chapter 12: multiple regression and model building

Chapter 12: Multiple Regression and Model Building

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

2

Where We’ve Been

Introduced the straight-line model relating a dependent variable y to an independent variable x

Estimated the parameters of the straight-line model using least squares

Assesses the model estimates Used the model to estimate a value of y

given x

Where We’re Going


3

Introduce a multiple-regression model to relate a variable y to two or more x variables

Present multiple regression models with both quantitative and qualitative independent variables

Assess how well the multiple regression model fits the sample data

Show how analyzing the model residuals can help detect problems with the model and the necessary modifications

12.1: Multiple Regression Models


4

0 1 1 2 2

1 2

0 1 1 2 2

The General Multiple Regression Model

where is the dependent variable, , , ... , are the independent variables, ( ) is the

de

k k

k

k k

y x x x

yx x xE y x x x

terministic portion of the model and

determines the contribution of the independent variable , which may be a quantitative variable of order one or higher or a qualitative variable

i

ix



5

Analyzing a Multiple-Regression ModelStep 1: Hypothesize the deterministic

portion of the model by choosing the independent variables x1, x2, … , xk.

Step 2: Estimate the unknown parameters 0, 1, 2, … , k .

Step 3: Specify the probability distribution of and estimate the standard deviation of this distribution.



6

Analyzing a Multiple-Regression ModelStep 4: Check that the assumptions

about are satisfied; if not make the required modifications to the model.

Step 5: Statistically evaluate the usefulness of the model.

Step 6: If the model is useful, use it for prediction, estimation and other purposes.



7

Assumptions about the Random Error 1. The mean is equal to 0.2. The variance is equal to 2.3. The probability distribution is a normal

distribution.4. Random errors are independent of one

another.

12.2: The First-Order Model: Estimating and Making Inferences about the Parameters

A First-Order Model in Five Quantitative Independent Variables

where x1, x2, … , xk are all quantitative variables that are not functions of other independent variables.


8

0 1 1 2 2 3 3 4 4 5 5( ) E y x x x x x



The parameters are estimated by finding the values for the ‘s that minimize


9

0 1 1 2 2 3 3 4 4 5 5( ) E y x x x x x

2ˆ( ) .SSE y y



The parameters are estimated by finding the values for the ‘s that minimize


10

0 1 1 2 2 3 3 4 4 5 5( ) E y x x x x x

2ˆ( )SSE y y

Only a truly talented mathematician (or geek) would choose to solve the necessary system of simultaneous linear

equations by hand. In practice, computers are left to do the

complicated calculation required by multiple regression models.

A collector of antique clocks hypothesizes that the auction price can be modeled as



11

0 1 1 2 2

1

2

whereauction price in dollarsage of clock in yearsnumber of bidders.

y x x

yxx

Based on the data in Table 12.1, the least squares prediction equation, the equation that minimizes SSE, is



12

1 2

2

ˆ 1,339 12.74 85.95516,727

516,727 17,8181 29

133.5 (the estimate for )

y x xSSE

SSEsn k

s




13

1 2

2

ˆ 1,339 12.74 85.95516,727

516,727 17,8181 29


y x xSSE

SSEsn k

s

The estimate for 1 is interpreted as the expected change in y given a one-unit change in x1 holding x2 constant

The estimate for 2 is interpreted as the expected change in y given a one-unit change in x2 holding x1 constant




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1 2

2

ˆ 1,339 12.74 85.95516,727

516,727 17,8181 29


y x xSSE

SSEsn k

s

Since it makes no sense to sell a clock of age 0 at an auction with no bidders, the intercept term has no meaningful interpretation in this example.

12.2: The First-Order Model:Estimating and Making Inferences about the Parameters

One-Tailed Test Two-Tailed Test


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Test of an Individual Parameter Coefficient in the Multiple Regression Model

0 : 0: ( )0

Rejection Region: ( )

i

a i

HH

t t t

0

/2

: 0: 0

Rejection Region:

i

a i

HH

t t

ˆ

/2

ˆTest statistic:

where and are based on ( 1) degrees of freedom and = number of observations

1 = number of parameters in the model

i

its

t t n knk



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Test of the Parameter Coefficient on the Number of Bidders

0 2

2

.05

: 0: 0

Rejection Region: 1.699a

HH

t t t

2

* 2

ˆ

ˆ 85.953Test statistic: 9.858.729

ts



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Test of the Parameter Coefficient on the Number of Bidders

0 2

2

.05

: 0: 0

Rejection Region: 1.699a

HH

t t t

2

* 2

ˆ

ˆ 85.953Test statistic: 9.858.729

ts

Since t* > t, reject the null hypothesis.



