The generalized method of moments

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1 Robert M. Kunst University of Vienna and Institute for Advanced Studies Vienna February 2008

2 Based on the book Generalized Method of Moments by Alastair R. Hall (2005), Oxford University Press.

3 The main motivation Following the publication of the seminal paper by Lars Peter Hansen in 1982, GMM (generalized method of moments) has been used increasingly in econometric estimation problems. Some econometrics textbooks have even switched from maximum likelihood (ML) to GMM in their basic introduction to estimation methods.

4 Why maximum likelihood? Given a parametric model for the statistical distribution of observed variables f θ (x) and so-called regularity conditions, the ML estimator ˆθ = argmax θ L(θ X), with the notation L(θ X) = f θ (x 1,...,x n ) is consistent and (at least asymptotically) efficient for estimating θ. Why then consider anything else?

5 Reasons for not using ML 1 Regularity conditions are violated (rare). 2 The researcher does not accept a parametric model frame. 3 The maximization of the likelihood is unattractive and time-consuming.

6 Why some do not accept parametric model frames In simple, data-driven models with little a priori information, a strong belief in the parametric model is acceptable. Bayesians recognize the likelihood models as subjective constructions. The question of believing in them is void. By contrast, theory-driven researchers believe in some parametric aspects of their models (parameters of interest) but have little belief in others (error process parameters). In a report on the forecasting performance of the Bank of England in 2003, Adrian Pagan views modelling as a trade-off between theoretical coherence and empirical coherence.

7 Pagan s efficiency frontier of econometric modelling All models on the frontier are efficient. Following a downward movement in the 1980s, empirical economics has been moving upward on the frontier. GMM usage is following this increased emphasis on theory.

8 A basic example: linear regression For a likelihood adept, analysis of the linear regression model y t = X t β + u t starts from the assumption of i.i.d. N(0, σ 2 ) u t. Given X t, y = (y 1,...,y n ) has a Gaussian distribution. Maximization of the likelihood L(β, σ 2 ; y, X) yields the familiar OLS estimates ˆβ = (X X) 1 X y and ˆσ 2 = n 1 (y X ˆβ) (y X ˆβ). Then, consequences of assumption violations are studied.

9 Methods of moments and OLS A GMM adept sees the regression model as defined by the population moments conditions E(u t ) = 0, E(u 2 t ) = σ 2, E(X t u t ) = 0. These are to be matched to sample moments conditions for û t = y t X t ˆβ = û t (β) and n n 1 û t = 0, t=1 n n 1 ût 2 = ˆσ 2, t=1 n n 1 X t û t = 0. t=1 Again, the solutions are the OLS estimates. Conditions on X and y are studied that yield good properties for the OLS estimates.

10 Methods of moments becomes GMM In the linear regression, k + 1 moments conditions yield k + 1 equations and thus k + 1 parameter estimates. If there are more moments conditions than parameters to be estimated, the moments equations cannot be solved exactly. This case is called GMM (generalized method of moments). In GMM, moments conditions are solved approximately. To this aim, single condition equations are weighted.

11 Population moment condition Definition (1.1 Population moment condition) Let θ 0 be a true unknown vector parameter to be estimated, v t a vector of random variables, and f (.) a vector of functions. Then, a population moment condition takes the form E{f (v t, θ 0 )} = 0, t T. Often, f (.) will contain linear functions only, then the problem essentially becomes one of linear regression. In other cases, f (.) may still be products of errors and functions of observed variables, then the problem becomes one of non-linear regression. The definition is even more general.

12 The GMM estimator Definition (1.2 GMM estimator) The Generalized Method of Moments estimator based on these population moments conditions is the value of θ that minimizes n n Q n (θ) = {n 1 f (v t, θ) }W n {n 1 f (v t, θ)}, t=1 where W n is a non-negative definite matrix that usually depends on the data but converges to a constant positive definite matrix as n. t=1

13 OLS as GMM revisited The k + 1 functions are f j (y t, X t, β, σ 2 ) = X jt û t (β), j = 1,...,k, f k+1 (y t, X t, β, σ 2 ) = û t (β) 2 σ 2. Choosing W = I k+1 yields Q n (β, σ 2 ) = n 2 {û(β)} XX {û(β)} + (n 1 û (β)û(β) σ 2 ) 2, which is minimized for the OLS estimate at Q n (ˆβ, ˆσ 2 ) = 0.

