Chapter 3. GMM: Selected Topics

Transcription

1 Chater 3. GMM: Selected oics Contents Otimal Instruments. he issue of interest Otimal Instruments under the i:i:d: assumtion he basic result Illustrative examles Otimal Instruments under the martingale di erence assumtion Otimal Instruments under general deendence Finite samle roerties 6 2. Introduction he size of GMM-based Wald tests Bootstra under the martingale di erence assumtion Bootstraing the GMM estimator Bootstra the test statistics Bootstra under general serial deendence Weak identi cation: a itfall and some ways around 2 3. Introduction Some consequences of weak instruments in linear models: a scary regression Robust Inference with weak instruments in linear models Robust Inference in nonlinear models

2 . Otimal Instruments.. he issue of interest When alying GMM in macroeconomics, a tyical but crucial ste is to transform conditional restrictions into unconditional ones. Seci cally, suose a model delivers the following conditional moment restrictions: E(d t ( 0 )jz t ) = 0; (t = ; 2; :::) () where d t ( 0 ) and z t are nite dimensional random vectors. (Here we have simli ed the matter by restricting z t to be nite dimensional.) his imlies, for any measurable integrable function f(:), we always have E (f(z t )d t ( 0 )) = 0 (t = ; 2; :::): (2) We end u with an in nite number of valid instruments, or equivalently, an in nitely number of unconditional moment restrictions. Question: which of these instruments should we use if the goal is to minimize the asymtotic variance of the GMM estimator?.2. Otimal Instruments under the i:i:d: assumtion.2.. he basic result Assumtion : (d t ( 0 ) 0 ; zt) 0 are indeendently and identically distributed in t = ; :::; : Proosition. Assume () and Assumtion hold. Suose 0 is an unknown q by arameter vector.. An otimal choice of the instruments (i.e, minimizing the asymtotic variance among all GMM estimators) in (2) is given by ( 0 ) 0 = z t ; (3)

3 where K is any q by q nonsingular matrix of nite constants, and = E d t ( 0 )d t ( 0 ) 0 jz t ; (4) 2. he resulting GMM estimator solves X zt d t (^ ) = 0; t= whose asymtotic covariance matrix is given by ( 0 ) 0 = E z ( 0 ) E t z :.2.2. Illustrative examles Examle. Consider the linear model y t = x t + u t where x t is a scalar random variable with E(x t u t ) 6= 0: Suose z t is a set of instruments satisfying E(u t jz t ) = 0; var(u t jz t ) = 2 ; and (y t ; x t ; z t ) are i:i:d. hen, aly (3) and (4), x t ) = z t 2 = k 2 E(x tjz t ); where k is an arbitrary constant. Since k can take any value, we are free to set k = 2 in which case the otimal instrument reduces to, zt = E(x t jz t ): Now we generalize the model to allow for heteroskedasticity, i.e., we assume var(u t jz t ) = 2 t ; 2

4 but leave all other asect of the seci cation the same. In this case, the otimal instrument takes the form x t ) = z t 2 t = k 2 t E(x t jz t ): In this case, it is not ossible to eliminate 2 t by judicious choice of k, although we can set k = to remove the minus sign. his gives zt = 2 E(x t jz t ): t Examle 2. Consider the following linear two-equation system: y ;t = x ;t + u ;t y 2;t = x 2;t 2 + u 2;t where x ;t and x 2;t are scalar random variables. Let u t = (u ;t ; u 2;t ) 0 : Suose z t is a set of variables satisfying E(u t jz t ) = 0; var(u t jz t ) = ; and (y t ; x t ; z t ) are iid. hen, aly (3) and (4), #! x ;t ) 0 = 0 2;t x 2;t 2 ) jz " 2 #! E(x ;t jz t ) 0 = KE 0 E(x 2;t jz t ) where K is a matrix of constants. In this case, the matrix weights E(x ;t jz t ) and E(x 2;t jz t ) to account for the correlation between the two equations. While the Proosition characterizes the otimal instruments, it does not fully resolve the roblem of instrument selection. he function zt deends on E (@d t ( 0 ) 0 =@j z t ) 3

