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Seasonal Models

This lecture covers seasonal analysis and forecasting methods, focusing on the importance of seasonal models in understanding price variations over time. Key topics include the identification of seasonal patterns in monthly, weekly, and quarterly data, the development of seasonal indices, and the use of various forecasting approaches such as dummy variable regression models and harmonic regression. Practical examples, including price variations for crops like tomatoes and avocados, highlight the real-world applications of these techniques in planning production and inventory management.

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Seasonal Models

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  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 Index for Forecasting • Seasonal index has an average of 1.0 • Each month’s value is an index of the annual mean • 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 SalesJun= 340,000 * 0.076 • 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 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 into the future • Note the pattern of 0s and 1s for months • December effect is in the intercept

  11. Seasonal Forecast with Dummy Variable Models Regression Results for Monthly Dummy Variable Model

  12. 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)

  13. Harmonic Regression for Seasonal Models • Sin and Cos functions 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 is Trend so it is: T = 1, 2, 3, 4, 5, … . X2 is Sin(2 * ρi() * T / SL) X3 is 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) • Only include T if a trend is present

  14. Harmonic Regression for Seasonal Models • If the seasonal variability increases or decreases over time include T • Create three variables T = Trend so it is 1 2 3 4 5 …. S = Sin((2 * ρi() * T) / SL) * T C =Cos((2 * ρi() * T) / SL) * T • Estimate OLS regression Ŷi = a + b1T + b2 S + b3 C + b4T2 + b5T3

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

  16. Harmonic Regression for Seasonal Models

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

  18. 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

  19. 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

  20. 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)

  21. Moving Average Forecasts

  22. Probabilistic MonthlyForecasts

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

  24. The Mathematics Behind Probabilistic Monthly Forecasts • The material on the next 4 slides is provided for completeness. It will NOT be on the exam. • The equations programmed into Simetar 2011 to simulate stochastic monthly indices are described.

  25. Probabilistic Monthly Price Forecasts • Seasonal Price Index • First simulate a stochastic annual price for year j Ỹj = NORM(Ῡi, STD) or =NORM(Ŷj , STD) • Calculate the Std Dev for each month’s (i) Price Index SDIi = SQRT(SDi2/ (Ῡ2* T)) where:SDi2 is the std dev of the Index, Ῡ2 is the overall mean of the data, and T is the number of years of data • Next simulate 12 values using the SDIi and the mean Price Index (PIi) for each month SPIi= NORM(PIi, SDIi) • Next scale the 12 SPIi stochastic values so they will sum to 12 (numerator is 4 if using quarterly data) Stoch PIi= SPIi * (12 / ∑(SPIi)) • Finally simulate stochastic monthly price (J) in year i PiJ = Ỹj* Stoch PIiJ

  26. Probabilistic Price Index Forecasts Line 36: Calculate the Std Dev for the index in each month SDIi = SQRT(SDi2/ (Ῡ2* T)) Line 40: Simulate stochastic index values for each of the 12 months SPIi = NORM(Ii , SD Ii) Line 43: Calculate adjusted Stoch Indices so they sum to 12 (or 4 if using quarterly data) Stoch PIi = SPIi * (12 / ∑(SPIi )) This is the final stochastic Price Index to be used for forecasting monthly Prices Lines 46: Simulate stochastic monthly price in year i PriceiJ = ỸYear j * Stoch PIiJ See Lecture 4 Demo worksheet Price Index worksheet

  27. Prob. Fract Contribution Index Forecasts • Seasonal Fractional Contribution Index • Simulate annual sales Ỹt = NORM(Ŷt, STD) • Calculate the Std Dev for each month’s Fractional Contribution Index. Divide by 12 if monthly and divide by 4 if quarterly data. See Line 53 in next slide. SDIi = SQRT(SDi2/ (Ῡ2* T))/ 12 Where: SDi2 is the std dev of the Index, Ῡ2 is the overall mean of the data, and T is the number of years of data • Next simulate 12 values using the SDIi and the mean Fractional Contribution Index (FCIi )for each month. See line 55. SFCIi = NORM(FCIi , SDIi) • Next scale the 12 SFCIi stochastic values so they sum to 1.0 See Line 59 Stoch FCIi = SFCIi * (1 / ∑(SFCIi )) • Finally simulate stochastic monthly (J) Sales in year I See line 65 PiJ = ỸYear j * Stoch FCIiJ

  28. Probabilistic Monthly Sales Forecasts

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