Three essays on bias, bias reduction and estimation in autoregressive time series models
This thesis consists of three essays on the subject of autoregressive time series of order one. The first essay derives an approximate bias of the ordinary least squares estimator (OLS) of the autoregressive parameter for series with moderate deviations from a unit root and for a fixed autoregressive coefficient. The result is used to derive the asymptotic distribution of the indirect inference method for (moderately) stationary, (moderately) explosive and explosive series with a fixed coefficient. The essay also shows how one can construct a jackknife and a simple bias-reduced estimator for stationary series by use of the bias function. A simple Monte Carlo experiment provides evidence that the three estimators outperform OLS in terms of their bias reduction capabilities. Given the derived discontinuity of the bias function around the vicinity of unity, the second essay proposes an optimal two-step local to unit root jackknife estimator to try and overcome the problem. This particular version of the jackknife requires knowledge of the variances of the full-sample and sub-sample estimators and the covariances between them. Hence, the essay derives their asymptotic counterparts. Via those asymptotic moments, the essay explains analytically why previous findings have found that using more sub-samples in the construction of the jackknife produces smaller variance. The third essay provides asymptotic theory for local to unit root autoregressive processes with a drift. It is shown that the limiting distribution is a joint normal with a mean zero and variance-covariance matrix which depends on the localising parameter. An interesting feature of this setup is that a consistent estimator of the localising parameter can be constructed. Hence, one can construct a t-statistic which has a standard normal limiting distribution to test the hypothesis of a unit root by directly testing the null of the localising parameter being equal to zero.
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