Ols Bias in a Nonstationary Autoregression

Abstract

An analytical formula is derived to approximate the finite sample bias of the ordinary least-squares (OLS) estimator of the autoregressive parameter when the underlying process has a unit root. It is found that the bias is expressible in terms of parabolic cylinder functions which are easy to compute. Numerical evaluation of the formula reveals that the approximation is very accurate. The derived formula inspires a heuristic approximation, obtained by leastsquares fitting of the asymptotic bias. More importantly, the formula proves analytically that the bias declines at a rate which is slower than the consistency rate, thus explaining some previous simulation findings. A case where the bias increases with the sample size is also given.

On Efficient Simulations in Dynamic Models*

“…Their appearance is in line with some analytical results on autoregressions; see Abadir (1993b). Powers of |α| T can be thought of as dummies representing unit roots asymptotically and near nonstationarity when T is finite; since:…”

“…For k = 1 and with DGP (1), this condition is violated for AV3 and AV4 when α = 0 as we have seen before. However, when α 6 = 0, moment generating functions (White (1958(White ( , 1961, Abadir (1993b) and references therein) show that the distribution of b β is the mirror image of that of b α, thus warranting the use of the technique of AV4 to simulate moments of any order or any other density-related properties such as quantiles. In addition, the other AVs considered in this work satisfy the condition for the first two moments.…”

Bias in the estimation of the mean reversion parameter in continuous time models

“…1 Standard estimation methods, such as least squares (LS), maximum likelihood (ML) or generalized method of moments (GMM), produce biased estimators for the mean reversion parameter. The bias is essentially of the Hurwicz type that Hurwicz (1950) developed in the context of dynamic discrete time models.…”

The finite-sample effects of VAR dimensions on OLS bias, OLS variance, and minimum MSE estimators

“…Abadir (1993) uses some results on moment generating functions to derive a high-order closed form (integral-free) analytical approximation to the univariate finite-sample bias of φ given Model A, k = p = 1, and with |φ| = 1. The final expression is based upon parabolic cylinder functions, and is computationally very efficient.…”

“…Dickey andFuller (1981, p. 1064), and we distinguish here between these approximations and the rigorous response surface approach that is used in this paper. Despite the fact that only 5 datapoints are used in the derivation of (2), it is accurate in-sample to 5 decimal places for bias, and is more accurate than the special function expression (see Abadir (1993 , Table 1)). We found that (2) also performs very well out-of-sample, at least to 1 decimal place of −100×bias.…”