Financial Analysis, Planning and
Forecasting
Theory and Application
Chapter 24
Time-Series: Analysis, Model, and Forecasting
By
Alice C. Lee
San Francisco State University
John C. Lee
J.P. Morgan Chase
Cheng F. Lee
Rutgers University
Outline

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24.1 Introduction
24.2 The Classical Time-Series Component Model
24.3 Moving Average and Seasonally Adjusted Time-Series
24.4 Linear and Log-Linear Time Trend Regressions
24.5 Exponential Smoothing and Forecasting
24.6 Autoregressive Forecasting Model
Appendix 24A. The X-11 Model for Decomposing Time-Series
Components
Appendix 24B. The Holt-Winters Forecasting Model for
Seasonal Series
24.2 The Classical Time-Series Component Model
Table 24.1 Earnings per share of Philip Morris
Year
EPS
1977
$1.399
1978
1.698
1979
2.043
1980
2.315
1981
2.635
1982
3.115
1983
3.59
1984
3.62
1985
5.235
1986
6.2
1987
7.84
24.2 The Classical Time-Series Component Model
Figure 24.1 Earnings per share of Philip Morris
24.2 The Classical Time-Series Component Model
Table 24.2 Quarterly Earnings per share of IBM Corporation
Quarter
Year
1
2
3
4
1995
$2.12
$2.97
$-0.96
$3.09
1996
1.41
2.51
2.45
3.93
1997
2.37
1.46
1.38
2.16
1998
1.08
1.54
1.6
2.55
1999
1.61
1.32
0.97
1.16
2000
0.85
1.1
1.11
1.52
2001
1
1.17
0.92
1.35
2002
0.69
0.03
0.78
0.6
2003
0.8
0.99
1.04
1.59
2004
0.81
1.03
0.93
1.7
2005
0.86
1.14
0.95
2.02
2006
1.09
1.31
1.47
2.35
24.2 The Classical Time-Series Component Model
Figure 24.2 Quarterly Earnings per share of IBM
24.2 The Classical Time-Series Component Model
Figure 24.3 S&P 500 Composite Index, 76/1-88/3
24.2 The Classical Time-Series Component Model
Figure 24.4 Three-Month Rate on Eurodollar Deposits, U.S. T-Bills,
1985-1988 (Quarterly Date)
24.2 The Classical Time-Series Component Model
Figure 24.5
Time-Series Decomposition
24.2 The Classical Time-Series Component Model
X t  Tt  Ct  St  I t
（24.1）
X t  Tt Ct St It
（24.2）
where
Tt = trend component
Ct = cyclical component
St = seasonal component
It = irregular component
24.3
Moving Average and Seasonally Adjusted Time-Series
1
zt     xt i
 3  i 0
2
k 1
zi  1 k   xt i
i 0
k 1
zi   wt i xt i
i 0
t  3,
, n
（24.3）
t  k ,
, n
（24.4）
t  k ,
, n
（24.5）
24.3
Moving Average and Seasonally Adjusted Time-Series
Table 24.3
24.3
Moving Average and Seasonally Adjusted Time-Series
1 3
zt     xt i
 4  i 0
t  4,
, n
（24.6）
0.3  0.2717  0.2967  0.215
 0.27085
4
0.2717  0.2967  0.215  0.325
 0.2771
4
24.3
Moving Average and Seasonally Adjusted Time-Series
24.3
Moving Average and Seasonally Adjusted Time-Series
zt .5
zt .5
1
1
 
    xt i
 4  i 2
1 2
    xt i
 4  i 1
t  3, 4,
t  2,3,
, n  2
, n  2
（24.7）
（24.7a）
zt .5  zt .5 t  3, 4, , n  2
（24.8）


z 
2
z2.5  z3.5 x1  2 x2  2 x3  2 x4  x5
*

t  3, z3 
2
8
*
t
24.3
Moving Average and Seasonally Adjusted Time-Series
Figure 24.6 Earnings per Share Versus Moving-Average EPS for
Johnson & Johnson
24.3
Moving Average and Seasonally Adjusted Time-Series
 xt  （24.9）
Percentage of moving average (PMA) = 100  * 
 zt 
 x3 
 0.2967 
100  *   100 
  108.2945
 0.273975 
 z3 
24.3
Moving Average and Seasonally Adjusted Time-Series
24.3
Moving Average and Seasonally Adjusted Time-Series


*
Figure 24.7 Trend of 100 xt z t Ratio for Johnson & Johnson
24.3
Moving Average and Seasonally Adjusted Time-Series
 xt  100Tt Ct St It
100  *  
 100St It
Tt Ct
 zt 
（24.10）
24.3
Moving Average and Seasonally Adjusted Time-Series
Figure 24.8 Adjusted Earnings per Share (EPS) of Johnson & Johnson
24.4 Linear and Log-Linear Time Trend Regressions
xt     t  t
xt  x0 e
（24.11）
gt
log e xt  log e  x0e
gt

