Filtering and forecasting with misspecified ARCH models. II: Making the right forecast with the wrong model.

*(English)*Zbl 0820.62098Summary: In part I of this paper, ibid. 52, No. 1/2, 61-90 (1992; Zbl 0761.62169), the first author showed that in data observed at high frequencies, an ARCH model may perform well in estimating the conditional variance of a process, even when the ARCH model is severely misspecified. While such models may perform reasonably well at filtering (i.e., at estimating unobserved instantaneous conditional variances), they may perform disastrously at medium- and long-term forecasting of the process and its volatility.

In this paper, we develop conditions under which a misspecified ARCH model sucessfully performs both tasks, filtering and forecasting. The key requirement (in addition to the conditions for consistent filtering) is that the ARCH model correctly specifies the functional form of the first two conditional moments of all state variables. We apply these results to a diffusion model employed in the options pricing literature, the stochastic volatility model of J. Hull and A. White [J. Finance 42, 281-300 (1987)], L. Scott [J. Financial Quant. Anal. 22, 419-437 (1987)], and J. Wiggins [J. Financial Econ. 19, 351-372 (1987)].

In this paper, we develop conditions under which a misspecified ARCH model sucessfully performs both tasks, filtering and forecasting. The key requirement (in addition to the conditions for consistent filtering) is that the ARCH model correctly specifies the functional form of the first two conditional moments of all state variables. We apply these results to a diffusion model employed in the options pricing literature, the stochastic volatility model of J. Hull and A. White [J. Finance 42, 281-300 (1987)], L. Scott [J. Financial Quant. Anal. 22, 419-437 (1987)], and J. Wiggins [J. Financial Econ. 19, 351-372 (1987)].

##### MSC:

62P20 | Applications of statistics to economics |

62M20 | Inference from stochastic processes and prediction |

91B84 | Economic time series analysis |

##### Keywords:

nonlinear filtering; smoothing; data-generalizing processes; stationary distributions; misspecified ARCH model; forecasting; conditional moments; diffusion model; options pricing; stochastic volatility model
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\textit{D. B. Nelson} and \textit{D. P. Foster}, J. Econom. 67, No. 2, 303--335 (1995; Zbl 0820.62098)

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##### References:

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