Interpretation and semiparametric efficiency in quantile regression under misspecification

Lee, Ying-Ying Article Interpretation and semiparametric efficiency in quantile regression under misspecification Econometrics Provided in Cooperation with: MDPI – Multidisciplinary Digital Publishing Institute, Basel Suggested Citation: Lee, Ying-Ying (2016) : Interpretation and semiparametric efficiency in quantile regression under misspecification, Econometrics, ISSN 2225-1146, MDPI, Basel, Vol. 4, Iss. 1, pp. 1-14, https://doi.org/10.3390/econometrics4010002 This Version is available at: https://hdl.handle.net/10419/171848 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by/4.0/ Article Interpretation and Semiparametric Efficiency in Quantile Regression under Misspecification Ying-Ying Lee Received: 4 October 2015/Accepted: 1 December 2015/Published: 24 December 2015 Academic Editor: Gabriel Montes-Rojas Department of Economics, University of Oxford, Manor Road Building, Manor Road, Oxford OX1 3UQ, UK; E-Mail: [email protected] Abstract: Allowing for misspecification in the linear conditional quantile function, this paper provides a new interpretation and the semiparametric efficiency bound for the quantile regression parameter β(τ)in Koenker and Bassett (1978). The first result on interpretation shows that under a mean-squared loss function, the probability limit of the Koenker–Bassett estimator minimizes a weighted distribution approximation error, defined as FY(X0β(τ)|X)−τ,i.e., the deviation of the conditional distribution function, evaluated at the linear quantile approximation, from the quantile level. The second result implies that the Koenker–Bassett estimator semiparametrically efficiently estimates the quantile regression parameter that produces parsimonious descriptive statistics for the conditional distribution. Therefore, quantile regression shares the attractive features of ordinary least squares: interpretability and semiparametric efficiency under misspecification. Keywords: semiparametric efficiency bounds; misspecification; conditional quantile function; conditional distribution function; best linear approximation JEL classification: C14; C21 1. Introduction This paper revisits the approximation properties of the linear quantile regression under misspecification ([1–3]). The quantile regression estimator, introduced by the seminal paper of Koenker and Bassett [4], offers parsimonious summary statistics for the conditional quantile function and is computationally tractable. Since the development of the estimator, researchers have frequently used quantile regression, in conjunction with ordinary least squares regression, to analyse how the outcome variable responds to the explanatory variables. For example, to model wage structure in labour economics, Angrist, Chernozhukov, and Fernández-Val [1] study returns to education at different points in the wage distribution and changes in inequality over time. A thorough review of recent developments in quantile regression can be found in [5]. The object of interest of this paper is the quantile regression (QR) parameter that is the probability limit of the Koenker–Bassett estimator without assuming the true conditional quantile function to be linear. Two results are presented: a new interpretation and the semiparametric efficiency bound for the QR parameter. The topic of interest is the conditional distribution function (CDF) of a continuous response variable Ygiven the regressor vector X, denoted as FY(y|X). An alternative for the CDF is the conditional quantile function (CQF) of Ygiven X, defined as Qτ(Y|X):=inf{y:FY(y|X)≥τ} for any quantile index τ∈(0, 1). Assuming integrability, the CQF minimizes the check loss function Qτ(Y|X)∈arg min q∈Q E[ρτ(Y−q(X))] Econometrics 2015,4, 2; doi:10.3390/econometrics4010002 www.mdpi.com/journal/econometrics Econometrics 2015,4, 2 2 of 14 where Qis the set of measurable functions of X,ρτ(u) = u(τ−1{u≤0})is known as the check function and 