Gabriele Fiorentini

Identification, estimation and testing of conditionally heteroskedastic factor models

We investigate several important inference issues for factor models with dynamic heteroskedasticity in the common factors. First, we show that such models are identified if we take into account the time-variation in the variances of the factors. Our results also apply to dynamic versions of the APT, dynamic factor models, and vector autoregressions. Secondly, we propose a consistent two-step estimation procedure which does not rely on knowledge of any factor estimates, and explain how to compute correct standard errors. Thirdly, we develop a simple preliminary LM test for the presence of ARCH effects in the common factors. Finally, we conduct a Monte Carlo analysis of the finite sample properties of the proposed estimators and hypothesis tests.

Maximum Likelihood Estimation and Inference in Multivariate Conditionally Heteroscedastic Dynamic Regression Models With Student t Innovations

We provide numerically reliable analytical expressions for the score, Hessian, and information matrix of conditionally heteroscedastic dynamic regression models when the conditional distribution is multivariatet. We also derive one-sided and two-sided Lagrange multiplier tests for multivariate normality versus multivariate t based on the first two moments of the squared norm of the standardized innovations evaluated at the Gaussian pseudo-maximum likelihood estimators of the conditional mean and variance parameters. Finally, we illustrate our techniques through both Monte Carlo simulations and an empirical application to 26 U.K. sectorial stock returns that confirms that their conditional distribution has fat tails.

Analytic derivatives and the computation of GARCH estimates

In the context of univariate GARCH models we show how analytic first and second derivatives of the log-likelihood can be successfully employed for estimation purposes. Maximum likelihood GARCH estimation usually relies on the numerical approximation to the log-likelihood derivatives, on the grounds that an exact analytic differentiation is much too burdensome. We argue that this is not the case and that the computational benefit of using the analytic derivatives (first and second) may be substantial. Furthermore, we make a comparison of various gradient algorithms that are used for the maximization of the GARCH Gaussian likelihood. We suggest the implementation of a globally efficient computation algorithm that is obtained by suitably combining the use of the estimated information matrix with that of the exact Hessian during the maximization process. As this would appear a straightforward extension, we then study the finite sample performance of the exact Hessian and its approximations (that is, the estimated information, outer products and misspecification robust matrices) in inference.

On the validity of the Jarque-Bera normality test in conditionally heteroskedastic dynamic regression models

We show that the Jarque-Bera test, originally devised for constant conditional variance models with no functional dependence between conditional mean and variance parameters, can be safely applied to a broad class of GARCH-M models, but not to all.

Estimation and empirical performance of Heston's stochastic volatility model: the case of a thinly traded market

Abstract This paper examines the stochastic volatility model suggested by Heston [Rev. Financ. Stud. 6 (1993) 327] in a thinly traded market context. We employ a time-series approach based on indirect inference to estimate the model parameters. We also discuss the potential effects of time-varying skewness and kurtosis on the performance of the model. It is found that the model tends to overprice out-of-the-money calls and underprice in-the-money calls. Moreover, the daily volatility risk premium shows a volatile behavior over time; however, our evidence suggests that the volatility risk premium has a negligible impact on the pricing performance of Hestons model.

Constrained Indirect Estimation

We develop generalized indirect estimation procedures that handle equality and inequality constraints on the auxiliary model parameters by extracting information from the relevant multipliers, and compare their asymptotic efficiency to maximum likelihood. We also show that, regardless of the validity of the restrictions, the asymptotic efficiency of such estimators can never decrease by explicitly considering the multipliers associated with additional equality constraints. Furthermore, we discuss the variety of effects on efficiency that can result from imposing constraints on a previously unrestricted model. As an example, we consider a stochastic volatility process estimated through a garch model with Gaussian or t distributed errors. Copyright 2004, Wiley-Blackwell.

Conditional Means of Time Series Processes and Time Series Processes for Conditional Means

We study the processes for the conditional mean and variance given a specification of the process for the observed time series. We derive general results for the conditional mean of univariate and vector linear processes, and then apply it to various models of interest. We also consider the joint process for a subvector and its expected value conditional on the whole information set. In this respect, we derive necessary and sufficient conditions for one of the variables in a bivariate VAR(l) to have a white noise univariate representation while its conditional mean follows an AR(l) with a high autocorrelation coefficient. We also compare the persistence of shocks to the conditional mean relative to the observed variable using mea sures of total and iterim persistence of shocks for stationary processes based on the impulse response function. We apply our results to post-war US monthly real stock market returns and dividend yields. Our findings seem to confirm that stock returns are very close to white noise, while expected returns are well represented by an AR(l) process with a firstorder autocorrelation of .9755. We also find that small unexpected variations in expected returns have a large negative immediate impact on observed returns, which is thereafter compensated by a slowly diminishing positive effect on expected returns.

Bayesian Analysis of the Output Gap

Our objective is to build output gap estimates that benefit from information provided by Phillips curve theory and business cycle studies. For this, we develop a Bayesian analysis of the bivariate Phillips curve model proposed by Kuttner for estimating potential output. Given our priors, we obtain samples from parameters and state variables joint posterior distribution following a Gibbs sampling strategy. We sample the state variables given parameters using the Carter–Kohn procedure, and we exploit a likelihood factorization to draw parameters given the state. A Metropolis–Hastings step is used to remove the conditioning on starting values. To accommodate the variance moderation that has been observed on U.S. gross domestic product, Kuttners model is extended for a change in variance parameters. We apply this methodology to the analysis of the output gap in the United States and in the European Monetary Union. Finally, some important extensions to the original Kuttner model are discussed.

Control Variates for Variance Reduction in Indirect Inference: Interest Rate Models in Continuous Time

Simulation estimators, such as indirect inference or simulated maximum likelihood, are successfully employed for estimating stochastic differential equations. They adjust for the bias (inconsistency) caused by discretization of the underlying stochastic process, which is in continuous time. The price to be paid is an increased variance of the estimated parameters. The variance suffers from an additional component, which depends on the stochastic simulation involved in the estimation procedure. To reduce this undesirable effect, one could increase the number of simulations (or the length of each simulation) and thus the computational cost. Alternatively, this paper shows how variance reduction can be achieved, at virtually no additional computational cost, by use of control variates. The Ornstein-Uhlenbeck process, used by Vasicek to model the short-term interest rate in continuous time, and the square-root process, used by Cox, Ingersoll and Ross, are explicitly considered and experimented with. Monte Carlo experiments show a global efficiency gain of almost 50% over the simple indirect estimator.

A tobit model with garch errors

In the context of time series regression, we extend the standard Tobit model to allow for the possibility of conditional heteroskedastic error processes of the GARCH type. We discuss the likelihood function of the Tobit model in the presence of conditionally heteroskedastic errors. Expressing the exact likelihood function turns out to be infeasible, and we propose an approximation by treating the model as being conditionally Gaussian. The performance of the estimator is investigated by means of Monte Carlo simulations. We find that, when the error terms follow a GARCH process, the proposed estimator considerably outperforms the standard Tobit quasi maximum likelihood estimator. The efficiency loss due to the approximation of the likelihood is finally evaluated.