Chap 13-1Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chapter 13
Multiple Regression
Statistics for Business and Economics
6th Edition
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-2
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. Chap 13-3
The Multiple Regression Model
Idea: Examine the linear relationship between 1 dependent (Y) & 2 or more independent variables (Xi)
εXβXβXββY kk22110
Multiple Regression Model with k Independent Variables:
Y-intercept Population slopes Random Error
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-4
Multiple Regression Equation
The coefficients of the multiple regression model are estimated using sample data
kik2i21i10i xbxbxbby ˆ
Estimated (or predicted) value of y
Estimated slope coefficients
Multiple regression equation with k independent variables:
Estimatedintercept
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. Chap 13-5
Two variable model
y
x1
x2
22110 xbxbby ˆ
Slope for v
ariable x 1
Slope for variable x2
Multiple Regression Equation(continued)
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-6
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)
n), 1,(i for σ]E[εand0]E[ε 22ii
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-7
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)
ji all for 0]εE[ε ji
0xcxcxcc KiK2i21i10
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-8
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 ($100’s)
Data are collected for 15 weeks
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-9
Pie Sales Example
Sales = b0 + b1 (Price)
+ b2 (Advertising)
WeekPie
SalesPrice
($)Advertising
($100s)
1 350 5.50 3.3
2 460 7.50 3.3
3 350 8.00 3.0
4 430 8.00 4.5
5 350 6.80 3.0
6 380 7.50 4.0
7 430 4.50 3.0
8 470 6.40 3.7
9 450 7.00 3.5
10 490 5.00 4.0
11 340 7.20 3.5
12 300 7.90 3.2
13 440 5.90 4.0
14 450 5.00 3.5
15 300 7.00 2.7
Multiple regression equation:
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-10
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. Chap 13-11
Multiple Regression Output
Regression Statistics
Multiple R 0.72213
R Square 0.52148
Adjusted R Square 0.44172
Standard Error 47.46341
Observations 15
ANOVA df SS MS F Significance F
Regression 2 29460.027 14730.013 6.53861 0.01201
Residual 12 27033.306 2252.776
Total 14 56493.333
Coefficients Standard Error t Stat P-value Lower 95% Upper 95%
Intercept 306.52619 114.25389 2.68285 0.01993 57.58835 555.46404
Price -24.97509 10.83213 -2.30565 0.03979 -48.57626 -1.37392
Advertising 74.13096 25.96732 2.85478 0.01449 17.55303 130.70888
ertising)74.131(Adv ce)24.975(Pri - 306.526 Sales
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-12
The Multiple Regression Equation
ertising)74.131(Adv ce)24.975(Pri - 306.526 Sales
b1 = -24.975: sales will decrease, on average, by 24.975 pies per week for each $1 increase in selling price, net of the effects of changes due to advertising
b2 = 74.131: sales will increase, on average, by 74.131 pies per week for each $100 increase in advertising, net of the effects of changes due to price
where Sales is in number of pies per week Price is in $ Advertising is in $100’s.
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-13
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
squares of sum total
squares of sum regression
SST
SSRR2
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-14
Regression Statistics
Multiple R 0.72213
R Square 0.52148
Adjusted R Square 0.44172
Standard Error 47.46341
Observations 15
ANOVA df SS MS F Significance F
Regression 2 29460.027 14730.013 6.53861 0.01201
Residual 12 27033.306 2252.776
Total 14 56493.333
Coefficients Standard Error t Stat P-value Lower 95% Upper 95%
Intercept 306.52619 114.25389 2.68285 0.01993 57.58835 555.46404
Price -24.97509 10.83213 -2.30565 0.03979 -48.57626 -1.37392
Advertising 74.13096 25.96732 2.85478 0.01449 17.55303 130.70888
.5214856493.3
29460.0
SST
SSRR2
52.1% of the variation in pie sales is explained by the variation in price and advertising
Coefficient of Determination, R2
(continued)
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-15
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
1Kn
SSE
1Kn
es
n
1i
2i
2e
iKiK2i21i10i εxβxβxββY
iii yye ˆ
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-16
Regression Statistics
Multiple R 0.72213
R Square 0.52148
Adjusted R Square 0.44172
Standard Error 47.46341
Observations 15
ANOVA df SS MS F Significance F
Regression 2 29460.027 14730.013 6.53861 0.01201
Residual 12 27033.306 2252.776
Total 14 56493.333
Coefficients Standard Error t Stat P-value Lower 95% Upper 95%
Intercept 306.52619 114.25389 2.68285 0.01993 57.58835 555.46404
Price -24.97509 10.83213 -2.30565 0.03979 -48.57626 -1.37392
Advertising 74.13096 25.96732 2.85478 0.01449 17.55303 130.70888
47.463se
The magnitude of this value can be compared to the average y value
Standard Error, se
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-17
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?
