Econometrica: Jan, 1988, Volume 56, Issue 1

The Student's t Approximation in a Stationary First Order Autoregressive Model

https://doi.org/0012-9682(198801)56:1<119:TSTAIA>2.0.CO;2-F
p. 119-145

J. C. Nankervis, N. E. Savin

The basic model is a Gaussian AR(1) model with an intercept, but no additional exogenous variables. The paper studies the distribution of the t statistic for testing the value of the autoregressive parameter when the model is estimated by least squares. The Monte Carlo estimates of the quantiles of the t statistic show that Student's t is not a satisfactory approximation for sample sizes typical in economic applications. The main problem is not the shape of the distribution of the t statistic, but its location. Adjusting the t statistic so that it has the same mean and standard deviation as Student's t, the distribution of this adjusted t statistic is accurately approximated by Student's t. Techniques are presented for accurately approximating the mean and standard deviation of the t statistic such that the adjusted t statistic can be readily calculated in practice. The analysis is extended in two directions. The first is to examine the effect of introducing an exogenous variable into the basic model. In the expanded model Student's t also accurately approximates the distribution of the t statistic for testing the autoregressive parameter and for testing the coefficient of the exogenous variable after these t statistics are adjusted for mean and standard deviation. The problem of obtaining a feasible adjustment procedure is that the moments of these t statistics now depend on nuisance parameters. The second is to examine the robustness of the results in the basic model to several nonnormal error distributions. For each nonnormal distribution the t statistic is adjusted using the mean and standard deviation appropriate for the case of normal errors and then the distribution of the modified t statistic is compared with that of Student's t.