Essentials of Time Series for Financial Applications

(5.22)

we have that by construction ${\overline{\sigma }}^2=\frac{1}{T}{\sum }_{t=1}^T{(R_t-{\hat{\mu }}_{t|t-1})}^2$, the sample variance of the data set. Such a restriction, called variance targeting, is often imposed in practical applications because it guarantees that, for a given data set, the ARCH(p) process will yield an unconditional variance that is exactly equal to the sample variance, which seems a sensible outcome. Note that this need stems from the common practice of estimating ARCH models using ML methods: under ML, the constrained optimization algorithm will aim at maximizing the probability ...