Moment Ratio Estimation of Autoregressive/Unit Root Parameters and Autocorrelation-Corrected Standard Errors

A Moment Ratio Estimator is proposed for the parameters of an Autoregressive (AR) model of the error in an Ordinary Least Squares (OLS) linear regression. Although it is computed from the conventional residual autocorrelation coefficients, it greatly reduces their bias, and provides corrected standard errors with far less bias and confidence intervals with far less size distortion than conventional alternatives. The estimator is in the spirit of the Median Unbiased estimator of Andrews (1993) and McCulloch (2008), but is more easily computed and provides smaller standard error bias in most cases. The presence of a unit root in the errors, and therefore the absence of a cointegrating relationship, requires reposing the problem, but does not by itself indicate that an OLS correlation between the variables is spurious. Hypothesis testing is standard, provided it is based on squared quasi-differenced residuals, and not on the squared residuals themselves. Although the present paper is restricted to the AR(1) case, the approach is readily extendable to higher-order AR processes. An exact unit root test similar to that of Andrews (1993) is implemented for the AR(1) case. The Moment Ratio estimator is applied to an income trend line regression, as well as to a monetary base demand function. In both cases, the Moment Ratio autoregressive coefficient estimate is quite close to unity, and a unit root in the errors cannot be rejected. However, the trend slope remains highly significant in the income trend line regression, and both the income elasticity and interest semi-elasticity remain highly significant in the base demand equation, even when a unit root is imposed. It is observed that despite their consistency, the popular HAC standard errors of Newey and West (1987) can greatly overstate the precision of OLS coefficient estimates with sample sizes and serial correlation commonly found in economic studies when, as has become standard, “automatic bandwidth selection” is employed.