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Business Forecasting Chapter 10 The Box–Jenkins Method of Forecasting
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Page 1: Box Jenkins Method

Business Forecasting

Chapter 10The Box–Jenkins Method of

Forecasting

Page 2: Box Jenkins Method

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

Page 3: Box Jenkins Method

Univariate DataUnivariate Data

• A majority of the real-world data show certain combinations of the above patterns.

• Stationary• Trend• Seasonality• Cyclical

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Box–Jenkins MethodBox–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

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Why Use The Box–Jenkins Why Use The Box–Jenkins Method?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

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Why Use the Box–Jenkins Why Use the Box–Jenkins Method?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.

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Pattern I: Stationary Pattern I: Stationary

Pattern 1: No Trend—Stationarydemand seems to cluster around a specific level.

Page 8: Box Jenkins Method

Pattern II: Trend Pattern II: Trend

Demand consistent

ly increases

or decreases over time.

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1 2 3 4 5 6 7 8 9 10 11 12 13 0

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1 2 3 4 5 6 7 8 9 10 11 12 13

TimeTime

Time

Time

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Pattern III: Seasonality Pattern III: Seasonality

U.S. Electricity Consumption

-

2,000,000

4,000,000

6,000,000

8,000,000

10,000,000

12,000,000

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90

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Bil

lio

n K

WH

Commercial Consumption

Industrial Consumption

Residential Consumption

(Source: Historical Electricity Data, Energy Information Association, http://www.eia.doe/gov)

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Pattern IV: Cyclical Pattern IV: Cyclical

U.S. 3-Month Treasury Bill Rate

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2.0

4.0

6.0

8.0

10.0

12.0

14.0

16.0

18.0

1964/1

1965/9

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2002/5

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Box–Jenkins Method Box–Jenkins Method AssumptionAssumption

• 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.

0

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1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85

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Box–Jenkins Method Box–Jenkins Method AssumptionAssumption

• 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.”

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Example of Differencing

Partial Data from Table 10.6 Time Actual Index Differences (t) tY )( 1 tt YY

(1) (2) (3)

86t 89.7 na

87t 90.1 0.4

88t 91.5 1.4

89t 92.4 0.9

90t 94.4 2.0

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Convert a trend time series to Convert a trend time series to stationary time series using the stationary time series using the

differencing methoddifferencing method

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U.S. Electricity Consumption

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4 ttt YYd

Convert a seasonal time Convert a seasonal time series to stationary time series to stationary time

seriesseries

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Differencing SummaryDifferencing Summary•To convert trend time series to stationary time series:

•To convert seasonal time series to stationary time series:

•Both of the above two methods can be applied/combined to remove the cyclical effects.

1 ttt YYd

4 ttt YYd

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How do we decide the How do we decide the model?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...

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AR Model FitAR 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.

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MA Model FitMA 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.

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ARIMA Model FitARIMA 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.

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Chapter SummaryChapter 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.

Page 22: Box Jenkins Method

Chapter Summary Chapter Summary (continued)(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.