Principles of Forecasting Principles of Forecasting: Applications in Revenue and Expenditure...

Principles of Forecasting

Principles of Forecasting: Applications in Revenue and Expenditure Forecasting

Michael L. Hand, Ph.D

Professor of Applied Statistics and Information Systems

Atkinson Graduate School of Management

Willamette University, 900 State Street, Salem, OR 97301

[email protected], 503.370.6056

Michael L. HandProfessor of Applied Statisticsand Information Systems

Atkinson Graduate School of ManagementWillamette University

October 18, 2006

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Presentation Overview

Philosophy/Perspective Taxonomy of Methods Forecasting Process with Special Attention to

Knowledge Acquisition Data Understanding Model Interpretation Model Assessment



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What is (a/to) Forecast?

A Forecast: (noun) a prophecy, estimate, or prediction of a future happening or condition

To Forecast: (verb) to calculate or predict some future event or condition, usually as a result of study and analysis of available pertinent data

The forecast process offers far greater potential return than merely the forecast; that is, the journey is more important than the destination.



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Challenges

Prediction is very difficult, especially if it's about the future.

Nils Bohr, Nobel laureate in physics

(though this sounds a lot more like Yogi Berra)



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Why Forecast?

The effectiveness of almost every human endeavor, every public initiative, depends in part upon unknown and uncertain future outcomes – the demand for services, the revenues to fund them.

The quality of decisions about whether or not to engage and at what level improves with the reliability of supporting forecasts.



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Why Forecast?

For every level of demand, there is a best level of service capacity.

0

20

40

60

80

100

120

140

0.0 5.0 10.0 15.0 20.0 25.0

Demand

Service Capacity



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Why Forecast?

In short, we forecast because we have little choice. A forecast is implied by essentially every decision that we make, every action that we take.

It is far better to foresee even without certainty than not to foresee at all.

Henri Poincare in The Foundations of Science



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Forecast Risks/Costs

Prophesy is a good line of business, but it is full of risks.

Mark Twain in Following the Equator

Forecast high Cost of excess capacity, misallocations

Forecast low Kicker



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Forecast Objective

Perfection? Forecasts that are without error? A naïve and unproductive view. Fails to properly manage expectations about the forecasting process. Preoccupation with being “right” is unhealthy and only serves to stifle the process.

Objective: Minimize Forecast Errors(and associated costs)

It is sufficient to develop forecasts that systematically reduce uncertainty (and thereby reduce the risks and costs associated with forecast errors.)



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A Brief Taxonomy of Forecasting Methods

Subjective

Expert Opinion

Survey Research

Historical Analogy

Objective/Data-Based

Associative

Multiple Regression

Econometric Models

Projective

DecompositionSmoothing

Time-Series Regr’n

Box-Jenkins/ARIMA



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Subjective Methods

Methods based primarily on judgment/expert opinion Generally little or no data to directly support forecast

requirementHistorical analogy may rely upon data from a comparable process

Best for long-range forecastsMore than two years out



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Data-Based Forecasting

In God we trust, all others bring data.W. Edwards Deming



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Associative Methods

“Causal”, multiple regression models relating response to a general set of predictors

Data/supporting forecast requirementIncreased model complexity and development effort

Assumes relationships among response and predictors are stable over time

Best for intermediate-term forecastsOne- to two-year forecast time horizon



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Associative Models

Oregon Personal Income Tax versus Unemployment

600

700

800

900

1000

1100

1200

1300

1400

1500

3.5 4.0 4.5 5.0 5.5 6.0

Unemployment Rate (%)

Personal Income Tax (in $Millions)



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Econometric ModelsLOG(GIwages) = 20.7 + 0.93*LOG(PIwages + PIother_lab) + [AR(1)=0.85]LOG(GIdividends) = 16.7 + 0.49*LOG(PIdir) + 0.30*LOG(MKTw5000)LOG(GIinterest) = 19.6 + 0.34*LOG(PIwages) + 0.04* IR3mo_tbill + 0.039* IR3mo_tbill (-1) + [AR(1)=0.65]LOG(GIcapgains) = 11.5 + 1.14*LOG(MKTw5000) + [MA(4) = -0.86]LOG(GIretirement) = -0.12 + 1.24*LOG(POP_OR65+) + 0.97*LOG(PItotal – PIwages) + 0.32*LOG(MKTw5000) +

[AR(1)=-0.50] LOG(GIproprietors) = -304.7 + 0.72*LOG(PIproprietors) + 2.10*LOG(EMPretail) + [AR(1)=1.0]LOG(GIschedule_e) = 14.4 + 1.1*LOG(CORP_PROFIT) + [AR(1)=0.78] LOG(GIother) = -2.1 + 4.14*LOG(EMPretail) Eff_tax_rate = 0.05 + 0.005* DMYtax_rate + 0.053* FDIST1mil + 0.04*(( GIschedule_e + GIproprietors)/ GIwages) +

[AR(1)=0.58]

