Christopher Dougherty EC220 - Introduction to econometrics (chapter 13) Slideshow: cointegration Original citation: Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 13). [Teaching Resource] © 2012 The Author This version available at: Available in LSE Learning Resources Online: May 2012 This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user credits the author and licenses their new creations under the identical terms

COINTEGRATION 1 In general, a linear combination of two or more time series will be nonstationary if one or more of them is nonstationary, and the degree of integration of the combination will be equal to that of the most highly integrated individual series.

COINTEGRATION 2 Hence, for example, a linear combination of an I(1) series and an I(0) series will be I(1), that of two I(1) series will also be I(1), and that of an I(1) series and an I(2) series will be I(2).

COINTEGRATION 3 However, if there is a long-run relationship between the time series, the outcome may be different.

COINTEGRATION 4 Consider, for example, Friedman’s Permanent Income Hypothesis and the consumption function C t P =  2 Y t P v t where C t P and Y t P are permanent consumption and income, respectively, and v t is a multiplicative disturbance term.

COINTEGRATION 5 In logarithms, the relationship becomes log C t P = log  2 + log Y t P + log v t where u t is the logarithm of v t.

COINTEGRATION 6 If the theory is correct, in the long run, ignoring short-run dynamics and the differences between the permanent and actual measures of the variables, consumption and income will grow at the same rate and the mean of the difference between their logarithms will be log  2.

COINTEGRATION 7 The figure shows plots of the logarithms of aggregate disposable personal income, DPI, and aggregate personal consumer expenditure, PCE, (left scale), and their difference (right scale) for the United States for the period 1959–2003.

COINTEGRATION 8 It can be seen that the gap between the two has been fairly stable, increasing a little in the first part of the period and declining a little thereafter. Thus, although the series for DPI and PCE are nonstationary, they appear to be wandering together.

COINTEGRATION 9 For this to be possible, ut must be a stationary process, for if it were not, the two series could drift apart indefinitely, violating the theoretical relationship.

COINTEGRATION 10 When two or more nonstationary time series are linked in such a way, they are said to be cointegrated. In this example, the slope coefficient of log Y t P is theoretically equal to 1, making it possible to inspect the divergence graphically in the figure.

COINTEGRATION 11 More generally, if there exists a relationship between a set of variables Y t, X 2t, …, X kt, the disturbance term u t can be thought of as measuring the deviation between the components of the model.

COINTEGRATION 12 In the short run the divergence between the components will fluctuate, but if the model is genuinely correct there will be a limit to the divergence. Hence, although the time series are nonstationary, u t will be stationary.

COINTEGRATION 13 If there are more than two variables in the model, it is possible that there may be multiple cointegrating relationships, the maximum number in theory being equal to k – 1.

COINTEGRATION 14 To test for cointegration, it is necessary to evaluate whether the disturbance term is a stationary process. In the case of the example of consumer expenditure and income, it is sufficient to perform a standard ADF unit root test on the difference between the two series.

============================================================ Augmented Dickey-Fuller Unit Root Test on Z ============================================================ Null Hypothesis: Z has a unit root Exogenous: Constant, Linear Trend Lag Length: 0 (Automatic based on SIC, MAXLAG=9) ============================================================ t-Statistic Prob.* ============================================================ Augmented Dickey-Fuller test statistic Test critical values1% level % level % level ============================================================ Augmented Dickey-Fuller Test Equation Dependent Variable: D(Z) Sample (adjusted): ============================================================ Variable CoefficientStd. Errort-Statistic Prob. ============================================================ Z(-1) C ============================================================ COINTEGRATION 15 The results are shown in the table, with the difference between the logarithms being denoted Z. The ADF test statistic is –1.63, which is less than –3.52, the critical value at the 5 percent level under the null hypothesis of nonstationarity.

============================================================ Augmented Dickey-Fuller Unit Root Test on Z ============================================================ Null Hypothesis: Z has a unit root Exogenous: Constant, Linear Trend Lag Length: 0 (Automatic based on SIC, MAXLAG=9) ============================================================ t-Statistic Prob.* ============================================================ Augmented Dickey-Fuller test statistic Test critical values1% level % level % level ============================================================ Augmented Dickey-Fuller Test Equation Dependent Variable: D(Z) Sample (adjusted): ============================================================ Variable CoefficientStd. Errort-Statistic Prob. ============================================================ Z(-1) C ============================================================ COINTEGRATION 16 This is a surprising result, for other studies have found the logarithms of consumer expenditure and income to be cointegrated. Part of the problem is the low power of the test against an alternative hypothesis of u t being a stationary process with high autocorrelation.

