FIGURE 17.8 Lag structure of the illustrative example.

Our illustrative example may be used to point out a few additional features of the Almon lag procedure:

1. The standard errors of the a coefficients are directly obtainable from the OLS regression (17.13.14), but the standard errors of some of the fa

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coefficients, the objective of primary interest, cannot be so obtained. But they can be obtained from the standard errors of the estimated a coefficients by using a well-known formula from statistics, which is given in exercise 17.18. Of course, there is no need to do this manually, for most statistical packages can do this routinely. The standard errors given in (17.13.15) were obtained from Eviews 4.

2. The ft's obtained in (17.13.16) are called unrestricted estimates in the sense that no a priori restrictions are placed on them. In some situations, however, one may want to impose the so-called endpoint restrictions on the ft's by assuming that ft0 and ftk (the current and kth lagged coefficient) are zero. Because of psychological, institutional, or technical reasons, the value of the explanatory variable in the current period may not have any impact on the current value of the regressand, thereby justifying the zero value for ft0. By the same token, beyond a certain time the kth lagged coefficient may not have any impact on the regressand, thus supporting the assumption that ftk is zero. In our inventory example, the coefficient of Xt_3 had a negative sign, which may not make economic sense. Hence, one may want to constrain that coefficient to zero.51 Of course, you do not have to constrain both ends; you could put restriction only on the first coefficient, called near-end restriction or on the last coefficient, called far-end restriction. For our inventory example, this is illustrated in exercise 17.28. Sometimes the ft's are estimated with the restriction that their sum is one. But one should not put such restrictions mindlessly because such restrictions also affect the values of the other (unconstrained) lagged coefficients.

3. Since the choice of the number of lagged coefficients as well as the degree of the polynomial is at the discretion of the modeler, some trial and error is inevitable, the charge of data mining notwithstanding. Here is where the Akaike and Schwarz information criteria discussed in Chapter 13 may come in handy.

4. Since we estimated (17.13.16) using three lags and the second-degree polynomial, it is a restricted least-squares model. Suppose, we decide to use three lags but do not use the Almon polynomial approach. That is, we estimate (17.13.11) by OLS. What then? Let us first see the results:

Yt = 26,008.60 + 0.9771X + 1.0139Xi_1 _ 0.2022 X_2 _ 0.3935Xt_3

If you compare these results with those given in (17.13.16), you will see that the overall R2 is practically the same, although the lagged pattern in (17.13.17) shows more of a humped shape than that exhibited by (17.13.16).

51For a concrete application, see D. B. Batten and Daniel Thornton, "Polynomial Distributed Lags and the Estimation of the St. Louis Equation," Review, Federal Reserve Bank of St. Louis, April 1983, pp. 13-25.

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As this example illustrates, one has to be careful in using the Almon distributed lag technique, as the results might be sensitive to the choice of the degree of the polynomial and/or the number of lagged coefficients.


Back in Section 1.4 we noted that, although regression analysis deals with the dependence of one variable on other variables, it does not necessarily imply causation. In other words, the existence of a relationship between variables does not prove causality or the direction of influence. But in regressions involving time series data, the situation may be somewhat different because, as one author puts it,

. . . time does not run backward. That is, if event A happens before event B, then it is possible that A is causing B. However, it is not possible that B is causing A. In other words, events in the past can cause events to happen today. Future events cannot.53 (Emphasis added.)

This is roughly the idea behind the so-called Granger causality test.54 But it should be noted clearly that the question of causality is deeply philosophical with all kinds of controversies. At one extreme are people who believe that "everything causes everything," and at the other extreme are people who deny the existence of causation whatsoever.55 The econometrician Edward Leamer prefers the term precedence over causality. Francis Diebold prefers the term predictive causality. As he writes:

. . . the statement "yi causes y" is just shorthand for the more precise, but long-winded, statement, "yi contains useful information for predicting yj (in the linear least squares sense), over and above the past histories of the other variables in the system." To save space, we simply say that yi causes y;.56

The Granger Test

To explain the Granger test, we will consider the often asked question in macroeconomics: Is it GDP that "causes" the money supply M (GDP ^ M)

52There is another test of causality that is sometimes used, the so-called Sims test of causality. We discuss it by way of an exercise.

53Gary Koop, Analysis of Economic Data, John Wiley & Sons, New York, 2000, p. 175.

54C. W. J. Granger, "Investigating Causal Relations by Econometric Models and Cross-Spectral Methods," Econometrica, July 1969, pp. 424-438. Although popularly known as the Granger causality test, it is appropriate to call it the Wiener-Granger causality test, for it was earlier suggested by Wiener. See N. Wiener, "The Theory of Prediction," in E. F. Beckenback, ed., Modern Mathematics for Engineers, McGraw-Hill, New York, 1956, pp. 165-190.