2R
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-18
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
(continued)
Adjusted Coefficient of Determination, 2R
1)(n/SST
1)K(n/SSE1R2
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-19
Regression Statistics
Multiple R 0.72213
R Square 0.52148
Adjusted R Square 0.44172
Standard Error 47.46341
Observations 15
ANOVA df SS MS F Significance F
Regression 2 29460.027 14730.013 6.53861 0.01201
Residual 12 27033.306 2252.776
Total 14 56493.333
Coefficients Standard Error t Stat P-value Lower 95% Upper 95%
Intercept 306.52619 114.25389 2.68285 0.01993 57.58835 555.46404
Price -24.97509 10.83213 -2.30565 0.03979 -48.57626 -1.37392
Advertising 74.13096 25.96732 2.85478 0.01449 17.55303 130.70888
.44172R2 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
2R
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-20
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
2Ry),yr(R ˆ
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-21
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. Chap 13-22
H0: βj = 0 (no linear relationship)
H1: βj ≠ 0 (linear relationship does exist between xi and y)
Test Statistic:
(df = n – k – 1)
jb
j
S
0bt
(continued)
Evaluating Individual Regression Coefficients
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-23
Regression Statistics
Multiple R 0.72213
R Square 0.52148
Adjusted R Square 0.44172
Standard Error 47.46341
Observations 15
ANOVA df SS MS F Significance F
Regression 2 29460.027 14730.013 6.53861 0.01201
Residual 12 27033.306 2252.776
Total 14 56493.333
Coefficients Standard Error t Stat P-value Lower 95% Upper 95%
Intercept 306.52619 114.25389 2.68285 0.01993 57.58835 555.46404
Price -24.97509 10.83213 -2.30565 0.03979 -48.57626 -1.37392
Advertising 74.13096 25.96732 2.85478 0.01449 17.55303 130.70888
t-value for Price is t = -2.306, with p-value .0398
t-value for Advertising is t = 2.855, with p-value .0145
(continued)
Evaluating Individual Regression Coefficients
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-24
d.f. = 15-2-1 = 12
= .05
t12, .025 = 2.1788
H0: βj = 0
H1: βj 0
The test statistic for each variable falls in the rejection region (p-values < .05)
There is evidence that both Price and Advertising affect pie sales at = .05
From Excel output:
Reject H0 for each variable
Coefficients Standard Error t Stat P-value
Price -24.97509 10.83213 -2.30565 0.03979
Advertising 74.13096 25.96732 2.85478 0.01449
Decision:
Conclusion:
Reject H0Reject H0
/2=.025
-tα/2
Do not reject H0
0 tα/2
/2=.025
-2.1788 2.1788
Example: Evaluating Individual Regression Coefficients
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-25
Confidence Interval Estimate for the Slope
Confidence interval limits for the population slope βj
Example: Form a 95% confidence interval for the effect of changes in price (x1) on pie sales:
-24.975 ± (2.1788)(10.832)
So the interval is -48.576 < β1 < -1.374
jbα/21,Knj Stb Coefficients Standard Error
Intercept 306.52619 114.25389
Price -24.97509 10.83213
Advertising 74.13096 25.96732
where t has (n – K – 1) d.f.
Here, t has
(15 – 2 – 1) = 12 d.f.
