Elsevier

Economic Modelling

Volume 29, Issue 4, July 2012, Pages 1450-1460
Economic Modelling

Testing for Granger non-causality in heterogeneous panels

https://doi.org/10.1016/j.econmod.2012.02.014Get rights and content

Abstract

This paper proposes a very simple test of Granger (1969) non-causality for heterogeneous panel data models. Our test statistic is based on the individual Wald statistics of Granger non causality averaged across the cross-section units. First, this statistic is shown to converge sequentially to a standard normal distribution. Second, the semi-asymptotic distribution of the average statistic is characterized for a fixed T sample. A standardized statistic based on an approximation of the moments of Wald statistics is hence proposed. Third, Monte Carlo experiments show that our standardized panel statistics have very good small sample properties, even in the presence of cross-sectional dependence.

Highlights

► We consider a heterogeneous panel data model. ► We propose a group mean Wald statistic. ► We derive the asymptotic and semi asymptotic distribution. ► We propose a second standardized average Wald statistic in short T sample. ► This approach does not require simulation.

Introduction

The aim of this paper is to propose a simple Granger (1969) non causality test in heterogeneous panel data models with fixed (as opposed to time-varying) coefficients. In the framework of a linear autoregressive data generating process, the extension of standard causality tests to panel data implies testing cross sectional linear restrictions on the coefficients of the model. As usual, the use of cross-sectional information may extend the information set on causality from a given variable to another. Indeed, in many economic matters it is highly probable that if a causal relationship exists for a country or an individual, it also exists for some other countries or individuals. In this case, the causality can be more efficiently tested in a panel context with NT observations. However, the use of cross-sectional information involves taking into account the heterogeneity across individuals in the definition of the causal relationship. As discussed in Granger (2003), the usual causality test in panel asks “if some variable, say Xt causes another variable, say Yt, everywhere in the panel [..]. This is rather a strong null hypothesis.” Consequently, we propose here a simple Granger non causality test for heterogeneous panel data models. This test allows us to take into account both dimensions of the heterogeneity present in this context: the heterogeneity of the causal relationships and the heterogeneity of the regression model used so as to test for Granger causality.

Let us consider the standard implication of Granger causality.1 For each individual, we say that variable x causes y if we are able to better predict y using all available information than in the case where the information set used does not include x (Granger, 1969). If x and y are observed on N individuals, gaging the presence of causality comes down to determining the optimal information set used to forecast y. Several solutions can be adopted. The most general one consists in testing the causality from variable x observed for the ith individual to the variable y observed for the jth individual, with j = i or j  i. The second solution is more restrictive and derives directly from the time series analysis. It implies testing the causal relationship for a given individual. The cross-sectional information is then used only to improve the specification of the model and the power of tests as in Holtz-Eakin et al. (1988). The baseline idea is to assume that there exists a minimal statistical representation which is common to x and y at least for a subgroup of individuals. In this paper we use such a model. In this case, causality tests can be implemented and considered as a natural extension of the standard time series tests in the cross-sectional dimension.

However, one of the main issues specific to panel data models refers to the specification of the heterogeneity between cross-section units. In this Granger causality context, the heterogeneity has two main dimensions. We hence distinguish between the heterogeneity of the regression model and that of the causal relationship from x to y. Indeed, the model considered may be different from an individual to another, whereas there is a causal relationship from x to y for all individuals. The simplest form of regression model heterogeneity takes the form of slope parameters' heterogeneity. More precisely, in a p order linear vectorial autoregressive model, we define four kinds of causal relationships. The first one, denoted Homogeneous Non Causality (HNC) hypothesis, implies that no individual causality relationship from x to y exists. The symmetric case is the Homogeneous Causality (HC) hypothesis, which occurs when N causality relationships exist, and when the individual predictors of y obtained conditionally on the past values of y and x are identical. The dynamics of y is then absolutely identical for all the individuals in the sample. The last two cases correspond to heterogeneous processes. Under the HEterogeneous Causality (HEC) hypothesis, we assume that N causality relationships exist, as in the HC case, but the dynamics of y is heterogeneous. Note, however, that the heterogeneity does not affect the causality result. Finally, under the HEterogeneous Non Causality (HENC) hypothesis, we assume that there is a causal relationship from x to y for a subgroup of individuals. Symmetrically, there is at least one and at most N-1 non causal relationships in the model. It is clear that in this case the heterogeneity deals with causality from x to y.