14 Hall s Example I: Asset Pricing This is a theory-based, a bit more involved example that is used throughout the book. A representative agent maximizes discounted utility E δ0u(c k t+k Ω t ). k=0 N assets j with maturity m j are held at prices p j,t and quantities q j,t. The budget constraint for consumption c t and saving is defined by the sum of the payoffs r j,t and wage income w t, N c t + p j,t q j,t = j=1 N r j,t q j,t mj + w t. j=1 The consumption good acts as the numeraire.

15 Euler s equation for Example I O.c.s. (= one can show ) that the optimal consumption path satisfies E{δ m r j j,t+mj U (c t+mj ) 0 p j,t U Ω t } = 1. (c t ) For the utility function U(c) = (c γ 0 1)/γ 0, this expression becomes E{δ m j 0 r j,t+mj p j,t say, which implies for any z t Ω t. ( c t+m j c t ) γ 0 1 Ω t } 1 = 0 = E{u j,t (γ 0, δ 0 ) Ω t }, E[E{u j,t (γ 0, δ 0 ) Ω t }z t ] = 0

16 Characteristics of Hall s Example I There are just two parameters to be estimated: δ and γ. The discount rate δ is easy to estimate, γ is difficult to estimate ( weakly identified ). Maximum Likelihood would require solving a complicated maximization under distributional assumptions. GMM is straight forward (though not trivial). For just one asset and five instruments z t = (1, x 1,t, x 1,t 1, x 2,t, x 2,t 1 ) with x 1,t = c t /c t 1 and x 2,t = r t /p t 1, Hall uses this model as a running example.

17 Short statistics review Hall closes his introduction with a review of some useful statistical theorems and definitions. Definition (1.3 Convergence in probability) The sequence of random variables {h n } converges in probability to the r.v. h iff for all ǫ > 0 lim P( h n h < ǫ) = 1, n in symbols plimh n = h or h n p h.

18 Orders in probability Definition (1.4 Orders in probability) 1 The sequence of r.v. {h n } is said to be of large order in probability c n, in symbols O p (c n ), if for every ǫ > 0 there are positive m ǫ, n ǫ such that P( h n /c n > m ǫ ) < ǫ for all n > n ǫ. 2 The sequence of r.v. {h n } is said to be of small order in probability c n, in symbols o p (c n ), if h n /c n p 0. These definitions extend the common mathematical notation O(x n ) and o(x n ) to random convergence. They may also be used for vectors or matrices. Often, c n will be a simple function of n, such as n α.

19 Consistency and distributional convergence Definition (1.5 Consistency of an estimator) Let {ˆθ n } be a sequence of estimators for the true parameter vector θ 0, then ˆθ n is said to be a consistent estimator of θ 0 if ˆθ n p θ0. Hall does not require strong consistency here. Definition (1.6 Convergence in distribution) The sequence of r.v. {h n } with distribution functions {F n (c)} converges in distribution to the r.v. h with distribution function F(c), in symbols h n d h, iff there exists nǫ for every ǫ > 0 such that F n (c) F(c) < ǫ for n > n ǫ at all continuity points c.

20 Slutzky and friends Lemma (Slutzky s Theorem) Let h n be a sequence of random vectors that converges in probability to the random vector h and let f (.) be a vector of continuous functions then f (h n ) p f (h). Lemma (Corollary) Let {M n } be a sequence of random matrices that converge in probability to a constant matrix M, and {h n } be a sequence of vector-valued r.v. that converge in distribution to N(0, Σ). Then M n h n d N(0, MΣM ).

