Seasonal Models

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

2. 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

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 Lecture 3

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

5. 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

6. 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

7. 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 deterministiclyforecast 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 Mar Price = 125 * 0.976

8. 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 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

9. 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

10. 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

11. Probabilistic MonthlyForecasts

12. Probabilistic MonthlyForecasts • Use the stochastic Indices to simulate stochastic monthly forecasts

13. 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

14. Probabilistic Forecast with Dummy Variable Models • Stochastic simulation to develop a probabilistic forecast of a random variable Ỹij= NORM(Ŷij, SEPi) Or use (Ŷij,StDv)

15. 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

16. Harmonic Regression for Seasonal Models This is what the X matrix looks like for a Harmonic Regression

17. Harmonic Regression for Seasonal Models

18. Harmonic Regression for Seasonal Models • Stochastic simulation used to develop a probabilistic forecast for a random variable Ỹi = NORM(Ŷi , SEPi)

19. 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

20. 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

21. 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, Std Dev) Simetar does a static Ŷiprobabilistic 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)

22. Moving Average Forecasts