Seasonal models
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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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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,

    • 450-550 pound Steers

  • Must explicitly incorporate the seasonal differences of prices to be able to forecast monthly prices


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


Seasonal Models

  • Seasonal indices

  • Composite forecast models

  • Dummy variable regression model

  • Harmonic regression model

  • Moving average model


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


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


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


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


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


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


Seasonal Forecast with Dummy Variable Models


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)


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


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


Harmonic Regression for Seasonal Models


Harmonic Regression for Seasonal Models


Harmonic Regression for Seasonal Models

  • Stochastic simulation can be used to develop probabilistic a forecast for a random variable

    Ỹi = NORM(Ŷi , SEPi)


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


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


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)


Moving Average Forecasts


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


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


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


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


Probabilistic Monthly Sales Forecasts


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