============================================================ Augmented Dickey-Fuller Unit Root Test on Z ============================================================ Null Hypothesis: Z has a unit root Exogenous: Constant, Linear Trend Lag Length: 0 (Automatic based on SIC, MAXLAG=9) ============================================================ t-Statistic Prob.* ============================================================ Augmented Dickey-Fuller test statistic Test critical values1% level % level % level ============================================================ Augmented Dickey-Fuller Test Equation Dependent Variable: D(Z) Sample (adjusted): ============================================================ Variable CoefficientStd. Errort-Statistic Prob. ============================================================ Z(-1) C ============================================================ COINTEGRATION 17 The coefficient of the lagged residual is –0.13, suggesting that the process is approximately AR(1) with autocorrelation 0.87, but the standard error is too large for the null hypothesis of nonstationarity to be rejected.

============================================================ Augmented Dickey-Fuller Unit Root Test on Z ============================================================ Null Hypothesis: Z has a unit root Exogenous: Constant, Linear Trend Lag Length: 0 (Automatic based on SIC, MAXLAG=9) ============================================================ t-Statistic Prob.* ============================================================ Augmented Dickey-Fuller test statistic Test critical values1% level % level % level ============================================================ Augmented Dickey-Fuller Test Equation Dependent Variable: D(Z) Sample (adjusted): ============================================================ Variable CoefficientStd. Errort-Statistic Prob. ============================================================ Z(-1) C ============================================================ COINTEGRATION 18 It is likely that persistence in the way that consumers behave is responsible for this. As consumers become more savings conscious, as they seem to have done from 1959 to about 1984, the gap between the logarithms widens.

============================================================ Augmented Dickey-Fuller Unit Root Test on Z ============================================================ Null Hypothesis: Z has a unit root Exogenous: Constant, Linear Trend Lag Length: 0 (Automatic based on SIC, MAXLAG=9) ============================================================ t-Statistic Prob.* ============================================================ Augmented Dickey-Fuller test statistic Test critical values1% level % level % level ============================================================ Augmented Dickey-Fuller Test Equation Dependent Variable: D(Z) Sample (adjusted): ============================================================ Variable CoefficientStd. Errort-Statistic Prob. ============================================================ Z(-1) C ============================================================ COINTEGRATION 19 As they become less savings conscious, as seems to have been the case since 1984, it narrows.

============================================================ Augmented Dickey-Fuller Unit Root Test on Z ============================================================ Null Hypothesis: Z has a unit root Exogenous: Constant, Linear Trend Lag Length: 0 (Automatic based on SIC, MAXLAG=9) ============================================================ t-Statistic Prob.* ============================================================ Augmented Dickey-Fuller test statistic Test critical values1% level % level % level ============================================================ Augmented Dickey-Fuller Test Equation Dependent Variable: D(Z) Sample (adjusted): ============================================================ Variable CoefficientStd. Errort-Statistic Prob. ============================================================ Z(-1) C ============================================================ COINTEGRATION 20 However, these changes evidently have long cycles, and so even over a period as long as 45 years it is difficult to discriminate between the hypothesis that the gap is a random walk and the alternative that it is stationary, with strong autocorrelation.

============================================================ Augmented Dickey-Fuller Unit Root Test on Z ============================================================ Null Hypothesis: Z has a unit root Exogenous: Constant, Linear Trend Lag Length: 0 (Automatic based on SIC, MAXLAG=9) ============================================================ t-Statistic Prob.* ============================================================ Augmented Dickey-Fuller test statistic Test critical values1% level % level % level ============================================================ Augmented Dickey-Fuller Test Equation Dependent Variable: D(Z) Sample (adjusted): ============================================================ Variable CoefficientStd. Errort-Statistic Prob. ============================================================ Z(-1) C ============================================================ COINTEGRATION 21 However, a sufficiently long time series would show that the gap is stationary, for it is not possible for it to decrease indefinitely.

COINTEGRATION 22 In the more general, where the cointegrating relationship has to be estimated, the test is an indirect one because it must be performed on the residuals from the regression, rather than on the disturbance term.

COINTEGRATION 23 In view of the fact that the least squares coefficients are chosen so as to minimize the sum of the squares of the residuals, and that the mean of the residuals is automatically zero, the time series for the residuals will tend to appear more stationary than the underlying series for the disturbance term.

COINTEGRATION 24 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.

COINTEGRATION 25 Asymptotic critical values for the case where the cointegrating relationship involves two variables are shown in the table. The test assumes that a constant has been included in the cointegrating relationship, and the critical values depend on whether a trend has been included as well. 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

COINTEGRATION 26 In the case of a cointegrating relationship, least squares estimators can be shown to be superconsistent.

COINTEGRATION 27 An important consequence is that OLS may be used to fit a cointegrating relationship, even if it belongs to a system of simultaneous relationships, for any simultaneous equations bias tends to zero asymptotically.

