Testing weak exogeneity in cointegrated panels

Abstract

For reasons of empirical tractability, analysis of cointegrated economic time series is often developed in a partial setting, in which a subset of variables is explicitly modelled conditional on the rest. This approach yields valid inference only if the conditioning variables are weakly exogenous for the parameters of interest. This article proposes a new test of weak exogeneity in panel cointegration models. The test has a limiting Gumbel distribution that is obtained by first letting $T\to \mathrm{\infty }$ and then letting $N\to \mathrm{\infty }$. We evaluate the accuracy of the asymptotic approximation in finite samples via simulation experiments. Finally, as an empirical illustration, we test weak exogeneity of disposable income and wealth in aggregate consumption.

Keywords:

• panel data
• cointegration
• weak exogeneity
• Monte Carlo methods

JEL Classification:

• C23
• C32

Acknowledgement

We thank A. Ludwig for kindly providing us with the data.

Notes

¹ See also Dolado (1992), Ericsson (1995) and Hendry (1995).

² See, e.g., Gemmell et al. (2012) and Acosta-Ormaechea and Yoo (2012) for recent examples of this approach.

³ Canning and Pedroni (2008) examined alternative panel testing strategies in a related setting, in which the concern is what they call ‘long-run causality’ rather than the validity of conditional inference.

⁴ The conventional joint Wald test can be expected to exhibit much worse behaviour in terms of both power and size in the types of samples considered here. For example, in a related panel setting Phillips and Sul (2003) found the Wald test for homogeneity highly unreliable owing to its unacceptable size distortions, which remain severe even with fairly large values of T. Furthermore, in a large N setting, the Wald test is not consistent because the number of restrictions to be tested increases with the sample size. However, it is also true that when the time dimension of the panel greatly exceeds its cross-sectional dimension, the testing strategy based on the joint Wald test could allow better accommodating flexible forms of cross-unit dependence, for example through SURE estimation of the parameters of the conditional model allowing for an unrestricted covariance matrix.

⁵ To avoid notational clutter, the model omits unit-specific constants and time-trends.

⁶ Larsson and Lyhagen (2000) discussed in detail how to test for a common cointegrating rank across units in the panel.

⁷ Note that the formulation in Equation 2 rules out cointegration across units in the panel and excludes also the possibility that deviations from the long-run equilibrium in a given unit could impact the behaviour of other units.

⁸ In this particular case, these estimates coincide with equation-by-equation estimates because right-hand-side variables are the same in all equations and there are no cross-equation restrictions.

⁹ The Westerlund and Hess (2011) test is based on the maximum across panel units of their individual Hausman test statistics for the null that their respective cointegrating vector parameters equal those of all the other cross-section units.

¹⁰ Strictly speaking, if the common factors are I(1), the CCE approach in Pesaran (2006), whose use is suggested below in the text, would still remove the common components (Kapetanios et al., 2011) and allow performing the panel weak exogeneity test. However, such a procedure would not take proper account of the possible presence of cross-unit cointegration.

¹¹ In a recent paper, Gengenbach et al. (2013) derived a Granger-type representation theorem, given the triangular representation of a panel cointegration model as in Equations 14 and 15 but including common factors in both the long-run relationship and the conditioning variables. In particular, they show that such a panel cointegration model also admits an error-correction representation. It can thus be shown that a marginal model for $\mathrm{\Delta }{z}_{it}$ as in Equation 4 can be derived from the error-correction representation with common factors entering in both the long-run relationship and the short-run dynamics (see Remark 2 in Gengenbach et al., 2013).

¹² An alternative approach could be the pooled mean group – PMG – estimator discussed by Pesaran et al. (1999). The PMG estimator can accommodate homogeneous long-run coefficients together with heterogeneous speed of adjustment and short-run coefficients.

¹³ In practical terms, this means that the subsequent estimation of ${\gamma }_i$ and the construction of the Wald statistics Equation 8 employ a generated regressor. In the time-series context (i.e., N = 1), this should be of little consequence for the behaviour of the variable-addition test, owing to the super-consistency of ${\hat{\theta }}_i$ with respect to T; see, e.g., Boswijk and Urbain (1997). In the panel context, however, the situation is somewhat less clear cut, because $\hat{\theta }$ based on the pooled approach converges at just the standard rate $\sqrt{N}$ with respect to the cross-sectional sample size. This might raise concerns about the performance of our testing procedure unless T is very large. Nevertheless, the results shown in the table reveal that the performance is little affected by the use of the pooled estimator of the long-run relationship.

¹⁴ Furthermore, closer comparison of panels A and B of Table 1 reveals that the decline in power by using $\hat{\theta }$ instead of $\theta$ is inversely related to $T/N$, as should be expected in the light of the theoretical argument mentioned in the previous footnote.

¹⁵ These results are not reported to save space, but they are available upon request.

¹⁶ Hence we do not test other aspects of their specification, such as the fact that cross-sectional dependence is not taken into account.

¹⁷ To avoid notational clutter, we present here an ARDL(1,1,1) version of the model. However, in practice, we allow for country-specific dynamics, with lag length determined by the Schwarz criterion.