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Seasonal Models

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,

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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

  12. Probabilistic Forecast with Dummy Variable Models • Stochastic simulation can be used 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) 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 • 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

  16. Harmonic Regression for Seasonal Models

  17. Harmonic Regression for Seasonal Models • Stochastic simulation can be used to develop probabilistic a 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. Stop HERE!! • The material on the next 4 slides is provided for completeness. It will NOT be on the exam. • If you ever have to do a stochastic seasonal index read this and see the demonstration program

  23. 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

  24. 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

  25. 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

  26. Probabilistic Monthly Sales Forecasts

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