Median-Unbiased Estimation of Higher Order Autoregressive/Unit Root Processes and Autocorrelation Consistent Covariance Estimation in a Money Demand Model

It is shown that the Newey-West (1987) Heteroskedasticity and Autocorrelation Consistent (HAC) covariance matrix estimator can greatly understate the standard errors of OLS regression coefficient estimates in finite samples, and therefore comparably overstate t-statistics. Although the bias vanishes in infinite samples and is tolerable in samples as small as 10, it can lead to t-statistics that are too high by a factor of 1.7-2.2 with a sample size in the range 65-1200 and first order autoregressive serial correlation with AR coefficient 0.9. The Exactly Median Unbiased estimator of Andrews (1993) for a directly observed AR(1) process is extended to the case of an AR(p) process that is only indirectly observed via OLS regression residuals. By allowing the maximum permitted order to increase without limit with the sample size, the estimator consistently estimates a stationary process with any autocovariance function. It also provides a unit root test (and therefore a test for cointegration of the regressors) that is exact up to median unbiased estimates of the higher order persistences. These Median Unbiased Autoregressive (MUAR) estimates of the autocovariance function are then used to construct an Autocorrelation Consistent (MUAR-AC) covariance matrix for the OLS coefficient estimates. Applied to a simple model of US demand for narrow money M1-S (official M1 + estimated Retail Sweep Accounts), it is found that a unit root in the errors and therefore absence of cointegration can be at least weakly rejected (at the 10% test size). The MUAR-AC standard errors are 128-153% higher than HAC standard errors, or equivalently, HAC t-statistics for any hypothesis concerning the coefficients are 128-153% too large. Despite the greatly increased MUAR-AC standard errors, the income elasticity and interest semielasticity of demand for M1-S remain highly significant, suggesting that narrow money (currency plus all checking accounts) may still be a useful indicator of monetary policy, and that the Fed should resume collecting direct data on it.