21 LLN Lemma (Weak law of large numbers) Assume v t is a sequence of r.v. with Ev t = µ, then any set of assumptions that imply n n 1 p v t µ t=1 is called a weak law of large numbers (WLLN). Strict stationarity together with some regularity conditions suffices.

22 CLT Lemma (Central limit theorem) Assume v t is a sequence of r.v. with Ev t = µ, then any set of assumptions that imply with n n 1/2 (v t µ) d N(0, Σ), t=1 Σ = lim n var{n 1/2 n (v t µ)} is called a central limit theorem (CLT). Strict stationarity together with some regularity conditions suffices. t=1

23 Consider y t = x tθ 0 + u t, t = 1,...,n, where x t collects p explanatory variables and θ 0 R p. Write u t (θ) = y t x tθ for the residual such that u t (θ 0 ) = u t. There is a q vector of observed instruments z t. We will also use the (n q) matrix Z = (z 1,...,z n ) and the (n p) matrix X. Assumption (2.1 Strict stationarity) The random vector v t = (x t, z t, u t ) is a strictly stationary process. Integrated processes can be handled (just take differences) but breaks etc. are excluded. Assumption (2.2 Population moment condition) The q vector z t satisfies E[z t u t (θ 0 )] = 0. θ 0 solves the basic problem but it may not be the only solution.

24 Assumption (2.3 Identification condition) rk{e(z t x t)} = p There must be at least as many instruments as regressors (q p) and these should be correlated with them. If assumption 2.3 holds and q > p, θ 0 is said to be over-identified. If q = p, it is just-identified. If the indicated rank is almost p 1, θ 0 is said to be weakly identified.

25 The estimator Given an appropriate weighting matrix W n, the GMM minimand is Q n (θ) = {n 1 u(θ) Z}W n {n 1 Z u(θ)}, and the GMM estimate is defined as ˆθ n = arg min θ Θ Q n(θ). Some algebraic manipulation yields ˆθ n = (X ZW n Z X) 1 X ZW n Z y. Clearly, X Z must have rank p, otherwise the estimate cannot be calculated. Assumption 2.3 is the large-sample counterpart of this condition.

26 In detail, the customary first-order condition Q n (θ) θ = 0 for ˆθ yields the equation (n 1 X Z)W n {n 1 Z u(ˆθ)} = 0, with its asymptotic counterpart E(x t z t)we{z t u t (θ 0 )} = 0.

27 The fundamental decomposition Define F = W 1/2 E(z t x t), with W 1/2 a root of W such that W = W 1/2 W 1/2 then re-write the population moment condition as F W 1/2 E{z t u t (θ 0 )} = 0. Multiplying these p equation from the left with a projection matrix yields the identifying restrictions of GMM: F(F F) 1 F W 1/2 E{z t u t (θ 0 )} = 0. These are q equations but only p are linearly independent. Conversely, the equation system {I q F(F F) 1 F }W 1/2 E{z t u t (θ 0 )} = 0 has q equations but the rank of the relevant matrix is only q p. This other system comprises the overidentifying restrictions of GMM.

28 Because the GMM estimator is defined in sample counterparts to population moment conditions, this decomposition is also valid in a finite-sample version. The identifying restrictions hold exactly. It is always possible to impose p restrictions if p parameters are to be estimated. Define F n = n 1 X ZWn 1/2. The equation F n (F nf n ) 1 F nw 1/2 n Z u(ˆθ) = 0 holds for finite n by definition. The overidentifying restrictions will usually not hold exactly for the finite-sample estimate ˆθ. The property that they should hold as n is the basis for customary GMM specification tests.

29 Asymptotic properties of the GMM estimator Like any good econometric estimator, the GMM estimator should be consistent (ˆθ n θ 0 ) and asymptotically normal (n 1/2 (ˆθ n θ 0 ) N(0, V) in distribution). Impose the following assumption: Assumption (2.4 Independence) The vector v t = (x t, z t, u t ) is independent of v t+s for all s 0. Explanatory variables, instruments, and errors taken together form an iid process. This defines a static model. It may be convenient to relax this assumption later.

