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

How Good is Your Forecast?

• Can your forecast beat a Moving Average?

• Business forecasters use Moving Average as a reference forecast– 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?

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

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 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

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

• Need enough observations to see several cycles in the data series

• 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

Cyclical Analysis Models• If you are using Annual data

– Define CL = Number of years * SL – 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 (SL = 1)

• 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

– Define CL = SL * No. Years for cycle length where SL = 4 number of quarters in a year

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 is possible number of years for a cycle

• Steps to estimate the best cycle length– Enter CL in a cell– Refer to the cell with CL to calculate the Sin() and

Cos() values in the X matrix– Estimate regression model in Simetar for a CL– Change the value for CL, observe the F ratio or

MAPE – 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

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

• Steps to estimate the best cycle length– Enter the No. Years in a cell– Calculate CL in a cell with CL = SL * Years– Refer to the cell with CL to calculate the second Sin() and

Cos() values in X matrix– Estimate regression model in Simetar for No. of Years– Change the no. of years in cycle (i.e., CL), observe the 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

Cyclical Analysis Models

• Part of the Y and X matrix for annual data

• Sin and Cos functions refer to CL in C49

Cyclical Analysis Models

• Y and X matrix for a monthly data series

• Sin and Cos functions refer to CL and SL

in C11 and F11

Lecture 4

Cyclical Analysis Models

• Sample table of R2 and MAPE for CL’s • CL = 9 for the chart and regression

shown here, based on maximum MAPE

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

• ES is a multiple stage forecast

Exponential Smoothing Models

1. No trend and additive seasonal variability (1,0)

2. Additive seasonal variability with an additive trend (1,1)

3. Multiplicative seasonal variability with an additive trend (2,1)

4. Multiplicative seasonal variability with a multiplicative trend (2,2)

Exponential Smoothing Models

• 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

5. Dampened trend with additive seasonal variability (1,1)

6. Multiplicative seasonal variability and dampened trend (2,2)

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

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 Simetar• Specify the data series to forecast• 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

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 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

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

Exponential Smoothing Models• ES Options• Season Method

– 0 No seasonal effects– 1 Additive seasonal effect– 2 Multiplicative seasonal

effect• Trend Method

– 0 No trend dampening– 1 Dampened Additive– 2 Dampened

Multiplicative• Stochastic Forecast

– TRUE– FALSE

Exponential Smoothing Forecast, 2/18/2012 9:45:35 PMLevel Smoothing Constant 0.008076 80.75694Trend Smoothing Constant 0.053853 538.526Season Smoothing Constant 0.067581 675.8104Dampening Parameter 0.943146 19431.46Periods in Season 4Trend Method (0=N,1=DA,2=DM) 1 (Dampened) Additive TrendSeason Method (0=N,1=A,2=M) 1 Additive SeasonConfidence Level for P.I.s 95% 1.644854Stochastic Forecast FALSEMean Abs. Percent Error 20.229Median Abs. Percent Error 17.609Weighted Abs. Percent Error 20.863Theil's U2 Statistic 0.260Root Mean Squared Error 3222.270Mean Abs. Error 2558.268Period Sales Predicted SalesError Level Trend Season ≈LPI

1 7057.62 #N/A #N/A #N/A #N/A #N/A2 8051.55 #N/A #N/A #N/A #N/A #N/A3 10897.3 #N/A #N/A #N/A #N/A #N/A4 12783.33 #N/A #N/A #N/A #N/A #N/A5 15798.51 12820.24 2978.266 14396.46 -11.1731 -1352.516 14332.79 13211.07 1121.721 14394.98 -10.05 -1099.667 16436.42 14881.78 1554.639 14398.05 -8.80252 600.49748 13643.48 16620.48 -2977 14365.71 -9.59675 2031.1669 12662.53 13004.15 -341.617 14353.9 -9.1997 -1375.41

10 15261.34 13245.57 2015.772 14361.5 -7.80001 -964.52811 12376.01 14954.64 -2578.63 14333.32 -8.47798 427.63812 25778.07 16356.49 9421.578 14401.41 -3.89856 2662.74413 8183.58 13022.32 -4838.74 14358.66 -5.78126 -1699.7814 8376.26 13388.68 -5012.42 14312.73 -7.63246 -1300.5415 11168.11 14733.17 -3565.06 14276.74 -8.74895 188.653416 11885.37 16931.23 -5045.86 14227.74 -10.446 2324.493

-20000

-10000

0

10000

20000

30000

40000

1 11 21 31 41 51 61 71 81 91 101111121131141151161171181191

Exponential Smoothing Forecast, 2/18/2012 9:45:35 PM

Sales Predicted Sales ≈LPI ≈UPI

Exponential Smoothing Models• Experiment with alternative settings for the

Trend and Seasonal Smoothing variables to see which combination is best

• Look for the lowest MAPEExponential Smoothing Forecast, 2/18/2012 9:45:35 PMLevel Smoothing Constant 0.008076 80.75694Trend Smoothing Constant 0 0Season Smoothing Constant 0.067581 675.8104Dampening Parameter 0.943146 19431.46Periods in Season 12Trend Method (0=N,1=DA,2=DM) 2 (Dampened) Multiplicative TrendSeason Method (0=N,1=A,2=M) 2 Multiplicative SeasonConfidence Level for P.I.s 95% 1.644854Stochastic Forecast TRUEMean Abs. Percent Error 11.918 Trend 0 1 2 1 2 1 2Median Abs. Percent Error 10.168 Season 0 0 0 1 1 2 2Weighted Abs. Percent Error 11.607 MAPE 35.62594 36.73564 35.62594 11.81451 11.9747 11.7713 11.91827

Exponential Smoothing Models

1. No trend and additive seasonal variability (1,0)

2. Additive seasonal variability with an additive trend (1,1)

3. Multiplicative seasonal variability with an additive trend (2,1)

4. Multiplicative seasonal variability with a multiplicative trend (2,2)

Exponential Smoothing Models

• 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

5. Dampened trend with additive seasonal variability (1,1)

6. Multiplicative seasonal variability and dampened trend (2,2)