1. Chapter 13 Multiple Regression
Statistics for
Business and Economics 6th Edition
Chapter 13
Multiple Regression
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

2. Chapter Goals After completing this chapter, you should be able to:
Apply multiple regression analysis to business decision-making situations
Analyze and interpret the computer output for a multiple regression model
Perform a hypothesis test for all regression coefficients or for a subset of coefficients
Fit and interpret nonlinear regression models
Incorporate qualitative variables into the regression model by using dummy variables
Discuss model specification and analyze residuals
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

3. The Multiple Regression Model
Idea: Examine the linear relationship between
1 dependent (Y) & 2 or more independent variables (Xi)
Multiple Regression Model with k Independent Variables:
Y-intercept
Population slopes
Random Error
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

4. Multiple Regression Equation
The coefficients of the multiple regression model are estimated using sample data
Multiple regression equation with k independent variables:
Estimated
(or predicted)
value of y
Estimated
intercept
Estimated slope coefficients
In this chapter we will always use a computer to obtain the regression slope coefficients and other regression summary measures.
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

5. Multiple Regression Equation
(continued)
Two variable model y
Slope for variable x1 x2
Slope for variable x2 x1
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

6. Standard Multiple Regression Assumptions
The values xi and the error terms εi are independent
The error terms are random variables with mean 0 and a constant variance, 2.
(The constant variance property is called
homoscedasticity)
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

7. Standard Multiple Regression Assumptions
(continued)
The random error terms, εi , are not correlated with one another, so that
It is not possible to find a set of numbers, c0, c1, , ck, such that
(This is the property of no linear relation for
the Xj’s)
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

8. Example: 2 Independent Variables
A distributor of frozen desert pies wants to evaluate factors thought to influence demand
Dependent variable: Pie sales (units per week)
Independent variables: Price (in $)
Advertising ($100’s)
Data are collected for 15 weeks
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

9. Pie Sales Example Sales = b0 + b1 (Price) + b2 (Advertising)
Week
Pie Sales
Price
($)
Advertising
($100s) 1
350
5.50
3.3 2
460
7.50 3
8.00
3.0 4
430
4.5 5
6.80 6
380
4.0 7
4.50 8
470
6.40
3.7 9
450
7.00
3.5 10
490
5.00 11
340
7.20 12
300
7.90
3.2 13
440
5.90 14 15
2.7
Multiple regression equation:
Sales = b0 + b1 (Price)
+ b2 (Advertising)
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

10. Estimating a Multiple Linear Regression Equation
Excel will be used to generate the coefficients and measures of goodness of fit for multiple regression
Excel:
Tools / Data Analysis... / Regression
PHStat:
PHStat / Regression / Multiple Regression…
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

11. Multiple Regression Output
Regression Statistics
Multiple R
R Square
Adjusted R Square
Standard Error
Observations 15
ANOVA   df SS MS F
Significance F
Regression 2
Residual 12
Total 14
Coefficients
t Stat
P-value
Lower 95%
Upper 95%
Intercept
Price
Advertising
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

12. The Multiple Regression Equation
where
Sales is in number of pies per week
Price is in $
Advertising is in $100’s.
b1 = : sales will decrease, on average, by pies per week for each $1 increase in selling price, net of the effects of changes due to advertising
b2 = : sales will increase, on average, by pies per week for each $100 increase in advertising, net of the effects of changes due to price
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

13. Coefficient of Determination, R2
Reports the proportion of total variation in y explained by all x variables taken together
This is the ratio of the explained variability to total sample variability
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

14. Coefficient of Determination, R2
(continued)
Regression Statistics
Multiple R
R Square
Adjusted R Square
Standard Error
Observations 15
ANOVA   df SS MS F
Significance F
Regression 2
Residual 12
Total 14
Coefficients
t Stat
P-value
Lower 95%
Upper 95%
Intercept
Price
Advertising
52.1% of the variation in pie sales is explained by the variation in price and advertising
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

15. Estimation of Error Variance
Consider the population regression model
The unbiased estimate of the variance of the errors is
where
The square root of the variance, se , is called the standard error of the estimate
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

