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.