Optimal bandwidth selection for robust generalized method of moments estimation

Transcription

1 Optimal bandwidth selection for robust generalized method of moments estimation Daniel Wilhelm he Institute for Fiscal Studies Department of Economics, UCL cemmap working paper CWP15/14

2 Optimal Bandwidth Selection for Robust Generalized Method of Moments Estimation Daniel Wilhelm UCL and CeMMAP March 22, 2014 Abstract A two-step generalized method of moments estimation procedure can be made robust to heteroskedasticity and autocorrelation in the data by using a nonparametric estimator of the optimal weighting matrix. his paper addresses the issue of choosing the corresponding smoothing parameter (or bandwidth so that the resulting point estimate is optimal in a certain sense. We derive an asymptotically optimal bandwidth that minimizes a higher-order approximation to the asymptotic meansquared error of the estimator of interest. We show that the optimal bandwidth is of the same order as the one minimizing the mean-squared error of the nonparametric plugin estimator, but the constants of proportionality are significantly different. Finally, we develop a data-driven bandwidth selection rule and show, in a simulation experiment, that it may substantially reduce the estimator s mean-squared error relative to existing bandwidth choices, especially when the number of moment conditions is large. JEL classification: C12; C13; C14; C22; C51 Keywords: GMM; higher-order expansion; optimal bandwidth; mean-squared error; long-run variance. Department of Economics, University College London, 30 Gordon St, London WC1H 0AX, United Kingdom; address: d.wilhelm@ucl.ac.uk. I thank Christian Hansen, Alan Bester, the co-editor and two referees for helpful comments. he author gratefully acknowledges financial support from the ESRC Centre for Microdata Methods and Practice at IFS (RES

3 1 Introduction Since the seminal paper by Hansen (1982 the generalized method of moments (GMM has become a popular method for the estimation of partially specified models based on moment conditions. In time series applications, two-step GMM estimators can be made robust to heteroskedasticity and autocorrelation (HAC by using a nonparametric plugin estimator of the optimal weighting matrix. he goal of this paper is to develop a selection rule for the corresponding smoothing parameter of the nonparametric estimator such that the resulting point estimator minimizes a suitably defined mean-squared error (MSE criterion. Many instances of poor finite sample performance of GMM estimators have been reported in the literature. See for example Hansen, Heaton, and Yaron (1996 and references therein. As an attempt to improve the properties different extensions and new estimators have been proposed, e.g. the empirical likelihood estimator introduced by Owen (1988, Qin and Lawless (1994, the exponential tilting estimator of Kitamura and Stutzer (1997 and Imbens, Spady, and Johnson (1998 and the continuous updating estimator by Hansen, Heaton, and Yaron (1996. Newey and Smith (2004 show that all these estimators are members of a larger class of generalized empirical likelihood (GEL estimators. A different approach termed fixed-b asymptotics is based on deriving more accurate approximations of estimators and test statistics based on an asymptotic sequence in which the HAC smoothing parameter tends to infinity at the same rate as the sample size. See for example Kiefer and Vogelsang (2002a,b, Instead of treating the smoothing parameter as proportional to the sample size, Sun and Phillips (2008 and Sun, Phillips, and Jin (2008 develop a higher-order asymptotic theory based on which they find the optimal rate at which the smoothing parameter (here a bandwidth minimizes the coverage probability error or length of confidence intervals. Similar in spirit, the present paper derives the optimal growth rate of the bandwidth to minimize an asymptotic mean-squared error (AMSE criterion. We approximate the MSE of the second-step GMM estimator by the MSE of the first few terms in a stochastic expansion. Since the proposed semiparametric estimator is first-order equivalent to ordinary GMM estimators in the iid case, the optimal bandwidth derived in this paper will minimize its second-order effects on the estimator and lead to second-order efficiency gains. In an unpublished dissertation, Jun (2007 independently develops a similar expansion and arrives at the same MSE-optimal bandwidth as derived in this paper, however under a slightly different set of assumptions. 1 Other bandwidth choices for HAC-robust estimation have been suggested by Andrews 2

4 (1991, Newey and West (1994 and Andrews and Monahan (1992, for example, and are very popular in applied research. In this paper, we show that these are suboptimal choices if MSE-optimal point estimation is of main interest. In finite samples, the existing methods can select bandwidths that are significantly different from the MSE-optimal bandwidth even though they share the same asymptotic order relative to the sample size. he difference is due to the other methods minimizing the AMSE of the weighing matrix estimator instead of minimizing the AMSE of the GMM estimator itself; they guarantee accurate estimates of the optimal weighting matrix, but not necessarily of the parameter of interest. In the linear regression framework with potential autocorrelation and heteroskedasticity, there are several papers (e.g. Robinson (1991, Xiao and Phillips (1998 and amaki (2007 that derive higher-order expansions of the MSE of semiparametric frequency domain estimators to determine an optimal bandwidth that minimizes higher-order terms of such expansions. In the present paper, however, we allow for nonlinear models and over-identification which significantly complicate the problem and require a different set of tools to derive such expansions. o approximate higher-order moments of the GMM estimator we develop a stochastic expansion of the estimator similar to the approach in Nagar (1959. See Rothenberg (1984 for an introduction to Nagar-type expansions and for further references. Several other authors have analyzed higher-order properties of GMM and GEL estimators using similar tools. Rilstone, Srivastava, and Ullah (1996 and Newey and Smith (2004 provide expressions for the higher-order bias and variance of GMM and GEL estimators when the data are iid. Anatolyev (2005 derives the higher-order bias in the presence of serial correlation. Finally, Goldstein and Messer (1992 present general conditions under which functionals of nonparametric plug-in estimators achieve the optimal rate of convergence. Depending on the functional under-smoothing the plugin estimator relative to the smoothing parameter used to optimally estimate the nonparametric quantity itself may be necessary. he paper is organized as follows. he first section introduces the econometric setup, derives a higher-order expansion of the two-step GMM estimator and the optimal bandwidth that minimizes an approximate MSE based on that expansion. he third section describes an approach to estimate the infeasible optimal bandwidths, followed by a simulation experiment that demonstrates the procedure s performance in finite samples. he paper concludes with an appendix containing all mathematical proofs. Let vec( denote the column-by-column stacking operation and vech( the columnby-column stacking operation of entries on and above the diagonal of a symmetric matrix. 3

