2. IDENTIFICATION OF β_α AND q_α

Before we consider estimation of β_α and q_α in (1), we have to find conditions under which the partially linear quantile regression model can be identified. The model is identified if β_α and q_α are uniquely determined by the population distribution of (Y,X,Z). The following result gives sufficient conditions for the identification of the model.

THEOREM 1. Suppose that Prob(Y ≤ q_α(X) + Z′β_α|X = x,Z = z) = α for all x and z. If

then q_α and β_α are identified.

The sufficient conditions can be relaxed but are stated in the form used to derive asymptotic properties of the proposed estimators. Condition (1) is standard but can be relaxed. For example, Knight (1998) derives the asymptotic distributions for linear median regression estimators under a more general assumption than condition (1). See also Smirnov (1952) for limiting distributions of sample quantiles under general assumptions. Condition (2) excludes a constant variable for Z. Furthermore, it requires that no components of Z be perfectly predictable by components of X. A similar but less stringent exclusion restriction is also needed for partially linear mean regression. For example, it is assumed in Robinson (1988) that E [var(Z|X)] is positive definite. See Robinson (1988) for detailed discussion. It is assumed throughout the remainder of this paper that β_α and q_α are identified.

3. DESCRIPTION OF THE AVERAGE QUANTILE REGRESSION ESTIMATOR

This section presents the main idea behind the estimation method and describes the AQR estimator of β_α. The estimation procedure involves two stages: in the first stage, β_α is estimated locally at each data point; in the second stage, these local estimates are averaged to obtain an n^−1/2-consistent estimator of β_α.

To describe the estimation procedure, we need some notation. Let {(Y_i,X_i,Z_i);1 ≤ i ≤ n} be a random sample of (Y,X,Z) in (1) with size n. Let C_n(X_i) be a cube in R^d_x centered at X_i with side length 2δ_n, where δ_n is a sequence of positive real numbers such that δ_n → 0. For u = (u₁,…,u_dx), a d_x-dimensional vector of nonnegative integers, let [u] = u₁ + ··· + u_dx. Let A_k be the set of all d_x-dimensional vectors u such that [u] ≤ k for some integer k ≥ 0 and let s(A_k) denote the number of elements in A_k. For z ∈ R^d_x with u ∈ A_k, let z^u =

. In addition, given X₁,X₂ ∈ R^d_x, define

where c = (c_u)_u∈Ak is a vector of dimension s(A_k). Define ρ_α(t) = |t| + (2α − 1)t. For each X_i, the first-stage estimator

of β_α is defined by

is the solution to the following minimization problem:

where 1(·) is a standard indicator function. Notice that the preliminary estimator defined here is a leave-one-out estimator and a multivariate uniform kernel is used in (2).

It is important to note that only local data points around X_i are used in (2). As a consequence,

converges in probability to β_α at a nonparametric rate. A more efficient estimator for β_α requires the use of all data points. Because

can be obtained at each data point X_i, the proposed estimation strategy in this paper is to average all

to estimate β_α. This averaging method ensures that all data points are used and thus leads to a faster rate of convergence.

In the second stage, the AQR estimator

is now obtained by

where τ_x(x) is a trimming function such that

with a compact subset

of R^d_x. The trimming function is introduced to estimate the parameters β_α without being overly influenced by the tail behavior of the distribution of X. It will turn out in Section 4 that this AQR estimator is n^−1/2-consistent for β_α and asymptotically normal.

The first-stage estimation procedure is a simplified version of the local polynomial estimation procedure developed in Chaudhuri (1991a, 1991b) and Chaudhuri et al. (1997). The estimation method considered here is different from that used in Chaudhuri (1991a, 1991b) and Chaudhuri et al. (1997) in that only a linear term with respect to Z is adopted, no cross products between X and Z are included in (2), and local weighting is carried out in terms of only X, not both X and Z, because the partially linear form of the regression model is assumed in this paper. A similar type of preliminary estimation procedure is also considered in Chen and Khan (2001). The main advantage of using this method is that the Bahadur-type expansion of the estimator is already well established and can be easily specialized for the purpose of this paper.

We conclude this section by mentioning the numerical algorithm of (2). Because the uniform kernel is independent of c and b, the problem (2) can be easily shown to have a linear programming representation. From the perspective of linear quantile regression, problem (2) can be understood as just weighted quantile regression with weight equal to the uniform kernel. All computational algorithms developed for linear quantile regression, therefore, can be used to solve problem (2) (see, e.g., Buchinsky, 1998, and references therein).

