Control for time trend in difference-in-differences? I'm running a difference-in-differences analysis on monthly repeated cross sectional data.
I've been told to add a linear time trend to the DiD model to adjust for underlying trends:
$$Y_{it} = B_0 + B_1 treatment_i + B_2 post_t + B_3 (treatment_i \times post_t) + B_4 month_t + \epsilon_{it}$$
where
month = number of months from the start of the study period
I'm wondering whether including a linear monthly time trend common to both groups is appropriate in a DiD analysis or whether the time trend may soak up trend changes in the post-period that are related to the treatment?
Note: I don't think this is relevant for the question but I will also adjust for calendar month for seasonality and a vector of individual level variables.
 A: In your setting you already control for aggregate time effects via the inclusion of time dummies ($\text{month}_t$), which are more flexible than a linear time trend. To probe for the robustness of their results, people typically include individual specific linear time trends. These help to rule out the possibility that treatment and control individuals were already on differential growth trajectories in their outcome variables, i.e. a change in the outcome that would have happened even in the absence of the treatment which is now picked up by $B_3$.
In your regression equation, this would amount to regressing
$$Y_{it} = B_0 + B_1 treat_i + B_2 post_t + B_3 (treat_i \times post_t) + B_4 month_t + B_5 (i\times t) + \epsilon_{it}$$
where you interact a linear time trend $t = 1,2,3,...,T$ with a dummy for each individual.
However, this is computationally and econometrically very demanding. These time trends might be "over controlling" and in addition to absorbing pre-existing trends, they may also absorb part of your treatment effect. This is particularly true in cases where the treatment kicks in with a lag or where it fades out over time (a dynamic treatment effect).
Justin Wolfers has a paper where he discusses the issues with the inclusion of these unit specific linear time trends. Also, if there are pre-existing differential trends across treated and untreated individuals, this would cast doubt on the "randomness" of this intervention/policy.
[I formatted your regression equation in the question. It wasn't clear what purpose the $j$ index was serving, so if this is not what you had in mind, please feel free to roll it back.]