5 By K m,n denote the mn mn commutation matrix so that, for any m n matrix M, K m,n vec(m = vec(m. Let be the Kronecker product and r F (β, with r Z and F (β a matrix being r times differentiable in β, denote the matrix of r-th order partial derivatives with respect to β, recursively defined as in Rilstone, Srivastava, and Ullah ( F (β := F (β. his notation for derivatives will sometimes be used to save space and simplify notation. denotes the Euclidean (matrix norm, M the transpose of a matrix M, with probability approaching one is abbreviated w.p.a. 1 and with probability one by w.p. 1. he notation x = O p (1 means that the sequence {x } =1 is uniformly tight. 2 Optimal Bandwidth In this section, we introduce the basic framework, define an appropriate MSE criterion and find the optimal bandwidth that minimizes it. he idea is to derive a higher-order approximation of the second-step estimator and to the MSE from the moments of this approximation. A higher-order analysis is required in this setup because first-order asymptotics do not depend on the smoothing parameter. Consider estimation of a parameter β 0 B from the moment equation Eg(X t, β 0 = 0 given a data sample {x t } t=1. If the dimension of the range of g is at least as large as the dimension of the parameter β 0, then a popular estimator of β 0 is the two-step GMM estimator defined as follows. First, estimate β 0 by some -consistent estimator, say β, that is then used to construct a consistent estimator ˆΩ( β of the long-run variance Ω 0 := s= Γ(s, Γ(s := E[g(X t+s, β 0 g(x t, β 0 ]. In a second step, compute the GMM estimator ˆβ of β 0 with weighting matrix ˆΩ( β 1, viz. ˆβ := arg min β B ĝ(β ˆΩ ( β 1 ĝ(β, (2.1 where ĝ(β := 1 t=1 g(x t, β. he second step improves the first-step estimator in terms of efficiency. In fact, ˆβ is optimal in the sense that it achieves the lowest asymptotic variance among all estimators of the form ˆβ W := arg min β B ĝ(β W ĝ(β for some positive definite weighting matrix W (see Hansen (1982. In the special case of an iid process {X t }, Ω 0 collapses to Ω 0 = E[g(X t, β 0 g(x t, β 0 ] and can simply be estimated by its sample analog ˆΩ ( β := 1 t=1 g(x t, βg(x t, β. When the iid assumption is not justified, one can perform inference robust to autocorrelated and/or heteroskedastic X t processes. Robustness here means that potential dependence and heteroskedasticity are treated nonparametrically and one does not have to 4

6 be explicit about the data generating process of the X t s. o that end, one needs to smooth the observations g(x t+s, βg(x t, β over s to ensure that ˆΩ ( β is consistent. In this paper, we use a nonparametric kernel estimator of the form ˆΩ ( β := 1 1 min{, s} s=1 t=max{1,1 s} ( s k g t+s ( S βg t ( β with g t (β := g(x t, β and k a kernel function. S, with S as, is a so-called bandwidth parameter that governs the degree of smoothing. Andrews (1991 derives a range of rates (in terms of at which S is allowed to diverge in order to guarantee consistency of ˆΩ ( β. hese conditions, however, do not suggest rules for choosing S for a fixed sample size. Small values of S imply averaging over only few observations which decreases the variability of the estimator ˆΩ ( β, but increases its bias. On the other hand, large bandwidths yield inclusion of more distant lags in the above sum, thereby increasing the variance, but decreasing the bias of the estimator. Below we show that the choice of S affects the bias and variance of the second-step estimator ˆβ in a similar way. his trade-off can be used to derive decision rules on how to pick S in finite samples. For example, Andrews (1991 derives the optimal bandwidth minimizing a truncated asymptotic mean-square error (AMSE criterion that balances bias and variance of ˆΩ ( β, thereby guaranteeing good properties of the estimator of the optimal weighting matrix. However, in the GMM estimation framework, the secondstep estimator ˆβ is the quantity of interest and, thus, the bandwidth should be chosen so as to take into account the bias and variance trade-off of ˆβ, rather than that of ˆΩ ( β. o that end the subsequent analysis develops a higher-order expansion of the MSE of the second-step estimator and then minimize the leading terms with respect to the bandwidth. Assumption 2.1. (a he process {X t } t= taking values in X R m is fourth-order stationary and α-mixing with mixing coefficients α(j satisfying j=1 j2 α(j (ν 1/ν < for some ν > 1. (b {x t } t=1 is an observed sample of {X t } t=. (c h(, β := (g(, β, vec( g(, β E g(, β, vec( 2 g(, β E 2 g(, β is a measurable function for every β B. (d sup t 1 E[ h(x t, β 0 4ν ] <. (e sup t 1 E[sup β B k g(x t, β 2 ] < for k = 1, 2, 3. (f here is a first-step estimator β satisfying β β 0 = O p ( 1/2. As a slight abuse of notation, in the remainder, x t represents the random variable X t as well as the observation x t, but the distinction should be clear from the context. Furthermore, dropping β as an argument of a function means that the function is evaluated at β 0, e.g. ˆΩ := ˆΩ (β 0 or g t := g t (β 0. Let G 0 := EG(x t, G(x t, β := g t (β/ β, G t (β := G(x t, β, and the sample counterpart G (β := 1 t=1 g t(β/ β. 5

