Alain Monfort

Simulation-based inference: A survey with special reference to panel data models

Abstract In this article we study the recent developments of inference methods based on simulations. In particular, we discuss the Simulated Generalized Method of Moments, the Simulated Maximum Likelihood Method, and the Simulated Pseudo Maximum Likelihood Methods. The asymptotic properties of the estimators are described when the number of observations n goes to infinity, and we distinguish the case where the number H of simulations per observation is fixed from the case where this number also goes to infinity. In the former case, the possible asymptotic bias is evaluated and, in the latter case, we carefully examine the consequences of the assumptions on the relative divergence rates of n and H . We also show how these methods apply to various contexts, in particular to the case of panel data models.

Qualitative threshold arch models

Abstract In this paper we consider a class of dynamic models in which both the conditional mean and the conditional variance are endogenous stepwise functions. We first consider the probabilistic properties of these models: stationarity conditions, leptokurtic effect, linear representation, optimal prediction. In this first part most results are based on Markov chains theory. Then we derive statistical properties of this class of models; pseudo-maximum likelihood estimators, conditional homoscedasticity tests, tests of weak or strong white noise, CAPM test, factors determination, ARCH-M effects. We also discuss the introduction of exogenous variables and the case of multiple lags. Finally, an application to the Paris Stock Index is proposed.

RATIONAL EXPECTATIONS IN DYNAMIC LINEAR MODELS: ANALYSIS OF THE SOLUTIONS

In this paper we analyze the solutions of linear econometric models with rational expectations. More precisely, we describe in detail the set of all the solutions; in particular this set is shown to be much larger than the sets previously considered. We also study various criteria of selection in this set of solutions and we examine to what extent these criteria redtiuce the set of the solutions.

Simulation Based Inference in Models with Heterogeneity

In this paper we discuss the usefulness, for models with heterogeneity, of simulation techniques in inference procedures, like maximum likelihood method, generalized moments method or pseudo maximum likelihood methods. These procedures are studied from the point of view of consistency, asymptotic normality, convergence rates and possible asymptotic bias. We carefully distinguish the case where the simulations are different for all the observations from the case where they are identical.

Testing non-nested hypotheses

Publisher Summary The comparison of different hypotheses, i.e. of competing models, is the basis of model specification. It may be performed along two main lines. The first one consists in associating with each model a loss function and in retaining the specification implying the smallest (estimated) loss. In practice, the loss function is defined either by updating some a priori knowledge on the models given the available observations (the Bayesian point of view), or by introducing some criterion taking into account the trade-off between the goodness of fit and the complexity of the model. The second approach is hypothesis testing theory. However, the determination of the decision rule is not done on the same basis as model choice. The basis of hypothesis testing theory is to introduce the probability of errors. This chapter focuses on the case where none of the hypotheses is a particular case of another one.

Testing nested or non-nested hypotheses

This paper presents a test procedure for nested or non-nested hypotheses. The test statistic is based on the difference between two estimators of the pseudo-true value as defined for instance by Sawa. This statistic is similar to the usual Wald statistic in the case of nested hypotheses and it can be replaced by an asymptotically equivalent one deduced from the score function.

Disequilibrium Econometrics in Simultaneous Equations Systems

This paper considers the econometric problems raised by multi-market disequilibrium models. It uses a specification which is derived from general disequilibrium theory and, therefore, provides a first bridge between the economic theory approach and the econometric theory approach of disequilibrium. The model is piecewise linear; the problem of the existence of a reduced form, which is a crucial issue in nonlinear models, is solved. Limited information estimators as well as full information estimators are proposed; a simple numerical algorithm is given for the computation of a FIML estimator.

A General Approach to Serial Correlation

In this paper the testing and estimation problems are discussed in the case of serial correlation. Various models are particular cases of the general framework considered: the nonlinear simultaneous equations models, the probit models, the tobit models, the disequilibrium models, the frontier models, etc. In this context, it is shown that the score test can be written explicitly and that the statistic obtained is a generalization of that of Durbin and Watson; moreover, the maximum likelihood estimation procedure is shown to be robust with respect to serial correlation.

Bayesian estimation of switching ARMA models

Abstract Switching ARMA processes have recently appeared as an efficient modelling to nonlinear time-series models, because they can represent multiple or heterogeneous dynamics through simple components. The levels of dependence between the observations are double: at a first level, the parameters of the model are selected by a Markovian procedure. At a second level, the next observation is generated according to a standard time-series model. When the model involves a moving average structure, the complexity of the resulting likelihood function is such that simulation techniques, like those proposed by Shephard (1994, Biometrika 81, 115–131) and Billio and Monfort (1998, Journal of Statistical Planning and Inference 68, 65–103), are necessary to derive an inference on the parameters of the model. We propose in this paper a Bayesian approach with a non-informative prior distribution developed in Mengersen and Robert (1996, Bayesian Statistics 5. Oxford University Press, Oxford, pp. 255–276) and Robert and Titterington (1998, Statistics and Computing 8(2), 145–158) in the setup of mixtures of distributions and hidden Markov models, respectively. The computation of the Bayes estimates relies on MCMC techniques which iteratively simulate missing states, innovations and parameters until convergence. The performances of the method are illustrated on several simulated examples. This work also extends the papers by Chib and Greenberg (1994, Journal of Econometrics 64, 183–206) and Chib (1996, Journal of Econometrics 75(1), 79–97) which deal with ARMA and hidden Markov models, respectively.

Econometric Asset Pricing Modelling

The purpose of this paper is to propose a general econometric approach to no-arbitrage asset pricing modelling based on three main ingredients: (i) the historical discrete-time dynamics of the factor representing the information, (ii) the Stochastic Discount Factor (SDF), and (iii) the discrete-time risk-neutral (R.N.) factor dynamics. Retaining an exponential-affine specification of the SDF, its modelling is equivalent to the specification of the risk sensitivity vector and of the short rate, if the latter is neither exogenous nor a known function of the factor. In this general framework, we distinguish three modelling strategies: the Direct Modelling, the Risk-Neutral Constrained Direct Modelling and the Back Modelling. In all the approaches we study the Internal Consistency Conditions (ICCs), implied by the absence of arbitrage opportunity assumption, and the identification problem. The general modelling strategies are applied to two important domains: security market models and term structure of interest rates models. In these contexts we stress the usefulness (and we suggest the use) of the Risk-Neutral Constrained Direct Modelling and of the Back Modelling approaches, both allowing to conciliate a flexible (non-Car) historical dynamics and a Car R.N. dynamics leading to explicit or quasi explicit pricing formulas for various derivative products. Moreover, we highlight the possibility to specify asset pricing models able to accommodate non-Car historical and non-Car R.N. factor dynamics with tractable pricing formulas. This result is based on the notion of (Risk-Neutral) Extended Car process that we introduce in the paper, and which allows to deal with sophisticated models like Gaussian and Inverse Gaussian GARCH-type models with regime-switching, or Wishart Quadratic Term Structure models.