time seres analysis

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2000 Prentice-Hall, Inc. Chap. 11- 1

The Least SquaresLinear Trend Model

Year Coded X Sales

95 0 2

96 1 5

97 2 2

98 3 2

99 4 7

00 5 6

0 1i iY b b X

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2000 Prentice-Hall, Inc. Chap. 11- 2

The Least SquaresLinear Trend Model (Continued)

i i i X ..X b b Y

743143210

Excel Output

C o e f f i c i e n t s

I n t e r c e p t 2 . 1 4 2 8 5 7 1 4

X V a r ia b l e 0 . 7 4 2 8 5 7 1 4

0

1

2

3

4

5

6

7

8

0 1 2 3 4 5 6X

S a l e s

Projected toyear 2001

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2000 Prentice-Hall, Inc. Chap. 11- 3

Year Coded X Sales

95 0 296 1 5

97 2 2

98 3 299 4 7

00 5 6

The Quadratic TrendModel

2

0 1 2i i iY b b X b X

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2000 Prentice-Hall, Inc. Chap. 11- 4

The Quadratic TrendModel (Continued)

2 2

0 1 2 2.857 .33 .214i i i i iY b b X b X X X

C o e ff i c i en t s I n t e r c e p t 2 . 8 5 7 1 4 2 8 6

X V a r ia b l e 1 - 0 . 3 2 8 5 7 1 4

X V a r ia b l e 2 0 . 2 1 4 2 8 5 7 1

Excel Output

0

1

2

3

4

5

6

7

8

0 1 2 3 4 5 6 X

S a l e s

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2000 Prentice-Hall, Inc. Chap. 11- 5

C o e f f i c i e n t s

In t e rc e p t 0 . 3 3 5 8 3 7 9 5X V a r ia b l e 0 . 0 8 0 6 8 5 4 4

The Exponential TrendModel

i X i b b Y

10 or 110 b lo g X b lo g Y

lo g i

Excel Output of Values in logs

i X i ) . )( .( Y

21172

Year Coded Sales

94 0 295 1 5

96 2 2

97 3 2

98 4 7

99 5 6

a n t ilo g ( . 3 3 5 8 3 7 9 5 ) = 2 . 1 7

a n t i lo g (. 0 8 0 6 8 5 4 4 ) = 1 . 2

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8/13/2019 Time Seres Analysis

7/22 2000 Prentice-Hall, Inc. Chap. 11- 7

Model Selection UsingDifferences

Use an Exponential Trend Model if thePercentage Differences Are More orLess Constant

3 2 12 1

1 2 1

100% 100% 100%n n

n

Y Y Y Y Y Y

Y Y Y

(continued)

8/13/2019 Time Seres Analysis

8/22 2000 Prentice-Hall, Inc. Chap. 11- 8

Autoregressive Modeling

Used for forecasting Takes advantage of autocorrelation

1st order - correlation between consecutivevalues2nd order - correlation between values 2

periods apart

Autoregressive model for pth order:

i p i p i i i Y AY AY AAY 22110

Random

Error

8/13/2019 Time Seres Analysis

9/22 2000 Prentice-Hall, Inc. Chap. 11- 9

Autoregressive Model:Example

The Office Concept Corp. has acquired a number of officeunits (in thousands of square feet) over the last 8 years.

Develop the 2nd order Autoregressive model. Year Units

93 494 3

95 296 397 298 299 400 6

8/13/2019 Time Seres Analysis

10/22 2000 Prentice-Hall, Inc. Chap. 11- 10

Autoregressive Model:Example Solution

Year Yi Yi-1 Yi-2 93 4 --- ---94 3 4 --- 95 2 3 4 96 3 2 3 97 2 3 2 98 2 2 3 99 4 2 2

00 6 4 2

C o e f f i c i e n t s

I n t e r c e p t 3 .5

X V a r ia b l e 1 0 . 8 1 2 5

X V a r ia b l e 2 -0 .9375

Excel Output

21 9375812553

i i i Y .Y ..Y

Develop the 2nd ordertable

Use Excel to run a

regression model

8/13/2019 Time Seres Analysis

11/22 2000 Prentice-Hall, Inc. Chap. 11- 11

Autoregressive Model Example:Forecasting

Use the 2nd order model to forecast number ofunits for 2001:

