 191.52 0.000
.|******| .|* | 2 0.877 0.104 363.05 0.000
.|******| .|** | 3 0.856 0.218 527.27 0.000
.|******| **|. | 4 0.797 -0.237 670.28 0.000
.|***** | .|. | 5 0.745 0.008 795.71 0.000
.|***** | *|. | 6 0.693 -0.130 904.86 0.000
.|***** | *|. | 7 0.626 -0.086 994.33 0.000
.|**** | .|* | 8 0.593 0.178 1074.9 0.000
TIME PLOTS OF THE VARIABLES
-12,000
-10,000
-8,000
-6,000
-4,000
-2,000
0
2,000
25 50 75 100 125 150 175 200
CGR
-12,000
-10,000
-8,000
-6,000
-4,000
-2,000
0
2,000
25 50 75 100 125 150 175 200
EPRR
-2
0
2
4
6
8
10
12
25 50 75 100 125 150 175 200
PCE
0
2
4
6
8
10
12
14
16
25 50 75 100 125 150 175 200
3TCM
Part 4: Replication
Table 4a: Unit Root Test Statistics, U.S. data, 1953:Q1-2007:Q3
The order of integration or level of difference used in the model forms the third column in
the table above where it is indicated that the 3-Month Treasury bills rate and Per capita
consumption growth are non-stationary at all levels.
Table 4b: Cointegration Test Statistics, U.S. 3-Month Treasury bill Rate, Inflation Rate,
EPRR and Per capita consumption growth (1953:Q1-2007:Q3)
Cointegration Test Variable (Dynamic OLS) Prob.* Significance
3-Month Treasury bill rate 0.0006 Yes
PCE deflator 0.0006 Yes
Ex post real interest rate 0.1304 No
Per capita consumption growth 0.0140 Yes
The 10 percent, 5 percent, and 1 percent critical values for the probability value of the
variables shows significance (p<0.05) except EPRR. The lag order for the cointegration model
used to compute the test statistic is 14 based on Schwarz criterion.
Summary of the Results and Discuss their Implications:
Table 4a and 4b provides a clear evidence that EPRR and 3-Month Treasury bill rate are
relatively not stochastic when compared to PCE deflator and Per capita consumption
growth. This relation is equally depicted in the time plots for the respective variables. To
test the hypothesis of no cointegration, we interpret the probability value from the Engle-
Granger cointegration- the equation rejects the null hypothesis of no cointegration at the
1% and 5% level. Table 4b also presents the dynamic ordinary least squares (OLS) which also
shows that the variables are significantly different from zero except EPRR
Variable ADF Order of Integration Level of significance
3-Month Treasury bill rate 0.886369 Non-stationary at all levels 1% & 5%
PCE deflator -15.5305 Stationary at 1st Difference Nil Ex post real interest rate -2.85439 Non-stationary at all levels Nil
Per capita consumption growth -15.5305 Stationary at 1st Difference 1% & 5%
A variety of economic models have been employed in the past to analyse the persistence of
real interest rate and how other variables play a part in the determination of interest rate
effects. Many empirical evidence have shown that real interest rate contain a unit root
A structural analysis might be important in revealing the cause of the persistent shock and
dependence. looking at the variables; 3-month treasury bills rate, consumption growth rate
,ex post real interest rate and personal consumption, it can be deciphered that all of the
variables respond to shock as shown in the fractional integration output. Hence, a test for
unit roots to further examine the past behaviour of real interest rates and its determinants.
Undoubtedly, inflation (personal consumption expenditure deflator) and per capita
consumption growth were stationary at order (I).
REFERENCES
Breeden, Douglas T. “An Intertemporal Asset Pricing Model with Stochastic Consumption and
Investment Opportunities.” Journal of Financial Economics, September 1979, 7(3), pp. 265-
96.
Cass, David. “Optimum Growth in an Aggregate Model of Capital Accumulation.” Review of
Economic Studies, July 1965, 32(3), pp. 233-40.
Clark, Gregory. A Farewell to Alms: A Brief History of the Economic World. Princeton, NJ: Princeton
University Press, 2007.
Granger, Clive W.J. and Joyeux, Roselyne. “An Introduction to Long-Memory Time Series Models and
Fractional Differencing.” Journal of Time Series Analysis, 1980, 1, pp. 15-39.
Hosking, J.R.M. “Fractional Differencing.” Biometrika, April 1981, 68(1), pp. 165-76.
