Primer on High-Order Moment Estimators

Transcription

1 Prmer on Hgh-Order Moment Estmators Ton M. Whted July 2007

2 The Errors-n-Varables Model We wll start wth the classcal EIV for one msmeasured regressor. The general case s n Erckson and Whted Econometrc Theory, Defntons: y = z α + χ β + u x = γ + χ + ε y s the dependent varable, z s a vector of perfectly measured regressors (whch ncludes an ntercept) χ s the msmeasured regressor u s the regresson dsturbance ε s the measurement error (α, β, γ) are coeffcents

3 Assumptons (u, ε, χ, z ),are..d. u, ε, χ, and z have fnte moments of every order u and ε are dstrbuted ndependently of each other and of (χ, z ) E(u ) = E(ε ) = 0 var (χ, z ) s postve defnte. β 0 and η s non-normally dstrbuted.

4 Partallng Because these estmators are based on polynomals, partallng out the perfectly measured regressors s essental: y z µ y = η β + u x z µ x = η + ε n whch [ ] µ x = E (z z ) 1 (z x ) [ ] µ y = E (z z ) 1 (z y ) η = χ z µ x

5 Moment Equatons Second-Order Moment Equatons [ E (y z µ y ) 2] = β 2 E ( η 2 ) ( ) + E u 2 E [(y z µ y ) (x z µ x )] = βe ( η 2 ) [ E (x z µ x ) 2] = E ( η 2 ) ( ) + E ε 2 Thrd-Order Moment Equatons [ E E ] (y z µ y ) 2 (x z µ x ) [ (y z µ y ) (x z µ x ) 2] = βe ( η 3 Thrd-Order Moment Estmator = β 2 E ( η 3 ) ) [ ] / β = E (y z µ y ) 2 (x z µ x ) E [(y z µ y ) (x z µ x ) 2]

6 More Moment Equatons Fourth Order Moment Equatons 3 Eh`y z µy `x z µx 2 2 Eh`y z µy `x z µx 3 Eh`y z µy `x z µx = β 3 E η 4 + 3βE η 2 E u 2 = β 2 h E η 4 + E η 2 E ε 2 + E u 2 h E η 2 + E ε 2 h = β E η 4 + 3E η 2 E ε 2. General Formula r m r Eh`y z µy `x z µx = E h`η β + u r `η + ε m r = rx m r X j=0 k=0 r «m r j k «β r j E u j E ε k E η m j k

7 Identfcaton Notce from the thrd-order moment equatons Eˆ(y z µ y) 2 (x z µ x) = β 2 E`η 3 Eˆ(y z µ y) (x z 2 µ x) = βe`η 3 that we can only solve for β f β 0 and f E `η 3 0. These are the two dentfyng assumptons. They can be tested smply by testng whether the two left-hand-sde quanttes are zero. In general Reersöl (1950, Econometrca) showed that ths model s dentfed as long as η s not normally dstrbuted. It s possble to work out dentfcaton tests for symmetrc data that use fourth order moments.

8 Dffcultes wth Usng : Hgh order moments cannot be estmated wth as much precson as the second order moments on whch conventonal regresson analyss s based. It s mportant that the hgh order moment nformaton be used as effcently as possble. Prevously, the use of hgh order moments has requred selectng an neffcent estmator. A more effcent estmator can be constructed va a mnmum varance combnaton of neffcent estmators A labor-ntensve technque No guarantee of effcency

9 Asde on GMM Let Let w be an (M 1) be an..d. vector of random varables for observaton. θ be an (P 1) vector of unknown coeffcents. g (w, θ) be an (L 1) vector of functons g : `R M R P R L, L P The functon g(w, θ) can be nonlnear. Let θ 0 be the true value of θ. Let ˆθ represent an estmate of θ. The hat notaton apples to anythng we mght want to estmate.

10 Moment Restrctons GMM s based on what are generally called moment restrctons and sometmes called orthogonalty condtons (The latter termnology comes from the ratonal expectatons lterature.) E (g (w, θ 0 )) = 0 Ths condton s expressed n terms of the populaton. The correspondng sample moment restrcton s 1 N N g (w, θ) = 0 =1 What we want to do s choose ˆθ to get N 1 N =1 g (w, θ) as close to zero as possble.

