Economist Writing Every Day

The Differences-in-Differences literature has blown up in the past several years. “Differences-in-Differences” refers to a statistical method that can be used to identify causal relationships (DID hereafter). If you’re interested in using the new methods in Stata, or just interested in what the big deal is, then this post is for you.

First, there’s the basic regression model where we have variables for time, treatment, and a variable that is the product of both. It looks like this:

The idea is that that there is that we can estimate the effect of time passing separately from the effect of the treatment. That allows us to ‘take out’ the effect of time’s passage and focus only on the effect of some treatment. Below is a common way of representing what’s going on in matrix form where the estimated y, yhat, is in each cell.

Each quadrant includes the estimated value for people who exist in each category.  For the moment, let’s assume a one-time wave of treatment intervention that is applied to a subsample. That means that there is no one who is treated in the initial period. If the treatment was assigned randomly, then β=0 and we can simply use the differences between the two groups at time=1.  But even if β≠0, then that difference between the treated and untreated groups at time=1 includes both the estimated effect of the treatment intervention and the effect of having already been treated prior to the intervention. In order to find the effect of the intervention, we need to take the 2^nd difference. δ is the effect of the intervention. That’s what we want to know. We have δ and can start enacting policy and prescribing behavioral changes.

Easy Peasy Lemon Squeezy. Except… What if the treatment timing is different and those different treatment cohorts have different treatment effects (heterogeneous effects)?*  What if the treatment effects change over time the longer an individual is treated (dynamic effects)**?  Further, what if the there are non-parallel pre-existing time trends between the treated and untreated groups (non-parallel trends)?*** Are there design changes that allow us to estimate effects even if there are different time trends?**** There’re more problems, but these are enough for more than one blog post.

For the moment, I’ll focus on just the problem of non-parallel time trends.

What if untreated and the to-be-treated had different pre-treatment trends? Then, using the above design, the estimated δ doesn’t just measure the effect of the treatment intervention, it also detects the effect of the different time trend. In other words, if the treated group outcomes were already on a non-parallel trajectory with the untreated group, then it’s possible that the estimated δ is not at all the causal effect of the treatment, and that it’s partially or entirely detecting the different pre-existing trajectory.

Below are 3 figures. The first two show the causal interpretation of δ in which β=0 and β≠0. The 3^rd illustrates how our estimated value of δ fails to be causal if there are non-parallel time trends between the treated and untreated groups. For ease, I’ve made β=0  in the 3^rd graph (though it need not be – the graph is just messier). Note that the trends are not parallel and that the true δ differs from the estimated delta. Also important is that the direction of the bias is unknown without knowing the time trend for the treated group. It’s possible for the estimated δ to be positive or negative or zero, regardless of the true delta. This makes knowing the time trends really important.

STATA Implementation

If you’re worried about the problems that I mention above the short answer is that you want to install csdid2. This is the updated version of csdid & drdid. These allow us to address the first 3 asterisked threats to research design that I noted above (and more!). You can install these by running the below code:

program fra
    syntax anything, [all replace force]
    local from "https://friosavila.github.io/stpackages"
    tokenize `anything'
    if "`1'`2'"==""  net from `from'
    else if !inlist("`1'","describe", "install", "get") {
        display as error "`1' invalid subcommand"
    }
    else {
        net `1' `2', `all' `replace' from(`from')
    }
    qui:net from http://www.stata.com/
end
fra install fra, replace
fra install csdid2
ssc install coefplot

Once you have the methods installed, let’s examine an example by using the below code for a data set. The particulars of what we’re measuring aren’t important. I just want to get you started with the an application of the method.

local mixtape https://raw.githubusercontent.com/Mixtape-Sessions
use `mixtape'/Advanced-DID/main/Exercises/Data/ehec_data.dta, clear
qui sum year, meanonly
replace yexp2 = cond(mi(yexp2), r(max) + 1, yexp2)

The csdid2 command is nice. You can use it to create an event study where stfips is the individual identifier, year is the time variable, and yexp2 denotes the times of treatment (the treatment cohorts).

csdid2 dins, time(year) ivar(stfips) gvar(yexp2) long2 notyet
estat event,  estore(csdid) plot
estimates restore csdid

The above output shows us many things, but I’ll address only a few of them. It shows us how treated individuals differ from not-yet treated individuals relative to the time just before the initial treatment. In the above table, we can see that the pre-treatment average effect is not statistically different from zero. We fail to reject the hypothesis that the treatment group pre-treatment average was identical to the not-yet treated average at the same time period. Hurrah! That’s good evidence for a significant effect of our treatment intervention. But… Those 8 preceding periods are all negative. That’s a little concerning. We can test the joint significance of those periods:

estat event, revent(-8/-1)

Uh oh. That small p-value means that the level of the 8 pretreatment periods significantly deviate from zero. Further, if you squint just a little, the coefficients appear to have a positive slope such that the post-treatment values would have been positive even without the treatment if the trend had continued. So, what now?

