C8D3 - MEASUREMENT ERROR v Z Y + + = 2 1 β β w Z X + = 1...

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Unformatted text preview: MEASUREMENT ERROR v Z Y + + = 2 1 β β w Z X + = 1 In this sequence we will investigate the consequences of measurement errors in the variables in a regression model. To keep the analysis simple, we will confine it to the simple regression model. MEASUREMENT ERROR v Z Y + + = 2 1 β β w Z X + = 2 We will start with measurement errors in the explanatory variable. Suppose that Y is determined by a variable Z , but Z is subject to measurement error, w . We will denote the measured explanatory variable X . MEASUREMENT ERROR v Z Y + + = 2 1 β β w Z X + = u X w v X v w X Y + + =- + + = +- + = 2 1 2 2 1 2 1 ) ( β β β β β β β 3 Substituting for Z from the second equation, we can rewrite the model as shown. MEASUREMENT ERROR v Z Y + + = 2 1 β β w Z X + = u X w v X v w X Y + + =- + + = +- + = 2 1 2 2 1 2 1 ) ( β β β β β β β 4 We are thus able to express Y as a linear function of the observable variable X , with the disturbance term being a compound of the disturbance term in the original model and the measurement error. w v u 2 β- = MEASUREMENT ERROR v Z Y + + = 2 1 β β w Z X + = u X w v X v w X Y + + =- + + = +- + = 2 1 2 2 1 2 1 ) ( β β β β β β β w 5 However if we fit this model using OLS, Assumption B.7 will be violated. X has a random component, the measurement error w . MEASUREMENT ERROR v Z Y + + = 2 1 β β w Z X + = 6 And w is also one of the components of the compound disturbance term. Hence u is not distributed independently of X . u X w v X v w X Y + + =- + + = +- + = 2 1 2 2 1 2 1 ) ( β β β β β β β w w MEASUREMENT ERROR v Z Y + + = 2 1 β β w Z X + = u X Y + + = 2 1 β β w v u 2 β- = 7 We will demonstrate that the OLS estimator of the slope coefficient is inconsistent and that in large samples it is biased downwards if β 2 is positive, and upwards if β 2 is negative. ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑--- + =--- + =-- +-- =--- = 2 2 2 2 2 2 2 2 1 1 ] [ X X n u u X X n X X u u X X X X u u X X X X X X Y Y X X b i i i i i i i i i i i i i β β β MEASUREMENT ERROR v Z Y + + = 2 1 β β w Z X + = u X Y + + = 2 1 β β w v u 2 β- = 8 We begin by writing down the OLS estimator and substituting for Y from the true model. In this case there are alternative versions of the true model. The analysis is simpler if you use the equation relating Y to X ....
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This note was uploaded on 05/26/2010 for the course ECON 301 taught by Professor Öcal during the Spring '10 term at Middle East Technical University.

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C8D3 - MEASUREMENT ERROR v Z Y + + = 2 1 β β w Z X + = 1...

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