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I'm running a diff-in-diff with model with years of education as an outcome variable.

$Y_{it}=\beta_0+\beta_1Treat_i+\gamma_t+\beta_2Treat_{i}*Post_t+X'\gamma+\epsilon_{it}$

The treatment dummy is based on a reform which had an age cutoff, and is therefore by construction negatively correlated with age. The sample runs from 2010-2017 meaning that the oldest individuals are only 38 in the last observed year. When I include Age as a control variable $\beta_2$ is almost completely unchanged. However, once i include Age^2 $\beta_2$ changes sign and becomes negative. Any suggestions on why this might be?

Also I'm aware that fitting a non-linear relationship is often plausible. E.g years of education, employment, wages etc. tend to exhibit decreasing marginal effects. Is this also necessary in a "young" sample since one might expect the turning point to be outside the range of the data?

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You would need to do some explorative data analysis to answer this. Look at the leverages (your linear regression tool should be able to produce diagnostic plots). If there are a few very old individuals, their squared age would dominate their expected response. Suppose, for example, age^2 has a negative coefficient. A very old individual therefore has a very low expected response, and since very old individuals are not treated, this would lower the estimates treatment effect and might well cause it to be negative.

You might also want to carry out an analysis on the subset of individuals who were eligible for treatment (i.e. only young individuals).

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