zbMATH — the first resource for mathematics

Asymptotically optimal smoothing with ARCH models. (English) Zbl 0906.62127
Summary: Suppose an observed time series is generated by a stochastic volatility model – i.e., there is an unobservable state variable controlling the volatility of the innovations in the series. As shown by the author [J. Econom. 52, No. 1/2, 61-90 (1992; Zbl 0761.62169)], and the author and D. P. Foster [Econometrica 62, No. 1, 1-41 (1994; Zbl 0804.62085)], a misspecified ARCH model will often be able to consistently (as a continuous time limit is approached) estimate the unobserved volatility process, using information in the lagged residuals. This paper shows how to more efficiently estimate such a voliatility process using information in both lagged and led residuals. In particular, this paper expands the optimal filtering results of the author and Foster (1994) and the author [Working Pap., Univ. Chicago, Grad. School Bus. (1994); see also J. Econom. 71, No. 1-2, 1-47 (1996; Zbl 0845.62080)] to smoothing and to filtering with a random initial condition.

62P20 Applications of statistics to economics
62M20 Inference from stochastic processes and prediction
91B84 Economic time series analysis
Full Text: DOI