Regression Co Relation

7/28/2019 Regression Co Relation

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Quantitative Methods

Regression & Correlation



• Contents.

• 1. Introduction: Example

• 2. A Simple Linear probabilistic model• 3. Least squares prediction equation

• 4. . Predicting y for a given x

• 7. Coefficient of determination• 8. Analysis of Variance

• 9. Computer Printouts

Piyoosh Bajoria 2



• Example.( Ad Sales) Consider the problem of predicting the gross

monthly sales volume y for a corporation that is not subject to substantial seasonal variation in its sales volume.For the predictor variable x we use thethe amount spent by the company on advertising during the month of interest.

We wish to determine whether advertising is worthwhile,that is whether

advertising is actually related to the firm’s sales volume. In addition we wishto use the amount spent on advertising to predict the sales volume. The data

in the table below represent a sample of advertising expenditures , x , and theassociated sales volume, y, for 10 randomly selected mont hs.

• Ad Sales Data

• Month y(yRs10,000) x(xRs10,000)

• 1 101 1.2

• 2 92 0.8

• 3 110 1.0

• 4 120 1.3

• 5 90 0.7• 6 82 0.8

• 7 93 1.0

• 8 75 0.6

• 9 91 0.9

• 10 105 1.1



.• Dependent(Response) variable (sales volume)

• Independent (Predictor) variable ( Ad expenditure)

• population regression line: y = A + Bx+ε • Estimated regression equation

ˆy = a+ bx

• Best fit straight line is obtained by using themethod of least squares; that is

minΣ(y - ˆy)2 which minimizes error sum of squares Σ ε 2

• (i) b: is Regression Coefficient. It is the slope of the estimated regression equation. It showschange in Y per unit change in X

• (ii) a is estimate of Y- intercept Pi oosh Ba oria

4



Least Squares Prediction Equation

• y ˆ = a + bx where

• b= n Σ x y - Σ x Σ y /nΣ x 2 – ( Σ x)2

• a = ¯y - b ¯x.

• or

• b= COV(X,Y)/S2x

• b= r (Sy/Sx)

Piyoosh Bajoria 5



Ad Sales Calculations• Month x y x2 xy y2

• 1 1.2 101 1.44 121.2 10,201

• 2 0.8 92 0.64 73.6 8,464

• 3 1.0 110 1.00 110.0 12,100

• 4 1.3 120 1.69 156.0 14,400

• 5 0.7 90 0.49 63.0 8,100

• 6 0.8 82 0.64 65.6 6,724

• 7 1.0 93 1.00 93.0 8,649

• 8 0.6 75 0.36 45.0 5,625

• 9 0.9 91 0.81 81.9 8,281

• 10 1.1 105 1.21 115.5 11,025

• Sum x y x2 xy y2 • 9.4 959 9.28 924.8 93,569

• ¯x = 0.94 ¯y = 95.9 S x =.2107 Sy =12.65 COV(x,Y)=2.334

Piyoosh Bajoria 6



• b = n Σ x y – ( Σ x )( Σ y) /nΣ x 2 – ( Σ x)2

= 10 ( 924.8 )-(9.4 )(959 )/10(9.28 )-(9.4)2

= 52.57

a = ¯y - b ¯x.= 95.9-(52.57) 0.94

= 46.49.

y ˆ = a+ bx = 46.49 + 52.57x

Piyoosh Bajoria 7



• Question. Predict sales volume,

y , fo r a g iven expend itu re level

o f Rs10, 000 (i.e. x = 1.0).• y ˆ = 46.49 + 52.57x = 46.49 +

(52.57)(1.0) = 99.06.

• So sales volume is Rs 990, 600.

Piyoosh Bajoria 8



standard error of est imate for a simp le

regress ion equat ion

• σ 2.

is a measure of spread of points (x, y)around the regression line.

• Se2 provides an estimate of σ 2.

• Se2

= Σ (y - ˆy)2 / (n -2)= Σ y 2 – a Σ y –b Σ xy / (n -2)

• Se=√ Se2

• Se is called standard error of estimate for a simple regression equation

Piyoosh Bajoria 9



• Question. Determine whether there is a

linear relationship between advertisingexpenditure, x, and sales volume, y.

