Fall 2008 under econometrics prof keunkwan ryu 13

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Fall 2008 under Econometrics Prof. Keunkwan Ryu 13 Large Sample Inference (cont) Easy to come up with examples for which this exact normality assumption will fail Any clearly skewed variable, like wages, arrests, savings, etc. can’t be normal, since a normal distribution is symmetric
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Fall 2008 under Econometrics Prof. Keunkwan Ryu 14 Skewed to the right
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Fall 2008 under Econometrics Prof. Keunkwan Ryu 15 Asymptotic Normality ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 1 , 0 Normal ~ ˆ ˆ (iii) of estimator consistent a is ˆ (ii) ˆ plim where , , 0 Normal ~ ˆ (i) s, assumption Markov - Gauss Under the 2 2 2 1 2 2 2 a j j j ij j j a j j se r n a a n β β β σ σ σ β β - = - -
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Fall 2008 under Econometrics Prof. Keunkwan Ryu 16 Asymptotic Normality (cont) Because the t distribution approaches the normal distribution for large df , we can also say that ( 29 ( 29 1 ~ ˆ ˆ - - - k n a j j j t se β β β Note that while we no longer need to assume normality with a large sample, we do still need homoskedasticity
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Fall 2008 under Econometrics Prof. Keunkwan Ryu 17 Asymptotic Standard Errors If u is not normally distributed, we sometimes will refer to the standard error as an asymptotic standard error, since ( 29 ( 29 ( 29 n c se R SST se j j j j j - = β σ β ˆ , 1 ˆ ˆ 2 2 So, we can expect standard errors to shrink at a rate proportional to the inverse of √ n
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Fall 2008 under Econometrics Prof. Keunkwan Ryu 18 Lagrange Multiplier statistic Once we are using large samples and relying on asymptotic normality for inference, we can use more that t and F stats The Lagrange multiplier or LM statistic is an alternative for testing multiple exclusion restrictions Because the LM statistic uses an auxiliary regression it’s sometimes called an nR 2 stat
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