Multiple Regression Analysis

Multiple Regression Model

Sections 16.1 - 16.6

The Model and Assumptions

If we can predict the value of a variable on the basis of one explanatory variable, we might make a better prediction with two or more explanatory variables Expect to reduce the chance component of our

model Hope to reduce the standard error of the estimate Expect to eliminate bias that may result if we

ignore a variable that substantially affects the dependent variable

The Model and Assumptions

The multiple regression model is

where yi is the dependent variable for the ith observation 0 is the Y intercept 1,.. ,k are the population partial regression coefficients x1i, x2i,…xki are the observed values of the independent variables,

X1, X2….Xk.

k = 1,2,3…K explanatory variables

ikikiiii xxxy ....22110

The Model and Assumptions

The assumptions of the model are the same as those discussed for simple regression

The expected value of Y for the given Xs is a linear function of the Xs

The standard deviation of the Y terms for given X values is a constant, designated as y|x

The observations, yi, are statistically independent The distribution of the Y values (error terms) is normal

kikiiii xxxyE ....)( 22110

Interpreting the Partial Regression Coefficients For each X term there is a partial regression

coefficient, k This coefficient measures the change in the

E(Y) given a one unit change in the explanatory variable Xk, holding the remaining explanatory variables

constant controlling for the remaining explanatory variables ceteris parabis Equivalent to a partial derivative in calculus

Method of Least Squares - OLS To estimate the population regression

equation, we use the method of least squares The model written in terms of the sample

notation is

The sample regression equation is

ikikiii exbxbxbby ...22110

kikiii xbxbxbby ...ˆ 22110

Method of Least Squares - OLS Goal is to minimize the distance between the

predicted values of Y, the , and the observed values, yi, that is, minimize the residual, ei

Minimize

iy

222110

22 )...()ˆ( kikiiiiii xbxbxbbySSEyye

Method of Least Squares - OLS Take partial derivatives of SSE with respect to each of the partial

regression coefficients and the intercept Each equation is set equal to zero

This gives us k+1 equations in k+1 unknowns The equations must be independent and non-homogeneous Using matrix algebra or a computer, this system of equations can

be solved With a single explanatory variable, the fitted model is a straight

line With two explanatory variables, the model represents a plane in a

three dimensional space With three or more variables it becomes a hyperplane in higher

dimensional space The sample regression equation is correctly called a regression

surface, but we will call it a regression line

An Example: The Human Capital Model Consider education as an investment in human capital There should be a return on this investment in terms of higher future

earnings Most people accept that earnings tend to rise with schooling levels, but

this knowledge by itself does not imply that individuals should go on for more schooling

More is usually costly Direct payments (tuition) Indirect payments (foregone earnings)

Thus the actual magnitude of the increased earnings with additional years of schooling is important

Can not simply calculate the average earnings for a sample of workers with different education levels Have to consider the effects on earnings of other factors, for example,

experience in the labor market, age, ability, race and sex

An Example: The Human Capital Model

Consider a first simple model (1) Earnings = 0 + 1education + Expect that the coefficient on education will be positive, 1 > 0

Realize that most people have higher earnings as they age, regardless of their education If age and education are positively correlated, the estimated

regression coefficient on education will overstate the marginal impact of education

A better model would account for the effect of age (2) Earnings = 0 + 1education +2age +

A Conceptual Experiment

Multiple regression involves a conceptual experiment that we might not be able to carry out in practice

What we would like to do is to compare individuals with different education levels who are the same age We would then be able to see the effects of

education on average earnings, while controlling for age

Current Population Survey, White Males, March 1991

What is the affect of an additional year of education?

