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ForecastingModelsWith Trend and Seasonal Effects

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Types of Seasonal Models Two possible models are: Additive Model y t = T t + S t + ε t Multiplicative Model y t = T t S t ε t Trend Effects Seasonal Effects Random Effects

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Additive Model Regression Forecasting Procedure Suppose a time series is modeled as having k seasons (Here we illustrate k = 4 quarters) –The following 4 equations represent time series value of 4 seasons Season 1: y t = β 0 + β 1 t + β 2 + ε t TtTt εtεt Season 2: y t = β 0 + β 1 t + β 3 + ε t StSt Season 3: y t = β 0 + β 1 t + β 4 + ε t Season 4: y t = β 0 + β 1 t + β 5 + ε t

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Additive Model Regression Forecasting Procedure Combining the 4 equations into one, we can use 4 dummy variables, S 1, S 2, S 3 and S 4 corresponding to seasons 1, 2, 3 and 4 respectively: The combination of 0s and 1s for each of the dummy variables at each period indicate the season corresponding to the time series value. –Season 1: S 1 = 1, S 2 = 0, S 3 = 0, S 4 = 0 –Season 2: S 1 = 0, S 2 = 1, S 3 = 0,S 4 = 0 –Season 3: S 1 = 0, S 2 = 0, S 3 = 1, S 4 = 0 –Season 4: S 1 = 0, S 2 = 0, S 3 = 0, S 4 = 0 We can simplified the above equation by removing β 5 S 4 y t = β 0 + β 1 t + β 2 S 1 + β 3 S 2 + β 4 S 3 + β 5 S 4 + ε t TtTt StSt εtεt

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Additive Model Regression Forecasting Procedure –Season 1: S 1 = 1, S 2 = 0, S 3 = 0 –Season 2: S 1 = 0, S 2 = 1, S 3 = 0 –Season 3: S 1 = 0, S 2 = 0, S 3 = 1 –Season 4: S 1 = 0, S 2 = 0, S 3 = 0 The combination of 0s and 1s for each of the dummy variables at each period indicate the season corresponding to the time series value. Multiple regression is then done on with t, S 1, S 2, and S 3 as the independent variables and the time series values y t as the dependent variable. y t = β 0 + β 1 t + β 2 S 1 + β 3 S 2 + β 4 S 3 + ε t TtTt StSt εtεt

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Example Troys Mobil Station Troy owns a gas station in a vacation resort city that has many spring and summer visitors. –Due to a steady increase in population Troy feels that average sales experience long term trend. –Troy also knows that sales vary by season due to the vacationers. Based on the last 5 years data below with sales in 1000s of gallons per season, Troy needs to predict total sales for next year (periods 21, 22, 23, and 24). YEAR SEASON 1 2 3 4 5 FALL 3497 3726 3989 4248 4443 WINTER 3484 3589 3870 4105 4307 SPRING 3553 3742 3996 4263 4466 SUMMER 3837 4050 4327 4544 4795

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Scatterplot of Time Series Fall Winter Spring Summer General Pattern: General Pattern: Winter less than Fall, Spring more than Winter, Summer more than Spring, Fall less than Summer

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The Model There is also apparent long term trend. The form of the model then is: y t = β 0 + β 1 t + β 2 F + β 3 W + β 4 S + ε t SpringWinterFall

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The Excel Input

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Add Dummy Variables Not Fall, In Winter, not Spring In Fall, not Winter, not Spring Not Fall, not Winter, In Spring Not Fall, not Winter, not Spring Pattern Repeats

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Regression Intput

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Regression Output Low p-value for F-test Low p-values for all t-tests Conclusion Good model – all factors significant

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The forecasting additive model is: F t = 3610.625 + 58.33t – 155F – 323W – 248.27S Forecasts for year 6 are produced as follows: F(Year 6, Fall) = 3610.625+58.33(21) – 155(1) – 323(0) – 248.27(0) F(Year 6, Winter) = 3610.625+58.33(22) – 155(0) – 323(1) – 248.27(0) F(Year 6, Spring) = 3610.625+58.33(23) – 155(0) – 323(0) – 248.27(1) F(Year 6, Summer) = 3610.625+58.33(24) – 155(0) – 323(0) – 248.27(0) Troys Mobil Station – Performing the forecast

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The Forecasts =$G$17+$G$18*B22+$G$19*C22+$G$20*D22+$G$21*E22 Drag F22 down to F25 =SUM(F22:F25)

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What if Some of the p-values are high? Would not just eliminate Spring or Winter A test exists to decide if adding the dummy variables add value to the model H 0 : 2 = 3 = 4 = 0 H A : At least one of these s 0 Run 2 models: –Full: Time + (3) Seasonal Variables –Reduced:Time Only Test --- Reject H 0 (Accept H A ) if F > F,3,DFE(Full) F = ((SSE REDUCED -SSE FULL )/3)/MSE FULL So if F >F,3,DFE(Full) ---Include seasonal variables

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Multiplicative Model Classical Decomposition Approach The time series is first decomposed into its components (trend, seasonal variation). After these components have been determined, the series is re-composed by multiplying the components.

