As indicated in class, the sign of the bias (positive or negative) of when omitting X₂ in the estimation of Y = ₀ + ₁X₁ + ₂X₂ + u can be summarized in the table below.

	Corr (X1,X2) >0	Corr (X1,X2) <0
2>0	+	-
2<0	-	+

Illustrate eachof the 4 cases in the table with an example that does not draw on the material in Section I of this problem set or specific examples we have discussed in class (in other words, please think of examples other than vehicle fatality rates; reperfusion treatment after a heart attack; training programs, education, and wages; and test scores and class size). Assume the regression equation you would like to estimate is Y = ₀ + ₁X₁ + ₂X₂ + u, but for some reason (lack of data or knowledge, for example), you end up omitting X₂ from the regression. (If need be you can use the same outcome variable Y and explanatory variable of interest X₁ in multiple examples, as long as you use different X₂ variables to illustrate the different cases.)

For each example:

1. Describe the variables Y, X₁, and X₂.

1. Indicate the sign of the correlation between X₁and X₂ and explain why you would expect this sign.
2. Indicate the sign of ₂ and explain why you would expect this sign.
3. Indicate the sign of the bias if you were to omit X₂ from the regression.
4. Indicate how the estimate of is likely to change when omitting X₂ from the regression, i.e. would the magnitude be smaller or larger when you estimate the full regression with both X₁ and X₂ as explanatory variables (i.e. is the effect understated or overstated when X₂ is omitted)? Explain your reasoning both in technical terms (using the formula for omitted variable bias and your answers to parts 1-4), and in terms a policymaker can understand.