5 and, in most cases, on as well (see the above examles); neither of these functions are tyically art of the seci cation of the underlying economic/statistical model. One natural solution is to estimate the comonents E (@d t ( 0 ) 0 =@j z t ) and from the data. In some cases this works in a straightforward manner, while in general it is comlicated. We now resent one examle in which the roosal works. Examle 3. Still consider the linear model studied in the rst examle, with the further assumtion that x t is generated by a linear model x t = z 0 t + v t : (5) With this seci cation, an otimal instrument is E(x t jz t ) = z 0 t: Hence, the otimal GMM estimator solves X zt(y 0 t x t^) = 0: t= Since is unknown, we relace it by the OLS estimate based on equation (5): ^ = (Z 0 Z) Z 0 X: Substituting in, we have Exlicitly X zt^(y 0 t x t^) = 0: t=! X X ^ = x t zt^ 0 y t zt^ 0 t= t= = (X 0 P Z X) X 0 P Z y: his is recisely the two-stage least square estimator (2SLS). herefore, 2SLS estimator can be interreted as the feasible otimal GMM estimator within this model. 4

6 In the receding examle, the construction of the otimal instruments rests crucially on the assumtion that E(x t jz t ) is linear. his seci cation may be natural in some contexts such as the linear simultaneous equations model but may not be so aroriate in others. In the general case, we can use non-arametric methods to estimate E(x t jz t ); e.g., aroximate the conditional exectation by a olynomial. For further discussions in such a direction, see Newey (990, 993)..3. Otimal Instruments under the martingale di erence assumtion We now relax Assumtion and resents a similar result for dynamic models. Suose the solution of a model delivers the following moment restrictions: E(d t ( 0 )ji t ) = 0 (6) where I t is the information set at t. Corollary. Assume (6) holds. Suose 0 is of dimension q.. hen an otimal choice of instruments in (2) is given by ( 0 ) 0 = I t ; where = E d t ( 0 )d t ( 0 ) 0 ji t ; and K is any q by q nonsingular matrix of nite constants. 2. his choice leads to a GMM estimator with asymtotic variance matrix V = E ( 0 ) I t ( 0 ).4. Otimal Instruments under general deendence I t Once serial correlation is introduced, which is the case if we have moment conditions E(d t ( 0 )ji t m ) = 0 with m >, 5

7 then the form of the otimal instrument will change, although the basic idea remains the same. he result is of limited use from a ractical oint of view, therefore we omit the details. You can refer to Hall (2005, ). 2. Finite samle roerties 2.. Introduction Our discussions so far have been asymtotic in nature. In ractice, we always face a nite samle. he following two issues are therefore of articular imortance:. Finite samle roerties of the GMM estimator, e.g., nite samle bias and MSE; 2. Finite samle roerties of GMM-based inference rocedures. Here we focus on the second issue. We will rst examine the nite samle roerties of the Wald tests when asymtotic critical values are used for inference. hen, we will discuss a bootstra rocedure, which can imrove the inference under some circumstances he size of GMM-based Wald tests he discussion below is based on the simulation analysis in Burnside and Eichenbaum (996, Henceforth BE). hey asked the following questions:. Does the size of the tests closely aroximate their asymtotic size? 2. Do joint tests of several restrictions erform as well or worse than tests of simle hyotheses, and what are resonsible for size distortions? 3. How can modelling assumtions, or restrictions imosed by hyothesis themselves, be used to imrove the erformance of these tests? 4. What ractical advice can be given to the ractitioner? 6

