How to clarify the description of common trend assumption?

Question

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"?

Answer 1

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.

Answer 2

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.