Econ 3780: Business and Economics Statisticsyuppal.people.ysu.edu › econ_3790 ›...

Econ 3790: Statistics Business and Economics

Instructor: Yogesh UppalEmail: [email protected]

Chapter 14

Covariance and Simple Correlation Coefficient

Simple Linear Regression

Covariance

Covariance between x and y is a measure of relationship between x and y.

1))((

1),cov(

n

xxyynSS

yx xy

Reed Auto periodically hasa special week-long sale. As part of the advertisingcampaign Reed runs one ormore television commercialsduring the weekend preceding the sale. Data from asample of 5 previous sales are shown on the next slide.

CovarianceCovariance

Example: Reed Auto SalesExample: Reed Auto Sales

CovarianceCovariance


Number ofNumber ofTV AdsTV Ads

Number ofNumber ofCars SoldCars Sold

1133221133

14142424181817172727

Covariancex y

1 14 -1 -6 6

3 24 1 4 4

2 18 0 -2 0

1 17 -1 -3 3

3 27 1 7 7

Total=10Total =

100 SSxy =20

xx yy ))(( yyxx

54

201

),cov(

nSS

yx xy

Simple Correlation Coefficient

11

),cov(

yx

yx• Simple Population Correlation Coefficient

• If

< 0, a negative relationship between x and y.

• If

> 0, a positive relationship between x and y.


Since population standard deviations of x and y are not known, we use their sample estimates to compute an estimate of .

11

),cov(

r

ssyxr

yx


x y SSx SSy

1 14 -1 -6 1 36

3 24 1 4 1 16

2 18 0 -2 0 4

1 17 -1 -3 1 9

3 27 1 7 1 49

Total=10 Total=98 Total=4 Total= 114

xx yy



936.034.5*1

5),cov(

34.54

1141)(

144

1)(

2

2

yxxy

y

x

ssyxr

nyys

nxxs

Chapter 14 Simple Linear Regression

Simple Linear Regression Model

Coefficient of Determination

Testing for Significance

Using the Estimated Regression Equationfor Estimation and Prediction

Residual Analysis

Simple Linear Regression Model

yy

= = 00

+ + 11

xx

++

where:where:00

and and 11

are called are called parameters of the modelparameters of the model,, is a random variable called theis a random variable called the

error termerror term..

The The simple linear regression modelsimple linear regression model

is:is:

The equation that describes how The equation that describes how yy

is related to is related to xx

andandan error term is called the an error term is called the regression modelregression model..

Simple Linear Regression Equation

Positive Linear RelationshipPositive Linear Relationship

yy

xxx

Slope Slope 11is positiveis positive

Regression lineRegression line

InterceptIntercept00

Simple Linear Regression EquationSimple Linear Regression Equation

Negative Linear RelationshipNegative Linear Relationship

yyy

xxx

Slope Slope 11is negativeis negative

Regression lineRegression lineInterceptIntercept00

Simple Linear Regression EquationSimple Linear Regression Equation

No RelationshipNo Relationship

yyy

xxx

Slope Slope 11is 0is 0

Regression lineRegression lineInterceptIntercept

00

Interpretation of 0

and 1

0 (intercept parameter): is the value of y when x = 0.

1 (slope parameter): is the change in y given x changes by 1 unit.

Estimated Simple Linear Regression EquationEstimated Simple Linear Regression Equation

The The estimated simple linear regression equationestimated simple linear regression equation

0 1y b b x 0 1y b b x

•• bb11

is the slope of the line.is the slope of the line.•• bb00

is the is the yy

intercept of the line.intercept of the line.•• The graph is called the estimated regression line.The graph is called the estimated regression line.

•• is the estimated value of y for a given value of x.y

Estimation Process

Regression ModelRegression Modelyy

= = 00

+ + 11

xx

++Regression EquationRegression Equation

EE((y|xy|x) = ) = 00

+ + 11

xxUnknown ParametersUnknown Parameters

00

, , 11

Sample Data:Sample Data:x yx yxx11

yy11. .. .. .. .

xxnn

yynn

EstimatedEstimatedRegression EquationRegression Equation

bb00

and band b11provide provide point estimatespoint estimates

ofof

00 and and

11

xbby 10ˆ

Least Squares Method

Slope for the Estimated Regression Equation

2

1

)(

))((

xxSSand

xxyySSWhere

SSSS

b

x

xy

y

xy

yy--Intercept for the Estimated Regression EquationIntercept for the Estimated Regression Equation

