Welcome to Econ 420 Applied Regression Analysis Study Guide Week Seven.

Welcome to Econ 420 Applied Regression Analysis

Study Guide

Week Seven

Answer Key to Assignment 5 Question 1- Part One

• Step 1– H0: B1 = B2 = B3 = 0– HA: At least one of these B’s is not zero

• Step 2:– Level of significance = 1%– Degrees of Freedom in Numerator = k = 3– Degrees of Freedom in Denominator = n – k

– 1 = 30 – 3 – 1 = 26– Critical F, Fc, = 4.64 (pg 319)

• Step 3:– Run regression and find F-statistic =

40.82042

• Step 4:– Because our F-statistic, 40.82 > 4.64,

the null hypothesis is rejected at the 1% significance level; it is 99% likely that at least one of these B’s is not zero.

Question 1- Part Two

• The estimated slope coefficient for income, is 0.022756.– SE = 0.005516– Degrees of Freedom = n – k – 1 = 30 –

3 – 1 = 26– tc = 2.056 (pg. 313)

– The 95% confidence interval for the coefficient on income is B^1 – tc• SE (B^1) < B1 < B^1 + tc• SE (B^1),

– The 95% confidence interval is 0.0114 < B1 < 0.0340.

– There is 95% chance that the true value of B1 is in the above range.

2. #17, Page 63

• a. Adjusted R2 = 1 – (1 – 0.7) • (9/5) = 0.46• b. Adjusted R2 = 1 – (1 – 0.7) • (19/15) = 0.62• c. Adjusted R2 = 1 – (1 – 0.7) • (99/95) = 0.69• d. With the same R2, when the sample number goes up,

adjusted R2 will increase. The implication here is that when you add more observations to your sample, the degrees of freedom goes up, and therefore the goodness of fit will increase.

• e. When the sample size is increased, R2 may increase, decrease or even stay the same. It depends on how well the new observations fit the regression line.

3. #4, PP 81-82

• a and b. Use the following formula to calculate the real values.

Inflation)for (Adjusted Value Real

100

Index Price

Value (raw) Nominal

Percentage change

• Is equal to (new value- old value) divided by the old value.

Year Real Tax Collection

% Change in Nominal Tax Collections

% Percent in Real Tax Collections

1990 419,561

1991 401,564 -0.25 -4.29

1992 402,363 3.25 0.20

1993 410,958 5.17 2.14

1994 421,968 5.34 2.68

1995 449,880 9.60 6.61

1996 481,243 10.12 6.97

1997 528,101 12.30 9.74

1998 576,898 10.93 9.24

1999 619,312 9.71 7.35

• The percent change in real tax collections tends to be much smaller than that of the nominal tax collections. This shows the importance of adjusting for inflation (see part c).

• c. If you didn’t adjust for inflation, the regression process would think tax collections increased a lot more than they did. Any regression results from a model that includes the nominal (unadjusted) tax collections are likely to be misleading.

4. #5, Page 82

• The model is a tautology, or it is very close to being a tautology. The right hand side simply adds up all the people who have left the nursing home for various reasons. The true value for each of the slope coefficients will always be 1. For example, if one more person leaves the nursing home to live with relatives, EXIT will always increase by 1, so the true value of B3 is 1. This is true for all the slope coefficients.

5. #6, Page 83

• a. HOUSE_EXP = 7 + 0.00017 INCOME• b. HOUSE_EXP = 7,000 + 170 INCOME• c. HOUSE_EXP = 7 + 0.17 INCOME• d. HOUSE_EXP = 0.7 + 0.17 INCOME• e. “b” is the easiest to interpret. You can say

that if someone has an additional 1,000 in income, on average, they will spend $170 more on housing that year.

• f. A measure of the price of housing, and the number of people in the household are two possible answers.

Chapter 5

• This week we will cover up to Page 94: Section 5-2 Interaction variables

Some elementary rules of partial differentiation

• Y = 2X1 + 3 X1X2 + 5 X33

• dY/dX1 measures change in Y as a result of one unit change in X1 assuming X2 and X3 are constant

• dY/dX1= 2 +3X2

• dY/dX2 = 3X1

• dY/dX3 = 15X32

Intercept Dummies

• Theory 1: Men’s earnings is ,in general, higher than women’s earnings

Graph of earnings versus experience

Years of work

Earnings

Female

Male

• How would a dummy variable capture this?

• Intercept dummy– Earnings = B0 + B1 (gender) + B2 (years of

work) + error• Where gender is dummy variable that takes a

value of 1 if the observation is a male and 0 otherwise.

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

workGender

1. female 0

2. male 1

3. male 1

3. female 0

4. male 1

Testing the theory

• You estimate your model as usual and get• Earnings^ = 1000+ 200 (gender) + 500 (years of

work) • Then you do a one sided t-test of significance on

the coefficient of gender– Ho: B1 ≤0– Ha: B1>0

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

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 female– Earnings^ = 1000+ 200 (0) + 500 (years of

work)

– Earnings^ = 1000+ 500 (years of work)

Graph of earnings versus experience

Years of work

Earnings

Female

Male

1000

1200

Slope Dummies

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

Graph of earnings versus experience

Years of work

Earnings

Female

Male

How would a dummy variable capture this?

• Slope dummy– Earnings = B0 + B1 (years of work) + B2 (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.

Suppose you have 5 observations in your data set, you will create 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

Testing the theory

• You estimate your model as usual and get• Earnings^ = 1000+ 500 (years of work)

+70(genwork)• Then you do a one sided t-test of significance on

the coefficient of genwork– Ho: B2 ≤0– Ha: B2>0

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

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 female– Earnings^ = 1000+ 500 (years of work) + 70

(years of work) * (0)

– Earnings^ = 1000+ 500 (years of work)

Graph of earnings versus experience

Years of work

Earnings

Female slope = 500

Male slope = 570

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.

Assignment 6 (20 points)Due: before 10 PM on Friday, October 12)

1. Suppose the theory suggests that advertising for sun blocks is more effective in summer than any other time of the year

– Formulate the model– What type of a data set will you use: time

series or cross sectional?– Set up a hypothesis to test the theory

2. Suppose we estimate a regression equation that sets the crime rate as a function of a state’s per capita income and the number of police officers in each state per 10,000 population. The estimated coefficient of per capita income happens to be positive. We suspect that the estimated coefficient of per capita income is biased positively because we have an omitted variable. Which of the following omitted variables is more likely to have caused the bias in our estimated coefficient of income and why?– Number of college educated individuals per 1000 population– Percentage of population living in poverty– State’s unemployment rate– Percentage of population who lives in urban areas.