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  1. Optimal Inference and Alternatives to the QMLE*
    1. Maximum Likelihood Estimator
      1. Asymptotic Behavior
      2. One Step Efficient Estimator
      3. Semiparametric Models and Adaptive Estimators
      4. Local Asymptotic Normality (LAN)
    2. MLE with Misspecified Density
      1. Condition for the Consistency of MLE
      2. Reparameterization Implying the Consistency of MLE
      3. Choice of the Instrumental Density h
      4. Asymptotic Distribution of the MLE
    3. Alternative Estimation Method
      1. Weighted LSE for the ARMA Parameters
      2. Self-Weighted QMLE
      3. Lp Estimators
      4. Estimators of the Least Absolute Values
      5. Whittle Estimator
    4. Bibliographical Notes
    5. Exercises

Christian Francq and Jean-Michel Zakoļan

Keywords: Adaptive GARCH Estimator, Asymptotic Behavior of the MLE, Comparison of QMLE and MLE of GARCH, LAN (Local Asymptotic Normality), Least Absolute Deviations, ML for GARCH, Misspecified MLE, Non Gaussian QMLE for GARCH, Optimal Wald, Lagrange Multiplier and Likelihood Ratio Tests, Optimality Condition for the QMLE, One Step ML Estimator, Semi-Parametric GARCH Model, Self-Weighted QMLE, Weighted Least Squares, Whittle Estimator

Description: The most commonly used estimation method of the GARCH models is that of the QML studied in Chapter 7. One of the attractive features of this method is that the asymptotic properties of the QML estimators are valid under mild assumptions. In particular, no moment assumption is required on the observed process in the pure GARCH case. The QML method has however several drawbacks, which motivate the introduction of alternative approaches. These drawbacks are the following: (i) the estimator is not explicit and requires a numerical optimization algorithm; (ii) the asymptotic normality of the estimator requires the existence of a moment of order 4 for the noise ηt; (iii) the QML estimator is inefficient in general; (iv) the asymptotic normality requires the existence of moments for εt in the general ARMA-GARCH case; (v) a complete parametric specification is required. In the ARCH case, the QLS estimator defined in Section 6.2 answers satisfactorily to the point (i), at the price of additional moment conditions. The maximum likelihood (ML) estimator studied in Section 9.1 of this chapter is an answer to the points (ii) and (iii), but it requires the knowledge of the density f of ηt. Indeed, it will be seen that adaptive estimators for the set of all the parameters do not exist in general semi-parametric GARCH models. Concerning the point (iii), it will be seen that the QML can sometimes be optimal outside of trivial case where f is Gaussian. In Section 9.2, the ML estimator will be studied in the (quite realistic) situation where f is misspecified. I will also be seen that the so-called LAN (Local Asymptotic Normality) property allows to show the local asymptotic optimality of test procedures based on the ML. In Section 9.3, less standard estimators are presented, in order to answer to some of the points (i)-(v). In this chapter, we just give the main principle of the estimation methods but do not state precise results. The mathematical details can be found in the references that are given along the text or at the end of the chapter.