11 Crteron Functon We mnmze a quadratc form: Q N (θ) = " N 1 N X g (w, θ) # b Ξ " N 1 N X =1 =1 g (w, θ) (1 L) (L L) (L 1) # that converges n probablty to {E [g (w, θ)]} Ξ{E [g (w, θ)]} If L = P, then the estmator s exactly dentfed, and we can fnd θ by solvng N 1 N X =1 g (w, θ) = 0 If L > P, the model s overdentfed and f t s nonlnear, you usually have to use numercal technques.

12 Our Alternatve GMM Estmator We combne the nformaton n the hgh order moments by usng GMM, whch s computatonally convenent and effcent. In our applcaton of GMM, we have: w (y z µ y, x z µ x ) θ (β, E(η 2 ), E(u 2 ), E(ε 2 ), E(η 3 ),...) g(w, θ) (y z µ y ) 2 β 2 E(η 2 ) + E(u 2 ). (y z µ y )(x z µ x ) 2 βe(η 3 ).

13 Our Alternatve GMM Estmator Because all of the observables are on the left and all of the unobservables are on the rght, we can just use the covarance of the observable moments as the weght matrx. The weght matrx does not depend on any parameters. No teratng! Two-step procedure: estmate µ x and µ y wth OLS, plug these estmates nto the above and apply GMM. (What else could you do?) Because we substtute OLS estmates of µ x and µ y n for ther true unknown values, we have to adjust the weghtng matrx.

14 An Asde on Two-Step GMM Estmators Suppose that you estmate a parameter vector δ of dmenson S va a dfferent procedure, and then plug ths estmate nto a GMM estmator. How do you calculate the varance of the orgnal parameter vector θ? The varance of the two-step estmator s ( GΩ 1 G ) 1 You can estmate Ω by bω 1 NX» g (w, θ) N =1 g(θ, w, δ) δ» φ δ (δ, w ) g (w, θ) n whch φ δ s the nfluence functon for δ. g(θ, w, δ) A clear dervaton of ths estmator s n Newey and McFadden s chapter n the 4 th volume of the Handbook of Econometrcs. δ φ δ (δ, w )

15 Other Thngs to Estmate The R 2 of the true regresson ρ 2 µ yv zz µ y + E ( ) η 2 β 2 = µ yv zz µ y + E(η 2) β2 + E(u 2 ) The R 2 of the measurement equaton τ 2 µ = xv zz µ x + E ( ) η 2 µ xv zz µ x + E(η 2) + E(ε2 ) The vector of perfectly measured regressors α = µ y µ x β

16 Standard Errors Because we substtute OLS estmates of µ x and µ y n for ther true unknown values, we have to adjust the weghtng matrx. You calculate the standard errors for these thngs by stackng the nfluence functons for ther ndvdual components and usng the delta method. Recall that the nfluence functon for a GMM estmator s (GΞG) 1 GΞE ˆg `w, θ Recall that many estmators fall under the umbrella of GMM. Ths formula can be used to calculate the nfluence functons of the components of the three thngs on the prevous slde µ y, V zz,..., and all of the GMM parameters. To calculate ther jont covarance matrx, stack ther nfluence functons and take the outer product. Because τ 2, α, and ρ 2 are nonlnear functons of ther components, use the delta method.

17 Generalzaton and Identfcaton Ths method can be used for multple msmeasured regressors. You need much more data for the multple msmeasured regressor case than for the sngle msmeasured regressor case. The moment condtons can be wrtten n general as: E 4(y z µy ) r JY 0 (x j z µ xj ) r r 3 JX 0 JY j 5 = E 4@ η j β j + u A (η j +ε j ) r j 5, j=1 j=1 j=1 n whch (r 0, r 1,..., r J ) are nonnegatve ntegers. Ths general model s dentfed (loosely) f all of the coeffcents on the msmeasured regressors are nonzero and f at least one of the msmeasured regressors has a skewed dstrbuton. The dentfcaton assumptons necessary for ths model are analogous to the assumpton of noncollnearty n an OLS model.

18 Fnte Sample Performance

19 Fnte Sample Performance