Score test of fit for composite hypothesis in the GARCH\((1,1)\) model. (English) Zbl 1375.62004
Summary: A score test of fit for testing the conditional distribution of the stationary GARCH\((1,1)\) model conceived by T. Bollerslev [J. Econom. 31, 307–327 (1986; Zbl 0616.62119)] is proposed. The null hypothesis asserting that the noise distribution belongs to the specified parametric class of distributions is considered.
Exploiting the pioneer idea of J. Neyman [Skand. Aktuarie Tidskr. 20, 149–199 (1937; Zbl 0018.03403, JFM 63.1092.02)] and the device proposed by T. Ledwina [J. Am. Stat. Assoc. 89, No. 427, 1000–1005 (1994; Zbl 0805.62022)], the efficient score statistic and its data-driven version are derived for this testing problem. The asymptotic null distribution of the score statistic is established. Replacing the nuisance parameters with their square-root consistent estimators results in the data-driven test statistic. It is proved that in that case the asymptotic behaviour of the test statistic remains unchanged under appropriate regularity conditions and under discretization of the estimators. Computer simulations of the critical value and the power performance of the test for several alternatives in the case of generalized error distribution family serving as a null distribution are also presented.
Exploiting the pioneer idea of J. Neyman [Skand. Aktuarie Tidskr. 20, 149–199 (1937; Zbl 0018.03403, JFM 63.1092.02)] and the device proposed by T. Ledwina [J. Am. Stat. Assoc. 89, No. 427, 1000–1005 (1994; Zbl 0805.62022)], the efficient score statistic and its data-driven version are derived for this testing problem. The asymptotic null distribution of the score statistic is established. Replacing the nuisance parameters with their square-root consistent estimators results in the data-driven test statistic. It is proved that in that case the asymptotic behaviour of the test statistic remains unchanged under appropriate regularity conditions and under discretization of the estimators. Computer simulations of the critical value and the power performance of the test for several alternatives in the case of generalized error distribution family serving as a null distribution are also presented.
MSC:
| 62G10 | Nonparametric hypothesis testing |
| 62M10 | Time series, auto-correlation, regression, etc. in statistics (GARCH) |
| 62E20 | Asymptotic distribution theory in statistics |
| 65C05 | Monte Carlo methods |
| 37M10 | Time series analysis of dynamical systems |
Keywords:
GARCH(1,1) model; conditional distribution; efficient score vector; data-driven test of fit; square-root consistent estimator; martingale difference array; central limit theorem; Monte Carlo simulations; GED familyReferences:
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