Small-sample Properties of Estimators in an ARCH(1) and GARCH(1,1) Model with a Generalized Error Distribution: a Robustness Study

GARCH Models have become a workhouse in volatility forecasting of financial and monetary market time series. In this article, we assess the small sample properties in estimation and the performance in volatility forecasting of four competing distribution free methods, including quasi-maximum likelihood and three regression based methods. The study is carried out by means of Monte Carlo simulations. To guarantee an utmost realistic framework, simulated time series are generated from a mixture of two symmetric generalized error distributions. This data generating process allow to reproduce the stylized facts of financial time series, in particular, peakedness and skewness. The results of the study suggest that regression based methods can be an asset in volatility forecasting, since model parameters are subject to structural change over time and the efficiency of the quasi- maximum likelihood method is confined to large sample sizes. Furthermore, the good performance of forecasts based on the historical volatility supports to use the variance targeting method for volatility forecasting.GARCH, volatility forecasting, Monte Carlo simulation, mixture of generalized error distributions, variance targeting.

Nonparametric estimation of mean and dispersion functions in extended generalized linear models.

In this paper the interest is in regression analysis for data that show possibly overdispersion or underdispersion. The starting point for modeling are generalized linear models in which we no longer admit a linear form for the mean regression function, but allow it to be any smooth function of the covariate(s). In view of analyzing overdispersed or underdispersed data, we additionally bring in an unknown dispersion function. The mean regression function and the dispersion function are then estimated using P-splines with difference type of penalty to prevent from overfitting. We discuss two approaches: one based on an extended quasi-likelihood idea and one based on a pseudo-likelihood approach. The choices of smoothing parameters and implementation issues are discussed. The performance of the estimation method is investigated via simulations and its use is illustrated on several data, including continuous data, counts and proportions.Double exponential family; Extended quasi-likelihood; Modeling; Overdispersion; Pseudo likelihood; P-splines; Regression; Variance estimation; Underdispersion;

Rheological effects in the linear response and spontaneous fluctuations of a sheared granular gas

The decay of a small homogeneous perturbation of the temperature of a dilute granular gas in the steady uniform shear flow state is investigated. Using kinetic theory based on the inelastic Boltzmann equation, a closed equation for the decay of the perturbation is derived. The equation involves the generalized shear viscosity of the gas in the time-dependent shear flow state, and therefore it predicts relevant rheological effects beyond the quasi-elastic limit. A good agreement is found when comparing the theory with molecular dynamics simulation results. Moreover, the Onsager postulate on the regression of fluctuations is fulfilled

Robust estimation of mean and dispersion functions in extended generalized additive models.

Generalized Linear Models are a widely used method to obtain parametric estimates for the mean function. They have been further extended to allow the relationship between the mean function and the covariates to be more flexible via Generalized Additive Models. However the fixed variance structure can in many cases be too restrictive. The Extended Quasi-Likelihood (EQL) framework allows for estimation of both the mean and the dispersion/variance as functions of covariates. As for other maximum likelihood methods though, EQL estimates are not resistant to outliers: we need methods to obtain robust estimates for both the mean and the dispersion function. In this paper we obtain functional estimates for the mean and the dispersion that are both robust and smooth. The performance of the proposed method is illustrated via a simulation study and some real data examples.Dispersion; Generalized additive modelling; Mean regression function; M-estimation; P-splines; Robust estimation;

Generalized quasi-likelihood ratio tests for varying coefficient quantile regression models

Quantile regression models which can track the relationship of predictive variables and the response variable in specific quantiles are especially useful in applications when extreme quantiles instead of the center of the distribution are interesting. Compared to classical conditional mean regressions, quantile regression models can provide a more comprehensive structure of the conditional distribution of the response variable. Also, they are more robust to skewed distributions and outliers. Therefore, quantile regression models have been applied extensively in many applied areas. Due to its greater flexibility, a varying coefficient regression technique has been extended to the quantile regression models recently. In this dissertation, my aim is to propose a new test procedure, termed as generalized quasi-likelihood (GQLR) test, to test whether all or partial coefficients are indeed constant or of some specific functions for the varying coefficient quantile regression models. The test statistics are constructed based on the comparison of the quasi-likelihood functions under null and alternative hypotheses. The asymptotic distributions of the proposed test statistics are also derived. First, the functional coefficients in a varying coefficient quantile regression model are estimated by applying local linear fitting technique with jackknife method. Then, I construct the generalized quasi-likelihood ratio test statistics to test whether the varying coefficients are of some specific functional forms, including two special cases: testing whether the varying coefficients are known or unknown constants. The asymptotic normality of the proposed test statistic is derived upon the Bahadur representation of the estimators. I also discuss how to estimate the asymptotic variance-covariance matrix and investigate the power of the proposed test procedures in Chapter 2. Secondly, I consider the similar testing procedure to test if partial coefficients in a varying coefficient quantile regression model are constant or of some specific form with other coefficients completely unspecified in Chapter 3. The corresponding generalized quasi-likelihood ratio test statistic is constructed based on comparing the quasi-likelihood functions under the null and alternative hypotheses. The asymptotic distributions of the proposed test statistics for both constancy and specific functional form are derived respectively and the power of the proposed test procedures is also investigated. Finally, to exam the finite sample performance of all test statistics proposed. In Chapters 2 and 3, Monte Carlo simulation studies are conducted respectively at the end of each chapter. I also apply the proposed test methodologies to test if the existing models in the literature used to analyze the Boston house price data are appropriate or not. The simulation results and the real example illustrate the effectiveness and practical usefulness of the proposed test statistics. Chapter 4 concludes the dissertation. I also discuss some future research topics related to this dissertation

Robust Estimation of Mean and Dispersion Functions in Extended Generalized Additive Models

Generalized Linear Models are a widely used method to obtain parametric es- timates for the mean function. They have been further extended to allow the re- lationship between the mean function and the covariates to be more flexible via Generalized Additive Models. However the fixed variance structure can in many cases be too restrictive. The Extended Quasi-Likelihood (EQL) framework allows for estimation of both the mean and the dispersion/variance as functions of covari- ates. As for other maximum likelihood methods though, EQL estimates are not resistant to outliers: we need methods to obtain robust estimates for both the mean and the dispersion function. In this paper we obtain functional estimates for the mean and the dispersion that are both robust and smooth. The performance of the proposed method is illustrated via a simulation study and some real data examples.dispersion;generalized additive modelling;mean regression function;quasilikelihood;M-estimation;P-splines;robust estimation

On the boundedness and nonmonotonicity of generalized score statistics

We show in the context of the linear regression model fitted by Gaussian quasi-likelihood estimation that the generalized score statistics of Boos and Hu and Kalbfleisch for individual parameters can be bounded and nonmonotone in the parameter, making i