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We thank Michael Bleaney, John McDermott, Sam Ouliaris, Eswar Prasad, Peter Wickham and seminar participants at the IMF for valuable comments and suggestions. Nada Mora, Grace Juhn and Ivan Guerra provided excellent research assistance.
The extension of these median-unbiased procedures to pth-order processes is considered in Andrews and Chen (1994); for AR(p) models the procedures are no longer exact, only approximate. Unbiased unit root tests have been previously developed by Stock (1991) and Rudebusch (1992), both for pth-order autoregressive processes with a time trend. See Andrews and Chen (1994) for a description of how these tests compare and contrast with the median-unbiased procedures.
In equation (1), ∈t~iid N(0,σ2) for σ2>0, Y0~N(μ, σ2/(l-α2)) if α ∈ (-1,1), and Y0 is an arbitrary constant or random variable if α =l.
If the distribution of the least squares estimator of α depends only on a and is monotone in α (as in the AR(1) case), then the resultant estimator will have the property of median unbiasedness (Andrews and Chen (1993)).
The size of the bias correction can be large, especially when α is close to one. For example, for a sample size of 60 observations using the model in equation (1), a least squares estimate of α = 0.85 or greater would correspond to a median-unbiased estimate of α = 1; that is, m(1)=0.85.