4. ASYMPTOTIC PROPERTIES OF THE AVERAGE QUANTILE REGRESSION ESTIMATOR

This section gives regularity conditions under which the AQR estimator of β_α is n^−1/2-consistent and asymptotically normal. Let ∥·∥ denote the Euclidean norm.

Assumption 1. {(Y_i,X_i,Z_i);1 ≤ i ≤ n} is a random sample of (Y,X,Z) in (1).

Let F_α(·|x,z) and f_α(·|x,z), respectively, denote the cumulative distribution function and the density function of U_α conditional on (X,Z) = (x,z). Moreover, let g(x) denote the density function of X and let g_xz(x,z) denote the joint density of X and Z with respect to an appropriate measure. Suppose that Z can be divided into Z = (Z^(c),Z^(d)), where Z^(c) denote the continuous components of Z and Z^(d) denote the remaining discrete components. Assume that Z^(d) has finitely many mass points. Let

be supports of X and Z = (Z^(c),Z^(d)) such that

are nonempty convex sets in R^d_x and R^d_zc.

Assumption 2.

(a) F_α(0|x,z) = α for all (x,z) in

(b) f_α(u|x,z) is bounded away from zero and continuously differentiable with respect to u in a neighborhood of zero for all (x,z) in

, and

(c) g is positive on

except on the boundary.

Following the nonparametric estimation literature, a function m: R^d → R will be said to have the order of smoothness p on a convex set

in R^d with p = l + γ, where l ≥ 0 is an integer and 0 < γ ≤ 1, and will be written as

, if (i) partial derivatives

exist and are continuous for all

and [u] ≤ l and (ii) there exists a constant M > 0 such that

In other words, a function m is l-times continuously differentiable and the lth derivative is Hölder continuous with exponent γ. The order of smoothness, if it applies to a vector or matrix-valued function, will be understood componentwise.

Assumption 3. The function q_α(·) has the order of smoothness

.

Assumption 4. There exists a γ, 0 < γ ≤ 1 such that

Assumption 5. The distribution of Z has bounded support.

Assumption 6. δ_n ∝ n^−κ, where κ is a positive real number satisfying 1/(2p_q) < κ < 1/(3d_x).

Assumption 7. The trimming function τ_x(x) has compact support

, where

has a nonempty interior and

.

Assumption 8. For all

, the matrix Σ(x) is nonsingular, where

Moreover, Σ(x), considered as a function of x, is in

) with some γ, 0 < γ ≤ 1.

It is necessary to make some comments regarding regularity conditions. Condition (a) of Assumption 2 imposes the conditional quantile restriction, and condition (b) is important for identification. Condition (c) ensures that there will be sufficiently many X_j's near X_i asymptotically as n → ∞. This condition with Assumption 4(b) guarantees that the marginal density of X is bounded away from zero and infinity on

. As done in Chen and Khan (2001), it is possible to include discrete random variables for X. This, however, is not explicitly done here for the sake of simplicity.

Assumption 3 requires that the order of smoothness p_q of q_α grow as the dimension of X increases.⁴

Assumption 6 restricts the range of the bandwidth.

that is too small can induce the loss of efficiency, whereas a

that is too large allows the estimator to be unduly influenced by the tail behavior of g(x).

Finally, Assumption 8 ensures that the variance of the asymptotic distribution of the estimator is well defined.⁷

The next theorem establishes the n^−1/2-consistency and asymptotic normality of the AQR estimator of β_α. Let e_z′ be the d_z × (d_z + 1) matrix such that e_z′ = (0,I_dz), where 0 denotes the d_z-dimensional zero vector and I_dz an identity matrix.

THEOREM 2. Suppose that the order of the polynomial in (2) is k = [p_q]. Let

denote the AQR estimator of β_α. Let Assumptions 1–8 hold. Then as n → ∞,

Although the n^−1/2-consistency and asymptotic normality of the AQR estimator are established, the variance V in Theorem 2 is somewhat complicated. It will be shown in Section 5 that in general this variance is different from the efficient variance bound. The variance V in Theorem 2 can be simplified under a stronger condition than in Assumption 2. The following corollary restates Theorem 2 under the assumption of homoskedasticity.

