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COINTEGRATION 1 The next topic is cointegration. Suppose that you have two nonstationary series X and Y and you hypothesize that Y is a linear function of X. If the hypothesis is correct, the divergence between Y and the linear function of X should be limited.

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2 To put this more technically, the disturbance term, which accounts for the discrepancy between Y and the linear function of X, should be a stationary series. If this is the case, Y and X are said to be cointegrated. COINTEGRATION

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3 Thus Y and X could both be I(1), and yet, if the model is correctly specified, one would expect u to be I(0). A requirement for cointegration is that all the variables in the relationship should be subject to the same degree of integration. COINTEGRATION

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4 To perform the test, you fit the relationship using OLS and examine the residuals. If the disturbance term is stationary, so will be the residuals. You test them for nonstationarity. COINTEGRATION

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============================================================ Dependent Variable: LGFOOD Method: Least Squares Sample: 1959 2003 Included observations: 45 ============================================================ Variable Coefficient Std. Error t-Statistic Prob. ============================================================ C 2.236158 0.388193 5.760428 0.0000 LGDPI 0.500184 0.008793 56.88557 0.0000 LGPRFOOD -0.074681 0.072864 -1.024941 0.3113 ============================================================ R-squared 0.992009 Mean dependent var 6.021331 Adjusted R-squared 0.991628 S.D. dependent var 0.222787 S.E. of regression 0.020384 Akaike info criter-4.883747 Sum squared resid 0.017452 Schwarz criterion -4.763303 Log likelihood 112.8843 F-statistic 2606.860 Durbin-Watson stat 0.478540 Prob(F-statistic) 0.000000 ============================================================ 5 We will see if the logarithmic model for expenditure on food is cointegrated. LGFOOD, LGDPI, and LGPRFOOD all appear to be I(1) when tested for unit roots. Here is the output from an OLS regression, which is adequate for the test. COINTEGRATION

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Residuals 6 Here are the residuals from the regression. They do look stationary, so it appears that we have found a cointegrating relationship. COINTEGRATION

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Augmented Dickey-Fuller Unit Root Test on ZFOOD ============================================================ t-Statistic Prob.* ============================================================ Augmented Dickey-Fuller test statistic -2.608869 0.0103 Test critical values1% level -2.619851 5% level -1.948686 10% level -1.612036 ============================================================ Dependent Variable: D(ZFOOD) ============================================================ Variable Coefficient Std. Error t-Statistic Prob. ============================================================ ZFOOD(-1) -0.293440 0.112478 -2.608869 0.0126 D(ZFOOD(-1)) 0.312003 0.158279 1.971221 0.0555 ============================================================ 7 The residuals have been saved with name ZFOOD. There is no reason to include either a constant or a time trend when testing the residuals for nonstationarity. COINTEGRATION

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Augmented Dickey-Fuller Unit Root Test on ZFOOD ============================================================ t-Statistic Prob.* ============================================================ Augmented Dickey-Fuller test statistic -2.608869 0.0103 Test critical values1% level -2.619851 5% level -1.948686 10% level -1.612036 ============================================================ Dependent Variable: D(ZFOOD) ============================================================ Variable Coefficient Std. Error t-Statistic Prob. ============================================================ ZFOOD(-1) -0.293440 0.112478 -2.608869 0.0126 D(ZFOOD(-1)) 0.312003 0.158279 1.971221 0.0555 ============================================================ 8 COINTEGRATION In view of the fact that the least squares coefficients are chosen so as to minimize the sum of the squares of the residuals, the time series for the residuals will tend to appear more stationary than the underlying series for the disturbance term.

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Augmented Dickey-Fuller Unit Root Test on ZFOOD ============================================================ t-Statistic Prob.* ============================================================ Augmented Dickey-Fuller test statistic -2.608869 0.0103 Test critical values1% level -2.619851 5% level -1.948686 10% level -1.612036 ============================================================ Dependent Variable: D(ZFOOD) ============================================================ Variable Coefficient Std. Error t-Statistic Prob. ============================================================ ZFOOD(-1) -0.293440 0.112478 -2.608869 0.0126 D(ZFOOD(-1)) 0.312003 0.158279 1.971221 0.0555 ============================================================ 9 COINTEGRATION To allow for this, the critical values for the test statistic are even higher than those for the standard test for nonstationarity of a time series. Asymptotic critical values for the case where the cointegrating relationship involves two variables are shown in the table. Asymptotic Critical Values of the Dickey-Fuller Statistic for a Cointegrating Relationship with Two Variables Regression equation contains: 5% 1% Constant, but no trend –3.34–3.90 Constant and trend–3.78–4.32

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Augmented Dickey-Fuller Unit Root Test on ZFOOD ============================================================ t-Statistic Prob.* ============================================================ Augmented Dickey-Fuller test statistic -2.608869 0.0103 Test critical values1% level -2.619851 5% level -1.948686 10% level -1.612036 ============================================================ Dependent Variable: D(ZFOOD) ============================================================ Variable Coefficient Std. Error t-Statistic Prob. ============================================================ ZFOOD(-1) -0.293440 0.112478 -2.608869 0.0126 D(ZFOOD(-1)) 0.312003 0.158279 1.971221 0.0555 ============================================================ 10 COINTEGRATION The original model (the regression of LGFOOD on LGDPI and LGPRFOOD) had a constant but no trend. Asymptotic Critical Values of the Dickey-Fuller Statistic for a Cointegrating Relationship with Two Variables Regression equation contains: 5% 1% Constant, but no trend –3.34–3.90 Constant and trend–3.78–4.32

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Augmented Dickey-Fuller Unit Root Test on ZFOOD ============================================================ t-Statistic Prob.* ============================================================ Augmented Dickey-Fuller test statistic -2.608869 0.0103 Test critical values1% level -2.619851 5% level -1.948686 10% level -1.612036 ============================================================ Dependent Variable: D(ZFOOD) ============================================================ Variable Coefficient Std. Error t-Statistic Prob. ============================================================ ZFOOD(-1) -0.293440 0.112478 -2.608869 0.0126 D(ZFOOD(-1)) 0.312003 0.158279 1.971221 0.0555 ============================================================ 11 COINTEGRATION Unfortunately, we are not able to reject the null hypothesis. The coefficient of the lagged variable is –0.29, which is not close to 0, but its t statistic is not large enough for us to reject nonstationarity, even at the 5 percent level. Asymptotic Critical Values of the Dickey-Fuller Statistic for a Cointegrating Relationship with Two Variables Regression equation contains: 5% 1% Constant, but no trend –3.34–3.90 Constant and trend–3.78–4.32

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12 COINTEGRATION However, this may simply be due to the low power of the test. The plot of the residuals did not look too bad. Residuals

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Copyright Christopher Dougherty 2000–2006. This slideshow may be freely copied for personal use. 21.08.06

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