（24.12）
 loge x0  gt
log e xt     t  
'
'
'
t
（24.13）
24.4 Linear and Log-Linear Time Trend Regressions
24.4 Linear and Log-Linear Time Trend Regressions
xˆt  ˆ  ˆt  3239.9485  3167.1018t
Figure 24.9 Ford’s Annual Sales (1968-1990)
24.4 Linear and Log-Linear Time Trend Regressions
Figure 24.10 SAS Printout for Least-Squares Fit (Straight-Line Method) to x = Sales
t
Model: MODEL1
Department Variable: SALES
Analysis of Variance
Source
DF
Sum of
Squares
Mean
Square
F Value
Prob＞F
Model
1
10150900144
10150900144
216.829
0.0001
Error
21
982834678.18
46801651.342
C Total
22
11133734822
Root MSE
6841.17324
R-square
0.9117
Dep Mean
41245.17000
Adj. R-sq
0.9075
C. V.
16.58660
24.4 Linear and Log-Linear Time Trend Regressions
Figure 24.10 SAS Printout for Least-Squares Fit (Straight-Line Method) to x = Sales
t
(Cont’d)
Parameter Estimates
Variable
DF
Parameter
Estimate
Standard
Error
T for H0:
Parameter=0
Prob ＞¦T¦
INTERCEP
1
3239.948538
2948.6230544
1.099
0.2843
PERIOD
1
3167.101789
215.05043838
14.727
0.0001
Durbin-Watson D
(For Number of Obs.)
1st Order Autocorrelation
0.405
23
0.751
24.4 Linear and Log-Linear Time Trend Regressions
Figure 24.11
Observation (Year 1-23) and Forecast (Year 24-30) Sales Using the Straight-Line Model
24.4 Linear and Log-Linear Time Trend Regressions
'
ˆ
loge xˆt  ˆ   t  9.4834  0.0827t
 0.0516  0.0038
R  0.958
2
24.5 Exponential Smoothing and Forecasting
s1  x1
s2   x2  1    s1
s3   x3  1    s2
st   xt  1    st 1
（24.14）
24.5 Exponential Smoothing and Forecasting
xˆt 1  st   xt  1    st 1
（24.15）
xˆ2  s1   0.35,000  1  0.35,100  5,070 units
xˆt 1  st  st 1    xt  st 1 
st 1   xt 1  1    st 2
st 2   xt 2  1    st 3
（24.16）
24.5 Exponential Smoothing and Forecasting
st   xt  1    st 1
  xt   1    xt 1  1    st  2
2
  xt   1    xt 1   1    xt  2  1    st 3
2
 t 1

k
t
st    1    xt k   1    s0 0    1
 k 0

3
(24.17)
24.5 Exponential Smoothing and Forecasting
24.5 Exponential Smoothing and Forecasting
Figure 24.12 Annual Earnings per Share of J&J (Simple
Exponential Smoothing)
24.5 Exponential Smoothing and Forecasting
Figure 24.13 Annual Earnings per Share of IBM (Simple
Exponential Smoothing)
24.5 Exponential Smoothing and Forecasting
n
MSE 
  xt  xˆt 
2
t 1
n
st   xt  1    st 1  Tt 1 
（24.18）
（24.19a）
Tt    st  st 1   1    Tt 1 （24.19b）
24.5 Exponential Smoothing and Forecasting
xt  a  bt  et
s1  x1
T1  0 or b
s2   x2  1    s1  T1 
s1   xt  1    st 1  Tt 1 
Tt    st  st 1   1    Tt 1
24.5 Exponential Smoothing and Forecasting
Figure 24.14 Annual Earnings per Share of J&J with Forecasts Based
on the Holt-Winters Model
24.5 Exponential Smoothing and Forecasting
Figure 24.15 Annual Earnings per Share of IBM with Forecasts Based
on the Holt-Winters Model
24.5 Exponential Smoothing and Forecasting
xt  m  st  mTt
（24.20）
24.6 Autoregressive Forecasting Model
xt  a0  a1 xt 1
（24.21）
xt  a0  a1 xt 1  a2 xt 2
xt  a0  a1 xt 1  a2 xt  2 
（24.22）
 a p xt  p （24.23）
24.6 Autoregressive Forecasting Model
24.6 Autoregressive Forecasting Model
Figure 24.16 Quarterly Sales Data for Johnson & Johnson
24.6 Autoregressive Forecasting Model
AR(1): Sales t = -17.7964 + 1.0333 sales t-1
(0.0353)
（24.24）
2
R  0.9564
AR(2): Sales t = -62.2730 + 0.7065 Sales t-1 +0.3608 Sales t-2
(0.1575)
R  0.9595
2
(0.1697)
（24.25）
AR(3): Sales t = - 88.6493 + 0.5945 Sales t-1 +0.0264 Sales t-2
(0.1555)
+ 0.4738 Salest-3
(0.1899)
R 2  0.9608
(0.2082)
（24.26）
24.6 Autoregressive Forecasting Model
Sales43  17.7964  1.0333  2825  2901.2761
Sales43  62.2730  0.7065  2825  0.3608  2838 2957.5399
Sales43  88.6493  0.5945  2825  0.0265  2838  0.4738  2471
 2836.78
MARPE 
S t  St
St
(24.27)
Summary
In this chapter, we examined time-series component
analysis and several methods of forecasting. The major
components of a time series are the trend, cyclical,
seasonal, and irregular components. To analyze these
time-series components, we used the moving-average
method to obtain seasonally adjusted time series. After
investigating the analysis of time-series components, we
discussed several forecasting models in detail. These
forecasting models are linear time trend regression, simple
exponential smoothing, the Holt-Winters forecasting model
without seasonality, the Holt-Winters forecasting model
with seasonality, and autoregressive forecasting.
Many factors determine the power of any forecasting model.
They include the time horizon of the forecast, the stability
of variance of data, and the presence of a trend, seasonal,
or cyclical component.
Appendix 24A. The X-11 Model for Decomposing TimeSeries Components
Ot2  It2  Ct2  St2  Pt 2  TDt2
Table 24A.1
（24A.1）
Appendix 24A. The X-11 Model for Decomposing TimeSeries Components
Figure 24A.1
Original Sales and the X-11
Final Component Series
of Caterpillar, 1969-1980
Source: J. A. Gentry and C. F. Lee, “Measuring and Interpreting Time, Firm and Ledger Effect,” in Cheng F. Lee(1983), Financial Analysis and
Planning: Theory and Application, A book of Readings
Appendix 24A. The X-11 Model for Decomposing TimeSeries Components
Table 24A.2
Appendix 24A. The X-11 Model for Decomposing TimeSeries Components
O 
'
2
2
 I C  S
I
I component =
2
' 2
C
2
(O )
S component =
（24A.2）
72.358 


