Re: cointegrated time series regression analysis

On Thu, 07 Jul 2005 11:10:17 EDT, Marvin <marvin.barron@xxxxxxxxx>
wrote:

>Hi Everybody,
>
>Let me thank you in advance for reading/thinking on my problem. I don't have a lot of formal math training, so my description of my problem is probably a little imprecise. I'd be really happy to get either some direction or reference to a book so I can explore the topic independently, because it seems like the sort of problem others would have encountered before.
>
>I'm working with two population sets, time series data, that I believe to be cointegrated. I'd like to further explore this relationship, which I generally believe to be linear. The first problem is that I don't like the nature of independent/dependent variables for an ordinary-least-square regression line. My understanding is that the OLS line is the line with the smallest sum-of-squares residual between the observed data and the line in terms of the dependent variable (y-axis). That means that it really matters which is the independent and which is the dependent variable. But in this case (where both variables have the same units), I'm not sure which to pick.
>
>Much better (I think) would be to figure out the equation for the line that minimizes the orthogonal residual/distance between the data and the line. It just seems that that line would be a truer description of the relationship between the data sets. Is that a good way to think? If so how would I figure out what that line was?
>
>That's problem #1.
>
>Problem #2 is this: The data sets can both be reasonably expected to grow over time. Both begin at about 100 and end at about 10,000. Because OLS minimizes the absolute residual, it gives much more weight to (and thus more attention to minimizing) residuals at the end of the data set, not at the beginning. For example, a 1% residual at the beginning is (1 squared) and easily minimized or ignored, but a 1% residual at the end is (100 squared) and could really dominate the solved coefficients. Is this as much a problem as I think or am I being silly? If it is a problem, is there a workaround (esp. in light of the previous problem)?
>
>Thanks so much for your feedback!

Marvin:

You've given a very nice description of two real problems. You'll find
discussions of both in most any econometrics book.

Ideally, to answer your first problem you need to define what you mean
by "relationship." Do you think that a change in one variable "causes"
the change in the other? Or is there an unobserved variable moving
both?

The orthogonal regression will probably give you an estimated
coefficient in between what you'd get by regressing the first on the
second versus the second on the first. This may or may not be the
right answer to your underlying question.

Your second problem may even help out with the first. The idea of
cointegration is that there is an underlying, nonstationary, variable
that drives the long run behavior of both series. If this is true,
then the regression coefficient of one series on the other (and it
doesn't matter which one is the dependent variable) describes the
long-run relation between the two. But, very much in line with your
intuition, two exploding series are likely to look correlated even
when the relation is spurious.

Here's the basic notion, although one can get more sophisticated.
First check that each series has a "unit root." If both do, then
regress one series on the other and check that the residuals are
stationary. If so, then the two series are cointegrated and the
regression coefficient is an estimate of the "cointegrating vector."

Just pulling stuff off my shelf, there are discussions of these issues
in
Ashenfelter, Levine, and Zimmerman: Statistics and Econometrics

Hill, Griffiths, & Lee: Undergraduate Econometrics

Wooldridge: Introductory Econometrics

-*** Startz
----------------------
Richard Startz RichardStartz@xxxxxxxxxxx
Lundberg Startz Associates
.

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