# Concerns about pre-trend stability testing

In this discussion, @Thomas Bilach well explains the equation to test pre-trend stability.

$$y_{kt} = \alpha_k + \lambda_t + \delta_{-2} d_{k,t-2} + \delta_{-1} d_{k,t-1} + \delta d_{kt} + \delta_{+1} d_{k,t+1} + \delta_{+2} d_{k,t+2} + \delta_{+\bar{3}} d_{k,t+\bar{3}} + \epsilon_{kt},$$

where I include some leads and lags of the policy dummy. The "coefficient values" refer to the estimates on each of the $$\delta$$'s. Note how $$\delta$$ is the immediate effect of a leniency law on your outcome(s) of interest; it's unsubscripted. I could've specified it as $$\delta_0$$ if I wanted to be explicit, but I think you get the basic idea.

I have two concerns here:

1. Do we run the regression without the intercept in this equation, and why do we run it without the intercept? I also read Autor (2003)'s paper, Table 7 suggested by @Thomas Bilach and it seems that they also do not include an intercept in their regression.

2. Why do we not add the independent variables but only group and period fixed effect in this equation?

1. Do we run the regression without the intercept in this equation, and why do we run it without the intercept? I also read Autor (2003)'s paper, Table 7 suggested by @Thomas Bilach and it seems that they also do not include an intercept in their regression.

Tabular results report the estimated coefficients on the leads and lags of treatment. The "intercept" is not of substantive interest, so it's not worth reporting. Technically, the fixed effects for each $$k$$ is akin to giving each unit (i.e., firm, county, state, country, etc.) its own intercept. Put very simply, each $$k$$ has its own starting point.

I think you want to estimate the following:

$$y_{k,t} = \mu + \alpha_k + \lambda_t + \delta_{-1} d_{k,t-1} + \delta d_{kt} + \delta_{+1} d_{k,t+1} + \delta_{+2} d_{k,t+2} + \epsilon_{k,t},$$

where $$\alpha_k$$ and $$\lambda_t$$ denote fixed effects for countries and years, respectively. To shorten the equation a bit, I included one lead and two lags of the policy variable. Here, $$\mu$$ denotes a "global" intercept. We usually drop it. In fact, it must dropped in a fixed effects equation. The fixed effects would absorb a global constant term. Note how estimating country fixed effects is equivalent to a model where we include a constant for each country.

Evaluators typically report (plot) the estimates on each of the $$\delta$$'s over time. The "global" constant term has no value in this context. Likewise, it's also customary to ignore the country- and year-specific effects; they nuisance. Tabular results will almost always note that fixed effects were estimated for unit and time, but omit the corresponding estimates. In short, focus on the policy variables.

1. Why do we not add independent variables but only group and period fixed effect in this equation?

The equation I specified is not the canonical approach. It is absolutely permissible to include covariates. In fact, it's desirable to do so. The following specification is a compact way of writing out the foregoing equation:

$$y_{k,t} = \alpha_k + \lambda_t + \sum^{m}_{\tau = -q} \delta_{\tau} d_{k,t+\tau} + X'_{kt}\theta + \epsilon_{k,t},$$

where, again, $$\alpha_k$$ and $$\lambda_t$$ denote fixed effects for countries and years, respectively. The constant term has now been dropped. The summation is a parsimonious way of specifying the lead/lag structure. It allows for any arbitrary number of $$q$$ leads and $$m$$ lags, where $$\tau = {-q,..., -2, -1, 0, 1, 2,..., m}$$.

The specification now includes a covariate vector $$X_{kt}$$. Including relevant time-varying regressors at the country level is appropriate. If you observe outcomes at the firm level then including firm level covariates is acceptable as well.

I recommend you modify the foregoing equation to suit your specific research goals.

• Thank you @Thomas Bilach, Now, I read your answer again and have one curiosity about "$\mu$". So, whether we still run the regression with intercept $\mu$ but we just not report, or we must run the regression without intercept. Sorry for this novice question but I hope I can be clarified about that. Thanks so much Jun 17, 2021 at 5:11
• Question for you. Say you include dummy variables for all countries. Software usually drops a country for you. What do you think the intercept represents at this point? Jun 17, 2021 at 5:19
• intercept now from my understanding, is a global constant term, not necessarily to be reported but I am wondering if we need to put into the regression in software( Stata in my case). I hope I did not fall into a fallacy here Jun 17, 2021 at 5:30
• The omitted country is now absorbed into the intercept. In short, you do not need to manually input an intercept. Jun 17, 2021 at 5:31
• I see no reason for you to do this. Jun 17, 2021 at 5:44