# Econometric test for detecting change from trend stationary to stationary process

I am stuck with the following problem from research. Is there an econometric test which detects change from trend stationary to stationary process.

I tried to search for this but the literature is quite vast. Can somebody point to the relevant literature ?

Any help is appreciated.

• "Trend stationary": Deterministic trend or stochastic (unit root)? Sep 6 '15 at 23:30
• I am looking for deterministic trend
– anup
Sep 7 '15 at 6:44

The OP is examining a deterministic trend, and whether it stops being present after a point in time. So the model could be something like

$$y_t = \alpha t + \beta y_{t-1} + u_t,\;\;\; t=1,...,T_A$$

$$y_t = \beta y_{t-1} + u_t,\;\;\;\; t=T_A+1,...,T$$

So we want to test whether $\alpha =0,\;\; t=T_A+1,...,T$

This is a test for "structural change". Interestingly, in the last 15 years there has been a lot of research around unit root testing if a stuctural break in deterministic components is present, but I could not find anything that tests directly for the disappearance of a deterministic trend.

At least "in spirit", it will be a Chow test (don't lookup the wiki page, it says really nothing), when the point of the structural break is assumed known -for example, we can "see it with our own eyes" because the effect of the trend was strong (steep slope) and "suddenly" the series levels off and starts to fluctuate around what appears to be a constant mean.

If we want to search for the point of structural change to a rather wide interval of observations (typically the middle 70%), then this is a Quandt Likelihood Ratio ("QLR") test.

...And the main question is: Do these tests need modification because a deterministic trend is assumed initially present and then absent?

Hamilton (1994), ch. 16.3 "Asymptotic Inference for an Autoregressive Process Around a Deterministic Time Trend" (pp. 463-472), analyzes in detail the first equation in the above model. By providing the asymptotic distribution of the coefficient estimators (it is not the same as when the autoregressive term is absent - for this last case, see Ch. 16.1) I believe it provides all that is necessary to execute validly a Chow or QLR test.