38For a detailed discussion of this, see Maddala et al., op. cit., Sec. 2.7.
822 PART FOUR: SIMULTANEOUS-EQUATION MODELS
21.11 COINTEGRATION: REGRESSION OF A UNIT ROOT TIME SERIES ON ANOTHER UNIT ROOT TIME SERIES
We have warned that the regression of a nonstationary time series on another nonstationary time series may produce a spurious regression. Let us suppose that we consider the PCE and PDI time series given in Table 21.1. Subjecting these time series individually to unit root analysis, you will find that they both are 7(1); that is, they contain a unit root. Suppose, then, that we regress PCE on PDI as follows:
Let us write this as:
Suppose we now subject ut to unit root analysis and find that it is stationary; that is, it is 7(0). This is an interesting situation, for although PCEt and PDIt are individually 7(1), that is, they have stochastic trends, their linear combination (21.11.2) is 7(0). So to speak, the linear combination cancels out the stochastic trends in the two series. If you take consumption and income as two 7(1) variables, savings defined as (income — consumption) could be 7(0). As a result, a regression of consumption on income as in (21.11.1) would be meaningful (i.e., not spurious). In this case we say that the two variables are cointegrated. Economically speaking, two variables will be cointegrated if they have a long-term, or equilibrium, relationship between them. Economic theory is often expressed in equilibrium terms, such as Fisher's quantity theory of money or the theory of purchasing parity (PPP), just to name a few.
In short, provided we check that the residuals from regressions like (21.11.1) are 7(0) or stationary, the traditional regression methodology (including the t and F tests) that we have considered extensively is applicable to data involving (nonstationary) time series. The valuable contribution of the concepts of unit root, cointegration, etc. is to force us to find out if the regression residuals are stationary. As Granger notes, "A test for cointegration can be thought of as a pre-test to avoid 'spurious regression' situations."39 In the language of cointegration theory, a regression such as (21.11.1) is known as a cointegrating regression and the slope parameter p2 is known as the cointegrating parameter. The concept of cointegration can be extended to a regression model containing k regressors. In this case we will have k cointegrating parameters.
Testing for Cointegration
A number of methods for testing cointegration have been proposed in the literature. We consider here two comparatively simple methods: (1) the DF
39C. W. J. Granger, "Developments in the Study of Co-Integrated Economic Variables, Oxford Bulletin of Economics and Statistics, vol. 48, 1986, p. 226.
CHAPTER TWENTY-ONE: TIME SERIES ECONOMETRICS: SOME BASIC CONCEPTS 823
or ADF unit root test on the residuals estimated from the cointegrating regression and (2) the cointegrating regression Durbin-Watson (CRDW) test.40
Engle-Granger (EG) or Augmented Engle-Granger (AEG) Test. We already know how to apply the DF or ADF unit root tests. All we have to do is estimate a regression like (21.11.1), obtain the residuals, and use the DF or ADF tests.41 There is one precaution to exercise, however. Since the estimated ut are based on the estimated cointegrating parameter p2, the DF and ADF critical significance values are not quite appropriate. Engle and Granger have calculated these values, which can be found in the refer-ences.42 Therefore, the DF and ADF tests in the present context are known as Engle-Granger (EG) and augmented Engle-Granger (AEG) tests. However, several software packages now present these critical values along with other outputs.
Let us illustrate these tests. We first regressed PCE on PDI and obtained the following regression:
PCEt = -171.4412 + 0.9672PDIt t = (-7.4808) (119.8712) (21.11.3)
Since PCE and PDI are individually nonstationary, there is the possibility that this regression is spurious. But when we performed a unit root test on the residuals obtained from (21.11.3), we obtained the following results:
The Engle-Granger 1 percent critical t value is -2.5899. Since the computed t (= t) value is much more negative than this, our conclusion is that the residuals from the regression of PCE on PDI are 7(0); that is, they are
40There is this difference between tests for unit roots and tests for cointegration. As David A. Dickey, Dennis W. Jansen, and Daniel I. Thornton observe, "Tests for unit roots are performed on univariate [i.e., single] time series. In contrast, cointegration deals with the relationship among a group of variables, where (unconditionally) each has a unit root." See their article, "A Primer on Cointegration with an Application to Money and Income," Economic Review, Federal Reserve Bank of St. Louis, March-April 1991, p. 59. As the name suggests, this article is an excellent introduction to cointegration testing.
41If PCE and PDI are not cointegrated, any linear combination of them will be nonstation-ary and, therefore, the ut will also be nonstationary.
42R. F. Engle and C. W. Granger, "Co-integration and Error Correction: Representation, Estimation and Testing," Econometrica, vol. 55, 1987, pp. 251-276.
Econometrics: Some Basic Concepts
824 PART FOUR: SIMULTANEOUS-EQUATION MODELS
stationary. Hence, (21.11.3) is a cointegrating regression and this regression is not spurious, even though individually the two variables are nonstation-ary. One can call (21.11.3) the static or long run consumption function and interpret its parameters as long run parameters. Thus, 0.9672 represents the long-run, or equilibrium, marginal propensity to consumer (MPC).
