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

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

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?

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, 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

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

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

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

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

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

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

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

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

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)

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

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 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

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

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 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

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

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)

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