16. Regression Statistics
Standard Error, se
Regression Statistics
Multiple R
R Square
Adjusted R Square
Standard Error
Observations 15
ANOVA   df SS MS F
Significance F
Regression 2
Residual 12
Total 14
Coefficients
t Stat
P-value
Lower 95%
Upper 95%
Intercept
Price
Advertising
The magnitude of this value can be compared to the average y value
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

17. Adjusted Coefficient of Determination,
R2 never decreases when a new X variable is added to the model, even if the new variable is not an important predictor variable
This can be a 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?
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

18. Adjusted Coefficient of Determination,
(continued)
Used to correct for the fact that adding non-relevant independent variables will still reduce the error sum of squares
(where n = sample size, K = number of independent variables)
Adjusted R2 provides a better comparison between multiple regression models with different numbers of independent variables
Penalize excessive use of unimportant independent variables
Smaller than R2
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

19. Regression Statistics
Multiple R
R Square
Adjusted R Square
Standard Error
Observations 15
ANOVA   df SS MS F
Significance F
Regression 2
Residual 12
Total 14
Coefficients
t Stat
P-value
Lower 95%
Upper 95%
Intercept
Price
Advertising
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
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

20. Coefficient of Multiple Correlation
The coefficient of multiple correlation is the correlation between the predicted value and the observed value of the dependent variable
Is the square root of the multiple coefficient of determination
Used as another measure of the strength of the linear relationship between the dependent variable and the independent variables
Comparable to the correlation between Y and X in simple regression
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

21. Evaluating Individual Regression Coefficients
Use t-tests for individual coefficients
Shows if a specific independent variable is conditionally important
Hypotheses:
H0: βj = 0 (no linear relationship)
H1: βj ≠ 0 (linear relationship does exist
between xj and y)
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

22. Evaluating Individual Regression Coefficients
(continued)
H0: βj = 0 (no linear relationship)
H1: βj ≠ 0 (linear relationship does exist
between xi and y)
Test Statistic:
(df = n – k – 1)
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

23. Evaluating Individual Regression Coefficients
(continued)
Regression Statistics
Multiple R
R Square
Adjusted R Square
Standard Error
Observations 15
ANOVA   df SS MS F
Significance F
Regression 2
Residual 12
Total 14
Coefficients
t Stat
P-value
Lower 95%
Upper 95%
Intercept
Price
Advertising
t-value for Price is t = , with p-value .0398
t-value for Advertising is t = 2.855, with p-value .0145
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

24. Example: Evaluating Individual Regression Coefficients
From Excel output:
H0: βj = 0
H1: βj  0
Coefficients
Standard Error
t Stat
P-value
Price
Advertising
d.f. = = 12
= .05
t12, .025 =
The test statistic for each variable falls in the rejection region (p-values < .05)
Decision:
Conclusion:
Reject H0 for each variable
a/2=.025
a/2=.025
There is evidence that both Price and Advertising affect pie sales at  = .05
Reject H0
Do not reject H0
Reject H0
-tα/2
tα/2
2.1788
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

25. Confidence Interval Estimate for the Slope
Confidence interval limits for the population slope βj
where t has
(n – K – 1) d.f.
Coefficients
Standard Error
Intercept
Price
Advertising
Here, t has
(15 – 2 – 1) = 12 d.f.
Example: Form a 95% confidence interval for the effect of changes in price (x1) on pie sales:
± (2.1788)(10.832)
So the interval is < β1 <
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

26. Confidence Interval Estimate for the Slope
(continued)
Confidence interval for the population slope βi
Coefficients
Standard Error …
Lower 95%
Upper 95%
Intercept
Price
Advertising
Example: Excel output also reports these interval endpoints:
Weekly sales are estimated to be reduced by between 1.37 to pies for each increase of $1 in the selling price
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

27. Test on All Coefficients
F-Test for Overall Significance of the Model
Shows if there is a linear relationship between all of the X variables considered together and Y
Use F test statistic
Hypotheses:
H0: β1 = β2 = … = βk = 0 (no linear relationship)
H1: at least one βi ≠ 0 (at least one independent
variable affects Y)
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

28. F-Test for Overall Significance
Test statistic:
where F has k (numerator) and
(n – K – 1) (denominator)
degrees of freedom
The decision rule is
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

