Lorenzo Panattoni

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.

Computational efficiency of FIML estimation

Abstract Computational efficiency is a crucial attribute for FIML estimators. Experiments reported in the literature suggest that a gradient approach, which uses the exact Hessian matrix (Newton method), converges rapidly when the values of the coefficients are near the optimum, while the use of suitable approximations to the Hessian improves the algorithm efficiency far from the optimum and hence enhances its robustness with respect to the initial choice of the parameters. In this paper a large number of Monte Carlo experiments have been performed in order to better understand the behavior of the Hessian and of some of its approximations. The results agree with the previous findings but show that the Hessian behaves better in an interval around the optimum that is narrower than commonly thought. They suggest that a globally efficient computation algorithm could be implemented by mixing the use of the approximations to the Hessian and of the exact Hessian during the maximization process in a proper way.

An implicit method to solve Saint Venant equations

Abstract An implicit numerical method for solving Saint Venant equations has been defined for an application relating to the river Arno. This method exploits the linearity in the discharge of the mass equation, by means of which it is possible to express the discharge as a function of the water level and to use this expression in the equation of motion. Then the resulting non-linear equation for a grid element on the (x, t) plane contains only the water levels as unknown quantities. The solution of the system of equations for the entire reach, by the use of the Newton iteration method, is facilitated by the particular form of the matrix of the coefficients. The rapid rate of convergence and the limited storage allocation are characteristics of this implicit scheme. This method has been compared with other implicit methods based on the same grid of points.

Mode predictors in nonlinear systems with identities

For a nonlinear system of simultaneous equations, the mode of the joint distribution of the endogenous variables in the forecast period is proposed as alternative to the more usual deterministic or mean predictors. A first method follows from maximizing the joint density of a subset of the endogenous variables, corresponding to stochastic equations only (analogously to FIML estimation, where identities are first substituted into stochastic equations). Then a more general approach is developed, which maintains the identities. The model with identities is viewed as a mapping between the space of the random errors and a hypersurface in the space of the endogenous variables; the probability density is defined, and maximization is performed on such a hypersurface. Experimental results on these two mode predictors (and comparisons with deterministic and mean predictors) are provided for a macro model of the Italian economy.

Gradient Methods in Fiml Estimation of Econometric Models

Through Monte Carlo experiments, this paper compares the performances of different gradient optimization algorithms, when performing full information maximum likelihood (FIML) estimation of econometric models. Different matrices are used (Hessian, outer products matrix, GLS-type matrix, as well as a mixture of them).

The behavior of trust-region methods in FIML-estimation

This paper presents a Monte-Carlo study on the practical reliability of numerical algorithms for FIML-estimation in nonlinear econometric models. The performance of different techniques of Hessian approximation in trust-region algorithms is compared regarding their “robustness” against “bad” starting points and their “global” and “local” convergence speed, i.e. the gain in the objective function, caused by individual iteration steps far off from and near to the optimum.Concerning robustness and global convergence speed the crude GLS-type Hessian approximations performed best, efficiently exploiting the special structure of the likelihood function. But, concerning local speed, general purpose techniques were strongly superior. So, some appropriate mixtures of these two types of approximations turned out to be the only techniques to be recommended.ZusammenfassungDiese Arbeit beschreibt eine Monte-Carlo-Studie über die praktische Verläßlichkeit numerischer Algorithmen zur FIML-Schätzung in nichtlinearen ökonometrischen Modellen. Dabei wird die Güte verschiedener Hessematrixnäherungen in trust-region Algorithmen vergleichen hinsichtlich der “Robustheit” gegenüber “schlechten” Starwerten und hinsichtlich “globaler” und “lokaler” Konvergenzgeschwindigkeit, d. h. der Größe der Verbesserung der Zielfunktion bei Iterationsschritten weit entfernt bzw. in der Nähe vom Optimum.Während sich GLS-Typ Näherungen der Hessematrix hinsichtlich Robustheit und globaler Konvergenz als deutlich überlegen erweisen wegen ihrer effizienten Ausnutzung der speziellen Struktur der Likelihood-Funktion, konvergieren Verfahren, die für allgemeine Zielfunktionen entwicklet wurden, wesentlich schneller in der Nähe des Optimums. Für die praktische Anwendung erweisen sich daher lediglich geeignete “Mischungen” dieser beiden Näherungstypen als empfehlenswert.

Alternative Estimators of a Diffusion Model of the Term Structure of Interest Rates. A Monte Carlo Comparison

We evaluate, through Monte Carlo experiments, the econometric performance of six alternative estimators of the basic parameters of the Cox—Ingersoll—Ross single-factor diffusion model of the term structure of interest rates. Different generating schemes are compared and the unobservability of the state-variable is taken into account. The effects of approximating interest rates, increasing frequency data and starting values are analyzed. A Monte Carlo evaluation of the effects on bond prices of biased parameter estimates is provided.

Optimal simulation with econometric models

Abstract The use of an econometric model for policy simulations is based on the analysis of the effects on some selected endogenous variables ( objectives ) due to shocks on some exogenous variables ( instruments ). In the commonly used simulation procedure these effects are examined by shocking the instruments one at a time. In this paper we suggest a criterion to combine the different shocks, on the basis of the concept of instruments effectiveness . It will be also shown how the resulting simulation procedure can be considered optimal from a certain point of view. A case study with an Italian econometric model is reported to somehow quantify the advantages of the suggested procedure with reference to the conventional one.

Uncertainty and Stability in a Macro-Econometric Model

One of the main techniques for determining the long term stability properties of a macro-econometric model has been for some time now to compute the eigenvalues of the linearized reduced form of the model. But these eigenvalues are affected by uncertainty, coming mostly from the error on the estimated coefficients. In this paper we study, using the French macro-economic model Mini-DMS, how taking into account the uncertainty can affect the conclusions. We shall give particular attention to the following points: are the conclusions concerning convergence certain, do the eigenvalues change significantly with the period at which the linearization is made, does the use of elasticities instead of multipliers produce significantly more stable eigenvalues.