Abstract
We analyze the properties of various methods for bias-correcting parameter estimates in both stationary and non-stationary vector autoregressive models. First, we show that two analytical bias formulas from the existing literature are in fact identical. Next, based on a detailed simulation study, we show that when the model is stationary this simple bias formula compares very favorably to bootstrap bias-correction, both in terms of bias and mean squared error. In non-stationary models, the analytical bias formula performs noticeably worse than bootstrapping. Both methods yield a notable improvement over ordinary least squares. We pay special attention to the risk of pushing an otherwise stationary model into the non-stationary region of the parameter space when correcting for bias. Finally, we consider a recently proposed reduced-bias weighted least squares estimator, and we find that it compares very favorably in non-stationary models.
Highlights
It is well-known that standard ordinary least squares (OLS) estimates of autoregressive parameters are biased in finite samples
Through a simulation experiment we investigate the properties of the analytical bias formula and we compare these properties with the properties of both standard OLS, Monte Carlo/bootstrap generated bias-adjusted estimates, and the weighted least squares approximate restricted likelihood (WLS) estimator recently developed by Chen and Deo [30], which should have reduced bias compared to standard least squares
- The analytical bias formula and the bootstrap approach both yield a very large reduction in bias compared to OLS, when the model is highly stationary
Summary
It is well-known that standard ordinary least squares (OLS) estimates of autoregressive parameters are biased in finite samples. In a multivariate context analytical expressions for the finite-sample bias in estimated vector autoregressive (VAR) models have been developed by Tjøstheim and Paulsen [26], Yamamoto and Kunitomo [27], Nicholls and Pope [28], Pope [29], and Bao and Ullah [23]. We analyze the finite-sample properties of bias-correction methods (both bootstrap and analytical methods) in the presence of skewed and fat-tailed data, and we compare a parametric bootstrap procedure, based on a normal distribution, with a residual-based bootstrap procedure when data are non-normal Among other things, this analysis will shed light on the often used practice in empirical studies of imposing a normal distribution when generating bootstrap samples from parameter values estimated on non-normal data samples.
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