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

3. 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

5. Seasonal Forecast Models Seasonal indices Composite forecast models Dummy variable regression model Harmonic regression model Moving average model

6. 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

7. 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

8. 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

9. 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 P Jan = Annual Avg Price * Index Jan P Mar = Annual Avg Price * Index Mar For an annual average price of $125 Jan Price = 125 * 0.600 = 75.0 Mar Price = 125 * 0.976 = 122.0

10. 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 Sales Jan = Total Annual Sales * Index Jan Sales Jun = Total Annual Sales * Index Jun For an annual sales forecast at 340,000 units Sales Jan = 340,000 * 0.050 = 17,000.0 Sales Jun = 340,000 * 0.076 = 25,840.0 This forecast is useful for planning production, input procurement, and inventory management The forecast can be probabilistic

11. 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 + b 1 Jan + b 2 Feb + … + b 11 Nov + b 13 T 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 + b 1 Qt1 + b 2 Qt2 + b 11 Qt3 + b 13 T

12. 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

13. Probabilistic Monthly Forecasts

14. Use the stochastic Indices to simulate stochastic monthly forecasts

15. 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 *T 2 +0.000 * T 3 note that Excel shows beta-hat on T 3 is 0.000 but in reality it is not zero, expanding decimals shows a value greater than zero

16. Probabilistic Forecast with Dummy Variable Models Stochastic simulation to develop a probabilistic forecast of a random variable Ỹ ij = NORM(Ŷ ij, σ)

17. 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 X 1 = Trend so it is: T = 1, 2, 3, 4, 5, …. X 2 = Sin(2 * ρi() * T / SL) X 3 = Cos(2 * ρi() * T / SL) Fit the regression equation of: Ŷ i = a + b 1 T + b 2 Sin((2 * ρi() * T) / SL) + b 3 Cos((2 * ρi() * T) / SL) + b 4 T 2 + b 5 T 3 –Only include T if a trend is present

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

19. Harmonic Regression for Seasonal Models

20. 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 Ŷ 4 = (Y 1 + Y 2 + Y 3 ) / 3 Ŷ 5 = (Y 2 + Y 3 + Y 4 ) / 3 Ŷ 6 = (Y 3 + Y 4 + Y 5 ) / 3

21. 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

22. Probabilistic Moving Average Forecasts Use the MA model with lowest MAPE but with a reasonable number of periods Simulate the forecasted values as Ỹ i = NORM(Ŷ i, σ ) Simetar does a static Ŷ i probabilistic forecast Caution on simulating to many periods with a static probabilistic forecast Ỹ T+5 = N((Y T+1 +Y T+2 + Y T+3 + Y T+4 )/4), σ ) For a dynamic simulation forecast Ỹ T+5 = N((Ỹ T+1 +Ỹ T+2 + Ỹ T+3 + Ỹ T+4 )/4, σ )

23. Moving Average Forecasts

24. Probabilistic Forecast for Seasonal Models Stochastic simulation used to develop a probabilistic forecast for a random variable Ỹ i = NORM(Ŷ i, σ) Ŷ i 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