Business Forecasting
Chapter 10
The Box–Jenkins Method of
Forecasting
Chapter Topics

The Box–Jenkins Models

Forecasting with Autoregressive Models (AR)

Forecasting with Moving Average Models (MA)



Autoregressive Integrated Moving Average
(ARIMA) models
Trends and Seasonality in Time Series

Trends

Seasonal Data
Chapter Summary
Univariate Data
•
•
•
•
•
Stationary
Trend
Seasonality
Cyclical
A majority of the real-world data show certain
combinations of the above patterns.
Box–Jenkins Method
Besides the smoothing techniques, what
other methods can we use to forecast
univariate data?
Using Box–Jenkins Methods
Capture the past pattern
Forecast the future
Why Use The Box–Jenkins Method?
• When facing very complicated data patterns
such as a combination of a trend, seasonal,
cyclical, and random fluctuations:
e.g. Earning data of a corporation
e.g. Forecasting stock price
e.g. Sales forecasting
e.g. Energy forecasting (electricity, gas)
e.g. Traffic flow of a city
Why Use the Box–Jenkins Method?
• When forecasting is the sole purpose of the
model.
• Very reliable especially in short term (0–6
months) prediction; reliable in short-to-mid
(6 months–1.5years) -term prediction.
• Confidence intervals for the estimates are
easily constructed.
Pattern I: Stationary
Pattern 1: No Trend—Stationary
demand seems to cluster around a specific level.
Pattern II: Trend
Demand
consistently
increases or
decreases
over time.
18
18
16
16
14
14
12
12
10
10
8
8
6
6
4
4
2
2
0
0
1
1
2
3
4
5
6
7
8
9
10
11
12
2
3
4
5
6
7
8
9
10
8
9
11
12
13
13
Time
Time
600
600
500
500
400
400
300
300
200
200
100
100
0
1
2
3
4
5
6
7
8
9
10
11
12
13
0
1
Time
2
3
4
5
6
7
Time
10
11
12
13
Pattern III: Seasonality
U.S. Electricity Consumption
12,000,000
(Source: Historical Electricity Data, Energy Information
Association, http://www.eia.doe/gov)
Billion KWH
10,000,000
8,000,000
6,000,000
Residential Consumption
4,000,000
Industrial Consumption
2,000,000
Commercial Consumption
2002/Q4
2002/Q2
2001/Q4
2001/Q2
2000/Q4
2000/Q2
1999/Q4
1999/Q2
1998/Q4
1998/Q2
1997/Q4
1997/Q2
1996/Q4
1996/Q2
1995/Q4
1995/Q2
1994/Q4
1994/Q2
1993/Q4
1993/Q2
1992/Q4
1992/Q2
1991/Q4
1991/Q2
1990/Q4
1990/Q2
-
2002/5
2000/9
1999/1
1997/5
1995/9
1994/1
1992/5
1990/9
1989/1
1987/5
1985/9
1984/1
1982/5
1980/9
1979/1
1977/5
1975/9
1974/1
1972/5
1970/9
1969/1
1967/5
1965/9
1964/1
Pattern IV: Cyclical
U.S. 3-Month Treasury Bill Rate
18.0
16.0
14.0
12.0
10.0
8.0
6.0
4.0
2.0
-
Box–Jenkins Method Assumption
•
•
In order to use the B/J method, the time
series should be stationary.
B/J main idea: Any stationary time series can
self-predict its own future from the past data.
10
9
8
7
6
5
4
3
2
1
0
1
5
9
13
17
21
25
29
33
37
41
45
49
53
57
61
65
69
73
77
81
85
Box–Jenkins Method Assumption
•
•
•
We know that not all time series are
stationary.
However, it is easy to convert a trend or a
seasonal time series to a stationary time
series.
Simply use the concept of “Differencing.”
Example of Differencing
Partial Data from Table 10.6
Time
(t)
(1)
t
86
t
87
t
88
t
89
t
90
Actual Index
Yt
Differences
(Yt  Yt 1 )
(2)
(3)
89.7
na
90.1
0.4
91.5
1.4
92.4
0.9
94.4
2.0
Convert a trend time series to stationary
time series using the differencing
method
-2.0
2003/Q1
2002/Q2
2001/Q3
2000/Q4
2000/Q1
1999/Q2
1998/Q3
1997/Q4
1997/Q1
1996/Q2
4.0
1995/Q3
6.0
1994/Q4
1994/Q1
1993/Q2
1992/Q3
1991/Q4
1991/Q1
1990/Q2
Convert a seasonal time series
to stationary time series
U.S. Electricity Consumption
12.0
10.0
8.0
dt  Yt  Yt 4
2.0
0.0
Differencing Summary
•
To convert trend time series to stationary
time series:
dt  Yt  Yt 1
•
To convert seasonal time series to
stationary time series:
dt  Yt  Yt 4
•
Both of the above two methods can be
applied/combined to remove the cyclical
effects.
How do we decide the model?
• Use Autocorrelation (AC) and Partial
Autocorrelation (PAC)
• First-order autocorrelation is a measure of how
correlated an observation is with an observation
one period away, that is: (yt,yt−1)
• Second-order autocorrelation is a measure of how
correlated an observation is with an observation
two periods away (yt,yt−2)
• etc...
AR Model Fit
• When the autocorrelation coefficients gradually fall to
0, and the partial correlation has spikes, an AR
model is appropriate. The order of the model
depends on the number of spikes.
• An AR(2) model is shown below.
MA Model Fit
• When the partial correlation coefficients gradually fall
to 0, and the autocorrelation has spikes, a MA model
is appropriate. The order of the model depends on
the number of spikes.
• An MA(1) model is shown below.
ARIMA Model Fit
• When both the autocorrelation and the partial
correlograms show irregular patterns, then an
ARIMA model is appropriate. The order of the
model depends on the number of spikes.
• An ARIMA(1,0,1) model is shown below.
Chapter Summary
• Box–Jenkins models capture a wide variety of time
series patterns.
• When faced with a complicated time series that
includes a combination of trend, seasonal factor,
cyclical, as well as random fluctuations, use of the
Box–Jenkins is appropriate.
• This methodology for forecasting is an iterative
process that begins by assuming a tentative
pattern that is fitted to the data so that error will
be minimized.
• The major assumption of the model is that the
data is stationary.
Chapter Summary
(continued)
• “Differencing” could be used to make the data
stationary.
• In using the different models of Box–Jenkins, we
depend on the autocorrelation (AC) and partial
autocorrelation (PAC) as diagnostic tools.
• Computer programs such as Minitab, and SPSS
provide all the analysis tools for using the Box–
Jenkins.