6 3 and 1 for the newest to oldest data respectively

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weights of .6, .3, and .1 for the newest to oldest data, respectively. Then, compare this Month 10 weighted moving average forecast with the Month 10 trend projection forecast. Month Jobs March 353 May 342 April 387 July 396 June 374 August 409 Septem. 399 October 412 Novem. 408 Month Jobs Example: Auger’s Plumbing Service Linear Trend Projection
Three-Month Weighted Moving Average = 422.27 (from earlier slide) Trend Projection = .1 Y Sep. + .3 Y Oct. + .6 Y Nov. = .1(399) + .3(412) + .6(408) = 408.3 The forecast for December will be the weighted average of the preceding three months: September, October, and November. Example: Auger’s Plumbing Service Linear Trend Projection
Due to the positive trend component in the time series, the trend projection produced a forecast that is more in line with the trend that exists. The weighted moving average, even with heavy (.6) weight placed on the current period, produced a forecast that is lagging behind the changing data. Conclusion Example: Auger’s Plumbing Service Linear Trend Projection
To the extent that seasonality exists, we need to incorporate it into our forecasting models to ensure accurate forecasts. We will first look at the case of a seasonal time series with no trend and then discuss how to model seasonality with trend. Seasonality without Trend
Example: Umbrella Sales Sometimes it is difficult to identify patterns in a time series presented in a table. Plotting the time series can be very informative. Seasonality without Trend Year Quarter 1 Quarter 2 Quarter 3 Quarter 4 1 125 153 106 88 2 118 161 133 102 3 138 144 113 80 4 109 137 125 109 5 130 165 128 96
Time Series Plot Example: Umbrella Sales Seasonality without Trend
The time series plot does not indicate any long-term trend in sales. However, close inspection of the plot does reveal a seasonal pattern. The first and third quarters have moderate sales, the second quarter the highest sales, and the fourth quarter tends to be the lowest quarter in terms of sales. Example: Umbrella Sales Seasonality without Trend
We will treat the season as a categorical variable. Recall that when a categorical variable has k levels, k – 1 dummy variables are required. If there are four seasons, we need three dummy variables. Qtr1 = 1 if Quarter 1, 0 otherwise Qtr2 = 1 if Quarter 2, 0 otherwise Qtr3 = 1 if Quarter 3, 0 otherwise Seasonality without Trend
General Form of the Equation is: Optimal Model is: The forecasts of quarterly sales in year 6 are: Qtr 1: Sales = 95 + 29(1) + 57(0) + 26(0) = 124 Qtr 2: Sales = 95 + 29(0) + 57(1) + 26(0) = 152 Qtr 3: Sales = 95 + 29(0) + 57(0) + 26(1) = 121 Qtr 4: Sales = 95 + 29(0) + 57(0) + 26(0) = 95 Example: Umbrella Sales Seasonality without Trend = b 0 + b 1 ( Qtr1 t ) + b 2 ( Qtr2 t ) + b 3 ( Qtr3 t ) Sales t = 95.0 + 29.0( Qtr1 t ) + 57.0( Qtr2 t ) + 26.0( Qtr3 t ) 95.0 29.0( 1 ) 57.0( 2 ) 26.0( 3 ) t t t t Sales Qtr Qtr Qtr
Example: Umbrella Sales We could have obtained the quarterly forecasts for next year by simply computing the average number of umbrellas sold in each quarter.

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