Business%20Forecasting

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Business Forecasting

• Chapter 10
• The BoxJenkins Method of Forecasting

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Chapter Topics

• The BoxJenkins 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

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Univariate Data

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

• Stationary
• Trend
• Seasonality
• Cyclical

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BoxJenkins Method
Besides the smoothing techniques, what other
methods can we use to forecast univariate data?
Using BoxJenkins Methods
Capture the past pattern
Forecast the future
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Why Use The BoxJenkins 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 BoxJenkins Method?

• When forecasting is the sole purpose of the
  model.
• Very reliable especially in short term (06
  months) prediction reliable in short-to-mid (6
  months1.5years) -term prediction.
• Confidence intervals for the estimates are easily
  constructed.

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Pattern I Stationary
Pattern 1 No TrendStationary demand seems to
cluster around a specific level.
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Pattern II Trend
Demand consistently increases or decreases over
time.
Time
Time
Time
Time
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Pattern III Seasonality
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Pattern IV Cyclical
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BoxJenkins 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.

Self-forecasting
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BoxJenkins 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.

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Example of Differencing
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Convert a trend time series to stationary time
series using the differencing method
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Convert a seasonal time series to stationary time
series
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Differencing 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.

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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...

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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.

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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.

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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.

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Chapter Summary

• BoxJenkins 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 BoxJenkins 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.

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Chapter Summary (continued)

• Differencing could be used to make the data
  stationary.
• In using the different models of BoxJenkins, 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
  BoxJenkins.