How to interpret ADF-test optimal lag model reduction?

Question

I was looking at howto figure out the optimal lag order for an ADF-test when I came across this PPT presentation. There are two examples of a ADF-test model reduction, one on p23 and one on p36.

I have two questions:

1. How do I come up with the F numbers of the reduction? For example, on p23 it says "F(1,487) = 0.92201 [0.3374]"? I understand what an F number is, but what calculation lies behind the 0.92201 number (and 0.3374)?
2. How do I interpret the numbers? For example, why is model 4 the preferred one on p23?

I'm not a complete noob, but I certainly wouldn't mind a nicely laid out pedagogical answer. Thanks!

Answer 1

For page 23 you have simple F statistics for regression (scroll down to regression problems in the link for the formula). So the number 0.92201 comes from this calculation (in R):

> (963.09-961.27)/(961.27/487)
[1] 0.922051

The value in the brackets is p-value of F-statistic with degrees of freedom 1 and 487:

> 1-pf((963.09-961.27)/(961.27/487),1,487)
[1] 0.3374135

Now all these statistic test whether the additional parameters in the model are zero or not. So the model 4 comes up as the best, because compared to it, hypothesis that additional parameters in models 1 to 3 are zero is not rejected. On the other hand the hypothesis that additional parameter in model 4 is zero compared to model 5 is not rejected.

Note under null hypothesis of unit root, the asymptotic statistics on coefficients of lagged differences are the same as in usual regression. The point of including lagged differences is to control for serial correlation, the question is how many. This example illustrates how you can choose how many to include.

In my opinion though it is rather futile exercise. Why recreate ADF test by hand, when there is readily available packages to do it for you? The goal of ADF is to test whether we have unit root or not. Based on that information some kind of model is built for the variable tested, which in practice never has the form used in ADF test. This of course is my personal opinion. As always your mileage may vary.