Statistical Inference For Several Types Of GARCH Models

In real data applications,financial time series data is a kind of data that researchers are very interested in,such as financial assets returns,options and asset pricing,stock index data.When analysing these data,the traditional linear method is often unable to accurately capture the characteristics of data.For instance,the data of stock return shows the characteristic of volatility,the variance of return is not a constant but changes over time.Researchers proposed nonlinear method to analyse these kind of data,one of a commonly used approach is the conditional heteroscedasticity models:the autore-gressive conditional heteroscedastic(ARCH)model(Engle,1982)and the generalized autoregressive conditional heteroscedastic(GARCH)model(Bollerslev,1986).In these models,the variance is changing over time.Since proposed,ARCH/GARCH models have earned well-deserved attention and are widely used in economics and finance.In this paper,we studied the issues of the statistical inference of several types of GARCH models.The main content is divided into three parts.In the first part,for analysing the influence of external factors on the volatility,we proposed the GARCH model with exogenous explanatory variables,gave the model parameter estimators based on quasi-maximum exponential likelihood estimation(QMELE),and studied the hypothesis testing problem.We presented the estimation performances of model parameters through the simulation studies,and applied our results to a real data exam-ple.In the second part,we focused on the asymmetry of volatility of the return yield series.We weaken the assumptions in the original DTGARCH model and improved the model,proposed the QMELE of model parameters,discussed the statistical properties of estimators.Through simulation studies,we compared the estimation performance of QMELE and standard quasi-maximum likelihood estimation(QMLE).Then,we applied our results to a real example of HSI data.In the third part,we proposed a random coefficient autoregressive model with conditional heteroscedasticity innovation,discussed the QMELE of model parameters and the statistical properties of estimators.Then,we studied the performance of the estimators through simulations and used the model to fit a set of real data.In the following part,we will introduce the main results of our studies.1.Statistical inference for the GARCH model with exogenous explanatory vari-ablesThrough the studies of financial time series data,researchers found that the volatil-ity of data not only change over time,but also influenced by external factors.Therefore,we consider the study of statistical inference for the GARCH models with exogenous explanatory variables.The definition of the model is following.Definition 1 The following model is called the generalized autoregressive condi-tional heteroskedasticity model with exogenous explanatory variables,denoted as GARCH-X model:yt=φYt-1+(?)t,(?)t=ηt(?),(1)ht=ω+(?)+(?)+ψXt,where Yt-1=(1,yt-1,yt-2,…,yt-p)’,Φ=(c,Φ1,…,Φp),ψ=(ψ1,ψ2,…,ψr),Xt=(x1t,x2t,…,xrt)’,i={1,2,……,n},p,q,s,and r are positive integers;{ηt} are i.i.d.noises with E|ηt|=1;Xt denotes the exogenous explanatory variables.Then we consider the parameters estimation problem of GARCH-X model.We use the quasi-maximum exponential likelihood estimation(QMELE)to estimate the parameters.Definition 2 Consider the following functions:Ln(θ)=(?)lt(θ),lt(θ)=log(?)+(?),we look for the minimizer of Ln(θ)on denoted by θrn:θen=arg mine Ln(θ).