Asymptotic conditional inference for regular nonergodic models with an application to autoregressive processes (Q797946)

Considering inference problems for parameters of a certain class of stochastic sequences constituting a so-called nonergodic family, the authors show that conditionally on a certain random variable the family can be reduced to a locally asymptotically normal one, and then under the conditional model the standard asymptotic results apply. A detailed application to explosive Gaussian autoregressive processes is given. Let \(\{X_ n\), \(n\geq 1\}\) be an AR(k) process: \((1-\theta B)\prod^{k- 1}_{j=1}(1-\theta_ jB)X_ n)=Z_ n,\) \(n\geq 1\), where \(\{Z_ n\), \(n\geq 1\}\) are i.i.d. N(0,1) variates, \(X_ n=0\), \(n\leq 0\), B is the backward shift operator, and \(|\theta | >\max (1,|\theta_ 1|...|\theta_{k-1}|)\). Conditionally on \(Y=\lim \theta^{- n}\sqrt{\theta^ 2-1}X_ n,\) the sequence \(W_ n=X_ n-E(X_ n| Y)\), \(n\geq 1\), is distributed as the AR(k): \((\theta -B)\prod^{k- 1}_{j=1}(1-\theta_ jB)W_ n=Z_ n,\) \(n\geq 1 (W_ n=0\), \(n\leq 0)\). This is an interesting result of AR processes in itself. Then conditionally on Y some optimality results for estimators and tests based on unconditional likelihood follows.