Slide Set 14 Inference Basded on the GMM Estimator Pietro Coretto pcoretto@unisa.it Econometrics Master in Economics and Finance (MEF) Università degli Studi di Napoli Federico II Version: Saturday 9 th March, 2019 (h09:01) P. Coretto MEF Inference Basded on the GMM Estimator 1 / 16 Summary Testing hypotheses on individual coefficients Testing linear restrictions Testing nonlinear restrictions Testing overidentifying restrictions Testing subsets of orthogonality conditions P. Coretto MEF Inference Basded on the GMM Estimator 2 / 16
Preamble The asymptotic variance of the GMM estimator is crucial to run hypothesis testing The general consistent estimator for AVar(Ŵ )) is given by AVar(Ŵ )) = (S xzŵ S xz) 1 (S xzŵ ŜŴ S xz) (S xzŵ S xz) 1 The best practice is to compute the 2-step GMM and get AVar ( Ŝ 1 ) ) 1 ) = (S xzŝ 1 S xz These are both consistent = the following test procedures don t depend on the particular choice. Notation: δ, and the same Ŵ. ) AVar are to denote any pair of estimates based on P. Coretto MEF Inference Basded on the GMM Estimator 3 / 16 As for the OLS, we did not restrict for conditional homoscedasticity. The estimates standard error ) AVar Se) = n [l,l] is called heteroscedasticity robust standard error. P. Coretto MEF Inference Basded on the GMM Estimator 4 / 16
Testing hypotheses on individual coefficients We want to test H 0 : δ l = δ l H 1 : δ l δ l for some l = 1, 2,..., L Under (C1) (C5) + H 0 we have that t l = nl δl ) ) AVar [l,l] d N (0, 1) Decision rule: reject H 0 if t l exceeds the (1 α 2 ) quantile of the N (0, 1) P. Coretto MEF Inference Basded on the GMM Estimator 5 / 16 Default test on individual coefficients { H0 : δ l = 0 H 1 : δ l 0 Decision rule at significance level α: reject H 0 if t l > q 1 α 2 where q 1 α is the (1 α 2 2 )-quantile of the N (0, 1) Confidence intervals on individual coefficients Let (1 α) be the confidence level. The confidence interval for δ k is given by δ l ± q 1 α Se(δ l) 2 P. Coretto MEF Inference Basded on the GMM Estimator 6 / 16
Testing linear restrictions Now we want to test a set of J linear restrictions H 0 : Rδ = r H 1 : Rδ r Where R is a (J L) matrix, rank(r)=j (full row rank), r R J The following Wald s statistic is similar to the OLS case W := n(r δ r) ( RAVar ) ) 1 R (R δ r) Under (C1) (C5) + H 0, it is easy to show that W d χ 2 (J) P. Coretto MEF Inference Basded on the GMM Estimator 7 / 16 Test { H0 : Rδ = r H 1 : Rδ r Decision rule at significance level α: reject H 0 if w > q 1 α where w is the observed value of the Wald s statistic W q 1 α is the (1 α)-quantile of χ 2 (J) P. Coretto MEF Inference Basded on the GMM Estimator 8 / 16
Testing nonlinear restrictions We want to test H 0 : a(δ) = 0 H 1 : a(δ) 0 Let a : R L R J, consider the matrix of first derivatives of a( ) computed at δ A(δ) = a(t) t t=δ A(δ) is of dimension J L, rank(a(δ)) = J (full row rank) Wald statistic (close to the OLS case) W := na) ( A) AVar )A) ) 1 a) P. Coretto MEF Inference Basded on the GMM Estimator 9 / 16 W d χ 2 (J) Under (C1) (C5) + H 0 Decision rule at significance level α: reject H 0 if w > q 1 α where w is the observed value of the Wald s statistic W q 1 α is the (1 α)-quantile of χ 2 (J) P. Coretto MEF Inference Basded on the GMM Estimator 10 / 16
Testing overidentifying restrictions (Hansen, 1982) Moment equations are over-identified when rank(σ xz ) = L < K. Then the GMM chooses δ in order to make J( ) small. Take Ŵ = Ŝ, and suppose to known the true parameter δ, denote it with δ 0. Define ḡ = ḡ n (δ 0 ) = 1 ni=1 n g i We are evaluating the sampling mean of g i at the true parameter, then for large enough n WLLN + CLT give as ḡ 0 p Ŝ S d nḡ N (0, S) Then nḡ Ŝ 1 ḡ = ( nḡ) Ŝ 1 ( nḡ) = J ( δ 0, Ŝ 1) d χ 2 (K L) P. Coretto MEF Inference Basded on the GMM Estimator 11 / 16 Let δ (Ŝ 1 ) the efficient GMM estimator, we know it s consistent for δ 0. Proposition (Hansen, 1982) Let δ (Ŝ 1 ) the efficient GMM estimator, under assumptions (C1) (C5) J (Ŝ 1 ), Ŝ 1) d χ 2 (K L) How to use this limit distribution? P. Coretto MEF Inference Basded on the GMM Estimator 12 / 16
Specification test: if J (Ŝ 1 ), Ŝ 1) is surprisingly large we can take this as strong failure of one or more assumptions. How large? Compare with a quantiles o χ 2 (K L). not robust against failures of orthogonality: the existence of the χ 2 (K L) limit distribution is obtained with (C3) being crucially true! not meant as orthogonality test: sometimes large values J (Ŝ 1 ), Ŝ 1) are used as evidence against the predetermidness of some of the K instruments in x i. This is only reasonable we are SURE that (C1), (C2), (C4) and (C5) hold. low power: it has been shown that tests based on J (Ŝ 1 ), Ŝ 1) have low power in not too large samples P. Coretto MEF Inference Basded on the GMM Estimator 13 / 16 Testing subsets of orthogonality conditions Divide the K instruments in two groups: K 1 instruments in x 1i : these are known to satisfy the orthogonality conditions K K 1 instruments in x 2i : suspected to violate the orthogonality conditions x i = ( ) x1i x 2i Assuming K 1 L, that is there are at least as many non-suspect instruments as there are regressors, we want to test H 0 : E[x 2i ε i ] = 0 H 1 : E[x 2i ε i ] 0 P. Coretto MEF Inference Basded on the GMM Estimator 14 / 16
Basic idea of the test: compute the efficient GMM estimator with the all instruments x i. Let J full be the corresponding J( ) statistic compute the efficient GMM estimator only based on the good subset of instruments x 1,i. Let J 1 be the corresponding J( ) statistic Intuition: if the inclusion of suspect instruments significantly increases the J( ), this is evidence against the predetermidness of x 2i. P. Coretto MEF Inference Basded on the GMM Estimator 15 / 16 Proposition (Newey, 1985) Assume K 1 L. Also assume (C1) (C5), where (C4) is strengthen by requiring that E[x 1,i z i] is of full rank, then (J full J 1 ) d χ 2 (K K 1 ) Decision rule: reject H 0 if (J full J 1 ) exceeds the (1 α) quantile of χ 2 (K K 1 )] P. Coretto MEF Inference Basded on the GMM Estimator 16 / 16