Higher-Order Conditional Moment Tests: A Simple Robust Approach Yi-Ting Chen Institute of Economics Academia Sinica (Incomplete draft) Abstract In this paper, we propose a simple approach to purge the estimation uncertainty effect on the higher-order conditional moment (CM) tests for checking the standardized error dis- tributions of the GARCH-type models. Our approach is different from the Newey (1985)- Tauchen (1985) and Wooldridge (1990) approaches in both the ideas and results. By utilizing the sample mean and variance of the standardized residuals, we establish a class √ of higher-order CM tests that are free of the estimation uncertainty and robust to the T - consistent estimators of the partially specified (conditional mean and variance) models. Importantly, our test statistics do not depend on the conditional mean and variance deriv- atives and hence they are model-invariant. This is a very attractive property in view of practical applications. As demonstrative examples, we apply our approach to generating some simple tests for the predictability of asymmetry and for the symmetry, normality, and some non-Gaussian distributions of standardized errors. Our approach encompasses the skewness-kurtosis-based normality test of Kiefer and Salmon (1983). Keywords : higher-order conditional moment test, estimation uncertainty, normality, symmetry, standardized error distribution. JEL Classifications : C12, G19 † Correspondence to: Yi-Ting Chen, Institute of Economics, Academia Sinica, Taipei 115, Taiwan. Tel:+886-2-27822791-622; Fax:+886-2-27853946; E-mail address: ytchen@gate.sinica.edu.tw ‡ This study is supported by the research grant NSC 94-2415-H-001-00.
1 Introduction The GARCH-type models have been widely used in modelling financial volatilities. Given their conditional mean specifications, such models interpret the return volatilities by using their conditional variance specifications. For a variety of well documented economic and statistical reasons, like Value-at-Risk evaluation, estimation efficiency, and density fore- cast, it is undoubtedly important to extend such a model from a partially specified (con- ditional mean and variance) model to a fully specified (conditional distribution) model. It should be very useful to have some simple and valid tests that can help us to explore the unknown standardized error distribution in this bottom-up model-building process. In the spirit of the conditional moment (CM) testing method of Newey (1985), Tauchen (1985), and Wooldridge (1990), we may establish such tests by using the higher-order moments, such as but not restricted to the skewness and kurtosis, of the standardized residuals of the estimated model, if the problem of estimation uncertainty is properly dealt with. This problem, also known as “Durbin’s problem”, is due to the fact that the parameter estimation generally makes the asymptotic null distribution of a standardized-residuals- based test statistic different from that of its standardized-errors-based counterpart. More- over, this difference also tends to be model-specific and estimation-method-specific. There- fore, we may not establish the asymptotically valid higher-order CM tests without consid- ering this effect in some proper way. This also reminds us that the existing tests designed for a specific model (estimation method) may not be always valid for general models (es- timation methods). Nonetheless, it is often overlooked in practice. For examples, we can see that the Jarque-Bera (1980, JB) normality test is routinely applied to the standardized residuals of GARCH-type models, even though this test was originally designed for the lin- ear regression with an intercept and conditionally homoskedastic errors and the ordinary least squares (OLS) method. Theoretically, it should be the Kiefer-Salmon (1983, KS) test, a variant of the JB test, rather than the conventional JB test, that is ensured to be valid in the presence of estimation uncertainty; see Bontemps and Meddahi (2005a). The classical Kolmogorov-type test in such applications also suffers from a similar problem; see Bai (2003). The Newey (1985)-Tauchen (1985) approach and the Wooldridge (1990) approach con- duct different strategies to deal with this problem. On the basis of the maximum likeli- hood (ML) method, the former constructs the asymptotically valid tests by calculating the estimation uncertainty. By contrast, the latter establishes the asymptotically valid tests by purging this undesirable effect based on a regression-type “orthogonal transformation”. Recently, Bontemps and Meddahi (2005b) applied a method similar to the Wooldridge 1
approach to establishing their moment-based tests for the standardized error distribution. Compared to the Newey-Tauchen approach, the Wooldridge approach has two appealing √ properties. First, it is robust to the T -consistent estimators of the partially specified model with T denoting the sample size. Second, it can check the predictability of the stan- dardized error distribution without assuming a fully specified model. These two properties are both due to the fact that this approach purges, rather than calculates, the estimation uncertainty. Nonetheless, these two approaches are both dependent on the conditional mean and variance derivatives in their practical applications, so that the resulting higher- order CM tests are not model-invariant. Indeed, the GARCH-type models typically imply certain recursive types of the conditional mean and variance derivatives that may not be simple to compute, especially when the models are complicated. It is therefore very desirable to have other simple tests that can be free of this inconvenience (model-variant non-robustness) in view of practical applications. In this paper, we propose a simple approach for this purpose. In should be noted that, unlike the Newey-Tauchen and Wooldridge approaches that may also be applied to establishing the conditional mean and variance tests, our approach is focused on the higher- order CM tests. Our idea is quite different from these two existing approaches. Intuitively, given the partially specified model, the sample mean and variance of standardized resid- uals should contain no information about the distributional shape of standardized errors, but they may contribute very important information about the estimation uncertainty. Consequently, we may utilize these two lower-order moments to purge this undesirable effect on the higher-order CM tests. Following this idea, we establish our tests for check- ing the (un-)predictability of the standardized error distribution and the adequacy of the postulated unconditional standardized error distribution. Testing these two hypotheses has quite important implications on building the full specified GARCH-type model, as we will discuss. Interestingly, the proposed tests share the same appealing properties with the higher-order CM tests generated by the Wooldridge approach. However, our tests are free of the conditional mean and variance derivatives, and hence they are model-invariant. This is a very attractive convenience (model-invariant robustness) not shared by the tests derived from the Newey-Tauchen and Wooldridge approaches. The remainder of this paper is organized as follows. In Section 2, we discuss the prob- lem of estimation uncertainty and the existing approaches. In Section 3, we demonstrate our approach and the proposed tests. In Section 4, we illustrate the applicability of our approach in generating some simple and valid tests for the predictability of asymmetry and the symmetry, normality, and non-normal distributions of standardized errors. As we will see, this includes the KS test as a particular case. (Sections 5 is the simulation 2
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