The ordinary least squares ols estimation procedure

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The ordinary least squares (OLS) estimation procedure for the coefficient value is the best linear unbiased estimation procedure (BLUE). Crucial Point: When the ordinary least squares (OLS) estimation procedure performs its calculations, it implicitly assumes that the standard ordinary least squares (OLS) regression premises are satisfied. In Chapter 16, we focused on the first standard ordinary least squares (OLS) premise. We shall now turn our attention to the second, error term/error term independence premise. We begin by examining precisely what the premise means. Subsequently, we investigate what problems do and do not emerge when the premise is violated and finally what can be done to address the problems that do arise.
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5 Covariance and Independence We introduced covariance to quantify the notions of correlation and independence. If two variables are correlated, their covariance is nonzero. On the other hand, if two variables are independent their covariance is 0. A scatter diagram allows us to illustrate how covariance is related to independence and correlation. To appreciate why, consider the equation we use to calculate covariance: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 2 2 1 Cov[ , ] N t t N N t x x y y x x y y x x y y x x y y N N = + + + = = x y Focus on one term in the numerator the covariance term, ( ) ( ) t t x x y y ; consider its sign in each of the four quadrants: ( x i x )( y i y ) > 0 ( x i - x )( y i - y ) > 0 ( x i x )( y i y ) < 0 ( x i x )( y i y ) < 0 ( x i - x ) ( y i y ) ( x i x )>0 ( y i y )>0 ( x i x )<0 ( y i y )>0 ( x i x )<0 ( y i y )<0 ( x i x )>0 ( y i y )<0 Quadrant I Quadrant II Quadrant III Quadrant IV Figure 17.1: Scatter Diagram and Covariance First quadrant. Dow growth rate is greater than its mean and Nasdaq growth is greater than its mean; the product of the deviations is positive in the first quadrant: ( ) ( ) ( ) ( ) 0and 0 0 t t t t x x y y x x y y > > > Second quadrant. Dow growth rate is less than its mean and Nasdaq growth is greater than its mean; the product of the deviations is negative in the second quadrant: ( ) ( ) ( ) ( ) 0and 0 0 t t t t x x y y x x y y < > <
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6 Third quadrant. Dow growth rate is less than its mean and Nasdaq growth is less than its mean; the product of the deviations is positive in the third quadrant: ( ) ( ) ( ) ( ) 0and 0 0 t t t t x x y y x x y y < < > Fourth quadrant. Dow growth rate is greater than its mean and Nasdaq growth is less than its mean; the product of the deviations is negative in the fourth quadrant: ( ) ( ) ( ) ( ) 0and 0 0 t t t t x x y y x x y y > < < Recall that we used precipitation in Amherst, the Nasdaq growth rate, and the Dow Jones growth rate to illustrate independent and correlated variables in Chapter 1: Figure 17.2: Precipitation versus Nasdaq Growth
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7 Figure 17.3: Dow Jones Growth versus Nasdaq Growth Precipitation in Amherst and the Nasdaq growth rate are independent; knowing one does not help us predict the other. Figure 17.2 shows that the scatter
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