1

Marietta College

Spring 2011Econ 420: Applied

Regression AnalysisDr. Jacqueline Khorassani

Week 12

2

Tuesday, March 29

Exam 3: Monday, April 25, 12- 2:30PM

The 2011-2012Fitzgerald Executive-in-Residence Project

• Focus– Employee Satisfaction in the Workplace

• Leader– Dale Wartluft ’63, a retired senior executive and member of

the Marietta College Board of Trustees • Two teams of students focus on two companies• Send application due electronically to Dr. Gama

Perruci (perrucig@marietta.edu) by Monday, April 11 • I have posted program information and application

form on my website

3

Leadership Q&A

David LeonhardtEconomics JournalistWashington BureauThe New York Times

TuesdayApril 5

7:30pm McDonough Gallery

Cosponsored by McDonough Center for Leadership & Business and the Economic Roundtable of the Ohio Valley

Topic:

“How Do We Grow From Here?

The Post-Crisis American Economy”

This is the last bonus opportunity of this semester

• 2 points for attending• 2-5 points per question• 2-10 points per summary

• Summaries are due before 5 pm on Friday, April 8 via an email attachment to me

• Total bonus points will be divided by 3 and added to your exams.

6

Intercept Dummies

• Theory 1: Men, in general, earn more than women

• How do we formulate a model to capture this difference?

7

Graph of earnings versus experience

Years of work

Earnings

Female

Male

8

How would a dummy variable capture this?

• Intercept dummy– Earnings = β0 + β1 (gender) + β2 (years of work) +

error• Where gender is dummy variable that takes a value of 1

if the observation is a male and 0 otherwise.

9

So you add one more variable to your data set. Suppose you have 5 observations in your data set, then it will look like this

Observation Earnings Years of work

Gender

1. female 0

2. male 1

3. male 1

3. female 0

4. male 1

10

Testing the theory• Say you estimate your model as usual and get• Earnings^ = 1000+ 200 (gender) + 500 (years of work) • How do you set up hypotheses to test the theory?

H0: β1 ≤0Ha: β1>0

• If you reject H0, then you have found significant evidence that men, in general earn more than women

11

How much more?• If your observation is a male

Earnings^ = 1000+ 200 (1) + 500 (years of work) Earnings^ = 1200+ 500 (years of work)

• If your observation is femaleEarnings^ = 1000+ 200 (0) + 500 (years of work) Earnings^ = 1000+ 500 (years of work)

12

Graph of earnings versus experience

Years of work

Earnings hat

Female

Male

1000

1200

13

Slope Dummies

• Theory 2: Men’s earnings grow at a higher rate than women’s earnings

14

Graph of earnings versus experience

Years of work

Earnings

Female

Male

15

How would a dummy variable capture this?

• Slope dummy– Earnings = β0 + β1 (years of work) + β2 (years of

work) *( gender) + error• Where gender is dummy variable that takes a value of 1

if the observation is a male and 0 otherwise.

16

Suppose you have 5 observations in your data set, you will generate a new variable (Genwork). Genwork is gender times years of work. your data set will look like this

Observation Earnings Years of work

Genwork

1. female 5 0

2. male 10 10

3. male 20 20

3. female 30 0

4. male 2 2

GENWORK is called an interactive variable

17

Testing the theory Let’s say you estimate your model as usual and get Earnings^ = 1000+ 500 (years of work) +70

(Genwork) How do you set up the hypotheses to test the

theory?H0: β2 ≤0Ha: β2>0

If you reject H0, then you have found significant evidence that men’s earnings grow at a higher rate with years of experience.

18

How much more?• If your observation is a male

Earnings^ = 1000+ 500 (years of work) + 70 (years of work)* (1)Earnings^ = 1200+ 570 (years of work)

• If your observation is femaleEarnings^ = 1000+ 500 (years of work) + 70 (years of work) * (0) Earnings^ = 1000+ 500 (years of work)

19

Graph of earnings versus experience

Years of work

Earnings

Female slope = 500

Male slope = 570

20

What if• The theory suggested that not only, in general, men’s

salaries are higher than women’s salaries but men also receive a higher rate of increases in their salaries compared to women over time.

• Then you are better off to estimate the model twice: once for male observations and once for female observations as the slope and the intercept must be allowed to vary across genders.

Asst 17 (in teams-20 minutes)

21

In Chapter 2, open the “FINAID” fileVariables are defined on page 43

Male is dummy that takes a value of 1 for male students1. At 10 percent level of significance test the hypothesis

that, in general, male students receive fewer dollars of financial aid than female students.

2. At 10 percent level of significance test the hypothesis that the effect of the amount that the parents of a student are judged to be able to contribute toward college expenses on the amount of financial aid awarded to a student depends on the student’s gender.

