© 2001 Prentice-Hall, Inc. Chap 13-1

BA 201

Lecture 21Autocorrelation

and Inferences about the Slope

© 2001 Prentice-Hall, Inc. Chap 13-2

Topics

Measuring Autocorrelation Inferences about the Slope

© 2001 Prentice-Hall, Inc. Chap 13-3

Autocorrelation

What is Autocorrelation? The error term in one time period is related

(correlated, autocorrelated) to the error term in a different time period

Can happen only in time-series data

© 2001 Prentice-Hall, Inc. Chap 13-4

Residual Analysis for Independence (Graphical

Approach)Not Independent Independent

e eTimeTime

Residual Is Plotted Against Time to Detect Any Autocorrelation

No Particular PatternCyclical Pattern

Time

Y

Time

Y

0 0

© 2001 Prentice-Hall, Inc. Chap 13-5

Residual Analysis for Independence

The Durbin-Watson Statistic Used when data is collected over time to

detect autocorrelation (residuals in one time period are related to residuals in another period)

Measures violation of independence assumption

21

2

2

1

( )n

i ii

n

ii

e eD

e

Should be close to 2 for independence of errors.

If not, examine the model for autocorrelation.

© 2001 Prentice-Hall, Inc. Chap 13-6

Durbin-Watson Statistic in PHStat

PHStat | Regression | Simple Linear Regression … Check the box for Durbin-Watson Statistic

An example in Excel spreadsheet Y is the DJIA (measured in % change from

previous day’s closing number to current day’s closing number)

X is the U.S. Treasury 30-year bond rates (measured in % change from previous day’s closing rate to current day’s closing rate)

Microsoft Excel

Worksheet

© 2001 Prentice-Hall, Inc. Chap 13-7

Accept H0

(no autocorrelatin)

Using the Durbin-Watson Statistic

: No autocorrelation (error terms are independent) : There is autocorrelation (error terms are not

independent)

0H

1H

0 42dL 4-dLdU 4-dU

Reject H0

(positive autocorrelation)

Inconclusive Reject H0

(negative autocorrelation)

© 2001 Prentice-Hall, Inc. Chap 13-8

Sample Linear Regression(continued)

Y

XObserved Value

|Y X iX

i

ii iY X

0 1i iY b b X

ie

0 1i iib bY X e 1b

0b

p.462

© 2001 Prentice-Hall, Inc. Chap 13-9

Inference about the Slope: t Test

t Test for a Population Slope Is there a linear dependency of Y on X ?

Null and Alternative Hypotheses H0: 1 = 0(No Linear Dependency) H1: 1 0 (Linear Dependency)

Test Statistic

Assumption Needed

Normality

1

1

1 1

2

1

where

( )

YXb n

bi

i

b St S

SX X

. . 2d f n

pp.483-484

© 2001 Prentice-Hall, Inc. Chap 13-10

Example: Produce StoreData for 7 Stores: Estimated

Regression Equation:

The slope of this model is 1.487.

Is Square Footage of the store affecting its Annual Sales at 5% level of significance?

Annual Store Square Sales

Feet ($000)

1 1,726 3,681

2 1,542 3,395

3 2,816 6,653

4 5,555 9,543

5 1,292 3,318

6 2,208 5,563

7 1,313 3,760

ˆ 1636.415 1.487i iY X

© 2001 Prentice-Hall, Inc. Chap 13-11

Inferences about the Slope: t Test Example

H0: 1 = 0

H1: 1 0

.05df 7 - 2 = 5Critical Value(s):

Test Statistic:

Decision:

Conclusion:There is evidence that square footage affects annual sales.t0 2.5706-2.5706

.025

Reject Reject

.025

From Excel Printout

Reject H0

Coefficients Standard Error t Stat P-valueIntercept 1636.4147 451.4953 3.6244 0.01515Footage 1.4866 0.1650 9.0099 0.00028

1b 1bS t

p-value

b0

© 2001 Prentice-Hall, Inc. Chap 13-12

Inferences about the Slope: Confidence Interval Example

Confidence Interval Estimate of the Slope:

11 2n bb t S Excel Printout for Produce Stores

At 95% level of confidence the confidence interval for the slope is (1.062, 1.911). Does not include 0.

Conclusion: There is a significant linear dependency of annual sales on the size of the store.

Lower 95% Upper 95%Intercept 475.810926 2797.01853X Variable 11.06249037 1.91077694

p.486

© 2001 Prentice-Hall, Inc. Chap 13-13

Inferences about the Slope: F Test

F Test for a Population Slope Is there a linear dependency of Y on X ?

Null and Alternative Hypotheses H0: 1 = 0 (No Linear Dependency) H1: 1 0 (Linear Dependency)

Test Statistic

Numerator d.f.=1, denominator d.f.=n-2 Assumption Needed

Normality

1 =

2

SSRMSR

FSSE MSEn

pp.484-485

© 2001 Prentice-Hall, Inc. Chap 13-14

ANOVAdf SS MS F Significance F

Regression 1 30380456.12 30380456.12 81.179 0.000281Residual 5 1871199.595 374239.919Total 6 32251655.71

Inferences about the Slope: F Test Example

Test Statistic:

Decision:

Conclusion:

H0: 1 = 0H1: 1 0 .05numerator df = 1denominator df 7 - 2 = 5

There is evidence that square footage affects annual sales.

From Excel Printout

Reject H0

0 6.61

Reject

= .05

1, 2nF

p-value

© 2001 Prentice-Hall, Inc. Chap 13-15

Relationship between a t Test and an F Test

Null and Alternative Hypotheses H0: 1 = 0 (No Linear Dependency) H1: 1 0(Linear Dependency)

The p –value of a t Test and the p –value of an F Test Are Exactly the Same

The Rejection Region of an F Test is always in the upper tail

2

2 1, 2n nt F

p.???

© 2001 Prentice-Hall, Inc. Chap 13-16

Summary

Addressed Measuring Autocorrelation Described Inference about the Slope