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A 100(1-)% Confidence Interval for a Parameter

ˆ/2ˆ ( )

where is based on ( 1) degrees of freedom and = number of observations

1 = number of parameters in the modelValid inferences about also require that the four assumptions abo

ii

i

t s

t n knk

ut are satisfied.



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A 100(1-)% Confidence Interval for 1

1

1

ˆ1 /2

ˆ1 .05

ˆ

ˆ

12.74 1.699(.905)12.74 1.54

t s

t s



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A 100(1-)% Confidence Interval for 1

1

1

ˆ1 /2

ˆ1 .05

ˆ

ˆ

12.74 1.699(.905)12.74 1.54

t s

t s

Holding the number of bidders constant, the result above tells us that we can be 90% sure that the auction price will rise between $11.20 and $14.28 for each 1-year increase in age.

12.3: Evaluating Overall Model Utility

Reject H 0 for i Evidence of a linear

relationship between y and xi

Do Not Reject H 0 for i There may be no

relationship between y and xi

Type II error occurred The relationship between

y and xi is more complex than a straight-line relationship


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The multiple coefficient of determination, R2, measures how much of the overall variation in y is explained by the least squares prediction equation.

2 Explained variability1Total variability

yy

yy yy

SS SSESSERSS SS


High values of R2 suggest a good model, but the usefulness of R2 falls as the number of observations becomes close to the number of parameters estimated.


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2 2

The Adjusted Multiple Coefficient of Determination

1 11 1 1( 1) ( 1)a

yy

n SSE nR Rn k SS n k

Ra2 adjusts for the number of observations

and the number of parameter estimates. It will always have a value no greater than R2.

0 1 2

2

2

The Analysis-of-Variance -Test: 0: At least one 0

( ) / /Test Statistic: / ( 1) (1 ) / ( 1)

Mean square (Model)Mean square (Error)

where is the sample size an

k

a i

yy

FHH

SS SSE k R kFSSE n k R n k

n

d is the number of terms in the model.Rejection region: , with numerator and ( 1) denominatordegrees of freedom.

kF F k n k



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0 1 2

2

2

The Analysis-of-Variance -Test: 0: At least one 0

( ) / /Test Statistic: / ( 1) (1 ) / ( 1)

Mean square (Model)Mean square (Error)

where is the sample size an

k

a i

yy

FHH

SS SSE k R kFSSE n k R n k

n

d is the number of terms in the model.Rejection region: , with numerator and ( 1) denominatordegrees of freedom.

kF F k n k

Rejecting the null hypothesis means that something in your model helps explain variations in y, but it may be that another model provides more reliable estimates and predictions.




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0 1 1 2 2

1

2

whereauction price in dollarsage of clock in yearsnumber of bidders

y x x

yxx

0 1 2: 0: At least one of the

two coefficients is nonzeroTest Statistic:

MS(Model) 2,141,531 120.19MSE 17,818

value: less than .00001

a

HH

F

p




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0 1 1 2 2

1

2

whereauction price in dollarsage of clock in yearsnumber of bidders

y x x

yxx

0 1 2: 0: At least one of the

two coefficients is nonzeroTest Statistic:

MS(Model) 2,141,531 120.19MSE 17,818

value: less than .00001

a

HH

F

p

Something in the model is useful, but the F-test can’t tell us which x-variables are individually useful.


Checking the Utility of a Multiple-Regression Model1. Use the F-test to conduct a test of the adequacy

of the overall model.2. Conduct t-tests on the “most important”

parameters. 3. Examine Ra

2 and 2s to evaluate how well the model fits the data.


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12.4: Using the Model for Estimation and Prediction

The model of antique clock prices can be used to predict sale prices for clocks of a certain age with a particular number of bidders.

What is the mean sale price for all 150-year-old clocks with 10 bidders?


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What is the mean auction sale price for a single 150-year-old clock with 10 bidders?