1{·} is the indicator function. A linear approximation to the CQF is provided by the QR parameter β(τ), which solves the population minimization problem β(τ):=arg min b∈RdEρτ(Y−X0b)(1) assuming the integrability and uniqueness of the solution, and dis the dimension of X. The QR parameter β(τ)provides a simple summary statistic for the CQF. The QR estimator introduced in [4] is the sample analogue ˆ β(τ)∈arg min b∈Rd 1 n n ∑ i=1 ρτ(Yi−X0 ib)(2) for the random sample (Yi,X0 i,i≤n)on the random variables (Y,X0). By the equivalent first-order condition, this estimator ˆ β(τ)is also the generalized method of moments (GMM) estimator based on the unconditional moment restriction ([6,7]) E[(τ−1{Y≤X0β(τ)})X] = 0 (3) This paper focuses on the population QR parameter defined by (1) or equivalently (3). If the CQF is modelled to be linear in the covariates Qτ(Y|X) = X0β(τ)or FY(X0β(τ)|X) = τ, the coefficient β(τ)satisfies the conditional moment restriction E[τ−1{Y≤X0β(τ)}|X] = 0 (4) almost surely. In the theoretical and applied econometrics literature, this linear QR model is often assumed to be correctly specified. Nevertheless, a well-known crossing problem arises: the CQF for different quantiles may cross at some values of X, except when β(τ)is the same for all τ. A logical monotone requirement is violated for Qτ(Y|X)or its estimator to be weakly increasing in the probability index τgiven X. The crossing problem for estimation could be treated by rearranging the estimator (for example, see [8] and the references therein.1). However, the crossing problem remains for the population CQF, suggesting that the linear QR model (4) is inherently misspecified. That is, there is no β(τ)∈Rdsatisfying the conditional moment (4) almost surely. Therefore, the parameter of interest in this paper is the QR parameter β(τ)defined by (1) or (3) without the linear CQF assumption in (4). We can view β(τ)as the pseudo-true value of the linear QR model under misspecification. As the Koenker–Bassett QR estimator is widely used, it is important to understand the approximation nature of the estimand. For the mean regression counterpart, ordinary least squares (OLS) consistently estimates the linear conditional expectation and minimizes mean-squared error loss for fitting the conditional expectation under misspecification. Chamberlain [9] proves the semiparametric efficiency of the OLS estimator, which provides additional justification for the widespread use of OLS. The attractive features of OLS, interpretability and semiparametric efficiency, under misspecification, motivate my investigation of parallel properties in QR. I study how this QR parameter approximates the CQF and the CDF and calculate its semiparametric efficiency bound. The first contribution of this paper is on how β(τ)minimizes the distribution approximation error, defined by FY(X0β(τ)|X)−τ, under a mean-squared loss function. The first-order condition (3) 1Chernozhukov, Fernández-Val, and Galichon [8] rearrange an estimator ˆ Qτ(Y|X)to be monotonic. The original estimator can be computationally tractable. The rearranged monotonic estimated CDF is ˆ FY(y|X) = R1 01{ˆ Qτ(Y|X)≤y}dτ. The rearranged quantile estimation is ˆ Q∗ τ(Y|X) = inf{y:ˆ FY(y|X)≥τ}. Econometrics 2015,4, 2 3 of 14 can be understood as the orthogonality condition of the covariates Xand the distribution approximation error in the projection model. I show that the QR parameter β(τ)minimizes the mean-squared distribution approximation error, inversely weighted by the conditional density function fY(X0β(τ)|X). Angrist, Chernozhukov, and Fernández-Val [1] (henceforth ACF) show that β(τ)minimizes the mean-squared quantile specification error, defined by Qτ(Y|X)−X0β(τ), using a weight primarily determined by the conditional density. ACF’s study, as well as my own results, suggests that QR approximates the CQF more accurately at points with more observations, but the corresponding CDF evaluated at the approximated point FY(X0β(τ)|X)is more distant