Asymptotic distribution of the sup-Wald statistic under specification errors

Abstract

Consider a simple structural break model where y t = α 1+ β 1 f( x t )+ u t for t≤ k 0 and y t = α 2+ β 2 f( x t )+ u t for t> k 0. The timing of break and the structural parameters are unknown. Suppose the true functional form of the regressor f(·) is misspecified as g(·). We do not place too many restrictions on the functional forms of f(·) and g(·). A frequently encountered example in economics is that the true model is measured in level, but we estimate a log-linear model, i.e. when f( x t )= x t and g( x t )=log( x t ) For any f(·) and g(·), we derive a nonstandard limiting null distribution of the sup-Wald test statistic under some very general regularity conditions. Monte Carlo simulations support our findings.