The method of moments approach to estimation implies

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The method of moments approach to estimation implies imposing the population moment restrictions on the sample moments What does this mean? Recall that for E(X), the mean of a population distribution, a sample estimator of E(X) is simply the arithmetic mean of the sample Fall 2008 under Econometrics Prof. Keunkwan Ryu
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16 More Derivation of OLS We want to choose values of the parameters that will ensure that the sample versions of our moment restrictions are true The sample versions are as follows: ( 29 ( 29 0 ˆ ˆ 0 ˆ ˆ 1 1 0 1 1 1 0 1 = - - = - - = - = - n i i i i n i i i x y x n x y n β β β β Fall 2008 under Econometrics Prof. Keunkwan Ryu
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17 More Derivation of OLS Given the definition of a sample mean, and properties of summation, we can rewrite the first condition as follows x y x y 1 0 1 0 ˆ ˆ or , ˆ ˆ β β β β - = + = Fall 2008 under Econometrics Prof. Keunkwan Ryu
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18 More Derivation of OLS ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 = = = = = - = - - - = - = - - - n i i i n i i n i i i n i i i n i i i i x x y y x x x x x y y x x x y y x 1 2 1 1 1 1 1 1 1 1 ˆ ˆ 0 ˆ ˆ β β β β Fall 2008 under Econometrics Prof. Keunkwan Ryu
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19 So the OLS estimated slope is ( 29 ( 29 ( 29 ( 29 0 that provided ˆ 1 2 1 2 1 1 - - - - = = = = n i i n i i n i i i x x x x y y x x β Fall 2008 under Econometrics Prof. Keunkwan Ryu
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20 Summary of OLS slope estimate The slope estimate is the sample covariance between x and y divided by the sample variance of x If x and y are positively correlated, the slope will be positive If x and y are negatively correlated, the slope will be negative Only need x to vary in our sample Fall 2008 under Econometrics Prof. Keunkwan Ryu
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21 More OLS Intuitively, OLS is fitting a line through the sample points such that the sum of squared residuals is as small as possible, hence the term least squares The residual, û , is an estimate of the error term, u, and is the difference between the fitted line (sample regression function) and the sample point Fall 2008 under Econometrics Prof. Keunkwan Ryu
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22 . . . . y 4 y 1 y 2 y 3 x 1 x 2 x 3 x 4 } } { { û 1 û 2 û 3 û 4 x y Sample regression line, sample data points and the associated estimated error terms x y 1 0 ˆ ˆ ˆ β β + = Fall 2008 under Econometrics Prof. Keunkwan Ryu
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23 Alternate approach to derivation Given the intuitive idea of fitting a line, we can set up a formal minimization problem That is, we want to choose our parameters such that we minimize the following: ( 29 ( 29 = = - - = n i i i n i i x y u 1 2 1 0 1 2 ˆ ˆ ˆ β β Fall 2008 under Econometrics Prof. Keunkwan Ryu
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24 Alternate approach, continued If one uses calculus to solve the minimization problem for the two parameters you obtain the following first order conditions, which are the same as we obtained before, multiplied by n ( 29 ( 29 0 ˆ ˆ 0 ˆ ˆ 1 1 0 1 1 0 = - - = - - = = n i i i i n i i i x y x x y β β β β Fall 2008 under Econometrics Prof. Keunkwan Ryu
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