from the targeted quantile level τ. This trade-off is controlled by the conditional density, which is distinct from OLS approximating the conditional mean, because the distribution and quantile functions are generally nonlinear operators. This observation is novel and increases the understanding of how the QR summarizes the outcome distribution. A numerical example in Figure 1in Section 4illustrates this finding. The second result is the semiparametric efficiency bound of the β(τ). Chamberlain’s results in [9] on the mean regression based on differentiable moment restrictions cannot be applied to semiparametric efficiency for QR, due to the lack of moment function differentiability in (3). Although Ai and Chen [10] provide general results for sequential moment restrictions containing unknown functions, which could cover the quantile regression setting, I calculate the efficiency bound accommodating regularity conditions specifically for the QR parameter β(τ)using the method of Severini and Tripathi [11]. It follows that the misspecification-robust asymptotic variance of the QR estimator ˆ β(τ)in (2) attains this bound, which means no regular2estimator for (3) has smaller asymptotic variance than ˆ β(τ). This result might be expected for an M-estimator, but, to my knowledge, the QR application has not been demonstrated and discussed rigorously in any publication. Furthermore, I calculate the efficiency bounds for jointly estimating QR parameters at a finite number of quantiles for both linear projection (3) and linear QR (4) models. Employing the widely-used method of Newey [12], Newey and Powell [13] find the semiparametric efficiency bound for β(τ)of the correctly-specified linear CQF in (4). Note that the efficiency bounds for (3) do not imply the bounds for (4); nor does the converse hold. In Section 2, I discuss the interpretation of the misspecified QR model in terms of approximating the CDF and the CQF. The theorems for the semiparametric efficiency bounds are in Section 3. In Section 4, I discuss the parallel properties of QR and OLS. The paper is concluded by a review of some existing efficient estimators for linear projection model (3) and linear QR model (4). 2. Interpreting QR under Misspecification Let Ybe a continuous response variable and Xbe a d×1 regressor vector. The quantile-specific residual is defined as the distance between the response variable and the CQF, ετ:=Y−Qτ(Y|X) with the conditional density fετ(e|X)at ετ=eor fY(y|X)at Y=y=e+Qτ(Y|X)for any τ∈(0, 1). This is a semiparametric problem in the sense that the distribution functions of ετand X, as well as the CQF, are unspecified and unrestricted other than by the following assumptions, which are standard in QR models. I assume the following regularity conditions, based on the conditions of Theorem 3 in ACF. (R1) (Yi,Xi,i≤n)are independent and identically distributed on the probability space (Ω,F,P)for each n; (R2) the conditional density fY(y|X=x)exists and is bounded and uniformly continuous in y, uniformly in xover the support of X; (R3) J(τ):=E[fY(X0β(τ)|X)XX0]is positive definite for all τ∈(0, 1), where β(τ)is uniquely defined in (1); 2See [12] for the definition of regular estimators. Econometrics 2015,4, 2 4 of 14 (R4) EkXk2+e<∞for some e>0; (R5) fY(X0β(τ)|X)to be bounded away from zero. The identification of the pseudo-true parameter β(τ)is assumed in (R3). The bounded conditional density function of the continuous response variable Ygiven Xin (R2) is needed for the existence of the CQF for any τ∈(0, 1). The uniform continuity guarantees the existence and differentiability of the distribution function, i.e.,dFY(y|X)/dy =fY(y|X)and FY(y|X) = Ry −∞fY(u|X)du with probability one. (R4) is used for the asymptotic normality of √n(ˆ β(τ)−β(τ)). The covariates X are allowed to contain discrete components. (R5) guarantees that the objective function defined below in Equation (6) is finite ∀β∈Rd, where β(τ)is the parameter of interest uniquely defined by Equation (1). The parameter of interest β(τ)is equivalent to solving EhXFYX0β(τ)X−τi=0 (5) by applying the law of iterated expectations on Equation (3). Equation (5) states that Xis orthogonal to the distribution approximation error FYX0β(τ)X−τ. The following theorem interprets QR via a weighed mean-squared error loss function on the distribution approximation error. Theorem 1. Assume (R1)–(R5). Then, ¯ β(τ) = β(τ)solves the equation ¯ β(τ) = arg min b∈RdEhfY(X0¯ β(τ)|X)−1FY(X0b|X)−τ2i. (6) Furthermore, if EfY(X0b|X)+(FY(X0b|X)−τ)f0 Y(X0b|X)/fY(X0b|X)XX0is positive definite at b=β(τ), then ¯ β(τ) = β(τ)is the unique solution to this problem (6). Proof of Theorem 1. The objective function in (6) is finite by the assumptions. Any fixed point b =¯ β(τ) would solve the first-order condition, EXFYX0bX−τ =0. By the law of iterated expectations, (3) implies the above first-order condition. Therefore, β(τ)solves (6). When the second-order condition holds, i.e., EfY(X0b|X) + (FY(X0b|X)−τ)f0 Y(X0b|X)/fY(X0b|X)XX0is positive definite at b =β(τ),β(τ) solves (6) uniquely. 2 Theorem 1states that β(τ)is the unique fixed point to an iterated minimum distance approximation, with a weight of a function of Xonly. The mean-squared loss makes it clear how the linear function matches the CDF to the targeted probability of interest. The loss function puts more weight on points where the conditional density fY(X0β(τ)|X)is small. As a result, the distribution approximation error is smaller at points with smaller conditional density. Now, I discuss the approximation nature of QR based on the distributional approximation error and quantile specification error. ACF interpret QR as the minimizer of the weighted mean-squared error loss function for quantile specification error, defined as the deviation between the approximation point X0β(τ)and the true CQF Qτ(Y|X),3 β(τ) = arg min β∈RdEh¯ wτ(X,β(τ))X0β−Qτ(Y|X)2i, where (7) ¯ wτ(X,β(τ)) = 1 2Z1 0fετuX0β(τ)−Qτ(Y|X)Xdu. (8) ACF define ¯ wτ(X,β(τ)) in (8) to be the importance weights that are the averages of the response variable over a line connecting the approximation point X0β(τ)and the true CQF. ACF note that 3For estimation, [14] studies different approaches based on the distribution regression and quantile regression. Econometrics 2015,4, 2 5 of 14 the regressors contribute disproportionately to the QR estimate and the primary determinant of the importance weight is the conditional density. Moreover, the first-order condition implied by (7)E¯ wτ(X,β(τ))XX0β(τ)−Qτ(Y|X) = 0 is a weighted orthogonal condition of the quantile specification error. A Taylor expansion provides intuition to connect the distribution approximation error and the quantile specification error: fY(X0β|X)−1FY(X0β|X)−τ2≈fY(X0β|X)Qτ(Y|X)−X0β2by fY(X0β|X) = fετ(X0β− Qτ(Y|X)|X). This observation implies the quantile specification error is smaller at points where the conditional density fY(X0β|X)is larger. On the other hand, the distribution approximation error is larger at points with larger fY(X0β|X). Comparing with the OLS, where the mean operator is linear, the CDF and its inverse operator, the CQF, are generally nonlinear. The distribution approximation error can be interpreted as the distance after a nonlinear transformation by the CDF, FY(X0β(τ)|X)−FY(Qτ(Y|X)|X). A Taylor expansion linearizes the distribution function to the quantile specification error multiplied by the conditional density function. The conditional density plays a crucial role on weighting the distribution approximation error and the quantile specification error. The above discussion provides additional insights to how the QR parameter approximates the CQF and fits the CDF to the targeted quantile level. Remark 1 (Mean-squared loss under misspecification).The linear function X0β(τ)is the best linear approximation under the check loss function in (1). While β(.5)is the least absolute derivations estimation, the QR parameter β(τ)for τ6=0.5 is the best linear predictor for a response