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8/13/2019 Time Seres Analysis
1/22
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
8/13/2019 Time Seres Analysis
2/22
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
8/13/2019 Time Seres Analysis
3/22
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
8/13/2019 Time Seres Analysis
4/22
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
8/13/2019 Time Seres Analysis
5/22
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
8/13/2019 Time Seres Analysis
6/22
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
8/13/2019 Time Seres Analysis
13/22
8/13/2019 Time Seres Analysis
14/22
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
8/13/2019 Time Seres Analysis
15/22
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
8/13/2019 Time Seres Analysis
16/22
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)
8/13/2019 Time Seres Analysis
17/22
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
8/13/2019 Time Seres Analysis
18/22
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
8/13/2019 Time Seres Analysis
19/22
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
8/13/2019 Time Seres Analysis
20/22
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
8/13/2019 Time Seres Analysis
21/22
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
8/13/2019 Time Seres Analysis
22/22
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)