I would really appreciate if some one could explain it to me (through econometric or intuitive sense)the importance of interacting control variables with linear time trends?
I was running a simple diff-in-diff model i.e:
outcome= B0*constant + B1*treat + B2time + B3* treat*time + B4*controls + error.
Where treat= 1 if the group receives the treatment, 0 otherwise time= 1 if treated in time period 2, 0 otherwise and the interaction term between them both gives us the treatment effect (through double differencing) and controls are all HH level characteristics.Since the treatment was given in time period 2 ( there are two time periods within the data set)
I was told to reform this model and to "control for the differences in time trends" by interacting each of the control variables with the time dichotomous variable. So now my model looks like this:
outcome= B0*constant + B1*treat + B2time + B3* treat*time + B4*control 1 + B5*control 2 + B5*control 3 + B6*control 4 + B7*controls 1*time + B8*controls 2*time + B9*control 3*time + B10*control 4*time + error term.
Basically, interact each control with the time variable- to quote un quote "control for the difference in time trends within these variables."
I understand the econometric sense behind interacting the treatment variable with the time variable (First this takes the difference across time within group- eliminating any group specific unobserved but time fixed effects. Then it takes the difference of the differences to rid of any time trends in the results. Thereby giving us the impact of the program onto the outcome by way of double differencing). And I would like to derive similar understanding/intuition for the interaction terms of my control variables as well.
But I fail to understand the econometric rational behind interacting the control variables with the time trend. For the following reasons:
Firstly: By including in the time dummy variable, are we not automatically controlling for the difference in the time trends within the specification? (as in the first model, where time was just introduced as a dummy and not interacted wit the controls).
Secondly: How does the interaction term (control x time) control for the differences in time trends between the two time periods? Intuitively speaking?
Lastly: How are the interaction terms with time (control x time; treatment x time) excludable? i.e: they only are able to affect our LHS through the intervention itself, and not by any other means. I kind of understand how the treatment x time interaction term does that (through the double difference mechanism described above). But how would interacting, let's say household wealth score with time period, make the entire term excludable onto child schooling outcome for time period 2?
Would really like some insight in this matter. Or any paper, citations, sources that have followed a similar empirical strategy to the second model shown above.