Lag Length Selection and the Construction of Unit Root Tests with Good Size and Power

It is widely known that when there are negative moving average errors, a high order augmented autoregression is necessary for unit root tests to have good size, but that information criteria such as the AIC and BIC tend to select a truncation lag that is very small. Furthermore, size distortions increase with the number of deterministic terms in the regression. We trace these problems to the fact that information criteria omit important biases induced by a low order augmented autoregression. We consider a class of Modified Information Criteria (MIC) which account for the fact that the bias in the sum of the autoregressive coefficients is highly dependent on the lag order k. Using a local asymptotic framework in which the root of an MA(1) process is local to -1, we show that the MIC allows for added dependence between k and the number of deterministic terms in the regression. Most importantly, the k selected by the recommended MAIC is such that both its level and rate of increase with the sample size are desirable for unit root tests in the local asymptotic framework, whereas the AIC, MBIC and especially the BIC are less attractive in at least one dimension. In monte-carlo experiments, the MAIC is found to yield huge size improvements to the DF(GLS) and the feasible point optimal P(t) test developed in Elliot, Rothenberg and Stock (1996). We also extend the M tests developed in Perron and Ng (1996) to allow for GLS detrending of the data. The M(GLS) tests are shown to have power functions that lie very close to the power envelope. In addition, we recommend using GLS detrended data to estimate the required autoregressive spectral density at frequency zero. This provides more efficient estimates on the one hand, and ensures that the estimate of the spectral density is invariant to the parameters of the deterministic trend function, a property not respected by the estimation procedure currently employed by several studies. The MAIC along with GLS detrended data yield a set of Mbar(GLS) tests with desirable size and power properties.