Least Squares MethodLeast Squares Method

0 1b y b x 0 1b y b x

__yy

= mean value for dependent variable= mean value for dependent variable

__xx

= mean value for independent variable= mean value for independent variable

Estimated Regression Equation

54

201

x

xy

SSSS

b

10)2(*52010 xbyb

Slope for the Estimated Regression Equation

y-Intercept for the Estimated Regression Equation

Estimated Regression Equation


xy *510ˆ

Scatter Diagram and Regression Line

xy 510ˆ

number of ads

3.53.02.52.01.51.0.5

num

ber o

f car

s so

ld

26

24

22

20

18

16

14

12

Estimate of Residuals

x y1 14 15 -1.03 24 25 -1.02 18 20 -2.01 17 15 2.03 27 25 2.0

y yye ˆ

Decomposition of total sum of squares

Relationship Among SST, SSR, SSE

where:where:SST = total sum of squaresSST = total sum of squaresSSR = sum of squares due to regressionSSR = sum of squares due to regressionSSE = sum of squares due to errorSSE = sum of squares due to error

SST = SSR + SSESST = SSR + SSE

2( )iy y 2( )iy y 2ˆ( )iy y 2ˆ( )iy y 2ˆ( )i iy y 2ˆ( )i iy y

-1 1 15 -5 25

-1 1 25 5 25

-2 4 20 0 0

2 4 15 -5 25

2 4 25 5 25SSE=14 SSR=100

yye ˆ 2)ˆ( yySSE yy ˆ 2)ˆ( yySSR

Decomposition of total sum of squares

Check if SST= SSR + SSE

y

The The coefficient of determinationcoefficient of determination

is:is:

Coefficient of DeterminationCoefficient of Determination

rr22

= SSR/SST= SSR/SST

rr22

= SSR/SST = 100/114 = 0.8772= SSR/SST = 100/114 = 0.8772

•• The regression relationship is very strong; about 88%The regression relationship is very strong; about 88%of the variability in the number of cars sold can beof the variability in the number of cars sold can beexplained by the number of TV ads.explained by the number of TV ads.

•• The coefficient of determination (rThe coefficient of determination (r22) is also the square of ) is also the square of the correlation coefficient (r).the correlation coefficient (r).

Sample Correlation Coefficient

21 ) of(sign rbr 21 ) of(sign rbr

ionDeterminat oft Coefficien ) of(sign 1br ionDeterminat oft Coefficien ) of(sign 1br

936.08772.0 r 936.08772.0 r

Sampling Distribution of b1

Estimate of σ2

The mean square error (MSE) provides the The mean square error (MSE) provides the estimate of estimate of σσ22..

ss

22

= MSE = SSE/(= MSE = SSE/(n n

2)2)

where:where:2)ˆ(SSE ii yy 2)ˆ(SSE ii yy

Interval Estimate of 1

:

Example: Reed Auto SalesReed Auto Sales

44.3508.1182.35

Testing for Significance: t Test

Hypotheses

Test Statistic

Where b1 is the slope estimate and SE(b1 ) is the standard error of b1 .

0 1: 0H 0 1: 0H

1: 0aH 1: 0aH

)(0

1

1

bSEbt

Rejection RuleRejection Rule

Testing for Significance: Testing for Significance: tt

TestTest

where: where: tt

is based on a is based on a tt

distributiondistributionwith with nn

--

2 degrees of freedom2 degrees of freedom

Reject Reject HH00

if if pp--value value <<

or or tt

<<

--tt

or or tt

>>

tt

1. Determine the hypotheses.1. Determine the hypotheses.

2. Specify the level of significance.2. Specify the level of significance.

3. Select the test statistic.3. Select the test statistic.

= .05= .05

4. State the rejection rule.4. State the rejection rule. Reject Reject HH00

if if pp--value value <<

.05.05or t or t ≤≤

3.182 or t 3.182 or t ≥≥

3.1823.182


TestTest

0 1: 0H 0 1: 0H

1: 0aH 1: 0aH

)( 1

1

bSEbt


TestTest

5. Compute the value of the test statistic.5. Compute the value of the test statistic.

6. Determine whether to reject 6. Determine whether to reject HH00

..

tt

= 4.63 > t= 4.63 > t/2/2

= 3.182. We can reject = 3.182. We can reject HH00

..

63.408.15

)( 1

1 bSE

bt

Some Cautions about the Interpretation of Significance Tests

Just because we are able to reject Just because we are able to reject HH00

: : 11

= 0 and= 0 anddemonstrate statistical significance does not enabledemonstrate statistical significance does not enableus to conclude that there is a us to conclude that there is a linear relationshiplinear relationshipbetween between xx

and and yy..

Rejecting Rejecting HH00

: : 11

= 0 and concluding that the= 0 and concluding that therelationship between relationship between xx

and and yy

is significant does is significant does

not enable us to conclude that a not enable us to conclude that a causecause--andand--effecteffectrelationshiprelationship

is present between is present between xx

and and yy..

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