8 BE considered two simulation exeriments. In the rst, the data are generated by Gaussian vector white noise, and in the second the DGP is Burnside and Eichenbaum (994). he ndings from the two exeriments are similar; we focus on the rst exeriment due to its simlicity. DGP: X it i:i:d:n(0; 2 i ); i = ; :::; n; t = ; :::; : n = 20; = 00; 2 = ::: = 2 n = : Parameters: Econometrician knows E(X it ) = 0 and is interested in estimating 2 i V ar(x it): Moment Conditions: E(Xit 2 2 i ) = 0; i = ; :::; n: GMM estimates: ^ i = P =2 t= X2 it Hyotheses of interest: H M : = ::: = M = ; M n: BE considered M 2 f; 2; 5; 0; 20g : Wald tests: W M = (^ ) 0 A 0 AV A 0 A (^ ) ; (7) where A = (I M 0 M(n M) ); ^ = (^ ; :::; ^ n ) 0 ; and V denotes a generic estimator of the asymtotic variance-covariance matrix of (^ lim V = G 0 0S0 G 0 :! ); i:e:; Note that Also note that the i th element of G 0 is it 2 i i = 2 i ; the ij th element of S 0 is E(X 2 it 2 i )(X 2 jt 2 j): W M! d 2 M under H M : Alternative Covariance Matrix Estimators V : 7

9 . Allow the data to be deendent, and estimate S 0 using the Newey and West (987) estimator; the bandwidth B = 4; 2. Allow the data to be deendent, and estimate S 0 using the Newey and West (987) estimator; the bandwidth B = 2; 3. Allow the data to be deendent, and estimate S 0 using the Newey and West (987) estimator, the bandwidth is determined using Andrews (99) ; 4. Exloit the assumtion that data are serially uncorrelated. hus, the ijth element of S 0 is estimated by P t= (X2 it ^ 2 i )(X 2 jt ^ 2 j): 5. Exloit the assumtion that data are serially uncorrelated and mutually indeendent. he iith element of S 0 is estimated by P t= (X2 it ^ 2 i ) 2 ; the o -diagonal elements are zero. 6. Imose Gaussianity. he ii th element of S 0 is estimated by 2^ 4 i ; the o -diagonal elements are zero. 7. Imose the null hyotheses on S 0. he iith element of S 0 is 2 for i n; the o -diagonal elements are zero. 8. Imose the null hyotheses on S 0 and G 0 : he iith element of S 0 is 2 for i n; the o -diagonal elements are zero. he ith element of G 0 is 2 for i n: he results are reorted in able. Hence, the conclusions and suggestions to ractitioner are:. he small samle size of the Wald tests tends to exceed the asymtotic size. he roblem becomes dramatically worse as the dimension of the joint tests being considered increases; 2. he bulk of the roblem has to do with di culty in estimating the variancecovariance matrix S 0. In their second simulation exeriment, BE further documents that the bias in estimating m ( 0 ) and the correlation between m (^) and ^S are not the main contributors to the size distortions. 8

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11 3. In ractice, to imrove the size roerty, it is useful to imose a riori information when estimating S 0 : wo imortant sources of such information are the economic theory being investigated and the null hyothesis being tested Bootstra under the martingale di erence assumtion Bootstra is an alternative way to aroximate the samling distribution of an estimator or a test statistic. Suose we have the following moment restrictions: E(m(X t ; 0 )) = 0; t = ; 2; :::: Assume m(x t ; 0 ) are serially uncorrelated. We now show how to use bootstra to aroximate the samling distribution of the two-ste GMM estimator and related test statistics Bootstraing the GMM estimator Recall where ^ = arg min! 0 X m(x t ; ) ^S (^ ) t= ^S (^ ) =! X m(x t ; ) ; t= X m(x t ; ^ )m(x t ; ^ ) 0 t= and ^ is some reliminary GMM estimator, say, GMM estimator using an identity weighting matrix. hen the bootstra aroximation to the samling distribution of ^ can be obtained as follows. Ste : Draw a samle of size with relacement from the observed samle fx ; :::; X g ; denote the samle of draws as fx ; :::; X g : Ste 2: Comute the GMM estimator using the random samle, i.e.,! 0! X ^ = arg min m (Xt ; ) ^S (^ ) X m (Xt ; ) ; t= 0 t=

