Noninvertibility and Pseudo-Maximum Likelihood Estimation of Misspecified ARMA Models

B.M. P~TSCHER
University of Maryland
Recently Tanaka and Satchell [ l l ] investigated the limiting properties of local
maximizers of the Gaussian pseudo-likelihood function of a misspecified
moving average model of order one in case the spectral density of the data
process has a zero at frequency zero. We show that pseudo-maximum likelihood
estimators in the narrower sense, that is, global maximizers of the Gaussian
pseudo-likelihood function, may exhibit behavior drastically different from
that of the local maximizers. Some general results on the limiting behavior of
pseudo-maximum likelihood estimators in potentially misspecified ARMA
models are also presented.

1. INTRODUCTION
Tanaka and Satchell [ll] recently investigated the limiting behavior of local
maximizers of the Gaussian pseudo-likelihood function for the moving average
model of order one (MA(1) model), emphasizing the case of a misspecified
model and the case where the spectral density of the data-generating
process has a zero at frequency zero. Their study has been motivated by recent
results in rational expectations models indicating that such models and
data-generating processes naturally occur in this context; see [ll] for further
discussion and references. Tanaka and Satchell [ l l ] show that at least one
local maximizer of the pseudo-likelihood function converges to unity (which
is the parameter value in an MA(1) model corresponding to a spectral density
with a zero at frequency zero) as sample size increases even if the model
is misspecified, as long as the spectral density of the data-generating process
has a zero at frequency zero and satisfies an additional condition (namely,
r > 4 in the notation of [l I]), which essentially states that the sum of the
autocorrelations of the data process after being passed through the filter
(1 - 1)-' is not too large. Of course, this result can be viewed as a "consistency"
result and the condition r > ;can be interpreted as saying that the
degree of misspecification is not too large. Tanaka and Satchell [ll] also conjecture
that no local maximizer converges to unity if r s ;.