Seasonal Models

• Materials for this lecture• Lecture 3 Seasonal Analysis.XLS• Read Chapter 15 pages 8-18• Read Chapter 16 Section 14

Uses for Seasonal Models

• Have you noticed a difference in prices from one season to another?– Tomatoes, avocados, grapes– Wheat, corn, – 450-550 pound Steers

• Must explicitly incorporate the seasonal differences of prices to be able to forecast monthly prices

Lecture 3

Seasonal and Moving Average Forecasts

• Monthly, weekly and quarterly data generally has a seasonal pattern

• Seasonal patterns repeat each year, as:– Seasonal production due to climate or

weather (seasons of the year or rainfall/drought)

– Seasonal demand (holidays, summer)• Cycle may also

be present

Seasonal Models

• Seasonal indices • Composite forecast models• Dummy variable regression model• Harmonic regression model• Moving average model

Seasonal Forecast Model Development

• Steps to follow for Seasonal Index model development– Graph the data– Check for a trend and seasonal

pattern– Develop and use a seasonal index if

no trend– If a trend is present, forecast the

trend and combine it with a seasonal index

– Develop the composite forecast

Seasonal Index Model• Seasonal index is a simple way to forecast a

monthly or quarterly data series• Index represents the fraction that each

month’s price or sales is above or below the annual mean

Using a Seasonal Price Index for Forecasting

• Seasonal index has an average of 1.0 – Each month’s value is an index (fraction) of

the annual mean price– Use a trend or structural model to forecast

the annual mean price – Use seasonal index to deterministicly

forecast monthly prices from annual average price forecastPJan = Annual Avg Price * IndexJan

PMar = Annual Avg Price * IndexMar

• For an annual average price of $125Jan Price = 125 * 0.600Mar Price = 125 * 0.976

Using a Fractional Contribution Index

• Fractional Contribution Index sums to 1.0 to represent total sales for the year– Each month’s value is the fraction of total

sales in the particular month– Use a trend or structural model for the

deterministic forecast of annual salesSalesJan = Total Annual Sales *

IndexJan

SalesJun = Total Annual Sales * IndexJun

• For an annual sales forecast at 340,000 units

SalesJan = 340,000 * 0.050 = 17,000.0

SalesJun = 340,000 * 0.076 = 25,840.0

• This forecast is useful for planning production, input procurement, and inventory management

• The forecast can be probabilistic

OLS Seasonal Forecast with Dummy Variable Models

• Dummy variable regression model can account for trend and season– Include a trend if one is present– Regression model to estimate is:

Ŷ = a + b1Jan + b2Feb + … + b11Nov + b13T

• Jan – Nov are individual dummy variable 0’s and 1’s

• Effect of Dec is captured in the intercept• If the data is quarterly, use 3 dummy

variables, for first 3 quarters and intercept picks up effect of fourth quarter

Ŷ = a + b1Qt1 + b2Qt2 + b11Qt3 + b13T

Seasonal Forecast with Dummy Variable Models

• Set up X matrix with 0’s and 1’s • Easy to forecast as the seasonal effects is

assumed to persist forever• Note the pattern of 0s and 1s for months• December effect is in the intercept

Probabilistic Monthly Forecasts

Probabilistic Monthly Forecasts

• Use the stochastic Indices to simulate stochastic monthly forecasts

Simualte a Monthly Stochastic Price give a Stochastic Annual Forecast

Stochastic Annual Price10.23 =NORM(20,6)

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Check10.06 9.98 10.67 11.03 10.47 10.73 9.97 9.52 10.28 9.86 10.07 10.14 10.23

=$F$43*G36 =AVERAGE(G45:R45)

Simulate a Stochastic Monthly Demand given a Stochastic Annual Sales Forecast

Stochastic Annual Sales Forecast902.78 =NORM(2000,600)

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec68.67 78.17 75.74 78.09 81.35 70.71 79.99 78.51 72.59 72.02 71.87 75.07 902.78

=$F$51*G37 =SUM(G53:R53)

Seasonal Forecast with Dummy Variable Models

• Regression Results for Monthly Dummy Variable Model

• May not have significant effect for each month• Must include all months when using model to

forecast• Jan forecast = 45.93+4.147 * (1) +1.553*T -0.017

*T2 +0.000 * T3

Probabilistic Forecast with Dummy Variable Models

• Stochastic simulation to develop a probabilistic forecast of a random variableỸij = NORM(Ŷij , SEPi) Or use (Ŷij,StDv)

Harmonic Regression for Seasonal Models

• Sin and Cos functions in OLS regression for isolating seasonal variation

• Define a variable SL to represent alternative seasonal lengths: 2, 3, 4, …

• Create the X Matrix for OLS regression X1 = Trend so it is: T = 1, 2, 3, 4, 5, … .X2 = Sin(2 * ρi() * T / SL)X3 = Cos(2 * ρi() * T / SL)Fit the regression equation of:Ŷi = a + b1T + b2 Sin((2 * ρi() * T) / SL)

+ b3 Cos((2 * ρi() * T) / SL) + b4T2 + b5T3

– Only include T if a trend is present

Harmonic Regression for Seasonal Models

This is what the X matrix looks like for a Harmonic Regression

Harmonic Regression for Seasonal Models

Harmonic Regression for Seasonal Models

• Stochastic simulation used to develop a probabilistic forecast for a random variableỸi = NORM(Ŷi , SEPi)

Moving Average Forecasts

• Moving average forecasts are used by the industry as the naive forecast – If you can not beat the MA then you can

be replaced by a simple forecast methodology

• Calculate a MA of length K periods and move the average each period, drop the oldest and add the newest value

3 Period MAŶ4 = (Y1 + Y2 + Y3) / 3Ŷ5 = (Y2 + Y3 + Y4) / 3Ŷ6 = (Y3 + Y4 + Y5) / 3

Moving Average Forecasts

• Example of a 12 Month MA model estimated and forecasted with Simetar

• Change slide scale to experiment MA length

• MA with lowest MAPE is best but still leave a couple of periods

Probabilistic Moving Average Forecasts

• Use the MA model with lowest MAPE but with a reasonable number of periods

• Simulate the forecasted values asỸi = NORM(Ŷi , Std Dev)

Simetar does a static Ŷi probabilistic forecast

• Caution on simulating to many periods with a static probabilistic forecast ỸT+5 = N((YT+1 +YT+2 + YT+3 + YT+4)/4), Std Dev)

• For a dynamic simulation forecast ỸT+5 = N((ỸT+1 +ỸT+2 + ỸT+3 + ỸT+4)/4, Std Dev)

Moving Average Forecasts