12 where m (X t ; ) = m(x t ; ) ^S () = X m(x t ; ^), t= X m (Xt ; )m (Xt ; ) 0 ; t= and ^ is some reliminary GMM estimator, say, GMM estimator with an identity weighting matrix. Ste 3: Reeating Stes and 2 many times (say B times) to obtain a set of estimates. Call them ^ () ; :::; ^ (B). We then use the distribution of ^(j) ^ as an aroximation to the samling distribution of (^ 0 ) Bootstra the test statistics We can aroximate samling distributions of commonly used test statistics using Bootstra. We focus on the t, the W ald and the J statistic. he t-statistic. Recall the that t-statistic for testing the r th comonent of 0 equal to some constant is given by (^ 0 ) q r (8) ^V (^) r;r where (^ 0 ) r denotes the r th comonent of ^ 0 and ^V (^) r;r denote the (r; r) th comonent of ^V ; with ^V () = h i ^G () 0 ^S () ^G () (9) he distribution of (8) can be aroximated by the emirical distribution of (^ ^)r q ^V (^ ) r;r where the formula of ^V (^ ) r;r is given in (9), with relaced by ^ :

13 he Wald statistic. (I will leave this as an exercise). As another exercise, bootstra the statistic (7) and comare with able (d to f only). he J statistic. Simly comute! 0 X J = m (Xt ; ^ ) ^S t= (^ )! X m (Xt ; ^ ) and reeat. Use the resulting emirical distribution to aroximate the distribution of J: t= 2.4. Bootstra under general serial deendence he extension to this case is not straightforward. he available rocedures are comlicated and do not work very satisfactory in ractice. Interested readers can read Hall, P., and J. L. Horowitz (996): Bootstra Critical Values for ests Based on Generalized-Method-of-Moment Estimators, Econometrica, 64, Weak identi cation: a itfall and some ways around 3.. Introduction Recall that for the linear instrumental variable regression to work, we need a set of instruments that are both valid (uncorrelated with the errors) and relevant (correlated with the endogenous regressors). For GMM, we say instruments z t are valid (or exogenous) if they satisfy moment restrictions E(z t d t ( 0 )) = 0. he requirement of "instrument relevance" is relaced by "identi cation", which is satis ed if E(z t d t ()) 6= 0 for 6= 0 : "Weak instruments" or "weak identi cation" will arise if instruments are only weakly correlated with included endogenous variables. his oses considerable challenges to inference using GMM and IV methods. Below, we 2

14 . Use a linear model to illustrate such consequences; 2. Discuss how to conduct inference for linear models with otential weak instruments; 3. Brie y discuss how to conduct inference for nonlinear models with weak identi - cation Some consequences of weak instruments in linear models: a scary regression Many aers have been written trying to measure the return to years of education. he setting usually involves estimating some wage equation as follows y i = Y i + x 0 2;i + " i ; where y i is some measure of the income for individual i and Y i is the years of education, and x 0 2;i are some covariates he di culty is that the education achievement Y i is endogenous. A oular solution is to use some instruments, which generate variations in Y i but otherwise are uncorrelated with y i : Angrist and Krueger (99) is a famous examle. hey argued that the quarter of birth is a valid instrument. he idea is that the comulsory school attendance law requires a student to start rst grade in the fall of the calendar year in which he or she turns age 6 and to continue attending school until he or she turns 6. hus an individual born in the early months of the year will usually enter rst grade when he or she is close to age 7 and will reach age 6 in the middle of tenth grade. An individual born in the third or fourth quarter will tyically start school either just before or just after turning age 6 and will nish tenth grade before reaching age 6. hey resented several tabulations to show that individuals born in the early month of the year on average have less years of education. hey estimated the wage equation and concluded that the educational attainment has a signi cant e ect on earnings and the magnitude is to the one estimated with OLS. While it can be argued that the quarter of birth itself may have a direct e ect on earnings, a more relevant concern is that the relationshi between quarter of birth and 3

