<p>If you have violations of the parallel trend assumption then you will have biased estimates of the causal effect (assuming you satisfy the other assumptions). If you know the maximum that the two series can drift apart over time or have a functional form that specifies how that difference can arise, then you can estimate the diff in diff in a bounded way. For example, if IN has a trend of 2% growth per year and OH has a trend growth of 3%, then adding a linear time variable to a log regression will result in parallel trends.</p>
<p>You don't need the trends to be identical. Effectively, you need them to be identical in expectation (see Lechner (2011) <a href="https://core.ac.uk/download/pdf/6387069.pdf" rel="nofollow noreferrer">The Estimation of Causal Effects by Difference-in-Difference Methods</a>).
<a href="https://i.sstatic.net/u34uD.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/u34uD.png" alt="Common Trend as an expectation from Lechner (2011)" /></a></p>
<p>What I did in my paper, <a href="https://www.sciencedirect.com/science/article/pii/S1042957318300184" rel="nofollow noreferrer">Competition and complementarities in retail banking: Evidence from debit card interchange regulation</a>, is look at the trends in the pre-treatment period and test if they were statistically distinguishable.
<a href="https://i.sstatic.net/qjmHd.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/qjmHd.jpg" alt="testing common trend in pre period" /></a>. Of course, you can never test that the common trend in the pre-period would persist in the counterfactual treatment period in the absence of the treatment. That can only be assumed.</p>
<p>Another approach is to do a <a href="https://en.wikipedia.org/wiki/Synthetic_control_method" rel="nofollow noreferrer">synthetic cohort</a> analysis. Basically, you construct a fake control for each treated unit out of linear combinations of control units to maximize the similarity between control and treated units. You can do that in a way that closely or even exactly matches the pre-treatment trend of the treatment and control groups.</p>

CC BY-SA 4.0
2022-11-03T01:45:29.670Z
Very helpful, thank you -- I think the first option is closer to what I am aiming at. Is there a way to get an upper bound on the bias if we observe a stochastic difference in trends? Something like, for $t < 0$ (i.e. pre-treatment) and $\sigma^2$ small: $$Y^1_t - Y^0_t = u_t \sim N(0,\sigma^2)$$ Where $Y^1_t$ is the treated group and $Y^0_t$ untreated? I'm somewhat interested in cases of papers where the previous trends look fairly parallel, but clearly differ in small ways. Thanks again!
wflewis
CC BY-SA 4.0
2022-11-02T21:17:23.927Z
2022-11-03T19:35:16.263Z
2022-11-03T19:35:16.263Z
BKay
<p>I am interested (for no particular reason) in estimating a hypothetical Differences-in-Differences model with one period of treatment. However, we observe small non-linear violations of the parallel trends assumption.</p>
<p>For instance, consider wages for IN vs. OH before and after IN implements a new tax policy. We see that prior wages in IN are equal to wages in OH for T = <span class="math-container">$-\infty, ..., -3,-2,-1$</span>. However, we see that prior wages are <span class="math-container">$\epsilon &gt; 0$</span> higher in OH over the period <span class="math-container">$T = -5-k, -5$</span>, with <span class="math-container">$k$</span> some small-ish natural number. Here, parallel trends does not hold, but we might still expect to be able to observe the impacts of a change in tax policy.</p>
<p>Is there a way to still properly estimate a Differences in Difference estimator under these situations? I imagine that we should see the same point estimate for treatment at time <span class="math-container">$T = 0$</span>, with a penalty term being added to the standard errors of the point estimate which is a function of <span class="math-container">$\epsilon$</span>. That said, I have been unable to find any papers which cover this issue.</p>
<p>Related papers include: <a href="https://jonathandroth.github.io/assets/files/HonestParallelTrends_Main.pdf" rel="nofollow noreferrer">https://jonathandroth.github.io/assets/files/HonestParallelTrends_Main.pdf</a></p>
<p>However, as far as I can tell, that paper deals with differences in trends after treatment, while I am interested in differences in trends prior to treatment.</p>

<econometrics><difference-in-difference>
Differences in Differences with Small Violations of Parallel Trends
wflewis
# Differences in Differences with Small Violations of Parallel Trends

(Asked by: username_0 on 2022-11-01)

I am interested (for no particular reason) in estimating a hypothetical Differences-in-Differences model with one period of treatment. However, we observe small non-linear violations of the parallel trends assumption.

For instance, consider wages for IN vs. OH before and after IN implements a new tax policy. We see that prior wages in IN are equal to wages in OH for T = $-\infty, ..., -3,-2,-1$. However, we see that prior wages are $\epsilon > 0$ higher in OH over the period $T = -5-k, -5$, with $k$ some small-ish natural number. Here, parallel trends does not hold, but we might still expect to be able to observe the impacts of a change in tax policy.

Is there a way to still properly estimate a Differences in Difference estimator under these situations? I imagine that we should see the same point estimate for treatment at time $T = 0$, with a penalty term being added to the standard errors of the point estimate which is a function of $\epsilon$. That said, I have been unable to find any papers which cover this issue.

Related papers include: <https://jonathandroth.github.io/assets/files/HonestParallelTrends_Main.pdf>

However, as far as I can tell, that paper deals with differences in trends after treatment, while I am interested in differences in trends prior to treatment.

**Tags:** econometrics, difference-in-difference

---

## 1 answers

### Answer by username_1 on 2022-11-02. Score: 1 | Accepted answer

If you have violations of the parallel trend assumption then you will have biased estimates of the causal effect (assuming you satisfy the other assumptions). If you know the maximum that the two series can drift apart over time or have a functional form that specifies how that difference can arise, then you can estimate the diff in diff in a bounded way. For example, if IN has a trend of 2% growth per year and OH has a trend growth of 3%, then adding a linear time variable to a log regression will result in parallel trends.

You don't need the trends to be identical. Effectively, you need them to be identical in expectation (see Lechner (2011) [The Estimation of Causal Effects by Difference-in-Difference Methods](https://core.ac.uk/download/pdf/6387069.pdf)).
[![Common Trend as an expectation from Lechner (2011)](https://i.sstatic.net/u34uD.png)](https://i.sstatic.net/u34uD.png)

What I did in my paper, [Competition and complementarities in retail banking: Evidence from debit card interchange regulation](https://www.sciencedirect.com/science/article/pii/S1042957318300184), is look at the trends in the pre-treatment period and test if they were statistically distinguishable.
[![testing common trend in pre period](https://i.sstatic.net/qjmHd.jpg)](https://i.sstatic.net/qjmHd.jpg). Of course, you can never test that the common trend in the pre-period would persist in the counterfactual treatment period in the absence of the treatment. That can only be assumed.

Another approach is to do a [synthetic cohort](https://en.wikipedia.org/wiki/Synthetic_control_method) analysis. Basically, you construct a fake control for each treated unit out of linear combinations of control units to maximize the similarity between control and treated units. You can do that in a way that closely or even exactly matches the pre-treatment trend of the treatment and control groups.

> **Comments:**
> **username_0**: Very helpful, thank you -- I think the first option is closer to what I am aiming at. Is there a way to get an upper bound on the bias if we observe a stochastic difference in trends? Something like, for $t < 0$ (i.e. pre-treatment) and $\sigma^2$ small: $$Y^1_t - Y^0_t = u_t \sim N(0,\sigma^2)$$ Where $Y^1_t$ is the treated group and $Y^0_t$ untreated? I'm somewhat interested in cases of papers where the previous trends look fairly parallel, but clearly differ in small ways. Thanks again!  

---

