Welcome to Econ 420 Applied Regression Analysis

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Welcome to Econ 420 Applied Regression Analysis

• Study Guide
• Week Seven

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Answer Key to Assignment 5 Question 1- Part One

• Step 1
• H0 B1 B2 B3 0
• HA At least one of these Bs 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)

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• Step 3
• Run regression and find F-statistic 40.82042
• Step 4
• Because our F-statistic, 40.82 gt 4.64, the null
  hypothesis is rejected at the 1 significance
  level it is 99 likely that at least one of
  these Bs is not zero.

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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)

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• The 95 confidence interval for the coefficient
  on income is B1 tc SE (B1) lt B1 lt B1 tc
  SE (B1),
• The 95 confidence interval is 0.0114 lt B1 lt
  0.0340.
• There is 95 chance that the true value of B1 is
  in the above range.

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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.

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3. 4, PP 81-82

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

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Percentage change

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

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• 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 didnt 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.

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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.

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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.

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Chapter 5

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

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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

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Intercept Dummies

• Theory 1 Mens earnings is ,in general, higher
  than womens earnings

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Graph of earnings versus experience
Earnings
Male
Female
Years of work
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• 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.

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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
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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 B1gt0
• If you reject Ho, then you have found significant
  evidence that men, in general earn more than
  women

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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)

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Graph of earnings versus experience
Earnings
Male
Female
1200
1000
Years of work
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Slope Dummies

• Theory 2 Men earnings grow at a higher rate
  than womens earnings

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Graph of earnings versus experience
Earnings
Male
Female
Years of work
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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.

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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
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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 B2gt0
• If you reject Ho, then you have found significant
  evidence that mens earnings grow at a higher
  rate with years of experience.

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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)

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Graph of earnings versus experience
Earnings
Male slope 570
Female slope 500
Years of work
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What if

• The theory suggested that not only, in general,
  mens salaries are higher than womens 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.

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

• 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

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• 2. Suppose we estimate a regression equation that
  sets the crime rate as a function of a states
  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
• States unemployment rate
• Percentage of population who lives in urban areas.