COROLLARY 3. Assume that the conditions in Theorem 2 hold. Furthermore, suppose that the conditional density of U_α given x and z, evaluated at zero, is independent of (x,z), namely, f_α(0|x,z) = f_α(0) for all (x,z) in

. Then as n → ∞,

The AQR estimator may be compared with other existing estimators in the literature. He and Shi (1996) consider M-type regression splines using bivariate tensor-product B-splines. They establish the asymptotic results under the assumption that U_α is independent of (X,Z). He and Liang (2000) develop estimators for linear and partially linear errors-in-varables models. Their asymptotic results are established under a stringent assumption that E(Z|X = x) = 0 for all x. Chen and Khan (2001) propose an estimation method for a partially linear censored quantile regression. Their estimator uses a two-stage estimation procedure. In the first stage, the conditional quantile function is nonparametrically estimated by the local polynomial method, which is also the case for the AQR estimator. In the second stage, they estimate β_α by a least-squares-type estimator using differenced values of the estimated conditional quantiles as dependent variables. The implementation of their estimator requires a kind of tuning parameter that they call a “selection function.” None of the existing estimators in the literature are asymptotically efficient under the conditional heteroskedasticity of U_α. An asymptotically efficient estimator will be constructed in Section 6.

We end this section by considering estimation of the nonparametric component q_α(·) of the model (1). Because the parametric component β_α can be estimated with an n^−1/2 rate, which is faster than the fastest possible rate of convergence for the nonparametric component, it is possible to estimate q_α(·) as asymptotically efficiently as if β_α were known. The function q_α(·) can be estimated by carrying out a local polynomial quantile regression of

on X. See Fan and Gijbels (1996, p. 202) and Yu and Jones (1998) for rule-of-thumb bandwidths.

5. THE SEMIPARAMETRIC EFFICIENCY BOUND

In this section, the semiparametric efficiency bound for β_α will be derived by adopting the method used in Newey and Powell (1993). The semiparametric efficiency bound may be viewed as the supremum of the Cramér-Rao-type bounds for regular parametric submodels. This bound can be calculated rigorously by a projection formula. More specifically, the efficiency bound V_B for β_α is the inverse of the expectation of the outer product of the efficient score for β_α, namely, V_B = {E [S_α S_α′]}⁻¹, where the efficient score S_α is defined by the projection of the score function for β_α onto the orthogonal complement of the tangent space in the nonparametric direction. See, for example, Newey (1990) and Bickel, Klaassen, Ritov, and Wellner (1993) for further discussion.

We consider the following parametric submodel for the nonparametric component q_α(·) of the regression function:

where h is an arbitrary function of X that satisfies E∥h∥² < ∞. As in Newey and Powell (1993), this paper does not attempt to specify an explicit parametric submodel for the distribution of (U_α,X,Z). Instead, we use the existing results of Newey and Powell (1990) to derive the efficient scores (S_βα,S_η) for (β_α′,η′)′. Then the efficient score S_α for β_α will be calculated by finding the projection of S_βα onto the orthogonal complement of the tangent space for η.

To begin, it is important to notice that the parametric submodel can be written as a linear quantile regression model with parameters (β_α′,η′)′. It then follows from Newey and Powell (1990) that the efficient scores for β_α and η have the form

By Proposition A.3.5 of Bickel et al. (1993, p. 433), the projection of S_βα onto the tangent space for η can be calculated by

Thus, the efficient score for β_α is

This yields the efficiency bound V_B

provided that E [S_α S_α′] is nonsingular. This result implies that the AQR estimator is not efficient in general. If the distribution of U_α is homoskedastic and X and Z are jointly normal, then the variance of the AQR estimator is the same as the efficiency bound, ignoring the effect of trimming. This does not necessarily mean that the AQR estimator is efficient in that special case because imposition of the additional restriction could change the efficiency bound.

6. EFFICIENT ESTIMATION OF β_α

This section constructs efficient estimators of β_α. When U_α is homoskedastic with respect to Z, more precisely if f_α(0|x,z) is independent of z, an optimally weighted AQR estimator will deliver asymptotic efficiency; if U_α permits general heteroskedasticity, then a one-step estimator will be constructed.

6.1. Homoskedastic U_α

In this section, we will show that the efficiency bound can be attained by considering a weighted version of the AQR estimator. Because the AQR estimator is just a simple average of the nonparametric estimators, it is plausible to conjecture that a weighted average can improve asymptotic efficiency. Indeed, an efficient estimator of β_α can be obtained by choosing a proper weighting function when U_α is homoskedastic. To show this, let w(x) be a d_z × d_z matrix-valued, weighting function such that E [τ_x(X)w(X)] is nonsingular and w(x) is in

for some γ, 0 < γ ≤ 1. A weighted AQR estimator

can be defined as

It is straightforward to show that when f_α(0|x,z) = f_α(0) for all x and z,

Clearly the optimal choice of the weight function is to set w(x) = var(Z|X = x). In practical applications, it is likely that var(Z|X = x) is unknown; however, it can be replaced by its uniformly consistent estimator. For instance, we can use the following weighting function:

where

denotes a kernel estimator of the corresponding expectation. Let

denote the weighted AQR estimator with weight function

. In the following theorems, |·| and o_p(1) will be understood componentwise.

