(1) Consider the panel data fixed effect model:
Yit = b1X1,it + …. + bkXk,it + αi + uit (1)
where the data have not been “entity-demeaned”, and i=1, …, N indicates the number of cross-sectional units or entities, and t = 1, …, T indicates the number of time periods observed. The same number of time periods is observed for each cross-sectional unit; that is, it is a balanced panel.
a. How many observations are used to estimate equation (1)?
b. What are the dimensions of the data matrix of right hand side variables used in the OLS regression?
c. Sketch the elements of the data matrix of right hand side variables.
d. Using equations, describe what entity-demeaning would entail.
e. Rewrite equation (1) for the entity-demeaned data.
f. Are the coefficient estimates from equation (1) the same as those from the demeaned equation in (e)?
g. What are the dimensions of the data matrix of right hand side variables for the entity-demeaned regression from question (e)?
h. Can the usual OLS homoscedastic or heteroscedastic consistent standard errors be used for inference with the model in equation (1)? If not, what type of standard errors should be employed? And, what is the motivation for any alternative standard errors that may be required (especially, what properties do they have)?
2) Using the logit regression expression Pr(deny = 1| P/I Ratio, black) = F(–2.26 + 2.74P/I ratio + 0.71black), what would be the effect on the probability of deny=1 from increasing the P/I ratio from 0.2 to 0.45, for
a. a person with black=1?
b. a person with black=0?
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