Economics 421 - Econometrics

Material for Lecture 7 on Tuesday 1/31/12

Today:

• Continue Chapter 12 - Autocorrelation

Next Time:

• Finish Chapter 12 - Autocorrelation
• Project Outline
• Begin Chapter 8 - Stochastic Regressors and Measurement Errors

Solution to Homeworks 1 and 2

Economics 421/521
Winter 2012
Solution to Homework #1

Part I. Hypothesis Testing

1. Suppose that you estimate a model of house prices to determine the impact of having beach frontage on the value of a house. You do some research, and you decide to use the size of the lot instead of the size of the house for a number of theoretical and data availability reasons. Your results (standard errors in parentheses) are:

PRICEi = 40 + 35.0 LOTi – 2.0 AGEi + 10.0 BEDi – 4.0FIREi + 100 BEACHi
        (29) (5.0)       (1.1)      (10.0)      (3.0)      (9.0)

n = 30, R² = .63

where,

PRICEi = the price of the ith house (in thousands of dollars)
LOTi = the size of the lot of the ith house (in thousands of square feet)
AGEi = the age of the ith house in years
BEDi = the number of bedrooms in the ith house
FIREi = a dummy variable for a fireplace (1 = yes for the ith house)
BEACHi = a dummy for having beach frontage (1 = yes for the ith house)

a) You expect the variables LOT, BED, and BEACH to have positive coefficients. Create and test the appropriate hypotheses to evaluate these expectations at the 5 percent level.

For LOT;

Ho: βLOT = 0
Ha: βLOT > 0

t – score: (35.0) / (5.0) = 7.0

t-critical: 1.711 because d.f. is 24 and 5% level of significance.

Since 7.0 > 1.711, we can reject the null hypothesis that the true coefficient of LOT is not positive.

For BED;

Ho: βBED = 0
Ha: βBED > 0

t – score: (10.0) / (10.0) = 1.0

t-critical: 1.711 because d.f. is 24 and 5% level of significance.

Since 1.0 < 1.711, we cannot reject the null hypothesis that the true coefficient of BED is not positive.

For BEACH;

Ho: βBEACH = 0
Ha: βBEACH > 0

t – score: (100) / (0.9) = 11.1

t-critical: 1.711 because d.f. is 24 and 5% level of significance.

Since 10.0 > 1.711, we can reject the null hypothesis that the true coefficient of BEACH is not positive.

b) You expect AGE to have a negative coefficient. Create and test the appropriate hypothesis to evaluate these expectations at the 10 percent level.

For AGE;

Ho: βAGE = 0
Ha: βAGE < 0

t – score: ( - 2.0) / (1.1) = - 1.81

t-critical: 1.318 because d.f. is 24 and 10% level of significance.

Since | - 1.81 | > |1.318|, we can reject the null hypothesis that the true coefficient of AGE is not negative.

c) At first you expect FIRE to have a positive coefficient, but one of your friends says that fireplaces are messy and are a pain to keep clean, so you are not sure. Run a two-sided t-test around zero to test these expectations at the 5 percent level.

For FIRE;

Ho: βFIRE = 0
Ha: βFIRE ≠ 0

t – score: ( - 4.0 ) / (3.0) = - 1.3

t-critical:: 2.064 because d.f. is 24 and 5% level of significance.

Since | - 1.3 | < 2.064, we cannot reject the null hypothesis that the true coefficient of FIRE is not different from zero.

2. Consider the following regression:

log(Qci) = 921.6 – 1.3log(Pci) + 0.7log(Pai) + 11.4log(Inci)
           (121)  (0.3)         (0.05)        (2.8)

n = 30, R2 = 0.82

where,

Qci = the total sales of CAMRY in the ith city, the year of 2003
Pci = the price of a CAMRY in the ith city, the year of 2003 (in thousands)
Pai = the price of a ACCORD in the ith city, the year of 2003 (in thousands)
Inci = the average income in the ith city, the year of 2003 (in thousands)

Numbers in the parentheses are standard errors.

a) What does the constant term (= 921.6) mean?

The constant term means is the value of the dependent variable when all the independent variables are zero.

b) How would you interpret the coefficient on log(Pci). Be explicit and explain, in terms of economic theory, the importance of its magnitude.

Totally differentiate the estimated equation w.r.t. Qci and Pci to obtain the elasticity:

Elasticity. |-1.3| > 1, so elastic. If the price of CAMRY increases by 1%, then the total sales of CAMRY decreases by 1.3%. (Thus, the coefficient gives the elasticity.)

c) Get t-values of each of all coefficients in the regression. Are all of our coefficients statistically significant at the 5% level of significance? How about at the 1% level of significance?

