Granger-causality analysis

The concept

The causality concept introduced by Granger (1969) is perhaps the most widely discussed form of causality in the econometrics literature. Granger defines a variable y1t to be causal for another time series variable y2t if the former helps predicting the latter. Formally, denoting by y2,t+h|iif the optimal h-step predictor of y2t at origin t based on the set of all the relevant information in the universe Qt, y1t may be defined to be Granger-noncausal for y2t if and only if

y2,t+hQt = y2,t+h Qt{y1s s<t}, h — 1, 2, . . . . (32.18)

Here Qt A denotes the set containing all elements of Q. t which are not in the set A. In other words, y1t is not causal for y2t if removing the past of y1t from the information set does not change the optimal forecast for y2t at any forecast hori­zon. In turn, y1t is Granger-causal for y2t if the equality in (32.18) is violated for at least one h and, thus, a better forecast of y2t is obtained for some forecast horizon by including the past of y1t in the information set. If Qt — {(y1/S, y2,s)’ s < t} and (y1t, y2t)’ is generated by a bivariate VAR(p) process,

p

I

a11, i

a12, i

yu-i

+ Ut,

(32.19)

i=1

a21, i

a22, i

У 2,t-i _

then (32.18) is easily seen to be equivalent to

a21,; — 0, i — 1, 2,…, p. (32.20)

Of course, Granger-causality can also be investigated in the framework of the VECM (see, e. g. Mosconi and Giannini, 1992).

Economic systems usually consist of more than two relevant variables. Hence, it is desirable to extend the concept of Granger-causality to higher dimensional systems. Different possible extensions have been considered (see, e. g. Lutkepohl, 1993; Dufour and Renault, 1998). One possible generalization assumes that the vector of all variables, yt, is partitioned into two subvectors so that yt = (y 1t, y2t)’. Then the definition in (32.18) may be used for the two subvectors, y1t, y2t, rather than two individual variables. If Qt = {ys | s < t} and yt is a VAR process of the form (32.19), where the aih/i are now matrices of appropriate dimensions, the restrictions for noncausality are the same as in the bivariate case so that y1t is Granger-noncausal for y2t if a21i = 0 for i = 1,…, p (Lutkepohl, 1991, sec. 2.3.1).

Подпись: + ut. Подпись: (32.21)

This approach is not satisfactory if interest centers on a causal relation between two variables within a higher dimensional system because a set of variables being causal for another set of variables does not necessarily imply that each member of the former set is causal for each member of the latter set. Therefore it is of interest to consider causality of y1t to y2t if there are further variables in the system. In this context, different causality concepts have been proposed which are most easily explained in terms of the three-dimensional VAR process

Within this system causality of y1t for y2t is sometimes checked by testing

H0 : a21/i = 0, i = 1,…, p. (32.22)

These restrictions are not equivalent to (32.18), however. They are equivalent to equality of the one-step forecasts, y2t+1|Qf = y2/t+1|Qf {yi ,|s<t}. The information in past y1t may still help improving the forecasts of y2t more than one period ahead if

(32.22) holds (Lutkepohl, 1993). Intuitively, this happens because there may be indirect causal links, e. g. y1t may have an impact on y3t which in turn may affect y2t. Thus, the definition of noncausality corresponding to the restrictions in (32.22) is not in line with an intuitive notion of the term. For higher dimensional pro­cesses the definition based on (32.18) results in more complicated nonlinear restrictions for the VAR coefficients. Details are given in Dufour and Renault (1998).

Testing for Granger-causality

Wald tests are standard tools for testing restrictions on the coefficients of VAR processes because the test statistics are easy to compute in this context. Unfortu­nately, they may have non-standard asymptotic properties if the VAR contains I(1) variables. In particular, Wald tests for Granger-causality are known to result in nonstandard limiting distributions depending on the cointegration properties of the system and possibly on nuisance parameters (see Toda and Phillips, 1993).

Dolado and Lutkepohl (1996) and Toda and Yamamoto (1995) point out a simple way to overcome the problems with these tests in the present context. As mentioned in Section 3.1, the non-standard asymptotic properties of the standard tests on the coefficients of cointegrated VAR processes are due to the singularity of the asymptotic distribution of the LS estimators. Hence, the idea is to get rid of the singularity by fitting a VAR process whose order exceeds the true order. It can be shown that this device leads to a nonsingular asymptotic distribution of the relevant coefficients, overcoming the problems associated with standard tests and their complicated non-standard limiting properties.

More generally, as mentioned in Section 3.1, Dolado and Lutkepohl (1996) show that whenever the elements in at least one of the complete coefficient matrices A; are not restricted at all under the null hypothesis, the Wald statistic has its usual limiting ^-distribution. Thus, if elements from all A„ i = 1,…, p, are involved in the restrictions as, for instance, in the noncausality restrictions in

(32.20) or (32.22), simply adding an extra (redundant) lag in estimating the para­meters of the process, ensures standard asymptotics for the Wald test. Of course, if the true DGP is a VAR( p) process, then a VAR( p + 1) with A p+1 = 0 is also an appropriate model. The test is then performed on the Ai, i = 1,…, p, only.

For this procedure to work it is not necessary to know the cointegration prop­erties of the system. Thus, if there is uncertainty with respect to the integration properties of the variables an extra lag may simply be added and the test may be performed on the lag augmented model to be on the safe side. Unfortunately, the procedure is not fully efficient due to the redundant parameters. A generali­zation of these ideas to Wald tests for nonlinear restrictions representing, for instance, other causality definitions, is discussed by Lutkepohl and Burda (1997).

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