Reply to B.M. Pötscher's comment on ‘adaptive estimation in time series regression models’

Abstract

Nearly all of Pbtscher’s comment on my 1992 paper is taken up with three accusations, each of which is wrong and easily dismissed. Potscher also finds and reports two typographical errors. As well, the reviewer of Potscher’s comment finds two typographical errors and one substantive but minor error, which I correct below. For brevity (P) refers to Potscher’s comment and (S) to my original paper. First, and to set the record straight, the two typographical errors Potscher noted are that a superscript T was omitted from 0: in Definition 2.1 and that a bracket was misplaced in Eq. (2.2). The reviewer of Potscher’s comment also noted that: (i) the quantity 2 was omitted from the right-hand side of the equation for the log-likelihood ratio; (ii) the incorrect words ‘and only if’ appeared in the sentence immediately preceding Definition 2.1; and (iii) my treatment of initial values in constructing the nonparametric density estimator was flawed. To correct the flaw, the definition of the estimator should be changed so that the support of the estimator includes the actual initial values. In the case of an AR(p) process, this means that (4.3) should be defined only for residuals from p + 1, . , T rather than from 1, . . . , T as written. Turning to Potscher’s three accusations, he first asserts that the principal theorem of(S), Theorem 4.1, is incorrect as it stands and that ‘(S) does not come close to providing a rigorous proof’ of the theorem. Potscher bases this on the observation that if one assumes the true order of the process is unknown, and estimates an ARMA model with p > 1 and CJ > 1 when the true order is p = 0 and 4 = 0 (that is, white noise), then the information matrix for 3 would be singular. This observation is certainly true but it has no bearing on Theorem 4.1, which is derived under the assumption that the true order of the process is known. Thus estimators of the ARMA model using values other than the true values of p and q are ruled out by assumption. That Theorem 4.1 takes p and