• H0 : B= 0 (no linear relationship)

• Ha : B ≠0 (there is linear relationship)• T.S. :t = b/( Se / √ Σ x 2 – n( ¯x)2 )

=(52.57 – 0)/ 6.84/√.444= 5.12

• critical value: t.025,8 = 2.306 • Reject H0 since 5.12> 2.306

• There is linear relationship between X &Y P i y o o s h B a j o r i a 11



• Question. Find a 95% confidence

interval for B

• b ± t α /2,n-2

(Sb )

• 52.57 ± 2.306( 6.84/√.444)

• 52.57 ± 2.306(10.25)

• 52.57 ± 23.57 = (28.90, 76.24)

Piyoosh Bajoria 12



Analysis of Variance

• Notation:

• TSS := Σ (y – ¯y)2 (Total SS of

deviations).

• SSR =Σ (y ˆ - ¯y)2 (SS of deviations

due to regression or explained deviations)

• SSE = Σ (y - ˆy)2 (SS of deviations for

the error or unexplained deviations)• TSS = SSR + SSE

Piyoosh Bajoria 13



ANOVA table For Regression• Question. Give the ANOVA table for the AD sales example.

• ANOVA Table• Source df SS MS F p-value

• Reg. 1 SSR MSR=SSR/(1) MSR/MSE

• Error n-2 SSE MSE=SSE/(n-2)

• Totals n-1 TSS

• ANOVA Table

• Source df SS MS F p-value

• Reg. 1 1,226.927 1,226.927 26.25 0.0001

• Error 8 373.973 46.747

• Totals 9 1,600.900

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• Question . Use ANOVA table to test for a significant

linear relationship between sales and advertising

expenditure.• Test.

• H0 : B = 0 (no linear relationship)

• Ha : B ≠ 0 (there is linear relationship)

• T.S.: F = MSR/MSE = 26.25 • RR: ( critical value: F .005,1,8 = 14.69)

• Reject H0 if F > 14.69

• Decision: Reject H0

• Conclusion: At 0.5% significance level there is sufficient statistical evidence to indicate a linear relationship

between advertising expenditure, x, and sales volume, y.

Piyoosh Bajoria 15



• Question Find the coefficient of correlation, r .

• r =COV(X,Y)/SxSy = 2.334/0.2107(12.65 )= 0.88

• Coefficient of determination

• R 2

= r 2

= ( 0.88 )2

= 0.77

It represents extent or proportion of variation in

Y which is explained by the regression equation

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• TSS = SSR + SSE

• 1=( SSR/ TSS)+(SSE/TSS)• ( SSR/ TSS) is proportion of total variation in Y explained by

regression equation.• It is coefficient of determinationR 2 =r 2

• 1=R 2 +(SSE/TSS)• R 2 = 1- (SSE/TSS)

17Piyoosh Bajoria



• R 2 =Σ (y ˆ - ¯y)2 / Σ (y – ¯y)2

• Σ (a+bX - ¯y)2 / Σ (y – ¯y)2 • Σ ( ̄ y - b x̄ +bX - ¯y)2 / Σ (y –

¯y)2

• Σ ( - b x̄ +bX )2 / Σ (y – ¯y)2

• b2 Σ (X- ̄X)2 / Σ (y – ¯y)2

• b2 S2 x / S2 Y

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• Y=Return on security index(Dependent)

• X =Return on market index(Independent)• y = A+ x • Company specific factors contribute to

unsystematic risk• Marketwide factors contribute to

systematic risks

• In investor’s portfolio systematic riskcannot be diversified away but

unsystematic risk can be diversified away

P i y o o s h B a j o r i a 19



• Investor would like to know exact proportion of

systematic & unsystematic risk in the total risk of

a security• Variance of returns of security is the total risk of

that security = V(Y)

• R2 represents proportion of variation in y

explained by the independent variable X

• Systematic Risk= R2 x Variance of returns of

security

• Systematic Risk= R2 x V(Y)

• Unsystematic Risk= Variance of returns of

security –Systematic Risk

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• Systematic Risk= 2 x Variance of market

returns• Systematic Risk= 2 x V(X)

• Unsystematic Risk= Variance of returns of

security –Systematic Risk• NOTE

• = r Sy /Sx

2= r 2 S2y /S2x

2= R2 V(y) /V(x)

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Model Assumptions• Assumptions About the Error Term = (y - ˆy)

– The error is a random variable with mean of

zero.

– The variance of , denoted by 2

– The values of are independent.

– The error is a normally distributed random

variable

22Piyoosh Bajoria

Regression Co Relation

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