lecture_note_091916

lecture_note_091916 - Econ 5280 Applied Econometrics...

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Econ 5280: Applied Econometrics Instructor: Professor Jin Seo Cho ©Jin Seo Cho, 2016, All Rights Reserved.
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Chapter 3 Classical Linear Model (CLM) 3.1 Introduction To Come. 3.2 Assumptions 1. Dataset is a sequence of IID observations: {( Y t , X t ) R k + 1 t = 1 , 2 , ,n } ; 2. For some unknown β * , E ( Y t X t ) = X t β * , where X t = X t 1 X t 2 X tk , and β = β 1 β 2 β k ; 3. n t = 1 ( X t X t ) is invertible; 4. U t X t N ( 0 , σ 2 * ) , where U t = Y t - E ( Y t X t ) . 3.3 OLS estimator • Definition of OLS estimator: ̂ β n = arg min β Q n ( β ) , 25
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where Q n ( β ) = n - 1 n t = 1 U t ( β ) 2 and U t ( β ) = Y t - X t β . ̂ σ 2 n = ( n - k ) - 1 k t = 1 U t ( ̂ β n ) 2 = n ( n - k ) - 1 Q n ( ̂ β n ) • Derivation of the OLS estimator: First-order conditions for arg min β Q n ( β ) = arg min β 1 n n t = 1 ( Y t - X t β ) 2 : [ β 1 ] - 1 n n t = 1 2 ( Y t - X t β ) X t 1 = 0 [ β 2 ] - 1 n n t = 1 2 ( Y t - X t β ) X t 2 = 0 [ β k ] - 1 n n t = 1 2 ( Y t - X t β ) X tk = 0 matrix form: n t = 1 ( Y t - X t β ) X t = 0 n t = 1 ( Y t X t - X t X t β ) = 0 n t = 1 Y t X t = n t = 1 ( X t X t ) β = n t = 1 X t X t β . derivation of the OLS estimator: ̂ β n = n t = 1 X t X t - 1 n t = 1 X t Y t by assumption 3, there exists the inverse of the matrix. 26
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• Another derivation of the OLS estimator: X = X 1 X 2 X n = X 11 X 12 X 1 k X 21 X 22 X 2 k X n 1 X n 2 X nk ; and Y = Y 1 Y 2 Y n . Then, X Y = X 11 X 21 X n 1 X 12 X 22 X n 2 X 1 k X 2 k X nk Y 1 Y 2 Y n = n t = 1 X t 1 Y t n t = 1 X t 2 Y t n t = 1 X tk Y t = n t = 1 X t Y t and X X = [ X 1 X 2 X n ] X 1 X 2 X n = X 1 X 1 + X 2 X 2 + + X n X n = n t = 1 X t X t . This implies that ̂ β n = n t = 1 X t X t - 1 n t = 1 X t Y t = ( X X ) - 1 X Y . 3.4 Properties of OLS estimator 1. Conditionally unbiased estimator: E [ ̂ β n X ] = β * and E σ 2 n X ] = σ 2 * . This also implies that they are unconditionally unbiased. That is, E [ ̂ β n ] = β * and E σ 2 n ] = σ 2 * . 2. Conditionally normally distributed estimator: ̂ β n ̂ U X N β * 0 2 * ( X X ) - 1 0 0 M , where ̂ U = ̂ U 1 ̂ U 2 ̂ U n = Y 1 - X 1 ̂ β n Y 2 - X 2 ̂ β n Y n - X n ̂ β n , 27
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and M = I - X ( X X ) - 1 X . 3. OLS estimator is a linear estimator. Definition (Linear Estimator) When an estimator is written as a linear function of dependent variable Y , it is called a linear estimator. 4. OLS estimator is the best (or most efficient) estimator. Definition (More Efficient Scalar Estimator) Suppose that θ * is to be estimated and that there are 2 estimators: ̂ θ n and ̃ θ n such that E [ ̂ θ n ] = θ * and E [ ̃ θ n ] = θ * . We say that one estimator, say ̂ θ n , is more efficient than another estimator ̃ θ n if var ( ̂ θ n ) var ( ̃ θ n ) . Definition (Most Efficient Scalar Estimator) If an unbiased estimator has the smallest variance than any other unbiased estimator, then we say that the estimator is the most efficient estimator.
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