Degrees of freedom in matching for calculating p-value of t-score

Question

I am running a matching analysis using the Match function from the Matching R-package. My sample includes 86048 firms, from which 611 have received treatment. After matching on sector(exact), number of employees, company age (both within .5 SD) and share of university graduates in workforce (within .10 SD), I find for 322 of the treated firms a match (415 unweighted matched observations).

The summary function of the output of the Match function provides me with a t-score(6.67) and a p-value(2.5e-11). My question regards the calculation of the p-value based on the t-score. Although this should be not that complicated, I am in doubt about the degrees of freedom to use in this.

Based on my intuition I would take Degrees of freedom treated group (322-1) plus the degrees of freedom matched group (322-1) to come to 642 degrees of freedom. However, when I re-calculated the p-value by hand, I found out that the package used the 86046 degrees of freedom of my original data set to calculate the p-value, which puzzles me since most of the data set is not used in the analysis.

For the significance of my results it doesn't really matter, but I want to understand the logic in this. Coming to my question: Should I indeed use the 86046 degrees of freedom of the original data set? If yes, why is this the case?

Answer 1

Matching assumes the t-statistic is z-distributed, so it essentially assumes infinite degrees of freedom. I would not report a degrees of freedom for this test, as it's not actually a t-test in that it doesn't use a t-distribution. For large data sets like the one you have, this approximation is fine and, as you've noticed, won't change your results.