To sum up, under the HNC hypothesis, no individual causality from x to y occurs. On the contrary, in the HC and HEC cases, there is a causality relationship for each individual of the sample. To be more precise, in the HC case, the same regression model is valid (identical parameters' estimators) for all individuals, whereas this is not the case for the HEC hypothesis. Finally, under the HENC hypothesis, the causality relationship is heterogeneous since the variable x causes y only for a subgroup of N  N1 units.

In this context, we propose a simple test of the Homogeneous Non Causality (HNC) hypothesis. Under the null hypothesis, there is no causal relationship for any of the units of the panel. Our contribution is three-fold. First, we specify the alternative as the HENC hypothesis. To put it differently, we do not test the HNC hypothesis against the HC hypothesis as Holtz-Eakin et al. (1988), which, as previously discussed, is a strong assumption. Indeed, we allow for two subgroups of cross-section units: the first one is characterized by causal relationships from x to y, but it does not necessarily rely on the same regression model, whereas there is no causal relationship from x to y in the case of the second subgroup. Second, we consider a heterogeneous panel data model with fixed coefficients (in time). It follows that both under the null and the alternative hypothesis the unconstrained parameters may be different from one individual to another. The dynamics of the variables may be thus heterogeneous across the cross-section units, regardless of the existence (or not) of causal relationships. Our framework hence relies on less strong assumptions than the ones in Holtz-Eakin et al. (1988), who assume the homogeneity of cross-section units, i.e. that the panel vector-autoregressive regression model is valid for all the individuals in the panel. Third, we adapt the Granger causality test-statistic to the case of unbalanced panels and/or different lag orders in the autoregressive process. Most importantly, we propose a block bootstrap procedure to correct the empirical critical values of panel Granger causality tests so as to account for cross-sectional dependence. To our knowledge, these issues have not been tackled before in this context.

Following the literature devoted to panel unit root tests in heterogeneous panels, and particularly Im et al. (2003), we propose a test statistic based on averaging standard individual Wald statistics of Granger non causality tests.2 Under the assumption of cross-section independence (as used in first generation panel unit root tests), we provide different results. First, this statistic is shown to converge sequentially in distribution to a standard normal variate when the time dimension T tends to infinity, followed by the individual dimension N. Second, for a fixed T sample the semi-asymptotic distribution of the average statistic is characterized. In this case, individual Wald statistics do not have a standard chi-squared distribution. However, under very general setting, it is shown that individual Wald statistics are independently distributed with finite second order moments. For a fixed T, the Lyapunov central limit theorem is sufficient to establish the distribution of the standardized average Wald statistic when N tends to infinity. The first two moments of this normal semi-asymptotic distribution correspond to the empirical mean of the corresponding theoretical moments of the individual Wald statistics. The issue is then to propose an evaluation of the first two moments of standard Wald statistics for small T samples. A first solution relies on Monte-Carlo or Bootstrap simulations. A second one consists in using an approximation of these moments based on the exact moments of the ratio of quadratic forms in normal variables derived from Magnus (1986) theorem for a fixed T sample, with T > 5 + 2K. Given these approximations, we propose a second standardized average Wald statistic to test the HNC hypothesis in short T sample. Then, contrary to Kónya (2006), our testing procedure does not require bootstrap critical values generated by simulations. However, a block bootstrap simulation approach similar to theirs is adapted to our framework (group mean Wald-statistic) so as to take into account cross-sectional dependencies.

