A Generalized Spectral Density Test Of Conditional Autoregressive Heteroscedasticity For Threshold Autoregressive Model

Threshold autoregressive models are widely used in time-series applications. When building or using such a model,it is important to know whether conditional heteroscedasticity exists. This thesis is composed of two sections in which we discuss generalized spectral density test of conditional autoregressive heteroscedasticity for threshold autoregressive model.In section I,we introduct generalized spectral density test.The generalized spectral density test is able to capture all pairwise dependencies including those with zero autocorrelation.The generalized spectral density and its derivative can be used to capture various specific aspects of serial dependence.These include tests of martingale, conditional homoscedasticity,conditional symmetry, and conditional homokur-tosis.In section II, We propose a generalized spectral density test of this hypothesis.We develop the asymptotic properties of the test statistics. The results indicate that the new test is consistent.The main framework of this thesis is following:In section I,we first propose Parzen's kernel-type for the generalized spectral density ,and its estimate is consistent,then we obtain the test statistic M(m,l,p) using this estimator and its derivative, M(m,l,p) has limit distribution N(0,1).The main results can be stated as the following.Lemma I Suppose that Assumptions(A.l),(A.2),(A.3)hold, and p = cn~λ for c ∈ (0, ∞), λ ∈ (0, l),then IMSE(fn, f) →0.Lemma II Suppose that Assumptions(A.2),(A.3),(A.4),(A.5)hold, and p — cn~λ for c ∈ (0, ∞), λ ∈ (0, 1),If {Xt} is iid,then M(m, l,p)→ N(0, 1)in distribution.