7 Following Parzen (1957, let q be the characteristic exponent that characterizes the smoothness of the kernel k at zero: q := max{α [0, : g α exists and 0 < g α < } 1 k(z with g α := lim z 0. For example, for the Bartlett, Parzen and ukey-hanning kernel z α the values of q are 1, 2 and 2, respectively. Assumption 2.2. Let the kernel k satisfy the following conditions: (a k : R [ 1, 1] satisfies k(0 = 1, k(x = k( x x R, k2 (xdx <, k(x dx <, k( is continuous at 0 and at all but a finite number of other points, and S, S 2 / 0, Sq / 0 for some q [0, for which g q, f (q [0, where f (q := 1 2π j= j q Γ(j. (b k(xdx < with { sup k(x := y x k(y, x 0 sup y x k(y, x < 0. Assumptions 2.1 and 2.2(a imply Assumptions A, B and C in Andrews (1991 applied to {g t } and {G t }, allowing us to use his consistency and rate of convergence results for HAC estimators. he necessity of Assumption 2.2(b is explained in Jansson (2002. Assumption 2.3. (a g : X B R l, l p, and β 0 int(b is the unique solution to Eg(x t, β 0 = 0, B R p is compact. (b rank(g 0 = p. (c For any x X, g(x, is twice continuously differentiable in a neighborhood N of β 0. (d here exists a function d : X X R with s= Ed(x t+s, x t < so that g satisfies the condition (g t+s (βg t (β (g t+s (β 0 g t (β 0 d(x t+s, x t w.p. 1 for β N. (e here exists a function b : X R with Eb(x t < such that k g(x, β k g(x, β 0 b(x β β 0 for k = 2, 3. (f Ω 0 is positive definite. he following proposition is the first main result of the paper, presenting an expansion of the second-step GMM estimator ˆβ up to the lowest orders involving the bandwidth S. his approximation constitutes a crucial ingredient for the computation of the optimal bandwidth. Proposition 2.1. Under Assumptions , ˆβ satisfies the stochastic expansion ˆβ = β 0 + κ 1, 1/2 + κ 2, S 1/2 1 + κ 3, S q 1/2 + o p (η 1/2 (2.2 with η := S 1/2 1/2 + S q, κ i, = O p (1 for i = 1, 2, 3, κ 1, := H 0 F κ 2, := H 0 /S (ˆΩ Ω P 0 F ( κ 3, := H 0 S q Ω Ω 0 P0 F 6

8 and F (β := ĝ(β, F := F (β 0, Ω := E ˆΩ, Σ 0 := (G 0Ω 1 0 G 0 1, H 0 := Σ 0 G 0Ω 1 0 and P 0 := Ω 1 0 Ω 1 0 G 0 H 0. Since the lowest-order term, H 0 F 1/2, does not depend on the bandwidth, the expansion (2.2 illustrates the well-known fact that nonparametric estimation of the GMM weighting matrix does not affect first-order asymptotics as long as that nonparametric estimator is consistent. he other two terms in the expansion involve the bandwidth and arise from the bias (S q (Ω Ω 0 and variance ( /S (ˆΩ Ω of the nonparametric estimator of the weighting matrix. In a similar expansion to (2.2 for the iid case, these two components do not appear and the next higher-order term after κ 1, 1/2 is of order 1 (see Newey and Smith (2004, which here is part of the remainder and plays no role in determining the optimal bandwidth derived below. Anatolyev (2005 does not explicitly present a stochastic expansion such as (2.2, but computes the higher-order bias B of ˆβ which turns out to be of order 1, i.e. E[ ˆβ β 0 ] = B 1 + o( 1, and therefore does not depend on the bandwidth. Interestingly, one can show that the two higher-order terms in (2.2 do not contribute to that bias (Jun (2007. In the class of GMM estimators defined by (2.1 and indexed by the bandwidth S, we now characterize the most efficient one under quadratic loss. Specifically, we rank estimators according to the MSE of the approximation ζ := β 0 +κ 1, 1/2 +κ 2, S 1/2 1 + κ 3, S q 1/2. heorem 2.1. Suppose Assumptions hold. Let W R p p be a weighting matrix. Define the weighted MSE MSE := E[(ζ β 0 W(ζ β 0 ]. hen with ν 1 := MSE = ν ν 2 S 2 + ν 3 S 2q 1 + o(η 2 1 s= ν 2 := lim E S 2 lim E [g th 0WH 0 g t+s ] [F P 0(ˆΩ Ω H 0WH 0 (ˆΩ Ω P 0 F ] S E [F P 0(ˆΩ Ω H 0WH 0 F ] ν 3 := lim S2q E [ F P 0(Ω Ω 0 H 0WH 0 (Ω Ω 0 P 0 F ] where all the limits exist and are finite. As explained above, the bias of the approximation ζ is zero so that the MSE expansion (2.1 represents only the variance of ζ. Despite the lack of a bias component of ζ, 7

9 the expansion displays the first-order tradeoff that is relevant for choosing a bandwidth: ν 2 S 2 increases in S and ν 3 S 2q 1 decreases in S. he terms have the standard order of squared bias and variance of HAC estimators as derived in Andrews (1991. Under additional conditions, the moments of the approximation ζ correspond to the moments of a formal Edgeworth expansion of the cdf of the second-step estimator ˆβ. he (finite moments of such an Edgeworth expansion can be used to approximate the distribution of ˆβ up to the specified order even when the corresponding moments of ˆβ do not exist (Götze and Hipp (1978, Rothenberg (1984, Magdalinos (1992. In this sense, we can regard (2.1 as an approximation of the MSE of ˆβ when ˆβ possesses second moments and as the MSE of an approximate estimator that shares the same Edgeworth expansion up to a specified order. Remark 2.1. For linear instrumental variable models with iid variables and normal errors, Kinal (1980 shows that ˆβ has finite moments up to order l p. Similar results have been conjectured for GMM and generalized empirical likelihood estimators (e.g. Kunitomo and Matsushita (2003, Guggenberger (2008. herefore, one should be careful in interpreting the MSE approximation in cases when the degree of over-identification is less than two. Other loss functions may then be more appropriate (see Zaman (1981, for example. Having established heorem 2.1, the calculation of an MSE-optimal bandwidth, S, becomes straightforward: for the second-order term to attain its fastest possible rate of convergence, the terms of order S 2 and S 2q 1 have to be balanced which is the case for S = c (q 1/(1+2q and some constant c (q. We refer to this bandwidth as the MSE( ˆβ-optimal bandwidth. he bandwidth minimizing the MSE of ˆΩ ( β as derived in Andrews (1991 we call the MSE(ˆΩ-optimal bandwidth. Corollary 2.1. Under the assumptions of heorem 2.1 and if l > p, minimizing the lowest order of MSE involving the bandwidth yields the optimal bandwidth growth rate 1/(1+2q. Moreover, S = c (q 1/(1+2q minimizes the higher-order AMSE defined as the limit of HMSE := (1+4q/(1+2q {MSE ν 1 1 } with ( { 1/(1+2q c c0 ν 3 2q, sign(ν 2 = sign(ν 3 (q :=, c 0 := 1, sign(ν 2 sign(ν 3. (2.3 ν 2 he expressions for ν 2 and ν 3 show that the optimal bandwidth growth rate is governed by the convergence rate of the covariances between the moment functions and the HAC estimator. he bias and variance of the HAC estimator itself only play an indirect role; in particular, the AMSE of ˆβ is not an increasing function of the AMSE of the HAC 8

