Adaptive inference for a semiparametric GARCH model
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This paper considers a semiparametric generalized autoregressive conditional heteroscedastic (S-GARCH) model, which has a smooth long run component with unknown form to depict time-varying parameters, and a GARCH-type short run component to capture the temporal dependence. For this S-GARCH model, we first estimate the time-varying long run component by the kernel estimator, and then estimate the non-time-varying parameters in short run component by the quasi maximum likelihood estimator (QMLE). We show that the QMLE is asymptotically normal with the usual parametric convergence rate. Next, we provide a consistent Bayesian information criterion for order selection. Furthermore, we construct a Lagrange multiplier (LM) test for linear parameter constraint and a portmanteau test for model diagnostic checking, and prove that both tests have the standard chi-squared limiting null distributions. Our entire statistical inference procedure not only works for the non-stationary data, but also has three novel features: first, our QMLE and two tests are adaptive to the unknown form of the long run component; second, our QMLE and two tests are easy-to-implement due to their related simple asymptotic variance expressions; third, our QMLE and two tests share the same efficiency and testing power as those in variance target method when the S-GARCH model is stationary.
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