# Difference in Difference with control - common trend interpretation

I have a question concerning the interpretation of the common trend assumption in a very specific case of diff-in-diff. I am using a panel to find the effect of a treatment (on houesehold level) on the number of years of schooling of individuals. The regression I use is $$\text{yrschl}_{it} = \beta_0 + \beta_1 \text{treatment}_h + \beta_2 \text{after}_t + \beta_3 (\text{treatment}_h \cdot \text{after}_t) + \gamma X_{iht} + \epsilon_{iht}$$ and I cluster the standard errors at the household level.

As there has been no randomization across the two groups, I use control variables as $\text{sex}_{it}$, $\text{age}_{it}$, $\text{age}^2_{it}$, $\text{mother's education}_{it}$, $\text{father's education}_{it}$ and $\text{urban/rural area}_{iht}$.

My question now is how to interpret the ceteris paribus condition jointly with the common trend assumption. In particular, my doubt concerns the ceteris paribus condition related to age.

How do I have to formulate the common trend assumption in this case?

Is it that, IN ABSENCE OF THE TREATMENT, the change in the average number of years of schooling of the children who are, e.g. 14 years old in $t=0$ and therefore 20 years old in $t=1$ (I only have data for 6 years later) and belong to the treatment group is the same as the one of those children who were 14 in $t=0$ and 20 in $t=1$ and are untreated?

Or does it rather mean that, IN ABSENCE OF THE TREATMENT, the change in the average years of schooling between treated 14 years old children in $t=0$ and treated 14 years old children in $t=1$ is the same as the one between untreated 14 years old children in $T=0$ and untreated 14 years old children in $t=1$? Does the ceteris paribus condition hold across time periods? Since I am using a balanced panel, there would be only observations for young individuals in $t=0$ and not in $t=1$ (as they got 6 years older in the meanwhile). This speaks against this second interpretation, isn't it?

• What is the treatment and when does it kick in? – Andy Aug 1 '15 at 10:19
• treatment is 1 if the individual belongs to a household in which there is a migration happening for the first time between t=0 and t=1 while it is 0 if until t=1 in households who never faced a migration. After_t is equal to 1 if t=1 and equal to 0 if t=0. P.s. I have edited my question because I forgot to write "IN ABSENCE OF THE TREATMENT" when interpreting the common trend assumption. – Gio Aug 1 '15 at 11:16

When your treatment status depends on (fully!) observed covariates the parallel trends assumption does not depend anymore on the unconditional but the conditional pre-treatment trends. If you were to state the identification assumption in a paper, you would write something like:

The identifying assumption is that pre-treatment trends in the outcome for treated and untreated households are parallel conditional on observables that determine treatment status. This means that treated and control households' outcomes with similar characteristics would have continued in a parallel way in the post-treatment period in the absence of the treatment.

Since you only have one pre-treatment period it is somewhat odd to speak about pre-treatment trends though. There is no way that you can infer those from a single data point but the required condition would be the one cited. For your case you are conditioning on the baseline period ($t=0$) characteristics.

Typically it is very difficult to sell a difference in differences framework if you cannot show these trends but if this is for an assignment or thesis work then it should be sufficient that you demonstrate your awareness of this problem. Even though it is not asked for in the question I still want to hint to one or two more points that you might want to think about:

• migration is a decision of the household and I highly doubt that this decision is fully captured by observables; in this case you might want to instrument the treatment status with some policy change (or another valid instrument) that affects the migration decision but not the educational outcomes
• depending on the country you are looking at, education is bounded below by minimum school leaving ages. Suppose your treatment would have a negative effect on years of education you may not be able to find this causal effect if observed educational outcomes cannot go any lower in response to treatment because of compulsory schooling laws

As a minor point you also may want to revisit the subscripts for mother's and father's education. Typically those do not change anymore after people have completed their formal education, i.e. they shouldn't have a time subscript.

• Thanks! In fact, I do have another period, which is 8 years before t=0. However, not all individuals were present/ at school in all three periods (Actually, when adopting the strategy described above, I limit the sample to those who were enrolled at school in t=0). So, even if I can test the common trend assumption, I only can do so for a subset of individuals which leads to a very low number of treated individuals. Therefore, even if I get a non-significant interaction coefficient when doing the placebo test (by changing after_t), it is likely to be due to the large standard errors, right? – Gio Aug 1 '15 at 16:19
• Moreover,if the interpretation of the common trend assumption is the above suggested one,then what is the difference between a simple OLS and a DiD?I know that the OLS assumes the same level of years of schooling across groups while the DiD only assumes the same trend,however,in the specific case of years of schooling,what does this difference exactly mean?Why should treated and control have different levels but the same trend?Is it like saying that we weaken the assumption by allowing, e.g. one group to start school later,but then, once started,the schooling behavior is similar? – Gio Aug 1 '15 at 16:29
• For that you might this other answer useful which explains the common trends assumption: stats.stackexchange.com/questions/152138/… – Andy Aug 1 '15 at 16:59
• Thanks for the link.Actually, I know the difference between the DiD and the OLS, but the problem is the specificity of my dependent variable years of schooling. Assuming that there is a common trend in years of schooling means that,even if the average of both groups is different, its development over time is similar.But how do you interpret the development over time of years of schooling?If this must be similar, then it is logic to think that their behavior with respect to education is similar and thus there should be no reason to have different levels of education(if there was no treatment!). – Gio Aug 1 '15 at 17:39
• It's like the issue I raised regarding the compulsory schooling laws. Suppose you follow two kids of equal age in the base period for 3 years where treatment occurs between year 2 and 3. Say from year one to two they increase schooling by 1. In year three the treated kid quits school because of the treated whereas the control kids increases again by 1 year, which would have happened to the treated kid too in the absence of the treatment. – Andy Aug 1 '15 at 17:44

Another thing:

I belive that put control variables that are invariant in the time (like sex and father/mother education) don´t have any sense. Your especification already considers time invariant factors.