10 estimator. In consequence, none of the existing procedures minimizing the AMSE of the HAC estimator (Andrews (1991, Newey and West (1994 and Andrews and Monahan (1992 among others are optimal in the above sense. he convergence rate of the MSE of ˆβ is not affected by the bandwidth choice because it is of order O( 1, only the second-order terms of the MSE, converging at an optimal rate of O( (1+4q/(1+2q, are. By choosing kernels of very high order q, this rate can be made arbitrarily close to O( 2. Nevertheless, kernels of order smaller than or equal to 2 are popular because, unlike kernels of higher order, they can produce positive definite covariance matrix estimates. Remark 2.2. Interestingly, the semiparametric estimator ˆβ converges at rate 1/2, but the optimal bandwidth minimizing MSE( ˆβ is of the same order as the optimal bandwidth minimizing MSE(ˆΩ. his result contrasts the findings in other semiparametric settings such as those studied by Powell and Stoker (1996 and Goldstein and Messer (1992, for example, in which under-smoothing the nonparametric plugin estimator leads to 1/2 - convergence rates of smooth functionals of that nonparametric plugin estimator. o gain more insight into which features of the data generating process determine the value of the optimal bandwidth and to be able to directly estimate the quantities involved, the following proposition derives more explicit expressions for the constants ν i. Proposition 2.2. Assume that {g t } follows a linear Gaussian process, viz. g t = Ψ s e t s s=0 for t = 1,...,, e t N(0, Σ e iid and Ψ s satisfies s=0 s4 Ψ s <. Define µ i := ki (xdx for i = 1, 2 and Ω (q 0 := 2πf (q. hen 2.1 Linear IV Model ν 1 = tr(ω 0 H 0WH 0, ν 2 = (2µ 1 + µ 2 (l ptr(σ 0 W, ( ν 3 = gqtr 2 Ω (q 0 H 0WH 0 Ω (q 0 P 0. In this sub-section, we specialize the expressions in Proposition 2.2 to a stylized instrumental variable model that allows us to analyze the difference between the MSE( ˆβ-optimal and the MSE(ˆΩ-optimal bandwidths and subsequently serves as the data generating process for the Monte Carlo simulations. Let y t and w t be random variables satisfying y t = β 0 w t + ε t 9

11 and z t an l-dimensional random vector of instruments such that w t = γι z t + v t where ι := (1,..., 1 R l and γ R. Define x t = (y t, w t, z t so that g(x t, β = (y t βw t z t. Further let {ε t } and {v t } be AR(1 processes with autocorrelation coefficient ρ (0, 1, viz. ε t = ρε t 1 + η t and v t = ρv t 1 + u t, where ( ( ( η t 1 σ 12 iid N 0,, σ σ 12 1 u t he instruments follow a VAR(1 process, z t = ρ z z t 1 + ɛ t with ρ z := diag(0, ρ,..., ρ, ρ (0, 1, and ɛ t iid N(0, I l independent of {(η t, u t }. hen, one can show that ( c 4c 0 g 2 (1 = 1ρ 2 ρ 2 (1 ρ 2 (2µ 1 + µ 2 (1 ρ ρ(1 + ρ ρ( l + (l 2ρ ρ + ρ 2 + ρ ρ 3 2 ( c 4c 0 g 2 (2 = 2ρ 2 ρ 2 (1 + ρ ρ(1 ρ 2 1/5 (2µ 1 + µ 2 (1 ρ ρ 3 ( l + (l 2ρ ρ + ρ 2 + ρ ρ 3 2 In this specific example, we can easily compare the MSE( ˆβ-optimal to the MSE(ˆΩ- optimal bandwidth derived in Andrews (1991, S = (qgq/µ 2 2 α(q 1/(1+2q with α(q = 2vec(Ω (q 0 W A vec(ω (q 0 /tr (W A (I l 2 + K ll (Ω 0 Ω 0. For W A = I l 2, the constants of proportionality become α(1 = 8(l 1ρρ 2 2/f(ρ, ρ 2, l and α(2 = 8(l 1ρρ 2 2(1 + ρρ 2 2 /((1 ρρ 2 2 f(ρ, ρ 2, l with f(ρ, ρ 2, l := (1 ρρ 2 2 [l(l (l 2 l 2ρρ 2 + (l 2 ρ 2 l(2 + 3ρ 2 + 4ρ 2 2ρ ρρ (1 + (l 3ρ 2 ρ 4 2 4ρρ ρ 2 ρ 6 2]. Notice that the MSE( ˆβ-optimal and the MSE(ˆΩ-optimal bandwidth both adapt to the persistence of the error processes (ρ, the persistence of the instruments ( ρ, and the number of instruments (l, but through very different functional forms. herefore, we expect there to be scenarios in which the MSE(ˆΩ-optimal bandwidth is clearly not MSE( ˆβ-optimal and the two bandwidths may differ significantly. he simulation evidence in Section 4 confirms these findings. 1/3 3 Data-driven Bandwidth Choice he optimal bandwidth S is infeasible because it depends on several unknown quantities. In the case in which {g t } is a linear Gaussian process, we require knowledge of Ω 0, Ω (q 0, and G 0. In this section, I describe a data-driven approach to select the optimal bandwidth by estimating the required quantities based on parametric approximating models for {g t }, 10

