1. Cycles and Exponential Smoothing Models
Materials for this lecture
Lecture 10 Cycles.XLS
Lecture 10 Exponential Smoothing.XLSX
Read Chapter 15 pages 18-30
Read Chapter 16 Section 14

2. How Does Regression Work?
Yt = a + b1 X1t + b2 Tt + b3 Yt-1 + b4 SIN()t + b4 COS()t

3. A Score for Your Forecast?
MAPE -- Mean absolute percent error
Statistic often used to determine how good your forecast is at forecasting the historical period
MAPE = ∑ [ (Ai – Fi) / Ai ] * (100/N)
Where Ai is the actual value in period i and
Fi is the forecasted value in period i
N is the number of historical periods
MAPE is the average percentage error for a forecast

4. How Good is Your Forecast?
Can your forecast beat a Moving Average?
Business forecasters use Moving Average as a reference forecast. They compare the:
MAPE for MA model
MAPE for your model
Example of two Data Series
X with a Moving Average MAPE of 23%
Your structural model’s MAPE of 15%
Y with a Moving Average MAPE of 12%
Your structural model’s MAPE of 10%
Which is the better model?

5. Cycles, Seasonal Decomposition and Exponential Smoothing Models
Business cycle
Beef cycle
Hog cycle
Weather cycle?
Cycles caused by over correction of an economic system
The Cob Web Theorem in action

6. Cycles and Exponential Smoothing Models
Cyclical analysis involves analyzing data for underlying cycles
Estimate the length of an average cycle
Forecast Y variable in part based on cycle length, may still include trend, seasonal, and structural variables
Exponential Smoothing is the most often used forecasting method in industry
Easy to use and update, very flexible
Only forecasts a few periods ahead is its major disadvantage

7. Cyclical Analysis Models
Harmonic regression model estimated with OLS regression used to estimate cycle length
Sin and Cos use CL variable
Recall Seasonal analysis used SL
Length of data needed: Enough observations to observe several cycles
Two considerations in estimating cycle length and specifying the OLS model
Annual data can easily exhibit a cycle
Monthly data can show a seasonal pattern around a multiple year cycle

8. Cyclical Analysis Models
Define CL = Number of years in the cycle
CL is used in both the Sin and Cos functions
If you are using Annual data CL equals the number of years for the cycle
If you are using Monthly data
Define CL = SL * No. Years for cycle length
where SL = 12 number of months in a year
If you are using Quarterly data
where SL = 4 number of quarters in a year

9. Cyclical Analysis Models
OLS regression model for annual data
Ŷ = a + b1T + b2 Sin(2*ρi()*T/CL) b3 Cos(2*ρi()*T/CL)
where: CL = possible number of years for a cycle
Steps to estimate best cycle length with Simetar
Enter CL in a cell
Reference the cell with CL to calculate all of the Sin() and Cos() values in the X matrix
Estimate regression model
Change the value for CL, observe the F ratio or MAPE
Repeat process for numerous CL values and find the CL associated with the largest F ratio or the lowest MAPE

10. Cyclical Analysis Models
OLS regression model for Monthly data
Ŷ = a +b1T+ b2 Sin(2*ρi*T/SL) + b3 Cos(2*ρi*T/SL)
+ b4 Sin(2*ρi*T/CL) + b5 Cos(2*ρi*T/CL)
where: SL = No. months (quarters, or weeks) in a year and CL = SL * No. years for a cycle to Test
Steps to estimate the best cycle length with Simetar
Enter the SL value in a cell
Calculate a value for CL where: CL = SL * Years
Refer to the cell with SL to calculate the first Sin() and Cos() values in your X matrix
Refer to the cell with CL to calculate the second Sin() and Cos() values in your X matrix
Estimate regression model in Simetar
Change the no. of years used to calculate CL, record F or MAPE
Repeat process for different CL values for no. of years and pick the CL associated with the highest F or the lowest MAPE

11. Cyclical Analysis Models with Annual Data
Part of the Y and X matrix for annual data
Sin and Cos functions refer to CL in C49

12. Cyclical Analysis Models with Monthly Data
Y and X matrix for a monthly data series
Sin and Cos functions refer to CL and SL
in C11 and F11
Lecture 4

13. Cyclical Analysis Model Results
Sample table of R2 and MAPE for CL’s
CL = 9 for the chart and regression shown here, based on maximum MAPE

