In a recent paper Engle and Sheppard (2001) have used a “target variance” approach to estimate a class of multivariate generalized autoregressive conditional heteroskedasticity (GARCH) models. The question we pose here is how to derive the asymptotic distribution of the estimators and find the correct standard errors for a univariate version of this. Suppose that

where ε_t is independent and identically distributed mean zero and variance one. The parameters are assumed to be positive, ω,β,γ > 0, and satisfy

Under (1), the unconditional variance, σ², is well defined and is given by

We can reparameterize the model in terms of (σ²,β,γ):

The target variance approach is to choose the estimator of θ = (β,γ) as

where

This method has a nice computational advantage over the usual GARCH estimation because the optimization is only with respect to the two parameters β,γ, and avoids the usual problem where there is a high covariance between the maximum likelihood estimates of ω and β.

Derive the asymptotic variance of

and suggest consistent standard errors.