A Hausman specification test based on root-\(N\)-consistent semiparametric estimators (Q1801817)

\textit{P. M. Robinson} [Econometrica 56, No. 4, 931-954 (1988; Zbl 0647.62100)] considered the model \(y_ i=z_ i\beta+\theta(x_ i)+u_ i\), where the functional form \(\theta(x_ i)\) is unknown to the researcher and he derived a \(\sqrt N\)-consistent semiparametric estimator of \(\beta\). Recently, the authors [Working Paper No. 8, Dpt. Econ., Univ. Guelph/Canada (1992)] considered estimation of a semiparametric dynamic panel data model \(y_{it}=\lambda y_{i,t- 1}+\theta(x_{it})+u_{it}\). Under the assumption that \(u_{it}\) is serially correlated, they used instrumental variable estimation to obtain a \(\sqrt N\)-consistent semiparametric estimator of \(\lambda\). In section 2 of this paper, the authors generalize Robinson's result to the case that \(E(u_ i| z_ i)\neq 0\) and obtain an instrumental variable \(\sqrt N\)-consistent estimator of \(\beta\). Then they propose a Hausman type specification test [\textit{J. A. Hausman}, Econometrika 46, 1251-1271 (1978; Zbl 0397.62043)] for testing the null hypothesis of \(E(u_ i| z_ i)=0\). Section 3 applies this Hausman test to the dynamic panel data model. The authors carry out some Monte Carlo experiments to study the finite sample properties of the proposed test statistic and conclude the paper in section 4.