# Pre-trend testing of DiD following Pischke (2005)?

This post simplifies how to conduct a pre-trend analysis using "coefficient plotting", which follows from lecture notes by Pischke 2005.

I simplified copying the description here:

A formal test which is also suitable for multivalued treatments or several groups is to interact the treatment variable with time dummies. Suppose you have 3 pre-treatment periods and 3 post-treatment periods, you would then regress

$$y_{it} = \lambda_i + \delta_t + \beta_{-2}D_{it} + \beta_{-1}D_{it} + \beta_1 D_{it} + \beta_2 D_{it} + \beta_3 D_{it} + \epsilon_{it}$$

where $$y$$ is the outcome for individual $$i$$ at time $$t$$, $$\lambda$$ and $$\delta$$ are individual and time fixed effects (this is a generalized way of writing down the diff-in-diff model which also allows for multiple treatments or treatments at different times). If the outcome trends between treatment and control group are the same, then $$\beta_{-2}$$ and $$\beta_{-1}$$ should be insignificant, i.e. the difference in differences is not significantly different between the two groups in the pre-treatment period.

So, it seems that one way to satisfy pre-trend testing is when the estimates of $$\beta_{-2}$$ and $$\beta_{-1}$$ are insignificant. But what about $$\beta_j$$, $$j \geq 0$$? From Pischke's notes, it seems that they should be identical but he did not mention whether they should be significant or not.

Apart from that, in situations where $$\beta_{-2}$$ and $$\beta_{-1}$$ are significant but close to zero, can it be counted as a parallel trend as Pischke (2005) as above?

So, it seems that one way to satisfy pre-trend testing is when the estimates of $$\beta_{−2}$$ and $$\beta_{−1}$$ are insignificant.

Correct.

It's evidence supporting parallel trends in the periods before a treatment. If it fails in practice, then we often turn to synthetic control methods to help us select from a pool of eligible controls with comparable outcome trajectories pre-shock.

But what about $$\beta_j$$, $$j \geq 0$$?

The estimates investigate whether treatment effects vary with time since exposure. For example, effects may accumulate or deteriorate with the passage of time. The lecture notes you cite make no claim about the significance of the $$\beta_j$$'s; the only assertion is that the effects in the post-period may not be identical. It depends entirely on how the intervention (i.e., treatment) affects your outcome of interest. It's not a requirement that the $$j$$ lags have any particular direction or magnitude. Sometimes a discovery that all of the $$\beta_j$$'s are zero may be an interesting finding as well. Not every evaluation returns a significant treatment effect.

Apart from that, in situations where $$\beta_{−2}$$ and $$\beta_{−1}$$ are significant but close to zero, can it be counted as a parallel trend as Pischke (2005) as above?

• (2)I thought "$\beta_{−2}$ close to 0 but significant" will be analogous to "$\beta_{−2}$ not close to 0" but insignificant. But from your explanation, it is not the case. So, "$\beta_{−2}$ close to 0 but significant" can be counted as anticipatory effects? I am surprised about that because the immediate difference between treatment and control group close by 0, so how come the anticipatory effect happen. Jul 7 at 8:25