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On stationarity and ergodicity of the bilinear model with applications to GARCH models. (English) Zbl 1224.62063
The author deals with the process \(\{y_t,\;t=1,2,\dots\}\) which is a solution of the so-called subdiagonal bilinear model introduced by C. W. J. Granger and A. P. Andersen [Angewandte Statistik und Ökonometrie. Heft 8. Göttingen: Vandenhoeck & Ruprecht. 94 S. (1978; Zbl 0379.62074)]: \[ y_t=a_0+\sum_{i=1}^{p}a_iy_{t-i}+ \sum_{i=1}^{q}b_i\varepsilon_{t-i}+\sum_{i=1}^{Q}\sum_{j=1}^{P}c_{i,j}y_{t-i-j}\varepsilon_{t-i}, \] where \(\{\varepsilon_{t}\}\) are i.i.d. random variables. This bilinear time series can be represented as \(y_t = Z_{1,t-1}+b_0\varepsilon_{t}\), where the process \(Z_t = (Z_{1,t},\dots, Z_{n,T})'\in\mathbb R^n\), \(n =\max\{p,P + q,P + Q\}\), solves the random coefficient autoregressive (RCA) equation \(Z_t=A_tZ_{t-1}+B_t\), where the random matrix \(A_t\) is of the form \(A_t=A_0+A_1\varepsilon_t\) and the random vector \(B_t=B_0+B_1\varepsilon_t+B_2\varepsilon^2_t\). For a sequence of i.i.d. matrices \(\{A_t\}\) with \(E[\log^+(| | A_t| | )]<\infty\) the associated Lyapunov exponent \(\gamma\) is defined by \(\gamma=\lim_{t\to\infty}t^{-1}E[\log\| A_t\cdots A_1\| ]\). Under the condition that the errors \(\{\varepsilon_{t}\}\) are i.i.d. random variables with \(E[\log^+(\varepsilon^2_{t})]<\infty\), the author proved that \(\gamma<0\) is a sufficient condition for the above bilinear model to have a unique stationary solution. If, furthermore, the model is irreducible, see P. Bougerol and N. Picard, Ann. Probab. 20, No. 4, 1714–1730 (1992; Zbl 0763.60015), then \(\gamma<0\) is a necessary condition for stationarity. For two special subcases of the general subdiagonal bilinear model precise necessary and sufficient conditions for stationarity are given. These results are used to give necessary and sufficient conditions for the existence of stationary solutions for two classes of GARCH\((p,q)\) models that can be written in the form of a bilinear time-series model that in turn can be written as an RCA model. These two classes include, among others, the linear GARCH model [R.F. Engle, Econometrica 50, 987–1007 (1982; Zbl 0491.62099); T. Bollerslev, J. Econom. 31, 307–327 (1986; Zbl 0616.62119)], Power GARCH [R. F. Engle and T. Bollerslev, Econ. Rev. 5, 1–50 (1986; Zbl 0619.62105)], LGARCH [P. M. Robinson, J. Econ. 47, No. 1, 67–84 (1991; Zbl 0734.62070)], Log- GARCH, EGARCH [D. B. Nelson, Econometrica 59, No. 2, 347–370 (1991; Zbl 0722.62069)], Asymmetric/GJR GARCH, T-GARCH [S. Taylor, Modelling financial time series. Chichester: Wiley (1986; Zbl 1130.91345); J.-M. Zakoïan, J. Econ. Dyn. Control 18, No. 5, 931–955 (1994; Zbl 0875.90197)]. Simple conditions are given for each of these specifications to have stationary solutions. Since these conditions are not only sufficient but also necessary, they are the weakest possible.

MSC:
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
60G10 Stationary stochastic processes
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