II

Chapter 15Chapter 15

Regression Analysis IRegression Analysis I

0

ObjectivesObjectives

(linear regression);(correlation)(linearregression);(correlation)

:

(l h d) ( ) ( l )(least squares method) (intercept) (slope)

( t d d f ti t ) (standarderrorofestimate)

( ffi i t f d t i ti ) (coefficientofdetermination)

1

IntroductionIntroduction

(dependent variable) vs. (independent variable) (dependentvariable)vs. (independentvariable)

_______:

_______:_______

vs.

2

ExampleExample

3

?

ExampleExample

=0.762

4

Ex) =17,000+500* + ;ExcelGraph

Regression ModelRegression Model

(probabilistic model) (probabilisticmodel)

,(randomness)

, ( +)

(deterministicmodel)

:

:

y=$100,000+(100$/ft2)(x);x

:

y=$100,000+(100$/ft2)(x)+ e;

e (randomness)

5

e (randomness)

Regression ModelRegression Model

contd cont d

Lower vs. HigherVariability

HousePrice

Variability

100K$

House sizex

House Price = 100,000 + 100(Size) +

6

House size

Regression ModelRegression Model

(simplelinearregressionmodel)

(multiplelinearregressionmodel)

, (logistic)

7

Regression ModelRegression Model

y- y

rise

run=slope (=rise/run)

x

=y-intercept

8

Regression ModelRegression Model

0 1 1 2 2 3 3Y X X X = + + + +2

0 1 1 2 2

logY X X

Y X X

= + + +

+ + +0 1 1 2 2logY X X = + + +

9

EstimationEstimation

Y X Y X + + +

(LeastSquareMethod)

0 1 1 0 1 1 Y X Y X = + + = +

(residual) =

10

EstimationEstimation

(Least Squares Line) (LeastSquaresLine)

n

0 1

20 1

1,

( ( ))minn

i ii

Y X

=

+

( )( )X X Y Y 1 0 12 2

( )( ) , ( )i i i i

i i

X X Y Y x yY X

X X x

= = =

, ,i i i ix X X y Y Y= = 2

1 , xy

x

ss

=

112

2( )( ) ( ), ,

1 1i i i

xy x

X X Y Y X Xs s

n n

= =

Exercise 1Exercise 1

1 1

6

x 1 2 3 4 5 6 ($1000) y 6 1 9 5 17 12

(1)

( ) y

(2)

(3) 4 5 (3) 4.5,?

(4)?

12

(4)?

Exercise 1Exercise 1

Example 15 1Example 15.1

(residual)

13

Exercise 1Exercise 1

1.:

2.

3. regression()

15.2

14

Exercise 2Exercise 2

15 02; Xm15 02: Toyota Camry Part I 1502;Xm1502:ToyotaCamry,PartI

1:

2:Excel

15

Exercise 2Exercise 2

Excel Excel

Xm1502

Y(A1:A101),X (B1:B101)

(line fit plot)(linefitplot)

16

Exercise 2Exercise 2

17

Test on the Regression CoefficientsTest on the Regression Coefficients

, ,.

!!

(standard error of estimate)

( ffi i t f d t i ti ) (coefficient of determination)

( f h l ) t- (t-test of the slope)

(sumofsquaresforerror)SSE

18

Test on the Regression CoefficientsTest on the Regression Coefficients

0 1 1 0 1 1 Y X Y X = + + = +

Step1::0 1

1 1

: 0: 0

HH

=

Step2::

Step3:

1

p

11

0 0 02 ( ) 2 2a ap value P a P P ts s s

= > = > = >

, t n-2

1 1 1

19

R-square (R2)R-square (R2)

(sum of squares for errors) (sumofsquaresforerrors)

2 2( )n n

i i iSSE e Y Y= = SSE?

1 1i i= =

()2R2( )

n

i iY Y2 21

2

( )1 1 , 0 1

( )

i iin

i i

Y YSSER RTSS Y Y

== =

1( )i i

iY Y

=

20

!2R

Exercise 2Exercise 2

Xm 1502; 152; Part II:Xm15 02; 15 2; PartII:

0.805167979 0.648295475 0.644706653 0 326488626 0.326488626 100

F F 1 19.25560737 19.25561 180.643 5.75E-24 98 10.44629263 0.106595 99 29 7019 99 29.7019

t P- 95% 95% 95.0% 95.0%Y 17.24872734 0.182092574 94.72505 3.57E-98 16.88737 17.61008 16.88737 17.61008

21

Odometer -0.066860885 0.004974639 -13.44035 5.75E-24 -0.076733 -0.056989 -0.076733 -0.056989

Exercise 2Exercise 2

?:?:

15.4&15.5

H1: 1 0, H0: 1 = 0 ,

5% 5%

1 2

.3265 .00497(99)(43.509)( 1)

eb

x

ssn s

= = =

1

1 13.46b

bts

= =

22

Exercise 2Exercise 2

p value13.49 1.984 0 p.

p-value

Compare

Variationiny=SSE+SSR

23

Exercise 2Exercise 2

24

Residual AnalysisResidual Analysis

,

0,

(residual)(residual)

(1)

25

Residual AnalysisResidual Analysis

(2)

(3)(Noautocorrelation)

26

Residual AnalysisResidual Analysis

27

Exercise 3Exercise 3

15 49 1549

S&P500

R X = + +0 1i i i MR X + +

(1) ( )

(2)

(3)

28

DiscussionDiscussion

(1)

(2)

(3)

(4)(outlier)

(5)

(6)

/ / / /

(7)

29