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I am studying a natural experiment that resulted in some cities randomly receiving different numbers of competitors in a certain sector. (Stores applied for licenses in a government-run lottery.) In particular, I'm interested in whether firms that, randomly, ended up as monopolies in their city received lower customer satisfaction scores than firms that ended up having competitors.

I understand that I need to control for the number of incumbents in a city, the number of applicants in a city to the lottery, and the number of winners the government would select for the broader region the city is in.

The equation I want to estimate is then: $satisfaction = \beta_0+\beta_1monopoly+\beta_2incumbents+\beta_3applicants+\beta_4winnersinregion+\epsilon$.

However, my dataset is very small, as I only observe data on 28 firms. Thus, my estimate of $\beta_1$ lacks precision and is statistically insignificant.

I am wondering if it is okay to combine my 'applicants' and 'winners in region' control variables into one control variable so as to increase the precision of my estimate. In particular, I would calculate the expected number of competitors in a city based on how many applicants the city had and how many licences were awarded in the city's region and estimate the following equation:

$satisfaction = \beta_0+\beta_1monopoly+\beta_2incumbents+\beta_3expectednumberofcompetitors+\epsilon$

Does this approach still adequately control for the potentially confounding variables 'applicants' and 'winners in region'?

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