Assignment
1. Consider the following linear regression model:
yi = β1 + β2xi2 + β3xi3 + εi = x'iβ + εi
a) Explain how the ordinary least squares estimator for β is determined and derive an expression for b.
b) Which assumptions are needed to make b an unbiased estimator for β?
c) Explain how a confidence interval for β2 can be constructed. Which additional assumptions are needed?
d) Explain how one can test the hypothesis that β3 = 1.
e) Explain how one can test the hypothesis that β2 = β3 = 0.
2. Using a sample of 545 full-time workers, a researcher is interested in the question as to whether women are systematically underpaid compared with men. First, a research estimates the average hourly wages in the sample for men and women, which are $5.91 and $5.09 respectively.
a) Do these numbers give an answer to the question of interest? Why not? How could one (at least partially) correct for this?
The researcher also runs a simple regression of an individual's wage on a male dummy, equal to 1 for males and 0 for females. This gives the results reported in
Hourly wages explained from gender: OLS results
|
Variable
|
Estimate
|
Standard error
|
t-ratio
|
|
constant
|
5.09
|
0.58
|
8.78
|
|
male
|
0.82
|
0.15
|
5.47
|
N = 545 s = 2.17 R2 = 0.26.
b) How can you interpret the coefficient estimate of 0.82? How do you interpret the estimated intercept of 5.09?
c) (How do you interpret the R2 of 0.26?
d) Explain the relationship between the coefficient estimates in the table and the average wage rates of males and females.
e) A student is unhappy with this model as ‘a female dummy is omitted from the model'. Comment upon this criticism.
f) Test, using the above results, the hypothesis that men and women have, on average, the same wage rate, against the one-sided alternative that women earn less. State the assumptions required for this test to be valid.
g) Construct a 95 % confidence interval for the average wage differential between males and females in the population.
3. Carefully read the following statements. Are they true or false? Explain.
a) Under the Gauss-Markov conditions, OLS can be shown to be BLUE. The phrase ‘linear' in this acronym refers to the fact that we are estimating a linear model.
b) In order to apply a t-test, the Gauss-Markov conditions are strictly required.
c) A regression of the OLS residual upon the regressors included in the model by construction yields R2 an of zero.
d) The hypothesis that the OLS estimator is equal to zero can be tested by means of a t-test.
e) From asymptotic theory, we learn that-under appropriate conditions-the error terms in a regression model will be approximately normally distributed if the sample size is sufficiently large.
f) If the absolute t-value of a coefficient is smaller than 1.96, we accept the null hypothesis that the coefficient is zero, with 95 % confidence.
g) Because OLS provides the best linear approximation of a variable y from a set of regressors, OLS also gives best linearunbiased estimators for the coefficients of these regressors.
h) If a variable in a model is significant at the 10 % level, it is also significant at the 5 % level.