Asst 18: Due Thursday

# 12 Page 240

22

23

Thursday, March 31• Asst 19 due Tuesday

# 5 Page 234– Including Part e (data is available online under

STOCK in Chapter 7) • Also on Tuesday there will be an in class

assignment (Asst 20) on Multicollineairty (Chapter 8)

– Need data set DRUGS (Chapter 5) • Exam 3: Monday, April 25, 12- 2:30PM

Collect Asst 18

# 12 Page 240

24

Leadership Q&A

David LeonhardtEconomics JournalistWashington BureauThe New York Times

TuesdayApril 5

7:30pm McDonough Gallery

Cosponsored by McDonough Center for Leadership & Business and the Economic Roundtable of the Ohio Valley

Topic:

“How Do We Grow From Here?

The Post-Crisis American Economy”

This is the last bonus opportunity of this semester

• 2 points for attending• 2-5 points per question• 2-10 points per summary

• Summaries are due before 5 pm on Friday, April 8 via an email attachment to me

• Total bonus points will be divided by 3 and added to your exams.

Return and discuss Asst 17

27

In Chapter 2, open the “FINAID” fileVariables are defined on page 43

Male is dummy that takes a value of 1 for male students1. At 10 percent level of significance test the hypothesis

that, in general, male students receive fewer dollars of financial aid than female students.

2. At 10 percent level of significance test the hypothesis that the effect of the amount that the parents of a student are judged to be able to contribute toward college expenses on the amount of financial aid awarded to a student depends on the student’s gender.

28

Included observations: 50 Dependent Variable: FINAID Variable Coefficient Std. Error t-Statistic Prob.

C 9813.022 1743.100 5.629638 0.0000HSRANK 83.26124 20.14795 4.132492 0.0002PARENT -0.342754 0.031505 -10.87921 0.0000MALE -1570.143 784.2971 -2.001975 0.0512

Coefficient of MALE is negative and significant at 10%Let’s see the graph of FINAID against PARENT holding HSRANK constant

At 10 percent level of significance test the hypothesis that, in general, male students receive fewer dollars of financial aid than female students.

29

Dependent Variable: FINAIDIncluded observations: 50

Variable Coefficient Std. Error t-Statistic Prob.

C 8741.326 1659.361 5.267887 0.0000HSRANK 91.13511 19.76721 4.610418 0.0000PARENT -0.308050 0.036411 -8.460356 0.0000MALEPAR -0.103247 0.043104 -2.395285 0.0207

Coefficient of MALEPAR is different from zero at 10%Let’s see the graph of FINAID against PARENT holding HSRANK constant

At 10 percent level of significance test the hypothesis that the effect of the amount that the parents of a student are judged to be able to contribute toward college expenses on the amount of financial aid awarded to a student depends on the student’s gender.

Let’s look at the Phillips Curve in economics

Inflation %

Unemployment (%)

The Phillips Curve shows an inverse relationship between inflation and unemployment.

1.5%

6%4%

2.5%

PC

How do we formulate the model?How about Y = β0 + β1 X+ є ?

Where Y is inflation and X is unemploymentNo, the line is non-linear Y = β0 + β1 (1/ X) + єOr Y = β0 + β1 X-1 + єWhat is the slope of a line showing the relationship between Y an X?dY/dX = - β1 X-2, ordY/dX = (- β1)* 1/ X2

Do we expect β1 to be positive or negative?Positive

Note that as X goes up slope declines

31

Sometimes the theory suggests that• when X is changed this period, Y is

expected to change next period.• Can you think of an example?• How do we formulate the model?• How do you test the theory?• What is the meaning of the estimated

coefficient?

32

3 problems with non-linear models

33

1. Hard to interpret the meaning of the estimated coefficients– Slope?

– Increasing at a declining rate or increasing at an increasing rate?

» Policy implications?– Elasticity?– Neither slope nor elasticity?

2. The adjusted R squared can be compared across two equations only if the dependent variables are the same.

3. An incorrect function may be a good fit for sample observations but not for population observations

Perfect Multicollinearity (Chapter 8) • When two or more independent variables have a

perfect (error free) linear relationship with each other

• Which assumption does this violate? (Page 94)– Violates Assumption 6

• Example– Unemployment = f (nominal interest, real interest,

inflation)• Consequences: OLS is not able to estimate the model• Remedy?

– Drop one variable

Imperfect Multicollinearity

• When two or more independent variables have an imperfect linear relationship with each other

• Example– Price of a house = f (number of bedrooms,

square footage)

1. β hats are unbiased2. β hats have higher than normal variance◦ Increases the chances of getting a β hat that has

the unexpected sign3. β hats have higher than normal standard errors

What does this imply with regards to t- test? Lower t-stats We may conclude that βs are not significantly different

from zero while they are significantly different from zero.4. Adding and subtracting variables and observations

will affect β hats significantly5. The adjusted R squared remains largely unaffected6. The β hats of uncorrelated variables remain

unaffected

Consequences of Imperfect Multicollinearity

• If you have high adjusted R squared but low t-stats suspect a multicollinearity problem

Informal detection of Multicollinearity

Calculate the correlation coefficients (r) between eachpair of independent variables and each independentvariable and the dependent variable.• EViews direction: quick group statistics

correlation type your variables (starting with your dependent variable… no need for constant)

– Two rules1) If |rX1, X2| > |rX1, Y | problem, or

2) If |rX1, X2 |> 0.8 problem

Formal test for Multicollinearity