On limiting distributions in explosive autoregressive processes
Let \[ Y_n= \beta'\widetilde{Y}_{n-1}+ \varepsilon_n, \quad\varepsilon_n\text{ i.i.d. } N(0,1), \quad n=1,2,\dots \] be the \(p\) th order purely explosive autoregressive process where \(\beta= (\beta_1,\dots, \beta_p)'\) is a real \(p\)-vector, which is unknown and \(\widetilde{Y}_j= (Y_j,\dots, Y_{j-p+1})'\). Let \(A_n= \sum_{j=1}^n \widetilde{Y}_{j-1} \widetilde{Y}_{j-1}'\). Then, the maximum likelihood estimator \(\widehat{\beta}_n\) is given by \[ \widehat{\beta}_n= A_n^{-1} \sum_{j=1}^n \widetilde{Y}_{j-1} \widetilde{Y}_j'. \] The limiting distributions of \(\widehat{\beta}_n\) (with a random or nonrandom normalization) are known, but the detailed proofs have been omitted in the literature. The author gives detailed proofs for these results.
scientific article; zbMATH DE number 1124489