singleequationgmmtestingslides

singleequationgmmtestingslides - Hypothesis Testing for...

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Hypothesis Testing for Linear Models The main types of hypothesis tests are Overidenti f cation restrictions Coe cient restrictions (linear and nonlinear) Subsets of orthogonality restrictions Instrument relevance.

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Remark: One should always f rst test the overidentifying restrictions before conducting the other tests. If the model speci f cation is rejected, it does not make sense to do the remaining tests.
Speci f cation Tests in Overidenti f ed Models An advantage of the GMM estimation in overidenti f ed models is the ability to test the speci f cation of the model. The J -statistic , introduced in Hansen (1982), refers to the value of the GMM objective function evaluated using an e cient GMM estimator: J = J ( ˆ δ ( ˆ S 1 ) , ˆ S 1 )= n g n ( ˆ δ ( ˆ S 1 )) 0 ˆ S 1 g n ( ˆ δ ( ˆ S 1 )) ˆ δ ( ˆ S 1 )= any e cient GMM estimator ˆ S p S Recall, If K = L, then J =0 ;i f K>L , then J>

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Under regularity conditions (see Hayashi, 2000, Chap. 3) and if the moment conditions are valid, then as n →∞ J d χ 2 ( K L ) Remarks 1. In a well-speci f ed overidenti f ed model with valid moment conditions the J -statistic behaves like a chi-square random variable with degrees of freedom equal to the number of overidentifying restrictions. 2. If the model is misspeci f ed and/or some of the moment conditions do not hold (e.g., E [ x it ε t ]= E [ x it ( y t z 0 t δ 0 )] 6 =0 for some i ), then the J -statistic will be large relative to a chi-square random variable with K L degrees of freedom.
3. The J -statistic acts as an omnibus test statistic for model misspeci f cation. Alarge J -statistic indicates a misspeci f ed model. Unfortunately, the J -statistic does not, by itself, give any information about how the model is misspeci f ed.

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The J-statistic and TSLS When E [ x i x 0 i ε 2 i ]= S = σ 2 Σ xx , e cient GMM reduces to TSLS. The J - stat ist icthentakesthefo rm J ( ˆ δ TSLS , ˆ σ 2 TSLS S 1 xx ) = n ( s xy S xz ˆ δ TSLS ) 0 S 1 xx ( s xy S xz ˆ δ TSLS ) ˆ σ 2 TSLS The TSLS J
Asymptotic Distribution of Sample Moments and J-statistic Result 1: The normalized sample moment evaluated at δ 0 is asymptotically normally distributed n g n ( δ 0 )= n S = 1 n n X t =1 x t ε t d N ( 0 , S ) S K × K = E [ g t g 0 t ]= E [ x t x 0 t ε 2 t ] This follows directly by the CLT for ergodic-stationary MDS.

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Result 2: The J-statistic evaluated at δ 0 and ˆ S 1 is asymptotically chi-square distributed with K degrees of freedom J = J ( δ 0 , ˆ S 1 )= n g n ( δ 0 ) 0 ˆ S 1 g n ( δ 0 ) d χ 2 ( K ) provided ˆ S p S . This follows directly from Result 1, Slutsky’s theorem and the CMT:
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This note was uploaded on 08/23/2010 for the course ECON 583 taught by Professor Zivot during the Fall '09 term at W. Alabama.

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singleequationgmmtestingslides - Hypothesis Testing for...

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