A time series is a sequence of measuments made at regular times, for example, daily temperature at noon, monthly earnings, number of babies born each year in Chicago.

Notation | Common Types | Features |

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The letter t will donote the time period, t = 1, 2, and so on. The measurement at time t will be written x[t]. The next measurement in the sequence would be x[t+1]. The preceding measurement would be x[t-1]. On paper the t would normally appear as a subscript rather than in brackets. Letters other than x can be used to denote distinct time series, for example, y[t]. Writing all the symbols on one base line is awkward, but the result is readable with any web browser.

The forecast made at time t of the measurement at time t+L will be written x^[t](L). Think of the notation this way. The first letter, x in this case, specifies a particular time series. The circumflex ^ identifies the value as an estimate rather than a known value. This is a common part of notation in statistics. The number in brackets is the time at which the estimate is made. The number in parentheses is the leadtime for making the forecast. For example, x^[2](3) is the forecast made at time 2 for the measurement that will be made at time 5.

In practice the most common leadtime is one period. For ecomony the one period ahead forecast x^[t](1) will be written simply x^[t].

The error of a forecast is x[t+L] - x^[t](L). When context identifies the time series under consideration, this error will be written e[t](L). When L = 1, the error will be written simply e[t].

Common Types of Time Series

Demographic
- population
- births
- marriages
- immigration
- emigration
- diseases
- accidents
Economic
- activity
  - gross domestic product
  - consumer spending
  - government spending
  - investment
  - exports and imports
- prices
  - price levels and inflation
  - stock prices
  - interest rates
- utilization rates
  - factory capacity
  - productivity
  - unemployment
Natural phenomena
- weather
- sunspots

Features of Time Series

Noise
Structure
- level
- trend
- repeating patterns
  - Regular repetition is called seasonality, for example, every 12 months, 4 quarters, 7 days, or 24 hours.
  - Business cycles are an example of irregular repetition.
Complexity
- univariate
- multivariate
- externalities

Loss Functions

Qualitative | Quantitative

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Qualitative

Areas That Forecasts Impact

Quantitative

Ultimate Variables

number of people served
profit
balance sheet amounts

Single Period Loss Measurements

simple difference, that is, the one period ahead error e[t]
absolute error, |e[t]|, which says the cost of forecasting high is the same as forecasting low
squared error, e[t] times itself, which emphasizes large errors
log absolute error, log( e[t] )
relative error, e[t] / x[t]

Multi Period Loss Measurements

distribution of differences
- maximum
- median
- minimum
sum of differences
- simple sums
- absolute
- squared
- relative
averages
- arithmetic
- geometric
general metrics
- "little ell two"
- weighted averages
  - Example Using Sum[t] |e[t-1] + 2e[t] + e[t+1]| / 4 as a loss function could fit an environment where being too high one period is offset by being too low the next. For example, with errors +1, -1, +1, -1, ... the loss function is always zero.
  - incorporate backorderig
  - incorporate loss carry forwards
  - What loss function would fit an enviroment in which it does not matter if forecasts can be one period late or one period early without any penalty?

If people behave consistently over time in a given situation requiring forecasts, that implies their acceptance of some loss function.

Forecast Methods

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Basic Concepts

Unchanging Features

All forecasting methods identify things that continue as they are, for example

value
trend
repeating seasonal pattern
influence of one time series on another

Averaging

The Law of Large Numbers says that averages over larger and larger samples are more and more likely to be accurate. Averaging lets certain types of changes cancel each other out. That is good if the changes are random noise. Noise is a succesion of changes that are unrelated. Noise can give the illusion of a pattern, like in Rorschach inkblots. [Herman Rorschach, born in Zurich, 1888-1922.] Averaging out a known pattern, such as seasonal change, can also let us look for other patterns.

Numerical averaging works with time series.
The Delphi Method averages the opinions of experts in qualitative forecasting.

Matching Procedure to Environment

Qualitative features and low frequency call for judgment.
Repeated numeric forecasts call for standardization and mathematics.

