Highly Insignificant F-Ratios

原文摘要

OCCASIONALLY IN EMPIRICAL WORK, a highly insignificant test statistic is encountered. A rule that the null hypothesis not be rejected when the value of a statistic falls below a threshold which is given by the tester's desired Type I error probability 8 < implies a fortiori that the null is not rejected when the probability that the statistic is smaller than the realized value is less than 8. We suggest an interpretation of this outcome in time series problems. For ease of exposition we focus on the usual F-ratio which tests that an intercept or a slope coefficient of a trending explanatory variable is zero, though similar results hold more generally. Conditions are given such that the F-ratio converges in probability to zero when the disturbances are I(d) for some d <0. A covariance stationary I(d) process is defined to have spectral density f(A) = 11 - eiA 2dg(A), where 0 <g(O) < oo. The I(d), d < 0, processes include noninvertible ones (that is, ones for which d S due possibly to misguided differencing of a stationary I(d + 1) series, where d < The I(d), d < 0, processes also include invertible ones (when - 2 <d < 0) which integrate to nonstationary series. With a suitable regression specification, the F-ratio will be seen to provide a consistent diagnostic for departures from unit root behavior in the direction of stationarity or less-nonstationarity. It has less power than many unit root tests, but has attractive features not all shared by these: it is exact in the Gaussian case and approximately valid more generally; it is a diagnostic which a regression package may automatically print out; it is compared with a null distribution of standard type; its critical regions are independent of explanatory variables. Like the usual unit root tests it can be robustified in large samples to permit short- or long-memory parametric or nonparametric autocorrelation under the null. It has a Lagrange multiplier (LM) interpretation against stationary autoregressive alternatives. The zero at the origin of the disturbance spectrum can be of fractional, as well as unit, order. To provide greater generality we go beyond the I(d) class by allowing for the presence of a slowly varying function, and quite general behavior of the spectrum away from the origin. The slowly varying function is reflected in a simple way in the results. Section 2 contains preliminary results on the variance of unweighted and weighted partial sums. In Section 3 these are applied to obtain either an exact rate or an upper bound for the F-ratio, and unit root testing implications are explored. The proofs are in appendices.

摘要来源：OpenAlex