THE REGRESSION ANALYSIS

1 The Simple Regression Model [30 pts]

1.1 Regression to the Origin [6 pts]

Consider the estimation of the following model

using a random sample {(y₁, ₁): i=1,2,...,n}, where B₁ is estimated using ordinary least squares (OLS).

a. Derive 3₁ using OLS. [1 pt]

b. Show under the necessary assumptions that 3₁ is unbiased if the population regression function is y₁ =

B₁+U₂. [2 pts]

c. Show -under the same assumptions used above that B₁ is biased if the population regression function is

Yi = Bo + B₁zi + Ui, with Bo 0. [1 pt]

d. Show that

1.2 Regression to a Constant [4 pts]

Consider the estimation of the following model

using a random sample {(yi): i = 1,2,...,n}.

a. Derive Bo using OLS. [2 pts]

b. Provide an interpretation of o. [2 pts]

Yi = Bo + ûi

1.3 Regression on a Binary Explanatory Variable and Average Treatment Effect [20 pts]

Let y be any response variable and a binary explanatory variable. Let {(ri, Yi) : i = 1,2,...,n} be a sample of

size n. Let no be the number of observations with ; = 0 and n₁ the number of observations with ₂ = 1. Let Yo

be the average of the y, with = 0, and ₁ the average with z; = 1.

a -

Explain why we can write

b. Argue that

c. Show that the average of y, in the entire sample, y, can be written as a weighted average:

y = (1-I)yo+zÿ₁.

d. Show that when z is binary,

e. Show that

f. Use parts d. and e. to show

g. Show, also,