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Seasonal Models. Materials for this lecture Lecture 3 Seasonal Analysis.XLSX Read Chapter 15 pages 8-18 Read Chapter 16 Section 14 NOTE: The completed Excel file for Lab 4 is on the Website with the Lecture Demos. Uses for Seasonal Models.

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Seasonal models
Seasonal Models

  • Materials for this lecture

  • Lecture 3 Seasonal Analysis.XLSX

  • Read Chapter 15 pages 8-18

  • Read Chapter 16 Section 14

  • NOTE: The completed Excel file for Lab 4 is on the Website with the Lecture Demos

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

  • You 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 forecast models
Seasonal Forecast 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

Two kinds of seasonal indices
Two kinds of Seasonal Indices

  • Price Index

    • The traditional index value shows the relative relationship of price between months or quarters

    • It is ONLY used with price data

  • Fractional Contribution Index

    • If the variable is a quantity we calculate a fractional contribution index to show the relative contribution of each month to the annual total quantity

    • It is ONLY used with qunatities

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 price index for forecasting
Using a Seasonal Price Index for Forecasting

  • Seasonal index has an average of 1.0

    • Each month’s seasonal index value is a fraction of the annual mean price

    • 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 = 75.0

    Mar Price = 125 * 0.976 = 122.0

Using a fractional contribution index
Using a Fractional Contribution Index

  • Fractional Contribution Index sums to 1.0 to represent annual quantity (e.g. sales)

    • 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 = 17,000.0

    SalesJun= 340,000 * 0.076 = 25,840.0

  • 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 andseason

    • 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

  • Affectof Dec is captured in the intercept

  • If the data is quarterly, use 3 dummy variables, for first 3 quarters and intercept picks up affect for 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 forever

  • Note the pattern of 0s and 1s for months

  • December affect is in the intercept

Probabilistic monthly forecasts
Probabilistic MonthlyForecasts

Probabilistic monthly forecasts1
Probabilistic MonthlyForecasts

  • Use the stochastic Indices to simulate stochastic monthly forecasts

Seasonal forecast with dummy variable models1
Seasonal Forecast with Dummy Variable Models

  • Regression Results for Monthly Dummy Variable Model

  • May not have significant effect for each month

  • Must include all months when using model to forecast

  • Jan forecast = 45.93+4.147 * (1) +1.553*T -0.017 *T2+0.000 * T3

Probabilistic forecast with dummy variable models
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)

Harmonic regression for seasonal models
Harmonic Regression for Seasonal Models

  • Sin and Cos functions in OLS regression used to isolate seasonal variation

  • Define a variable SL to represent alternative seasonal lengths: 2, 3, 4, …

  • Create the X Matrix for OLS regression

    X1 =Trend so it is: T = 1, 2, 3, 4, 5, … .

    X2 =Sin(2 * ρi() * T / SL)

    X3 =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) + b4T2+ b5T3

    • Only include T if a trend is present

Harmonic regression for seasonal models1
Harmonic Regression for Seasonal Models

This is what the X matrix looks like for a Harmonic Regression

Harmonic regression for seasonal models3
Harmonic Regression for Seasonal Models

  • Stochastic simulation used to develop a probabilistic 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)