This paper analyzes autoregressive time series models where
the errors are assumed to be martingale difference sequences
that satisfy an additional symmetry condition on their fourth-order
moments. Under these conditions quasi maximum likelihood estimators
of the autoregressive parameters are no longer efficient in
the generalized method of moments (GMM) sense. The main result
of the paper is the construction of efficient semiparametric
instrumental variables estimators for the autoregressive
parameters. The optimal instruments are linear functions of
the innovation sequence.
It is shown that a frequency domain approximation of the optimal
instruments leads to an estimator that only depends on the data
periodogram and an unknown linear filter. Semiparametric methods
to estimate the optimal filter are proposed.
The procedure is equivalent to GMM estimators where lagged
observations are used as instruments. As a result of the additional
symmetry assumption on the fourth moments the number of instruments
is allowed to grow at the same rate as the sample. No lag
truncation parameters are needed to implement the estimator,
which makes it particularly appealing from an applied point
of view.