Wouldn’t it be cool if we knew the alternative scenario in which the treated individuals had not been treated? That’s the standard against which we’d test the observed post-treatment effects. Alas, we can’t see what didn’t happen. BUT, asserting some premises makes the job easier. Let’s say that the pre-treatment trend, whatever it is, would have continued had the treatment not been applied. That’s where the honestdid stata package comes in. Here’s the installation code:

local github https://raw.githubusercontent.com
net install honestdid, from(`github'/mcaceresb/stata-honestdid/main) replace
honestdid _plugin_check

What does this package do? It does exactly what we need. It assumes that the pre-treatment trend of the prior 8 periods continues, and then tests whether one or more post-treatment coefficients deviate from that trend. Further, as a matter of robustness, the trend that acts as the standard for comparison is allowed to deviate from the pre-treatment trend by a multiple, M, of the maximum pretreatment deviations from trend. If that’s kind of wonky – just imagine a cone that continues from the pre-treatment trend that plots the null hypotheses. Larger M’s imply larger cones. Let’s test to see whether the time-zero effect significantly differs from zero.

estimates restore csdid
matrix l_vec=1\0\0\0\0\0
local plotopts xtitle(Mbar) ytitle(95% Robust CI)
honestdid, pre(5/12) post(13/18) mvec(0(0.5)2) coefplot name(csdid2lvec,replace) l_vec(l_vec)

What does the above table tell us? It gives us several values of M and the confidence interval for the difference between the coefficient and the trend at the 95% level of confidence. The first CI is the original time-0 coefficient. When M is zero, then the null assumes the same linear trend as during the pretreatment. Again, M is the ratio by which maximum deviations from the trend during the pretreatment are used as the null hypothesis during the post-treatment period.  So, above, we can see that the initial treatment effect deviates from the linear pretreatment trend. However, if our standard is the maximum deviation from trend that existed prior to the treatment, then we find that the alpha is just barely greater than 0.05 (because the CI just barely includes zero).

That’s the process. Of course, robustness checks are necessary and there are plenty of margins for kicking the tires. One can vary the pre-treatment periods which determine the pre-trend, which post-treatment coefficient(s) to test, and the value of M that should be the standard for inference. The creators of the honestdid seem to like the standard of identifying the minimum M at which the coefficient fails to be significant. I suspect that further updates to the program will come along that spits that specific number out by default.

I’ve left a lot out of the DID discussion and why it’s such a big deal. But I wanted to share some of what I’ve learned recently with an easy-to-implement example. Do you have questions, comments, or suggestions? Please let me know in the comments below.

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The above code and description is heavily based on the original author’s support documentation and my own Statalist post. You can read more at the above links and the below references.

*Sun, Liyang, and Sarah Abraham. 2021. “Estimating Dynamic Treatment Effects in Event Studies with Heterogeneous Treatment Effects.” Journal of Econometrics, Themed Issue: Treatment Effect 1, 225 (2): 175–99. https://doi.org/10.1016/j.jeconom.2020.09.006.

**Sant’Anna, Pedro H. C., and Jun Zhao. 2020. “Doubly Robust Difference-in-Differences Estimators.” Journal of Econometrics 219 (1): 101–22. https://doi.org/10.1016/j.jeconom.2020.06.003.

***Callaway, Brantly, and Pedro H. C. Santa Anna. 2021. “Difference-in-Differences with Multiple Time Periods.” Journal of Econometrics, Themed Issue: Treatment Effect 1, 225 (2): 200–230. https://doi.org/10.1016/j.jeconom.2020.12.001.

****Rambachan, Ashesh, and Jonathan Roth. 2023. “A More Credible Approach to Parallel Trends.” The Review of Economic Studies 90 (5): 2555–91. https://doi.org/10.1093/restud/rdad018.