This work is devoted to the study of the parameter test for the Generalized Autoregressive Conditional Heteroskedasticity (GARCH) model. Based on the daily GARCH model, using the parameter estimator obtained by intraday high-frequency data, the adjusted Likelihood Ratio test statistic and Wald test statistic are provided. Asymptotic distributions of the two adjusted test statistics are deducted and a way to select the optimal sampling frequency is also discussed. Simulation studies show that the proposed test statistics have better size and power than traditional ones (without using intraday high-frequency data). An empirical study is given to illustrate the potential applications of the proposed tests. The results show the idea of this article is of certain superiority and it can be extended to other GARCH type models.

Three common tests (Lagrange multiplier test, Likelihood ratio test and Wald test) are considered to test the GARCH effect of the volatility proxy model, proposed by Visser in 2011. Under reasonable assumptions, asymptotic distribution of the three test statistics and their asymptotic equivalence are established. The impact of different proxies to the test performance is also discussed. Numerous simulation studies are carried out to assess the performance of the three tests under different volatility proxies. An empirical study is given to show a potential application of the proposed tests.

The portmanteau test is an effective tool for testing the goodness of fit of models. Motivated by the fact that high-frequency data can improve the estimation accuracy of models, a modified portmanteau test using high-frequency data is proposed for ARCH-type models in this paper. Simulation results show that the empirical size and power of the modified test statistics of the model using high-frequency data are better than those of the daily model. Three stock indices (CSI 300, SSE 50, CSI 500) are taken as an example to illustrate the practical application of the test.