31

The average value of all clocks with these characteristics can be found by using the statistical software to generate a confidence interval. (See Figure 12.7)

In this case, the confidence interval indicates that we can be 95% sure that the average price of a single 150-year-old clock sold at auction with 10 bidders will be between $1,154.10 and $1,709.30.



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What is the mean sale price for a single 50-year-old clock with 2 bidders?


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What is the mean sale price for a single 50-year-old clock with 2 bidders?


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Since 50 years-of-age and 2 bidders are both outside of the range of values in our data set, any prediction using these values would be unreliable.

12.5: Model Building: Interaction Models

In some cases, the impact of an independent variable xi on y will depend on the value of some other independent variable xk.

Interaction models include the cross-products of independent variables as well as the first-order values.


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0 1 1 2 2 3 1 2

1 3 2

1 2

2

An Interaction Model Relating ( ) to Two Quantitative Independent Variables

( )where represents the change in ( ) for every one-unit change in holding fixedand

E y

E y x x x xx E y

x x

3 1

2 1

represents the change in ( ) for every one-unit change in holding fixed.

x E yx x



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In the antique clock auction example, assume the collector has reason to believe that the impact of age (x1) on price (y) varies with the number of bidders (x2) .

The model is nowy = 0 + 1x1 + 2x2 + 3x1x2 + .



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39



0 1 2 3

The Global -Test:

The test statistic is = 193.04-value = 0

Reject the null hypothesis

FH

Fp

0 3

The -Test on the Interaction Parameter: 0

The test statistic is = 6.11 (two-tailed)-value = 0 (= 0/2 = 0 for a one-tailed test)

Reject the null hypothesis

tH

tp



40



The MINITAB results are reported in Figure 12.11 in the text.

1 2 1 2

2

2

The Estimated Model is ˆ 320.5 0.878 ( 93.26) 1.2978To estimate the change in the price of 150-year-old clock given a one-unit change in , we must include the interction term.

Estimated s

y x x x x

x

x

2 3 1ˆ ˆlope 93.26 1.30(150) 101.74x



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Once the interaction term has passed the t-test, it is unnecessary to test the individual

independent variables.

12.6: Model Building: Quadratic and Other Higher Order Models

A quadratic (second-order) model includes the square of an independent variable:

y = 0 + 1x + 2x2 + .This allows more complex relationships to be modeled.


43


A quadratic (second-order) model includes the square of an independent variable:

y = 0 + 1x + 2x2 + .

1 is the shift parameter and

2 is the rate of curvature.


44



45

Example 12.7 considers whether home size (x) impacts electrical usage (y) in a positive but decreasing way.

The MINITAB results are shown in Figure 12.13.



46



47

According to the results, the equation that minimizes SSE for the 10 observations is

2

2

ˆ 1, 216.14 2.3989 .00045

.9767a

y x x

R



48



49

Since 0 is not in the range of the independent variable (a house of 0 ft2?), the estimated intercept is not meaningful.

The positive estimate on 1 indicates a positive relationship, although the slope is not constant (we’ve estimated a curve, not a straight line).

The negative value on 2 indicates the rate of increase in power usage declines for larger homes.



50

The Global F-Test H0: 1= 2= 0 Ha: At least one of the coefficients ≠ 0

The test statistic is F = 189.71, p-value near 0. Reject H0.



51

t-Test of 2 H0: 2= 0

Ha: 2< 0 The test statistic is t = -7.62, p-value = .0001

(two-tailed). The one-tailed test statistic is .0001/2 = .00005 Reject the null hypothesis.


McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building 52

Complete Second-Order Model with Two Quantitative Independent Variables

E(y) = 0 + 1x1 + 2x2 + 3x1x2 + 4x12 + 5x2

2

y-intercept

Changing 1 and 2 causes the surface to shift along the x1 and x2 axes

Controls the rotation of the surface

Signs and values of these parameters control the type of surface and the rates of curvature


McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building 53

12.7: Model Building: Qualitative (Dummy) Variable Models

Qualitative variables can be included in regression models through the use of dummy variables.

Assign a value of 0 (the base level) to one category and 1, 2, 3 … to the other categories.