29. F-Test for Overall Significance
(continued)
Regression Statistics
Multiple R
R Square
Adjusted R Square
Standard Error
Observations 15
ANOVA   df SS MS F
Significance F
Regression 2
Residual 12
Total 14
Coefficients
t Stat
P-value
Lower 95%
Upper 95%
Intercept
Price
Advertising
With 2 and 12 degrees of freedom
P-value for the F-Test
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

30. F-Test for Overall Significance
(continued)
Test Statistic:
Decision:
Conclusion:
H0: β1 = β2 = 0
H1: β1 and β2 not both zero
 = .05
df1= df2 = 12
Critical Value:
F = 3.885
Since F test statistic is in the rejection region (p-value < .05), reject H0
 = .05 F
There is evidence that at least one independent variable affects Y
Do not
reject H0
Reject H0
F.05 = 3.885
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

31. Tests on a Subset of Regression Coefficients
Consider a multiple regression model involving variables xj and zj , and the null hypothesis that the z variable coefficients are all zero:
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

32. Tests on a Subset of Regression Coefficients
(continued)
Goal: compare the error sum of squares for the complete model with the error sum of squares for the restricted model
First run a regression for the complete model and obtain SSE
Next run a restricted regression that excludes the z variables (the number of variables excluded is r) and obtain the restricted error sum of squares SSE(r)
Compute the F statistic and apply the decision rule for a significance level 
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

33. Prediction Given a population regression model
then given a new observation of a data point
(x1,n+1, x 2,n+1, , x K,n+1)
the best linear unbiased forecast of yn+1 is
It is risky to forecast for new X values outside the range of the data used to estimate the model coefficients, because we do not have data to support that the linear model extends beyond the observed range. ^
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

34. Using The Equation to Make Predictions
Predict sales for a week in which the selling price is $5.50 and advertising is $350:
Note that Advertising is in $100’s, so $350 means that X2 = 3.5
Predicted sales is pies
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

35. Predictions in PHStat PHStat | regression | multiple regression …
Check the “confidence and prediction interval estimates” box
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

36. Predictions in PHStat Predicted y value Input values (continued) <
Confidence interval for the mean y value, given these x’s
<
Prediction interval for an individual y value, given these x’s
<
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

37. Residuals in Multiple Regression
Two variable model y
Sample observation yi
Residual = ei = (yi – yi)
< yi
<
x2i x2
x1i x1
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

38. Nonlinear Regression Models
The relationship between the dependent variable and an independent variable may not be linear
Can review the scatter diagram to check for non-linear relationships
Example: Quadratic model
The second independent variable is the square of the first variable
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

39. Quadratic Regression Model
Model form:
where:
β0 = Y intercept
β1 = regression coefficient for linear effect of X on Y
β2 = regression coefficient for quadratic effect on Y
εi = random error in Y for observation i
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

40. Linear vs. Nonlinear FitY Y X X X X
residuals
residuals
Linear fit does not give random residuals
Nonlinear fit gives random residuals 
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

41. 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
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

42. Testing for Significance: Quadratic Effect
Testing the Quadratic Effect
Compare the linear regression estimate
with quadratic regression estimate
Hypotheses
(The quadratic term does not improve the model)
(The quadratic term improves the model)
H0: β2 = 0
H1: β2  0
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

43. Testing for Significance: Quadratic Effect
(continued)
Testing the Quadratic Effect
Hypotheses
(The quadratic term does not improve the model)
(The quadratic term improves the model)
The test statistic is
H0: β2 = 0
H1: β2  0
where:
b2 = squared term slope
coefficient
β2 = hypothesized slope (zero)
Sb = standard error of the slope 2
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

44. Testing for Significance: Quadratic Effect
(continued)
Testing the Quadratic Effect
Compare R2 from simple regression to
R2 from the quadratic model
If R2 from the quadratic model is larger than R2 from the simple model, then the quadratic model is a better model
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

45. Example: Quadratic Model
Purity increases as filter time increases:
Purity
Filter
Time 3 1 7 2 8 15 5 22 33 40 10 54 12 67 13 70 14 78 85 87 16 99 17
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