THEOREM 4. Assume that the conditions in Corollary 3 hold. Moreover, suppose that

Sufficient conditions for uniform consistency of

on a compact set can be easily obtained using the results of the literature, for example, Bierens (1983, 1987) and Andrews (1995). It is easy to see that when f_α(0|x,z) = f_α(0) for all (x,z), the variance

is the same as the efficiency bound except for the existence of the trimming function. The estimator constructed in this paper is not efficient in a strict sense because it does not use all observations. It is expected that the loss of efficiency due to the existence of the trimming function could be eliminated by letting the support of τ_x grow very slowly as the sample size increases. For example, Robinson (1988) considers the trimming function

(in his notation), where

is a kernel estimator of the probability density function of X_i and b is a positive constant. The effect of trimming is eliminated in Robinson (1988) by letting b converge to zero very slowly. In addition, Klein and Spady (1993) use elaborate trimming procedures to obtain an efficient semiparametric estimator for binary response models. Details are not worked out here, however.

If U_α permits restricted heteroskedasticity, in other words, U_α is homoskedastic with respect to Z, it is also possible to construct an efficient estimator via optimal weighting. More specifically, if f_α(0|x,z) = f_α(0|x) for all z, then the asymptotic variance

for the weighted AQR estimator

has the following form:

This reveals that the optimal weighting function is w(x) = f_α²(0|x)var(Z|X = x). The efficiency bound, therefore, can be attained by using a consistent estimator of f_α²(0|x)var(Z|X = x) as the weighting function. The conditional density f_α(0|x) can be consistently estimated using estimated U_α's. On the other hand, if U_α permits general heteroskedasticity, then no weighted AQR estimator can deliver asymptotic efficiency. In the next section, an efficient estimator will be obtained by the one-step method.

6.2. Heteroskedastic U_α

In this section, a one-step asymptotically efficient estimator of β_α is constructed by taking one step from the AQR estimator of β_α.

⁸

In fact, any n^−1/2-consistent estimator of β_α can be used as an initial estimator.

Let

denote the AQR estimator defined in Section 3.

If S_α were known except for β_α, a one-step asymptotically efficient estimator

would be obtained by

Of course, this estimator is not feasible because S_α contains unknown population quantities such as q_α(X),f_α(0|X,Z), and T(X). Moreover, as pointed out by Newey and Powell (1990), the score function is not continuous in parameters β_α. As a result of this discontinuity, Newey and Powell (1990) make use of a sample splitting method for the efficient estimation of a (censored) linear quantile regression model. The sample splitting method adopted in Newey and Powell (1990) consists of using each half of the observations to estimate the efficient score for the other half. As a result, the ordering of the data may matter for the estimation of β_α.

Instead of using the technique of Newey and Powell (1990), this paper smooths the score function. Differentiability of the score function enables us to use standard Taylor series methods to obtain the asymptotic properties of the one-step estimator. The smoothing method requires the introduction of an additional tuning parameter, but we feel that this is acceptable because it is very hard to find any reasonable rule to choose the ordering of the data in practical applications. In addition, it will be shown in Section 7 that the simulation results are somewhat insensitive to the choice of the tuning parameter.⁹

Another alternative could be to invoke stochastic equicontinuity arguments in Andrews (1994). Unlike the mean regression case,

is a step function and therefore is not smooth enough to apply the existing results in the literature.

