Multiplier bootstrap specification tests for conditional variance functions in heteroskedastic regression models

In this paper, the problem of testing the parametric forms of the conditional variance functions in heteroskedastic regression models is studied. Two residual-marked empirical processes are used to construct test statistics, and two orthogonal projections onto the corresponding tangent spaces of nuisance parameters are proposed to eliminate the “parameter estimation effect” often observed in the literature but challenging to address. The asymptotic properties of the test statistics under the null, the alternative, and a sequence of local alternatives converging to the null at the parametric rate are established. Furthermore, a computationally efficient multiplier bootstrap procedure is proposed to obtain the critical values, and its asymptotic validity is formally justified. The favorable finite-sample performance confirms several superior properties and helps select the weighting function for constructing the residual-marked empirical process. Finally, the broad applicability of the proposed methodology is highlighted, and the heteroskedasticity test is presented as a special example.