Marine Carrasco

MIXING AND MOMENT PROPERTIES OF VARIOUS GARCH AND STOCHASTIC VOLATILITY MODELS

This paper first provides some useful results on a generalized random coefficient autoregressive model and a generalized hidden Markov model. These results simultaneously imply strict stationarity, existence of higher order moments, geometric ergodicity, and I²-mixing with exponential decay rates, which are important properties for statistical inference. As applications, we then provide easy-to-verify sufficient conditions to ensure I²-mixing and finite higher order moments for various linear and nonlinear GARCH(1,1), linear and power GARCH(p,q), stochastic volatility, and autoregressive conditional duration models. For many of these models, our sufficient conditions for existence of second moments and exponential I²-mixing are also necessary. For several GARCH(1,1) models, our sufficient conditions for existence of higher order moments again coincide with the necessary ones in He and Terasvirta (1999, Journal of Econometrics 92, 173–192).

Generalization of GMM to a continuum of moment conditions

This paper proposes a version of the generalized method of moments procedure that handles both the case where the number of moment conditions is finite and the case where there is a continuum of moment conditions. Typically, the moment conditions are indexed by an index parameter that takes its values in an interval. The objective function to minimize is then the norm of the moment conditions in a Hilbert space. The estimator is shown to be consistent and asymptotically normal. The optimal estimator is obtained by minimizing the norm of the moment conditions in the reproducing kernel Hilbert space associated with the covariance. We show an easy way to calculate this estimator. Finally, we study properties of a specification test using overidentifying restrictions. Results of this paper are useful in many instances where a continuum of moment conditions arises. Examples include efficient estimation of continuous time regression models, cross-sectional models that satisfy conditional moment restrictions, and scalar diffusion processes.

Linear Inverse Problems in Structural Econometrics Estimation Based on Spectral Decomposition and Regularization

Inverse problems can be described as functional equations where the value of the function is known or easily estimable but the argument is unknown. Many problems in econometrics can be stated in the form of inverse problems where the argument itself is a function. For example, consider a nonlinear regression where the functional form is the object of interest. One can readily estimate the conditional expectation of the dependent variable given a vector of instruments. From this estimate, one would like to recover the unknown functional form. This chapter provides an introduction to the estimation of the solution to inverse problems. It focuses mainly on integral equations of the first kind. Solving these equations is particularly challenging as the solution does not necessarily exist, may not be unique, and is not continuous. As a result, a regularized (or smoothed) solution needs to be implemented. We review different regularization methods and study the properties of the estimator. Integral equations of the first kind appear, for example, in the generalized method of moments when the number of moment conditions is infinite, and in the nonparametric estimation of instrumental variable regressions. In the last section of this chapter, we investigate integral equations of the second kind, whose solutions may not be unique but are continuous. Such equations arise when additive models and measurement error models are estimated nonparametrically.

Tests for Unit-Root versus Threshold Specification With an Application to the Purchasing Power Parity Relationship

We consider modeling the real exchange rate by a stationary three-regime self-exciting threshold autoregressive (SETAR) model with possibly a unit root in the middle regime. This representation is consistent with purchasing power parity in the presence of trading costs. Our main contribution is to provide statistical tools for testing unit root versus a SETAR. First, we show that a SETAR with a unit root in the middle regime is stationary and mixing under reasonable assumptions. Second, we derive analytically the asymptotic distribution of our unit-root test under the null. Using monthly real exchange rate data, our test rejects the null of unit-root against a threshold process for five European series.

Misspecified Structural Change, Threshold, and Markov-switching models

Sudden perturbations of a large amplitude occur frequently in macroeconomic and financial time series. A usual practice is to test linearity against a permanent structural change. However, changes can also be captured by nonlinear stationary models such that Threshold and Markov-switching models. In this paper, we show that tests designed for a threshold alternative have also power against parameter instability originating from Structural Change or Markov-switching models. On the other hand, it is shown that tests for structural change have no power if the data are generated by a Markov-switching or Threshold model. Therefore, it appears that testing the null of parameter stability against a threshold alternative is a robust way to detect parameter instability in economic and financial time series. A Monte Carlo analysis based on several models studied in the literature illustrates how the tests perform in practice.

