Time Seres Analysis

8/13/2019 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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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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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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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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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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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)



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



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



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


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



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



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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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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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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Measuring Errors

Mean absolute deviation (MAD)

Not sensitive to extreme observations

1

n

ii

i

Y Y

MADn

(continued)


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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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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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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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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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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)

Time Seres Analysis

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