2001 2000 19993.5 .8125 .9375Y Y Y

3.5 .8125 6 .9375 4

4.625

1 23.5 .8125 .9375

i i iY Y Y

8/13/2019 Time Seres Analysis

12/22 2000 Prentice-Hall, Inc. Chap. 11- 12

Autoregressive ModelingSteps

1. Choose p : note that df = n - p - 12. Form a series of lag predictor variables

Y i-1

, y i-2

, y i-p

3. Use excel to run regression model using all p variables

4. Test significance of a pIf null hypothesis rejected, this model isselectedIf null hypothesis not rejected, decrease p by 1

and repeat

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2000 Prentice-Hall, Inc. Chap. 11- 14

Residual Analysis

Random errors

Trend not accounted for

Cyclical effects not accounted for

Seasonal effects not accounted for

T T

T T

e e

e e

0 0

0 0

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2000 Prentice-Hall, Inc. Chap. 11- 15

Measuring Errors

Choose a model that gives the smallest

measuring errors

Sum square error (SSE)

Sensitive to outliers

2

1

n

ii

i

SSE Y Y

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2000 Prentice-Hall, Inc. Chap. 11- 16

Measuring Errors

Mean absolute deviation (MAD)

Not sensitive to extreme observations

1

n

ii

i

Y Y

MADn

(continued)

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2000 Prentice-Hall, Inc. Chap. 11- 17

Principal of Parsimony

Suppose 2 or more models providegood fit for data

Select the simplest model Simplest model types: Least-squares linear Least-square quadratic 1st order autoregressive

More complex types: 2nd and 3rd order autoregressive

Least-squares exponential

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2000 Prentice-Hall, Inc. Chap. 11- 18

Forecasting WithSeasonal Data

Use categorical predictor variables with least-square trending fitting

Exponential model with quarterly data:

The b i provides the multiplier for the ith quarterrelative to the 4th quarter

Q i = 1 if ith quarter and 0 if not

X j = the coded variable denoting the time period

321

43210

Q Q Q X b b b b b Y

i

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2000 Prentice-Hall, Inc. Chap. 11- 19

Forecasting With QuarterlyData: Example

4 4 5 .7 7

4 4 4 .2 7

4 6 2 .6 9

4 5 9 .2 7

5 0 0 . 7 1

5 4 4 . 7 5

5 8 4 . 4 1

6 1 5 . 9 3

6 4 5 . 5

6 7 0 . 6 3

6 8 7 . 3 1

7 4 0 . 7 4

7 5 7 . 1 2

8 8 5 . 1 4

9 4 7 . 2 8

9 7 0 . 4 3

I

23

4

Quarter 1994 1995 1996 1997

Standards and Poors Composite Stock Price Index:

R e g r e s si o n S t a ti s ti c s

M u l ti p l e R 0.99005245R S q u a r e 0 .980203854

Adjuste d R Sq ua re 0 .973005256

S tanda rd E r ro r 0 .04361558

Obse rva t i ons 16

Excel Output

Appears to be

an excellent fit.

r 2 is .98

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2000 Prentice-Hall, Inc. Chap. 11- 20

Quarterly Data:Example

CoefficientsIntercept 6.029403386X Variable (Trend) 0.055222261X Variable (Q1) -0.006892656

X Variable (Q2) 0.011566505X Variable (Q3) -0.019380022

Excel Output

2110 b ln Q b ln X b ln Y

ln i i Regression Equation for the first quarter:

100690550296 Q .X .. i

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2000 Prentice-Hall, Inc. Chap. 11- 21

Chapter Summary

Discussed the importance of forecasting Addressed component factors of the time-

series model Performed smoothing of data series

Moving averages Exponential smoothing

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2000 Prentice-Hall Inc Chap 11- 22

Chapter Summary

Described least square trend fitting andforecasting

Linear, quadratic and exponential models Addressed autoregressive models Described procedure for choosing

appropriate models Discussed seasonal data (use of dummy

variables)

(continued)