Functional form misspecification in regressions with integrated time series

Abstract

This thesis addresses the issue of functional form misspecification in models with nonstationary covariates. In particular we assume that the variables of the model are unit root processes. First we examine the asymptotic behaviour of the least squares estimator, under functional form misspecification, in regression models like those analysed by Park and Philips (2001) in their recent paper in Econometrica. In contrast to the stationary case, we find that convergence to some pseudo-true value does not always hold. In some cases of the estimator diverges. Whenever the estimator converges the order of consistency is usually slower and the limit distribution theory different than the one under correct specification. Moreover a conditional moment test for functional form is considered within the theoretical framework of Park and Philips (2001). In contrast to the stationary case, under nonstationarity the test may be two-sided. The asymptotic power of the test is derived against a set of alternatives where each alternative is characterised by the asymptotic order of the true specification. Moreover it is shown that the use of integrable weighting functions in the construction of the test statistic improves asymptotic power against a set of alternatives. Next the test for functional form is extended to cointegrating relationships. Our framework allows for a fitted model that is possibly nonlinear in variables and in view of this the linear specifications commonly used in practice constitute a special case. The test is consistent under both functional form misspecification and no cointegration. So the functional form test can be also used as a cointegration test. In both cases the divergence rate attained equals n/M, with n and M being the sample size and the bandwidth used in the estimation of long-run covariance matrices respectively.

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