## Saturday, March 3, 2012

### You had to be there!

It was the Fall of 1983. I was on Study Leave from Monash University in Australia, and I was spending about 6 months in the Economics Department  at the University of Western Ontario. My good friend, Aman Ullah, had arranged the visit and I ha a wonderful time.

There were several seminar programs running on a weekly basis, including one in econometrics. Some time that term there was an econometrics seminar presented by two young Assistant Professors in the department. Here is how it went down.

The research that was presented suggested that the usual Wald test has an unfortunate property when the parametric restrictions that are being tested are non-linear. Specifically, the authors claimed that the value taken by the Wald test statistic could change, according to how the restrictions are written. In other words, the value of the test statistic, and possibly the outcome of the test itself, are not invariant to the way in which the restrictions are expressed.

For example, consider the following single non-linear restriction on the coefficients of a linear regression model: (β1 / β2) = β3. An alternative (and absolutely equivalent) way of expressing this restriction is: (β1 - β2*β3) = 0. You'll get different values for the Wald test statistic depending on which of these equivalent restrictions you test. Moreover, the actual size and the power of the Wald test in finite samples can be very different depending on the choice you make.

Now, remember that this was 1983. What was being put forward in that seminar was something of a shock! Indeed, those of us who had "been around a while" were - to put it mildly - somewhat skeptical.

How could this be? The Wald test had been in widespread use by statisticians and econometricians since its introduction in 1943. Here we were, forty years later, finding out that it had a major flaw - some might say a fatal flaw - if applied to a really important class of problems.

After all, this "problem" doesn't arise with competing testing principles, such as the Likelihood Ratio test or the Lagrange Multiplier (Score) test. Surely these two guys couldn't be right?

There was the usual lively discussion, and then we retired to the faculty club for a jug or two of Labatt's IPA. The discussion continued, and after a few more beers the claim about the flawed Wald test looked more and more convincing!

Of course, as most of you will know already, Alan Gregory and Mike Veall were perfectly correct in their claim, and their paper subsequently appeared in Econometrica. The 1983 seminar was acknowledged in the opening footnote of the paper. Other, related, contributions followed. These included papers by Phillips and Park (), Bresusch and Schmidt (1985) and Lafontaine and White (1986). Alan and Mike have gone on to extremely successful academic careers at Queen's University and McMaster University, respectively.

Today, thanks to their insight and persistence, we're justifiably cautious about our use of the Wald test. Some of us who were at that 1983 seminar should have been equally cautious in our initial reactions to the initial airing of their early draft paper!

It was a privilege to be there!

Note: The links to the following references will be helpful only if your computer's IP address gives you access to the electronic versions of the publications in question. That's why a written References section is provided.

References

Breusch, T. S., and P. Schmidt, 1985. Alternative forms of the Wald test: How long is a piece of string? Mimeo., Department of Economics, University of Southampton.

Gregory, A. and M. Veall, 1985. Formulating Wald tests of nonlinear restrictions. Econometrica, 53, 1465-1468.

Lafontaine, F. and K. J. White. 1986. Obtaining any Wald statistic you want. Economics Letters, 21, 35-40.

Phillips, P. C. B.. and J. Y. Park, 1988. On the formulation of Wald tests of nonlinear restrictions. Econometrica, 56, 1065-1083.

Wald, A., 1943. Tests of statistical hypotheses concerning several parameters when the number of observations is large. Transactions of the American Mathematical Society, 54, 426-482.