LINREG2 - THE TWO-VARIABLE LINEAR REGRESSION MODEL Herman J...

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1 These lecture notes are based on lecture notes that I wrote while teaching at the University of California, San Diego, in the winter of 1987. 1 THE TWO-VARIABLE LINEAR REGRESSION MODEL Herman J. Bierens Pennsylvania State University April 30, 2012 1. Introduction Suppose you are an economics or business major in a college close to the beach in the southern part of the US, for example southern California 1 , where the weather is almost always nice the whole year around. In order to support yourself through college, you have started your own (weekend) business: an ice cream parlor on the beach. You have experienced that on hot weekends you usually sell more ice cream than on cold weekends. Also, you have recorded the average temperature and the sales of ice cream during eight weekends. Let Y j be the sales of ice cream on weekend j , measured in $100, and let X j be the average temperature on weekend j , measured in units of 10 degrees Fahrenheit: Table 1 : Ice cream data Sales (unit = $100) Temperature (unit = 10 degrees) Y 1 = 8 X 1 = 5 Y 2 = 10 X 2 = 7 Y 3 = 8 X 3 = 6 Y 4 = 13 X 4 = 8 Y 5 = 15 X 5 = 10 Y 6 = 14 X 6 = 9 Y 7 = 11 X 7 = 7 Y 8 = 9 X 8 = 8 You want to use this information to forecast next weekend's sales of ice cream, given a good forecast of next weekend's temperature. Such a forecast of the sales will enable you to
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2 reduce your cost by adjusting your purchase of ice cream to the expected demand, because the ice cream you don't sell has to be thrown away. Let your forecasting scheme be ˆ Y ' ˆ α % ˆ β . X , i.e., given the temperature of X times 10 degrees and given the values of α ^ and β ^ , times $100 ˆ Y will be your forecast of the sales of ice cream. This forecasting scheme together with the points is plotted in Figure 1: ( X j , Y j ), j ' 1,2,...,8, Figure 1 Scatter plot of together with the line ( X j , Y j ), j ' 1,2,...,8, ˆ Y ' ˆ α % ˆ β . X . The best values for α ^ and β ^ are those for which the forecast error (= actual sales minus forecasted sales) is minimal. However, you do not know yet the actual sales in the next weekend, but you do know the actual sales in the eight weekends for which you have recorded your sales and the corresponding temperature. So what you could do is to forecast the sales of ice cream on each of these eight weekends and to determine α ^ and β ^ such that the forecast errors are minimal. Because forecast errors can be positive and negative, as can be seen from Figure 1, the sum of the forecast errors is not a good measure of the performance of your forecasting
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3 scheme, because large positive errors can be offset by large negative errors. Therefore, use the sum of squared errors as your measure of the accuracy of your forecasts: Q α , ˆ β ) ' j n j ' 1 ( Y j & ˆ Y j ) 2 ' j n j ' 1 ( Y j & ˆ α & ˆ β X j ) 2 , where n is the sample size ( n = 8 in our example), and minimize Q ( α ^ , β ^ ) to It can be ˆ α and ˆ β .
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