Horowitz (1998a) uses a smoothed least-absolute-deviations estimator for a linear median-regression model to obtain asymptotic refinements of bootstrap tests. Following his idea, we replace the indicator function in S_α with a smooth function. Specifically, let J be a bounded, differentiable function satisfying J(t) = 0 if t ≤ −1 and J(t) = 1 if t ≥ 1. The function J can be regarded as the integral of a kernel function. Let {j_n} be a sequence of positive real numbers that converges to zero. In addition, let

, where

are some compact subsets of R^d_x and R^d_zc, respectively. The trimming function τ(x,z) is introduced for the same reason as before. For a given b, a smoothed feasible score function

is then defined as

where

, denote consistent estimators of q_α(x),f_α(0|x,z), and T(x), respectively. Notice that 1 − J(·/j_n) can be arbitrarily close to 1(· ≤ 0) for sufficiently large n. An actual one-step estimator proposed here is

To complete the description of the one-step efficient estimator, we need to specify the nonparametric estimators of q_α(X_i),f_α(0|X_i,Z_i), and T(X_i). First of all, q_α(X_i) can be estimated by the first element of

of (2). For f_α(0|x,z), a standard kernel density estimator may be used. Observe that f_α(0|x,z) can be written as f_α(0|x,z) = f₁(0,x,z)/f₂(x,z), where f₁ and f₂ are joint densities of (U_α,X,Z) and (X,Z), respectively. This suggests that the conditional density of U_α at zero can be estimated consistently by obtaining the ratio of the kernel estimator of f₁ to the kernel estimator of f₂. More specifically, the kernel estimator

is defined as

where

kernel function with a bandwidth ν_1n, and K_xz is a (d_x + d_z)-dimensional kernel function with a bandwidth ν_2n. Finally, T(X_i) can be estimated by

where K_x is a d_x-dimensional kernel function with a bandwidth γ_n.

The following additional regularity conditions are useful to derive the asymptotic properties of the one-step estimator of β_α.

Assumption 9. The trimming function τ(x,z) has compact support

.

Assumption 10. The conditional density f_α(u|x,z) is continuously twice differentiable with respect to u in a neighborhood of zero for all (x,z) in

.

Assumption 11. Let J⁽ⁱ⁾(v) = dⁱJ(v)/dvⁱ.

(a) J(·) is bounded, J(v) = 0 if v ≤ −1, and J(v) = 1 if v ≥ 1.

(b) J is twice differentiable, J⁽¹⁾(v) is symmetrical about v = 0, J⁽²⁾ is Lipschitz continuous, and J⁽ⁱ⁾(v) is bounded for i = 1,2.

(c)

Assumption 12. j_n ∝ n^−η, where η is a positive real number satisfying ¼ < η < 5/18.

Assumption 13.

Assumptions 10–12 are necessary to make smoothing have no effect on the asymptotic distribution of the one-step estimator. Just like the assumption for

in Theorem 4, Assumption 13 is a high-level assumption that requires that kernel estimators be uniformly consistent on the compact sets. It is easy to obtain sufficient conditions for Assumption 13 using the results of Bierens (1983, 1987) and Andrews (1995). The main result of this paper is as follows.

THEOREM 5. Let

denote the one-step estimator defined in (3). Let Assumptions 1–13 hold. Then as n → ∞ (assuming that V_* is well defined),

The variance of Theorem 5 is the same as the efficiency bound except for the effect of the trimming function τ. Just like the case of homoskedastic U_α, the effect of the trimming function is likely to be eliminated by letting the support of τ grow.

For statistical inference, it is necessary to obtain a consistent estimator of the variance V_*. There may be several ways to estimate this variance matrix, but a simple consistent estimator of V_* that is a by-product of the estimation procedure is

It is shown in the proof of Theorem 5 in the Appendix that

converges to V_* in probability. An alternative way to estimate V_* is to replace the components of V_* with its empirical counterparts. A consistent estimator can be given by

where

denote consistent nonparametric estimators of f_α(0|x,z) and E [·|x] .

Adopting the same idea, one can obtain a consistent estimator of the asymptotic variance V of the AQR estimator. Specifically, the consistent estimator

has the form

and

is a nonparametric estimator of Σ(x) using the estimated f_α(0|X,Z). It is also straightforward to obtain consistent estimators of the variances of the AQR estimator and the weighted AQR estimator when U_α is homoskedastic.

7. MONTE CARLO EXPERIMENTS

This section presents the results of a Monte Carlo investigation of the finite sample performance of the proposed estimators in the previous sections. In all experiments α = 0.5 and n = 100. Following Robinson (1988) and Chen and Khan (2001), we considered the following model:

where X_i and Z_i were drawn from a bivariate standard normal distribution with correlation 0.5 and ε_i was drawn from the standard normal distribution that is independent of X and Z. Three different functions for q and two different functions for σ were simulated:

where C is a constant that was chosen to make σ(X_i,Z_i) have standard deviation 1/3. The function q₂, which is taken from Härdle (1990, p. 122), has a bell-shaped hump around zero. The parameter β was set to be 1. The trimming function used in the experiments was τ_x(x) = 1(|x| ≤ 2).¹⁰