Tradeoffs

Methods good in one environment can be bad in another environment. For example forecast methods that are good a detecting trends in stock prices often miss turning points between up trends and down trends.

Judgment Based Methods

Extrapolation

naive: identify one or a few simple features as unchanging. "The more things change, the more things stay the same."
complex, such as all historical factors
successive scenarios

Composite Opinion

bottom up, such as sales force opinions
top down, such as executive group opinion
Delphi technique, to control human interactions

Survey Based Methods

What potential customers want. Potential customer reactions to proposed products

Time Series Methods

Naive : x^[t+1] = x[t]
Moving Average : x^[t+1] = (1/n)(x[t]+x[t-1]+...+x[t-n+1])
or in update form : x^[t+1] = x^[t] + (1/n)(x[t]-x[t-n])
The value n is a whole number parameter, 2 or greater.
Exponential Smoothing. Exponential smoothing is a family of efficient formulas that often are as accurate as more sophisticated methods.
Simple exponential smoothing has the update form : x^[t+1] = x^[t] + (alpha)( x[t]-x^[t] )
The value alpha is a parameter, typically between 0.1 and 0.2, but possibly between 0.0 and 2.0.
naive trend : x^[t+1] = x[t] + (x[t] - x[t-1])
naive seasonal : x^[t+s] = x[t] where there are s periods per cycle
naive seasonal trend: x^[t+s] = x[t] + ( x[t+s-1] - x[t-1] )
Fitted Parametric Models
- regression
- decomposition
  additive : x[t] = B + T[t] + S[t] + C[t] + N[t]
  or
  multiplicative : x[t] = B * T[t] * S[t] * C[t] * N[t]
  - B is the base level
  - T[t] is the trend component
  - S[t] is the seasonal component, also called seasonal index
  - C[t] is a long term cylce such as a business cycle
  - N[t] is the noise component
  Averaging techniques separate the components. For example, simple linear regression averages out noise, seasonality, and long term cycles, leaving only level and trend. Centered moving averages, each over one circle of seasons, leave level, trend, and long term cycles.
Bayesian updating of parameters
Statistical methods
statistical techniques not only produce forecasts but also quantify precision and reliability.
- autoregressive
- moving average
- Box-Jenkins
- ARIMA (p,d,q)
- filters

Causal Models

Regression
- correlated variables
- leading indicators
Econometric Models
- input-output matrices
- general econometrics
- nonlinear systems

Combination Forecasts

Integration of Judgment and Quantitative Forecasts
Simple and Weighted Averages

References

Books | Journals | Articles

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Books

Anderson, O. D., editor,
Forecasting, Proceedings of the Institute of Statisticians, Annual Conference, Cambridge, 1976
North-Holland Publishing, Amsterdam-New York, 1979
QA, 279.2, I 56, 1979

See pages 43-166 for the article by
Jenkins, Gwylim M.,
Practical experience with modeling and forecasting time series

Analyze decision making system served by forecasts.
- policies and style
- types of decisions and actions
Define forecasts needed to serve decision making system.
- variables to be forecast
- planning horizon
- frequency
- accuracy
- level of aggregation
Develop conceptual model.
- mechanisms
- environment
Define data available and not available.
Decide what to do about gaps.
Develop method for generating forecasts, in light of
- accuracy needed
- availability and reliability of data
- policy and environmental variables
- cost
- skills available
Conduct experiments to assess accuracy of forecasts over varying leadtimes.
Determine how judgments are to be incorporated for untypical events.
Implement the forecasting system.
Appraise retrospectively its effectiveness.

Granger, C. W. J.
Some Recent Developments in Forecasting techniques and strategy

What information set is to be used?
- numeric
- non-numeric
- related series
How is this information set to be utilized?
- use time series values as originally recorded
- use some function of original values
- use pre-whitened values
What criterion function should be used?

Combinations of forecasts are good.

