lec27v1_1up

lec27v1_1up - Stat 104 Quantitative Methods for Economists...

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Stat 104: Quantitative Methods for Economists Class 27: Regression Redux 1
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Regression Analysis… Regression analysis is used to predict the value of one variable (the dependent variable ) on the basis of other variables (the independent variables ). 2 Dependent variable: denoted Y Independent variables: denoted X 1 , X 2 , …, X k
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Some Notes and Terms s In Simple Linear Regression, one X variable is used to explain the variable Y s In Multiple Regression, more than one X variable is used to explain the variable Y. s For now we will concentrate on simple regression. 3
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Basically, we want to fit a line to our data set. The equation of our line is given by ɵ Y b b X = + 0 1 we use the symbol (Yhat) to stand for the fitted line; Y will always stand for the observed observations. ɵ Y 4
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* ɵ Y i Y i The fitted value: X i ɵ Y b b X i i = + 0 1 5
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For the ith observation the residual is defined to be: e Y Y i i i = - ɵ residual Y * X i ɵ Y i i } 6
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The most popular criterion for fitting a line is called the least squares method. This method says to 2 2 n n n Find 0 b 1 b and that makes this sum as small as possible These two values define a line The farther away a point is from the estimated line, the more serious the error. By squaring the errors, we “penalize” large residuals so that we can avoid them. 2 0 1 1 1 1 ˆ ( ) ( ) i i i i i i i i Y b bX Y Y e = = = - - = - = 7
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X X Y Y i i i n n 1 1 2 = - - = ( )( ) The values of b 0 and b 1 which minimize the residual sum of squares are: X X i i 1 - = ( ) b Y b X 0 1 = - These formulas can be derived using calculus- we pass. These formulas are the intercept and slope for the “best fitting line”. 8
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This note was uploaded on 03/27/2012 for the course STATS 104 taught by Professor Michaelparzen during the Fall '11 term at Harvard.

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lec27v1_1up - Stat 104 Quantitative Methods for Economists...

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