sda 3e chapter 6 (1)

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© 2007 Pearson Education

Chapter 6: Regression Analysis

Part 1: Simple Linear Regression

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Regression Analysis Building models that characterize the

relationships between a dependent variable

and one (single) or more (multiple)independent variables, all of which arenumerical, for example:

Sales = a + b*Price + c*Coupons +

d*Advertising + e*Price*Advertising Cross-sectional data

Time series data (forecasting)

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Simple Linear Regression Single independent variable

Linear relationship

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SLR ModelY = 0 + 1 X + e

Intercept slope error

E(Y X)

f(Y X)

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Error Terms (Residuals)e

i = Y

i - 0 - 1 X

i

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Estimation of the Regression

LineTrue regression line (unknown):

Y = 0 + 1 X + e

Estimated regression line:

Y = b0

+ b1 X

Observed errors:

ei= Y i - b

0- b

1 X i

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Least Squares Regression=

+-n

i

ii X bbY 1

2

10])[(

2

1

2

1

1

X n X

Y X nY X

bn

ii

n

i

ii

-

-

=

=

=

b0

= Y - b1 X

minimize

a

bc

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Excel Trendlines Construct a scatter diagram

Method 1: Select Chart/Add Trendline

Method 2: Select data series; right click

= 0.0854(6000) – 108.59

= 403.816000ˆY

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Regression and

Investment Risk Systematic risk – variation in stock price

explained by the market

Measured by beta

Beta = 1: perfect match to marketmovements

Beta < 1: stock is less volatile than market Beta > 1: stock is more volatile than

market

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Systematic Risk (Beta)

Beta is the slope of theregression line

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Theory Without Regression

Y

Best estimate for Y is the

mean; independent of the

value of X

A measure of total variation is

SST = (Y i - Y)2

Unexplained

variation

Xi

Yi

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Theory With RegressionObserved values Yi

Fitted values Yi

Xi

Yi

Variation unexplained

after regression, Y - Y

Fitted line

Y = b0 + b1X

Y

Variation explained

by regression, Y - Y

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Sums of SquaresSST = (Y

i- Y)2

= ( – Y )

2

+ (Y - )2

Y ˆ

Y ˆ

= SSR + SSE

Explained variation Unexplained variation

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Coefficient of Determination R 2 = SSR/SST = (SST – SSE)/SST = 1 – SSE/SST =

coefficient of determination: the proportion of variation explained by the independent variable

(regression model)

0 R 2 1

Adjusted R 2 incorporates sample size and number of explanatory variables (in multiple regression models).

Useful for comparing against models with differentnumbers of explanatory variables.

-

---=

2

111

22

n

n R Radj

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Correlation Coefficient Sample correlation coefficient

R = R 2

Properties

-1 R 1

R = 1 => perfect positive correlation

R = -1 => perfect negative correlation

R = 0 => no correlation

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Standard Error of the Estimate MSE = SSE/(n-2) = an unbiased

estimate of the variance of the errors

about the regression line Standard error of the estimate is

Measures the spread of data about the line

SYX =

2-n

SSE

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Confidence Interval for Mean

Value of Y Confidence intervals depend on the value of

the independent variable

t /2, n-2 SYXhi iY ˆ

=

-

-+= n

i

i

ii

X X

X X

nh

1

2

2

)(

)(1

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Prediction Intervals Prediction intervals apply to individual

(not mean) values of the dependent

variable

t /2, n-2 SYX1+hi iY ˆ

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PHStat Simple Linear

Regression

Define range of dependentand independent variables

Choose regression options

Choose output options

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Regression Statistics and

Confidence Interval Outputs

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Regression as ANOVA Testing for significance of regression

SST = SSR + SSE

The null hypothesis implies that SST = SSE, or SSR = 0

MSR = SSR/1 = variance explained by regression

F = MSR/MSE

If F > critical value, it is likely that 1 0, or that theregression line is significant

H0: 1 = 0

H1: 1 0

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t-test for Significance of

Regression

= -

-=

n

iiYX X X S

bt

1

2

11

)( /

with n-2 degrees of freedom

This allows you to test

H0: slope = 1

H1: slope 1

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Excel Regression Tool Excel menu > Tools > Data Analysis >

Regression

Input variable ranges

Check appropriateboxes

Select output options

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Regression OutputCorrelation coefficient

SYX

b0 b1

p-value for

significance of

regression

t- test for slope Confidence

interval for slope

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Residuals

Standard residuals are residuals divided by their standard

error, expressed in units independent of the units of the

data.

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Assumptions Underlying

Regression Linearity

Check with scatter diagram of the data or the residual plot

Normally distributed errors for each X with mean 0 and constant

variance Examine histogram of standardized residuals or use goodness-of-fit

tests

Homoscedasticity – constant variance about the regression linefor all values of the independent variable

Examine by plotting residuals and looking for differences invariances at different values of X

No autocorrelation. Residuals should be independent for eachvalue of the independent variable. Important if the independentvariable is time (e.g., forecasting models).

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Residual Plot

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Histogram of Residuals

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Evaluating HomoscedasticityOK

Heteroscadastic

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Autocorrelation Durbin-Watson statistic

D < 1 suggest autocorrelation

D > 1.5 suggest no autocorrelation

D > 2.5 suggest negative autocorrelation

PHStat tool calculates this statistic

=

=

--

=n

i

i

n

i

ii

e

ee

D

1

2

2

2

1 )(

sda 3e chapter 6 (1)

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