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Seasonal Models

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- 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

- 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 indices
- Composite forecast models
- Dummy variable regression model
- Harmonic regression model
- Moving average model

- 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

- 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 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

- 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

- 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

- 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

- 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

- Use the stochastic Indices to simulate stochastic monthly forecasts

- 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

- Stochastic simulation to develop a probabilistic forecast of a random variable
Ỹij= NORM(Ŷij, SEPi) Or use (Ŷij,StDv)

- 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

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

- Stochastic simulation used to develop a probabilistic forecast for a random variable
Ỹi = NORM(Ŷi , SEPi)

- 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

- 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

- 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)