COINTEGRATION 28 As an example, consider the model shown. Y t and X t are endogenous variables, Z t is exogenous, and  Yt,  Xt, and  Zt are iid N(0,1) disturbance terms. We expect OLS estimators to be inconsistent if used to fit either of the first two equations. If |  | < 1, Z is stationary, and X and Y are also stationary.

COINTEGRATION 29 However, if  = 1, Z is nonstationary, and X and Y will also be nonstationary. So, if we fit the second equation, for example, the OLS estimator of  2 will be superconsistent. If  = 1, Z is a random walk, and X and Y are also nonstationary. The OLS estimator of  2 will be superconsistent.

COINTEGRATION 30 This will be illustrated by a simulation. where the first two equations are as shown. If  = 1, Z is a random walk, and X and Y are also nonstationary. The OLS estimator of  2 will be superconsistent.

COINTEGRATION 31 The distributions in the figure are for the case  = 0.5. Z is stationary, and so are Y and X. You will have no difficulty in demonstrating that plim a 2 OLS =  = 0.5

COINTEGRATION 32 The distributions to the left of the figure are for  = 1, and you can see that in this case the estimator shows signs of being consistent. But is it superconsistent? The variance seems to be decreasing relatively slowly, not fast, especially for small sample sizes.  = 0.5  = 1.0

COINTEGRATION 33 The explanation is that the superconsistency becomes apparent only for very large sample sizes. Comparing the distributions for T = 1,600 and T = 3,200, the sample size has doubled and so has the height.  = 1.0

============================================================ Dependent Variable: LGFOOD Method: Least Squares Sample: Included observations: 45 ============================================================ Variable Coefficient Std. Error t-Statistic Prob. ============================================================ C LGDPI LGPRFOOD ============================================================ R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criter Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) ============================================================ COINTEGRATION 34 We will now consider an empirical example, a logarithmic regression of expenditure on food on DPI and the relative price of food using the Demand Functions data set.

COINTEGRATION 35 The residuals are shown in the figure. The pattern is mixed and it is not possible to say whether it looks stationary or nonstationary.

Augmented Dickey-Fuller Unit Root Test on ELGFOOD ============================================================ Lag Length: 0 (Automatic based on SIC, MAXLAG=9) ============================================================ t-Statistic Prob.* ============================================================ Augmented Dickey-Fuller test statistic Test critical values1% level % level % level ============================================================ *MacKinnon (1996) one-sided p-values. Augmented Dickey-Fuller Test Equation Dependent Variable: D(ELGFOOD) Method: Least Squares Sample(adjusted): Included observations: 44 after adjusting endpoints ============================================================ Variable CoefficientStd. Errort-Statistic Prob. ============================================================ ELGFOOD(-1) C ============================================================ COINTEGRATION 36 The Engle–Granger statistic is –1.93, not significant even at the 5 percent level. The failure to reject the null hypothesis of nonstationarity suggests that the variables are not cointegrated. 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

Augmented Dickey-Fuller Unit Root Test on ELGFOOD ============================================================ Lag Length: 0 (Automatic based on SIC, MAXLAG=9) ============================================================ t-Statistic Prob.* ============================================================ Augmented Dickey-Fuller test statistic Test critical values1% level % level % level ============================================================ *MacKinnon (1996) one-sided p-values. Augmented Dickey-Fuller Test Equation Dependent Variable: D(ELGFOOD) Method: Least Squares Sample(adjusted): Included observations: 44 after adjusting endpoints ============================================================ Variable CoefficientStd. Errort-Statistic Prob. ============================================================ ELGFOOD(-1) C ============================================================ COINTEGRATION 37 Nevertheless, the coefficient of the lagged residuals is –0.21, suggesting an AR(1) process with  equal to about 0.8. 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

COINTEGRATION 38 Thus the failure of the test to reject the null hypothesis of nonstationarity may merely reflect its low power against the alternative hypothesis that the disturbance term is a highly autocorrelated stationary process. Consequently, it is possible that the variables are in fact cointegrated.

Copyright Christopher Dougherty These slideshows may be downloaded by anyone, anywhere for personal use. Subject to respect for copyright and, where appropriate, attribution, they may be used as a resource for teaching an econometrics course. There is no need to refer to the author. The content of this slideshow comes from Section 13.5 of C. Dougherty, Introduction to Econometrics, fourth edition 2011, Oxford University Press. Additional (free) resources for both students and instructors may be downloaded from the OUP Online Resource Centre Individuals studying econometrics on their own and who feel that they might benefit from participation in a formal course should consider the London School of Economics summer school course EC212 Introduction to Econometrics or the University of London International Programmes distance learning course 20 Elements of Econometrics