1. Lecture 9 Seasonal Models
Materials for this lecture
Lecture 9 Seasonal Analysis.XLSX
Read Chapter 15 pages 8-18
Read Chapter 16 Section 14

2. Uses for Seasonal Models
Have you noticed a difference in prices from one season to another?
Tomatoes, avocados, grapes, lettuce
Wheat, corn, hay
pound Steers
Gasoline
You must explicitly incorporate the seasonal differences of prices to be able to forecast monthly prices

3. Seasonal and Moving Average Forecasts
Monthly, weekly and quarterly data generally have 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 and the seasonal pattern is mapped over the top of the cycle
Lecture 3

5. Econometric Seasonal Forecast Models
Seasonal indices
Composite forecast models
Dummy variable regression model
Harmonic regression model
Moving average model

6. 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 is present
If a trend is present, forecast the trend and combine it with a seasonal index
Develop the composite forecast
Trend and seasonal components

7. 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 quantities

8. 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
Steps to calculate a seasonal index

9. 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 deterministicly forecast 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 * = 75.0
Mar Price = 125 * = 122.0

10. Seasonal Index Model

12. 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 * = 17,000.0
SalesJun = 340,000 * = 25,840.0
This forecast is useful for planning production, input procurement, and inventory management
The forecast can be probabilistic

13. 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 be estimated is:
Ŷ = a + b1Jan + b2Feb + … + b11Nov + b13T
Jan – Nov are individual dummy variable 0’s and 1’s
Effect of Dec is captured in the intercept
If the data are quarterly, use 3 dummy variables, for first 3 quarters and intercept picks up affect for fourth quarter
Ŷ = a + b1Qt1 + b2Qt2 + b11Qt3 + b13T

14. Seasonal Forecast with Dummy Variable Models
Set up X matrix with 0’s and 1’s
Easy to forecast as the seasonal effect is assumed to persist forever
Note the pattern of 0s and 1s for months
December affect is captured in the intercept

15. 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 = * (1) *T *T * T3

16. Probabilistic Monthly Forecasts

17. Probabilistic Monthly Forecasts
Use the stochastic Indices to simulate stochastic monthly forecasts

18. Probabilistic Forecast with Dummy Variable Models
Stochastic simulation to develop a probabilistic forecast of a random variable
Ỹij = NORM(Ŷij , σ)

19. 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

20. Harmonic Regression for Seasonal Models
This is what the X matrix looks like for a Harmonic Regression

21. Harmonic Regression for Seasonal Models

22. 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

23. 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

24. 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 , σ )
Simetar does a static Ŷi probabilistic forecast
Caution on simulating to many periods with a static probabilistic forecast
ỸT+5 = N((YT+1 +YT+2 + YT+3 + YT+4)/4), σ )
For a dynamic simulation forecast
ỸT+5 = N((ỸT+1 +ỸT+2 + ỸT+3 + ỸT+4)/4, σ )

25. Moving Average Forecasts

26. Probabilistic Forecast for Seasonal Models
Stochastic simulation used to develop a probabilistic forecast for a random variable
Ỹi = NORM(Ŷi , σ)
Ŷi is the forecast for each future period based on your expectations of the exogenous variables for future periods
σ is a constant and is calculated as the std dev of the residuals