Chapter 3_Regression Analysis1

8/8/2019 Chapter 3_Regression Analysis1

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³Regression is a statistical

technique which establish afunctional relationship between

two or more variables in theform of an equation to

estimate the value of one

variable based on the value of another variable´



Regression Analysis

Simple Linear Regression Model

y = F0 + F1 x + I

Simple Linear Regression Equationy = F0 + F1 x

Estimated Simple Linear Regression Equation

x b byÖ 10 !



Principle of least squares technique



Case 1:

Observed points : (4,8); (8,1); (12,6)

Estimated points : (4,6); (8,5); (12,4)

Observed points : (4,8); (8,1); (12,6)

Estimated points : (4,2); (8,5); (12,8)



Error (graph 1) Error (graph 2)8-6=-2 8-2=6

1-5=-4 1-5=-4

6-4=2 6-8=-2

Total error=0 Total error=0



Absolute error Absolute error

I8-6I=2 I8-2I=6

I1-5I=4 I1-5I=4

I6-4I=2 I6-8I=2

Total Absolute error=8 Total Abs error=12



Case 2:

Observed points: (2,4); (6,7); (10,2)

Estimated points: (2,4); (6,3); (10,2)

Observed points: (2,4); (6,7); (10,2)

Estimated points: (2,5); (6,4); (10,3)



Abs Error Abs Error

I4-4I=0 I4-5I=1I7-3I=4 I7-4I=3

I2-2I=0 I2-3I=1

Total Abs error=4 Total Abs error=5



Error Square ErrorSquare

(4-4)2 =0 (4-5) 2=1(7-3) 2=16 (7-4) 2=9

(2-2) 2=0 (2-3) 2=1

Sum of error square=16 (Graph 1)

Sum of error square=11 (Graph 2)



Least Squares Method

Least Squares Criterion

where:

y i = observed value of the dependent variable

for the i th observation

? A§ 2)Ö(min

iiy y

nobservatioithfor the

variabledependenttheof valueestimatedyÖ i !



Slope for the Estimated Regression Equation

x = value of independent variable for i th observation

y = value of dependent variable for i th observation

n = total number of observations

y -Intercept for the Estimated Regression Equation

221

§§

§ §§

! x xn

y x x yn

b

xb yb 10 !

variabledependentor mean valuey

t variableindependenor mean valuex

!

!



Simple Linear Regression

Reed Auto periodically has a special week-longsale. As part of the advertising campaign Reedruns one or more television commercials duringthe weekend preceding the sale. Data from a

sample of 5 previous sales are shown below.

Number of TV Ads Number of Cars Sold

1 143 24

2 18

1 17

3 27



The HRD manager of a company wants to find a

measure which he can use to fix the monthly

income of persons applying for a job in the

production department. As an experimental

project, he collected data on 7 persons from that

department referring to years of service and their

monthly income (in 000¶s).

Years of

experience 11 7 9 5 8 6 10Income 10 8 6 5 9 7 11



Find the regression equation of income onyears of service.

What initial start would you recommend for a person applying for the job after having

served in a similar capacity in another company for 13 years?

Do you think other factors are to beconsidered (in addition to the years of service) in fixing the income? Explain.



Properties of regression lines andtheir coefficients:

1. Correlation coefficient is thegeometric mean between theregression coefficient

2. The sign of correlation coefficient isthe same as that of regressioncoefficient.

3. Regression coefficients aredependent of the change origin but

not of scale.



In partially destroyed laboratory recordof an analysis of correlation data, the

following results only are available.Variance of X is 9Regression equations :

8x-10y+66=040x-18y=214Find

1. The mean values of x and y

2. The correlation coefficientbetween x and y

3. The standard deviation of y



In finance, it is of interest to look at the relationship

between Y, a stock¶s average return, and X, the

overall market return. The slope coefficient computed

by linear regression is called the stock¶s beta by

investment analysts. A beta greater than 1 indicates

that the stock is relatively sensitive to changes in the

market; a beta less than 1 indicates that the stock is

relatively insensitive. For the following data, compute

the beta and suggest market trend.

X(%)

10 12 8 15 9 11 8 10 13 11

Y

(%)11 15 3 18 10 12 6 7 18 13



Multiple regression Analysis



A linear regression equation with morethan one independent variable is called a

mul t ipl e regression mod el.



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where

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k

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y

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!



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minimizeswhichxb.......xbxbbybeequationregressionlinearfittedtheLet

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y

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Let us consider the case where two

independent variables and a dependent

variable.



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A marketing manager of a company wants

to predict demand for the product. He isbelieving strongly demand is highly

influenced by annual average price of the

product (in units) & advertising

expenditure (Rs in lakh).He has collectedpast data to know the effect of these

factors on demand and given below:

Y 4 6 7 9 13 15

X1 15 12 8 6 4 3

X2 30 24 20 14 10 4



cc.52.5isvolumeandmm58islengthwhose

egganof weighttheestimateHencevolume.andlengthitson

weighteggof equationregressionlineartheObtain

0.974r 0.581r 0.578r

4.41 4.39 2.26

56.03y 51.48x 55.95x

eggs.300of gm)(inweightand

cc)(involumemm),(inlengthontmeasuremen

fromobtainedareresultsfollowingThe

12y2y1

y21

21

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The Federal Reserve is performing a

preliminary study to determine the

relationship between certain economic

indicators and annual percentage change

in the gross national product (GNP). Two

such indicators being examined are theamount of the federal government¶s deficit

(in billions of dollars) and the Dow Jones

Industrial Average (the mean value over

the year). Data for 6 years follow:



Change in GNP 2.5 -1.0 4.0 1.0 1.5 3.0

Federal Deficit 100.0 400.0 120.0 200.0 180.0 80.0Dow Jones 2850 2100 3300 2400 2550 2700

i) Calculate the least squares equation that bestdescribes the data.

ii) What % change in GNP would be expected in a year in which the federal deficit was $240 billion and themean Dow Jones value was 3000?



Multiple correlation analysis:

It is a measure of association

between a dependent variable and severalindependent variables taken together.



The coefficient of multiple correlation is given

by,

1.a0et eeilieal aysval eItsr1

rr2rrr2

12

12y2y1

2

y2

2

y1

y.12

!



Coefficient of multiple determination:

It is the proportion of the total variation

in the multiple values of dependent

variable y, accounted for or explained bythe independent variables in the multiple

regression model.

The square of coefficient of multiple

correlation is called Coefficient of multiple

determination.

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