variable under the asymmetric loss function ρτ(·)in (1). ACF note that the prediction under the asymmetric check loss function is often not the object of interest in empirical work, with the exception of the forecasting literature, for example [15]. For the mean regression counterpart, OLS consistently estimates the linear conditional expectation and minimizes mean-squared error loss for fitting the conditional expectation under misspecification. The robust nature of OLS also motivates research on misspecification in panel data models. For example, Galvao and Kato [16] investigate linear panel data models under misspecification. The pseudo-true value of the fixed effect estimator provides the best partial linear approximation to the conditional mean given the explanatory variables and the unobservable individual effect.4 3. The Semiparametric Efficiency Bounds Section 3.1 presents the semiparametric efficiency bound for the unconditional moment restriction (3). Section 3.2 discusses the existing results on the semiparametric efficiency bound for the conditional moment restriction (4). 3.1. QR under Misspecification I calculate the semiparametric efficiency bound for the unconditional moment restriction (3) by the approach of Severini and Tripathi [11]. Theorem 2. Assume (R1)–(R4). The semiparametric efficiency bound for estimating the population QR parameter β(τ), defined in (1) or equivalently (3), is J(τ)−1Γ(τ,τ)J(τ)−1, where J(τ)is defined in (R3) and Γ(τi,τj):=Ehτi−1{Y<X0β(τi)}τj−1{Y<X0β(τj)}XX0i for any τi,τj∈ T :=a closed subset of [e, 1 −e]for e>0. In general, the semiparametrically-efficient joint asymptotic covariance of the estimators for (β0(τ1),β0(τ2), ..., β0(τm)0is J(τi)−1Γ(τi,τj)J(τj)−1, for any τi,τj∈ T , i,j=1, 2, ..., m, for a finite integer m≥1. 4Galvao and Kato [16] show that misspecification affects the bias correction and convergence rate of the estimator and provide a misspecification-robust inference procedure. In panel models under time series misspecification, Lee [17] proposes bias reduction methods for the incidental parameter. Econometrics 2015,4, 2 6 of 14 Proof of Theorem 2. See the Appendix. 2 My proof accommodates the regularity assumptions for quantile regression and modifies Section 9 of [11]. For example, the covariate Xcan contain discrete components, by constructing two tangent spaces for the conditional density of Ygiven Xand the marginal density of X, respectively. In the efficiency bound, J(τ):=E[fY(X0β(τ)|X)XX0]is obtained by assuming the interchangeability of integration and differentiation for the nonsmooth check function.5 The method in [11] has been used in the monotone binary model in [18], Lewbel [19] latent variable model in [20] and the partial linear single index model in [21], for example. I work in the Hilbert space of tangent vectors of the square-root density functions and using the Riesz–Fréchet representation theorem. Another equivalent approach in [12] works in a Hilbert space of random variables and uses the projection on the linear space spanned by the scores from the one-dimensional subproblems to find the efficient influence function. The efficiency bound is then the second moment of the efficient influence function, J(τ)−1X(τ−1{Y≤X0β}). Newey’s efficient influence function is the score function evaluated at the unique representers by the Riesz–Fréchet theorem used in [11]; a more detailed comparison of these two approaches is discussed in [11]. ACF show that the QR process ˆ β(·)is asymptotically mean-zero Gaussian with the covariance function J(τ1)−1Γ(τ1,τ2)J(τ2)−1for any τ1,τ2∈ T , which is the semiparametric efficiency bound in Theorem 2. This asymptotic covariance under misspecification for a single quantile, J(τ)−1Γ(τ,τ)J(τ)−1, has been presented in [2] and [3]. Hahn [3] further shows the QR estimator is well approximated by the bootstrap distribution, even when the linear quantile restriction is misspecified. An alternative estimator for the misspecification-robust