Lecture 9 Seasonal Models Materials for this lecture Lecture 9 Seasonal Analysis.XLSX 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, lettuce Wheat, corn, hay 450-550 pound Steers Gasoline You must explicitly incorporate the seasonal differences of prices to be able to forecast monthly prices

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 Lecture 3

Seasonal Forecast 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

Two kinds of Seasonal Indices Price Index The traditional index value shows the relative relationship of price between months or quarters It is ONLY used with price data Fractional Contribution Index If the variable is a quantity we calculate a fractional contribution index to show the relative contribution of each month to the annual total quantity It is ONLY used with quantities

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 Steps to calculate a seasonal index

Using a Seasonal Price Index for Forecasting Seasonal index has an average of 1.0 Each month’s seasonal index value is a 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 forecast PJan = Annual Avg Price * IndexJan PMar = Annual Avg Price * IndexMar For an annual average price of $125 Jan Price = 125 * 0.600 = 75.0 Mar Price = 125 * 0.976 = 122.0

Using a Fractional Contribution Index Fractional Contribution Index sums to 1.0 to represent annual quantity (e.g. sales) 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 sales SalesJan = 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 are quarterly, use 3 dummy variables, for first 3 quarters and intercept picks up affect for 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 affect is in the intercept

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 note that Excel shows beta-hat on T3 is 0.000 but in reality it is not zero, expanding decimals shows a value greater than zero

Probabilistic Monthly Forecasts

Probabilistic Monthly Forecasts Use the stochastic Indices to simulate stochastic monthly forecasts

Probabilistic Forecast with Dummy Variable Models Stochastic simulation to develop a probabilistic forecast of a random variable Ỹij = NORM(Ŷij , σ)

Harmonic Regression for Seasonal Models Sin and Cos functions in OLS regression used to isolate 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

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 MA Ŷ4 = (Y1 + Y2 + Y3) / 3 Ŷ5 = (Y2 + Y3 + Y4) / 3 Ŷ6 = (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 as Ỹi = NORM(Ŷi , σ ) 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), σ ) For a dynamic simulation forecast ỸT+5 = N((ỸT+1 +ỸT+2 + ỸT+3 + ỸT+4)/4, σ )

Moving Average Forecasts

Probabilistic Forecast for Seasonal Models Stochastic simulation used to develop a probabilistic forecast for a random variable Ỹi = NORM(Ŷi , σ) Ŷi is the forecast for each future period based on your expectations of the exogenous variables for future periods σ is a constant and is calculated as the std dev of the residuals