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Diagnostic Checking For Regression Models In Time Series

Posted on:2008-12-09Degree:DoctorType:Dissertation
Country:ChinaCandidate:J H WuFull Text:PDF
GTID:1100360212991441Subject:Probability theory and mathematical statistics
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Time series is a data series observed over time which can be found in almost all fields such as economic, finance, technology, astronomy, geography, meteorology, medicine, biology and etc. Statistical analysis and inference of time series is named time series analysis. In recent, financial time series analysis have received a growing amount of attention. Engle (1982) argued originally that the conditional variance of the error may be time-varying in studying the variance of UK inflation and then proposed the autoregressive conditional heteroscedasticity (ARCH(p)) model which is well known for all. Since his seminal work, time series with conditional heteroscedasticity have received a great of attention in both the theory and the applications. In the literature, various GARCH models have been proposed to fit the actual economic/finance data with different properties. Meantime, all various parametric or nonparametric estimation methods have been suggested and investigated. However, is the model valid? that is, is the data really from the model? The problem has not received more attention which it deserves. In fact, this is a problem on model checking. In the well-known Box-Jenkins time series modeling three steps: modeling, parameter estimation and model checking, their roles should be the same. As argued in the monograph by Li , however, a lot has been done for the first two steps, and the third (model checking) has not received the attention it deserves. For the GARCH models which have received a great of attention in the last two decades, there are a few papers focusing on model checking.In classical regressions, there are largely two major classes of techniques used for model diagnostic checking which are locally smoothing and globally smoothing methods. For the former, the tests require nonparametric estimation of mean regression function and then often suffer the curse of dimensionality. Many tests were proposed to avoid the severe dimensionality problems by globally smoothing methods. The tests do not need nonparametric smoothing, but are less sensitive to the high frequency alternative. Therefore, the pros and cons of the two methodologies are fairly clear. Moreover, the above two methods is used in the case in which the response is an univariate variable. Therefore, the thesis proposes some new approaches to constructing tests for checking the adequacy of regression models in time series. It is worthwhile to point out that the time series considered in this thesis is the univariate or multivariate, the form of the regression function is general, and the regressors can be several lagged variables.We first study model checking for the general autoregressive models (including mean regression models and variance regression models in the time series with heteroscedastic-ity). By averaging the weighted residuals, we construct a score type test statistic. The tests have the following feathers: in the null model, they are asymptotically chi-squared and then they are trackable, they are sensitive to alternative and can detect the directional alternative converging to the null with the parameter rate, they involve weight functions, which provides us with the flexibility to choose scores for enhancing power performance, especially under directional alternatives. For a directional alternative, the optimal score type test is investigated. And for a class of alternatives, we construct asymptotically distribution-free maximin test which has many desirable properties. A possibility to construct score-based omnibus tests is discussed when the alternative is saturated. It is worthwhile to point out that this is the first paper to study theoretically power properties in diagnostic checking for GARCH-type models. As a product, we also study the asymptotically properties of parameter estimation when the parameter model is not correctly specified.Note that when the sample is small, the power of the tests maybe is not high, which is resulted from the plug-in estimation in the construction of our score-type test. Therefore, we develop the nonparametric Monte Carlo test (NMCT) approach in dependent data case. By the developed NMCT approach, we can determine the reject value without the plug-in estimation and enhanced the power of the test when the sample is small. Simulation results show that when the sample is large or moderate, the NMCT approach is not better than the test with plug-in estimation. The reason is that our score test performs well when the sample is not small. When the sample is very small, the NMCT shows its usefulness. That is, when the sample is small, the power of the tests with the critical values determined by the NMCT approach is higher than that of the limiting distribution.Moreover, to avoid the plug-in estimation, we construct the empirical likelihood based score type test which is self-invariant. The test shares some desirable properties with parametric likelihood such as Bartlett correctability and Wilks' theorem. On the other hand, the resulted test shares many desirable feathers of score-type tests: it is asymptotically chi-squared under the null hypothesis and can detect the alternatives converging to the null at a parametric rate. It is worthwhile to point out that the naive EL-based tests are found to be not of Wilks' phenomenon in the study. So, a bias correction technique is proposed in the construction of the EL-based score type tests and thenthe adjusted ones are of Wilks' phenomenon.In the actual applications, multivariate time series have been found more and more useful. The univariate conditional heteroscedasticity models were extended to the multivariate ones almost as soon as the original paper on ARCH was published. For both parameter estimation and model checking, it is more difficult to do in multivariate GARCH-type models than in univariate GARCH-type models. It is reported by the paper published in the Journal of Applied Econometrics (2006) that further development of multivariate diagnostic tests is one of ten open issues/research topics in multivariate GARCH-type models and that progress in this issue would greatly contribute to the theory and practice of multivariate GARCH-type models. Usually, any direct extension of existing methodologies cannot construct powerful tests. In fact, we should pay particular attention on the correlation between the components of the vector variables in both the theory and the applications. In the thesis, we study the model checking for the vector autoregressive models and the multivariate GARCH-type models through various techniques. Specifically, in checking the adequacy of vector autoregressive models, we construct the test statistics by averaging each weighted component of the (vector) residuals. To avoid the plug-in estimation, we develop the empirical likelihood based score type test by corrected-bias techniques, and the test is self-invariant. In checking the adequacy of variance model of multivariate GARCH-type models, we construct the test statistics by averaging the weighted function of standardized residuals. Also we study theoretically the power of all the above tests.Some simulation studies are carried through and the applications to some real data set are illustrated, which show the usefulness of our results in the thesis.
Keywords/Search Tags:Conditional Heteroscedasticity, Empirical Likelihood, Empirical Likelihood Ratio Test, Model Checking, Multivariate GARCH-type Models, Nonparametric Monte Carlo Test Method, Quasi-maximum Likelihood Estimation, Score-type Test, Time Series
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