asymptotic covariance matrix of ˆ β(τ)is the nonparametric kernel method in ACF. 3.2. QR for Linear Specification Assuming the linear QR model in (4) is correctly specified, i.e.,Qτ(Y|X) = X0β(τ)almost surely, the asymptotic covariance for the QR process ˆ β(·)derived by ACF is simplified to J(τ1)−1Γ0(τ1,τ2)J(τ2)−1, where Γ0(τ1,τ2):=(min {τ1,τ2}−τ1τ2)E[XX0]for any τ1,τ2∈(0, 1). The asymptotic covariance J(τ)−1Γ0(τ,τ)J(τ)−1for a single quantile τ, first derived by Powell [7], is widely used for inference in most empirical studies, which implicitly assume correct specification. The semiparametric efficiency bound for the correctly-specified quantile regression (4) is τ(1−τ)EXX0f2 ετ(0|X)−1, where fY(X0β(τ)|X) = fετ(0|X)a.s. and EXX0f2 ετ(0|X)is assumed to be finite and nonsingular. This is first calculated in [13] using the method developed in [12]. If, in addition, the conditional density function of ετgiven Xis independent of X, i.e.,fY(Qτ(Y|X)|X) = fετ(0|X) = fετ(0), and fετ(0)>0, the semiparametric efficiency bound becomes τ(1−τ)E[XX0]−1/f2 ετ(0). This asymptotic covariance is attained by ˆ β(τ), first shown in [4]. This has an interesting resemblance to the fact that the OLS estimator is semiparametrically efficient in a homoskedastic regression model, i.e.,e=Y−X0β,E[e|X] = 0, and E[e2|X] = E[e2]. I further show, in general, that the semiparametrically-efficient joint asymptotic covariance of the estimators for (β0(τ1), ..., β0(τm))0is min τi,τj−τiτjEhXX0fετi(0|X)fετj(0|X)i−1(9) for any τi,τj∈ T ,i,j=1, 2, ..., m, for any finite integer m≥1. The regularity conditions imposed, (R1), (R2) and (R4), are weaker than the assumptions in [13]; for example, they assume f(ε,X)is 5Severini and Tripathi construct the tangent space for the continuous and bounded joint density f(X,Y)in Section 9 of [11]. Additionally, they define Jon the derivative of the moment restriction. Econometrics 2015,4, 2 7 of 14 absolutely continuous in ε, which implies uniform continuity in (R2). See the Appendix for the detailed proof for (9). 4. Discussion and Conclusions Misspecification is a generic phenomenon; especially in quantile regression (QR), the true conditional quantile function (CQF) might be nonlinear or different functions of the covariates at different quantiles. Table 1summarizes the parallel properties of QR and OLS. Under misspecification, the pseudo-true OLS coefficient can be interpreted as the best linear predictor of the conditional mean function, E[Y|X], in the sense that the coefficient minimizes the mean-squared error of the linear approximation to the conditional mean. The approximation properties of OLS have been well studied (see, for example, [22]). With respect to the QR counterpart, I present the inverse density-weighted mean-squared error loss function based on the distribution approximation error FY(X0β|X)−τ. This result complements the interpretation based on the quantile specification error in [1]. My results imply that the Koenker–Bassett estimator is semiparametrically efficient for misspecified linear projection models and correctly specified linear quantile regression models when fY(Qτ(Y|X)|X) = fετ(0|X)does not depend on X. Alternatively, the smoothed empirical likelihood estimator using the unconditional moment restriction in [23] has the same asymptotic distribution as the Koenker-Bassett estimator and, hence, attains the efficiency bound. Table 1. Summary properties of OLS and quantile regression (QR). OLS QR Linear Projection Model objective minimized E[(Y−X0β)2]E[ρτ(Y−X0β(τ))] (interpretation) E[(E[Y|X]−X0β)2]E[¯ wτ(Qτ(Y|X)−X0β(τ))2] E[fY(X0β(τ)|X)−1(FY(X0β(τ)|X)−τ)2] unconditional moment E[X(Y−X0β)] = 0E[X(1{Y≤X0β(τ)}−τ)] = 0 (interpretation) E[X(E[Y|X]−X0β)] = 0E[X(FY(X0β(τ)|X)−τ)] = 0 E¯ wτXX0β(τ)−Qτ(Y|X)=0 efficient estimators arg minβ∈Rd1 n∑n i=1(Yi−X0 iβ)2arg minβ∈Rd1 n∑n i=1ρτ(Yi−X0 iβ) = (∑n i=1XiX0 i)−1(∑n i=1XiYi)(Koenker–Bassett) (OLS) asymptotic covariance Q−1ΩQ−1∗J−1ΓJ−1 efficiency bounds Chamberlain (1987) [9] Theorem 2 Linear Regression Model conditional moment E[Y|X] = X0βQτ(Y|X) = X0β(τ) or FY(X0β(τ)|X) = τ efficiency bounds Chamberlain (1987) [9]†Newey and Powell (1990) [13] homoscedasticity-type var[Y|X] = σ2fετ(0|X) = fετ(0) condition efficient estimators OLS Koenker–Bassett ∗Q=E[XX0]and Ω=E[XX0e2]where e=Y−X0β;†The feasible generalized least squares estimator is semiparametrically efficient, for example. Under the linear quantile regression model, the Koenker–Bassett estimator consistently estimates the true β(τ), although it is not semiparametrically efficient given heteroskedasticity. Researchers have proposed many efficient estimators for the correctly-specified linear quantile regression parameter, for example the one-step score estimator in [13], the smoothed conditional empirical likelihood estimator in [24] and the sieve minimum distance (SMD) estimator in [25,26]. However, for all of these estimators, the pseudo-true values under misspecification are different, and their interpretations have not been thoroughly studied. Therefore, the semiparametric efficiency bounds Econometrics 2015,4, 2 8 of 14 of these pseudo-true values are also different. For example, an unweighted SMD estimator converges to a pseudo-true value βSMD that minimizes Eh(FY(X0β|X)−τ)2i.6The first-order condition is E[X(FY(X0βSMD|X)−τ)fY(X0βSMD|X)]=0, which is the unconditional moment used in [13] for the semiparametrically efficient GMM estimator under correct specification. The conditional density weight is similar to the generalized least squares in the mean regression in that it uses a weight function of the conditional variance to construct an efficient estimator. It is interesting to note that the pseudo-true value of the SMD estimator minimizes Eh(FY(X0β|X)−τ)2i≈Ehf2 Y(Qτ(Y|X)|X)(X0β−Qτ(Y|X))2i. The distribution approximation error is weighted evenly over the support of Xfor βSMD, in contrast to the QR parameter, which is weighted more at points with smaller conditional density in Theorem 1. Therefore, the SMD estimator might have more desirable and reasonable approximation properties than QR. Nevertheless, the SMD estimator is computationally more demanding than the Koenker–Bassett estimator. A numerical example in Figure 1illustrates how the Koenker and Bassett (KB) and SMD estimators approximate the CQF and the CDF. 1.0 1.2 1.4 1.6 1.8 2.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Quantile Approx x CQF KB SMD TRUE CQF 1.0 1.2 1.4 1.6 1.8 2.0 0.2 0.4 0.6 0.8 1.0 Distribution Approx x CDF KB SMD Figure 1. This numerical example is constructed by X∼Uni f orm[1, 2],e|X=x∼Uni f orm[0, x] and Y=cos(2X) + e. Therefore, fY(y|X) = 1/X,FY(y|X) = (y−cos(2X))/Xand Qτ(Y|X) = τX+cos(4X). Set τ=0.5 for the median. The red solid line is for the QR parameter βKB defined in (3) and estimated by the Koenker-Bassett (KB) estimator. The blue dashed line is the approximation by the SMD estimator βSMD minimizing E[(FY(X0β|X)−τ)2]. The approximations are X0βKB =−0.324 +0.161Xand X0βSMD =−0.204 +0.078X. The left panel shows the linear approximations X0βKB,X0βSMD and the true CQF. The green circles are 300 random draws from the DGP. The right panel shows the corresponding CDFs FY(X0βKB|X)and FY(X0βSMD|X). For smaller x where the conditional density is larger, the quantile specification error of SMD is smaller than that of KB in the left panel. For the distribution approximation error in the right panel, SMD weights more evenly over the support of X, while KB has smaller distribution approximation error at larger xwith smaller density. This discussion leads to open-ended questions: What is an appropriate linear approximation or a meaningful summary statistic for the nonlinear CQF? How should economists measure the 6The conditional moment restriction in (4) can be expressed as m(X,β) = τ−FY(X0β|X) = 0. In [26], an unweighted penalized sieve minimum distance estimator minimizes a possibly penalized consistent estimate of the minimum distance criterion, E[m(X,β)2].