1. Advanced Forecasting
Copyright (c) 2008 by The McGraw-Hill Companies. This material is intended solely for educational use by licensed users of LearningStats. It may not be copied or resold for profit.

2. Copyright Notice Portions of MINITAB Statistical Software input and output contained in this document are printed with permission of Minitab, Inc. MINITABTM is a trademark of Minitab Inc. in the United States and other countries and is used herein with the owner's permission.

3. Trendless Data
For an observed time series y1, y2, ..., yt with no consistent trend, here are three common ways to forecast one period ahead (Ft is the forecast and yt is the actual).
Forecast Method
Characteristics
Ft+1 = yt
Same as Last Period Very simple, but accuracy depends solely on the most recent actual data point.
Ft+1 = (yt + yt yt-n)/n
Average of Past n Periods Simple, but each past data point has equal weight.
Ft+1 = a yt + (1-a) Ft
Exponential Smoothing Assigns a given weight to the most recent actual data point, and its reciprocal weight to the most recent forecast.

4. Single Smoothing
The single-smoothing updating model is Ft+1 = a yt + (1-a) Ft where
yt = actual data in period t.
Ft = forecast for period t.
a = weight given to the current data (0  a  1)
But how do we “seed” the initial forecast for period 1?
Minitab sets F1 to the average of the first six values of yt.
Excel's Tools > Data Analysis > Exponential Smoothing sets F1 to y1.
Unless n is small, it doesn’t make very much difference how we seed F1.

5. Trended Data
For an observed time series y1, y2, ..., yt with consistent trend, here are three common ways to forecast one period ahead (Ft is the forecast and yt is the actual for period t).
Forecast Method
Characteristics
Graph extrapolation
Eyeball Method Visually fitting a trend and projecting it. Simple, but imprecise.
Ft+1 = yt + (yt  yt-n)/n
Same Average Change Simple, but assumes linearity and does not consider variation in trend.
Ft+1 = Lt + Tt
Double Smoothing Uses an updating mechanism to calculate level (Lt) and trend (Tt) components for period t.

6. Double Smoothing The double-smoothing updating model is:
Lt = a yt + (1a) Ft
Tt = b (Lt  Lt-1) + (1  b) Tt-1
Ft+1 = Lt + Tt
where
yt is the actual data value in period t
Lt is the level component in period t
Tt is the trend component in period t
Ft is the forecast for period t
a and b are constants between 0 and 1.

7. Double Smoothing: Example
Living donor transplants in California have a clear upward trend, so double exponential smoothing is appropriate. This example uses a = 0.20 and b = MINITAB’s smoothed (“Fits”) forecasts (shown in red) track the data well for The one-year forecast for 2003 (shown in green) is believable, ceteris paribus. Note MINITAB’s addition of 95% confidence limits.

8. Seasonal Data
For an observed time series y1, y2, ..., yt with both trend and seasonal periodicity p, here are three common ways to forecast one period ahead (Ft is forecast, yt is actual,).
Forecast Method
Characteristics
Ft+1 = yt-p+1
Same as p periods earlier For example, forecast next July the same as last July. Easy, but ignores trend.
Extrapolate trend by month (or quarter or whatever)
p-Trends For example, project fitted trend for all past Julys. Simple, attractive way to handle seasonality.
Ft+m = (Ft + mTt) St
Winters’ Method Forecast m periods ahead based on smoothed forecasts (Ft), trend (Tt), and seasonality (St). Widely used, but hard to explain.

9. Note Smoothing constants follow MINITAB’s notation.
Winters’ Method
To obtain a forecast Ft+m for m periods ahead with seasonal data, the updating equations are:
Ft+m = (Ft + mTt)St
Ft = a yt/St-p + (1a)(Ft-1 + Tt-1)
St = d yt/Ft + (1d)St-p
Tt = g(FtFt-1) + (1g)Tt-1
where
Ft = smoothed series in period t
yt = actual value in period t
Tt = trend estimate in period t
St = seasonality estimate in period t
a = smoothing constant for level
g = smoothing constant for trend
d = smoothing constant for seasonal
Note Smoothing constants follow MINITAB’s notation.
See J. Holton Wilson and Barry Keating, Business Forecasting 2/e, Irwin, 1994), p. 116.