Cointegrating Regression Durbin-Watson (CRDW) Test. An alternative, and quicker, method of finding out whether PCE and PDI are cointe-grated is the CRDW test, whose critical values were first provided by Sargan and Bhargava.43 In CRDW we use the Durbin-Watson d obtained from the cointegrating regression, such as d = 0.5316 given in (21.11.3). But now the null hypothesis is that d = 0 rather than the standard d = 2. This is because in Chapter 12 we observed that d ~ 2(1 — p), so if there is to be a unit root, the estimated p will be about 1, which implies that d will be about zero.
On the basis of 10,000 simulations formed from 100 observations each, the 1, 5, and 10 percent critical values to test the hypothesis that the true d = 0 are 0.511, 0.386, and 0.322, respectively. Thus, if the computed d value is smaller than, say, 0.511, we reject the null hypothesis of cointegration at the 1 percent level. In our example, the value of 0.5316 is above this critical value, suggesting that PCE and PDI are cointegrated, thus reinforcing the finding on the basis of the EG test.44
To sum up, our conclusion, based on both the EG and CRDW tests, is that PCE and PDI are cointegrated.45 Although they individually exhibit random walks, there seems to be a stable long-run relationship between them; they will not wander away from each other, which is evident from Figure 21.1.
Cointegration and Error Correction Mechanism (ECM)
We just showed that PCE and PDI are cointegrated; that is, there is a lon term, or equilibrium, relationship between the two. Of course, in the short run there may be disequilibrium. Therefore, one can treat the error term in (21.11.2) as the "equilibrium error." And we can use this error term to tie the short-run behavior of PCE to its long-run value. The error correction mechanism (ECM) first used by Sargan46 and later popularized by Engle
43J. D. Sargan and A. S. Bhargava, "Testing Residuals from Least-Squares Regression for being Generated by the Gaussian Random Walk," Econometrica, vol. 51, 1983, pp. 153-174.
44There is considerable debate about the superiority of CRDW over DF, which can be found in the references. The debate revolves around the power of the two statistics, that is, the probability of not committing a Type II error. Engle and Granger, for example, prefer the ADF to the CRDW test.
45The EG and CRDW tests are now supplemented (supplanted?) by more powerful tests developed by Johansen. But the discussion of the Johansen method is beyond the scope of this book because the mathematics involved is quite complex, although several software packages now use the Johansen method.
46J. D. Sargan, "Wages and Prices in the United Kingdom: A Study in Econometric Methodology," in K. F. Wallis and D. F. Hendry, eds., Quantitative Economics and Econometric Analysis, Basil Blackwell, Oxford, U.K., 1984.
CHAPTER TWENTY-ONE: TIME SERIES ECONOMETRICS: SOME BASIC CONCEPTS 825
and Granger corrects for disequilibrium. An important theorem, known as the Granger representation theorem, states that if two variables Y and X are cointegrated, then the relationship between the two can be expressed as ECM. To see what this means, let us revert to our PCE-PDI example. Now consider the following model:
where A as usual denotes the first difference operator, st is a random error term, and ut—1 = (PCEt—1 — fa1 — faPDIt—1), that is, the one-period lagged value of the error from the cointegrating regression (21.11.1).
ECM equation (21.11.5) states that APCE depends on APDI and also on the equilibrium error term.47 If the latter is nonzero, then the model is out of equilibrium. Suppose APDI is zero and ut—1 is positive. This means PCEt—1 is too high to be in equilibrium, that is, PCEt—1 is above its equilibrium value of (a0 + a1PDIt—1). Since a2 is expected to be negative, the term a2ut—1 is negative and, therefore, APCEt will be negative to restore the equilibrium. That is, if PCEt is above its equilibrium value, it will start falling in the next period to correct the equilibrium error; hence the name ECM. By the same token, if ut—1 is negative (i.e., PCE is below its equilibrium value), a2ut—1 will be positive, which will cause ACPEt to be positive, leading PCEt to rise in period t. Thus, the absolute value of a2 decides how quickly the equilibrium is restored. In practice, we estimate ut—1 by ut— 1 = (PCEt — — fa2PDIt).
Returning to our illustrative example, the empirical counterpart of (21.11.5) is:
Statistically, the equilibrium error term is zero, suggesting that PCE adjusts to changes in PDI in the same time period. As (21.11.6) shows, short-run changes in PDI have a positive impact on short-run changes in personal consumption. One can interpret 0.2906 as the short-run marginal propensity to consume (MPC); the long-run MPC is given by the estimated (static) equilibrium relation (21.11.3) as 0.9672.
Before we conclude this section, the caution sounded by S. G. Hall is worth remembering:
While the concept of cointegration is clearly an important theoretical underpinning of the error correction model there are still a number of problems surrounding its
47The following discussion is based on Gary Koop, op. cit., pp. 159-160 and Kerry Peterson, op. cit., Sec. 8.5.
826 PART FOUR: SIMULTANEOUS-EQUATION MODELS
practical application; the critical values and small sample performance of many of these tests are unknown for a wide range of models; informed inspection of the cor-relogram may still be an important tool.48
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