12 similarly as proposed in Andrews (1991. he idea is to first construct the first-step estimator β, then fit a parsimonious auto-regressive (AR model to {g t ( β} t=1 and, finally, to substitute its parameter estimates into analytical formulae for Ω 0 and Ω (q 0 assuming that the AR is the true model. ogether with the usual sample average estimator for G 0, these estimates, are then substituted into the expressions of ν 2 and ν 3 in Proposition 2.2 to yield estimates of the optimal bandwidths. We focus on estimating a univariate AR(1 model for each component of {g t ( β}, although other approximating models like vector autoregressions or models with more lags could be considered. Let ˆρ i and ˆσ i be the estimated coefficient and the residual variance of the i-th estimated AR(1 process. We can construct estimators of Ω 0 and Ω (q 0 as ˆΩ (q 0 := diag(ˆω 1,..., ˆω l and ˆΩ 0 := diag(ˆω (q 1,..., ˆω (q l with ˆω i := ˆσ i 2 /(1 ˆρ i 2, ˆω (1 i := 2ˆσ i 2 ˆρ i /[(1 ˆρ i 3 (1 + ˆρ i ] and ˆω (2 i Ĥ 0 := ˆΣ 0 G ( β ˆΩ 1 0, ˆΣ 0 := (G ( := 2ˆσ i 2 ˆρ i /[(1 ˆρ i 4 ]. hen, estimate H 0, Σ 0 and P 0 by β ˆΩ 1 0 G ( β 1 and ˆP := ˆΩ 0 ˆΩ 0 G ( βĥ0. Finally, substitute all these expressions into the formulae of Proposition 2.2 to get estimates ˆν 2 and ˆν 3 of ν 2 and ν 3, and the estimator of the optimal bandwidth, ( 1/(1+2q c0ˆν 3 Ŝ := 1/(1+2q. ˆν 2 he difference in performance one incurs by using Ŝ instead of the infeasible bandwidth minimizing the finite-sample MSE of ˆβ has four sources: the error made by replacing the MSE of ˆβ by the MSE of the first terms in the higher-order expansion (ζ, the error due to the large sample approximation of the MSE, the estimation error in ˆΩ 0 and ˆΩ (q 0, and the error made by potential misspecification of the approximating parametric model for {g t }. In practice, one hopes that these errors are small. As mentioned in the discussion after heorem 2.1, the first type of error vanishes with the sample size under additional assumptions. he second and third type also disappear as. he fourth type of error can typically be conjectured to be negligible because the MSE of ˆβ tends to be relatively flat around its minimum (as is the case in the Monte Carlo simulations, for example, so that misspecification in the approximating model is not expected to have a large impact on the properties of the resulting GMM estimator. Nevertheless, the applied researcher should bear in mind that the plugin procedure is not automatic and some thought has to go into selecting an appropriate approximating model and the potential impact of the aforementioned types of errors has to be considered. Remark 3.1. As in Andrews and Monahan (1992 one may want to consider pre-whitening the series {g t ( β} t=1 before fitting the AR process. he reported increases in accuracy of 11

13 test statistics in Andrews and Monahan (1992 and Newey and West (1994 are expected to occur with the procedure presented here as well. 4 Simulations In this section, we discuss a small simulation experiment that illustrates the theoretical findings from the previous sections and, in particular, shows that the MSE( ˆβ-optimal bandwidth may lead to a substantially lower finite-sample MSE of ˆβ relative to choosing the MSE(ˆΩ-optimal bandwidth. We simulate the model from Section 2.1, denoted by AR(1-HOM, for different degrees of serial correlation (ρ {0.01, 0.1, 0.5, 0.9, 0.99}, weak and strong instruments (γ {0.1, 2} and increasing number of instruments (l {2, 3, 4, 5, 10, 15, 25}. We also consider two variants of the model, one in which the outcome equation is replaced by a model with heteroskedastic errors, y t = β 0 w t + w t ε t (referred to as AR(1-HE, and one in which the AR(1 error process is replaced by an MA(1, i.e. ε t = ρη t 1 + η t and v t = ρu t 1 + u t. We simulate 1, 000 samples and, to save space, present only results for sample size = 64, ρ = 0.9, σ 12 = 0.9, β 0 = 1 and the Bartlett kernel. Other parameter combinations yield similar results. For each of the three different data generating processes, ables 1, 3, and 5 report four different bandwidths ( bw averaged over the simulation samples: optimal, Andrews, naive and sim, referring to the MSE( ˆβ-optimal, the MSE(ˆΩ-optimal, the naive choice S = 1/(1+2q and to the (infeasible bandwidth that minimizes the simulated MSE( ˆβ over a grid of bandwidths, respectively. he optimal bandwidth is estimated based on the procedure in Section 3. he table also shows the bias, standard deviation ( SD and MSE of ˆβ. In almost all cases considered here, the MSE( ˆβ-optimal bandwidth is closer to the one minimizing the simulated MSE than the MSE(ˆΩ-optimal bandwidth. In all scenarios, the MSE( ˆβ-optimal bandwidth is smaller than the MSE(ˆΩ- optimal bandwidth one, in some cases substantially smaller. ables 2, 4, and 6 show the ratios of MSE ( MSE ratio and higher-order MSE ( HMSE ratio, as defined in Corollary 2.1, based on the MSE( ˆβ-optimal bandwidth divided by those based on the MSE(ˆΩ-optimal bandwidth. he number of instruments is fixed at l = 10, but other numbers yield qualitatively the same results. µ 2 /l denotes the standardized concentration parameter measuring the strength of the instruments (Stock, Wright, and Yogo (2002. he MSE ratios demonstrate that the MSE( ˆβ-optimal bandwidth may lead to substantial MSE gains relative to the MSE(ˆΩ-optimal bandwidth. he gains are particularly large, up to more than 20%, when the number of instruments is large 12

14 and the estimator of the optimal weighting matrix becomes less precise. As predicted by the theoretical results in the previous sections, the MSE( ˆβ-optimal bandwidth may also lead to dramatic higher-order MSE gains relative to the MSE(ˆΩ-optimal bandwidth of up to more than 90%. Unlike the MSE(ˆΩ-optimal bandwidth defined by Andrews (1991, the MSE( ˆβ- optimal bandwidth is formally not defined for the case l = p in which the estimator ˆβ is independent of the weighting matrix. o study scenarios in which l/p is close to this boundary, we conclude this section by considering a robustness check in which l/p approaches one. able 7 reports the same MSE and HMSE ratio as ables 2, 4, and 6, but for a sequence l/p {5/1, 4/2, 3/2, 4/3, 5/4, 8/7, 10/9} that approaches one., ρ, ρ, and γ are fixed at values 128, 0.5, 0.9, and 2, respectively, but other values yield similar results. he MSE based on the MSE( ˆβ-optimal bandwidth stays close to or slightly smaller than the one based on the MSE(ˆΩ-optimal bandwidth for all values of l/p. Similarly, the HMSE is significantly smaller for the optimal bandwidth. In the case of the AR(1-HE model, the HMSE gains are even up to 67%. 5 Conclusion his paper develops a selection procedure for the bandwidth of a HAC estimator of the optimal GMM weighting matrix which minimizes the asymptotic MSE of the resulting two-step GMM estimator. We show that it is of the same order as the bandwidth minimizing the MSE of the nonparametric plugin estimator, but the constants of proportionality differ significantly. he simulation study suggests that the data-driven version of the selection procedure works well in finite samples and may substantially reduce the firstand second-order MSE of the GMM estimator relative to existing, sub-optimal choices, especially when the number of moment conditions is large. Notes 1 I thank Michael Jansson for making me aware of this work. References Anatolyev, S. (2005: GMM, GEL, Serial Correlation, and Asymptotic Bias, Econometrica, 73(3,