             Pci      Pai     Inci
t-score     -4.3      14.     4.08

Two-tailed test at the 5% level of significance, t-critical = 2.056 because d.f. is 26.
Two-tailed test at the 1% level of significance, t-critical = 2.779 because d.f. is 26.

Therefore, all of our coefficients statistically significant at the both 1% and 5% level of significances.

d) Interpret R2. What does it mean?

R2 tells how well the sample regression line fits the data.

(TSS = ESS + RSS)

Thus, it is the ratio of the explained variation in the dependent variable divided by the total variation and hence measures the percentage of the total variation that is explained by the variable sin the model.

 

Part II. Short Answer

1. State the Gauss-Markov Theorem and explain the term BLUE.

Given the classical assumptions (model is linear and correctly specified, X's are exogenous, no perfect multicollinearity, error has zero mean, homoskedasticity, errors are independent, errors and X's are uncorrelated, errors normally distributed), the OLS estimator is the minimum variance estimator from among the set of all linear unbiased estimators. OLS is BLUE. This means it is the Best (minimum variance) Linear Unbiased Estimator.

Part III. Estimation

1. Given data on M2, real GDP, and the T-bill rate, estimate the following regression and test whether the coefficients differ from zero. Do the coefficients have the expected signs?:

M_t = β₀ + β₁RGDP_t + β₂Tbill_t + e_t

To do this problem, first create a new workfile by following these steps (some of the figures are pop-ups that bring up larger, clearer versions):

Next, read in the data set (save it to your computer first by right clicking on the link in the homework set):

Finally, run the regression of M2 on a constant, RGDP and the TBillRate:

Running this regression gives:

The critical value at the 1% level is 2.576, thus the coefficients are significant. They also have the expected sign: People tend to demand more money as income (RGDP) increases. They also tend to demand less money as the T-bill rate increases – this is because as the T-bill rate goes up, the opportunity cost to holding money also goes up.

 

Economics 421/521
Winter 2012
Solution to homework #2

1. Using the EAEF data set, regress LGEARN on S, EXP, and ASVABC. Use F-tests to determine whether the coefficients on S and EXP are (a) jointly significant, and (b) equal. [Parts (a) and (b) are two separate tests.]

First, read in the data:

 

Then, take the log of earnings:

 

Next, estimate the unrestricted model This is needed for both parts (a) and (b). To estimate the UR model, regress lnearnings on a constant, exper, and asvabc:

 

 

The results are:

 

(a) The restricted model for the null hypothesis that the coefficients on s and exper are zero is:

 

F= [(149.31-130.35)/2]/[130.35/(540-4) =38.98

The critical value for this test is F(2,536)= 3.05 (approx. for 536),  so reject that both coefficients are zero.

(b) For this part, the null hypothesis is that the coefficients on s and exper are equal. To impose this on the model, start with:

lnsalary = b₁ + b₂*s + b₃*exper + b₄*asvabc + u

Then impose that b₂=b₃:

lnsalary = b₁ + b₂*s + b₂*exper + b₄*asvabc + u

Group terms:

lnsalary = b₁ + b₂*(s + exper) + b₄*asvabc + u

This is the model we need to estimate. Thus, the first step is to obtain data on the sum of s and exper:

 

Run the restricted regression:

Here are the results:

 

Finally, use these to calculate the F-statistic:

F= [(138.29-130.35)/1]/[130.35/(540-4) = 32.65

The critical value for this test is F(1,536)= 3.90 (approx.),  so reject that the coefficients are equal.

2. Problem 7.1 in the text.

In this case the F statistic is simply the ratio of the residual sum of squares (more generally it is the rss divided by n-k, i.e. F = [rss₂/(n₂-k)]/ [rss₁/(n₁-k)], where n₁ and n₂ are the number of observations in each sample, but when n₁=n₂ the terms cancel):

F=28,101/321 = 87.54

The critical value (5%) for and F(8,8) = 3.44, so the null of homoskedasticity is rejected.

3. Moved to the next homework.

Homework 4

Economics 421/521
Winter 2012
Homework #4
Due in lab next week

1. Perform a Durbin-Watson test at the 5% level of significance for positive first-order autocorrelation using the following regression output (standard errors in parentheses):

Y_t = 2.0 + 3.7*X_1t - 4.4*X_2t,    T = 42
       (.7)   (1.1)          (2.8)           DW = 1.22

2. Recall the model from homework 1:

Given data on M2, real GDP, and the T-bill rate, estimate the following regression...:

M_t = β₀ + β₁RGDP_t + β₂Tbill_t + e_t

Don't be surprised if the fit is very good - we'll explain why that may be misleading later in the course.

Does model suffer from serial correlation? Use a Durbin-Watson test to answer the question. Is the fit as good as the R² and t-statistics indicate?

3. Regress the change in the log of real consumption (C) on the change in the log of real disposable income (DI) and test for serial correlation using a Durbin-Watson test. The data are here (the data are quarterly, and span the time period 1947:Q1 - 2007:Q3).