Koopmans, Tjalling C. “On the Concept of Optimal Economic Growth,” in the Economic Approach to
Development Planning. Amsterdam: Elsevier, 1965, pp. 225-300.
Lucas, Robert E. “Asset Prices in an Exchange Economy.” Econometrica, November 1978, 46(6), pp.
1429-45.
Rose, Andrew K. “Is the Real Interest Rate Stable?” Journal of Finance, December 1988, 43(5), pp.
1095-112.
Sun, Yixiao and Phillips, Peter C.B. “Understanding the Fisher Equation.” Journal of Applied
Econometrics, November-December 2004, 19(7), pp. 869-86.
Taylor, John B. “Discretion versus Policy Rules in Practice.” Carnegie-Rochester Conference Series on
Public Policy, December 1993, 39, pp. 195-214.
APPENDIX
Fractional Integration
Dependent Variable: DLOG(CGR)
Method: Least Squares
Date: 06/09/17 Time: 00:14
Sample (adjusted): 3 219
Included observations: 217 after adjustments
Convergence achieved after 13 iterations
MA Backcast: 2 Variable Coefficient Std. Error t-Statistic Prob. C 0.006381 0.008713 0.732341 0.4648
PCE 9.36E-05 0.002800 0.033427 0.9734
EPRR 0.000894 2.54E-05 35.20569 0.0000
_3TCM -0.001816 0.002714 -0.669117 0.5042
AR(1) 0.198698 0.070268 2.827727 0.0051
MA(1) -0.999822 0.028257 -35.38283 0.0000 R-squared 0.581020 Mean dependent var 0.038775
Adjusted R-squared 0.571091 S.D. dependent var 1.245687
S.E. of regression 0.815815 Akaike info criterion 2.458001
Sum squared resid 140.4318 Schwarz criterion 2.551455
Log likelihood -260.6932 Hannan-Quinn criter. 2.495753
F-statistic 58.52071 Durbin-Watson stat 2.063847
Prob(F-statistic) 0.000000 Inverted AR Roots .20
Inverted MA Roots 1.00
Null Hypothesis: CGR has a unit root
Exogenous: Constant
Lag Length: 0 (Automatic - based on SIC, maxlag=14) t-Statistic Prob.* Augmented Dickey-Fuller test statistic 0.886369 0.9952
Test critical values: 1% level -3.460313
5% level -2.874617
10% level -2.573817 *MacKinnon (1996) one-sided p-values.
Augmented Dickey-Fuller Test Equation
Dependent Variable: D(CGR)
Method: Least Squares
Date: 06/09/17 Time: 01:16
Sample (adjusted): 2 219
Included observations: 218 after adjustments Variable Coefficient Std. Error t-Statistic Prob.
CGR(-1) 22.88248 25.81598 0.886369 0.3764
C -94.17414 71.24199 -1.321891 0.1876 R-squared 0.003624 Mean dependent var -45.87443
Adjusted R-squared -0.000989 S.D. dependent var 677.2536
S.E. of regression 677.5883 Akaike info criterion 15.88409
Sum squared resid 99171204 Schwarz criterion 15.91514
Log likelihood -1729.366 Hannan-Quinn criter. 15.89663
F-statistic 0.785650 Durbin-Watson stat 1.005772
Prob(F-statistic) 0.376404
Null Hypothesis: D(CGR) has a unit root
Exogenous: Constant
Lag Length: 0 (Automatic - based on SIC, maxlag=14) t-Statistic Prob.* Augmented Dickey-Fuller test statistic 0.734888 0.9927
Test critical values: 1% level -3.460453
5% level -2.874679
10% level -2.573850 *MacKinnon (1996) one-sided p-values.
Augmented Dickey-Fuller Test Equation
Dependent Variable: D(CGR,2)
Method: Least Squares
Date: 06/09/17 Time: 01:16
Sample (adjusted): 3 219
Included observations: 217 after adjustments Variable Coefficient Std. Error t-Statistic Prob. D(CGR(-1)) 16.59933 22.58757 0.734888 0.4632
C -45.99642 46.12289 -0.997258 0.3198 R-squared 0.002506 Mean dependent var -46.08332
Adjusted R-squared -0.002134 S.D. dependent var 678.7066
S.E. of regression 679.4303 Akaike info criterion 15.88956
Sum squared resid 99249499 Schwarz criterion 15.92071
Log likelihood -1722.017 Hannan-Quinn criter. 15.90214
F-statistic 0.540060 Durbin-Watson stat 1.012028
Prob(F-statistic) 0.463209
Null Hypothesis: D(CGR,2) has a unit root
Exogenous: Constant
Lag Length: 0 (Automatic - based on SIC, maxlag=14) t-Statistic Prob.* Augmented Dickey-Fuller test statistic 0.165636 0.9698
Test critical values: 1% level -3.460596
5% level -2.874741
10% level -2.573883 *MacKinnon (1996) one-sided p-values.
