A while back I posted (here, here, and here) about constructing confidence bands to go with the Hodrick-Prescott filter. Subsequently, I wrote up the material more formally, and that paper is to appear in Applied Economics Letters.
Alfredo drew my attention to Steve Pische's reply to a question raised by Mark Schaffer in the Mostly Harmless Econometrics blog. The post was titled, Probit Better than LPM? The question related to my own posts (here, here, and here, in reverse order) on this blog concerning the choice between OLS (the Linear Probability Model - LPM) or the Logit/Probit models for binary data.
Thanks, Alfredo, as this isn't a blog I follow.
Alfredo asked: "Would you care to respond? I feel like this is truly an exchange from which a lot of people can learn".
When we think of the power curve associated with some statistical test, we usually envisage a curve that looks something like (half or all of) an inverted Normal density. That is, the curve rises smoothly and monotonically from a height equal to the significance level of the test (say 1% or 5%), until eventually it reaches its maximum height of 100%.
The latter value reflects the fact that power is a probability.
But is this picture that invariably comes to mind - and that we see reproduced in all elementary econometrics and statistics texts - really the full story?
Approximating unknown (continuously differentiable) functions by using a Taylor (MacLaurin) series expansion is common-place in econometrics. However, do you ever pause to recall that such approximations are only locally valid - that is, valid only in a neighbourhood of the (possibly vector) point about which the approximation is made?
Unlike some other types of approximations - such as Fourier approximations - they are not globally valid.
Does this matter? Is it something we should be concerning ourselves with?
Let's think about a standard result from regression analysis that we're totally familiar with. Suppose that we have a linear OLS regression model with non-random regressors, and normally distributed errors that are serially independent and homoskedastic. Then, the usual F-test statistic, for testing the validity of a set of linear restrictions on the model's parameters, is exactly F-distributed in finite samples, if the null hypothesis is true.
In fact, the F-test is Uniformly Most Powerful Invariant (UMPI) in this situation. That's why we use it! If the null hypothesis is false, then this test statistic follows a non-central F-distribution.
It's less well-known that all of these results still hold if the assumed normality of the errors is dropped in favour of an assumption that the errors follow any distribution in the so-called "elliptically symmetric" family of distributions. On this point, see my earlier post here.
What if I were now to say that some of the regressors are actually random, rather than non-random? Is the F-test statistic still exactly F-distributed (under the null hypothesis)?
With the scientific world abuzz today over the (possible) confirmation of the existence of the Higgs Boson, this postfrom David Smith on the SmartData Collective is a must-read for anyone with an interest in statistics.