Why including individual level variables help to control for confounding events?

Question

In Pischke,2005, p.7's note, he documented

Including individual level variables may not only help to control for confouning trends, but may also reduce the variance of $\epsilon_{ist}$ , which may reduce the standard errors of the estimate of $\beta$.

It is clear about the part "reduce the variance of $\epsilon_{ist}$ , which may reduce the standard errors of the estimate of $\beta$."

But I do not understand why he states"Including individual-level variables may help to control for confounding trends". Normally, I need to perform an event study test to make sure the confounding event has no impact on my result, so the way he makes this conclusion is quite a bit strange to me.

Answer 1

Because even if you include in diff-in-diff fixed effects your model can still suffer from omitted variable bias.

Fixed individual effects only help to control for time invariant individual effects. For example, IQ or innate ability that is often though to be time invariant can be controlled by fixed effects. However, fixed individual effects do not help to control for time variant fixed effects.

Now you might think that you can solve this issue by including time fixed effects, but that is still not perfect solution because time fixed effects control for time variant omitted variables that affect all panel members in a homogenous way.

However, there can be many variables that can be heterogenous both across time and across individuals.

For example, if you estimate the following model:

$$y_{it} = \alpha_i + \gamma_t + \beta_1 T_{it} + \epsilon_{it} \tag{A}$$

where $y$ is dependent variable (lets say output), $\alpha_i$ fixed effects, $\gamma_t$ time fixed effects, $T$ some treatment and $\epsilon$ error.

But the true model should look like this:

$$y_{it} = \alpha_i + \gamma_t + \beta_1 T_{it} + \beta_2 X_{it} + \epsilon_{it} \tag{B}$$

and $X_{it}$ is some variable that is heterogenous both across individuals and time, lets say that it is the amount of stress that naturally fluctuates over time and all individuals might have different levels of stress.

If you estimate model A you will still suffer from omitted bias problem because the fixed effects cannot adequately capture the effect of omitted $X$ and thus your $\beta_1$ will still be biased in A. Including $X$ explicitly controls for the confounding trend in $X$.

Consequently, just because you include individual and time fixed effects that does not mean you can just ignore the issue of confounders or omitted variable bias. It will be just less of an issue than in standard cross-sectional or time series regression since you are able to control at least for individual and time invariant, or time varying but homogenous effects. However, you are still left with potential omitted variable bias stemming from non-included time varying heterogenous trends.