θen is called the quasi-maximum exponential likelihood estimator(QMELE).Next,we discuss the asymptotic properties of estimators.We need following as-sumptions.(C1)(?)satisfying Equation(1)is strictly stationary and ergodic.The pa-rameter space Θ is compact.ω>0,αi≥0,βi≥0,(?)βi<1,where i=1,2,...,s.(C2)Let A(z)=(?)aizi,B(z)=1-(?)βizi.For each θ∈Θ,A(z)and B(z)have no common root,A(1)≠0,αq+βs≠0.(C3)ηt2 has a nondegeGerate distribution with Eηt2<∞.{ηt} is i.i.d with median 0,E|ηt|=1.Its density function f(x)is continuous and satisfies f(0)>0,(?)f(x)<∞,(?)|f’(x)|<∞.(C4)E(?)<∞ for any ρ∈(0,1),where ξρt=1+(?)ρi|yt-i|.Under assumptions,we give the strong consistency and asymptotic normality ofθen.Theorem 1 Under assumptions(C1)-(C4)} θen→θ0,a.s.,n→∞.Theorem 2 Under assumptions,we have:(1)(?)(θen-θ0)=Op(1),as n→∞,(2)(?)(θen-θ0)(?)N(0,(?)),as n→∞ where ∑0=diag(∑(1),E(2)),To study the influence of exogenous explanatory variables,we consider the hypothesis testing problem and propose the WALD test statistic.The following proposition gives the specific hypothesis test and the test statistic.Proposition 1 Consider the following hypothesis testing problem:H0:ψ=0 v.s.H1:ψ≠0.Null hypothesis H0:ψ0 implies that the model doesn’t have exogenous explanatory variables,while alternative hypothesis H1:ψ≠0 shows the model is influenced by exogenous explanatory variables.Based on the Wald test and let C=(0,0,...,1)1×(2+p+q+s+r),we obtain the Wald test statistic:The test statistic satisfy:Wn(?)x(1)2,n→∞.The simulation results show that the estimation performance of QMELE is much better than QMLE.We use the model to analyse the data of S&P500,and the results imply that the volatility of return series is influenced by external factors.2.Statistical inference for the double threshold generalized autoregressive condi-tional heteroskedasticity modelIn real data applications,financial time series data shows the characteristics of asymmetry.For capturing this asymmetry,Brooks(2001)proposed the double thresh-old generalized autoregressive conditional heteroskedasticity(DTGARCH)model,and the definition is as follows.Definition 3 The following model is called the double threshold generalized au-toregressive conditional heteroskedasticity model,denoted as DTGARCH model:εt～N(0,ht),(?),where It(j)=I(rj-1≤xt-d<rj).In several empirical literature,the model commonly has one threshold parame-ter,namely the model has two regimes,hence we consider the following two-regime DTGARCH model:where Yt-1=(1,Yt-1,Yt-2,…,yt-p)’,φj=(cj,φj1,…,φjp)’,j={1,2},i={1,2,...,n},p,q,s,and d are positive integers,and d≤p;{ηt} are i.i.d.noises with E|ηt|＝1;r is the threshold parameter.For clarity,In the following text,the DTGARCH model refers to the above two-regime DTGARCH model.Then we consider the parameters estimation problem of DTGARCH model.We use the quasi-maximum exponential likelihood estimation(QMELE)to estimate the parameters.For the sake of representation,we give the following notation.Letθ=(γ’,δ’,r’)’be the model parameters with true value θ0=(γ’0,δ’0,r’0)’,whereγ=(φ’1,φ’2),δ(ω1,α11,…,α1q,β11,…,β1s,ω2,α21,...,α2q,β21,...,β2s)’.The pa-rameter space is Θ=Θγ×Θδ×Θr,where Θγ(?)R2(p+1),Θ8(?)R02(q+s+1),Θr,(?)R,R=(-∞,∞)and R0=[0,∞).Assume that Θγ,Θδ,and Θd are compact and θ0 is an interior point in Θ.Then we give out the definition of QMELE.Definition 4 Consider the following functions:where I1t=I(yt-d≤r),I2t=I(yt-d>r).We look for the minimizer of Ln(θ)on Θ,denoted by θn:θn=arg(?)Ln(θ).