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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
slide3
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
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 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)

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

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