2
99.508


 42.051

2
 99.508
 52.88%
2
' 2
S
2
2
(O )
C component =
2
2
' 2
(O )
 17.86%
53.831


2
99.508


2
 29.27%
Appendix 24A. The X-11 Model for Decomposing TimeSeries Components
Table 24A.3 Relative Contributions of Components to Changes in Caterpillar Sales for 1-, 2-, 3-, and 4Quarter Time Spans
Relative Contribution (in percent)
Span in Quarters
Trend-Cycle
Seasonal
Irregular
Total
1
17.86
29.27
52.88
100
2
46.94
28.44
24.62
100
3
68.50
13.08
18.42
100
4
82.58
0.15
17.27
100
Appendix 24B. The Holt-Winters Forecasting Model for
Seasonal Series
 xt 
st   
  1    st 1  Tt 1 
 Ft  L 
Tt    st  st 1   1    Tt 1
 xt
Ft   
 st

  1    Ft  L

xt  m   st  mTt  Ft  m  L 
（24B.1）
（24B.2）
（24B.3）
（24B.4）
Appendix 24B. The Holt-Winters Forecasting Model for
Seasonal Series
Table 24B.1
Appendix 24B. The Holt-Winters Forecasting Model for
Seasonal Series
24B.2
Quarter 1 = 1.130064
Quarter 2 = 1.128077
Quarter 3 = 1.037570
Quarter 4 = .704288
24B.3
Appendix 24B.
The Holt-Winters
Forecasting Model for
Seasonal Series
Appendix 24B. The Holt-Winters Forecasting Model for
Seasonal Series
d t  0.275977  0.003482t
s0   a  b  0    initial seasonal index for fourth quarter 
  0.275977  0.704288  0.194367
Appendix 24B. The Holt-Winters Forecasting Model for
Seasonal Series
 x10 
s10  0.2 
  0.8  s9  T9 
 F10 4 
 x10 
 0.2    0.8  s9  T9 
 F6 
 0.38 
 0.2 
  0.8  0.341176  0.015552 
 1.161223 
 0.35083
Appendix 24B. The Holt-Winters Forecasting Model for
Seasonal Series
T10  0.3  s10  s9   0.7T9
 0.3  0.35083  0.341176  0.7 .015552
 0.013783
 x10 
F10  0.3    0.7 F6
 s10 
 0.38 
 0.3 
  0.7 1.161223
 0.35083 
 1.137799
Appendix 24B. The Holt-Winters Forecasting Model for
Seasonal Series
Figure 24B.1
Quarterly Earnings per Share of J&J (Actual and Smoothed EPS)
Appendix 24B. The Holt-Winters Forecasting Model for
Seasonal Series
Figure 24B.2
Quarterly Earnings per Share of J&J (Actual and Forecasted EPS)
Appendix 24B. The Holt-Winters Forecasting Model for
Seasonal Series
x11   s10  T10  F7    0.350830  0.0137831.099288 
 0.400815