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Seasonal Models

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- Materials for this lecture
- Lecture 3 Seasonal Analysis.XLS
- Read Chapter 15 pages 8-18
- Read Chapter 16 Section 14

- 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

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

- 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 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

- 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

- 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

- 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

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

- 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

- 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

- Stochastic simulation can be used to develop probabilistic a 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)

- 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

- 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

- First simulate a stochastic annual price for year j

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

- 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

- Simulate annual sales