54



55

A Qualitative Independent Variable with k Levels

where xi is the dummy variable for level i + 1 and0 1 1 2 2 1 1 k ky x x x

0 1

0 1 2

0 2 3

0 1

1 if is observed at level 10 otherwise

i

A B A

B C A

C D A

j j j j A

y ix



56

For the golf ball example from Chapter 10, there were four levels (the brands).Testing differences in brands can be done with the model

0 1 1 2 2 3 3

1 2 3

( )where

1 if Brand B 1 if Brand C 1 if Brand D, and 0 otherwise 0 otherwise 0 otherwise

E y x x x

x x x



57

Brand A is the base level, so 0 represents the mean distance (A) for Brand A, and

1 = B - A2 = C - A3 = D - A



58

Testing that the four means are equal is equivalent to testing the significance of the s:

H0: 1 = 2 = 3 = 0

Ha: At least of one the s ≠ 0



59

Testing that the four means are equal is equivalent to testing the significance of the s:H0: 1 = 2 = 3 = 0


The test statistic is the F-statistic.Here F = 43.99, p-value .000. Hence we reject the null hypothesis that the golf balls all have the same mean driving distance.



60

Testing that the four means are equal is equivalent to testing the significance of the s:H0: 1 = 2 = 3 = 0


The test statistic if the F-statistic.Here F = 43.99, p-value .000. Hence we reject the null hypothesis that the golf balls all have the same mean driving distance.

Remember that the maximum number of dummy variables is one less than the number of levels for the qualitative variable.

12.8: Model Building: Models with Both Quantitative and Qualitative Variables

Suppose a first-order model is used to evaluate the impact on mean monthly sales of expenditures in three advertising media: television, radio and newspaper. Expenditure, x1, is a quantitative variable Types of media, x2 and x3, are qualitative

variables (limited to k levels -1)McClave: Statistics, 11th ed. Chapter 12: Multiple

Regression and Model Building61



62

0 1 1 2 2 3 3 4 1 2 4 1 3

1

2

3

( )where x advertising expenditure

1 if radio 0 otherwise1 if television 0 otherwise

Newspaper is the base level.

E y x x x x x x x

x

x



63

0 1 1 2 2 3 3 4 1 2 4 1 3

Main effects, Main effects, Interactionadvertising type of mediumexpenditure

0 1 1

0 2

Intercept

( )

Newspaper medium line: ( )

Radio medium line: ( ) ( )

E y x x x x x x x

E y x

E y

1 4 1

Slope

0 3 1 5 1

Intercept Slope

( )

Television medium line: ( ) ( ) ( )

x

E y x


Suppose now a second-order model is used to evaluate the impact of expenditures in the three advertising media on sales.

The relationship between expenditures, x1, and sales, y, is assumed to be curvilinear.


64


In this model, each medium is assumed to have the save impact on sales.


65

20 1 1 2 1

1

( )where advertising expenditureE y x x

x



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20 1 1 2 1 3 2 4 3

1

2

3

( )where x advertising expenditure

1 if radio 0 otherwise1 if television 0 otherwise

Newspaper is the base level.

E y x x x x

x

x

In this model, theintercepts differ but the shapes of the curves are the same.



67

20 1 1 2 1 3 2 4 3

2 25 1 2 6 1 3 7 1 2 8 1 3

( )E y x x x x

x x x x x x x x

In this model, the response curve for each media type is different – that is, advertising expenditure and media type interact, at varying rates.

12.9: Model Building: Comparing Nested Models

Two models are nested if one model contains all the terms of the second model and at least one additional term. The more complex of the two models is called the complete model and the simpler of the two is called the reduced model.


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69

Recall the interaction model relating the auction price (y) of antique clocks to age (x1) and bidders (x2) :

0 1 1 2 2 3 1 2( ) .E y x x x x



70

If the relationship is not constant, a second-order model should be considered:

2 20 1 1 2 2 3 1 2 4 1 5 2

Reduced model

Complete model

( ) .E y x x x x x x



71

If the complete model produces a better fit, then the s on the quadratic terms should be significant.

