Intermediate Econometrics

1. (20 points total) A research team wants to know how much does education affect wage rates. The team collected 1000 observations on hourly wage rates, education and other variables from the 2008 Current Population Survey. Wage is measured in earnings per hour (WAGE) and education (EDUC) denotes years of schooling. The following equation is estimated by least squares. The estimates and standard errors are
(WAGE) ̂ = 6.08 + 0.07〖EDUC〗^2 (1.02) (0.005)
(a) (5 points) Sketch the estimated regression function for EDUC= 0 to 20 years (in 5-yearly intervals).
(b) (5 points) Predict the wage of a person with 10 years of schooling.
(c) (5 points) Using each model, find the marginal effect of another year of experience on the expected worker rating for a worker with 10 years’ experience.
(d) (5 points) Construct a 95% interval estimate for the marginal effect found in part c.
  2. (30 points total) The life-cycle pattern of wages can be explained by MODEL 1 below
MODEL 1
Wage=β_1+β_2 EDUC+β_3 EXPER+β_4 EXPER^2+ e (1)
The STATA output from estimating the equation using 1000 observations is
Source | SS df MS Number of obs = 1000 -------------+------------------------------ F( 3, 996) = 104.25 Model | 34973.3163 3 11657.7721 Prob > F = 0.0000 Residual | 111382.245 996 111.829563 R-squared = 0.2390 -------------+------------------------------ Adj R-squared = 0.2367 Total | 146355.561 999 146.502063 Root MSE = 10.575 ------------------------------------------------------------------------------ wage | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- educ | 2.277391 .1394284 16.33 0.000 2.003784 2.550999 exper | .6820989 .1048198 6.51 0.000 .476406 .8877919 exper2 | -.0100907 .0018645 -5.41 0.000 -.0137495 -.006432 _cons | -13.43025 2.028486 -6.62 0.000 -17.41084 -9.449648 ------------------------------------------------------------------------------
The variance-covariance matrix is
educ exper exper2 _cons
educ .01944028
exper -.00021758 .01098718
exper2 .00001547 -.00018926 0.000003476
_cons -.21550584 -.12402316 .00182269 4.1147573
(a) (5 points) After how may years of experiece do wages start to decline? (Express your answer in terms of β’s?
(b) (5 points) What is the marginal effect of education on wages? Is it statistically significant at 5%?
(c) (5 points) Find the elasticity of wages with respect to experience when EXPER = 4. Is it statistically significant?
(d) (5 points) Find the 95% confidence interval for the marginal effect of experience on wages when EXPER = 4.
  After estimating Eq. (1), the residuals are obtained and plotted in the histogram below.
(e) (5 points) What is the reason for assuming that the error term e is normally distributed? Do you have evidence that this is true?
(e) (5 points) Another possible model of wages is
MODEL 2
log⁡(Wage)=β_1+β_2 EDUC+β_3 EXPER+β_4 EXPER^2+ e (2)
Carefully explain, how would you evaluate if Model 1 or Model 2 is a better fit of the data?
3. (15 points total) Consider the model
y=β_1+β_2 x_2+ β_3 x_3+e
and suppose that application of least squares to 63 observations on these variables yields the following results ((cov(b)) ̂ denotes the estimated variance-covariance matrix.
[■(b_1@b_2@b_3 )]=[■(2@3@-1)], (cov(b)) ̂=[■(3&-2&1@-2&4&0@1&0&3)], σ ̂^2=2.5193 R^2=0.9
(a) (5 points) Test the hypothesis that β_2=0 using a 95% confidence interval.
(b) (5 points) Use a t-test to test the hypothesis H_0:β_1+2β_2=5 against the alternative H_1:β_1+2β_2≠5 at 10% significance level.
(c) (5 points) Use p-values to test the hypothesis H_0:β_1-β_2+β_3=4 against the alternative H_1:β_1-β_2+β_3≠4 at 5% significance level.
  4. (10 points total) Consider a model of wheat yield that allows for the yield response to be different for the three different periods
y=β_1+β_2 t+ β_3 rg+β_4 rd+β_5 rf+e (3)
Where
y is the wheat yield in tonnes per hectare,
t is the trend term to allow for technological change,
rg is rainfall at germination (May-June),
rd is rainfall at development stage (July-August), and
rf is rainfall at flowering (September- October).
You estimated this model using 48 annual observations on a number of variables related to wheat yield in the Toodyay Shire of Western Australia, for the period 1950-1997. The unit of measurement for rainfall is centimeters.
The estimated results are below
Test the hypothesis that the response of yield to rainfall is the same irrespective of whether the rain falls during germination, development, or flowering. The results of the restricted model are:
5. (15 points total) Let us investigate if taking econometrics affect starting salary. Let SAL = salary in dollars, GPA= grade point average on a 4.0 scale (the higher one’s GPA is, the better is his/her academic performance), METRICS = 1 if student took econometrics and METRICS = 0 if otherwise. Using a sample of 50 recent graduates, we obtain the estimated regression
(a) (5 points) Interpret the estimated equation.
(b) (5 points) How would you modify the equation to see whether double international students had a higher starting salaries than local students?
(c) (5 points) How would you modify the equation to see if the value of econometrics was the same for international and local students?
END OF EXAMINATION.
STATISTICAL TABLES FOLLOW.