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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 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)