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Article Excerpt
1. INTRODUCTION
This article studies the local robustness properties of estimation and testing procedures for the conditional location and scale parameters of a strictly stationary time series model. The class of conditional location and scale time series models is quite broad and includes several well-known dynamic models largely applied in economics and empirical finance, such as pure conditional scale models [like autoregressive conditional heteroscedasticity (ARCH) models] (Engle 1982) or models that jointly parameterize the conditional location and the scale of the given time series (as, e.g., ARCH in mean models) (Engle, Lilien, and Robins 1987). Typically, classical (nonrobust) estimation of the parameters of such models is obtained by means of a pseudo-maximum likelihood (PML) approach based on the nominal assumption of a conditionally Gaussian log-likelihood (see also Bollerslev and Wooldridge 1992). Such PML estimators (PMLEs) are based on an unbounded conditional score function, implying--as shown later--an unbounded time series influence function (IF) (Kunsch 1984; Hampel 1968, 1974). Consequently, PMLEs for conditional location and scale models are not robust under local departures from conditional normality. In this article we propose a new class of inference procedures that are robust to local nonparametric misspecifications of a parametric, conditionally Gaussian, location and scale model. More specifically, we consider the class of robust, conditionally unbiased M-estimators for the parameters of conditional location and scale models and derive the optimal (i.e., most efficient) robust estimator within this class. Based on such estimators, we then obtain several maximum likelihood (ML)-type bounded-influence tests for parametric hypotheses on the parameters of the conditional location and scale equations following the general approach of Heritier and Ronchetti (1994) and Ronchetti and Trojani (2001).
The need for robust procedures in estimation and testing has been stressed by many authors and is now widely recognized in both the statistical and the econometric literature (see, e.g., Hampel 1974; Koenker and Bassett 1978; Huber 1981; Koenker 1982; Hampel, Ronchetti, Rousseeuw, and Stahel 1986; Peracchi 1990; Markatou and Ronchetti 1997; Krishnakumar and Ronchetti 1997; Ronchetti and Trojani 2001; Ortelli and Trojani 2004; Gagliardini, Trojani, and Urga 2004). However, the problem of robust estimation for the parameters of conditional location and scale models has been considered by only a few authors and only from the specific perspective of high-breakdown estimation. Even less attention has been devoted to robust inference within conditional location and scale models. High-breakdown estimators resistant to large amount of contamination have been proposed by Sakata and White (1998) and Muler and Yohai (1999). These estimators are very useful at the exploratory and estimation stages. Here we focus on the inference stage, where we typically have an approximate model and can expect small deviations from the model. Alternatives to high-breakdown estimators are also needed because these estimators are computationally intensive and cannot be applied to estimate the parameters of a class of broadly applied models, such as, for instance, threshold ARCH or ARCH in mean models.
This article derives optimal bounded-influence estimation and testing procedures for a general conditional location and scale model, which are computationally only slightly more demanding than those required by classical PML estimation of such models. We discuss the more specific contributions to the current literature next.
First, we characterize the robustness of conditionally unbiased M-estimators for nonlinear conditional location and scale models by computing the time series IF for the implied asymptotic functional estimator. This has been defined by Kunsch (1984), who applied this concept to AR(p) processes. We extend this result to our general model (1). This is a first necessary step that allows us to construct robust statistical procedures that can control for both (a) local asymptotic bias on the parameter estimates and (b) local asymptotic distortion on the level and the power of ML-type tests.
Second, we derive the optimal bounded-influence estimator for the parameters of conditional location and scale models under a conditionally Gaussian reference model. This extends the optimality result of Kunsch (1984) [obtained for AR(p) models] and the application of optimal conditionally unbiased M-estimators of Kunsch, Stefanski, and Carroll (1989) [obtained for generalized linear models] to general nonlinear second-order dynamic models. Based on these results, optimal bounded-influence versions of the classical Wald, score, and likelihood ratio tests can be derived along the general lines proposed by Heritier and Ronchetti (1994) and Ronchetti and Trojani (2001).
Third, we propose a feasible algorithm for the computation of our optimal robust estimators, which can be easily implemented in a standard package such as Matlab. This procedure is based on a truncating procedure that uses a set of Huber's weights to downweight the impact of influential observations. Fisher consistency at the model is preserved by means of some auxiliary recentering vectors, which in a time series setting generally must be computed by simulations--as in, for instance, a robust generalized method of moments (RGMM) (Ronchetti and Trojani 2001) setting. Using the conditional unbiasedness of our estimator, we provide analytical Laplace approximations for such vectors that strongly reduce computation time by avoiding simulation of multidimensional integrals.