H0: 4 = 5 = 0

Ha: At least one of 4 and 5 is non-zero

2 20 1 1 2 2 3 1 2 4 1 5 2

Reduced model

Complete model

( ) .E y x x x x x x



72

F-Test for Comparing Nested Models

0 1 1

0 1 1 1 1

0 1 2

0

Reduced model: ( )

Complete model: ( )

: 0

: At least one of the parameters in is nonzeroTest Statistic:

g g

g g g g k k

g g k

a

E y x x

E y x x x x

H

H H

F

0( ) / ( ) ( ) / # s in / [ ( 1)]

R C R C

C C

SSE SSE k g SSE SSE HSSE n k MSE



73

F-Test for Comparing Nested ModelswhereSSER = sum of squared errors for the reduced model

SSEC = sum of squared errors for the complete model

MSEC = mean square error (s2) for the complete model

k – g = number of parameters specified in H0

k + 1 = number of parameters in the complete model n = sample sizeRejection region: F > F, with k – g numerator and n – (k + 1)

denominator degrees of freedom.



74

The growth of carnations (y) is assumed to be a function of the temperature (x1) and the amount of fertilizer (x2).

The data are shown in Table 12.6 in the text.



75


The complete second order model is

The least squares prediction equation from Table 12.6 isrounded to

2 20 1 1 2 2 3 1 2 4 1 5 2

2 21 2 1 2 1 2

( )

ˆ 5,127.90 31.10 139.75 .146 .133 1.14

E y x x x x x x

y x x x x x x



76


To test the significance of the contribution of the interaction and second-order terms, useH0: 3 = 4 = 5 = 0

Ha: At least one of 3, 4 or 5 ≠ 0

This requires estimating the complete model in reduced form, dropping the parameters in the null hypothesis.Results are given in Figure 12.31.



77

0 3 4 5

0

0

: 0: At least one of the parameters in is nonzero

Test Statistic: ( ) / ( ) ( ) / # s in

/ [ ( 1)](6,671.50852 59.17832) / 3 782.15

2.81802Rejection region:

a

R C R C

C C

HH H

SSE SSE k g SSE SSE HFSSE n k MSE

F

F

.05 3.07



78

0 3 4 5

0

0

: 0: At least one of the parameters in is nonzero

Test Statistic: ( ) / ( ) ( ) / # s in

/ [ ( 1)](6,671.50852 59.17832) / 3 782.15

2.81802Rejection region:

a

R C R C

C C

HH H

SSE SSE k g SSE SSE HFSSE n k MSE

F

F

.05 3.07 Reject the null hypothesis: the complete model seems to provide better predictions than the reduced model.


A parsimonious model is a general linear model with a small number of parameters. In situations where two competing models have essentially the same predictive power (as determined by an F-test), choose the more parsimonious of the two.


79


A parsimonious model is a general linear model with a small number of parameters. In situations where two competing models have essentially the same predictive power (as determined by an F-test), choose the more parsimonious of the two.


80

If the models are not nested, the choice is more subjective, based on Ra

2, s, and an understanding of the theory behind the model.

12.10: Model Building: Stepwise Regression

It is often unclear which independent variables have a significant impact on y.

Screening variables in an attempt to identify the most important ones is known as stepwise regression.


81



82


Stepwise regression must be used with caution Many t-tests are conducted, leading to

high probabilities of Type I or Type II errors.

Usually, no interaction or higher-order terms are considered – and reality may not be that simple.


83

12.11: Residual Analysis: Checking the Regression Assumptions

Regression analysis is based on the four assumptions about the random error considered earlier.

1. The mean is equal to 0.2. The variance is equal to 2.3. The probability distribution is a normal

distribution.4. Random errors are independent of one another.


84



85

If these assumptions are not valid, the results of the regression estimation are called into question.

Checking the validity of the assumptions involves analyzing the residuals of the regression.



86

A regression residual is defined as the difference between an observed y-value and its corresponding predicted value:

0 1 1 2 2ˆ ˆ ˆ ˆˆ ˆ( ) ( )k ky y y x x x



87

Properties of the Regression Residuals1. The mean of the residuals is equal to 0.

2. The standard deviation of the residuals is equal to the standard deviations of the fitted regression model.

ˆ(Residuals) ( ) 0y y

2 2

2

ˆ(Residuals) ( ) 0

(Residuals)( 1) ( 1)

y y

SSEs MSEn k n k



88

If the model is misspecified, the mean of will not equal 0. Residual analysis may reveal this

problem. The home-size electricity usage example

illustrates this.


The plot of the first-order model shows a curvilinear residual pattern …

while the quadratic model shows a more random pattern.


89



90

A pattern in the residual plot may indicate a problem with the model.



