One-step Semiparametric Estimation of the GARCH Model

In maximum likelihood estimation, the real but unknown innovation distribution is often replaced by a nonparametric estimate, and thus the estimation procedure becomes semiparametric. These semiparametric approaches generally involve two steps: the first step that incorporate an initial estimate of the model parameter to produce a residual sample, and the second step that uses the residuals to estimate the likelihood, which is subsequently maximized to obtain the final estimate of the model parameter. Therefore, the characteristics of the initial input estimator may be carried over to the final semiparametric estimator, and the performance of the semiparametric estimator will be impaired if the input estimate is deficient. In this article we have studied a one-step semiparametric estimator where no initial input is necessary. The estimation procedure is illustrated via a generalized autoregressive conditional heteroskedasticity (GARCH) model. Asymptotic properties of the estimator are established, and finite sample performance of the estimator is evaluated via simulation. The results suggest that the proposed one-step semiparametric estimator avoids significant drawbacks of its two-step counterparts.