Sunday, August 21, 2011

Granger Causality Testing

More from my paper in progress. It's for an audience that isn't so familiar with econometrics but has a reasonable background in statistics. This is very rough, comments are very welcome!

A variable x is said to Granger cause another variable y if past values of x help predict the current level of y given all other appropriate information. This definition is based on the concept of causal ordering. Two variables may be contemporaneously correlated by chance but it is unlikely that the past values of x will be useful in predicting y, given all the past values of y, unless x does actually cause y in a philosophical sense. Similarly, if y in fact causes x, then given the past history of y it is unlikely that information on x will help predict y. Granger causality is not identical to causation in the classical philosophical sense, but it does demonstrate the likelihood of such causation or the lack of such causation more forcefully than does simple contemporaneous correlation (Geweke, 1984). However, where a third variable, z, drives both x and y, x might still appear to drive y though there is no actual causal mechanism directly linking the variables. The simplest test of Granger causality requires estimating the following two regression equations:

where p is the number of lags that adequately models the dynamic structure so that the coefficients of further lags of variables are not statistically significant and the error terms e are white noise. The error terms may, however, be correlated across equations. If the p parameters are jointly significant then the null that x does not Granger cause y can be rejected. Similarly, if the p parameters are jointly significant then the null that y does not Granger cause x can be rejected. This test is usually refereed to as the Granger causality test. There are several variants including the Sims (1972) causality test and the Toda and Yamamoto (1995) procedure discussed below.

There has been much criticism of Granger causality testing in the econometrics literature. Roberts and Nord (1985) found that the functional form of the time series affected the sensitivity of both Granger's and Sims' tests. Data that had undergone logarithmic transformation showed no sign of causality while the untransformed data yielded significant results. This stands to reason, as logarithmic transformation tends to reduce heteroscedasticity and increase the stationarity of the variables. However Chowdhury (1987) found more disturbing results that give support to those who have doubted whether Granger causality was related to philosophical causality or economic exogeneity in any meaningful way. He found that a Granger test indicated that GNP caused sunspots! A Sims test showed that prices caused sunspots! None of the alternative hypotheses were validated. Prices and income may be exogenous in the sunspot equations, but sunspots are not endogenous in any meaningful philosophical or economic way. But because sunspots are quite predictable prices and income might have anticipated them. The forward-looking behavior of human agents can be an obstacle to Granger causality testing.

Sargent (1979) and Sims (1980) introduced the vector autoregression or VAR modeling approach as a method of carrying out econometric analysis with a minimum of a priori assumptions about economic theory (Qin, 2011). The VAR model generalizes the model given by equations (1) and (2) to a multivariate setting. A multivariate Granger causality test can be identical to that described above but simply with more control variables in the regression but tests can also be constructed to exclude the lags of variables from multiple equations (Sims, 1980). The VAR approach to econometrics has been much criticized, but the critics, such as Epstein (1987) and Darnell and Evans (1990), argue that multivariate Granger causality tests are a (or the only) useful application of VARs. The advantage of multivariate Granger tests over bivariate Granger tests is that they can help avoid spurious correlations and can aid in testing the general validity of the causation test. This is through adding additional variables that may be responsible for causing y or whose effects might obscure the effect of x on y (Lütkepohl, 1982; Stern, 1993). There may also be indirect channels of causation from x to y, which VAR modeling could uncover.

Though a VAR cannot, due to limits on degrees of freedom, include all variables that may be causally related to the principal variable under investigation, some attempt can be made to include as many as possible. Of course, failure to reject the null hypothesis that x does not cause y, does not necessarily mean that there is in fact no causality. A lack of sensitivity could be due to a misspecified lag length, insufficiently frequent observations, too small a sample, or the lack of Granger causality even if philosophical causation occurs.

Engle and Granger (1987) introduced the notion of cointegration and tied it closely to the VAR model. Time series that must be differenced in order to render them stationary are referred to as integrated or stochastically trending series. The simplest case is the classic random walk where the current value of a variable is equal to its previous value plus a white noise error term. Typically, linear combinations of integrated process also are integrated. The residual from a regression of the two variables will be non-stationary. This violates the classical conditions for a linear regression. Such a regression is known as a spurious regression (Granger and Newbold, 1974). However, if a group of integrated variables share a common stochastic trend the linear combination will be non-integrated. This phenomenon - the elimination of a stochastic trend by an appropriate linear function - is known as cointegration (Engle and Granger, 1987). If two variables share a common trend, there will be Granger causality in one or more directions between them (Cuthbertson et al., 1992). Cointegration tests themselves cannot establish the direction of causality but tests can be applied to cointegrating VARs such as those estimated using the Johansen procedure (Johansen and Juselius, 1990).

An advantage of cointegration analysis is that if any integrated variables are omitted from the cointegrating relationship, which should be included in it, then the remaining variables will fail to cointegrate. Thus, if we can reject the null of non-causality in a cointegrated model, we can be more confident that this is not a spurious causality due to omitted variables.

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