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Adaptive shrinkage for autocorrelation
Abstract
It has long been known that the sample autocorrelation (ACF) and partial autocorrelation (PACF) functions underestimate the magnitude of correlation in stationary time series. Although finite bias correction formulae can be found their use typically increases estimator variability. On the other hand, shrinking estimators toward zero can reduce variance but increase bias resulting in higher mean squared error for some correlation structures. This paper introduces a novel, computationally efficient, penalised M-estimator for (partial) autocorrelation, with the penalty pushing the estimator toward a target selected from the data. This both encapsulates and differs from previous attempts at penalised estimation for autocorrelation, which shrink the estimator toward the target value of zero. We provide data-driven target and tuning parameters which improve estimation of the ACF and PACF in finite samples and the limit. The estimators of the PACF can be used to efficiently fit autoregressive models, with the resulting estimators satisfying an oracle property. Additionally, we introduce an AIC statistic using the penalised PACF, which provides consistent estimation of the true order within the autoregressive class, a result sorely amiss from this literature to date.
