Parameter stability and semiparametric inference in time varying auto-regressive conditional heteroscedasticity models

We develop a complete methodology for detecting time varying or non-time-varying parameters in auto-regressive conditional heteroscedasticity (ARCH) processes. For this, we estimate and test various semiparametric versions of time varying ARCH models which include two well-known non-stationary ARCH-type models introduced in the econometrics literature. Using kernel estimation, we show that non-time-varying parameters can be estimated at the usual parametric rate of convergence and, for Gaussian noise, we construct estimates that are asymptotically efficient in a semiparametric sense. Then we introduce two statistical tests which can be used for detecting non-time-varying parameters or for testing the second-order dynamics. An information criterion for selecting the number of lags is also provided. We illustrate our methodology with several real data sets.