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Smooth the time series to remove random effects and seasonality and isolate trend. Calculate moving averages to get values for T t for each period t. Determine period factors to isolate the (seasonal) (error) factors. Calculate the ratio y t /T t. Determine the un adjusted seasonal factors to eliminate the random component from the period factors Classical Decomposition Average all the y t /T t that correspond to the same season.

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Determine the adjusted seasonal factors. Calculate: [Unadjusted seasonal factor] [Average seasonal factor] Determine Deseasonalized data values. Calculate: y t [Adjusted seasonal factors] t Determine a deseasonalized trend forecast. Classical Decomposition (Contd) Use linear regression on the deseasonalized time series. Calculate: (Desesonalized values) [Adjusted seasonal factors]). Determine an adjusted seasonal forecast.

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The CFA is the exclusive bargaining agent for public Canadian college faculty. Membership in the organization has grown over the years, but in the summer months there was always a decline. To prepare the budget for the 2001 fiscal year, a forecast of the average quarterly membership covering the year 2001 was required. CANADIAN FACULTY ASSOCIATION (CFA)

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CFA - Solution Membership records from 1997 through 2000 were collected and graphed. 1997199819992000 The graph exhibits long term trend The graph exhibits seasonality pattern

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First moving average period is centered at quarter (1+4)/ 2 = 2.5 Centered moving average of the first two moving averages is [7245.01 + 7380.75]/2 = 7312.875 Smooth the time series to remove random effects and seasonality. Calculate moving averages. Step 1: Isolating the Trend Component Average membership for the first 4 periods = [7130+6940+7354+7556]/4 = 7245.01 Second moving average period is centered at quarter (2+5)/ 2 = 3.5 Average membership for periods [2, 5] = [6940+7354+7556+7673]/4 = 7380.75 Centered location is t = 3 Trend value at period 3, T 3

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=AVERAGE(C3:C6,C4:C7) Drag down to D16

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period factorS t ε t Since y t =T t S t ε t, then the period factor, S t ε t is given by S t t = y t /T t Step 2 Determining the Period Factors Determine period factors to isolate the (Seasonal) (Random error) factor. Calculate the ratio y t / T t. Example: In period 7 (3rd quarter of 1998): S 7 ε 7 = y 7 /T 7 = 7662/7643.875 = 1.002371

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=C5/D5 Drag down to E16

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This eliminates the random factor from the period factors, S t ε t This leaves us with only the seasonality component for each season. Example: Unadjusted Seasonal Factor for the third quarter. S 3 = {S 3,97 + S 3,98 3,98 + S 3,99 3,99 } / 3 = {1.0056+1.0024+1.0079} / 3 = 1.0053 Step 3 Unadjusted Seasonal Factors Determine the un adjusted seasonal factors to eliminate the random component from the period factors Average all the y t / T t that correspond to the same season.

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= AVERAGE(E3,E7,E11,E15) Drag down to F6 Paste Special(Values) Copy F3:F6

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Average seasonal factor = (1.01490+.96580+1.00533+1.01624)/4=1.00057 Step 4 Adjusted Seasonal Factors Determine the adjusted seasonal factors so that average adjusted factor is 1 Calculate: Unadjusted seasonal factors Average seasonal factor Quarter 1 2 3 4 Unadjusted Seasonal Factor 1.01490.96580 1.00533 1.01624 Adjusted Seasonal Factor 1.014325.965252 1.004759 1.015663 Unadjusted Seasonal Factors/1.00057

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F3/AVERAGE($F$3:$F$6) Drag down to G18

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Step 5 The Deseasonalized Time Series Deseasonalized series value for Period 6 (2 nd quarter, 1998) 7595.94 y 6 /(Quarter 2 Adjusted Seasonal Factor) = 7332 / 0.965252 = 7595.94 Determine Deseasonalized data values. Calculate: y t [Adjusted seasonal factors] t

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=C3/G3 Drag to cell H18

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Step 6 The Time Series Trend Component Regress on the Deseasonalized Time Series Determine a deseasonalized forecast from the resulting regression equation (Unadjusted Forecast) t = 7069.6677 + 78.4046t Period (t) 17 18 19 20 Unadjusted Forecast (t) 8402.55 8480.95 8559.36 8637.76

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Run regression Deseason vs. Period =$L$18+$L$19*B19 Drag to cell I22

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Step 7 The Forecast Re-seasonalize the forecast by multiplying the unadjusted forecast by the adjusted seasonal factor for each period. Unadjusted Forecast (t) 8402.55 8480.95 8559.36 8637.76 Period 17 18 19 20 Adjusted Forecast (t) 8522.928186.268600.098773.06 Adjusted Seasonal Factor 1.014325.965252 1.004759 1.015663

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=I19*G3 Drag down to J22 Seasonally Adjusted Forecasts

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Review Additive Model for Time Series with Trend and Seasonal Effects –Use of Dummy Variables 1 less than the number of seasons –Use of Regression Modified F test if all p-values not <.05 Multiplicative Model for Time Series with Trend and Seasonal Effects –Determine a set of adjusted period factors to deseasonalize data –Do regression to obtain unadjusted forecasts –Reseasonalize results to give seasonally adjusted forecasts. Excel

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