15 Andrews, D. W. K. (1991: Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimation, Econometrica, 59(3, Andrews, D. W. K., and J. C. Monahan (1992: An Improved Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimator, Econometrica, 60(4, Goldstein, L., and K. Messer (1992: Optimal Plug-in Estimators for Nonparametric Functional Estimation, he Annals of Statistics, 20(3, Götze, F., and C. Hipp (1978: Asymptotic Expansions in the Central Limit heorem under Moment Conditions, Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete, 42, Guggenberger, P. (2008: Finite Sample Evidence Suggesting a Heavy ail Problem of the Generalized Empirical Likelihood Estimator, Econometric Reviews, 27(4, Hall, P., and C. C. Heyde (1980: Martingale Limit heory and Its Applications. Academic Press, New York. Hansen, L. P. (1982: Large Sample Properties of Generalized Method of Moments Estimators, Econometrica, 50(4, Hansen, L. P., J. Heaton, and A. Yaron (1996: Finite-Sample Properties of Some Alternative GMM Estimators, Journal of Business and Economic Statistics, 14(3, Imbens, G. W., R. H. Spady, and P. Johnson (1998: Information heoretic Approaches to Inference in Moment Condition Models, Econometrica, 66(2, Jansson, M. (2002: Consistent Covariance Matrix Estimation for Linear Processes, Econometric heory, 18, Jun, B. H. (2007: Essays in Econometrics, Ph.D. thesis, University of California, Berkeley. Kiefer, N., and. Vogelsang (2002a: Heteroskedasticity-Autocorrelation Robust Standard Errors Using he Bartlett Kernel Without runcation, Econometrica, 70(5,

16 (2002b: Heteroskedasticity-Autocorrelation Robust esting Using Bandwidth Equal to Sample Size, Econometric heory, 18, (2005: A New Asymptotic heory for Heteroskedasticity-Autocorrelation Robust ests, Econometric heory, 21, Kinal,. W. (1980: he Existence of Moments of k-class Estimators, Econometrica, 48(1, pp Kitamura, Y., and M. Stutzer (1997: An Information-heoretic Alternative to Generalized Method of Moments Estimation, Econometrica, 65(4, Kunitomo, N., and Y. Matsushita (2003: Finite Sample Distributions of the Empirical Likelihood Estimator and the GMM Estimator, Discussion Paper F-200, CIRJE. Magdalinos, M. A. (1992: Stochastic Expansions and Asymptotic Approximations, Econometric heory, 8(3, Nagar, A. L. (1959: he Bias and Moment Matrix of the General k-class Estimators of the Parameters in Simultaneous Equations, Econometrica, 27(4, Newey, W. K., and D. McFadden (1994: Large Sample Estimation and Hypothesis esting, in Handbook of Econometrics, ed. by R. F. Engle, and D. L. McFadden, vol. IV, pp Elsevier Science B.V. Newey, W. K., and R. J. Smith (2004: Higher Order Properties of GMM and Generalized Empirical Likelihood Estimators, Econometrica, 72(1, Newey, W. K., and K. West (1994: Automatic Lag Selection in Covariance Matrix Estimation, he Review of Economic Studies, 61(4, Owen, A. B. (1988: Empirical Likelihood Ratio Confidence Intervals for a Single Functional, Biometrika, 75(2, Parzen, E. (1957: On Consistent Estimates of the Spectrum of a Stationary ime Series, he Annals of Mathematical Statistics, 28(2, Powell, J. L., and. M. Stoker (1996: Optimal Bandwidth Choice for Densityweighted Averages, Journal of Econometrics, 75, Qin, J., and J. Lawless (1994: Empirical Likelihood and General Estimating Equations, he Annals of Statistics, 22(1,

17 Rilstone, P., V. K. Srivastava, and A. Ullah (1996: he Second-order Bias and Mean Squared Error of Nonlinear Estimators, Journal of Econometrics, 75(2, Robinson, P. M. (1991: Automatic Frequency Domain Inference on Semiparametric and Nonparametric Models, Econometrica, 59(5, Rothenberg,. (1984: Approximating the Distributions of Econometric Estimators and est Statistics, in Handbook of Econometrics, ed. by Z. Griliches, and M. D. Intriligator, vol. II, pp Elsevier Science Publishers B.V. Stock, J. H., J. H. Wright, and M. Yogo (2002: A Survey of Weak Instruments and Weak Identification in Generalized Method of Moments, Journal of Business and Economic Statistics, 20(4. Sun, Y., and P. C. B. Phillips (2008: Optimal Bandwidth Choice for Interval Estimation in GMM Regression, Discussion Paper 1661, Cowles Foundation, Yale University. Sun, Y., P. C. B. Phillips, and S. Jin (2008: Optimal Bandwidth Selection in Heteroskedasticity-Autocorrelation Robust esting, Econometrica, 76(1, amaki, K. (2007: Second Order Optimality for Estimators in ime Series Regression Models, Journal of Multivariate Analysis, 98, White, H., and I. Domowitz (1984: Nonlinear Regression with Dependent Observations, Econometrica, 52(1, Xiao, Z., and P. C. B. Phillips (1998: Higher-order Approximations for Frequency Domain ime Series Regression, Journal of Econometrics, 86, Zaman, A. (1981: Estimators Without Moments: he Case of the Reciprocal of a Normal Mean, Journal of Econometrics, 15(2, A Proofs Lemma A.1. Under Assumptions and 2.3(a (c, ˆβ β 0 = ψ 1/2 + o p ( 1/2 with ψ = G 1 0 ĝ = Op (1. 16

