# Standard error implications when combining IPTW and difference-in-differences

My question is about combining Inverse Probability of Treatment Weighting (IPTW) with a difference-in-difference regression with two periods (pre and post treatment). Basically, I first computed the IPTW’ weights from a propensity score model and now I need to apply those weights to the observations in order to get the treatment effect. As explained by @Dimitriy V. Masterov, the following regressions (fixed effects model or the first-difference model) are equivalent.

Fixed effects model $$\text{Y}_{it} = α_1 + α_2\text{Treatment}_i + α_3\text{Time}_t + β(\text{Treatment}_i\times\text{Time}_t) + ε_{it}$$

Where $\text{Treatment}_i$ is a dummy which equals 1 for treated, and $\text{Time}_t$ is a dummy which equals 1 if the observation refers to post treatment time. $β$ is the treatment effect.

First-difference model $$\text{ΔY}_{i} = \text{Y}_{it(t=1)} - \text{Y}_{it(t=0)} = α + β\text{Treatment}_i + ε_{i}$$

When applying the IPTW’ weights in a Weighted least squares (WLS) regression, I get the same estimate of $β$ in the fixed effects model and the first-difference approach but the standard errors are different.

In the fixed effects model, the treatment effect $\widehat{β}$ is 0.034554 with a standard error of 0.019219 while in the first-difference model, $\widehat{β}$ remains the same (0.034554) but the standard error is now 0.016835.

Please, could you help me figure out why the standard errors are different?

Many thanks.

• Interesting question (+1)! Can you please give a minimal reproducible example to go along-side it? The estimates for the standard errors are suspiciously close so some further info might be helpful. Maybe sample sizes have a role to play here... – usεr11852 says Reinstate Monic Aug 10 '18 at 21:44
• Sorry @usεr11852, unfortunately I'm not able to provide a reproducible example. The analysis is carried out on a secured server with no access to internet. Concerning the sample size, I have 1050 observations in the first-difference model and 2100 in the fixed effects model (each observation at t0 and t1) – nghauran Aug 11 '18 at 17:02
• I appreciate what you mean by a secure server but as it stands one cannot tell if it is the weighting scheme from the IPTW that causes the difference, is it the sample size difference or is it something else all together. To be poetic about it: While searching for the tallest peak (MLE) we arrived at the same spot in the map ($\hat{\beta}$) but when we look around the mountain curvature ($\frac{d^{2}L}{d\beta^{2}}$) is different; it is just slightly steeper now (making the SE a bit smaller). So, the question: Why is this mountain different? IPTWs? Sample size asymptotics? What? :) – usεr11852 says Reinstate Monic Aug 11 '18 at 19:11