PaperlayerView exact revision on arXiv

Source-linked AI summary

Minimum Integrated Distance Estimation in Simultaneous Equation Models

Zhengyuan Gao, Antonio Galvao

arXiv:1412.2143v1math.STmath.OCstat.TH

TL;DR

The paper addresses estimation with an unspecified practical weighting measure by proposing a weighted minimum-distance estimator that uses Kantorovich’s optimal-transport formulation to endogenize the measure. It reports greater efficiency than other available estimators and standard convergent asymptotic results for empirical estimates.

  • Problem

    The practical weighting measure of the product is left unspecified, although estimating functions, distributions, and primitives in the system is of interest.

    Source: §1 · pp. 2, 5

  • Method

    The paper proposes a weighted minimum-distance estimator from independence, using Kantorovich’s optimal-transport formulation to endogenize the weighting measure.

    Source: §1 · pp. 3–4, §7 · p. 35

  • Results

    The proposed estimator is more efficient than others available in the literature, and empirical estimates have standard convergent asymptotic results for different statistical analyses.

    Source: §7 · p. 35

  • Takeaways & Limitations

    The approach estimates parameters of interest simultaneously without assuming the existence of densities.

    Source: §7 · p. 35

    Without further restrictions, γ(·) is an infeasible element with infinite dimensions.

    Source: §3.2 · p. 14

Abstract

from arXiv · show

This paper considers estimation and inference in semiparametric econometric models. Standard procedures estimate the model based on an independence restriction that induces a minimum distance between a joint cumulative distribution function and the product of the marginal cumulative distribution functions. This paper develops a new estimator which generalizes estimation by allowing endogeneity of the weighting measure and estimating the optimal measure nonparametrically. The optimality corresponds to the minimum of the integrated distance. To accomplish this aim we use Kantorovich's formulation of the optimal transportation problem. The minimizing distance is equivalent to the total variation distance and thus characterizes finer topological structures of the distributions. The estimation also provides greater generality by dealing with probability measures on compact metric spaces without assuming existence of densities. Asymptotic statistics of the empirical estimates have standard convergent results and are available for different statistical analyses. In addition, we provide a tractable implementation for computing the estimator in practice.

1 Introduction

The introduction presents a weighted minimum-distance estimator for independence in semiparametric models, with a nonparametrically estimated weighting measure. It situates the estimator among related approaches and outlines its optimal-transport basis, identification, inference, and practical estimation.

1 Introduction

  • The estimator endogenizes and nonparametrically estimates the weighting measure, enabling automatic optimal measure selection without external restrictions on that measure.The paper also claims the estimator is more efficient than existing estimators.Source: §1 · p. 2
  • The framework covers a broad range of models, including nonlinear simultaneous equation models that do not require knowledge of the reduced form.Source: §1 · p. 2
  • The paper establishes consistency and limiting distributions, and provides both asymptotically biased (root-n inconsistent) and unbiased estimation statistics.It also develops practical inference procedures.Source: §1 · p. 4
  • The paper claims the proposed method is more efficient than existing estimators and provides a tractable implementation for computing it.Source: §1 · p. 4
  • Using Kantorovich optimal transport and its dual, the paper represents the criterion through a kernel-based distance and establishes identification when that criterion is zero.The dual distance characterizes finer topologies of probability measures.Source: §1 · p. 4

Related work

  • Prior work left the practical weighting measure on the product space of ε and x unspecified.That work provided a foundation for estimating θ0 in general nonlinear simultaneous equation models.Source: §1 · p. 5
  • Komunjer and Santos (2010) develop a semiparametric estimator for invertible nonseparable models with scalar latent variables and an infinite-dimensional component.Source: §1 · p. 5
  • Santos (2011) proposes an M-estimator assuming strict monotonicity in the scalar error term and derives corresponding asymptotic properties.Source: §1 · p. 5

Identification

  • For θ ≠ θ0, ε generally is not independent of X, and an integrated criterion reflects that dependence through the weighted measure.The passage attributes this identification point to Benkard and Berry (2006).Source: §1 · p. 5

2 Model Framework

The section presents a framework for models on compact metric spaces that does not require densities or a reduced form, using independence between model errors and exogenous variables to identify the true parameter. Its examples cover separable and nonseparable triangular models, and an estimator with a dual representation that preserves identifiability under an appropriate integrated distance.