18 Proof. We need to check the assumptions of Newey and McFadden (1994, heorem 3.2 with Ŵ replaced by ˆΩ ( β. First of all, by Assumptions and Andrews (1991, heorem 1(b ˆΩ ( β p Ω 0. (i, (ii and (v hold by assumption. (iii and (iv hold by Assumption 2.1 and White and Domowitz (1984, heorem 2.3, 2.4. Q.E.D. Proof of Proposition 2.1. Step I: Expansion of the optimal weighting matrix. A aylor expansion of ˆΩ ( β around β 0 yields vec (ˆΩ ( β = vec (ˆΩ + ˆΩ ( β( β β 0 = vec(ω 0 + vec (ˆΩ Ω 0 + Ω 0 ( β β ( ˆΩ Ω 0 ( β β 0 ( ˆΩ ( β ˆΩ ( β β 0 (A.1 where β lies on the line segment joining β and β 0. By Assumptions and Andrews (1991, Proposition 1(a,(b, ˆΩ Ω 0 = ω 1, S 1/2 1/2 + ω 2, S q with ω i, = O p (1, i = 1, 2, and ˆΩ Ω 0 = O p (η. Next, we show that ˆΩ ( β ˆΩ = O p ( β β 0. o this end, let ḡ t := g t ( β. Notice that, by Assumption 2.1(f, β N w.p.a. 1 and, thus, β N w.p.a. 1. From Assumption 2.3(d, we get ˆΩ ( β ˆΩ 1 1 min{, s} s=1 t=max{1,1 s} 1 s= min{, s} ( s k [ ḡ t+s ḡ t g t+s g t] S C d(x t+s, x t β β 0 s=1 t=max{1,1 s} ( = C Ed(x t+s, x t + o p (1 β β 0 = O p ( β β 0 (A.2 which holds w.p.a. 1 and for some constant C. (A.1 together with (A.2 and the firstorder asymptotics in Assumption 2.1(f then imply ˆΩ ( β = Ω 0 + ω 1, S 1/2 1/2 + ω 2, S q + R (A.3 with R = O p (η 1/2, ω 1, := /S (ˆΩ Ω = O p (1 and ω 2, := S q (Ω Ω 0 = O(1. Step II: Expansion of the second-step GMM estimator. Write the second-stage estimator ˆθ := ( ˆβ, ˆλ of θ 0 := (β 0, 0 B Λ, Λ := [0, l, as the solution to ( G ( ˆβ ˆλ ĝ( ˆβ + ˆΩ ( βˆλ = 0. (A.4 17

19 Further, define ˆm(θ := 1 t=1 m t(θ with ( m t (θ := G(x t, β λ g t (β + Ω 0 λ for some θ := (β, λ B Λ. hen use the expansion in (A.3 to rewrite (A.4 as ( 0 0 = ˆm(ˆθ (ω 1, S 1/2 1/2 + ω 2, S q + R ˆλ. (A.5 Next, consider ˆλ = ˆΩ ( β 1 ĝ( ˆβ. By Assumptions and Andrews (1991, Proposition 1(a,(b, heorem 1(b, ˆΩ ( β Ω 0 = O p (η. Also, by an expansion of ĝ( ˆβ around β 0, Lemma A.1, Assumption 2.1 and the CL, ĝ( ˆβ = (I l G 0 H 0 ĝ + o p ( 1/2, and thus ˆλ = [Ω 0 + O p (η ] 1 ( (I l G 0 H 0 F 1/2 + o p ( 1/2 = P 0 F 1/2 + O p (η 1/2 + o p ( 1/2. (A.6 Consider the following expansion of ˆm(ˆθ around θ 0 : ˆm(ˆθ = ˆm(θ 0 + ˆm(θ 0 (ˆθ θ [ ] 2 2 ˆm( θ (ˆθ θ 0 (ˆθ θ 0. where θ lies on the line segment joining ˆθ and θ 0. By Assumptions 2.1(e and 2.3(c (d, 2 ˆm( θ is O p (1. Lemma A.1, Assumption 2.1 and the CL for mixing sequences then imply ˆm(ˆθ = ˆm(θ 0 + M 1 (ˆθ θ 0 + O p ( 1. Substituting (A.6 and (A.7 into (A.5 and solving for ˆθ θ 0 yields ( ˆθ θ 0 = M1 1 ˆm(θ 0 M1 1 0 (ω 1, S1/2 1/2 + ω 2, S q P 0F 1/2 + O p ( 1 + O p (η 2 1/2 + o p (η 1/2 (A.7 where ( ( M 1 = 0 G 0 G 0 Ω 0, M 1 1 = Σ 0 H 0 H 0 P 0. herefore, ( ˆβ β0 = H 0 F + H 0 ω 1, P 0 F S 1/2 1/2 + H 0 ω 2, P 0 F S q + o p(η. Since F, ω 1, and ω 2, are O p (1, we also have that κ i, = O p (1 for i = 1, 2, 3. Q.E.D. 18

20 Proof of heorem 2.1. We need to derive the order of E[κ i, Wκ j, ], i, j {1, 2, 3}. Consider the case i = j = 1: E[κ 1, Wκ 1, ] = E[F H 0WH 0 F ] = 1 = 1 s= ( 1 E[g th 0WH 0 g s ] s,t=1 ( 1 s E[g th 0WH 0 g t s ] s= E[g th 0WH 0 g t s ]. he limiting sum can be shown to be finite using Assumption 2.1, the Hölder Inequality and the mixing inequality of Hall and Heyde (1980, Corollary A.2. Similarly, we can show that E[κ 2, Wκ 2, ] and E[κ 3, Wκ 3, ] are O(1, E[κ 1, Wκ 2, ] = O(S 1/2 1/2, but E[κ 1, Wκ 3, ] = o(η and E[κ 2, Wκ 3, ] = o(1. Q.E.D. Proof of Proposition 2.2. From the proof of heorem 2.1, E[κ 1, Wκ 1, ] E[g th 0WH 0 g t s ] = vec(ω 0 vec (H 0WH 0. s= By Andrews (1991, Proposition 1(b we have S q (Ω Ω 0 g q Ω (q 0 and thus E[κ 3, Wκ 3, ] = E[F P 0S q (Ω Ω 0 H 0WH 0 S q (Ω Ω 0 P 0 F ] [ ] gq 2 E g tp 0Ω (q 0 H 0WH 0 Ω (q 0 P 0 g t s s= ( = gqvec(ω 2 0 vec P 0Ω (q 0 H 0WH 0 Ω (q 0 P 0 ( = gqtr 2 Ω (q 0 H 0WH 0 Ω (q 0 P 0 because, for conformable matrices A, B, tr(ab = tr(ba, and P 0 G 0 H 0 = 0. Next, consider the terms E [ ] [ ] κ 1, Wκ 2, = E F H 0WH 0 /S (ˆΩ Ω P 0 F [ = E (F F vec (H 0WH ] 0 /S (ˆΩ Ω P 0 [ ( ] = E (F F (P 0 H 0WH 0 vec /S (ˆΩ Ω [ ( ( ] = E F F vec /S (ˆΩ Ω vec (P 0 H 0WH 0 and, similarly, [ ] E[κ 2, Wκ 2, ] = E F P 0 /S (ˆΩ Ω H 0WH 0 /S (ˆΩ Ω P 0 F = [ ] vec(p 0 P S 0 E F F (ˆΩ Ω (ˆΩ Ω vec(h 0WH 0. 19

