Econometrics questions help

Question

1. For two generic random variables Zi and Wi the Law of Iterated Expectations (LIE) says that:

E[Wi] = EZ [E[Wi|Zi]] E[Zi] = EW [E[Zi|Wi]]

You are given the following regression model:

Yi = β0 + β1Xi + ui

and you are told that E[ui|Xi] = 0.

a) Does A1 hold?

b) Is β1 the causal effect of Xi on Yi?

c) Show that E(ui) = 0. Interpret what it means.

d) Show that Cov(ui, Xi) = 0. Interpret what it means.

2. Consider two random variables Ti and Gi. To answer the following questions a simple sketch is enough.

a) Draw a scatter plot such that Cov(Ti,Gi) is positive and R2 is high and close to 1.

b) Draw a scatter plot such that Cov(Ti,Gi) is positive and R2 is low.

c) Draw a scatter plot such that Cov(Ti,Gi) is negative and R2 is high and close to 1.

d) Draw a scatter plot such that Cov(Ti,Gi) = 0. What would be the value of the R2 in this case? Explain in detail. (Hint: use the OLS estimator for the slope of a simple linear regression to answer)

3. Suppose that the true relationship between the variables Yi and Xi is given by: Yi = α + βXi + ui

with ui = Wi + Zi, where Wi, Zi are additional variables that you are ignoring and that end up in the error term.

a) Suppose that Cov(Xi, Wi) > 0 and that Cov(Xi, Zi) > 0. Is it possible that A1 holds in this situation? Please explain why using both formality and intuition.

1

b) Suppose that Cov(Xi, Wi) > 0 and that Cov(Xi, Zi) < 0. Is it possible that the regressor is uncorrelated with the error term? Can you say precisely when this is the case?

c) Does Cov(Xi, ui) = 0 imply that A1 holds?

Answer 1

I just went through this entire problems set yesterday with another student at NYU. It would take me 45 minutes to type out all of these proofs. but they are very simple and intuitive.