GI - Gross Income from the source indicated PItotal – Total Oregon Personal Income PIwages – Wage and Salary Component of Personal Income PIother_lab – Other labor component of Personal Income PIdir – Dividends, Interest and Rent component of Personal Income PIproprietors – Proprietors’ Income component of Personal Income MKTw5000 – Wilshire 5000 stock indexEMPretail – Oregon Retail Employment CORP_PROFIT – U.S. Corporate Profits POP_OR65+ – Oregon 65 and older population IR3mo_tbill – Discount rate of 3 month Treasury Bill FDIST1mil - Filer Distribution Model, Ratio of $1 million-plus filers to Total filers DMYtax_rate – Dummy variable for 1982 through 1984 tax rate increase

http://egov.oregon.gov/DAS/OEA/docs/revenue/pit_forecastmethod.pdf

Personal Income Tax ModelOffice of Economic AnalysisDepartment of Administrative Services



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Projection/Extrapolation

I have seen the future and it is very much like the present, only longer.

Kehlog Albran, The Profit



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Projective Methods

Simple extrapolation in time Predictors are time and functions of time

Trend, seasonal, cyclical factors Minimal data/supporting forecast requirement Assumes current conditions will persist Best for short-term forecasts

One year out (two if we stretch) or less



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Projective Models

Oregon Personal Income Tax Revenues (in $ Millions)

600

700

800

900

1000

1100

1200

1300

1400

1500

1600

1996:011996:021996:031996:041997:011997:021997:031997:041998:011998:021998:031998:041999:011999:021999:031999:042000:012000:022000:032000:042001:012001:022001:032001:042002:012002:022002:032002:042003:012003:022003:032003:042004:012004:022004:032004:042005:012005:022005:032005:042006:012006:022006:032006:042007:012007:022007:032007:04

Time

Data/Forecasts/Level

DataForecastLevel

Winters’ Seasonal Exponential Smoothing



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Forecasting Process

Enterprise Understanding Data Understanding Alternative Model Identification Model Estimation Model Assessment – Adequacy, Quality Model Selection Model Interpretation Forecasting

Important (oft overlooked) knowledge acquisition stages(see Class_Tools:Hand_Outs:Forecasting:NNG_Paper.pdf)



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600

700

800

900

1000

1100

1200

1300

1400

1500

1996:011996:021996:031996:041997:011997:021997:031997:041998:011998:021998:031998:041999:011999:021999:031999:042000:012000:022000:032000:042001:012001:022001:032001:042002:012002:022002:032002:042003:012003:022003:032003:042004:012004:022004:032004:042005:012005:022005:032005:04Period

Data/Forecast

Example: Oregon Personal Income Taxes, 1996 – 2005

Data Understanding

(see Class Tools resource Hand_Outs:Forecasting:MultDecompPITFull.xls)

Note dramatic shift in level and nature of seasonal variation



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600

700

800

900

1000

1100

1200

1300

1400

1500

1996:011996:021996:031996:041997:011997:021997:031997:041998:011998:021998:031998:041999:011999:021999:031999:042000:012000:022000:032000:042001:012001:022001:032001:04Period

Data/Forecasts

Example: Oregon Personal Income Taxes, 1996 – 2001

Data Understanding

(see Class Tools resource Hand_Outs:Forecasting:MultDecompPIT.xls)

For simplicity, we restrict our initial view to the fairly stable period

from 1996 – 2001



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Example: Classical Multiplicative Decomposition

[ ]⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

×+=

×=

×××=

L

t

ttt

ttttt

s

s

s

tbby

SeasonalTrendy

IrregularSeasonalCycleTrendy

M2

1

10ˆ

ˆ

Conceptual Decomposition:

Conceptual Forecast:

Forecasting Model:

Trend: Long-term growth/declineCycle: Long-term slow, irregular oscillationSeasonal: Regular, periodic variation w/in calendar yearIrregular: Short-term, erratic variation



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Example: Classical Multiplicative Decomposition

ttttt IrregularSeasonalCycleTrendy ×××=Conceptual Decomposition:

600

700

800

900

1000

1100

1200

1300

1400

1500

1600

1996:011996:021996:031996:041997:011997:021997:031997:041998:011998:021998:031998:041999:011999:021999:031999:042000:012000:022000:032000:042001:012001:022001:032001:04Period

Data



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Example: Classical Multiplicative Decomposition Visual Representation

600

800

1000

1200

1400

1600

Trend

0.85

0.95

1.05

1.15

1.25

Seasonal

0.85

0.95

1.05

1.15

1.25

Cyclical

0.85

0.95

1.05

1.15

1.25

Irregular



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Example: Classical Multiplicative Decomposition, Model Interpretation

[ ] [ ]⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

×+=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

×+=

2057.1

8913.0

9236.0

9794.0

5017.189291.731ˆ,ˆ 2

1

10 ty

s

s

s

tbby t

L

t M

Model Interpretation

Initial, time-zero (1995:Q4) level is $731.92 millionIncreasing at $18.5 million per quarter Seasonal pattern

Peak in Q4 21% over trendTrough in Q3 11% below trend



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Example: Classical Multiplicative Decomposition, Forecasts

