## Saturday, November 5, 2016

### Snakes in a Room

Teachers frequently use analogies when explaining new concepts. In fact, most people do. A good analogy can be quite eye-opening.

The other day my wife was in the room while I was on the 'phone explaining to someone why we often like to apply BOTH the ADF test and the KPSS test when we're trying to ascertain whether a partcular time-series is stationary or non-stationary. (More specifically, whether it is I(0) or I(1).) The conversation was, not surprisingly, relatively technical in nature.

After the call was over, it occurred to me that my wife (who is an artist, and not an econometrician) might ask me what the heck all that gobbly-gook was all about. As it happened, she didn't - she had more important things on her mind, no doubt. But it forced me to think about a useful analogy that one might use in this particular instance.

I'm not suggesting that what I came up with is the best possible analogy, but for what it's worth I'll share it with you.

First, though, here's a brief summary of the technical issue(s).
• We want to test if our time-series is I(1) or I(0).
• When we apply the ADF test the null hypothesis is that the series is I(1), while the alternative hypothesis is that it is I(0). On the other hand, the KPSS test is based on a null hypothesis that the series is I(0), and the alternative hypothesis is that it is I(1).
• So, the null and alternative hypotheses are reversed when we move from one test to the other.
• We also know that both of these tests can lack power in many situations. They're not all that reliable.
• If we wrongly conclude that the series is I(0), when in fact it is I(1), this is pretty serious. We don't want to end up working with a non-stationary time-series unwittingly. For instance, we may end up fittng a "spurious regression".
•  On the other hand, if we conclude that the series is I(1), when in fact it is I(0), this may not be quite so serious. For instance, if we were to (unnecessarily) difference the I(0) series it would still be stationary - although not I(0). That's not so bad in istself, but it may mean that we don't go on to test for possible cointegration between this series and any others that we may be working with.
• So, there's a strong incentive to come to the correct conclusion when we test the data. For that reason, a "second opinion" might be a good idea. If both the ADF and KPSS tests lead us to the same conclusion, we might take some comfort from that.

Now, here's my analogy. Suppose we have a dimly lit room with two windows, and in that room there are some snakes. They may be a completely harmless ones, or they may be deadly. Before entering the room, we'd like to try and figure what sort they are!

I take a flashlight, shine it in one wndow, and try to decide what I'm facing. I could go ahead on the basis of what I see through the one window using one flashlight and take a chance. Or, I could go around to the other window, perhaps use a different flashlight, and see what I can conclude from that different perspective.

If I come to the same conclusion about the nature of the snakes, whichever window and flashlight I use, then I may feel much better informed and prepared when I enter the room than if I come to diferent conclusions from the two viewings.

Wouldn't you want to look through both wndows before proceeding?

1. Such a wonderful analogy there, Professor.
But I still ask if there are Bayesian methods of unit root test and their relative unit root models?

1. Thanks. Yes, there's quite a literature on that - Peter Phillips had some work. I'll check out the references and post them.

2. Really a nice post, Sir. However, in your many blog posts you have clarified this and explained many times the rationale of doing two tests. One thing, if two tests give conflicting results would it be safe to conclude that series X is I(1)?
Thank you.

1. That's what I'd do, for the most part. The loss structure here is somewhat asymmetric - the "costs" of failing to detect a unit root are generally higher than those associated with "detecting" one when really the data are stationary.

3. The general rule is that two independent tests are more reliable than one -- and in fact, the more independent tests we have, the more robust our conclusion if they lead to the same conclusion. There comes a point of diminishing returns. The cost (opportunity or otherwise) of another test may not be worth is.

I'm not sure how good the snake example is. For one, the two tests aren't truly independent. They'd be more independent if two different persons performed both the two different tests. Then the tests depend on one's reliability in distinguishing between venomous and nonvenomous snakes.

In that last issue, one really wants to know how or why the snakes got there. For example, if the snakes are known to be indigenous to the US, a large black snake is pretty much guaranteed to be harmless (and a nice pet, to boot). However, I would not bet my life on the large black snake not being a black mamba imported from Africa, or some other deadly species.

One should take a third option, and hire a pest removal service with snake expertise. Or take a fourth option and stay out of the room.

4. Great post and an even greater analogy!

If I may add, in Maddala and Kim's "Unit Roots, Cointegration and Structural Breaks" there is an excellent (and perhaps dated) discussion on what they call "confirmatory analysis" (Section 4.6 on page 126), that is exactly what the above post is about.

Referencing to a paper by Bruke (1994), Madalla and Kim conclude that:

"The overall conclusion is that if the true model is stationary, the
proportion of correct confirmations is low. It is thus, more important
to consider better unit root tests and stationary tests (as discussed in
section 4.3-4.5) than to use confirmatory analysis with defective tests."