10. Winters’ Method: Example
U.S. oil imports are recorded per 4-week period (13 “months” per year). Using periodicity 13 with data from , MINITAB uses Winters’ method to forecast 2004 (shown in green with 95% confidence limits in blue).
Source:

11. Decomposition
Time series decomposition seeks to separate a time series Y into four components: trend (T), cycle (C), seasonality (S), and irregular (I). These components are assumed to follow either an additive or a multiplicative model:
Model
Components
Used For
Additive
Y = T + C + S + I
Data of similar magnitude (short run or trend-free data) with constant absolute growth or decline.
Multiplicative
Y = T × C × S × I
Data of increasing or decreasing magnitude (long run or trended data) with constant percent growth or decline.

12. Decomposition
We ignore cycles on grounds that there is no accepted theory of cycles. To decompose a monthly* time series Y:
Fit the trend T (assumed linear)
Estimate the seasonal factor Sj for each month:
If additive, average YT for each month
If multiplicative, average Y/T for each month
If forecasts are desired:
extrapolate the trend
add (or multiply by) the seasonal factor for each month
*Quarters or other sub-periods follow similar steps

13. Decomposition: Example
U.S. oil imports are recorded per 4-week period (13 “months” per year). Here is MINITAB’s decomposition using periodicity 13 with data from , and forecasts for 13 “months” in 2004:
Source:

14. Decomposition: Other Exhibits
Here are some additional MINITAB graphs from decomposition of the oil import data.

15. ARIMA Models Basic Model Notation
Original series yt = y1, y2, ... , yn  n observations
Differenced series zt = yt  yt-1 = z1, z2, ... , zn-d  nd observations
AR(1) model (autoregressive of order 1): zt = d + f1zt-1 + et
AR(p) model (autoregressive of order q): zt = d + f1zt-1+ f2zt ft-pzt-p + et
MA(1) model (moving average of order 1): zt = d + et  q1et-1
MA(q) model (moving average of order q): zt = d + et  q1et-1  q2et-2  ...  qt-qet-q
ARMA (p,q) models: zt = d + f1zt-1+ f2zt ft-pzt-p
+ et - q1et-1  q2et-2  ...  qt-qet-q
ARIMA(p,d,q) models (d is number of differences in working series)
See J. Holton Wilson and Barry Keating, Business Forecasting 2/e, Irwin, 1994).

16. ARIMA
Model identification depends on the data. It is rarely necessary to go beyond AR(1) or MA(1), although seasonal data will require fancier models. Criteria for model adequacy can be found in any forecasting textbook. Time series modeling using ARIMA is an advanced topic that requires additional study*.
* See J. Holton Wilson and Barry Keating, Business Forecasting 2/e, Irwin, 1994).

17. AR(1) Patterns
See J. Holton Wilson and Barry Keating, Business Forecasting 2/e, Irwin, 1994).

18. MA(1) Patterns
See J. Holton Wilson and Barry Keating, Business Forecasting 2/e, Irwin, 1994).

19. ARIMA: Example
Here is MINITAB’s ARIMA for U.S. Oil Imports using an ARIMA model with one seasonal difference on data from with forecasts for 13 “months” in The forecasts have very wide confidence bands.

20. ARIMA: Model Fit
MINITAB’s results show that the model needs refinement (p-values for B-P statistics are too small) and the ACF and PACF plots (next screen) suggest possible need for seasonal terms.
Type Coef SE Coef T P AR MA
Constant
Differencing: 0 regular, 1 seasonal of order 13
Number of observations: Original series 104, after differencing 91
Residuals: SS = (backforecasts excluded)
MS = DF = 88
Modified Box-Pierce (Ljung-Box) Chi-Square statistic
Lag
Chi-Square DF
P-Value

21. ARIMA: ACF and PACF Residual Plots
Spikes at 7 and 13 suggest model revision.

22. Other Methods Seasonal binaries in regression
Moving average to remove trend (instead of linear OLS trend) then decomposition on residuals
Ad hoc methods

23. Final Advice Remember Occam’s Razor Don’t be dazzled by equations
Occam's Razor Given two sufficient explanations, we prefer the simpler. Named for the English philosopher William of Occam (d. 1349).
Remember Occam’s Razor
Don’t be dazzled by equations
Think about underlying causes
Are your forecasts credible?
Clean, simple graphs help