Transcript
Business Forecasting
Chapter 10The 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 DataUnivariate Data
• A majority of the real-world data show certain combinations of the above patterns.
• Stationary• Trend• Seasonality• Cyclical
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
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
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.
Pattern I: Stationary Pattern I: Stationary
Pattern 1: No Trend—Stationarydemand seems to cluster around a specific level.
Pattern II: Trend Pattern II: Trend
Demand consistent
ly increases
or decreases over time.
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TimeTime
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Pattern III: Seasonality Pattern III: Seasonality
U.S. Electricity Consumption
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Commercial Consumption
Industrial Consumption
Residential Consumption
(Source: Historical Electricity Data, Energy Information Association, http://www.eia.doe/gov)
Pattern IV: Cyclical Pattern IV: Cyclical
U.S. 3-Month Treasury Bill Rate
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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.
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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.”
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
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
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
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
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...
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.
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.
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.
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.
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.
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