A Consistent Heteroskedasticity Robust LM Type Specification Test for Semiparametric Models

This paper develops a consistent heteroskedasticity robust Lagrange Multiplier (LM) type specification test for semiparametric conditional mean models. Consistency is achieved by turning a conditional moment restriction into a growing number of unconditional moment restrictions using series methods. The proposed test statistic is straightforward to compute and is asymptotically standard normal under the null. Compared with the earlier literature on series-based specification tests in parametric models, I rely on the projection property of series estimators and derive a different normalization of the test statistic. Compared with the recent test in Gupta (2018), I use a different way of accounting for heteroskedasticity. I demonstrate using Monte Carlo studies that my test has superior finite sample performance compared with the existing tests. I apply the test to one of the semiparametric gasoline demand specifications from Yatchew and No (2001) and find no evidence against it.