Can You Actually TEST for Multicollinearity?

When you're undertaking a piece of applied econometrics, something that's always on your mind is the need to test the specification of your model, and to test the validity of the various underlying assumptions that you're making. At least - I hope it's always on your mind!

This is an important aspect of any modelling exercise, whether you're working with a linear regression model, or with some nonlinear model such Logit, Probit, Poisson regression, etc. Most people are pretty good when it comes to such testing in the context of the linear regression model. They seem to be more lax once they move away from that framework. That makes me grumpy, but that's not what this particular post is about.

It's actually about a rather silly question that you sometimes encounter, namely: "Have you tested to see if multicollinearity is a problem for your results?"

I'll explain why this isn't really a sensible question, and why the answer to the question in the title for this post is a resounding "No!"

First of all, let's stop and think for a moment what is actually going on when we perform a statistical test f some hypothesis. The null and alternative hypotheses are statements, or conjectures, about some feature of the underlying population. That is, they're associated with the data-generating process that supposedly gave rise to the sample data that we actually observe. For example, the hypothesis that we're testing may be a statement to the effect that one of the parameters in the population takes a particular value.

Hypothesis testing is an example of statistical inference. We use specific sample information to try and learn (infer) something about the unobserved characteristics of the population at large.

To state the obvious, in the context of a regression model, it's meaningful to have null and alternative hypotheses of the form, H₀: β₂ = 0 and H₁: β₂  > 0. Under standard conditions we might then test the validity of H₀ by using the t-statistic, t₂ = (b₂ - β₂) / s.e.(b₂). We'd reject H₀ in favour of H₁ if t₂ > c_α, where c_α is the 100(1 - α)'^th percentile of Student's t-distribution with (n - k) degrees of freedom.

On the other hand, it's nonsensical to write down null and alternative hypotheses of the form, H₀: b₂ = 0 and H₁: b₂ >0. These "hypotheses" are actually statements about the random variable, b₂, which is a function of the observed sample data. They are not statements about a fixed, unobserved, population parameter! They don;t form any basis for making inferences.

Now, let's return to the "problem" of multicollinearity.

What do we mean by this term, anyway? This turns out to be the key question!

Multicollinearity is a phenomenon associated with our particular sample of data when we're trying to estimate a regression model. Essentially, it's a situation where there is insufficient information in the sample of data to enable us to enable us to draw "reliable" inferences about the individual parameters of the underlying (population) model.

I'll be elaborating more on the "informational content" aspect of this phenomenon in a follow-up post. Yes, there are various sample measures that we can compute and report, to help us gauge how severe this data "problem" may be. But they're not statistical tests, in any sense of the word

Because multicollinearity is a characteristic of the sample, and not a characteristic of the population, you should immediately be suspicious when someone starts talking about "testing for multicollinearity". Right?

Apparently not everyone gets it!

There's an old paper by Farrar and Glauber (1967) which, on the face of it might seem to take a different stance. In fact, if you were around when this paper was published (or if you've bothered to actually read it carefully), you'll know that this paper makes two contributions. First, it provides a very sensible discussion of what multicollinearity is all about. Second, the authors take some well known results from the statistics literature (notably, by Wishart, 1928; Wilks, 1932; and Bartlett, 1950) and use them to give "tests" of the hypothesis that the regressor matrix, X, is orthogonal.

How can this be? Well, there's a simple explanation if you read the Farrar and Glauber paper carefully, and note what assumptions are made when they "borrow" the old statistics results. Specifically, there's an explicit (and necessary) assumption that in the population the X matrix is random, and that it follows a multivariate normal distribution.

This assumption is, of course totally at odds with what is usually assumed in the linear regression model! The "tests" that Farrar and Glauber gave us aren't really tests of multicollinearity in the sample. Unfortunately, this point wasn't fully appreciated by everyone.

There are some sound suggestions in this paper, including looking at the sample multiple correlations between each regressor, and all of the other regressors. These, and other sample measures such as variance inflation factors, are useful from a diagnostic viewpoint, but they don't constitute tests of "zero multicollinearity".

So, why am I even mentioning the Farrar and Glauber paper now?

Well, I was intrigued to come across some Stata code (Shehata, 2012) that allows one to implement the Farrar and Glauber "tests". I'm not sure that this is really very helpful. Indeed, this seems to me to be a great example of applying someone's results without understanding (bothering to read?) the assumptions on which they're based!

Be careful out there - and be highly suspicious of strangers bearing gifts!

References

Bartlett, M. S., 1950. Tests of significance in factor analysis. British Journal of Psychology, Statistical Section, 3, 77-85.

Farrar, D. E. and R. R. Glauber, 1967. Multicollinearity in regression analysis: The problem revisited.  Review of Economics and Statistics, 49, 92-107.

Shehata, E. A. E., 2012. FGTEST: Stata module to compute Farrar-Glauber Multicollinearity Chi2, F, t tests.

Wilks, S. S., 1932. Certain generalizations in the analysis of variance. Biometrika, 24, 477-494.

Wishart, J., 1928. The generalized product moment distribution in samples from a multivariate normal population. Biometrika, 20A, 32-52.

© 2013, David E. Giles