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singleequationgmmtestingslides

# singleequationgmmtestingslides - Hypothesis Testing for...

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

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Remark: One should always fi rst test the overidentifying restrictions before conducting the other tests. If the model speci fi cation is rejected, it does not make sense to do the remaining tests.
Speci fi cation Tests in Overidenti fi ed Models An advantage of the GMM estimation in overidenti fi ed models is the ability to test the speci fi 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 ; if K > L, then J > 0 .

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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 fi ed overidenti fi 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 fi 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 fi cation. A large J -statistic indicates a misspeci fi ed model. Unfortunately, the J -statistic does not, by itself, give any information about how the model is misspeci fi 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 - statistic then takes the form 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 -statistic is also known as Sargan’s statistic (see Sargan, 1958).
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: n g n ( δ 0 ) 0 ˆ S 1 g n ( δ 0 ) d N ( 0 , S ) 0 S 1 N ( 0 , S ) ³ S 1 / 2 z ´ 0 S 1 ( S 1 / 2 z ) = z 0 z χ 2 ( K ) where z N ( 0 , I k ) and S 1 = S 1 / 2 0 S 1 / 2 .
Result 3: The J-statistic evaluated at δ 0 and ˆ W p W is not asymptotically chi-square distributed with K degrees of freedom J = J ( δ 0 , ˆ W ) = n g n ( δ 0 ) 0 ˆ Wg n ( δ 0 ) d 9 χ 2 ( K )

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singleequationgmmtestingslides - Hypothesis Testing for...

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