The Wald Test of Common Factors in Spatial Model Specification Search Strategies

Sebastian Juhl

doi:10.1017/pan.2020.23

Abstract

AbstractDistinguishing substantively meaningful spillover effects from correlated residuals is of great importance in cross-sectional studies. Both forms of spatial dependence not only hold different implications for the choice of an unbiased estimator but also for the validity of inferences. To guide model specification, different empirical strategies involve the estimation of an unrestricted spatial Durbin model and subsequently use the Wald test to scrutinize the nonlinear restriction of common factors implied by pure error dependence. However, the Wald test’s sensitivity to algebraically equivalent formulations of the null hypothesis receives scant attention in the context of cross-sectional analyses. This article shows analytically that the noninvariance of the Wald test to such reparameterizations stems from the application of a Taylor series expansion to approximate the restriction’s sampling distribution. While asymptotically valid, Monte Carlo simulations reveal that alternative formulations of the common factor restriction frequently produce conflicting conclusions in finite samples. An empirical example illustrates the substantive implications of this problem. Consequently, researchers should either base inferences on bootstrap critical values for the Wald statistic or use the likelihood ratio test which is invariant to such reparameterizations when deciding on the model specification that adequately reflects the spatial process generating the data.

Highlights

Many empirical specification search procedures rely on the Wald test to assess the nonlinear common factor restriction, the test’s lack of invariance to algebraically equivalent formulations of the null hypothesis poses a serious problem for the accuracy of inferences
This study investigates the consequences of the Wald test’s sensitivity to alternative and algebraically equivalent expressions of the common factor hypothesis for its ability to guide the empirical model specification search
By presenting analytical evidence and using Monte Carlo simulations as well as an empirical example, it shows that the necessity to approximate the sampling variability of a nonlinear function by a Taylor series expansion causes the Wald test’s sensitivity to algebraically equivalent reparameterizations of the null hypothesis

Summary

Motivation

The correct specification of the inherently unknown spatial process generating observable patterns of interrelatedness among the units of analysis constitutes a considerable challenge in crosssectional studies. A violation of the common factor restriction is indicative of the existence of indirect spillover effects that need to be included in the systematic part of the regression model As this discussion suggests, the spatially lagged exogenous variables in the SDM model specification can be understood as instruments for omitted variables that are correlated with included regressors (e.g., Elhorst b, ). Phillips and Park ( ), for example, show that an Edgeworth expansion of the Wald statistic provides additional information on the statistic’s distribution which can be used to obtain corrected critical values and modified test statistics for each functional representation of the null hypothesis (e.g., de Paula and Cribari-Neto ; King and Goh ) Besides these corrections, simulation techniques allow researchers to generate the empirical distribution under the null hypothesis for each specification of the common factor restriction and base inferences on these reference distributions (e.g., Lafontaine and White ; Goh and King ). The Wald test based on H0(I I ) remains valid as its asymptotic distribution is obtained under the null hypothesis which precludes the problematic value (Gregory and Veall )

Conclusion

Findings

Data Availability Statement