Likelihood-Based Estimation of Latent Generalized ARCH Structures

Abstract

GARCH models are commonly used as latent processes in econometrics, Þnancial economics and macroeconomics. Yet no exact likelihood analysis of these models has been provided so far. In this paper we outline the issues and suggest a Markov chain Monte Carlo algorithm which allows the calculation of a classical estimator via the simulated EM algorithm or a Bayesian solution in O(T ) computational operations, where T denotes the sample size. We assess the performance of our proposed algorithm in the context of both artiÞcial examples and an empirical application to 26 UK sectorial stock returns, and compare it to existing approximate solutions.

Indirect estimation of large conditionally heteroskedastic factor models, with an application to the Dow 30 stocks

“…In addition, we assess the efficiency of our estimators relative to the Bayesian estimators proposed by Fiorentini, Sentana and Shephard (2004) under the assumption of constant idiosyncratic variances. Interestingly, we find that their estimators exhibit the usual trade-off between bias and variance: prior information leads to lower variation in the estimated parameters, but unless the prior is centred around the true values, it introduces a finite sample bias.…”

“…In addition, we conduct an empirically realistic Monte Carlo experiment to assess the finite sample performance of our two proposed indirect estimators relative to the approximate methods of HRS and SF. We also compare them to the Bayesian estimators of Fiorentini, Sentana and Shephard (2004) in their restricted case.…”

“…Following Fiorentini, Sentana and Shephard (2004), we use independent beta priors on φ + ρ and ρ/(φ + ρ) with parameters (6,1.25) and (2.5,1.25) for the φ 0 + ρ 0 = .95 designs in an attempt to reflect the typical estimates obtained with weekly data. We also use a standard inverted gamma prior for the unconditional variance of the common factor, δ, with mean 1 and standard deviation 1/2.…”

Volatility Forecasting

“…Harvey, Ruiz and Shephard (1994) and King, Sentana and Wadhwani (1994) were among the first to estimate multivariate stochastic volatility models. More recent empirical studies and numerically efficient algorithms for the estimation of latent multivariate volatility structures include Aguilar and West (2000), Fiorentini, Sentana and Shephard (2004) and Liesenfeld and Richard (2003). Issues related to identification within heteroskedastic factor models have been studied by Sentana and Fiorentini (2001).…”

Volatility and predictability of the Russian ruble exchange rate

“…In contrast to standard ARCH and GARCH models, for which the likelihood functions are readily available through a prediction error decomposition type argument (see ARCH), the likelihood functions for latent GARCH models are generally not available in closed form. General estimation and inference procedures for latent GARCH models based on Markov Chain Monte Carlo methods have been developed by Fiorentini, Sentana and Shephard (2004) (see also SV).…”