θn is called the quasi-maximum exponential likelihood estimator(QMELE)of θ0.Next,we discuss the asymptotic properties of estimators.Let λ=(γ’,δ’)’denotes the parameters in our model except the threshold parameter r,then we have Θλ=Θγ×Θδ;and θsndenotes the minimizer of Ln(θ)on Θλfor fixed r.And we need following assumptions.(Cl){yt} is strictly stationary and ergodic,and the parameter space Θ is compact.ωj>0,αji≥0,βji≥0,and(?)βji1,where i=1,2,…,s,j=1,2.(C2)Let Aj(z)=(?)αjizi,and Bj(z)=1-(?)βjizi.For each θ∈Θ,Aj(z)and Bj(z)have no common root,Aj(1)≠0,αq+βs≠0,and let Aj=(aj1,…,αjq)’,Aj≠Aj’if j≠j’.A similar condition holds for Bj=(βj1,...,βis)’,Φj=(Φj1,...,Φjq)’andωj.(C3)ηt2 has a nondegenerate distribution with Eηt2<∞.{ηt}is i.i.d with median 0,E|ηt|=1.Its density function f(x)is continuous and satisfy f(0)>0,(?)f(x)<∞,and(?)|f(x)|<∞.(C4)(?)for any ρ∈(0,1),where(?)Under assumptions,we give the strong consistency and asymptotic normality ofθsn.Theorem 3 If assumptions(C1)-(C4)hold,QMELE is consistent estimation,then,θsn,→θ0,a.s.,asn→∞.Theorem 4 Under assumptions(C1)-(C4),we have:(1)(?),as n→∞,(2)(?)as n→∞;where ∑0=diag(E1,E2),Ω0=diag{Ω1,Ω2),∑i=diag{∑i(1),∑i(1)},i=1,2,Next,the simulation studies compare the estimation performances of QMELE and traditional QMLE.The results show that QMELE has a much better performance than QMLE.Finally,we applied our research to an empirical investigation of stock returns in Hong Kong stock market.In this real data analysing,the QMELE also has a better performance than QMLE,hence the fitting performance of our model is considerable.3.Statistical inference for a random coefficients autoregressive model with condi-tional heteroscedasticity innovationWe study the problem of statistical inference for random coefficients autoregressive model with generalized autoregressive conditional heteroscedasticity innovation.Definition 5 The following model is the random coefficients auto’regressive pro-cesses with generalized autoregressive conditional heteroscedasticity,denoted as RCAR-GARCH model:Then we consider the parameters estimation problem of RCAR-GARCH model.We use the quasi-maximum exponential likelihood estimation(QMELE)to estimate the parameters.Definition 6 Consider the following functions:where(?)We look for the minimizer of on Θ,denoted byθren:θren=argminΘLn(θ).θren is called the quasi-maximum exponential likelihood estimator(QMELE)of θ0.Next,we discuss the asymptotic properties of estimators.We need following as-sumptions.(C1){yt}t=1n satisfying the model is strictly stationary and ergodic.The parameter space Θ is compact.ω>0,αi≥0,βi≥0,(?)βi<1,where i=1,2,…,s.(C2)Let A(z)=(?)αizi,B(z)=1-(?)βizi.For each θ∈Θ,A(z)and B(z)have no common root,A(1)≠0,αc+βs≠0.(C3)ηt2 has a nondegenerate distribution with Eηt2<∞.{ηt} is i.i.d with median 0,E|ηt|=1.Its density function f(x)is continuous and satisfies f(0)>0,(?)f(x)<∞,(?)|f’(x)|<∞.(C4)(?)for any ρ∈(0,1),where ξρt=1+(?)ρi|yt-i|.Under assumptions,we give the strong consistency and asymptotic normality ofθrcn.Theorem 5 Under assumptions(C1)-(C4),θrcn→θ0,a.s.,n→∞,Theorem 6 Under assumptions(C1)-(C4),we have:(1)(?),n→∞,(2)(?),n→∞,where ∑0=diag(∑(1),∑(2)),Next,the simulation studies compare the estimation performances of QMELE and traditional QMLE.The results show that QMELE has a much better performance than QMLE.Finally,we applied our research to an empirical investigation.In this real data analysing,the QMELE also has a better performance than QMLE,hence the fitting effect of our model is considerable.