Augmented Dickey-Fuller Test Equation
Dependent Variable: D(CGR,3)
Method: Least Squares
Date: 06/09/17 Time: 01:16
Sample (adjusted): 4 219
Included observations: 216 after adjustments Variable Coefficient Std. Error t-Statistic Prob. D(CGR(-1),2) 2.259183 13.63943 0.165636 0.8686
C -46.24234 46.38905 -0.996838 0.3200 R-squared 0.000128 Mean dependent var -46.26560
Adjusted R-squared -0.004544 S.D. dependent var 680.2301
S.E. of regression 681.7738 Akaike info criterion 15.89649
Sum squared resid 99470526 Schwarz criterion 15.92774
Log likelihood -1714.821 Hannan-Quinn criter. 15.90912
F-statistic 0.027435 Durbin-Watson stat 1.006721
Prob(F-statistic) 0.868600
Null Hypothesis: PCE has a unit root
Exogenous: Constant
Lag Length: 2 (Automatic - based on SIC, maxlag=14) t-Statistic Prob.* Augmented Dickey-Fuller test statistic -2.386041 0.1469
Test critical values: 1% level -3.460596
5% level -2.874741
10% level -2.573883 *MacKinnon (1996) one-sided p-values.
Augmented Dickey-Fuller Test Equation
Dependent Variable: D(PCE)
Method: Least Squares
Date: 06/09/17 Time: 01:19
Sample (adjusted): 4 219
Included observations: 216 after adjustments Variable Coefficient Std. Error t-Statistic Prob. PCE(-1) -0.084611 0.035461 -2.386041 0.0179
D(PCE(-1)) -0.362200 0.068943 -5.253631 0.0000
D(PCE(-2)) -0.227295 0.066559 -3.414915 0.0008
C 0.298905 0.146956 2.033982 0.0432 R-squared 0.189227 Mean dependent var -0.002495
Adjusted R-squared 0.177754 S.D. dependent var 1.345736
S.E. of regression 1.220283 Akaike info criterion 3.254388
Sum squared resid 315.6873 Schwarz criterion 3.316893
Log likelihood -347.4739 Hannan-Quinn criter. 3.279640
F-statistic 16.49300 Durbin-Watson stat 1.995809
Prob(F-statistic) 0.000000
Null Hypothesis: D(PCE) has a unit root
Exogenous: Constant
Lag Length: 1 (Automatic - based on SIC, maxlag=14) t-Statistic Prob.* Augmented Dickey-Fuller test statistic -15.53051 0.0000
Test critical values: 1% level -3.460596
5% level -2.874741
10% level -2.573883 *MacKinnon (1996) one-sided p-values.
Augmented Dickey-Fuller Test Equation
Dependent Variable: D(PCE,2)
Method: Least Squares
Date: 06/09/17 Time: 01:20
Sample (adjusted): 4 219
Included observations: 216 after adjustments Variable Coefficient Std. Error t-Statistic Prob. D(PCE(-1)) -1.669831 0.107519 -15.53051 0.0000
D(PCE(-1),2) 0.256747 0.066121 3.882970 0.0001
C 0.009632 0.083963 0.114713 0.9088 R-squared 0.685379 Mean dependent var -0.019160
Adjusted R-squared 0.682425 S.D. dependent var 2.189122
S.E. of regression 1.233654 Akaike info criterion 3.271629
Sum squared resid 324.1650 Schwarz criterion 3.318508
Log likelihood -350.3359 Hannan-Quinn criter. 3.290568
F-statistic 232.0026 Durbin-Watson stat 2.011034
Prob(F-statistic) 0.000000
Null Hypothesis: EPRR has a unit root
Exogenous: Constant
Lag Length: 0 (Automatic - based on SIC, maxlag=14) t-Statistic Prob.* Augmented Dickey-Fuller test statistic -0.574873 0.8722
Test critical values: 1% level -3.460313
5% level -2.874617
10% level -2.573817 *MacKinnon (1996) one-sided p-values.
Augmented Dickey-Fuller Test Equation
Dependent Variable: D(EPRR)
Method: Least Squares
Date: 06/09/17 Time: 01:21
Sample (adjusted): 2 219
Included observations: 218 after adjustments Variable Coefficient Std. Error t-Statistic Prob.