30 Consistency is straight forward ˆθ n = (X ZW n Z X) 1 X ZW n Z y = θ 0 + {(n 1 X Z)W n (n 1 Z X)} 1 (n 1 X Z)W n (n 1 Z u) p θ 0 + {E(z t x t)we(x t z t)} 1 E(z t x t)we(z t u t ) = θ 0, using the population moment condition, LLN and Slutzky s Theorem.

31 Asymptotic distribution Start from the representation n 1/2 (ˆθ n θ 0 ) = {(n 1 X Z)W n (n 1 Z X)} 1 (n 1 X Z)W n (n 1/2 Z u), where everything converges to constants, except for the last term n n 1/2 Z u = n 1/2 z t u t, which follows a CLT (iid!), such that n 1/2 Z u d N(0, S). The interesting part is S defined by S = var(z t u t ). Hall calls the complicated constant limit matrix M (see above), such that the Corollary of Slutzky s Theorem yields t=1 n 1/2 (ˆθ n θ 0 ) d N(0, MSM ).

32 Estimating S In order to construct asymptotic confidence intervals or hypothesis tests, S must be estimated. One can show (o.c.s.) that n Ŝ n = n 1 u t (ˆθ n ) 2 z t z t t=1 p S. This is not trivial, as u(ˆθ n ) are not true errors. Of course, under the so-called classical assumptions Assumption (2.5 Classical assumptions on errors) Eu t = 0, Eu 2 t = σ 2 0,u t and z t independent, one may use with ˆσ 2 n = n 1 u(ˆθ n ) u(ˆθ n ). Ŝ CIV = ˆσ 2 nn 1 Z Z,

33 Asymptotic distribution of moments For technical reasons, one may be interested in the asymptotic distribution of n 1/2 Z u(ˆθ n ), the sample counterpart to the E(z t u t ) that is supposed to be zero. Consider the representation in F notation Therefore, ˆθ n θ 0 = (F nf n ) 1 F nw 1/2 n n 1 Z u. u(ˆθ n ) = u X ˆθ n + Xθ 0 = u X(F nf n ) 1 F nw 1/2 n n 1 Z u.

34 Thus, residuals are expressed as a function of errors. This representation can be multiplied by n 1/2 W 1/2 Z, so it yields n 1/2 W 1/2 Z u(ˆθ n ) = (I q P n )W 1/2 n n 1/2 Z u, with the short notation P n = F n (F nf n ) 1 F n, and finally n 1/2 W 1/2 Z u(ˆθ n ) p N(0, NSN ), with a slightly complicated but tractable matrix N. This asymptotic property is convenient for developing specification tests.

35 Optimal choice of the weighting matrix The asymptotic variance of the GMM estimator is MSM, where M = {E(x t z t)we(z t x t)} 1 E(x t z t)w. O.c.s. (L.P. Hansen) that the variance becomes minimal for W 0 = S 1. Then, the asymptotic variance becomes, by straight forward insertion, V 0 = {E(x t z t)s 1 E(z t x t)} 1.

36 Two-step GMM In practice, S must be replaced by a consistent estimator, for example by Ŝ n, which however is infeasible, as it uses a GMM estimate ˆθ n. Solution is two-step: 1 Estimate θ by a simple and not optimal weighting matrix, for example the identity matrix I. Estimate S based on the residuals. 2 Estimate θ again by using the weighting matrix according to step 1. The thus defined estimator is called the optimal two-step GMM estimator. Iterations may continue and then define the optimal iterated GMM estimator. Under the classical assumptions 2.5, the two-step GMM estimator becomes the familiar two-stage least-squares estimator (2SLS).