15 educational attainment maybe very weak. Bound, Jaeger and Baker (995) shows this may well be the case (he following table is from Bound, Jaeger and Baker (995), able ): Hence, the question is how misleading the results can be in such a situation. o address this roblem Bound, Jaeger and Baker (995) did something very clever (following the suggestion of Krueger). hey relaced each individual s real quarter of birth by a fake quarter of birth, randomly generated by a comuter. What they found was amazing: It didn t matter whether you used the real quarter of birth or the fake one as the instrument 2SLS gave basically the same answer! he detailed results are reorted in the following table.he intuition behind the results can be illustrated by a simle examle. y i = x i + " i x i = z i + v i where z i are xed, v i is a zero mean error term: For simlicity, assume the variables " i he examle is taken from Hansen (2006), with some change. 4

16 and v i are normally distributed and that " i is indeendent of v i. We have ^ IV 0 = P n i= z i" i P n i= z ix i Now, suose the instrument and the endogenous variable are not correlated, i.e., = 0:hen, for a given n, therefore n nx i= n nx z i " i N = N(0; E(zi 2 " 2 i )) i= z i x i = n nx z i v i N 2 = N(0; E(zi 2 vi 2 )) i= ^ IV 0 N N 2 : he above result holds for any n, it also holds when n! : he distribution is drastically di erent from the standard normal aroximation and the standard inference is invalid. In articular, in the resence of identi cation failure, n(^ IV 0 ) diverges, hence severe over-rejection of the null hyothesis will occur if standard critical values are used. 5

17 3.3. Robust Inference with weak instruments in linear models A artial solution to the above roblem is to use test statistics that are not sensitive to the strength of instruments (this excludes the t and F statistics). hree statistics have attracted wide attention: the Anderson-Rubin (AR) statistic, Kleibergen s LM statistic, Moreira s conditional likelihood ratio statistic. Asymtotically, those statistics all have distributions that do not deend on the strength of the instruments. We consider a linear regression model with a single endogenous regressor and no included exogenous variable y = Y + u with Y = Z + v where y and Y are by vector of observations on endogenous variables, Z is a by k matrix of instruments. It is useful to de ne the following two quantities: S = (Z0 Z) =2 (Z 0 Ȳb 0 ) b0 b 0 (0) and where = (Z0 Z) =2 (Z 0 Ȳ a 0 ) a0 a 0 () Ȳ = [y; Y ] ; b 0 = [; 0 ] 0 ; a 0 = [ 0 ; ] 0 and is the variance of the reduced form errors. () is a su cient statistic for : Let ^ and ^S denote and S evaluated with ^ =Ȳ 0 M Z Ȳ=( K) relacing : he Anderson-Rubin statistic. null hyothesis 0 = 0 using the statistic AR( 0 ) = Anderson and Rubin (949) roosed testing the (y Y 0 ) 0 P Z (y Y 0 )=k (y Y 0 ) 0 M Z (y Y 0 )=( k) = ^S 0 ^S k With xed instruments and normal errors, the quadratic forms in the numerator and denominator of are indeendent chi-squared random variables under the null hyothesis, 6

18 and AR( 0 ) has exact F k; have k null distribution. Droing the Gaussian assumtion, we AR( 0 )! d 2 k =k: Because the numerator and denominator of the Anderson Rubin statistic are evaluated at the true arameter value, it has an asymtotic chi-square distribution even if the unknown arameters are oorly identi ed. Kleibergen s Statistic. Kleibergen (200) roosed the statistic ^S0 ^ 2 K( 0 ) = ^ 0 ^ If k =, then K( 0 ) = AR( 0 ). Kleibergen showed that under either conventional or weak-instrument asymtotics, K( 0 ) has 2 as its null distribution. Moreira s Statistic. Moreira (2003) roosed testing = 0 using the conditional likelihood ratio test statistic s M( 0 ) = ^S 0 ^S ^ 0 ^ + ^S0 ^S + ^ 0 2 ^ 4 ^S0 ^S! 2 ^ 0 ^ ^S0 ^ 2 he (weak instruments) asymtotic distribution of M( 0 ) is non-standard. However, conditional on the value of ^ ; it does not deend on the strength of the instruments and the null distribution can be obtained by Monte Carlo simulation. Remark. Due to the duality between hyothesis tests and con dence sets, these tests can be used to construct con dence sets robust to weak instruments. For examle, a fully robust 95% con dence set can be constructed as the set of 0 for which the AR statistic, AR( 0 ), fails to reject at the 5% signi cance level Robust Inference in nonlinear models Nonlinear Anderson Rubin Statistic. Recall that because the numerator and denominator of the Anderson Rubin statistic are evaluated at the true arameter value, 7