The finite sample properties of our test statistics are examined using Monte-Carlo methods. The simulation results clearly show that our panel based tests have very good properties even in samples with very small values of T and N. The size of our standardized statistic based on the semi-asymptotic moments is reasonably close to the nominal size for all the values of T and N considered. Besides, the power of our panel test statistic substantially exceeds that of Granger non Causality tests based on single time series in all experiments and in particular for very small values of T, e.g. T = 10, provided that there are at least a few cross-section units in the panel (e.g. N = 5). Furthermore, approximated critical values are proposed for finite T and N samples, as well as a block-bootstrap procedure to compute empirical critical values when taking into account cross-section dependence.

The rest of the paper is organized as follows. Section 2 is devoted to the definition of the Granger causality test in heterogeneous panel data models. Section 3 sets out the asymptotic distribution of the average Wald statistic. Section 4 derives the semi-asymptotic distribution for fixed T sample and Section 5 presents the main results obtained from Monte Carlo experiments. Section 6 extends the results to a fixed N sample and discusses the case with cross-sectional dependence as well as the unbalanced panel framework. The last section provides some concluding remarks.

Section snippets

A non causality test in heterogeneous panel data models

Let us denote by x and y, two stationary variables observed for N individuals on T periods. For each individual i = 1,.., N, at time t = 1,.., T, we consider the following linear model:yi,t=αi+k=1Kγikyi,tk+k=1Kβikxi,tk+εi,twith KN and βi = (βi(1), …, βi(K))′. For simplicity, the individual effects αi are supposed to be fixed in the time dimension. Initial conditions (yi,  K, …, yi, 0) and (xi,  K, …, xi, 0) of both individual processes yi, t and xi, t are given and observable. We assume that lag orders K are

Asymptotic distribution

We propose to derive the asymptotic distribution of the average statistic WN, THnc under the null hypothesis of non causality. For that, we consider the case of a sequential convergence, i.e. when T tends to infinity and then N tends to infinity. This sequential convergence result can be deduced from the standard convergence result of the individual Wald statistic Wi, T in a large T sample. In a non dynamic model, the normality Assumption in A1 would be sufficient to establish the fact for all T,

Fixed T samples and semi-asymptotic distributions

Asymptotically, individual Wald statistics Wi, T converge toward an identical chi-squared distribution for each i = 1,.., N,. Nonetheless, this convergence result cannot be generalized to any time dimension T, even if we assume the normality of residuals. We then seek to show that, for a fixed T dimension, individual Wald statistics have finite second order moments even if they do not have the same distribution and this distribution is not a standard one.

Let us consider the expression (5) of Wi, T

Monte Carlo simulation results

In this section, we propose three sets of Monte Carlo experiments to examine the finite sample properties of the alternative panel-based non causality tests. The first set focuses on the benchmark model5:yi,t=α+γiyi,kk+βixi,tk+εi,t

The parameters of the model are calibrated as follows. The auto-regressive

Fixed T and fixed N distributions

If N and T are fixed, the standardized statistic ZNHnc and the average statistic WN, THnc do not converge to standard distributions under the HNC hypothesis. Two solutions are then envisageable: the first consists in using the mean Wald statistic WN, THnc and to compute the exact empirical critical values, denoted cN, T(α), for the corresponding sizes N and T via stochastic simulations. The upper panel in Table 4 reports the results of an example of such a simulation. As in Im et al. (2003), the

Conclusion

In this paper, we propose a simple Granger (1969) non-causality test for heterogeneous panel data models. Under the null hypothesis of Homogeneous Non Causality (HNC), there is no causal relationship for any of the cross-section units of the panel. Under the alternative, there are two subgroups of cross-section units: one characterized by causal relationships from x to y (even though the regression model is not necessarily the same) and another subgroup for which there is no causal relationship

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A substantial part of the work for this paper was undertaken in the Department of Economics of the University Paris IX Dauphine, EURIsCO. I am grateful for the comments received from the participants of the econometric seminars at University Paris I, Paris X Nanterre, Orléans, Aix-Marseille, Marne-la-Vallée, Maastricht University and University of Geneva, EC 2.

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