21 In order to find expressions for these two cross-products involving /S (ˆΩ Ω, we make use of the BN-decomposition of the linear process {g t }, viz. g t = Ψe t + ẽ t 1 ẽ t where Ψ := j 0 Ψ j, ẽ t := j 0 Ψ j e t j and the tail sums Ψ j := k j+1 Ψ k. With this representation of {g t } we can calculate limiting variances and covariances of g t based only on Ψe t and disregard the transient part of the process, ẽ t 1 ẽ t. Since e t N(0, Σ e, third and fourth moments are zero. herefore, [ ( ] S 1/2 3/2 E F F vec /S (ˆΩ Ω 1 min{, v} ( = E g t g s v k vec (g u+v g u E[g u+v g S u] s,t=1 v=1 u=max{1,1 v} = E Ψe t Ψe t vec (Ψe u Ψe u E[Ψe u Ψe u] + o(1 = = + E s,t=1 t=1 u=1 u t Ψe t Ψe s s,t=1 1 v=1 v 0 ( t s k s,t=1 + s,t=1 min{, v} u=max{1,1 v} S S 1 v=1 v 0 ( v k S min{, v} u=max{1,1 v} E [Ψe t Ψe s Ψe s Ψe t ] ( v k vec (Ψe u+v Ψe S u E [Ψe t Ψe s Ψe u Ψe u+v ] + o(1 ( t s k E [Ψe t Ψe s Ψe t Ψe s ] + o(1 One can show that E[Ψe t Ψe s Ψe s Ψe t ] = E[Ψe t Ψe s Ψe t Ψe s ] = vec(ω 0 Ω 0 when s t and 0 otherwise. Noticing 1 S s,t=1 k((t s/s µ 1, we then have S E [ κ 1, Wκ 2, ] 2µ1 vec(ω 0 Ω 0 vec (P 0 H 0WH 0 = 2µ 1 tr((ω 0 Ω 0 (P 0 H 0WH 0 = 2µ 1 tr(ω 0 P 0 Ω 0 H 0WH 0 = 2µ 1 tr(i l G 0 H 0 tr(ω 0 Ω 1 0 G 0 Σ 0 WΣ 0 G 0Ω 1 0 = 2µ 1 (l tr(h 0 G 0 tr(σ 0 WΣ 0 G 0Ω 1 0 G 0 = 2µ 1 (l ptr(σ 0 W 20

22 which uses the fact that H 0 G 0 = I p. By a similar derivation, E[κ 2, Wκ 2, ] µ 2 (l ptr(σ 0 W so that ν 2 = (2µ 1 + µ 2 (l ptr(σ 0 W. Q.E.D. AR(1-HOM, = 64, l = 10 γ = 0.1 γ = 2 ρ optimal Andrews naive sim optimal Andrews naive sim 0.01 bw bias sd MSE bw bias sd MSE bw bias sd MSE bw bias sd MSE bw bias sd MSE able 1: Bandwidths ( bw, bias, standard deviation ( SD and MSE of ˆβ when computed based on the MSE(ˆΩ-optimal ( optimal, the MSE(ˆΩ-optimal ( Andrews, S = 1/(1+2q ( naive or the simulated MSE-minimizing ( sim bandwidth. 21

23 AR(1-HOM, = 64 γ = 0.1 γ = 2 ρ l MSE ratio HMSE ratio µ 2 /l MSE ratio HMSE ratio µ 2 /l able 2: Ratios of MSE ( MSE ratio and higher-order MSE ( HMSE ratio based on the MSE( ˆβ- optimal bandwidth divided by those based on the MSE(ˆΩ-optimal bandwidth. µ 2 /l is the standardized concentration parameter measuring the strength of the instruments. 22

24 AR(1-HE, = 64, l = 10 γ = 0.1 γ = 2 ρ optimal Andrews naive sim optimal Andrews naive sim 0.01 bw bias sd MSE bw bias sd MSE bw bias sd MSE bw bias sd MSE bw bias sd MSE able 3: Bandwidths ( bw, bias, standard deviation ( SD and MSE of ˆβ when computed based on the MSE(ˆΩ-optimal ( optimal, the MSE(ˆΩ-optimal ( Andrews, S = 1/(1+2q ( naive or the simulated MSE-minimizing ( sim bandwidth. 23

25 AR(1-HE, = 64 γ = 0.1 γ = 2 ρ l MSE ratio HMSE ratio µ 2 /l MSE ratio HMSE ratio µ 2 /l able 4: Ratios of MSE ( MSE ratio and higher-order MSE ( HMSE ratio based on the MSE( ˆβ- optimal bandwidth divided by those based on the MSE(ˆΩ-optimal bandwidth. µ 2 /l is the standardized concentration parameter measuring the strength of the instruments. 24

26 MA(1, = 64, l = 10 γ = 0.1 γ = 2 ρ optimal Andrews naive sim optimal Andrews naive sim 0.01 bw bias sd MSE bw bias sd MSE bw bias sd MSE bw bias sd MSE bw bias sd MSE able 5: Bandwidths ( bw, bias, standard deviation ( SD and MSE of ˆβ when computed based on the MSE(ˆΩ-optimal ( optimal, the MSE(ˆΩ-optimal ( Andrews, S = 1/(1+2q ( naive or the simulated MSE-minimizing ( sim bandwidth. 25

27 MA(1, = 64 γ = 0.1 γ = 2 ρ l MSE ratio HMSE ratio µ 2 /l MSE ratio HMSE ratio µ 2 /l able 6: Ratios of MSE ( MSE ratio and higher-order MSE ( HMSE ratio based on the MSE( ˆβ- optimal bandwidth divided by those based on the MSE(ˆΩ-optimal bandwidth. µ 2 /l is the standardized concentration parameter measuring the strength of the instruments. 26