37 Several types of mis-specification Hall distinguishes among three cases: 1 Correct specification. The specified model M is the true model. Ez t u t (θ 0 ) = 0 for a unique θ 0 Θ. 2 The true model M A is not the specified model M but θ + is still a unique solution to the moment conditions. θ + is a pseudo-true value. 3 The true model M B is not M, and there is no θ such that Ez t u t (θ) = 0. In most aspects, item # 2 is innocuous. M A (θ + ) can be treated like M(θ 0 ), these models are observationally equivalent w.r.t. moment conditions.

38 The idea of the J test In the case of item #3, identifying restrictions will hold by definition but overidentifying restrictions will be violated even in the limit for large n. One may consider the statistic J n = nq n (ˆθ n ) = n 1 u(ˆθ n ) ZŜ 1 n Z u(ˆθ n ), which, under H 0 of Ez t u t (θ 0 ) = 0, converges to a χ 2 q p in distribution. Note that the asymptotic properties of the moment condition are used in deriving the χ 2 limit.

39 A few assumptions to define well behaved models. Assumption (3.1 Strict stationarity) The r dimensional random vectors {v t, t Z} form a strictly stationary process with sample space V R r. Assumption (3.2 Regularity conditions for f ) The function f : V Θ R q satisfies: 1 f is continuous on Θ for all v t V; 2 Ef (v t, θ) < for all θ Θ; 3 Ef (v t, θ) is continuous on Θ. The continuity assumption is restrictive and excludes some cases of empirical interest.

40 Population moment condition and identification Assumption (3.3 Population moment condition) The r.v. v t and the parameter θ 0 R p satisfy the q vector of population moment conditions Ef (v t, θ 0 ) = 0. Assumption (3.4 Global identification) Ef (v t, θ) 0 for all θ Θ with θ θ 0. These two conditions exclude the misspecified case M B as well as the case of multiple solutions. In some applications, local identification may suffice.

41 An example for a non-identified model: partial adjustment In the model y t y t 1 = β 0 (y 0 y t 1 ) + u t, u t = ρ 0 u t 1 + e t, y represents a target or desired level for y t, and e t is i.i.d. with mean 0. θ = (β, ρ,y ) and p = 3. Assume there are some q > p variables z that serve as instruments such that Ez t e t (θ) = 0, e t (θ) = y t β(1 ρ)y (1 + ρ β)y t 1 (β 1)ρy t 2, where the residuals just follow from the model definition by transformation. The model is nonlinear in θ. An AR(2) model with a constant is identified and linear and would have parameters µ = ( µ, φ 1, φ 2 ), say. O.c.s. that θ as a function of given µ has multiple solutions due to a quadratic function. The model is not globally identified.

42 Conditions for local identification Local identification, which may be more relevant in practice, requires that derivatives exist and that for every θ an entire ǫ neighborhood is contained in the parameter space. Assumption (3.5 Regularity condition on f (v t, θ)/ θ) 1 the derivative matrix f (v t, θ)/ θ exists and is continuous on Θ for all v t V; 2 θ 0 is an interior point of Θ; 3 E f (v t, θ)/ θ <. Assumption (3.6 Local identification) rk(e f (v t, θ)/ θ ) = p. This condition naturally generalizes Assumption 2.3 from the linear model.

43 The partial adjustment model is locally identified With ỹ t = (1, y t 1, y t 2 ), the matrix of Assumption 3.6 can be written as E f (v t, θ)/ θ ) = E(z t ỹ t)m(θ 0 ), where (1 ρ)y βy β(1 ρ) M(θ) = ρ (β 1) 0 Clearly, in general this matrix is of full rank, and the local identification condition holds.

44 Technical issues of GMM estimation in nonlinear models The GMM estimate ˆθ is the minimum of the function n n Q n (θ) = {n 1 f (v t, θ)} W n {n 1 f (v t, θ)}. t=1 For the large-sample behavior of the weighting matrix we assume t=1 Assumption (3.7 Properties of the weighting matrix) W n is a non-negative definite matrix that converges in probability to the positive definite constant matrix W. Typically, ˆθ does not have a closed-form solution. It will be obtained via an iterative numerical algorithm.

