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Seasonal Models

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

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Seasonal Models

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    1. Seasonal Models Materials for this lecture Lecture 4 Seasonal Analysis.XLS Read Chapter 15 pages 8-18 Read Chapter 16 Section 14

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

    4. 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 seasonal index Develop the composite forecast

    5. 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 year’s mean

    6. 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 annual price Use seasonal index to deterministically 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 Mar Price = 125 * 0.976

    7. 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 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 if the annual sales forecast is stochastic

    8. Probabilistic Monthly Price Forecasts Seasonal Price Index First simulate a stochastic annual price ?Year j =NORM(?i , STD) or =NORM(Yj , STD) Calculate the Std Dev for each month’s Price Index SDIi = SQRT(SDi2 / (?2 * T)) where:SDi2 is the std dev of the Index, ?2 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 , SD Ii) 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 = ?Year j * Stoch PIiJ

    9. Probabilistic Price Index Forecasts Line 36: Calculate the Std Dev for the index in each month SDIi = SQRT(SDi2 / (?2 * 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 = ?Year j * Stoch PIiJ See Lecture 4 Demo worksheet Price Index worksheet

    10. Prob. Fract Contribution Index Forecasts Seasonal Fractional Contribution Index Simulate annual sales ?t = NORM(Yt, 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 / (?2 * T)) / 12 Where: SDi2 is the std dev of the Index, ?2 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 = ?Year j * Stoch FCIiJ

    11. Probabilistic Monthly Sales Forecasts

    12. OLS Seasonal Forecast with Dummy Variable Models Dummy variable regression model can account for seasonal difference Can include a trend if one is present Regression model to estimate is: Y = a +b1Jan +b2Feb+ …+ b12T+ b13T2+ b14T3 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

    13. 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

    14. Seasonal Forecast with Dummy Variable Models

    15. Probabilistic Forecast with Dummy Variable Models Stochastic simulation can be used to develop probabilistic forecast of random variable ?i = NORM(Yi , SEPi)

    16. 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 * ?i() * T / SL) X3 is Cos(2 * ?i() * T / SL) Yi = a + b1T + b2 Sin((2 * ?i() * T) / SL) + b3 Cos((2 * ?i() * T) / SL) Include T if a trend is present

    17. 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 * ?i() * T) / SL) * T C = Cos((2 * ?i() * T) / SL) * T Estimate OLS regression Yi = a + b1T + b2 S + b3 C Include T and T2 if a trend is present

    18. Harmonic Regression for Seasonal Models

    19. Harmonic Regression for Seasonal Models

    20. Harmonic Regression for Seasonal Models Stochastic simulation can be used to develop probabilistic forecast of a random variable ?i = NORM(Yi , SEPi)

    21. 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 the 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 Data 3 Period MA Y4 Y = (Y1 + Y2 + Y3) / 3 Y5 Y = (Y2 + Y3 + Y4) / 3 Y6 Y = (Y3 + Y4 + Y5) / 3

    22. Moving Average Forecasts 12 Month MA model estimated and forecast in Simetar Change slide scale to experiment MA length MA with lowest MAPE is best

    23. Probabilistic Moving Average Forecasts Use the MA model with lowest MAPE Simulate the forecasted vales as ?i = NORM(Yi , Std Dev) Simetar does a static Yi probabilistic forecast Caution on simulating to many periods with a static probabilistic forecast ?T+5 = N((YT+1 +YT+2 + YT+3 + YT+4)/4), Std Dev) For a dynamic simulation it should be ?T+5 = N((?T+1 +?T+2 + ?T+3 + ?T+4)/4, Std Dev)

    24. Moving Average Forecasts

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