On the Origins of Conditional Heteroscedasticity in Time Series
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The volatility clustering frequently observed in financial/economic time series is often ascribed to GARCH and/or stochastic volatility models. This paper demonstrates the usefulness of reconceptualizing the usual definition of conditional heteroscedasticity as the (h = 1) special case of h-step-ahead conditional heteroscedasticity, where the conditional volatility in period t depends on observable variables up through period t – h. Here it is shown that, for h > 1, h-stepahead conditional heteroscedasticity arises – necessarily and endogenously - from nonlinear serial dependence in a time series; whereas one-step-ahead conditional heteroscedasticity (i.e., h = 1) requires multiple and heterogeneously-skedastic innovation terms. Consequently, the best response to observed volatility clustering may often be to model the nonlinear serial dependence which is likely causing it, rather than ‘tacking on’ an ad hoc volatility model. Even where such nonlinear modeling is infeasible – or where volatility is quantified using, say, a model-free implied volatility measure rather than squared returns – these results suggest a re-consideration of the usefulness of lag-one terms in volatility models. An application to observed daily stock returns is given.
Abstract
The volatility clustering frequently observed in financial/economic time series is often ascribed to GARCH and/or stochastic volatility models. This paper demonstrates the usefulness of reconceptualizing the usual definition of conditional heteroscedasticity as the (h = 1) special case of h-step-ahead conditional heteroscedasticity, where the conditional volatility in period t depends on observable variables up through period t – h. Here it is shown that, for h > 1, h-stepahead conditional heteroscedasticity arises – necessarily and endogenously - from nonlinear serial dependence in a time series; whereas one-step-ahead conditional heteroscedasticity (i.e., h = 1) requires multiple and heterogeneously-skedastic innovation terms. Consequently, the best response to observed volatility clustering may often be to model the nonlinear serial dependence which is likely causing it, rather than ‘tacking on’ an ad hoc volatility model. Even where such nonlinear modeling is infeasible – or where volatility is quantified using, say, a model-free implied volatility measure rather than squared returns – these results suggest a re-consideration of the usefulness of lag-one terms in volatility models. An application to observed daily stock returns is given.
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