[ ] [ ]⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

×+=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

×+=

2057.1

8913.0

9236.0

9794.0

5017.189291.731ˆ,ˆ 2

1

10 ty

s

s

s

tbby t

L

t M

Forecasts

( ) [ ] [ ] [ ] [ ]

( ) [ ] [ ] [ ] [ ]

( ) [ ] [ ] [ ] [ ]

( ) [ ] [ ] [ ] [ ]

( ) [ ] [ ] [ ] [ ] 1507.071249.9779ytQ

1097.601231.4762ytQ

1120.321212.9744ytQ

1169.901194.4727ytQ

735.00750.4309ytQ

=×=×+==

=×=×+==

=×=×+==

=×=×+==

=×=×+==

2057.12057.1)28(5017.189291.731ˆ284:2002

8913.08913.0)27(5017.189291.731ˆ273:2002

9236.09236.0)26(5017.189291.731ˆ262:2002

9794.09794.0)25(5017.189291.731ˆ251:2002

9794.09794.0)1(5017.189291.731ˆ11:1996

28

27

26

25

1

M



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Forecast Model Assessment

Residual analysis: A somewhat scatological endeavor, whereby we assess forecast quality through an analysis of residuals or what the forecast process leaves unexplained.

Residual (Error) = Actual – Forecast

Assessment possible for any type of forecasting process – statistical, organizational, ad hoc, arbitrary.



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Example: Classical Multiplicative Decomposition, Residuals/Errors

Time Year Qtr Period Tax Forecast Error1 1996 1 1996:01 700.38 735.00 -34.622 1996 2 1996:02 694.46 710.19 -15.743 1996 3 1996:03 731.49 701.83 29.664 1996 4 1996:04 933.63 971.70 -38.07

…21 2001 1 2001:01 1075.97 1097.42 -21.4522 2001 2 2001:02 1011.88 1051.96 -40.0823 2001 3 2001:03 1063.42 1031.64 31.7924 2001 4 2001:04 1399.33 1417.84 -18.5025 2002 1 2002:01 1169.9026 2002 2 2002:02 1120.3227 2002 3 2002:03 1097.6028 2002 4 2002:04 1507.07



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Example: Classical Multiplicative Decomposition, Time Series Plot of Residuals


-60

-40

-20

0

20

40

60

80

100

1996:011996:021996:031996:041997:011997:021997:031997:041998:011998:021998:031998:041999:011999:021999:031999:042000:012000:022000:032000:042001:012001:022001:032001:04

Period

Prediction Errors (in $ Millions)



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Desirable Properties of Residuals

Small aggregate error measure Independent/random

No remaining pattern Mean zero, Unbiased Constant variance Normality

Required for many statistical assessments

These properties can be tested with a variety of charts and graphs too numerous to mention here.



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Measures of Forecast Accuracy

Error Summary Measures Mean Squared Error, MSE Mean Absolute Deviation, MAD Mean Absolute Percentage Error, MAPE Mean Percentage Error, MPE (Bias)

R2 = (SSTO – SSE)/SSTO Proportion (or percentage) of variation

“explained by” (or “attributable to”) forecast model

Prediction Intervals



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Example: Classical Multiplicative Decomposition, Measures of Forecast Accuracy

Error Summary Measures Mean Squared Error, MSD Std Deviation of Residuals, s ≈ √MSD Mean Absolute Deviation, MAD Mean Absolute Pct Error, MAPE Mean Pct Error, MPE (Bias)

R2 = (SSTO – SSE)/SSTO

SSTO 870280SSE 32390

MSD 1349.60s 36.74MAD 30.08MAPE 3.07%MPE -0.15%

R2 96.28%

Summaries

600

700

800

900

1000

1100

1200

1300

1400

1500

1600

1996:011996:021996:031996:041997:011997:021997:031997:041998:011998:021998:031998:041999:011999:021999:031999:042000:012000:022000:032000:042001:012001:022001:032001:042002:012002:022002:032002:042003:012003:022003:032003:04

Data/Forecast



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Conclusion

Forecasting process can be about far more than mere forecasts, it can also provide for essential Knowledge Acquisition Data Understanding Model Interpretation Model Assessment



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Basic Forecasting References

Armstrong. Long-Range Forecasting: From Crystal Ball to Computer. Wiley-Interscience, 1978.

(Also available in .pdf form at: http://www-marketing.wharton.upenn.edu/forecast/Long-Range%20Forecasting/contents.html

Bowerman, O'Connell, Hand. Business Statistics in Practice, 2nd Edition. McGraw-Hill/Irwin, 2001.

Bowerman, O'Connell, Koehler. Forecasting, Time Series and Regression, Fourth Edition. Duxbury Press, 2005.

Hanke and Wichern. Business Forecasting, 8th Edition. Prentice-Hall, 2005.

Makridakis, Wheelwright, Hyndman. Forecasting Methods and Applications, 3rd Edition. John Wiley and Sons, 1998

Principles of Forecasting Principles of Forecasting: Applications in Revenue and Expenditure...

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