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When reading a DID paper, I faced a common phrase "unmeasured covariates", for example

In a similar manner, the coefficient on $P_t$ captures the combined effects of any unmeasured covariates that change between the two periods but affect outcomes the same way in both groups

or

A key point is that the group effects and time trends stem from underlying differences in unmeasured covariates across groups and time periods. The DID design is meant to control for these unmeasured confounders even though the underlying variables are not measured explicitly

I cannot fully understand these sentence due to the lack of understanding "unmeasured covariates" phrase

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Unmeasured covariate is simply some omitted variable. For example, suppose that the true model that explains wages is given by:

$$w_i = \beta_0 + \beta_1 E_i + \beta_2 A_i + e_i$$

where $w_i$ is wage, $E_i$ your education and $A_i$ innate ability or talent.

Unfortunately in real life we have no statistics about talent/ability of an individual. Consequently in real life we can only estimate the following regression:

$$w_i = \beta_0 + \beta_1 E_i +e_i$$

However, the model above suffers from omitted variable bias (e.g. problem of having unmeasurable covariates see discussion in Verbeek Guide to Modern Econometrics pp 55).

The Diff-in-Diff solves this issue because as explained in the previous question you asked here, even if the outcome is different due to unobserved variable between groups or individuals, 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, since for example if your level of ability grants you \$1000 extra dollars in your monthly wage, it should do so before and after some intervention and so any increase in your wage above what would be given by the trend (for which you use control group) can be argued to be attributed to the intervention.

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  • $\begingroup$ Thanks @1muflon1. Of my curiousity, what does $R_i$ stands for in your second equation, I mean, which variable it is in theory or applied model. $\endgroup$ Jun 2 at 23:51
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    $\begingroup$ @Knowledge-chaser sorry that was a typo it was supposed to be E_i, R is next key right of E I accidentally must have hit it - I already corrected the text $\endgroup$
    – 1muflon1
    Jun 3 at 0:01
  • $\begingroup$ And the very last point is "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". I assume outcome is dependent variable, so you mean that all differences in time invariant unobservable reflected in error term (normally epsilon) would be reflected already in the value of dependent variable? $\endgroup$ Jun 3 at 0:54
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    $\begingroup$ @Knowledge-chaser 1. in response to the first question, yes by outcomes I mean dependent variable. 2. No, the second sentence would not be correct. There are different ways how DiD can be specified a popular one is via fixed effects. If you use fixed effects specifications the unobservable are captured by fixed effects, fixed effects regression would look like this $w_{it}=β_i+β_1E_{it}+e_{it}$, note now each panel member will have its own intercept and it is this 'individualized intercept' (in proper terminology fixed effect) that captures the effect of unobservables not error term $\endgroup$
    – 1muflon1
    Jun 3 at 1:27
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    $\begingroup$ if the unobservables would still in error term there would still be bias $\endgroup$
    – 1muflon1
    Jun 3 at 1:28

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