garch模型英文讲解.pptVIP

Volatility and ARCH/GARCH Models:A Review Example: ARCH(1) model. Taking derivatives with respect to θ=(ω,α,γ), where γ=K mean pars: Assuming normality, we maximize with respect to θ the function: Note that the δ?/δγ=0 (K f.o.c.’s) will give us GLS.Denote δ?/δθ=S(yt,θ)=0 (S(.) is the score vector) We have a (K+2xK+2) system. But, it is a non-linear system. We will need to use numerical optimization. ? Gauss-Newton or BHHH can be easily implemented. ? Given the AR structure, we will need to make assumptions about σ0 (and ε0,ε1 , ..εp if we assume an AR(p) process for the mean). Alternatively, we can take σ0 (and ε0,ε1 , ..εp) as parameters to be estimated (it can be computationally more intensive and estimation can lose power.) Note: The appeal of MLE is the optimal properties of the resulting estimators under ideal conditions.Crowder (1976) gives one set of sufficient regularity conditions for the MLE in models with dependent observations to be consistent and asymptotically normally distributed. Verifying these regularity conditions is very difficult for general ARCH models - proof for special cases like GARCH(1,1) exists. For GARCH(1,1) model: if E(ln α1,zt2 +β1] 0, the model is strictly stationary and ergodic. See Lumsdaine (1992). If the conditional density is well specified and θ0 belongs to Ω, then ? Common practice in empirical studies: Assume the necessary regularity conditions are satisfied. ? Under the correct specification assumption, A0=B0, where The estimator B0 has a computational advantage over A0.: Only first derivatives are needed. But A0=B0 only if the distribution is correctly specified. This is very difficult to know in practice. We estimate A0 and B0 by replacing θ0 by its estimated MLE value. ? Block-diagonality In many applications of ARCH, the parameters can be partitioned into mean parameters, θ1, and variance parameters, θ2. Then, δμt(θ)/δθ2=0 and, although, δσt(θ)/δθ1≠0, the Information matrix is block-diagonal (un