Giorgio Calzolari

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.

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.

Antithetic variates to estimate the simulation bias in non-linear models

Abstract This paper describes a Monte Carlo experiment, which makes use of antithetic variate sampling, to get an accurate estimate of the deterministic simulation bias in the non-linear Klein—Goldberger model. The computational efficiency is more than 500 times greater than in case of simple random sampling.

Indirect estimation of α-stable stochastic volatility models

The @a-stable family of distributions constitutes a generalization of the Gaussian distribution, allowing for asymmetry and thicker tails. Its many useful properties, including a central limit theorem, are especially appreciated in the financial field. However, estimation difficulties have up to now hindered its widespread use among practitioners. The authors introduce an indirect estimation approach to stochastic volatility models with @a-stable innovations that exploits, as auxiliary model, a GARCH(1, 1) with t-distributed innovations. The approach is illustrated by means of a detailed simulation study and an application to currency crises.

A Note on the Variance of Ex-Post Forecasts in Econometric Models

IT IS WELL KNOWN that in an econometric model, under general assumptions, the ex-post forecast error (conditional on predetermined variables) can be decomposed into the sum of two independent terms; the former is the component due to error in estimated coefficients, while the latter depends on the random error terms. The asymptotic covariance matrix of the first component can be analytically derived through intermediate computation of the covariance matrix of restricted reduced form coefficients. However, in the case of overidentified models, this may lead to a great increase in the problems dimensions, the number of reduced form parameters being larger (often much larger) than the number of structural parameters. This intermediate step is, however, unnecessary; this note discusses a straightforward analytic derivation of the desired matrix directly from the estimated structural parameters, without any increase in the dimensions of the problem, thus facilitating the computation even for medium to large econometric models. Let

A Program for Stochastic Simulation of Econometric Models

model. Many results (observed value, deterministic solution, computed mean among the replications, minimum and maximum of the stochastic solution, first relative differences of the observed, deterministic and mean stochastic values) are displayed and several empirical indicators of goodness of fit are computed: (1) the mean over the simulation period of the actual, deterministic and mean stochastic values; (2) the root mean square error (RMSE) of the deterministic and mean stochastic solutions; (3) the mean absolute percentage error (MAPE) of the deterministic and mean stochastic solutions; (4) Theils inequality coefficient (U) of the deterministic and mean stochastic solutions; (5) the coefficients and standard errors of the regression (with intercept) of the observed values on the deterministic or mean stochastic solutions; (6) the coefficients and standard errors of the regression (without intercept) of the first relative differences of the observed values over those of the deterministic or mean stochastic solutions. The package is written in FORTRAN IV and ASSEMBLER 370 languages. It consists of approximately one thousand statements, in addition to the statements necessary to formalize the model. The required storage for the program is 60 kilobytes. A large work matrix is then required to hold intermediate and final results of the computation; its dimensions depend on the parameters specified in the input file (number of equations, simulation period, number of variables, etc.). The stochastic simulation of the Klein I model requires about 13 seconds of CPU time for 100 replications over 21 years of the sample period.

Indirect Estimation of α-Stable Distributions and Processes

The alpha-stable family of distributions constitutes a generalization of the Gaussian distribution, allowing for asymmetry and thicker tails. Its practical usefulness is coupled with a marked theoretical appeal, as it stems from a generalized version of the central limit theorem in which the assumption of the finiteness of the variance is replaced by a less restrictive assumption concerning a somehow regular behavior of the tails. Estimation difficulties have however hindered its diffusion among practitioners. Since simulated values from alpha-stable distributions can be straightforwardly obtained, the indirect inference approach could prove useful to overcome these estimation difficulties. In this paper we provide a description of how to implement such a method by using a skew-t distribution as an auxiliary model. The indirect inference approach will be introduced in the setting of the estimation of the distribution parameters and then extended to linear time series models with alpha-stable disturbances. The performance of this estimation method is then assessed on simulated data. An application on time-series models for the inflation rate concludes the paper.

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.