EPRR(-1) -11.96557 20.81428 -0.574873 0.5660
C -25.41724 58.11868 -0.437333 0.6623 R-squared 0.001528 Mean dependent var -45.87359
Adjusted R-squared -0.003095 S.D. dependent var 677.4201
S.E. of regression 678.4676 Akaike info criterion 15.88668
Sum squared resid 99428738 Schwarz criterion 15.91773
Log likelihood -1729.648 Hannan-Quinn criter. 15.89922
F-statistic 0.330479 Durbin-Watson stat 1.001954
Prob(F-statistic) 0.565975
Null Hypothesis: D(EPRR,2) has a unit root
Exogenous: Constant
Lag Length: 1 (Automatic - based on SIC, maxlag=14) t-Statistic Prob.* Augmented Dickey-Fuller test statistic -2.854388 0.0526
Test critical values: 1% level -3.460739
5% level -2.874804
10% level -2.573917 *MacKinnon (1996) one-sided p-values.
Augmented Dickey-Fuller Test Equation
Dependent Variable: D(EPRR,3)
Method: Least Squares
Date: 06/09/17 Time: 01:22
Sample (adjusted): 5 219
Included observations: 215 after adjustments Variable Coefficient Std. Error t-Statistic Prob. D(EPRR(-1),2) -111.9242 39.21128 -2.854388 0.0047
D(EPRR(-1),3) 56.97723 21.94793 2.596018 0.0101
C -45.76877 45.89062 -0.997345 0.3197 R-squared 0.037050 Mean dependent var -46.52843
Adjusted R-squared 0.027966 S.D. dependent var 682.4875
S.E. of regression 672.8766 Akaike info criterion 15.87486
Sum squared resid 95985738 Schwarz criterion 15.92189
Log likelihood -1703.547 Hannan-Quinn criter. 15.89386
F-statistic 4.078449 Durbin-Watson stat 1.002930
Prob(F-statistic) 0.018280
Null Hypothesis: D(PCE) has a unit root
Exogenous: Constant
Lag Length: 1 (Automatic - based on SIC, maxlag=14) t-Statistic Prob.* Augmented Dickey-Fuller test statistic -15.53051 0.0000
Test critical values: 1% level -3.460596
5% level -2.874741
10% level -2.573883 *MacKinnon (1996) one-sided p-values.
Augmented Dickey-Fuller Test Equation
Dependent Variable: D(PCE,2)
Method: Least Squares
Date: 06/09/17 Time: 01:25
Sample (adjusted): 4 219
Included observations: 216 after adjustments Variable Coefficient Std. Error t-Statistic Prob. D(PCE(-1)) -1.669831 0.107519 -15.53051 0.0000
D(PCE(-1),2) 0.256747 0.066121 3.882970 0.0001
C 0.009632 0.083963 0.114713 0.9088 R-squared 0.685379 Mean dependent var -0.019160
Adjusted R-squared 0.682425 S.D. dependent var 2.189122
S.E. of regression 1.233654 Akaike info criterion 3.271629
Sum squared resid 324.1650 Schwarz criterion 3.318508
Log likelihood -350.3359 Hannan-Quinn criter. 3.290568
F-statistic 232.0026 Durbin-Watson stat 2.011034
Prob(F-statistic) 0.000000
Date: 06/09/17 Time: 01:46
Series: CGR EPRR _3TCM PCE
Sample: 1 219
Included observations: 219
Null hypothesis: Series are not cointegrated
Cointegrating equation deterministics: C
Automatic lags specification based on Schwarz criterion (maxlag=14)
Dependent tau-statistic Prob.* z-statistic Prob.*
CGR -4.260599 0.0377 -52.07718 0.0006
EPRR -4.262521 0.0375 -52.09807 0.0006
_3TCM -3.498753 0.1990 -25.32218 0.1304
PCE -4.129776 0.0524 -37.67988 0.0140 *MacKinnon (1996) p-values.
Intermediate Results:
CGR EPRR _3TCM PCE
Rho - 1 -0.481032 -0.481192 -0.261841 -0.299493
Rho S.E. 0.112902 0.112889 0.074838 0.072520
Residual variance 4.404948 4.404056 1.910262 2.007643
Long-run residual variance 1.137966 1.137889 0.386494 0.681122
Number of lags 5 5 3 2
Number of observations 213 213 215 216
Number of stochastic trends** 4 4 4 4 **Number of stochastic trends in asymptotic distribution