Econometrics: One Thing That I Have Never, Ever Understood. Never. And Probably Never Will…: Very Late Thursday Focus for September 18, 2014

The smart David Giles writes things that thousands of time-series econometricians out of the San Diego tradition have written, and continue to write:

David Giles: The (Non-) Standard Asymptotics of Dickey-Fuller Tests: “One of the most widely used tests in econometrics…

…is the (augmented) Dickey-Fuller (DF) test. We use it in the context of time series data to test the null hypothesis that a series has a unit root (i.e., it is I(1)), against the alternative hypothesis that the series is I(0), and hence stationary…

A stationary time series is one in which (once a deterministic non-stochastic trend is removed) you are willing to bet at very heavy odds that if you look far into the future the variable will still be close to what you calculate as its past sample average. A non-stationary time series is one that is not. This is kinda important: whether (and how much) radical uncertainty there is about the long run is a thing. This is a big issue if a piece of what enters your model is not what the time series has been over the course of your sample but rather what the agents in your model anticipate that the time series will be in the distant future. Things that look very well-behaved from today’s perspective and in the past may well have underlying generating processes that are not so well-behaved in the future.

But consider for the white-noise innovation ε the time series:

x_t = x_t-1 + ε_t – (1-θ)ε_t-1

• For θ = 0 the time series is stationary: it is x_t = ε_t.

• For θ > 0 the time series is non-stationary: it is x_t = ε_t + θ(ε_t-1 + ε_t-2 + ε_t-3 + … + ε₁)

Because the null contains generating processes that are close in distribution over any finite sample to the alternative, no test can ever have any power against the generalized alternative. It can have power over the alternative that the time series is non-stationary with θ > 1/2, or θ > 1/10. But not against the alternative that time series is non-stationary with θ > 0. When the variable that enters one’s model is the present value of expected future x’s, for a real interest rate of r the impact on that present value of an innovation ε is:

(1 + θ/r)ε_t

For small values of r and time series less than centuries in length, points in the null against which Giles’s tests have no power at all are economically very different indeed from the stationary alternative.

Dave Giles might respond that the null hypothesis is, obviously, not that the time series is non-stationary but that the time series is non-stationary and has a low-order ARIMA representation in which all of the moving-average coefficients are identically zero.

But where does the assumption that all of the time series we are interested in have (or ought to have, or can be modeled as having) low order ARI representations come from? And why should we believe it?