1. Consider the simple linear regression model without an intercept, y = β1x + u, with

the assumption E(u|x)=0. Also assume that E(x)=0

a. Show that E(y)=0 and using this as well as E(x)=0 show that the covariance

between x and y is given by E(xy) and that the variance of x is given by ( )

2 E x

b. Using the results in a. and the assumptions above show that E(xu)=0 and hence

that

β1 = E(xy) / ( )

2 E x

c. What is the Least Squares estimate of β1 (say 1

~

β ) in this model without an

intercept? How does it relate to the population value β1 ? Do they have to be the

same – Why? Why not?

2. Wooldridge 2.4 (i.e. Chapter 2 Question 4)

3. Wooldridge 2.6

Computer problems (show any relevant Stata output):

1. Wooldridge C2.4 (i.e. Chapter 2 Question C4), parts (i) and (ii)

For the same dataset (WAGE2.DTA), also answer the following questions:

(i) Plot wage versus iq with the fitted regression line shown.

(scatter y x || lfit y x gives the scatter of y vs x with a fitted line.)

(ii) Verify that the regression slope estimate is equal to (i) the ratio between the

sample covariance (between wage and iq) and the variance of iq, and (ii) the

sample correlation (between wage and iq) times the ratio between the standard

deviations of the two variables.

(corr y x, covariance gives the covariance matrix for x and y, whereas the

corr command without the “covariance” option gives the correlation matrix.)

(iii) Form the fitted values for the dependent variable. (After the regression

command, do predict wagehat. This command will create a new variable

wagehat.) How does the sample average of wagehat compare to the sample

average of wage? What is the correlation between wagehat and iq? (Can you

figure these out without using Stata?)

(iv) Form the OLS residuals for this regression. (Do predict uhat, resid, which

will create a new variable uhat with the estimated residuals.) Verify that (a) the

sample average of uhat is equal to zero and (b) the correlation between uhat and

iq is equal to zero.

(v) Now do the reverse regression by reversing the roles of wage and iq (so that iq is

now the dependent variable and wage the independent variable).

a. What is the estimated slope for this regression?

b. Explain why it is not surprising to see the same sign for the slope as in the

original regression.

c. For which regression model do you think the zero conditional-mean

assumption on the error is more believable? You now have two models:

wage = β0 + β1iq + u, with the assumption E(u|iq)=0

iq = γ0 + γ1wage + v, with the assumption E(v|wage)=0

2. Wooldridge C2.6 (note that log(expend) is in the data as the lexpend variable)

For the same dataset (MEAP93.DTA), also answer the following questions:

(vi) Plot math10 versus lexpend with the fitted regression line shown.

(vii) To visualize the non-linearity described by this model, create the fitted values

from the regression and then do a scatter plot of the fitted values versus expend

(not lexpend). Explain how the effect of expenditures changes at higher values of

expenditures.

(viii) Suppose that we wanted to measure math10 as a fraction (a number between 0

and 1) rather than on a 0-to-100 scale. Specifically, we could do the following in

Stata to re-scale the math10 variable:

. replace math10 = math10 / 100

(This replaces the original math10 values with the values divided by 100.) If you

re-ran the regression (now regressing the re-scaled math10 upon lexpend), how

would the slope and intercept estimates change (compared to the original results)?

[end]

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