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On residual empirical processes of GARCH-SM models: application to conditional symmetry tests. (English) Zbl 1199.62011

One of the earliest models where the conditional variance depends upon the past realizations is the so-called autoregressive conditional heteroskedastic (ARCH) model introduced by R.F. Engle [Econometrica 50, 987–1007 (1982; Zbl 0491.62099)] to model the changing volatility of economic time series. For more details see the review article of T. Bollerslev, R.Y. Chou and K.F. Kroner [J. Econom. 52, No. 1-2, 5–59 (1992; Zbl 0825.90057)], the comprehensive monograph of R.F. Engle [ARCH, Selected readings. NY: Oxford Univ. Press (1995)] and the references therein. The ARCH model has been extended in a number of directions. The most important of these is the extension to include moving-average parts, namely the generalized ARCH (GARCH) model introduced by T. Bollerslev [J. Econ. 31, 307–327 (1986; Zbl 0616.62119)]. More recently, L. Giraitis, P. Kokoszka and R. Leipus [Econom. Theory 16, No. 1, 3–22 (2000; Zbl 0986.60030)] discussed the \(ARCH(\infty)\) class, which includes the ARCH and GARCH models as special cases. They established sufficient conditions for the existence of a stationary solution and gave its explicit representation. Based on L. Le Cam [Asymptotic methods in statistical decision theory. N.Y.: Springer (1986; Zbl 0605.62002)], S. Lee and M. Taniguchi [Stat. Sin. 15, No. 1, 215–234 (2005; Zbl 1059.62014)] established the local asymptotic normality (LAN) and studied the residual empirical process (REP) for the \(ARCH(\infty)\) class with stochastic mean \([ARCH(\infty)-SM]\). As an application of the results, they constructed an asymptotically optimal test for the \([ARCH(\infty)-SM]\) models. W. Stute [Test 10, No. 2, 393–403 (2001; Zbl 1109.62351)] discussed and derived consistency and distributional convergence results for functions of the residuals of ARCH(p)-models. His strong representation result is then used to build symmetry test statistics. In this article, his results are extended to the GARCH framework.
The authors consider the GARCH-SM model (also known as the AR-GARCH process) defined by the sequence \((Y_t)_{t\in\mathbb Z}\) as \[ Y_t=\theta Y_{t-1}+X_t,\quad X_t=h_t\varepsilon_t, h_t^2=\omega+\alpha X_{t-1}^2+\beta h^2_{t-1}, \] where the innovations \(\varepsilon_t\) are i.i.d. with mean zero and variance one and continuous distribution function \(F\). Moreover, the \(\varepsilon_t\) are independent of the \(X_{t-1},X_{t-2},\dots\). The \(Y_t\) are observations and \(X_t=h_t\varepsilon_t\) is an unobserved GARCH(1,1) sequence. The authors establish the consistency and limiting functional law of the residual empirical process from a GARCH(1,1)-SM model. The results are then applied to derive a symmetry test for GARCH-SM models. Simulations are given to show the asymptotic behaviour and normality of the test statistic.

MSC:

62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62G10 Nonparametric hypothesis testing
62G20 Asymptotic properties of nonparametric inference
65C60 Computational problems in statistics (MSC2010)
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References:

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