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

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

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 models

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 models1

Seasonal Models

  • Seasonal indices

  • Composite forecast models

  • Dummy variable regression model

  • Harmonic regression model

  • Moving average model


Seasonal forecast model development

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 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

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

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

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

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 models1

Seasonal Forecast with Dummy Variable Models


Probabilistic 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

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 models1

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 models2

Harmonic Regression for Seasonal Models


Harmonic regression for seasonal models3

Harmonic Regression for Seasonal Models


Harmonic regression for seasonal models4

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

  • 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 forecasts1

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

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 forecasts2

Moving Average Forecasts


Stop here

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

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

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

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

Probabilistic Monthly Sales Forecasts


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