55For an excellent discussion of this topic, see Arnold Zellner, "Causality and Econometrics," Carnegie-Rochester Conference Series, 10, K. Brunner and A. H. Meltzer, eds., North Holland Publishing Company, Amsterdam, 1979, pp. 9-50.

56Francis X. Diebold, Elements of Forecasting, South Western Publishing, 2d ed., 2001, p. 254.

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or is it the money supply M that causes GDP (M ^ GDP), where the arrow points to the direction of causality. The Granger causality test assumes that the information relevant to the prediction of the respective variables, GDP and M, is contained solely in the time series data on these variables. The test involves estimating the following pair of regressions:

i=i j=i where it is assumed that the disturbances uit and u2t are uncorrelated. In passing, note that, since we have two variables, we are dealing with bilateral causality. In the chapters on time series econometrics, we will extend this to multivariable causality through the technique of vector autoregression (VAR).

Equation (17.14.1) postulates that current GDP is related to past values of itself as well as that of M, and (17.14.2) postulates a similar behavior for M. Note that these regressions can be cast in growth forms, GDP and M, where a dot over a variable indicates its growth rate. We now distinguish four cases:

1. Unidirectional causality from M to GDP is indicated if the estimated coefficients on the lagged M in (17.14.1) are statistically different from zero as a group (i.e., Y. a = 0) and the set of estimated coefficients on the lagged GDP in (17.14.2) is not statistically different from zero (i.e., Y. Sj = 0).

2. Conversely, unidirectional causality from GDP to M exists if the set of lagged M coefficients in (17.14.1) is not statistically different from zero (i.e., YJai = 0) and the set of the lagged GDP coefficients in (17.14.2) is statistically different from zero (i.e., Y Sj = 0).

3. Feedback, or bilateral causality, is suggested when the sets of M and GDP coefficients are statistically significantly different from zero in both regressions.

4. Finally, independence is suggested when the sets of M and GDP coefficients are not statistically significant in both the regressions.

More generally, since the future cannot predict the past, if variable X (Granger) causes variable Y, then changes in X should precede changes in Y. Therefore, in a regression of Y on other variables (including its own past values) if we include past or lagged values of X and it significantly improves the prediction of Y, then we can say that X (Granger) causes Y. A similar definition applies if Y (Granger) causes X.

The steps involved in implementing the Granger causality test are as follows. We illustrate these steps with the GDP-money example given in Eq. (17.14.1).

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1. Regress current GDP on all lagged GDP terms and other variables, if any, but do not include the lagged M variables in this regression. As per Chapter 8, this is the restricted regression. From this regression obtain the restricted residual sum of squares, RSS#.

2. Now run the regression including the lagged M terms. In the language of Chapter 8, this is the unrestricted regression. From this regression obtain the unrestricted residual sum of squares, RSSur.

3. The null hypothesis is H0: E ai = 0, that is, lagged M terms do not belong in the regression.

4. To test this hypothesis, we apply the F test given by (8.7.9), namely,

which follows the F distribution with m and (n - k) df. In the present case m is equal to the number of lagged M terms and k is the number of parameters estimated in the unrestricted regression.

5. If the computed F value exceeds the critical F value at the chosen level of significance, we reject the null hypothesis, in which case the lagged M terms belong in the regression. This is another way of saying that M causes GDP.

6. Steps 1 to 5 can be repeated to test the model (17.14.2), that is, whether GDP causes M.

Before we illustrate the Granger causality test, there are several things that need to be noted:

1. It is assumed that the two variables, GDP and M, are stationary. We have already discussed the concept of stationarity in intuitive terms before and will discuss it more formally in Chapter 21. Sometimes taking the first differences of the variables makes them stationary, if they are not already stationary in the level form.

2. The number of lagged terms to be introduced in the causality tests is an important practical question. As in the case of the distributed lag models, we may have to use the Akaike or Schwarz information criterion to make the choice. But it should be added that the direction of causality may depend critically on the number of lagged terms included.

3. We have assumed that the error terms entering the causality test are uncorrelated. If this is not the case, appropriate transformation, as discussed in Chapter 12, may have to be taken.57

4. Since our interest is in testing for causality, one need not present the estimated coefficients of models (17.14.1) and (17.14.2) explicitly (to save space); just the results of the F test given in (8.7.9) will suffice.

57For further details, see Wojciech W. Charemza and Derek F. Deadman, New Directions in Econometric Practice: General to Specific Modelling, Cointegration and Vector Autoregression, 3d ed., Edward Elgar Publisher, 1997, Chap. 6.

17. Dynamic Econometric Models: Autoregressive and Distributed-Lag Models


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