Exact predictive densities for linear models with ARCH disturbances.

*(English)*Zbl 0668.62080It is shown how exact predictive densities may be formed in the ARCH linear model by means of Monte Carlo integration with importance sampling. Several improvements in computational efficiency over earlier implementations of this procedure are developed, including use of the exact likelihood function rather than an asymptotic approximation to construct the importance sampling distribution, and antithetic acceleration of convergence. A numerical approach to the formulation of posterior odds ratios and the combination of non-nested models is also introduced.

These methods are applied to daily quotations of closing stock prices. Forecasts are formulated using linear models, ARCH linear models and an integrated model constructed from the posterior probabilities of the respective models. The use of the exact predictive density in a decision- theoretic context is illustrated by deriving the optimal day-to-day portfolio adjustments of a trader with constant relative risk aversion.

These methods are applied to daily quotations of closing stock prices. Forecasts are formulated using linear models, ARCH linear models and an integrated model constructed from the posterior probabilities of the respective models. The use of the exact predictive density in a decision- theoretic context is illustrated by deriving the optimal day-to-day portfolio adjustments of a trader with constant relative risk aversion.

##### MSC:

62P20 | Applications of statistics to economics |

62M20 | Inference from stochastic processes and prediction |

65C99 | Probabilistic methods, stochastic differential equations |

##### Keywords:

time series; exact predictive densities; ARCH linear model; Monte Carlo integration; exact likelihood function; importance sampling distribution; antithetic acceleration of convergence; posterior odds ratios; combination of non-nested models; linear models; portfolio adjustments; risk aversion
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##### References:

[1] | Box, G.; Jenkins, G.: Time series analysis, forecasting, and control. (1976) · Zbl 0363.62069 |

[2] | Diebold, F. X.; Nerlove, M.: The dynamics of exchange rate volatility: A multivariate latent factor ARCH model. (1986) · Zbl 1126.91365 |

[3] | Doan, T.; Litterman, R.; Sims, C. A.: Forecasting and conditional projection using realistic prior distributions. Econometric reviews 3, 1-100 (1984) · Zbl 0613.62142 |

[4] | Engle, R.: Autoregressive conditional heteroscedasticity with estimates of the variance of united kingdom inflation. Econometrica 50, 987-1008 (1982) · Zbl 0491.62099 |

[5] | Engle, R.: Estimates of the variance of U.S. Inflation based on the ARCH model. Journal of money, credit, and banking 15, 286-301 (1983) |

[6] | Engle, R.; Bollerslev, T.: Modeling the persistence of conditional variances. Econometric reviews 5, 1-50 (1986) · Zbl 0619.62105 |

[7] | Geweke, J.: Exact inference in the inequality constrained normal linear regression model. Journal of applied econometrics 1, 127-141 (1986) |

[8] | Geweke, J.: Exact inference in models with autoregressive conditional heteroscedasticity. Dynamic econometric modeling (1986) |

[9] | Geweke, J.: Bayesian inference in econometric models using Monte Carlo integration. (1986) · Zbl 0683.62068 |

[10] | Geweke, J.: The secular and cyclical behavior of real GDP in nineteen OECD countries. Journal of business and economic statistics (1986) |

[11] | Geweke, J.: Antithetic acceleration of Monte Carlo integration in Bayesian inference. Journal of econometrics 38, 73-90 (1987) · Zbl 0667.62079 |

[12] | Geweke, J.; Marshall, R. C.; Zarkin, G.: Exact inference for continuous time Markov chain models. Review of economic studies 53, 653-669 (1986) · Zbl 0595.62097 |

[13] | Hammersly, J. M.; Handscomb, D. C.: Monte Carlo methods. (1964) · Zbl 0121.35503 |

[14] | Kloek, T.; Van Dijk, H. K.: Bayesian estimates of equation system parameters: an application of integration by Monte Carlo. Econometrica 46, 1-20 (1978) · Zbl 0376.62014 |

[15] | Thompson, P. A.; Miller, R. B.: Bayesian analysis of univariate time series: forecasting via simulation. (1984) |

[16] | Thompson, P. A.; Miller, R. B.: Bayesian analysis of univariate time series: extensions of the basic procedure. (1985) |

[17] | Zellner, A.: An introduction to Bayesian inference in econometrics. (1971) · Zbl 0246.62098 |

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