ECMT2150 INTERMEDIATE ECONOMETRICS Week 2 Tutorial
Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit
ECMT2150 INTERMEDIATE ECONOMETRICS
Week 2 Tutorial
OLS & Properties of OLS
Stata 1 Continuing on with working in Stata on our analysis of house prices. Open the do file you created for Tutorial 1 last week, and add the necessary commands and notes to your do file to complete the following questions. We are also going to see how to save your estimation results and produce a neat table of the 3 sets of results.
a) Consider the following model:
y = F0 + F1x1 + F2x2 + F3x3 + 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. Store your results using the command estimates store.
ii. Interpret your coefficient estimates (
j,j = 0, 1, 2,3).
b) Now, consider instead the following related model:
ln y = F0 + F1x1 + F2 ln x2 + F3 ln x3 + u
where y, x1, x2, and x3 are defined as above.
i. Estimate this model using OLS. First, you will need to create some new variables
– taking the natural log of house prices, lot size and house size. Store your estimation results.
ii. How would you interpret your coefficient estimates now?
c) Now, modify your model of (b) by including the (natural log of) the assessed value of the house in the model. In particular, consider the model:
ln y = F0 + F1x1 + F2 ln x2 + F3 ln x3 + F4 ln x4 + u
where x4 is the assessed value of the house (in $000’s) and the other variables are as before.
i. Estimate this model using OLS. Store your estimation results.
ii. Interpret the coefficients of your model.
iii. Produce a table of the results you have obtained for the 3 different models in a), b) and c). Use the package estout and the command esttab.
iv. What impact has the introduction of the assessed value variable had on the estimated coefficients, 
1, 
2, and 
3 ? Can you explain this change?
v. How would you describe the causal relationship between y and x4?
Q1 Which of the following models are (or can be transformed into) linear regression models?
a. yi = F0 + F1x i(2) + ui
b. yi = F0 + F1 ln xi + ui
c. ln yi = F0 + F1xi + ui
d. yi = F0 exp(F1xi + ui )
e. yi = F0 + F1(3)xi + ui
f. yi = F0 + F1 (1/xi ) + ui
Q2 (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
metTicS = F0 + F1 Study + F2 Sleep + F3 wOTk + F4 leiSuTe + 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?
Q3 (Wooldridge Question 2.7)
Consider the savings function:
Sav = F0 + F1 inc + u, u = √inc . e,
where e is a random variable with E(e) = 0 and VaT(e) = ae(2) . Assume that e is independent of inc.
a. 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). ]
b. Show that VaT(u|inc) = ae(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.]
c. Provide a discussion that supports the assumption that the variance of savings increases with family income.
Q4 (Wooldridge Question 2.2)
In the simple linear regression model y = F0 + F1x + 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.
Extra problems (more practice if you would like it)
1) Suppose someone has givenyou the following regression results: yˆt =2.6911 − 0.4795xt
where y is the coffee consumption in Australia (cups per person per day); x is the retail price of coffee ($ per kilo); and t is the time period.
[Let us assume for simplicity that this is a demand curve. Note that demand and supply side factors will jointly determine the relationship between price and quantity, so estimating a demand equation can be complicated.]
a. What is the interpretation of the intercept in this example? Does it make economic sense?
b. How would you interpret the slope coefficient?
c. Is it possible to tell what the true least squares line is? That is, can you find β0 and β 1 ?
d. The price elasticity of demand is defined as the percentage change in the quantity demanded for a percentage change in the price. That is, the elasticity of y with respect to x is defined as 7 = 
. Note that 
is just the slope of y with respect to x. From the above regression results, can you determine the elasticity of demand for coffee? If not, what additional information do you need?
2) (Computer Exercise) Use the data in WAGE2 to estimate a simple regression explaining monthly salary (wage) in terms of IQ score (IQ). IQ (intelligence quotient) tests were developed over 100 years ago and attempt to measure a person’s innate cognitive ability (IQ tests are sometimes referred to as tests of ‘general intelligence’). There is a substantial body of research which examines whether IQ is related to a range of outcomes such as occupational status, income and even criminal activity. In this exercise we consider whether and how IQ affect the wage people earn in the labour market.
a. Report the average, minimum and maximum values, and the standard deviation for wage, education and IQ in the sample (IQ scores are standardized so that the average in the population is 100 with a standard deviation equal to 15).
b. Estimate a simple regression model where a one-point increase in IQ changes
wage by a constant dollar amount. Use this model to find the predicted increase in wage for an increase in IQ of 15 points. Does IQ explain most of the variation in wage?
c. Now, estimate a model where each one-point increase in IQ has the same percentage effect on wage. If IQ increases by 15 points, what is the approximate percentage increase in predicted wage?
d. Do you think the simple regression captures a causal effect of IQ on the wage? Explain.
3) (Wooldridge Question 3.4) The median starting salary for new law school graduates is determined by:
log(salary) = F0 + F1 LSAT + F2 GPA + F3 log(libvol) + F4 log(cost) + F5rank + u,
where LSAT is the median LSAT score for the graduation class, GPA is the median college GPA for the class, libvol is the number of volumes in the law school library, cost is the annual cost of attending law school, and rank is a law school ranking (with rank = 1 being the best).
a. Explain why we expect F5 ≤ 0.
b. What signs do you expect for the other slope parameters? Justify your
answers.
c. Using the data in LAWSCH86 (you do not need to do any regression), the estimated equation is
log(
lary) = 8.34 + .0047 LAST + .248 GPA + .095 log(libvol) +.038 log(cost) − .0033 rank
n = 136, R2 = .842
What is the predicted ceteris paribus difference in salary for schools with a median GPA different by one point? (Report your answer as a percentage.)
d. Interpret the coefficient on the variable log(libvol).
e. Would you say it is better to attend a higher ranked law school? How much is a difference in ranking of 20 worth in terms of predicted starting salary?
2023-06-06
OLS & Properties of OLS