MathematicalForecasting_Box-Jenkins.doc - MATHEMATICAL...

MATHEMATICAL FORECASTING USING THE BOX-JENKINS METHODOLOGY
TECHNICAL BRIEFING
JOSEPH GEORGE CALDWELL, PHD
© 2007 JOSEPH GEORGE CALDWELL. ALL RIGHTS RESERVED. POSTED AT INTERNET
WEBSITES AND . MAY BE
COPIED OR REPOSTED FOR NONCOMMERCIAL USE, WITH ATTRIBUTION.
Note: The document The Box-Jenkins Forecasting Technique , posted at
, presents a nontechnical description of the
Box-Jenkins methodology. For a technical description of the Box-Jenkins approach, see the
document, TIMES Box-Jenkins Forecasting System , posted at
. A computer program
that can be used to develop a broad class of Box-Jenkins models is posted at the Foundation
website, (6 February 2009).
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BRIEFING ROAD MAP
 1. MATHEMATICAL FORECASTING CONCEPTS (20-30 MINUTES)
 2. TECHNICAL INTRODUCTION TO THE BOX-JENKINS METHODOLOGY (20-30)
 3. FORECASTING ACCURACY COMPARISONS (5-10)
 4. MODEL BUILDING WITH THE BOX-JENKINS METHODOLOGY (40-60)
 5. APPLICATION TO ECONOMETRIC AND CONTROL PROBLEMS (10-15)
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1. MATHEMATICAL FORECASTING CONCEPTS
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MATHEMATICAL FORECASTING METHODOLOGY (“FORECASTER”)
BASED ON A MATHEMATICAL MODEL OF THE PROCESS
TWO APPROACHES
 MODEL FITTING (INTUITIVE, HEURISTIC)
 MODEL BUILDING (THEORETICAL FOUNDATION)
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HEURISTIC FORECASTERS
A PARTICULAR MODEL IS FITTED TO DATA
EXAMPLES:
 MOVING AVERAGE
 EXPONENTIAL SMOOTHING
 TRENDS, CURVES, HARMONICS, PATTERNS
GOOD FIT → GOOD FORECAST?
GOOD MODEL → GOOD FORECAST
NEED TO BUILD A GOOD MODEL
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MODEL BUILDING
CHOOSE A COMPREHENSIVE CLASS OF MODELS
 IDENTIFICATION
 FITTING
 DIAGNOSTIC CHECKING
REPEAT UNTIL ADEQUATE MODEL CONSTRUCTED
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CLASSES OF MODELS (FOR PREDICTION AND CONTROL)
ECONOMETRIC MODELS
 DYNAMIC CAUSAL MODEL (MANY VARIABLES)
 ECONOMIC THEORY
 EXAMPLE: MODEL OF THE ECONOMY
 USUAL METHODOLOGY: ECONOMETRICS (E.G., MULTIPLE REGRESSION
ANALYSIS, TWO-STAGE LEAST SQUARES)
PHYSICAL MODELS
 DYNAMIC CAUSAL MODEL (EQUATIONS OF MOTION)
 PHYSICS
 EXAMPLE: RADAR TRACKING OF A MISSILE
 USUAL METHODOLOGY: KALMAN FILTERING
PURELY STOCHASTIC MODELS (UNIVARIATE, NO EXOGENOUS VARIABLES)
 DESCRIBE STOCHASTIC BEHAVIOR
 TIME SERIES ANALYSIS (EMPIRICAL: NO UNDERLYING ECONOMIC OR PHYSICAL
MODEL)
 EXAMPLE: FORECASTING PRODUCT DEMAND
 USUAL METHODOLOGY: BOX-JENKINS (ARIMA) METHODOLOGY
COMBINATION DYNAMIC-STOCHASTIC MODELS
 FEW VARIABLES (BUT MORE THAN ONE)
 SIMPLE (EMPIRICAL) MODEL OF RELATIONSHIP
 EXAMPLES: TRANSFER-FUNCTION MODEL, FUEL-MIXTURE CONTROL
 USUAL METHODOLOGY: BOX-JENKINS; KALMAN FILTERING, “STATE-SPACE”
MODELS
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FORECASTING ACCURACY
STOCHASTIC VS. HEURISTIC
FORECAST ERROR VARIANCE
LEAD TIME 1 2 3 4 5 6 7 8 9 10
MSE (BROWN) 102 158 218 256 363 452 554 669 799 944
MSE(B-J) 42 91 136 180 222 266 317 371 427 483
ECONOMETRIC VS. STOCHASTIC
THEIL COEFFICIENT
MODEL PRICE QUANTITY
ECONOMETRIC 0.80 0.65
BOX-JENKINS 1.00 0.70
RANDOM WALK 1.00 1.00
MEAN 18.23 0.96
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FORECASTING DIFFICULTY
STOCHASTIC MODEL
