# How to clarify the description of common trend assumption?

From reading a paper, I saw the way they describe the common trend is

Specifically, DID designs assume that confounders varying across the groups are time invariant, and time-varying confounders are group invariant. Researchers refer to these twin claims as a common trend assumption

The confounding variable is a variable affecting both independent and dependent variable, causing spurious trend, I understand. But I cannot clearly understand the concept the paper above mention about the common trend assumption.

Update:

I just recognized that I did not fully understand some terms here: "time invariant", "group invariant", "time varying confounders", and is there any example for "confounders varying across the groups"?

The previous +1 answer by tdm already provides mathematical argument but let me give you intuition based on graphical argument.

Consider the picture below taken from Columbia Public Health. As the graph shows, the common trend is important because if outcome in two places evolves in the same way (there is constant difference in outcomes i.e. there is common trend), even if the outcome is different you can argue that you can get treatment effect from observing change in trend between treatment and control. This controls for any time invariant unobservable, like for example innate ability or IQ between groups because the differences in these time invariant unobservable is already reflected in difference in outcomes, but it should not affect trend after intervention (if there was common trend before intervention). By adding time fixed effects (which is difficult to visualize) you can control also for time variant factors (e.g. some cyclical seasonality) that affect both groups. is there any example for "confounders varying across the groups"?

Yes, for example above mentioned innate ability or IQ. This will can very well vary by groups even though it should be time invariant.

• Thanks 1muflon1. I have a question. From my understanding, outcome is dependent variable. I get confused in this sentence "*even if the outcome is different *". do you mean the dependent variable difference between control and treatment during pre-treatment period? Jun 3 at 0:26
• But if what I mentioned above is the case, so the sentence "even if the outcome is different you can argue that you can get treatment effect from observing change in trend between treatment and control" does not make sense to me then. Jun 3 at 0:29
• But yeah, if we say the "IQ" it should be more convinced then. I deem Jun 3 at 0:45
• @Knowledge-chaser 1. yes by different outcome I mean different value of dependent variable. 2 what part of it does not make sense? I can try to elaborate but I first need to know why you think its unreasonable
– 1muflon1
Jun 3 at 1:29
• @Knowledge-chaser I mean for a rigorous logic why you can do that defer to the mathematical derivation by tdm. In my answer I just offer additional intuition which is not rigorous but can be easier to understand. The point here is that if the trend between two variables is exactly the same then you can assume that if there is no intervention the variable would just follow its trend, so you can use control to know what the trend in treatment would be and then just measure treatment effect as deviation from the trend. Again this is mere intuition for rigor refer to the answer by tdm
– 1muflon1
Jun 3 at 9:39

Consider a setting with two time periods, $$t = 0,1$$ and a control $$c = 0$$ and treatment $$c = 1$$ group.

Let $$T = 1[t = 1]$$ be the time dummy and $$C = 1[c = 1]$$ be the treatment dummy. Then the DiD regressing is given by: $$y_{c,t} = \alpha_0 + \alpha_1 T + \alpha_2 C + \alpha_3 T\times C + \varepsilon_{c,t}$$ If there are unobserved cofounders $$z_{c,t}$$ then these are part of the error term $$\varepsilon_{c,t}$$. We can then write: $$\varepsilon_{c,t} = \eta_{c,t} + z_{c,t},$$ where $$\eta_{c,t}$$ is just noise (e.g. measurement error), so we assume it satisfies the condition $$\mathbb{E}[\eta_{c,t}|T,C] = 0$$. The DiD estimator is given by: $$\mathbb{E}[y_{c,t}|T = 1, C = 1] - \mathbb{E}[y_{c,t}|T = 1, C = 0] -\left(\mathbb{E}[y_{c,t}|T = 0, C = 1] - \mathbb{E}[y_{c,t}|T = 0, C = 0]\right)$$ Computing this gives:

\begin{align*} &\alpha_3 +\mathbb{E}[z_{c,t}|T = 1, C = 1] - \mathbb{E}[z_{c,t}|T = 1, C = 0],\\ &-\left(\mathbb{E}[z_{c,t}|T = 0, C = 1] - \mathbb{E}[z_{c,t}|T = 0, C = 0]\right),\\ =&\alpha_3 + \mathbb{E}[z_{11} - z_{0,1} + z_{1,0} - z_{0,0}]. \end{align*} For identification, we would like this to be equal to $$\alpha_3$$, which requires that: $$\mathbb{E}[z_{1,1} + z_{0,0} - z_{0,1} - z_{1,0}] = 0.$$ One assumption that will lead to this condition is if cofounders are either time variant but group invariant (i.e. don't depend on $$c$$) or group variant but time invariant (i.e. don't depend on $$c$$). Denoting the first by $$x_t$$ and the second by $$y_c$$, we can then write:
$$z_{c,t} = x_c + y_t,$$ So: $$\mathbb{E}[z_{1,1} + z_{0,0} - z_{0,1} - z_{1,0}] = \mathbb{E}[(x_1 + y_1) + (x_0 + y_0) - (x_0 + y_1) - (x_1 + y_0)] = 0.$$ Which means that $$\alpha_3$$ is identified using the DiD estimator.

• Thank you so much for your help but I get lost in the mathematic equation, I am wondering that if you can have a look on the update part of the question that I asked some terms in this sentence. I get lost mainly because I did not understand this sentence thoroughly. Jun 2 at 10:23
• @Knowledge-chaser Time invariant means that the variable does not depend on time. Group invariant means that the variable does not vary across groups. For example, education might be a time invariant factor if it does not change over time (but might be different between control and treatment groups). Macro factors like inflation can be seen as group invariant as it is the same for both treatment and control, (but it might change over time).
– tdm
Jun 2 at 10:58
• Thanks @tdm . I am ambiguous about the part "For example, education might be a time invariant factor if it does not change over time (but might be different between control and treatment groups)." I am wondering why education does not change over time, I am sorry if it is a stupid question, but one's education should be accumulated over time. I imagine it seems like the firm age variable, which increases by time. The example for inflation is totally amazing and understandable Jun 3 at 0:12
• I mean, whether the assumption that education does not change over time is reasonable tho, I am wondering if there is any other great example for time invariant factor. Thank you Jun 3 at 0:15