A SPECTRAL METHOD FOR DECONVOLVING A DENSITY

We propose a new estimator for the density of a random variable observed with an additive measurement error. This estimator is based on the spectral decomposition of the convolution operator, which is compact for an appropriate choice of reference spaces. The density is approximated by a sequence of orthonormal eigenfunctions of the convolution operator. The resulting estimator is shown to be consistent and asymptotically normal. While most estimation methods assume that the characteristic function (CF) of the error does not vanish, we relax this assumption and allow for isolated zeros. For instance, the CF of the uniform and symmetrically truncated normal distributions have isolated zeros. We show that, in the presence of zeros, the density is identified even though the convolution operator is not one-to-one. We propose two consistent estimators of the density. We apply our method to the estimation of the measurement error density of hourly income collected from survey data.

Detecting Mean Reversion in Real Exchange Rates from a Multiple Regime STAR Model

Recent studies on general equilibrium models with transaction costs show that the dynamics of the real exchange rate are necessarily nonlinear. Our contribution to the literature on nonlinear price adjustment mechanisms is threefold. First, we model the real exchange rate by a Multi-Regime Logistic Smooth Transition AutoRegression (MR-LSTAR), allowing for both ESTAR-type and SETAR-type dynamics. This choice is motivated by the fact that even the theoretical models, which predict a smooth behavior for the real exchange rate, do not rule out the possibility of a discontinuous adjustment as a limit case. Second, we propose two classes of unit-root tests against this MR- LSTAR alternative, based respectively on the likelihood and on an auxiliary model. Their asymptotic distributions are derived analytically. Third, when applied to 28 bilateral real exchange rates, our tests reject the null hypothesis of a unit root for eleven series bringing evidence in favor of the purchasing power parity.

OPTIMAL TEST FOR MARKOV SWITCHING PARAMETERS

This paper proposes a class of optimal tests for the constancy of parameters in random coefficients models. Our testing procedure covers the class of Hamiltons models, where the parameters vary according to an unobservable Markov chain, but also applies to nonlinear models where the random coefficients need not be Markov. We show that the contiguous alternatives converge to the null hypothesis at a rate that is slower than the standard rate. Therefore, standard approaches do not apply. We use Bartlett-type identities for the construction of the test statistics. This has several desirable properties. First, it only requires estimating the model under the null hypothesis where the parameters are constant. Second, the proposed test is asymptotically optimal in the sense that it maximizes a weighted power function. We derive the asymptotic distribution of our test under the null and local alternatives. Asymptotically valid bootstrap critical values are also proposed.

Simulation-Based Method of Moments and Efficiency

The method of moments is based on a relation Eθ0(h(Xt, θ)) = 0, from which an estimator of θ is deduced. In many econometric models, the moment restrictions can not be evaluated numerically due to, for instance, the presence of a latent variable. Monte Carlo simulations method make possible the evaluation of the generalized method of moments (GMM) criterion. This is the basis for the simulated method of moments. Another approach involves defining an auxiliary model and finding the value of the parameters that minimizes a criterion based either on the pseudoscore (efficient method of moments) or the difference between the pseudotrue value and the quasi-maximum likelihood estimator (indirect inference). If the auxiliary model is sufficiently rich to encompass the true model, then these two methods deliver an estimator that is asymptotically as efficient as the maximum likelihood estimator.

Efficient Estimation Using the Characteristic Function

The method of moments proposed by Carrasco and Florens (2000) permits to fully exploit the information contained in the characteristic function and yields an estimator which is asymptotically as efficient as the maximum likelihood estimator. However, this estimation procedure depends on a regularization or tuning parameter a that needs to be selected. The aim of the present paper is to provide a way to optimally choose a by minimizing the approximate mean square error (AMSE) of the estimator. Following an approach similar to that of Newey and Smith (2004), we derive a higher-order expansion of the estimator from which we characterize the finite sample dependence of the AMSE on a. We provide a data-driven procedure for selecting the regularization parameter that relies on parametric bootstrap. We show that this procedure delivers a root T consistent estimator of a. Moreover, the data-driven selection of the regularization parameter preserves the consistency, asymptotic normality and efficiency of the CGMM estimator. Simulation experiments based on a CIR model show the relevance of the proposed approach.