1. BA 275 Quantitative Business Methods
Agenda
Residual Analysis
Multiple Linear Regression
Adjusted R-squared
Prediction

2. Simple Linear Regression Model
population
True effect of X on Y
Estimated effect of X on Y
sample
Key questions:
1. Does X have any effect on Y?
2. If yes, how large is the effect?
3. Given X, what is the estimated Y?

3. Hypothesis Testing Key Q1: Does X have any effect on Y?b0
H0: b1 = 0
Ha: b1 ≠ 0 b1
SEb1
SEb0
Degrees of freedom = n – p – 1
p = # of independent variables used.

4. Interval Estimation Key Q2: If so, how large is the effect?b0 b1
SEb1
SEb0
Degrees of freedom = n – p – 1
p = # of independent variables used.

5. Prediction and Confidence Intervals Key Q3: Given X, what is the estimated Y?
What is your estimated price of that 2000-sf house on the 9th street?
Quick answer: estimated price = (2) =
What is the average price of a house that occupies 2000 sf?
What is the difference?

6. Prediction and Confidence Intervals
Prediction interval
Confidence interval

7. Model Comparison: A Good Fit?
SS = Sum of Squares = ???

8. Residual Analysis
The three conditions required for the validity of the regression analysis are:
the error variable is normally distributed.
the error variance is constant for all values of x.
the errors are independent of each other.
How can we diagnose violations of these conditions?

9. Residual Analysis
We do not have e (random error), but we can calculate residuals from the sample.
Residual = actual Y – estimated Y
Examining the residuals (or standardized residuals), help detect violations of the required conditions.

10. Residuals, Standardized Residuals, and Studentized Residuals

11. The random error e is normally distributed

12. The error variance se is constant for all values of X and estimated Y
Constant spread !

13. The spread increases with y
Constant Variance
When the requirement of a constant variance is violated we have a condition of heteroscedasticity.
Diagnose heteroscedasticity by plotting the residual against the predicted y.
Residual + + + + + + + + + + + + + ^ + + + y + + + + + + + +
The spread increases with y ^

14. The errors are independent of each other
Do not want to see any pattern.

15. Non Independence of Error Variables
Residual
Residual + + + + + + + + + + + + + + +
Time
Time + + + + + + + + + + + + +
Note the runs of positive residuals,
replaced by runs of negative residuals
Note the oscillating behavior of the
residuals around zero.

16. Multiple Regression Model

17. Correlations

18. Fitted Model Q: Effect of AGE? H0: bAGE = 0 Ha: bAGE ≠ 0
Multiple Regression Analysis
Dependent variable: Price
Standard T
Parameter Estimate Error Statistic P-Value
CONSTANT
Age
Bidder ? ? ? ?
Q: Effect of AGE?
H0: bAGE = 0
Ha: bAGE ≠ 0
Q: Effect of BIDDER?
H0: bBIDDER = 0
Ha: bBIDDER ≠ 0
Degrees of freedom = ?

19. Fitted Model Fitted Model:
Multiple Regression Analysis
Dependent variable: Price
Standard T
Parameter Estimate Error Statistic P-Value
CONSTANT
Age
Bidder
Fitted Model:
Estimated price = AGE BIDDER

20. Analysis of Variance ? ?

21. Model Selection

22. Using the Model
What is the total variation of auction prices? How much has been explained by the model?
If there are 10 bidders and the age of the clock is 100 years old, what is the expected auction price?
If AGE is held fixed and the number of bidders increases from 10 to 11, how much does PRICE increase?

23. Prediction and Confidence Intervals
Fitted Model:
Estimated price = AGE BIDDER
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