46. Example: Quadratic Model
(continued)
Simple regression results:
y = Time ^
Coefficients
Standard Error
t Stat
P-value
Intercept
Time
2.078E-10
t statistic, F statistic, and R2 are all high, but the residuals are not random:
Regression Statistics
R Square
Adjusted R Square
Standard Error F
Significance F
2.0778E-10
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

47. Example: Quadratic Model
(continued)
Quadratic regression results:
y = Time (Time)2 ^
Coefficients
Standard Error
t Stat
P-value
Intercept
Time
Time-squared
1.165E-05
Regression Statistics
R Square
Adjusted R Square
Standard Error F
Significance F
2.368E-13
The quadratic term is significant and improves the model: R2 is higher and se is lower, residuals are now random
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

48. The Log Transformation
The Multiplicative Model:
Original multiplicative model
Transformed multiplicative model
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

49. Interpretation of coefficients
For the multiplicative model:
When both dependent and independent variables are logged:
The coefficient of the independent variable Xk can be interpreted as
a 1 percent change in Xk leads to an estimated bk percentage change in the average value of Y
bk is the elasticity of Y with respect to a change in Xk
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

50. Dummy Variables
A dummy variable is a categorical independent variable with two levels:
yes or no, on or off, male or female
recorded as 0 or 1
Regression intercepts are different if the variable is significant
Assumes equal slopes for other variables
If more than two levels, the number of dummy variables needed is (number of levels - 1)
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

51. Dummy Variable Example
Let:
y = Pie Sales
x1 = Price
x2 = Holiday (X2 = 1 if a holiday occurred during the week) (X2 = 0 if there was no holiday that week)
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

52. Dummy Variable Example
(continued)
Holiday
No Holiday
Different intercept
Same slope
y (sales)
If H0: β2 = 0 is rejected, then
“Holiday” has a significant effect on pie sales
b0 + b2
Holiday (x2 = 1) b0
No Holiday (x2 = 0)
x1 (Price)
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

53. Interpreting the Dummy Variable Coefficient
Example:
Sales: number of pies sold per week
Price: pie price in $
Holiday:
1 If a holiday occurred during the week
0 If no holiday occurred
b2 = 15: on average, sales were 15 pies greater in weeks with a holiday than in weeks without a holiday, given the same price
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

54. Interaction Between Explanatory Variables
Hypothesizes interaction between pairs of x variables
Response to one x variable may vary at different levels of another x variable
Contains two-way cross product terms
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

55. Effect of Interaction Given:
Without interaction term, effect of X1 on Y is measured by β1
With interaction term, effect of X1 on Y is measured by β1 + β3 X2
Effect changes as X2 changes
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

56. Slopes are different if the effect of x1 on y depends on x2 value
Interaction Example
Suppose x2 is a dummy variable and the estimated regression equation is y 12
x2 = 1:
y = 1 + 2x1 + 3(1) + 4x1(1) = 4 + 6x1 ^ 8 4
x2 = 0:
y = 1 + 2x1 + 3(0) + 4x1(0) = 1 + 2x1 ^ x1
0.5 1
1.5
Slopes are different if the effect of x1 on y depends on x2 value
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

57. Significance of Interaction Term
The coefficient b3 is an estimate of the difference in the coefficient of x1 when x2 = 1 compared to when x2 = 0
The t statistic for b3 can be used to test the hypothesis
If we reject the null hypothesis we conclude that there is a difference in the slope coefficient for the two subgroups
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

58. Multiple Regression Assumptions
Errors (residuals) from the regression model:
ei = (yi – yi)
<
Assumptions:
The errors are normally distributed
Errors have a constant variance
The model errors are independent
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

59. Analysis of Residuals in Multiple Regression
These residual plots are used in multiple regression:
Residuals vs. yi
Residuals vs. x1i
Residuals vs. x2i
Residuals vs. time (if time series data)
<
Use the residual plots to check for violations of regression assumptions
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

60. Chapter Summary Developed the multiple regression model
Tested the significance of the multiple regression model
Discussed adjusted R2 ( R2 )
Tested individual regression coefficients
Tested portions of the regression model
Used quadratic terms and log transformations in regression models
Used dummy variables
Evaluated interaction effects
Discussed using residual plots to check model assumptions
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.