4. Explain why the Durbin Watson statistic is always between 0 and 4. Also explain why the Durbin-Watson statistic is between 0 and 2 when there is positive serial correlation, between 2 and 4 when there is negative serial correlation, and equal to 2 when there is no correlation at all.

5. Continuing with the model we used in problem 2, test for the presence of fourth order serial correlation.

6. Continuing with the model we used in problem 3, use the AR(1) procedure in EViews to correct the model for the presence of first-order serial correlation.

Material for Lecture 6 on Thursday 1/26/12

Today:

• Finish Heteroskedasticity
• Begin Chapter 12 - Autocorrelation

Next Time:

• Continue Chapter 12 - Autocorrelation

LM Tests for Heteroskedasticity

Here is an outline of the LM tests for Heteroskedasticity:

Material for Lecture 5 on Tuesday 1/24/12

Today:

• Continue Chapter 7 - Heteroskedasticity
• Begin Chapter 12 - Autocorrelation

Next Time:

• Continue Chapter 12 - Autocorrelation

Homework 3

[Note: The assignment will be discussed in lab this week, and will be due in lab next week.]

Economics 421/521
Winter 2012
Homework #3

1. Problem 3 from Homework 2 (the problem that was canceled on the last set).

2. Using the first model of heteroskedasticity, i.e. that resid² = α₀ + α₁*years + α₂*years², correct the salary model in problem 3 from Homework 2 for heteroskedasticity and reestimate.

3. Problem 7.2 in the text.

4. Test the salary model in problem 3 from Homework 2 for heteroskedasticty using White's test. Correct the standard errors using White's correction. How do the coefficients and corrected standard errors compare to those obtained in problem 3 of Homework 2?

5. What are the consequences of estimating an autoregressive model using OLS?

[Note: pdf of problems 7.1 and 7.2, problem 7.1 was on the last homework.]

Material for Lecture 4 on Thursday 1/19/12

Today:

• Continue Chapter 7 - Heteroskedasticity

Next Time:

• Finish Chapter 7 - Heteroskedasticity, perhaps start Chapter 12 on Autocorrelation

Material for Lecture 3 on Tuesday 1/17/12

Today:

• Chapter 7 - Heteroskedasticity

Next Time:

• Continue Chapter 7 - Heteroskedasticity

Material for Lecture 2 on Thursday 1/12/12

Today:

• Review of hypothesis testing
• Chapter 7 - Heteroskedasticity

Next Time:

• Continue Chapter 7 - Heteroskedasticity

Homeworks 1 and 2

[Note: Homework 1 and homework 2 will both be due in lab in week 3.]

Economics 421/521
Winter 2012
Homework #1

Part I. Hypothesis Testing

1. Suppose that you estimate a model of house prices to determine the impact of having beach frontage on the value of a house.  After researching the problem, you decide to use the size of the lot instead of the size of the house as your explanatory variable for a number of theoretical and data availability reasons.  The results (standard errors in parentheses) are:

PRICEi = 40 + 35.0 LOTi – 2.0 AGEi + 10.0 BEDi – 4.0 FIREi + 100 BEACHi
        (29) (5.0)       (1.1)      (10.0)      (3.0)       (9.0)

where n = 30, R2 = .63, and PRICE_i = the price of the i^th house (in thousands of dollars), LOT_i = the size of the lot of the i^th house (in thousands of square feet), AGE_i = the age of the i^th house in years, BED_i = the number of bedrooms in the i^th house, FIRE_i = a dummy variable for a fireplace (1 = yes for the ith house), and BEACH_i = a dummy for having beach frontage (1 = yes for the ith house).

a) You expect the variables LOT, BED, and BEACH to have positive coefficients.  Test each of these hypotheses at the 5 percent level.

b) You expect AGE to have a negative coefficient.  Test this hypothesis at the 10 percent level.

c) At first you expect FIRE to have a positive coefficient, but one of your friends says that fireplaces are messy and are a pain to keep clean, so you are not sure.  Run a two-sided t-test around zero to test the two-sided hypothesis at the 5 percent level.

2. Consider the following regression:

log(Qci) = 921.6 – 1.3 log(Pci) + 0.7 log(Pai) + 11.4 log(Inci) 
           (121)  (0.3)          (0.05)          (2.8)

where n = 30, R2 = 0.82, and  where Qc_i = the total sales of CAMRY in the ith city in 2003, Pc_i = the price of a CAMRY in the i^th city in 2003 (in thousands), Pa_i = the price of an ACCORD in the i^th city in 2003 (in thousands), and Inc_i = the average income in the i^th city, the year of 2003 (in thousands).  The numbers in the parentheses are standard errors.

a) How is the constant term interpreted?

b) How would you interpret the coefficient on log(Pc_i).  Be explicit and explain, in terms of economic theory, the importance of its magnitude.

c) Get t-values for the coefficients in the regression.  Are all of our coefficients statistically significant at the 5% level of significance?  How about at the 1% level of significance?

d) Interpret R². Can we have a negative R²?

