### Establishing Relationships Between Two Variables

Another important application of OLS is the comparison of two different data sets. In this case, we can think of one of the time series as constituting the independent variable *x* and the other constituting the independent variable *y*. The methods that we discussed in the previous section for estimating trends in a time series generalize readily, except our predictor is no longer time, but rather, some variable. Note that the correction for autocorrelation is actually somewhat more complicated in this case, and the details are beyond the scope of this course. As a general rule, *even* if the residuals show substantial autocorrelation, the required correction to the statistical degrees of freedom (*N' *), will be small as long as either one of the two time series being compared has low autocorrelation. Nonetheless, any substantial structure in the residuals remains a cause for concern regarding the reliability of the regression results.

We will investigate this sort of application of OLS with an example, where our independent variable is a measure of El Niño — the so-called *Niño 3.4 index* — and our dependent variable is December average temperatures in State College, PA.

The demonstration is given in three parts below (click each link to open a new window and then the arrow to begin the demonstration):

You can play around with the data set used in this example using this link: Explore Using the File testdata.txt