Box, George E. P. and Gwilym M. Jenkins
Time Series Analysis: Forecasting and Control. Revised Edition
Holden-Day, Oakland California, 1976

classic text
model building
1. identification of the model
2. estimation of parameters
3. fitting
4. analysis of residuals
5. diagnostic checking
multivariate series

Brown, Robert G. Holt, Rinehart and Winston, New York, 1967
HD, 55, B7

The author is a pioneer in computer assisted inventory management.
Warmdot Company is an extended example throughout.

fitting demand patterns
exchange curves

Granger, C. W. J. and Paul Newbold
Forecasting Economic Time Series
Academic Press, New York, 1986
HB, 3730, G67, 1986

Model building
Box-Jenkins, exponential smoothing, Holt-Winters
stepwise autoregression
state space
comparison of methods over many time series

Pankratz, Alan
Forecasting with Univariate Box-Jenkins Models
John Wiley & Sons, New York, 1983
QA, 280, P37, 1983

basic exposition
detailed examples of particular series

Smith, Bernard T.
Focus Forecasting: Computer Techniques for Inventory Control
CBI Publishing Company, Boston, 1978
HD, 55, S48

forecasts next three months by
- picking best simple predictor for past three months
- based on prior 15 months
empirical tradeoffs between turnover and service
target inventory level is a multiple of forecast alone
salesman style
technical details omitted

Wagner, Harvey M.
Statistical Management of Inventory Systems
John Wiley & Sons, New York, 1962
HD, 55, W3

Mathematical treatment
stationary reorder point and order quantity models
aggregate versus item by item rules
aggregate performance indices
multi-echelon inventories
interaction between levels of management as in agency theory

Webster, Charles E.
The Executives Guide to Business and Economic Forecasting
Probus Publishing Company, Chicago,Illinois, 1986
HD, 30.27, W4, 1986

conceptual presentation
lots of general economic series discussed

Wheelwright, Steven C. and Spyros Makridakis
Forecasting Methods for Management, fourth edition
John Wiley & Sons, New York, 1985
HD, 30.27, W46, 1985

broad management text
limitations
simple methods and judgment criteria
exponential smoothing variations
decomposition: seasonality, trend, cycle, error
Box-Jenkins, Parzen, Kalman filters, Lewandowski, multivariate
regression
econometric modeling
monitoring
data development and computers
- variables
  - time period covered by each data value
  - level of detail required
  - frequency with which the data are required
  - unit of measurement
  - required level of accuracy
- data sources
- data accuracy
  - sampling error
  - measurement error
  - deliberately hidden or distorted information
  - poorly designed questionnaires
  - data aggregates, omission, double counting
  - classification and definition
  - time lags
- data bases, updating, auditing
selection of methods
qualitative and technological methods
judgmental methods

Wight, Oliver W.
Production and Inventory Management in the Computer Age
Cahners Books International, Boston, 1974

cautions against complex forecasting methods, for example,
- double exponential smoothing
- any computer forecasting for lumpy demands (page 152)
Forecast Characteristics (page 156)
- Forecasts are more accurate for larger groups of items.
- Forecasts are more accurate over shorter leadtimes.

Journals

Management Science
Operations Research
Operational Research Quarterly
Journal of Forecasting
Journal of Time Series Analysis
Biometrika

Articles

Deborah Adamson, Off the Mark: Earnings Estimates Miss More Than They Hit, Daily News, April 14, 1996, Business pages 1 and 3.

Peter Drucker, Wall Street Journal, December 1, 1992.

Everett S. Gardner, Jr., Exponential Smoothing: the state of the Art, Journal of Forecasting, Vol. 4, pages 1-28, 1985.

David M Georgoff and Robert G. Murdick, Manager's Guide to Forecasting, Harvard Business Review, January-February, 1986, pages 110-120 plus table.

A. C. Harvey, A Unified View of Statistical Forecasting Procedures, Journal of Forecasting, Vol. 3, pages 245-275, 1984.

Dana Wechsler Linden, Dreary Days in the dismal Science, Forbes, Jan. 21, 1991, pages 68-70.

S. Makridakis et al., The Accuracy of Extrapolation (Time Series) Methods: Results of a Forecasting Competition, Journal of Forecasting, Vol. 1, pages 111-153, 1982.

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