Wooldridge PPT ch9

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
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
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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