Adjusted profile likelihood applied to estimation and testing of unit roots
A short review of unit-root econometrics is given from the point of view of testing. The adjusted likelihoods of Cox and Reid (1987, 1993) are presented and applied to the usual AR(1) with constant, an AR(1) process suggested by Bhargava (1986), and an AR(2) process. Biases of the associated maximum-likelihood estimates (MLEs) are pondered briefly. A Wald statistic based on adjusted profile likelihood is proposed. The Cox-Reid adjusted estimate (AE) for the autoregressive coefficient of the unit-root AR(1) model with zero constant is even asymptotically more accurate, in terms of mean-square error (MSE), than the MLE. The derived tests are more powerful than the corresponding Dickey-Fuller tests if the starting value of the process deviates sufficiently from the unconditional mean. An iteratively adjusted estimate is introduced which can also be more accurate than the MLE. We obtain also an estimate and a Wald statistic which are asymptotically distributed compactly and symmetrically around zero under a unit root but the estimate is not consistent in general. The MLE and the AE are consistent not only as the sample size tends to infinity but also when the (absolute value of the) deviation of the starting value from the unconditional mean of the time series is tuned towards infinity. The finding exposes why Wald-kind of tests are more powerful than tests based on standardised coefficients when the starting value lies far from the unconditional mean. The AE and the corresponding Wald statistic are derived for the Bhargava AR(1) model. We obtain the asymptotic distributions of them and simulate the previously unknown finite sample distributions of the MLE and the usual Wald statistic under a unit root. Again the AE is the more accurate estimate. Distortion towards a unit root is pointed out. The adjusted estimate and the Wald statistic follow their asymptotic distributions better than the unadjusted when the process is a unit-root AR(1) with drift or the Bhargava AR(1). Accuracy is gained also under the unit-root AR(2) model. A practical advice is to apply a unit-root test based on the Bhargava model when the process can be assumed to have started from the unconditional mean under the alternative and otherwise a test based on the ordinary AR(1) with constant model. The adjustment often decreases the bias at the cost of variance but it can yield a reduction in both, too, which happens under the Bhargava model and 'typically' under the unit-root AR(2) model. The two most distinctive findings are perhaps that the AE can be asymptotically more accurate than the corresponding MLE or in finite samples when the AE is calculated from an embedding model.