# Econometrics: difference-in-differences basic proof

At the most rudimentary level, how is a fixed effects model equal to first differencing, and how is first differencing equal to the regression

$$\text{Outcome}_{it} = α_1 + α_2\text{Treat}_i + α_3\text{Post}_t + β(\text{Treat}_i\times\text{Post}_t) + ε_{it}$$

Where $\text{Treat}_i$ is a dummy which equals 1 if individual $i$ received treatment, and $\text{Post}_t$ is a dummy which equals 1 if the observation refers to post treatment time.

Under what circumstances would you use the simple first difference, and when would you have to resort to the regression? Also, what are the standard error implications of using one over the other?

• I find the first part a bit unclear. Are you saying that estimating the equation you stated in first differences and using the fixed effects estimator gives the same result, and now you wonder why that is? – ekvall Jan 28 '14 at 14:22
• YES, EXACTLY WHAT I AM SEEKING. – EconvsHumans Jan 28 '14 at 14:42
• Rather than an answer, here is a hint: Frisch-Waugh-Lovell Theorem – Bill Jan 28 '14 at 19:58

Here's the intuition (dropping any covariates and error terms). The expected values are:

1. Treatment guy in the pre period: $y=\alpha_1 + \alpha_2$
2. Treatment guy in the post period: $y=\alpha_1 + \alpha_2 + \alpha_3 +\beta$
3. Control gal in the pre period: $y=\alpha_1$
4. Control gal in the post period: $y=\alpha_1 + \alpha_3$

A first difference fixed effects model would difference each person's pre and post like this:

1. Treatment guy: $y=\alpha_1 + \alpha_2 + \alpha_3 +\beta-(\alpha_1 + \alpha_2)=\alpha_3 + \beta$
2. Control gal: $y=\alpha_1 + \alpha_3-\alpha_1=\alpha_3$

Now if you run a regression of the change in outcome $y$ on treatment status dummy, you recover the interaction coefficient.

I always use the regression since it spits out the standard errors, but it is particularly handy when you have more than 2 periods in your panel.

• Many thanks @Dimitriy V. Masterov for this nice explanation. I added the weights from IPTW (inverse probability of treatment weighting) to the regression. I got the same coefficients in the fixed effects model and the first-difference approach but the standard errors are different. Please, could you give me some hint to figure out what happen? Also, as OP asked before, what are the standard error implications of using one over the other? Thanks! – nghauran Aug 8 '18 at 17:42
• @ANG This would make for a good separate question. Please provide some detail on your data/setting and show exact commands you used in the body of the question. – Dimitriy V. Masterov Aug 8 '18 at 17:47