Fall 2008 under econometrics prof keunkwan ryu 7

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Fall 2008 under Econometrics Prof. Keunkwan Ryu 7 Nonnested Alternatives (cont) More difficult if one model uses y and the other uses ln( y ) Can follow same basic logic and transform predicted ln( y ) to get ŷ for the second step In any case, Davidson-MacKinnon test may reject neither or both models rather than clearly preferring one specification
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Fall 2008 under Econometrics Prof. Keunkwan Ryu 8 Proxy Variables What if model is misspecified because no data is available on an important x variable? It may be possible to avoid omitted variable bias by using a proxy variable A proxy variable must be related to the unobservable variable – for example: x 3 * = δ 0 + δ 3 x 3 + v 3 , where * implies unobserved Now suppose we just substitute x 3 for x 3 *
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Fall 2008 under Econometrics Prof. Keunkwan Ryu 9 Proxy Variables (continued) What do we need for for this solution to give us consistent estimates of β 1 and β 2 ? E( x 3 * | x 1 , x 2 , x 3 ) = E( x 3 * | x 3 ) = δ 0 + δ 3 x 3 That is, u is uncorrelated with x 1 , x 2 and x 3 * and v 3 is uncorrelated with x 1 , x 2 and x 3 So really running y = ( β 0 + β 3 δ 0 ) + β 1 x 1 + β 2 x 2 + β 3 δ 3 x 3 + ( u + β 3 v 3 ) and have just redefined intercept, error term x 3 coefficient
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Fall 2008 under Econometrics Prof. Keunkwan Ryu 10 Proxy Variables (continued) Without out assumptions, can end up with biased estimates Say x 3 * = δ 0 + δ 1 x 1 + δ 2 x 2 + δ 3 x 3 + v 3 Then really running y = ( β 0 + β 3 δ 0 ) + ( β 1 + β 3 δ 1 ) x 1 + ( β 2 + β 3 δ 2 ) x 2 + β 3 δ 3 x 3 + ( u + β 3 v 3 ) Bias will depend on signs of β 3 and δ j This bias may still be smaller than omitted variable bias, though
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Fall 2008 under Econometrics Prof. Keunkwan Ryu 11 Lagged Dependent Variables What if there are unobserved variables, and you can’t find reasonable proxy variables? May be possible to include a lagged dependent variable to account for omitted variables that contribute to both past and current levels of y Obviously, you must think past and current y are related for this to make sense
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Fall 2008 under Econometrics Prof. Keunkwan Ryu
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