Statistical Inference For Heavy-tailed GARCH Models And Partially Linear Additive Models

Volatility clustering and autocorrelation are the most significant characters that fi-nancial time series data often show.To capture these features,most of the statistical model assumes that the data are dependent on the conditional variance or calibration parameters,of which the most famous and frequently quoted one is autoregressive con-ditional heteroscedasticity(ARCH)model proposed by Engle in 1982.Many schol-ars have carried on the thorough research and extension on the ARCH model,thereby derived a series of volatility models,one of the most widely used is generalized au-toregressive conditional heteroscedasticity(GARCH)model proposed by Boller rslev in 1986.While parameter estimation and diagnostic test for GARCH-type models are two major problems that scholars focused on,the accumulation of a series of theoreti-cal research results in this field has also played an important role in the empirical data analysis of exchange rate and stock price.This paper aims at the smooth GARCH model.We hope to construct an integrated estimation and test system for stationary GARCH series under heavy-tailed distribu-tion situations.By extending Fan’s NGQMLE,we build a new two-step NGQMELE through changing the estimator within the first step from QMLE to QMELE.And then we established it’s consistency and asymptotic normality under the condition of limit-ed first moment and secondary moment of the residuals respectively.This will great-ly lower the moment requirements for the error term.In addition,for the new two-step NGQMELE we proposed two goodness-of-fit test statistics Q(M),Q(2)(M)based on the autocorrelation function of the residuals’ absolute value and square value,and proved their asymptotic properties in limited secondary moment and fourth-order mo-ment cases respectively.The results of numerical simulation and real-data analysis both show that Q(2)(M)will lose power in thick tail situations,and Q(M)is always a better test through thick and thin.Although Q(M)showed good effect on test power under the condition of limit-ed second moment of error term,the actual financial yield rate data is usually fairly thick-tailed(Eεt2= ∞).So we are still looking for a goodness-of-fit test based on our two-step NGQMELE that can be applied to thick tail situation.Until recent years,how-ever,there are few scholars begin to study test of goodness-of-fit under the condition EEεt1 = ∞ and most are based on LAD-type estimators.Chen proposed a sign-based portmanteau test for GARCH model with LAD estimator,which lower the moment con-dition to E|εt|2τ<∞,τ>0.Following this,we proposed a sign-based portmanteau test statistics S(M)based on the pre-mentioned two-step NGQMELE,and under the strictly stationary assumption we proved the asymptotic properties of the test.The sign-based test is suitable for thick tail with limited first moment.The results of numerical simulation shows the perfect performance of S(M),we also carried on the comparative analysis with Q(M),Q(2)(M).Case analysis on the USD to CNY exchange rate reflects the actual application effect of our proposed three statistics.Notice that application of S(M)only need to meet the conditionE|εt|<∞,and therefore is suitable for more financial time series data with thick tail.Besides,the three kinds of test statistics we put forward adapt to different moment conditions respectively,so a comprehensive consid-erations could lead to a more accurate test of goodness-of-fit.So far,we have finished the estimation and test construction for stationary GARCH models under heavy-tailed distribution situations.Using NGQMELE and S(M),Q(M),Q(2)(M)we can derive an appropriate GARCH model to fit most stationary time series data.Besides,there are lots of parametric and nonparametric models fitting time series data,of which GARCH is a popular parametric method,and nonparametric methods include partially linear time series models,nonparametric additive models and semi-parametric single index models.In the second part of this paper,we also focus on fitting time series data by partially linear additive model.Combining the orthogonal series approximation and the adaptive sparse group LASSO regularization,we select the important variables between and within the groups simultaneously.Specially,we propose a two-step algorithm to obtain the grouped sparse estimators.Numerical s-tudies show that the proposed method outperforms LASSO method in both fitting and forecasting.An empirical analysis is used to illustrate the methodology.