Part II. Short Answer

1. State the Gauss-Markov Theorem and explain the term BLUE.

Part III. Estimation

1. Given data on M2, real GDP, and the T-bill rate, estimate the following regression and test whether the coefficients differ from zero. Do the coefficients have the expected signs?:

M_t = β₀ + β₁RGDP_t + β₂Tbill_t + e_t

Don't be surprised if the fit is very good.  We'll explain why the good fit is misleading in this model later in the course.

---

Economics 421/521
Winter 2012
Homework #2

1. Using the EAEF data set, regress LGEARN on S, EXP, and ASVABC. Use F-tests to determine whether the coefficients on S and EXP are (a) jointly significant, and (b) equal. [Parts (a) and (b) are two separate tests.]

Note: Here are the variable definitions (see pages. 443-444 in Appendix B of the text):

• EARNINGS = current hourly earnings in $ reported in 2002 interview.
• S = Years of schooling (highest grade completed as of 2002).
• EXP = Total out-of-school work experience (years) as of the 2002 interview.
• ASVABC = Scaled standardized test score.

2. Problem 7.1 in the text.

3. Using this data set, repeat the example from class for the first of the three cases we discussed, i.e. first regress the log of salary on a constant and the two variables proxying for experience, years and years²:

     log(salary) = β₀ + β₁*years + β₂*years² + u_t

Then, form the estimated residual squared (resid²) and perform the LM test for heteroskedasticity (note: resid is the estimated value of u_t).

Material for Lecture 1 on Tuesday 1/10/12

Today:

• Review of regression models
• Assumptions required for estimates to be BLUE ( pgs. 111-114 and 159-160)
• Guass-Markov Theorem (pg. 132-133)

Next Time:

• Hypothesis Tests
• Heteroskedasticity (Chapter 7)

Syllabus for Winter 2012

Introduction to Econometrics
Course: Economics 421/521
Professor: Mark Thoma
Office/Hours: PLC 471 on T/Th 3:30-4:30 p.m.
Phone/Email: (541) 346-4673, mthoma@uoregon.edu
Web Page: http://economistsview.typepad.com/economics421/

Course Description: This course is a continuation of the econometrics sequence. The first course, EC 420/520, introduces the linear regression model and discusses estimation and testing under (mostly) ideal conditions. This course looks at what happens when the conditions are less than ideal due to departures from the assumptions necessary for ordinary least squares to be the best linear unbiased estimator, and then provides alternative regression techniques that address  problems arising from the violations of the basic assumptions.

Text: Dougherty, Christopher, Introduction to Econometrics, (Oxford: University Press)

Prerequisites: Economics 320 or the equivalent.

GTFs, Office Hours, Location, and Email Address:

Gulcan Cil	M 1-3	PLC 431	gcil@uoregon.edu
Colin Corbett	T/Th 11-11:50	PLC 504	corbett@uoregon.edu

Lab Times:

CRN	Time	Day	Room
22318	1600-1720	M	442 MCK	Gulcan Cil
22319	1730-1850	M	442 MCK	Gulcan Cil
22320	1600-1720	W	442 MCK	Gulcan Cil
22321	1730-1850	W	442 MCK	Gulcan Cil

Tests and Grading: There will be a midterm exam and a final. The midterm will be given Tuesday, February 14th. The final will be given on Monday, March 19th at 8:00 a.m. No make-up exams will be given. The midterm is worth 30% and the final is worth 40%. Grades will be assigned according to your relative standing in the class.

Empirical Project: There will be an empirical paper that will comprise 15% of your grade. The paper is due no later than Thursday, March 15 at the beginning of class. Details will be given during lecture.

Computer Labs: The statistical software package EViews will be used for estimation and testing. Labs will consist of instruction and examples helpful in completing the homework assignments, and other activities. The homework is worth 15% of your grade.

*Tentative* Course Outline:

We will cover the following chapters:
Review of Multiple Regression and Hypothesis Testing
Heteroscedasticity	Ch. 7
Autocorrelation	Ch.12
Stochastic Regressors and Measurement Errors	Ch. 8
Simultaneous Equations Estimation	Ch. 9
And, as time permits:
Binary Choice Models and Maximum Likelihood Estimation	Ch. 10
Models Using Time Series Data	Ch. 11

More details on the readings, homework, homework due dates, etc. will be posted here on an ongoing basis, so please check back regularly.

Course Materials for Winter 2011

Course materials for the last time the course was offered (Winter 2010) are available here.