Omitted/irrelevant variables

Question 1

How does omitting a relevant variable impact the subsequent regression coefficients?

Answer 1

They will be biased and the standard errors will be invalid, meaning any t/f-tests will be invalid

Question 2

How do we determine the direction of the bias?

Answer 2

We multiply the likely sign of the coefficient on the omitted variable by the likely sign of the correlation between the included and the omitted variable.

Question 3

How do we determine the direction of the bias on R^2?

Answer 3

If the bias is the same direction as the true effect of the included variable then R^2 is biased upwards and vice versa.

Question 4

What is the impact on coefficients if we include an irrelevant variable?

Answer 4

The variance will increase and will become less efficient, however, the standard errors remain valid.

Question 5

How do we solve omitted variable bias?

Answer 5

We use a proxy variable: it is linearly related to the missing variable, e.g: x2= lambda + mu Z.
For x1, the estimated coefficient, standard error and t statistic will remain the same, along with R^2.