This may sound like a rudimentary question, but I am curious if dummy variables reduce or eliminate the need for certain other controls. For instance, if I am looking at the impact of some variable X_1 that is at a CITY level (across several countries) and want to use some X_2 as a control - if X_2 is only available at a COUNTRY level, then wouldn't a COUNTRY dummy variable capture all relevant variance anyway and eliminate the need for X_2 as a control in estimating X_1?
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$\begingroup$ The general answer is "No". If you want a more specific or detailed answer you should give all the details on the framework. Do you speak about the linear model, and ask about under which circumstances $\beta$ is significant in $$ y_{nt} = \alpha + x_{nt}\beta + u_{nt} $$ but no longer significant in $$ y_{nt} = \alpha_n + x_{nt}\beta + v_{nt},$$ where the error term $ v_{nt} $ is assumed to be orthogonal to $X$ ? $\endgroup$– BertrandMay 27, 2022 at 16:22
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$\begingroup$ Per the below, the data is a cross-section rather than a panel. So I am looking at the difference between: ( y = a + x_1*B_1 + x_2*B_2 + countrydummyB_n + u ) vs. ( y = a + x_1*B_1 + countrydummyB_n + v ) I'm not good with latex but you can imagine a subscript of _i for each of these representing city i. Would the estimate for B_1 be consistent across both? $\endgroup$– JKim59817May 27, 2022 at 16:26
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2 Answers
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No it is not guaranteed it will solve the problem. The fixed country dummy would capture all relevant effects that are time invariant.
Now in principle you could also add time fixed effects but that’s still no panacea because time fixed effects assume the time effect is spatially homogenous.
If you believe that $X_2$ is something that varies across time and has heterogenous effect (e.g. output of firm, number of pupils per school etc) then you still need to control for $X_2$.
answered May 27, 2022 at 16:20
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$\begingroup$ The data is a cross section rather than a panel. So let's say X_1 is percentage of people in each city in poverty, X_2 is the country gdp per capita (which is the same for every city in a country in the cross-section) and then we also have a dummy variable for each country. If all I care about is understanding the impact of X_1 on some given Y, do I need to include X_2 or will the dummy suffice to capture all relevant country-level effects? $\endgroup$ May 27, 2022 at 16:25
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$\begingroup$ @JKim59817 dummy would suffice for the regression to be unbiased but A) you will need a lot of observations in each sub unit in your cross section since every country dummy is another non-deterministic parameter. B) it would be strange not to use easily findable GDP. Even if reg with dummies would be unbiased you would get more from the reg with actual variable interpretation-wise. If I would be conference/workshop participant, and you would say you did this for easily measured variable like GDP I would raise that as an issue $\endgroup$– 1muflon1 ♦May 27, 2022 at 17:04
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My answer is "Yes" for your specific question with $X_2$ being available only at the country level. (No if $X_2$ is a general variable.)
Country dummies eliminate the need for $X_2$ as a control because $X_2$ is perfectly predicted by the country dummies. More technically, $X_2$ and $m-1$ country dummies ($m$ = number of countries) are perfectly collinear. If you regress $y$ on $X_1$, $X_2$ and country dummies, $X_2$ or one country dummy will be omitted depending on which of x2 and i.country is specified first.
That said, please make sure that the two models (one with $X_2$ and without country dummies, and the other with country dummies) are two different models because one controls for only $X_2$ while the other controls for all cross-country heterogeneity. You can also compare the $X_1$ coefficients from both models for whether $X_2$ is sufficient as a control, which can be interesting.
