Econometrics Question, see description (was too long)

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:

𝑡𝑖𝑐𝑘𝑒𝑡𝑠𝑖 = 𝛽0 + 𝛽1𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒𝑖 + 𝛽2𝑥1𝑖 + 𝛽3𝑥2𝑖 + ⋯ + 𝛽𝑘+1𝑥𝑘𝑖 + 𝜀𝑖

Where 𝑡𝑖𝑐𝑘𝑒𝑡𝑠𝑖 is the average number of bus tickets person 𝑖 purchases per month, 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒𝑖 is the distance of person 𝑖’s residence from the nearest bus station, and the 𝑥 variables are other variables that you think should be held fixed when estimating the effect of 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒𝑖 on 𝑡𝑖𝑐𝑘𝑒𝑡𝑠𝑖. We do not observe a person’s income so it cannot be included as a regressor. If it tends to be less expensive to live near bus stations, how do you expect the omission of income from the regression to bias the estimate of 𝛽1?

Answer 1

If it tends to be less expensive to live near the bus station, we may view the distance variable as a proxy to the income (when income variable is omitted). As such: smaller distance could mean lower income and using bus more frequently. This perhaps means a weak and negative correlation between distance variable and average number of tickets purchased. In other words, β₁ is probably a small values and has a negative sign.