$31,523.24 - 27,970.59 = $3,552.65

All workers are 40 years old n

Average Annual Earnings

Educ = 12 227 $27,970.59

Educ = 13 132 $31,523.24

A Conceptual Experiment

Frequently we do not have large enough data sets to be able to ask this type of question

Multiple regression analysis allows us to perform the conceptual exercise of comparing individuals with the same age and different education levels, even if the sample contains no such pairs of individuals

Sample Data

Data was obtained for the March 1992 Current Population Survey The CPS is the source of the official Government statistics on employment

and unemployment A very important secondary purpose is to collect information such as age,

sex, race, education, income and previous work experience. The survey has been conducted monthly for over 50 years About 57,000 households are interviewed monthly, containing approximately

114,500 persons 15 years and older; based on the civilian non-institutional population

For multiple regression question, sample consists of white male respondents 18-65 years old, who spent at least one week in the labor force in the preceding year and who provided information on wage earnings during the preceding year.

Sample size is 30,040 Students download Multiple Regression Human Capital Hand-out

Sample Statistics

age earn educ

Mean 37.50 27561.92 13.02

Standard Error 0.070 119.610 0.017

Median 36 24000 13

Mode 35 30000 12

Standard Deviation 12.19 20730.89 2.92

Sample Variance 148.54 429769891.23 8.54

Minimum 18 2 0

Maximum 65 199998 20

Count 30040 30040 30040

In 1991, the average white male in the sample was 37.5 years old, had 13.0 years of education and earned $27,561.92.

Correlation Matrix

Second, consider the correlation matrix, which shows the simple correlation coefficients for all pairs of variables

There is a small, but positive correlation between education and age A simple regression of earnings on education will overstate the

effect of education because education is positively correlated with age and age has a strong positive effect on earnings

  age earn educ

age 1

earn 0.365051 1

educ 0.072856 0.413496 1

Earnings = 0 + 1education +

b0 =b1 =

Sb0 =

Sb1 =

= Se

educnear 78.293339.10622ˆ Multiple R 0.41349589R Square 0.170978851Adjusted R Square 0.170951252Standard Error 18875.91561Observations 30040

ANOVAdf SS MS F Significance F

Regression 1 2.21E+12 2.21E+12 6195.092532 0Residual 30038 1.07E+13 356300190.2Total 30039 1.29E+13

Coefficients Standard Error t Stat P-valueIntercept -10622.38757 497.2073501 -21.36410005 1.62E-100educ 2933.783915 37.27384753 78.70891012 0

Regression Statistics

Is Education a Significant Explanatory Variable? Use t-test

H0: 1 ≤ 0 No relationship

H1: 1> 0 Positive relationship

t-test statistic = 78.709 and the p-value is 0.000

Reject the H0: 1 ≤ 0 There is a significant positive relationship between

education and earnings

Additional Information from the Analysis For each additional year of schooling,

average earnings increase by $2,933.78 The R2 = .1710

Find that 17.1% of the variation in earnings across workers is explained by variation in education levels

The standard error of the estimate, Se equals $18,876

Earnings = 0 + 1education +2age +

ageeducnear 74.57273.275904.29834ˆ

b1 =

b2 =

b0 =

Multiple R 0.532684023R Square 0.283752268Adjusted R Square 0.283704577Standard Error 17545.43262Observations 30040

ANOVAdf SS MS F Significance F

Regression 2 3.66E+12 1.83E+12 5949.803748 0Residual 30037 9.25E+12 307842206Total 30039 1.29E+13

Coefficients Standard Error t Stat P-valueIntercept -29834.04464 540.0326891 -55.24488654 0educ 2759.729904 34.73889338 79.44207877 0age 572.7382481 8.328296328 68.77015725 0

Regression Statistics

Sb0 =

Sb1 =

Sb2 =

=Se

Interpret the Coefficients

In terms of this problem

For each additional year of schooling, average earnings increase by $2,759.73, controlling for age

For each additional year of age, average earnings increase by $572.74, controlling for schooling

ageeducnear 74.57273.275904.29834ˆ

Prediction

Predict the mean earnings for white male workers who are 30 old and have a college degree

The standard error of the estimate, Se = $17,545

where k = no. of explanatory variables

84.503,31)30(74.572)16(73.275904.29834ˆ near

1

)ˆ( 2

kn

yyS ie

Assessing the Regression as a Whole

Want to assess the performance of the model as a whole H0: 1 = 2 = 3 = …= k = 0

The model has no worth

H1: At least one regression coefficient is not equal to zero The model has worth

If all the b’s are close to zero, then the SSR will approach zero 0

SSE

SSR

Assessing the Regression as a Whole Test Statistic

where k = the number of explanatory variables If the null hypothesis is true, the calculated

test statistic will be close to zero; if the null hypothesis is false, the F test statistic will be “large”