Computing the AQR estimates requires choosing the order of polynomial k and the bandwidth δ_n in (2). In the experiments k = 3. The asymptotic results of Section 3 only provide the range of δ_n in terms of the asymptotic order. A higher order asymptotic theory is required to obtain an asymptotically optimal δ_n. However, there is a simple, informal selection rule based on the rule-of-thumb bandwidth for the estimation of the nonparametric component q_α. Let

be the rule-of-thumb bandwidth for the estimation of q when k = 3, suggested by Fan and Gijbels (1996, p. 202). Specifically,

is of the form

where w₀(·) is a weight function,

is obtained from a global polynomial fit,

is a kernel density estimate of the residuals of the global polynomial fit, and

is the αth sample quantile of the residuals. Also,

denotes the fourth derivative of

. The weight function was set to be w₀(x) = 1(|x| ≤ 2). The global polynomial fit was obtained by carrying out the median regression of Y on the constant term, X,X²,…,X⁵, and Z. The bandwidth

converges at rate n^−1/9, which is optimal for the estimation of q. Undersmoothing is required for the estimation of β. A simple bandwidth such as

converges at rate n^−1/5 and satisfies Assumption 6 when p_q ≥ 3. By some preliminary simulations, the averages of ad hoc bandwidths

ranged between 0.7 and 1.1 across the designs considered in the experiments. In the experiments, δ_n ∈ {0.5, 0.6,…,1.5}, which includes the range of the averages of

. There were 1,000 replications in each experiment. The computations were carried out in GAUSS with GAUSS pseudo-random number generators.

Figure 1 shows the asymptotic (dashed lines) and empirical (solid lines) root mean squared errors (RMSEs) of the AQR estimates of β. The asymptotic RMSEs were calculated (ignoring the effect of trimming) by the formula of the asymptotic variance given in Section 4. For all designs, the empirical RMSEs are quite close to the asymptotic RMSEs over a wide range of bandwidths including the range of the average

. It can be seen that the results for q₃ are more sensitive to the bandwidth than those for q₁ and q₂, but the results for all designs are quite insensitive to the bandwidth in the range of the rule of thumb. The empirical biases of the AQR estimates were also computed and were negligible relative to the empirical standard deviations, so they are not reported here.

Results of the Monte Carlo experiments for the AQR estimates of β.

More tuning parameters are required to compute the one-step efficient estimates. The trimming function was set to be τ(x,z) = 1(|x| ≤ 2, |z| ≤ 2). Gaussian kernels were used to estimate f_α(0|X_i,Z_i). Bandwidths were ν_1n = n^−1/7 and ν_2n = n^−1/6 with respect to standardized designs. For the estimation of T(X_i), γ_n = s_x n^−1/5, where s_x is the sample standard deviation of X. The smoothing function J is the integral of the quartic kernel such that

The AQR estimates of β and the estimates of q(·) were computed using bandwidth δ_n = 0.8 Finally, the bandwidth j_n has to be chosen, but the asymptotic theory in Section 5 provides only qualitative restriction for the choice of j_n. The experiments focused on the sensitivity to the choice of j_n. In the experiments, j_n = {0.1, 0.2,…,2.0}.

Figure 2 shows the asymptotic (dashed lines) and empirical (solid lines) RMSEs of the one-step estimates of β. In addition, it also shows the empirical RMSEs (dotted lines) of the AQR estimates. Asymptotic results given in previous sections indicate that the AQR estimator is as efficient as the one-step estimator for homoskedastic designs. Furthermore, the optimally weighted AQR estimator is basically the same as the AQR estimator because var(Z|X) is a constant in the experiments. On the other hand, it can be checked by some calculation that the asymptotic RMSE of the AQR estimator exceeds that of the one-step estimator by a factor of 1.4 for heteroskedastic designs. One noteworthy result is that the empirical RMSEs of the one-step estimates are somewhat larger than asymptotic counterparts for the heteroskedastic designs. This is not too surprising because the one-step estimates use several nonparametric estimates, which can be inaccurate for small sample size such as n = 100. The one-step estimates perform better than the AQR estimates for most of the values of bandwidth j_n. It also appears that the results are somewhat insensitive to the choice of the bandwidth j_n as long as j_n is not too small. In summary, the results of Monte Carlo experiments indicate that our proposed estimators work reasonably well in the finite samples.

Results of the Monte Carlo experiments for the one-step estimates of β.