How to read a pooled quarter-to-quarter test

A pooled quarter-to-quarter test takes every company's consecutive-quarter log changes, stacks them into one sample, and fits a single regression line through all the points. The line describes how the log change in cost of revenue moves with the log change in revenue, read across quarters rather than across companies.

The case here uses 176 quarter-pairs drawn from 12 companies. Each pair holds one company's log change in revenue and its log change in cost of revenue from one quarter to the next. All 176 pairs sit in one sample. No company is set apart from the others.

We fit an ordinary least squares line to those 176 points. The slope of the line is 0.681. That number is the change in the log change of cost of revenue for each unit change in the log change of revenue, read across all quarters of all companies at once.

The fit accounts for 0.34 of the total variation in the log change of cost of revenue. We can check that figure against the sums of squares. The total sum of squares is 51.7. The regression sum of squares is 17.8. Their ratio gives R squared:

17.8 / 51.7 = 0.34

The Pearson correlation between the two log changes, across all 176 pairs, is 0.586. Squaring it gives the same R squared.

The p-value attached to the slope is under 0.0001. That value tests the null that the slope is zero. The test treats the 176 pairs as if each were an independent draw from the same population.

Here is where the stacking matters, and it looks simpler than it is. A company contributes more than one pair. On average, each company supplies 14.7 quarter-pairs. One pair shares a boundary value with the pair before it. The log change from quarter two to quarter three is measured from the same level that the log change from quarter one to quarter two arrived at. Two pairs from the same company are not separate observations. They are linked through the level that sits between them.

When the test treats 176 linked pairs as 176 independent draws, the standard error of the slope is computed on the assumption that each pair carries its own independent information. The p-value of under 0.0001 rests on that assumption. The R squared of 0.34 and the Pearson r of 0.586 describe the scatter of the stacked points. That scatter includes the within-company links.

The pooled fit reads a relation across quarters. The 176 points are 12 sequences of linked quarter-pairs stacked into one sample. The slope, the R squared, and the p-value all describe that stacked sample as it is, with the links inside it.