Fourth, we study the efficiency and the robustness properties of our estimator by Monte Carlo simulation. We estimate a simple AR(1)-ARCH(1) process under several models of local contamination of a conditionally Gaussian process. Under the Gaussian reference model, the classical MLE and our robust estimator have essentially the same efficiency. In contrast, under local deviations from conditional normality, classical PMLEs, tests, and confidence intervals are found to be highly inefficient, whereas robust procedures perform very satisfactorily.
Finally, we present an application to robust testing for ARCH where robust procedures help identify ARCH structures that could not be detected using the classical inference approach.
The article is organized as follows. Section 2 introduces conditional location and scale models and the corresponding classical M-estimation procedure. Section 3 computes the time series IF for conditionally unbiased M-estimators. It then approximates the asymptotic bias on the parameter estimates induced by local deviations from the conditional Gaussian reference model. In a second step, the optimal robust estimator is derived, and the optimality of robust inference procedures based on such estimators is discussed. The section concludes by deriving analytic approximations for the auxiliary recentering vectors in our robust estimation and by presenting an algorithm to compute our robust estimator in applications. Section 4 discusses robust inference procedures based on our robust estimators. Section 5 presents the Monte Carlo experiments where the performance of our robust estimation and inference approach is evaluated in the setting of an AR(1)-ARCH(1) model. It also provides some advice for applications. Section 6 presents the empirical application to testing for ARCH. Section 7 summarizes and concludes.
2. CONDITIONALLY UNBIASED M-ESTIMATORS
Let Y := ([y.sub.t])[.sub.t[member of]Z] be a real-valued strictly stationary random sequence on the probability space ([R.sup.[infinity]], F, [P.sub.*]) and let P := {[P.sub.[theta]], [theta] [member of] [THETA] [??] [R.sup.p]} be some parametric model for [P.sub.*]. Under model [P.sub.[theta].sub.0], the random variable [y.sub.t] has a conditionally Gaussian distribution, [y.sub.t]|[F.sub.t-1] [approximately] N([[mu].sub.t]([[theta].sub.0]), [[sigma].sub.t.sup.2]([[theta].sub.0])). Specifically,
[y.sub.t] = [[mu].sub.t]([[theta].sub.0]) + [[epsilon].sub.t]([[theta].sub.0]), [[epsilon].sub.t.sup.2]([[theta].sub.0]) = [[sigma].sub.t.sup.2]([[theta].sub.0]) + [v.sub.t]([[theta].sub.0]), (1)
where [[mu].sub.t]([[theta].sub.0]) and [[sigma].sub.t.sup.2]([[theta].sub.0]) parameterize the conditional mean and the conditional variance of [y.sub.t] given the information [F.sub.t-1] up to time t - 1. We let [y.sub.1.sup.m] := ([y.sub.1],..., [y.sub.m]) denote the finite random sequence of Y and by [P.sub.[[theta].sub.0].sup.m] ([P.sub.*.sup.m]) the m-dimensional marginal distribution of [y.sub.1.sup.m] induced by [P.sub.[theta].sub.0] ([P.sub.*]). Model (1) covers a broad class of well-known parametric models for time series. A general example is the following.
Example 1. Double-threshold AR(1)-ARCH(1) models with volatility asymmetries (see, e.g., Li and Li 1996; Glosten, Jagannathan, and Runkle 1993) assume the specification
[[mu].sub.t]([[theta].sub.0]) = [[rho].sub.0] + ([[rho].sub.1] + [[rho].sub.2][d.sub.1,t-1])[y.sub.t-1],
[[sigma].sub.t.sup.2]([[theta].sub.0]) = [[alpha].sub.0] + ([[alpha].sub.1] + [[alpha].sub.2][d.sub.2,t-1]) X ([y.sub.t-1] - [[rho].sub.0] - ([[rho].sub.1] + [[rho].sub.2][d.sub.1,t-2])[y.sub.t-2])[.sup.2] + [[alpha].sub.3][d.sub.1,t-1], (2)
with the dummy variables [d.sub.1,t-1] = 1 if [[rho].sub.0] + [[rho].sub.1][y.sub.t-1] > and otherwise and [d.sub.2,t-1] = 1 if [[epsilon].sub.t-1]([[theta].sub.0] < and otherwise.
Model (1) includes linear autoregressive (AR) models as straightforward special cases. Kunsch (1984) defined a time series IF in this context and derived an optimal bounded-influence estimator for the parameters of an AR(p) model. Martin and Yohai (1986) provided bounded-influence estimators for AR and moving average (MA) models and studied the asymptotic bias implied by additive outliers....
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