)1(1,

knSSE

kSSR

F knk

Assessing the Regression as a Whole The calculated F test

statistic is compared with the critical F to determine whether the null hypothesis should be rejected If Fk,n-k-1 > F,k,n-k-1 (cv)

reject the H0

⍺

cv F

reject

ANOVA Table in Regression

Multiple R 0.532684023R Square 0.283752268Adjusted R Square 0.283704577Standard Error 17545.43262Observations 30040

ANOVAdf SS MS F Significance F

Regression 2 3.66E+12 1.83E+12 5949.803748 0Residual 30037 9.25E+12 307842206Total 30039 1.29E+13

Coefficients Standard Error t Stat P-valueIntercept -29834.04464 540.0326891 -55.24488654 0educ 2759.729904 34.73889338 79.44207877 0age 572.7382481 8.328296328 68.77015725 0

Regression Statistics

)1(1,

knSSE

kSSR

F knk

SSRSSE

Finally note the p-value, written as Significance F, which equals 0.0000. This tells us that we have a zero probability of observing a test statistic as large as

5,949.8 if the null hypothesis is true. The model has worth.

P-value

Inferences Concerning the Population Regression Coefficients Which explanatory variables have

coefficients significantly different from zero? Perform a hypothesis test for each

explanatory variable Essentially the same t-test used for simple

regression Hypotheses

H0: k = 0

H1: k 0

Inferences Concerning the Population Regression Coefficients The test statistic is

where K = number of independent variables The denominator, , is the standard error of the

regression coefficient, bk Take the standard errors of the regression coefficients

from the computer output

kb

kKn S

bt

01

kbS

Inferences Concerning the Population Regression Coefficients In our model, there are

two explanatory variables There will be two tests

about population regression coefficients

Test whether Education is a significant variable H0: educ ≤ 0

H1: educ > 0

Test whether Age is a significant variable H0: age ≤ 0

H1: age > 0

Let ⍺ = 0.01 t,.01 = 2.326 from the t

tables

T-test

Test statistic: educ Test statistic: age

442.797389.34

73.275930040 t

738.34educbS

77.68328.8

74.57230040 t

328.8agebS

Reject the null hypothesis, one tail test, = .01. Find that education is significantly and positively related to earnings.

Again, we reject the null hypothesis and conclude that age is significantly and positively related to earnings.

Coefficients Standard Error t Stat P-valueIntercept -29834.04464 540.0326891 -55.24488654 0age 572.7382481 8.328296328 68.77015725 0Educ 2759.729904 34.73889338 79.44207877 0

p-values < 0.01

The Coefficient of Determination and the Adjusted R2 The R2 value is still defined as the ratio of the SSR to the SST We see that 28.38% of the variation in earnings is explained by

variation in education and in age The simple regression has an R2 = 0.1710

Appears that adding the new explanatory variable improved the “goodness of fit”

This conclusion can be misleading As we add new explanatory variables to our model, the R2 always

increases, even when the new explanatory variables are not significant

The SSE always decreases as more explanatory variables are added This is a mathematical property and doesn’t depend on the relevance of

the additional variables

The Coefficient of Determination and the Adjusted R2 If we take into account the degrees of freedom SSE/(n-k-1) can

increase or decrease Depending on whether the additional variables are significant explanatory

variables or not Adjust the R2 statistic as follows:

Adjusted R2 can increase if the additional explanatory variables are important Can decrease if the additional explanatory variables are not significant

When comparing regression models with different numbers of explanatory variables, you should compare the adjusted R2 to decide which is the best model

The adjusted R2 1, but can take on a value less than zero if the model is very poor

)1(

)1(1

)1(

)1(12

knSST

nSSE

nSST

knSSEadjR

Online Homework - Chapter 16 Multiple Regression CengageNOW sixteenth assignment