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-26
Confidence Interval Estimate for the Slope
Confidence interval for the population slope βi
Example: Excel output also reports these interval endpoints:
Weekly sales are estimated to be reduced by between 1.37 to 48.58 pies for each increase of $1 in the selling price
Coefficients Standard Error … Lower 95% Upper 95%
Intercept 306.52619 114.25389 … 57.58835 555.46404
Price -24.97509 10.83213 … -48.57626 -1.37392
Advertising 74.13096 25.96732 … 17.55303 130.70888
(continued)
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-27
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. Chap 13-28
F-Test for Overall Significance
Test statistic:
where F has k (numerator) and
(n – K – 1) (denominator)
degrees of freedom The decision rule is
1)KSSE/(n
SSR/K
s
MSRF
2e
α1,Knk,0 FF if H Reject
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-29
6.53862252.8
14730.0
MSE
MSRF
Regression Statistics
Multiple R 0.72213
R Square 0.52148
Adjusted R Square 0.44172
Standard Error 47.46341
Observations 15
ANOVA df SS MS F Significance F
Regression 2 29460.027 14730.013 6.53861 0.01201
Residual 12 27033.306 2252.776
Total 14 56493.333
Coefficients Standard Error t Stat P-value Lower 95% Upper 95%
Intercept 306.52619 114.25389 2.68285 0.01993 57.58835 555.46404
Price -24.97509 10.83213 -2.30565 0.03979 -48.57626 -1.37392
Advertising 74.13096 25.96732 2.85478 0.01449 17.55303 130.70888
(continued)
F-Test for Overall Significance
With 2 and 12 degrees of freedom
P-value for the F-Test
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-30
H0: β1 = β2 = 0
H1: β1 and β2 not both zero
= .05
df1= 2 df2 = 12
Test Statistic:
Decision:
Conclusion:
Since F test statistic is in the rejection region (p-value < .05), reject H0
There is evidence that at least one independent variable affects Y
0
= .05
F.05 = 3.885Reject H0Do not
reject H0
6.5386MSE
MSRF
Critical Value:
F = 3.885
F-Test for Overall Significance(continued)
F
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-31
Consider a multiple regression model involving variables xj and zj , and the null hypothesis that the z variable coefficients are all zero:
r)1,...,(j 0α of one least at :H
0ααα:H
j1
r210
irir1i1KiK1i10i εzαzαxβxββy
Tests on a Subset of Regression Coefficients
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-32
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
Tests on a Subset of Regression Coefficients
(continued)
α1,rKnr,2e
0 Fs
r/)SSESSE(r)(F if H Reject
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-33
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.
n),1,2,(iεxβxβxββy iKiK2i21i10i
1nK,K1n2,21n1,101n xbxbxbby ˆ
^
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-34
Using The Equation to Make Predictions
Predict sales for a week in which the selling price is $5.50 and advertising is $350:
Predicted sales is 428.62 pies
428.62
(3.5) 74.131 (5.50) 24.975 - 306.526
ertising)74.131(Adv ce)24.975(Pri - 306.526 Sales
Note that Advertising is in $100’s, so $350 means that X2 = 3.5
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-35
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. Chap 13-36
Input values
Predictions in PHStat(continued)
Predicted y value
<
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. Chap 13-37
Two variable model
y
x1
x2
22110 xbxbby ˆyi
yi
<
x2i
x1i
Sample observation
Residuals in Multiple Regression
Residual =
ei = (yi – yi)
<
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-38
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
Nonlinear Regression Models
εXβXββY 212110
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-39
Quadratic Regression Model
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
i21i21i10i εXβXββY
Model form:
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-40
Linear fit does not give random residuals
Linear vs. Nonlinear Fit
Nonlinear fit gives random residuals
X
resi
dua
ls
X
Y
X
resi
dua
ls
Y
X
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-41
Quadratic Regression Model
Quadratic models may be considered when the scatter diagram takes on one of the following shapes:
X1
Y
X1X1
YYY
β1 < 0 β1 > 0 β1 < 0 β1 > 0
β1 = the coefficient of the linear term
β2 = the coefficient of the squared term
X1
β2 > 0 β2 > 0 β2 < 0 β2 < 0
i21i21i10i εXβXββY
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-42
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)
2 12110 xbxbby ˆ
110 xbby ˆ
H0: β2 = 0
H1: β2 0
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-43
Testing for Significance: Quadratic Effect
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
(continued)
2b
22
s
βbt
3nd.f.
