1.  (Wooldridge  Question  2.2)    In  the  simple  linear  regression  model  y  = β0  + β1x + u suppose that E(u) ≠ 0. Letting a0  = E(u), show that the model can always be written with the same slope, but a new intercept and error, where the new error has a zero expected value.

2.  (Wooldridge Question 2.7) Consider the savings function:

sav =  β0  +  β1 inc + u,     u  = √inc ∙ e,

where e is a random variable with E(e) = 0 and vaT(e) = σe(2) . Assume that e is independent of inc.

i.          Show that  E(u |inc) = 0, so that the key zero conditional mean assumption

(Assumption SLR.4) is satisfied. [Hint: If e is independent of inc, then

E(e |inc) = E(e). ]

ii.          Show that vaT(u |inc) = σe(2)inc, so that the homoscedasticity Assumption

SLR.5 is violated. In particular, the variance of sav increases with inc. [Hint:

vaT(e |inc) = vaT(e) if e and inc are independent.]

iii.          Provide a discussion that supports the assumption that the variance of savings

increases with family income.

3.  (adapted from Wooldridge Question 3.5) In a study relating marks obtained by students

in undergraduate econometrics (metrics) in Australian universities to time spent in

various activities, a survey is conducted among several students. The students are given questionnaires and asked to write down how many hours they spend each week in four activities: studying, sleeping, working, and leisure. Any activity is put into one of the four categories, so that for each student, the sum of hours in the four activities must be 168.

(a)  In the model

metrics =  β0  +  β1 study + β2 sleep + β3 work +  β4 leisure + u

does it make sense to hold sleep, work, and leisure fixed, while changing study?

(b)  Explain why this model violates assumption MLR.3.

(c)  How could you reformulate the model so that its parameters have a useful interpretation and it satisfies assumption MLR.3?

4.  For each of the following, state whether it can cause OLS estimators to be biased?

(a)  Heteroskedasticity.

(b) Omitting an important variable.

(c)  A sample correlation coefficient of .95 between two independent variables both included in the model.

For each, if your answer is no, then say why it does not cause bias in the OLS estimator.

If yes, explain the source of the bias.

5.  For each of the following, state whether (and how or why) it can cause the usual OLS t-

statistics to be invalid (that is, not to have t distributions under H0)?

(a)  Heteroskedasticity.

(b) A sample correlation coefficient of .95 between two independent variables that are in the model.

(c)  Omitting an important explanatory variable

6.  (Computer Exercise) The data file hprice1 (hprice1.dta) contains a small sample of house prices.

(a)    Use   this   dataset   to   examine   the    relationship   between    house   prices,    lot

(property) sizes, house sizes, and the number of bedrooms.  In particular, consider the following model:

y = β0 +β1x1 +β2x2 +β3x3 +u,

where y is the house price (in $000’s), x1 the number of bedrooms, x2 the lot size (in square feet), and x3 the house size (in square feet).

i.  Estimate this model using OLS.