Shown see the appendix that q α β is minimal for n

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shown (see the Appendix) that Q ( α ^ , β ^ ) is minimal for $ $ ' ' n j ' 1 ( X j & ¯ X )( Y j & ¯ Y ) ' n j ' 1 ( X j & ¯ X ) 2 ' ' n j ' 1 ( X j & ¯ X ) Y j ' n j ' 1 ( X j & ¯ X ) 2 $ " ' ¯ Y & $ $ ¯ X , where ¯ X ' (1/ n ) ' n j ' 1 X j and ¯ Y ' (1/ n ) ' n j ' 1 Y j . (1) In the ice cream parlor case we have n ' 8, ¯ X ' 7.5, ¯ Y ' 11, ' n j ' 1 X 2 j ' 468, ' n j ' 1 X j Y j ' 687, ' n j ' 1 ( X j & ¯ X )( Y j & ¯ Y ) ' ' n j ' 1 X j Y j & n . ¯ X . ¯ Y ' 27, ' n j ' 1 ( X j & ¯ X ) 2 ' ' n j ' 1 X 2 j & n . ¯ X 2 ' 18, so that ˆ β ' 1.5, ˆ α ' & 0.25. Thus, our best forecasting scheme is This is the straight line in Figure 1. ˆ Y ' & 0.25 % 1.5 X . Now suppose that the forecast of next weekend's temperature is 75 degrees. Then X = 7.5, hence = 11. Therefore, the best forecast of next weekend's sales is: ˆ Y ' & 0.25 % 1.5 (7.5) = $1,100. ˆ Y x $100 2. The two-variable linear regression model. In order to answer the question how good this forecast is, we have to make assumptions about the true relationship between the dependent variable Y j and the independent variable X j , (also called the explanatory variable ). The true relationship we are going to assume is the two- variable linear regression model: Y j ' " % $ . X j % U j , j ' 1,2, ..... , n . (2)
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4 The U j 's are random error variables, called error terms , for which we assume: Assumption I : The U j 's are independent and identically distributed ( i.i.d ) random variables . Assumption II : The mathematical expectation of U j equals zero : E ( U j ) = 0 for j = 1,2,..., n . Assumption III : The variance of the U j 's is constant σ 2 ' var( U j ) ' E [( U j & E ( U j )) 2 ] ' E [ U 2 j ] and finite. Regarding the explanatory variables X j we shall assume for the time being that Assumption IV : The independent variables X j are non-random . This assumption is not strictly necessary, and is actually quite unrealistic in economics, but will be made for the sake of convenience, as it will ease the argument. Finally, we will assume that the errors are normally distributed: Assumption V : The errors U j 's are N (0, σ 2 ) distributed . In particular, we shall need the latter assumption in order to say something about the reliability of the forecast. These assumptions will be relaxed later on. 3. The properties of and ˆ α ˆ β . Although we have motivated model (2) by the need to forecast out-of-sample values of the dependent variables a linear regression model is more often used for testing economic Y j , hypotheses. For example, let be the hourly wage of wage earner j in a random sample of size n Y j of wage earners, and let be a gender indicator, say if person j is a female, and if X j X j ' 1 X j ' 0 person j is a male. If you suspect gender discrimination in the workplace, you can test this suspicion by testing the null hypothesis that β = 0 (no gender discrimination) against one of
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2 The estimators and are called "Ordinary" least squares estimators to distinguish ˆ α ˆ β them from "Nonlinear" least squares estimators. 5 three possible alternative hypotheses: (a) β 0: women are paid different hourly wages than men, either higher or lower; (b) β > 0: women are paid higher hourly wages than men; (c) β < 0: women are paid lower hourly wages than men.
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