Change-point estimation in ARCH models.

*(English)*Zbl 0997.62068From the paper: This paper studies ARCH models defined by the equations
\[
r_k=\sigma_k \varepsilon_k, \quad\sigma^2_k= a+\sum^\infty_{j=1} b(j)r^2_{k-j}, \tag{1}
\]
where the \(\varepsilon_k\) are independent and identically distributed (i.i.d.) errors and \(a\) and the \(b(j)\) are non-negative constants. Since their introduction ARCH type models have become perhaps the most popular and extensively studied financial econometric models. We study the estimation of a change-point in (1). Note first that if the \(\varepsilon_k\) have mean zero and unit variance and the sequence \(\{r_k\}\) is weakly stationary, then
\[
\text{var} [r_k]= a\left/ \left(1-\sum^\infty_{j=1} b(j) \right.\right). \tag{2}
\]
Suppose now that the parameters \(a\) and \(b(j)\) change at an unknown point \(k^*\) in such a way that the variance given by (2) changes. We are interested in estimating the change-point \(k^*\). This problem is different from the problem of testing for structural changes where \(k^*\) is assumed to be known and one tests whether a change in model structure has taken place at \(k^*\).

Besides the change-point problem we study the cross-covariance function for ARCH models. Bounds for the cross-covariance function are derived and explicit formulae are obtained in special cases. Consistency of a CUSUM type change-point estimator is proved and its rate of convergence is established. A Hájek-Rényi type inequality is also proved. The results are obtained under weak moment assumptions.

Besides the change-point problem we study the cross-covariance function for ARCH models. Bounds for the cross-covariance function are derived and explicit formulae are obtained in special cases. Consistency of a CUSUM type change-point estimator is proved and its rate of convergence is established. A Hájek-Rényi type inequality is also proved. The results are obtained under weak moment assumptions.

##### MSC:

62M10 | Time series, auto-correlation, regression, etc. in statistics (GARCH) |