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Is it possible to construct a difference-in-difference regression with only 4 data points, representing Group 1/2 pre/post values?

For example, Wooldridge describes a setup where there is a random sample of units (e.g. firms) in two periods of time under two different treatments. Say, firms in California in 2012 and 2013 and firms in Wisconsin in the same years. Further, there is a change whereby, say California in 2013 changed their tax policy. So as he describes, it sounds like we are not tracking the same units per state in the two years but a different sample per state per year (i.e. 4 distinct samples).

This leads to my question: using the example above, can this same setup be used where you have a summary of the outcome variable for each state and year (not samples from each)? So, there are only 4 data points, say, total revenue of fast food restaurants in California in 2012 and 2013 (and the same for Wisconsin).

Thus the regression uses 4 data points, but clearly a standard error on the interaction term cannot be estimated.

dat<-data.frame(sales=c(1234,2232,1530,2500),year=c(0,0,1,1),state=c(0,1,0,1))
summary(lm(sales~year*state,data=dat))
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    $\begingroup$ There is not regression here, just $(X_{1t+1}-X_{1t}) - (X_{2t+1}-X_{2t})$. $\endgroup$ – tchakravarty Dec 20 '13 at 22:56
  • $\begingroup$ Can there be a standard error? $\endgroup$ – B_Miner Dec 20 '13 at 23:59
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    $\begingroup$ It looks like there's no degrees of freedom for error. $\endgroup$ – Glen_b -Reinstate Monica Dec 21 '13 at 0:19
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As pointed out in the comments, you can simply calculate the difference-in-differences, and you can't get a standard error. However, notice that when you have samples from each group and time period, your standard errors only concern the uncertainty in inferring the group x time aggregates from your samples. That means that if your four data point are actual aggregates, this estimator comes with less uncertainty than the one based on samples, no matter how big those samples are.

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