Keunkwan ryu 12 measurement error sometimes we have

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Prof. Keunkwan Ryu 12 Measurement Error Sometimes we have the variable we want, but we think it is measured with error Examples: A survey asks how many hours did you work over the last year, or how many weeks you used child care when your child was young Measurement error in y different from measurement error in x
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Fall 2008 under Econometrics Prof. Keunkwan Ryu 13 Measurement Error in a Dependent Variable Define measurement error as e 0 = y y * Thus, really estimating y = X β + u + e 0 When will OLS produce unbiased results? If e 0 and x j , u are uncorrelated is unbiased If E( e 0 ) ≠ 0 then β 0 will be biased, though While unbiased, larger variances than with no measurement error
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Fall 2008 under Econometrics Prof. Keunkwan Ryu 14 Measurement Error in an Explanatory Variable Define measurement error as e 1 = x 1 x 1 * Assume E( e 1 ) = 0 , E( y | x 1 *, x 1 ) = E( y | x 1 *) Really estimating y = β 0 + β 1 x 1 + ( u β 1 e 1 ) The effect of measurement error on OLS estimates depends on our assumption about the correlation between e 1 and x 1 Suppose Cov( x 1 , e 1 ) = 0 OLS remains unbiased, variances larger
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Fall 2008 under Econometrics Prof. Keunkwan Ryu 15 Measurement Error in an Explanatory Variable (cont) Suppose Cov( x 1 * , e 1 ) = 0, known as the classical errors-in-variables assumption, then Cov( x 1 , e 1 ) = E( x 1 e 1 ) = E( x 1 * e 1 ) + E( e 1 2 ) = 0 + σ e 2 x 1 is correlated with the error so estimate is biased ( 29 ( 29 ( 29 + = + - = + - = - + = 2 2 * 2 * 1 2 2 * 2 1 2 2 * 2 1 1 1 1 1 1 1 1 1 , ˆ plim e x x e x e e x e x Var e u x Cov σ σ σ β σ σ σ β σ σ σ β β β β β
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Fall 2008 under Econometrics Prof. Keunkwan Ryu 16 Measurement Error in an Explanatory Variable (cont) Notice that the multiplicative error is just Var( x 1 *)/Var( x 1 ) Since Var( x 1 *)/Var( x 1 ) < 1, the estimate is biased toward zero – called attenuation bias It’s more complicated with a multiple regression, but can still expect attenuation bias with classical errors in variables
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Fall 2008 under Econometrics Prof. Keunkwan Ryu
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