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Asymptotic filtering theory for multivariate ARCH models. (English) Zbl 0845.62080
Summary: ARCH models are widely used to estimate conditional variances and covariances in financial time series models. How successfully can ARCH models carry out this estimation when they are misspecified? How can ARCH models be made robust to misspecification? D. B. Nelson and D. P. Foster [Econometrica 62, No. 1, 1-41 (1994; Zbl 0804.62085)] employed continuous record asymptotics to answer these questions in the univariate case. This paper considers the general multivariate case. Our results allow us, for example, to construct an asymptotically optimal ARCH model for estimating the conditional variance or conditional beta of a stock return given lagged returns on the stock, volume, market returns, implicit volatility from options contracts, and other relevant data. We also allow for time-varying shapes of conditional densities (e.g., ‘heteroskewticity’ and ‘heterokurticity’). Examples are provided.

MSC:
62P20 Applications of statistics to economics
62M20 Inference from stochastic processes and prediction
Software:
Mathematica; GAUSS
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