# Difference in Difference method: how to test for assumption of common trend between treatment and control group?

Following a comment from a previous thread, I want to know how one can test for the assumption of common trend between the treatment and control group in the Difference in Difference method?

Can I test that assumption with data of two time points (for example, baseline survey in 2002, treatment happens from 2002 to 2006 and follow-up survey in 2006)?

Thank you very much!

Edited: After posting this question, the "related" panel leads me to this unanswered question, in which the asker wanted to understand intuitions behind a method to account for time trends in DID method. I want to link it here as that question is very interesting to me too. Thank you!

The typical thing to do is visual inspection of the pre-treatment trends for the control and treatment group. This is particularly easy if you only have those two groups given a single binary treatment. Ideally the pre-treatment trends should look something like this: This graph was taken from a previous answer to the question why we need the common trends assumption. This includes also an explanation of the blue-dashed line which is the counterfactual outcome for the treated that can be assumed if we can reasonably verify the parallel trends assumption.

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).

The idea is the following. You include the interactions of the time dummies and the treatment indicator for the first two pre-treatment periods and you leave out the one interaction for the last pre-treatment period due to the dummy variable trap. Also now all the other interactions are expressed relative to the omitted period which serves as the baseline. 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.

An attractive feature of this test is that also the interactions of the time dummies after the treatment with the treatment indicator is informative. For instance, $\beta_{1}, \beta_2, \beta_3$ show you whether the treatment effect fades out over time, stays constant, or even increases. An application of this approach is Autor (2003).

Note that the literature generally refers to $\beta_{-2}, \beta_{-1}$ as "leads" and $\beta_{1}, \beta_2, \beta_3$ as "lags", even though they are merely interactions of the treatment indicator with time dummies and are not actually leads and lags of the treatment indicator in a time-series jargon sense. A more detailed explanation of this parallel trends test is provided in the lecture notes by Steve Pischke (here on page 7, or here on page 9).

• The answer is really helpful and thank you for linking to the excellent notes of Pischke. The notes provide helpful explanations for my both questions, particularly on DiD for panel data. I am sorry for the late acceptance. For the case when there are only two time points (as in my question), is it true that the only way to justify the common trend is to provide a reasonable assumption? (As in Card and Krueger 1994 and rejusitify when more data are available, Card and Krueger, 2000)
– Khan
Jul 9, 2015 at 11:20
• It's very hard to sell a difference in differences story with only two time periods because you cannot show anything regarding the evolution of the pre-treatment trends. You would need to make a very strong argument as to why those trends should have been parallel to begin with if you cannot show it graphically. For the regression based test you also need at least 3 time periods.
– Andy
Jul 9, 2015 at 11:30
• Thank you very much. I wanted to to see if my understanding is right after a day of reading. I think I would still use DiD and complement with Rubin causal model (so I have two estimators). I am only a masters student so I think as long as I provide both advantages and limitations of the estimators, I would be able to draw some inference (I haven't been taught these methods so hopefully I would be fine). Thank you very much!
– Khan
Jul 9, 2015 at 11:35
• I disagree with this terminology of "lead/lag". A "lag" is an adjustment for the previous value of the outcome. So if ones systolic blood pressure last year was 180, the value of 180 would be the covariate value for the present year as a one year lag. Following this convention, it makes no sense to me to adjust for a "lead" ever. This is different from a fixed effect adjustment for time-period. To estimate the model you depict in the graphic, I would use that type of time series: fixed effect time adjustment, and a pre/post indicator. May 7, 2018 at 16:35
• The terminology is standard in the econometrics literature. See Angrist and Pischke (2009) Mostly Harmless Econometrics.
– Andy
May 7, 2018 at 17:18

There is a good manner to verify if the pre-trend common assumption is reasonable in a difference-in-difference framework with two time and two periods. But is necessary to have some data for more than one pre-treatment period (Sometimes, the DiD with two periods performs better than the DiD with multiple periods).

Considering your example, you can run a DiD with the period of 2002 like a post-treatment and another pre treatment period (Suppose 2001). If the ATT be statistically significant its an evidence against the pre-trend common assumption, in other words, in the period of 2001-2002 the effect was already happening.

The following papers use this approach:

Beatty and Shimshack, 2011

Lima and Silveira-Neto, 2015

• Hi thanks for the links and your interest. However, because I don't have data in 2001 (or any previous year other than 2002), I don't think the papers are useful in guiding me testing for the common trends. Thank you very much anyway.
– Khan
Jul 15, 2015 at 22:54