19 it has an asymtotic chi-square distribution even if the unknown arameters are oorly identi ed. his observation suggests tests of = 0 based on the nonlinear analog of the AR statistic, which is the so-called continuous-udating GMM objective function in which the weight matrix is evaluated at the same arameter value as the numerator J CU ( 0 ) = r! 0 r X m(x t ; 0 ) ^S(0 ) t= If there is no serial correlation, then ^S( 0 ) =! X m(x t ; 0 ) t= X ~m(x t ; 0 ) ~m(x t ; 0 ) 0 t= X with ~m(x t ; 0 ) = m(x t ; 0 ) m(x t ; 0 ) If m(x t ; 0 ) is serially correlated, then ^S( 0 ) is relaced by the estimate of the long run variance using some kernel based method. Under the null hyothesis, J CU ( 0 ) has a 2 K limiting distribution where K is the number of moment restrictions. Notice that we need to re-center m(x t ; 0 ) when estimating ^S( 0 ): Otherwise ^S( 0 ) diverges under the alternative hyothesis and the test does not have ower. t= Kleibergen s Statistic. Kleibergen (2005) roosed testing the hyothesis = 0 using a generalization of K( 0 ) and showed that the roosed statistic has a chi-square limiting distribution. You can refer to his aer for details. 8

20 References [] Anderson,. W., and Rubin, H. (949), Estimation of the Parameters of a Single Equation in a Comlete System of Stochastic Equations, Annals of Mathematical Statistics, 20, [2] Donald W. K. Andrews, (999). "Consistent Moment Selection Procedures for Generalized Method of Moments Estimation," Econometrica, vol. 67(3), ages [3] Angrist, J. D., and Krueger, A. B. (99), Does Comulsory School Attendance A ect Schooling and Earnings, Quarterly Journal of Economics, 06, [4] Bound, J., Jaeger, D. A., and Baker, R. (995), Problems With Instrumental Variables Estimation When the Correlation Between the Instruments and the Endogenous Exlanatory Variables Is Weak, Journal of the American Statistical Association, 90, [5] Burnside, C. and Eichenbaum, M. (996), "Small-Samle Proerties of GMM- Based Wald ests", Journal of Business and Economic Statistics, 4, [6] Chamberlain, G. (987). Asymtotic E ciency in Estimation with Conditional Moment Restrictions, Journal of Econometrics, 34, [7] Hall (2005), Generalized Method of Moments, Oxford Press. [8] Kleibergen, F. (2002), Pivotal Statistics for esting Structural Parameters in Instrumental Variables Regression, Econometrica, Vol. 70, No. 5, [9] - (2005), esting Parameters in GMM Without Assuming hat hey Are Identi ed, Econometrica, Vol. 73, No. 4 (July, 2005), [0] Moreira, M. J. (2003), A Conditional Likelihood Ratio est for Structural Models", Econometrica, Vol. 7, No. 4, [] W. K. Newey, (990). "E cient Instrumental Variables Estimation of Nonlinear Models, Econometrica 58,

21 [2] - (993). "E cient Estimation of Models with Conditional Moment Restrictions, in G.S. Maddala, C.R. Rao, and H.D. Vinod, eds., Handbook of Statistics, Volume : Econometrics. Amsterdam: North-Holland. [3] Stock, J. H., Wright, J.H. and Yogo, M. (2002), "A Survey of Weak Instruments and Weak Identi cation in Generalized Method of Moments," Journal of Business & Economic Statistics, Volume 20,

