Semester 1, 2022
Practice Questions Solutions Guide
Question 1.
(i) The population regression model is y = β0 + β1 x1 + β2 x2 + u (1) but we estimate y =
βˆ 0 + βˆ 1 x1 + v (2) by OLS. The OLS estimator for model (2):
(a) will provide an unbiased estimate of the true population parameter β1 when either β2 = 0
(x2 has no effect on y) or E (x2 | x1) = 0 (that is, x1 and x2 are independent).
(b) will provide an estimate of β1 that has positive (or upward) bias when (β2 > 0 and
Corr(x1, x2) > 0) or (β2 < 0 and Corr(x1, x2) < 0).
(ii) Advantages of using larger samples of data in regression analysis:
• More information / more variation in the x’s
• More accurate inferences → smaller standard errors
• Able to rely on asymptotic or large sample results for inference – i.e. can drop the nor
mality assumption and still use t, F tests.
• For time series, consistency of OLS requires much less restrictive assumptions than those
required for unbiasness (especially w.r.t. ZCM assumption and exogeneity).
(iii) Model: log (price) = β0 + β1 area + β2 bdrms + β3 area × bdrms + u
The partial effect of area on log\
(price) is ∆
\
log
(price)
∆
area
= β1 + β3 bdrms.
(iv) ‘Contemporaneous exogeneity’ means that the expected variable of the error term for time
period t, is unrelated to the value of the explanatory variables in the same time period t. That
is: E (ut | xt) = 0 for all t. In contrast, strict exogeneity requires that the expected value of
the error term in time period t is unrelated to the value of the explanatory variables in every
time period. That is, E (ut | X) = 0 for all t. (This much stronger assumption is required for
OLS to be unbiasness).
Question 2.
Model: log (wage) = β0 + β1 educ + β2 exper + β3 tenure + u
(i) Interpretation of β1:
The coefficient on educ, i.e. β1, ×100, is the approximate percentage increase in the predicted
wage from an extra year of education, other things equal. If educ is increased by 1 year, other
things equal, the expected wage will increase by approximately β1 × 100%.
1(ii) Calculate the exact percentage effect of another year of education on the predicted wage level.
The exact percentage effect of educ on the predicted wage level = 100 h exp βˆ 1 − 1 i =
100 [exp (0.074864) − 1] = 100 [1.077737569 − 1] = 7.77%.
(iii) Test the null hypothesis that all the slope parameters in the model are jointly equal to zero
using a 1% significance level. What do you conclude?
Test:
H0 : β1 = 0, β2 = 0, β3 = 0
H1 : H0 is false
Test Statistic:
F =
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