45 GMM as a solution to first-order conditions If derivatives exist (Assumption 3.5), the minimization can rely on Q n (θ)/θ = 0, or, in detail, {n 1 n t=1 f (v t, ˆθ n ) θ } W n {n 1 n f (v t, ˆθ n )} = 0. t=1 This expression has a population counterpart E{ f (v t, θ 0 ) θ } WEf (v t, θ 0 ) = 0.

46 Numerical optimization Numerical minimization has three aspects: 1 the starting value θ(0); 2 the iterative search method, i.e. how to find θ(j + 1) given θ(j); 3 the convergence criterion. Hall recommends to vary starting values as a sensitivity check. Convergence criteria can be absolute ( θ(j + 1) θ(j) < ǫ), relative ( θ(j + 1) θ(j) / θ(j) < ǫ), or a mixture of both. They can check convergence of θ, of the function Q, or of a gradient in particular, if gradients of Q are used in the search iterations.

47 Fundamental decomposition in the nonlinear model Consider the first-order condition Again, this permits a form E{ f (v t, θ 0 ) θ } WEf (v t, θ 0 ) = 0. F(θ 0 ) W 1/2 E{f (v t, θ 0 )} = 0, and a decomposition into the identifying restrictions F(θ 0 ){F(θ 0 ) F(θ 0 )} 1 F(θ 0 ) W 1/2 E{f (v t, θ 0 )} = 0, and the overidentifying restrictions.

48 The overidentifying restrictions The overidentifying restrictions [I q F(θ 0 ){F(θ 0 ) F(θ 0 )} 1 F(θ 0 ) ]W 1/2 E{f (v t, θ 0 )} = 0 are not fulfilled in sample (in estimation) but only asymptotically if the model is specified correctly (or misspecified of the M A type?). Note that the two projection matrices sum to I and therefore Q n (ˆθ) shows directly how well the overidentifying restrictions are satisfied. This is the basis for a popular specification test statistic.

49 Asymptotic properties of the GMM estimator Assumption (3.8 Ergodicity) The random process {v t, t Z} is ergodic, meaning its sample moments converge to the population moments. There exist non-ergodic stationary processes, and minimal conditions to exclude them are complex. We just assume. Assumption (3.9 Compactness) The parameter space Θ is compact. No sequence of admissible parameter values should converge to a non-admissible one. No sequence should escape to infinity. Assumption (3.10 Domination of f ) Esup θ Θ f (v t, θ) <. Because Θ is now bounded anyway, this assumption excludes local areas and points with undefined expectation.

50 Consistency of GMM O.c.s. the following Lemma. Lemma Assumptions 3.1,3.2, imply that p sup θ Θ Q n (θ) Q 0 (θ) 0, where Q 0 (θ) is introduced for the population analog to Q n (θ). Using this lemma, it is not difficult to show Theorem (3.1 Consistency of the GMM estimator) Assumptions and imply that ˆθ n p θ0. Note that this theorem assumes global identification. Q n converges to Q 0, and its minimum converges to the minimum of Q 0, which is unique according to assumptions.

51 Assumptions for the asymptotic normality of GMM Application of a CLT requires some more technical assumptions. Introduce g n (θ) = n 1 n t=1 f (v t, θ), the sample analog to Ef (v t, θ). Assumption (3.11 Variance of sample moment) 1 E{f (v t, θ 0 )f (v t, θ 0 ) } < ; 2 var(n 1/2 g n (θ 0 )) S; 3 S is finite and positive definite. This assumption is needed for a CLT: Lemma Assumptions 3.1,3.3,3.8, and 3.11 imply n 1/2 g n (θ 0 ) d N(0, S).

52 Now g converges but so must its derivatives G n (θ) = n 1 n t=1 f (v t, θ)/ θ. Denote its population analog E f (v t, θ 0 )/ θ by G 0. Assumption (3.12 Continuity of expected derivatives) E f (v t, θ)/ θ is continuous in a neighborhood N ǫ of θ 0. Assumption (3.13 Uniform convergence of G n (θ)) sup θ Nǫ G n (θ) E f (v t, θ)/ θ p 0. With these assumptions, G n (ˆθ n ) finally converges to G 0 in probability and the desired property follows.