91

A residual larger than 3s (in absolute value) is considered an outlier. Outliers will have an undue influence on

the estimates.1. Mistakenly recorded data2. An observation that is for some reason truly

different from the others3. Random chance



92

A residual larger than 3s (in absolute value) is considered an outlier. Leaving an outlier that should be

removed in the data set will produce misleading estimates and predictions (#1 & #2 above).

So will removing an outlier that actually belongs in the data set (#3 above).



93

Residual plots should be centered on 0 and within ±3s of 0.

Residual histograms should be relatively bell-shaped.

Residual normal probability plots should display straight lines.



94

Slight departures from normality will not seriously harm the validity of the estimates, but as the departure from normality grows, the validity falls.

REGRESSION ANALYSISREGRESSION ANALYSIS IS ROBUST WITH IS ROBUST WITH

RESPECT TO (SMALL) RESPECT TO (SMALL) NONNORMAL ERRORS.NONNORMAL ERRORS.



95

If the variance of changes as y changes, the constant variance assumption is violated.



96

A first-order model is used to relate the salaries (y) of social workers to years of experience (x).



97

1

0 1

2

( )ˆ 11,368.72 2141.38

.78713.31; value 0

E y xy x

Rt p



98

The model seems to provide good predictions, but the residual plot reveals a non-random pattern:

The residual increases as the estimated mean salary increases, violating the constant variance assumption

Transforming the dependent variable often stabilizes the residual Possible transformations of y Natural logarithm Square root sin-1y1/2



99



100



101



102



103



104



105

12.12: Some Pitfalls: Estimability, Multicollinearity and Extrapolation


106



107



108

Problem 1: Parameter Estimability

If x does not take on a sufficient number of different values, no single unique line can be estimated.



109

Problem 2: Multicollinearity

Multicollinearity exists when two or more of the independent variables in a regression are correlated.

If xi and xj move together in some way, finding the impact on y of a one-unit change in either of them holding the other constant will be difficult or impossible.



110

Problem 2: Multicollinearity

Multicollinearity can be detected in various ways.A simple check is to calculate the correlation coefficients (rij)for each pair of independent variables in the model. Any significant rij may indicate a multicollinearity problem.

If severe multicollinearity exists, the result may be1.Significant F-values but insignificant t-values 2.Signs on s opposite to those expected3.Errors in estimates, standard errors, etc.


The Federal Trade Commission (FTC) ranks cigarettes according to their tar (x1), nicotine (x2), weight in grams (x3) and carbon monoxide (y) content .

25 data points (see Table 12.11) are used to estimate the model


111

0 1 1 2 2 3 3( ) .E y x x x



112


F = 78.98, p-value < .0001 t1= 3.97, p-value = .0007

t2= -0.67, p-value = .5072

t3= -0.3, p-value = .9735


113

0 1 1 2 2 3 3

1 2 3

( )( ) 3.202 .963 ( 2.63) ( 0.13)

(See Figure 12.49)

E y x x xE y x x x


F = 78.98, p-value < .0001 t1= 3.97, p-value = .0007

t2= -0.67, p-value = .5072

t3= -0.3, p-value = .9735


114

0 1 1 2 2 3 3

1 2 3

( )( ) 3.202 .963 ( 2.63) ( 0.13)

(See Figure 12.49)

E y x x xE y x x x

The negative signs on two variables and the insignificant t-values are suggestive of multicollinearity .


The coefficients of correlation, rij, provide further evidence: rtar, nicotine = .9766 rtar, weight = .4908 rweight, nicotine = .5002

Each rij is significantly different from 0 at the = .05 level.


115



116

Possible Responses to Problems Created by Multicollinearity in Regression Drop one or more correlated independent

variables from the model. If all the xs are retained,

Avoid making inferences about the individual parameters from the t-tests.

Restrict inferences about E(y) and future y values to values of the xs that fall within the range of the sample data.



117

Problem 3: Extrapolation

The data used to estimate the model provide information only on the range of values in the data set. There is no reason to assume that the dependent variable’s response will be the same over a different range of values.



118

Problem 3: Extrapolation



119

Problem 4: Correlated Errors

If the error terms are not independent (a frequent problem in time series), the model tests and prediction intervals are invalid. Special techniques are used to deal with time series models.

chapter 12: multiple regression and model building

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