22 Aendix Proof of Proosition. Let ^ denote the GMM estimator using zt as instruments and V its asymtotic variance. Let ^ denote an alternative GMM estimator using x t as instruments, where x t = f(z t ) for some vector-valued function f(:). Let V denote its asymtotic variance of ^. It su ces to show that (V V ) is a ositive semi-de nite matrix. Write ^ = ^ + (^ ^ ); hen, herefore, V ar( ^) = V ar( ^ ) + V ar( (^ ^ )) V ar( ^) V ar( ^ ) +Cov ^ ; h^ ^ i + Cov h^ ^ i ; ^ : = V ar( (^ ^ )) + Cov ^ ; h^ ^ i + Cov h^ ^ i ; ^ : Because the rst term on the right hand side is ositive semi-de nite, the roof will be comlete if we can show Or, equivalently, to show lim Cov( ^ ; (^ ^ )) = 0:! lim Cov( ^ ; ^) = lim V ar( ^ ):!! o establish (A.), exlicit formulae for ^ and ^ are needed. satis es X zt d t (^ ) = 0: ake a rst order aylor s exansion around the true value 0 ; t= (A.) First consider ^. It 0 = X t= z t d t ( 0 ) + X t= z t ( 0 ) (^ 0 ) + o (): A-

23 Because lim! X t= z t ( 0 ) = E t ( 0 ) = E E t ( 0 ) jz ( 0 ) 0 = KE z t KD ; ( 0 ) z t where we have de ned we have, herefore D = E (^ ( 0 ) z t ( 0 ) z t ; X 0 ) = D K =2 zt d t ( 0 ) + o (): lim V ar( (^ 0 )) = D K V ar (zt d t ( 0 )) K D = D :! Note that the last equality follows because t= V ar (zt d t ( 0 ( 0 ) 0 = V ar z t d t ( 0 ( 0 ) 0 = KE z t d t ( 0 )d t ( 0 ) 0 ( 0 ) 0 = KE z ( 0 ) E z t = KD K 0 ( 0 ) K 0 z t Now consider ^ and aly similar arguments. We have! 0! X ^ = arg min X x t d t () ^S x t d t () ; (A.2) t= where ^S is a consistent estimate of the otimal weighting matrix S 0, with S 0 = V ar(x t d t ( 0 )): A-2 t= K 0

24 he rst order condition of (A.2) imlies X t=! t (^) x t ^S X t= x t d t (^) ake a rst order aylor s exansion of P t= x td t (^) around the true value 0 ; we have 0 = + X t=! t (^) x t ^S X t= Because ^! 0, we have x t (^)! 0 ^S X t= X t= x t d t ( 0 )!! = t ( 0 ) x t (^ 0 ) + o (): lim! X t= x t (^) = lim! X t= x t ( 0 ) D and herefore, And lim ^S = S0 :! (^ 0 ) = (D 0 S 0 D) D 0 S 0 =2 X x t d t ( 0 ): t= lim Cov( (^ 0 ); (^ 0 ))! 0! 0 = lim 0 S0 D) D 0 S0 X X x t d t ( 0 ) zt d t ( 0 ) K 0 D A! t= t= (due to iid) = E (D 0 S0 D) D 0 S0 x t d t ( 0 )d t ( 0 ) 0 zt 0 K 0 D = E (D 0 S0 D) D 0 S0 x t E d t ( 0 )d t ( 0 ) 0 0 jz t z 0 t K 0 D (A.3) For the term in the middle, x t E [d t ( 0 )d t ( 0 )jz t ] 0 z 0 t = x t ( 0 ) 0 t z K 0 = E A-3 x t ( 0 ) z t K 0 :

25 Hence, (A.3) equals E(D 0 S0 D) D 0 S0 = (D 0 S 0 D) D 0 S 0 E E = (D 0 S0 D) D 0 S0 t ( 0 ) x t = D t ( 0 ) E x t z t K t ( 0 ) x t z t K 0 D K 0 D K 0 D A-4