where:
b2 = squared term slope coefficient
β2 = hypothesized slope (zero)
Sb = standard error of the slope2
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-44
Testing for Significance: Quadratic Effect
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
(continued)
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-45
Example: Quadratic Model
Purity increases as filter time increases:
PurityFilterTime
3 1
7 2
8 3
15 5
22 7
33 8
40 10
54 12
67 13
70 14
78 15
85 15
87 16
99 17
Purity vs. Time
0
20
40
60
80
100
0 5 10 15 20
Time
Pu
rity
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-46
Example: Quadratic Model(continued)
Regression Statistics
R Square 0.96888
Adjusted R Square 0.96628
Standard Error 6.15997
Simple regression results:
y = -11.283 + 5.985 Time
CoefficientsStandard
Error t Stat P-value
Intercept -11.28267 3.46805 -3.25332 0.00691
Time 5.98520 0.30966 19.32819 2.078E-10
F Significance F
373.57904 2.0778E-10
^
Time Residual Plot
-10
-5
0
5
10
0 5 10 15 20
Time
Resid
uals
t statistic, F statistic, and R2 are all high, but the residuals are not random:
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-47
CoefficientsStandard
Error t Stat P-value
Intercept 1.53870 2.24465 0.68550 0.50722
Time 1.56496 0.60179 2.60052 0.02467
Time-squared 0.24516 0.03258 7.52406 1.165E-05
Regression Statistics
R Square 0.99494
Adjusted R Square 0.99402
Standard Error 2.59513
F Significance F
1080.7330 2.368E-13
Quadratic regression results:
y = 1.539 + 1.565 Time + 0.245 (Time)2^
Example: Quadratic Model(continued)
Time Residual Plot
-5
0
5
10
0 5 10 15 20
Time
Res
idua
ls
Time-squared Residual Plot
-5
0
5
10
0 100 200 300 400
Time-squared
Res
idua
lsThe 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. Chap 13-48
Original multiplicative model
Transformed multiplicative model
The Log Transformation
The Multiplicative Model:
εXXβY 21 β2
β10
)log(ε)log(Xβ)log(Xβ)log(βlog(Y) 22110
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-49
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
i1i10i ε logX log ββ log Ylog
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-50
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. Chap 13-51
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)
210 xbxbby21
ˆ
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-52
Same slope
Dummy Variable Example (continued)
x1 (Price)
y (sales)
b0 + b2
b0
1010
12010
xb b (0)bxbby
xb)b(b(1)bxbby
121
121
ˆ
ˆHoliday
No Holiday
Different intercept
Holiday (x2 = 1)No Holiday (x
2 = 0)
If H0: β2 = 0 is rejected, then“Holiday” has a significant effect on pie sales
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-53
Sales: number of pies sold per weekPrice: pie price in $
Holiday:
Interpreting the Dummy Variable Coefficient
Example:
1 If a holiday occurred during the week0 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
)15(Holiday 30(Price) - 300 Sales
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-54
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
)x(xbxbxbb
xbxbxbby
21322110
3322110
ˆ
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-55
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
21322110
1231220
XXβXβXββ
)XXβ(βXββY
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-56
x2 = 1:y = 1 + 2x1 + 3(1) + 4x1(1) = 4 + 6x1
x2 = 0: y = 1 + 2x1 + 3(0) + 4x1(0) = 1 + 2x1
Interaction Example
Slopes are different if the effect of x1 on y depends on x2 value
x1
44
88
1212
00
00 110.50.5 1.51.5
y
Suppose x2 is a dummy variable and the estimated regression equation is 2121 x4x3x2x1y ˆ
^
^
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-57
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
0β0,β|0β:H
0β0,β|0β:H
2131
2130
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-58
Multiple Regression Assumptions
Assumptions: The errors are normally distributed Errors have a constant variance The model errors are independent
ei = (yi – yi)
<
Errors (residuals) from the regression model:
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-59
Analysis of Residualsin 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. Chap 13-60
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