Econometrics Problem

Question

Determine the sign of the expected bias introduced by omitting a relevant variable. State your assumptions and provide a very brief rationale for your assumptions.

We plan to estimate the following regression:

𝑙𝑒𝑎𝑟𝑛𝑖𝑛𝑔𝑖 = 𝛽0 + 𝛽1𝑐𝑜𝑚𝑝𝑢𝑡𝑒𝑟𝑠𝑖 + 𝛽2𝑥1𝑖 + 𝛽3𝑥2𝑖 + ⋯ + 𝛽𝑘+1𝑥𝑘𝑖 + 𝜀𝑖

Where 𝑙𝑒𝑎𝑟𝑛𝑖𝑛𝑔𝑖 is a measure of student learning at school 𝑖, 𝑐𝑜𝑚𝑝𝑢𝑡𝑒𝑟𝑠𝑖 is a dummy variable indicating whether or not students have access to computers at school 𝑖, and the 𝑥 variables are other variables that you think should be held fixed when estimating the effect of 𝑐𝑜𝑚𝑝𝑢𝑡𝑒𝑟𝑠𝑖 on 𝑙𝑒𝑎𝑟𝑛𝑖𝑛𝑔𝑖. We do not observe the amount of funding for school 𝑖 so it cannot be included as a regressor. How do you expect the omission of funding from the regression to bias the estimate of 𝛽1?



Answer 1

Omitted variable bias occurs when we leave a variable, Z, out of our regression which is correlated both with the dependent variable (Y) and main regressor (X) in our model. The sign of the bias can, loosely, be determined with the following formula:

Sign(Bias) = Sign(Correlation between Y and Z)*Sign(Correlation between X and Z).

In this example, student learning is Y, computer access is X, and school funding is Z. I would assume that school funding is positively correlated with student learning and that school funding is also positively correlated with computer access. Hence, the bias should have a positive sign. In other words, OLS estimates a relationship between computer access and learning which is larger than the "true" causal effect.

This makes intuitive sense. Part of the estimated effect of computer access on student learning really just reflects the fact that schools with computers tend to be well-funded in general.