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, – pound Steers Must explicitly incorporate the seasonal differences of prices to be able to forecast monthly prices

Lecture 3 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

Seasonal Models Seasonal indices Composite forecast models Dummy variable regression model Harmonic regression model Moving average model

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 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 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 deterministicly forecast monthly prices from annual average price forecast P Jan = Annual Avg Price * Index Jan P Mar = Annual Avg Price * Index Mar For an annual average price of $125 Jan Price = 125 * Mar Price = 125 * 0.976

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 Sales Jan = Total Annual Sales * Index Jan Sales Jun = Total Annual Sales * Index Jun For an annual sales forecast at 340,000 units Sales Jan = 340,000 * Sales Jun = 340,000 * This forecast is useful for planning production, input procurement, and inventory management The forecast can be probabilistic

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 + b 1 Jan + b 2 Feb + … + b 11 Nov + b 13 T 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 + b 1 Qt1 + b 2 Qt2 + b 11 Qt3 + b 13 T

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

Seasonal 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 Ỹ ij = NORM(Ŷ ij, SEP i ) Or use (Ŷ ij,StDv)

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 X 1 is Trend so it is: T = …. X 2 is Sin(2 * ρi() * T / SL) X 3 is Cos(2 * ρi() * T / SL) Ŷ i = a + b 1 T + b 2 Sin((2 * ρi() * T) / SL) + b 3 Cos((2 * ρi() * T) / SL) –Only include T if a trend is present

Harmonic Regression for Seasonal Models If the seasonal variability increases or decreases over time Create three variables T = Trend so it is …. S = Sin((2 * ρi() * T) / SL) * T C = Cos((2 * ρi() * T) / SL) * T Estimate OLS regression Ŷ i = a + b 1 T + b 2 S + b 3 C + b 4 T 2 + b 5 T 3

Harmonic Regression for Seasonal Models

Stochastic simulation can be used to develop probabilistic a forecast for a random variable Ỹ i = NORM(Ŷ i, SEP i )

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 Ŷ 4 = (Y 1 + Y 2 + Y 3 ) / 3 Ŷ 5 = (Y 2 + Y 3 + Y 4 ) / 3 Ŷ 6 = (Y 3 + Y 4 + Y 5 ) / 3

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 Use the MA model with lowest MAPE but with a reasonable number of periods Simulate the forecasted values as Ỹ i = NORM(Ŷ i, Std Dev) Simetar does a static Ŷ i probabilistic forecast Caution on simulating to many periods with a static probabilistic forecast Ỹ T+5 = N((Y T+1 +Y T+2 + Y T+3 + Y T+4 )/4), Std Dev) For a dynamic simulation forecast Ỹ T+5 = N((Ỹ T+1 +Ỹ T+2 + Ỹ T+3 + Ỹ T+4 )/4, Std Dev)

Moving Average Forecasts

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 Seasonal Price Index –First simulate a stochastic annual price for year j Ỹ j = NORM(Ῡ i, STD) or =NORM(Ŷ j, STD) –Calculate the Std Dev for each month’s (i) Price Index SDI i = SQRT(SD i 2 / (Ῡ 2 * T)) where:SD i 2 is the std dev of the Index, Ῡ 2 is the overall mean of the data, and T is the number of years of data –Next simulate 12 values using the SDI i and the mean Price Index (PI i ) for each month SPI i = NORM(PI i, SDI i ) –Next scale the 12 SPI i stochastic values so they will sum to 12 (numerator is 4 if using quarterly data) Stoch PI i = SPI i * (12 / ∑(SPI i )) –Finally simulate stochastic monthly price (J) in year i P iJ = Ỹ j * Stoch PI iJ

Probabilistic Price Index Forecasts Line 36: Calculate the Std Dev for the index in each month SDI i = SQRT(SD i 2 / (Ῡ 2 * T)) Line 40: Simulate stochastic index values for each of the 12 months SPI i = NORM(I i, SD I i ) Line 43: Calculate adjusted Stoch Indices so they sum to 12 (or 4 if using quarterly data) Stoch PI i = SPI i * (12 / ∑(SPI i )) This is the final stochastic Price Index to be used for forecasting monthly Prices Lines 46: Simulate stochastic monthly price in year i Price iJ = Ỹ Year j * Stoch PI iJ See Lecture 4 Demo worksheet Price Index worksheet

Prob. Fract Contribution Index Forecasts Seasonal Fractional Contribution Index –Simulate annual sales Ỹ t = NORM(Ŷ t, 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. SDI i = SQRT(SD i 2 / (Ῡ 2 * T)) / 12 Where: SD i 2 is the std dev of the Index, Ῡ 2 is the overall mean of the data, and T is the number of years of data –Next simulate 12 values using the SDI i and the mean Fractional Contribution Index (FCI i )for each month. See line 55. SFCI i = NORM(FCI i, SDI i ) –Next scale the 12 SFCI i stochastic values so they sum to 1.0 See Line 59 Stoch FCI i = SFCI i * (1 / ∑(SFCI i )) –Finally simulate stochastic monthly (J) Sales in year I See line 65 P iJ = Ỹ Year j * Stoch FCI iJ

Probabilistic Monthly Sales Forecasts