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### Economics 105: Statistics

GH 22 & 23 due Thursday, 17th

GH 24 (last one! Please rejoice silently.) due Thur 24th

Unit 3 Review will be due Tuesday, 29th (I’ll hand it out Thur 24th). It’ll cover what we get through.

I’ve decided NOT to squeeze in the individual RAP presentations. I will move that % of your grade to RAP & Review 3.

Quadratic Regression Model

Quadratic models may be considered when the scatter diagram takes on one of the following shapes:

Y

Y

Y

Y

X1

X1

X1

X1

β1 < 0

β1 > 0

β1 < 0

β1 > 0

β2 > 0

β2 > 0

β2 < 0

β2 < 0

β1 = the coefficient of the linear term

β2 = the coefficient of the squared term

Testing the Overall Model

- Estimate the model to obtain the sample regression equation:

- The “whole model” F-test

H0: β1 = β2 = β3 = … = β15 = 0

H1: at least 1 βi ≠ 0

- F-test statistic =

Testing the Overall Model

Critical value = 2.082= F.INV(0.99,15,430-15-1)

p-value = 0 = 1-F.DIST(120.145,15,430-15-1,1)

Coefficient of Determination for Multiple Regression

- Reports the proportion of total variation in Y explained by all X variables taken together
- Consider this model

Multiple Coefficient of Determination

(continued)

52.1% of the variation in pie sales is explained by the variation in price and advertising

Adjusted R2

- R2 never decreases when a new X variable is added to the model
- disadvantage when comparing models
- What is the net effect of adding a new variable?
- We lose a degree of freedom when a new X variable is added
- Did the new X variable add enough explanatory power to offset the loss of one degree of freedom?

Adjusted R2

(continued)

- Penalizes excessive use of unimportant variables
- Smaller than R2and can increase, decrease, or stay same
- Useful in comparing among models, but don’t rely too heavily on it – use theory and statistical signif

Adjusted R2

(continued)

44.2% of the variation in pie sales is explained by the variation in price and advertising, taking into account the sample size and number of independent variables

Log Functional Forms

- Linear-Log
- Log-linear
- Log-log
- Log of a variable means interpretation is a percentage change in the variable
- (don’t forget Mark’s pet peeve)

Log Functional Forms

- Here’s why: ln(x+x) – ln(x) =
- calculus:
- Numerically: ln(1.01) = .00995 = .01
- ln(1.10) = .0953 = .10 (sort of)

Dummy Variables

- A dummy variable is a categorical explanatory variable with two levels:
- yes or no, on or off, male or female
- coded as 0’s and 1’s
- Regression intercepts are different if the variable is significant
- Assumes equal slopes for other explanatory variables (i.e., equal marginal effects!)
- “Dummy Variable Trap”
- If more than two categories, the number of dummy variables included is (number of categories - 1)

Dummy Variable Example (with 2 categories)

- E[ GPA | EconMajor = 1] = ?
- E[ GPA | EconMajor = 0] = ?
- Take the difference to interpret EconMajor

Dummy Variable Example (More than 2 categories)

- Model the effect of class year on GPA, controlling for study hours

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