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?
 A: 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.
A: 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.
