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

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


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


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.


Two variable model

y

x1

x2

22110 xbxbby ˆ

Slope for v

ariable x 1

Slope for variable x2

Multiple Regression Equation(continued)


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


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


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


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:


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…


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


The Multiple Regression Equation


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.


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



Multiple R 0.72213

R Square 0.52148



Observations 15


Regression 2 29460.027 14730.013 6.53861 0.01201

Residual 12 27033.306 2252.776

Total 14 56493.333


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)


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



Multiple R 0.72213

R Square 0.52148



Observations 15


Regression 2 29460.027 14730.013 6.53861 0.01201

Residual 12 27033.306 2252.776

Total 14 56493.333


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


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


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



Multiple R 0.72213

R Square 0.52148



Observations 15


Regression 2 29460.027 14730.013 6.53861 0.01201

Residual 12 27033.306 2252.776

Total 14 56493.333


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


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


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)


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)




Multiple R 0.72213

R Square 0.52148



Observations 15


Regression 2 29460.027 14730.013 6.53861 0.01201

Residual 12 27033.306 2252.776

Total 14 56493.333


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)



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


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.


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)


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)


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


6.53862252.8

14730.0

MSE

MSRF


Multiple R 0.72213

R Square 0.52148



Observations 15


Regression 2 29460.027 14730.013 6.53861 0.01201

Residual 12 27033.306 2252.776

Total 14 56493.333


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


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


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


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


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

^


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


Note that Advertising is in $100’s, so $350 means that X2 = 3.5


Predictions in PHStat

PHStat | regression | multiple regression …

Check the “confidence and prediction interval estimates” box


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

<


Two variable model

y

x1

x2

22110 xbxbby ˆyi

yi

<

x2i

x1i

Sample observation

Residuals in Multiple Regression

Residual =

ei = (yi – yi)

<


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


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:


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


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


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



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



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)


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


Example: Quadratic Model(continued)


R Square 0.96888



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:


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


R Square 0.99494



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


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


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


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)


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


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


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


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


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


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

^

^


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


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:


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


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

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

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