14. Exponential Smoothing Models
ES is the most popular forecasting method
Very good for forecasting a few periods
Like moving average, but greater weights placed on more recent observations
MA assumes equal weights for each lagged value, i.e., XT+I =(XT-3 + XT-2 + XT-1 ) / 3
ES assumes weights are different
i.e., XT+I =((1-β) * XT-2 + β * XT-1 ) / 3

15. Exponential Smoothing Models
2. Additive seasonal variability with an additive trend (1,1)
1. No trend and additive seasonal variability (1,0)
3. Multiplicative seasonal variability with an additive trend (2,1)
4. Multiplicative seasonal variability with a multiplicative trend (2,2)

16. Exponential Smoothing Models
5. Dampened trend with additive seasonal variability (1,1)
6. Multiplicative seasonal variability and dampened trend (2,2)
Select the type of model to fit based on the presence of
Trend – additive or multiplicative, dampened or not
Seasonal variability – additive or multiplicative
Do this prior to the estimation if not using Simetar. With Simetar you can experiment with different specifications after the model is estimated
Can select 3 seasonal effects: none, additive, multiplicative
Can select 3 trend effects: none, additive, multiplicative

17. Exponential Smoothing Models
Different forms of ES models (options in Simetar)
1. Simple exponential smoothing, additive seasonal and no trend (1 seasonal ,0 trend)
2. Additive seasonal and additive trend (1,1)
3. Additive trend and multiplicative seasonal variability (2,1)
4. Multiplicative trend and multiplicative seasonal variability (2,2)
5. Dampened trend ES with additive seasonal variability (1,1)
6. Dampened trend ES with multiplicative seasonal variability (2,2)
Numbers match chart numbers in last two slides
Numbers in ()’s match Simetar ES option settings

18. Exponential Smoothing Forecasts
Using the Forecasting Icon for ES
Data on the Excel toolbar to get Data Ribbon
Select Solver
Close Solver
Select the “Exponential Smoothing” tab in the menu Specify the data series to forecast (see next slide for the menu)
Provide initial guesses for
Dampening Factor (0.25),
Optional Trend Factor (0.5), and
Optional Season Factor (0.5) if monthly or quarterly data
Indicate the Optional Seasons per Period as 12 if monthly data
Forecast Periods of 1 to 6

19. Exponential Smoothing Models
Simetar estimates many different forms of ES models
Provides deterministic forecasts
Provides probabilistic forecasts
Parameters for ES model estimated by Solver to minimize MAPE for residuals
PRIOR to running ES, You MUST open Solver and close it so Simetar can Optimize Parameters
Provide starting guesses for parameters 0.25 to 0.50
Enter no. of periods/year if monthly or quarterly data

20. Exponential Smoothing Models
Initial Parameters for ES
Dampening Factor is required for all models – good guess is 0.25
Optional Trend factor entered as 0.5 if the data have any trend
Optional Seasonal factor, 0.5, if the data are monthly or you have >30 years annual data (with annual data you have a cycle)
Optional Seasons per Period
Indicate the number of months
for seasonal effect as 12
Indicate cycle length if using annual data, say 9 years

21. Exponential Smoothing Models
ES Options
Trend Method
0 No trend dampening
1 Dampened Additive
2 Dampened Multiplicative
Season Method
0 No seasonal effects
1 Additive seasonal effect
2 Multiplicative seasonal effect
Stochastic Forecast
TRUE
FALSE

22. Exponential Smoothing Models
Experiment with alternative settings for the Trend and Seasonal Smoothing variables to see which combination is best
All possible combinations are listed below
Look for the model formulation with the lowest MAPE

23. Exponential Smoothing Models
2. Additive seasonal variability with an additive trend (1,1)
1. No trend and additive seasonal variability (1,0)
3. Multiplicative seasonal variability with an additive trend (2,1)
4. Multiplicative seasonal variability with a multiplicative trend (2,2)

24. Exponential Smoothing Models
5. Dampened trend with additive seasonal variability (1,1)
6. Multiplicative seasonal variability and dampened trend (2,2)
Select the type of model to fit based on the presence of
Trend – additive or multiplicative, dampened or not
Seasonal variability – additive or multiplicative
Do this prior to the estimation. With Simetar you can experiment with different specifications after model is estimated
Can select 3 seasonal effects: none, additive, multiplicative
Can select 3 trend effects: none, additive, multiplicative