53 Asymptotic normality of GMM Theorem (3.2 Asymptotic normality of parameter estimator) Assumptions and imply n 1/2 (ˆθ n θ 0 ) d N(0, MSM ), where M = (G 0 WG 0) 1 G 0 W. Clearly, G 0 must be estimated by some finite-sample analog, and so must S. Unfortunately, estimation of S is not trivial. Theorem (3.3 Asymptotic normality of sample moments) Assumptions and imply Wn 1/2 n 1/2 g n (ˆθ n ) d N(0, NW 1/2 SW 1/2 N ) where N = I q F(θ 0 ){F(θ 0 ) F(θ 0 )} 1 F(θ 0 ).

54 Estimating S S is the limit of the variances of cumulative sums ( ) n S = lim var n 1/2 f t, n where (f t ) is stationary but not i.i.d. In time-series analysis, this variance is known as the spectrum at zero, while econometricians call it the long-run variance. In population, it is simply given by S = j= where Γ j are the autocovariance matrices of f t. Plugging in autocovariance estimates (for ˆf t ) and chopping of the sum at some J and J yields an unattractive estimate. Γ j, t=1

55 Nonparametric estimation of S In time-series analysis, a preferred method for estimating the spectral density is kernel or window estimation. At frequency zero, this amounts to n 1 Ŝ HAC = Γ 0 + ω j,n (ˆΓ j + ˆΓ j), with the kernel function ω j,n that downweights autocovariance estimates with large j. Usually, ω j,n = 0 for j > b(n), with the bandwidth b(n) as n. The most common kernel function is the Bartlett or triangular kernel ω j,n = { j=1 1 j b(n)+1, j b(n), 0, else.

56 What does HAC mean? HAC stands for heteroskedasticity and autocorrelation consistent covariance estimation, a technique developed by Newey and West, not for heteroscedastic autocorrelation covariance (Hall). The HAC method requires spectral estimation at frequency zero, and this is done via kernel estimation. For this reason, some econometricians call the triangular Bartlett window a Newey and West window.

57 Parametric estimation of S As an alternative way, one may fit a vector autoregression (VAR) to f t (or rather ˆf t : f t = A 1 f t 1 + A 2 f t A k f t k + e t, selecting k according to Schwarz BIC, for example, and estimate S via k Ŝ VAR = (I q A k ) 1ˆΣ(e)(I k q A k ) 1, j=1 with ˆΣ(e) = n 1 ê ê estimated from the residuals of the VAR. One may even combine parametric and nonparametric stages (Andrews and Monahan). j=1

58 Optimal choice of the weighting matrix In analogy to the linear model, the optimal weighting matrix W is obtained by W = S 1. Theorem (3.4 Optimal weighting matrix) Assumptions , imply that the minimum asymptotic variance matrix of ˆθ is (G 0 S 1 G 0 ) 1, which is obtained by W = S 1. In practice, S must be estimated. One may start by weighting W = I and use the obtained ˆθ to obtain Ŝ to update ˆθ (two-step GMM) or one may iterate these steps to convergence (iterated GMM). A variant solves for ˆθ and Ŝ jointly, leaving iterations to the computer (continuously updated GMM).

59 A technical independence result Theorem (3.5 Asymptotic independence of estimate and moment) Assume and hold and W = S 1, then n 1/2 (ˆθ n θ 0 ) and S 1/2 n 1/2 g n (ˆθ n ) are asymptotically independent. This theorem guarantees that iterated GMM converges to the exact solution as n and thus is comparable to continuous updating. It also guarantees that J test specification statistics on overidentifying restrictions will be independent from the parameter estimates. The details of its proof (in Hall s book) show that this independence does not hold if W S 1.

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