2.1 Model Setup and Assumptions

  • The model allows compact metric spaces without densities and does not require a reduced form, with non-invertibility at θ ≠ θ0 providing flexibility.The authors state that their approach also generalizes beyond a specified bounded measure and compact parameter space by estimating an optimal measure.Source: §2.1 · pp. 6, 8, §2.2 · p. 10
  • Independence of ε = ρ(X, Y, θ) and X holds if and only if θ = θ0, providing the identification condition for the model parameter.The framework establishes identification in the dual problem as well as in the primal problem.Source: §2.1 · pp. 7–8

2.2 An Example

  • For a bounded measure µ(x, ε) of (x, ε) and compact parameter space Θ, the integrated distance function is continuous on Θ and θ0 is its unique global minimum.Source: §2.2 · p. 9
  • A nonseparable supply-and-demand example with a triangular structure satisfies C9, but its triangular system cannot retrieve the true reduced form.The setup assumes neither a derivative condition nor the existence of a reduced form.Source: §2.2 · pp. 9–10
  • The proposed estimator generalizes (4) and induces a dual representation.The paper uses this dual-representation setup to establish identifiability.Source: §2.2 · p. 10
  • Identifiability relies on choosing an integrated distance that places Hθ(x, ε) and Pθ(x, ε) in a complete separable metric space on Q.The identification procedure uses this condition to distinguish parameter values by their associated distance values.Source: §2.2 · p. 10
  • The true parameter θ0, which gives a unique integrated-distance value, is well separated from other parameter values.The passage relates this separation to an integrated distance that induces a complete separable metric space for Hθ(x, ε) and Pθ(x, ε) on Q.Source: §2.2 · p. 10

3 Criterion Function

This section develops an independence-based minimum-distance criterion for dissimilarity between Hθ and Pθ, reformulating it through optimal transport and dual and kernel representations. It endogenizes the weighting measure by estimating it jointly with the parameter, and presents WH(θ) as a distance equivalent to total variation that characterizes weak topology on probability measures.

3 Criterion Function

  • The section establishes the validity of the dual problem and proves a kernel representation theorem.Source: §3 · p. 11

3.1 Primal Objective Function

  • The criterion endogenizes the measure μ and estimates it jointly with θ₀ under minimal assumptions, because the choice of μ affects parameter estimates.The authors also establish that an optimal transport measure exists and can be represented using kernels.Source: §3.1 · p. 12

3.2 Transport Measure Representation

  • The reformulated criterion uses a 1-Wasserstein distance between the joint distribution and its independence-based product measure, with a transport measure coupling their marginals.Because the transport measure is infinite-dimensional without further restrictions, the section introduces a dual representation to make estimation tractable.Source: §3.2 · pp. 12–14

3.3 The Dual Problem

  • The dual problem shifts the focus from sample distances to cumulative distribution functions, and the paper conducts both identification and estimation in that dual problem.Source: §3.3 · p. 15
  • The dual criterion modifies the integral-distance estimation problem to use a distance similar to total variation between empirical measures.Evaluating the distance by comparing mean function values across the two samples makes estimation more feasible; resampling can reduce false equality risk in small samples such as n = 10.Source: §3.3 · p. 16

3.4 Kernel Representation

  • A kernel representation embeds the infinite-dimensional criterion function into a tractable space and expresses it as a kernel-based distance.Source: §3.4 · p. 17

3.4.1 Kernel Representation Theorem

  • For a continuous kernel on a compact metric space, the representation theorem gives conditions under which its feature functions densely represent continuous functions.The stated conditions include an injective feature map and an algebra formed by the span of its feature functions.Source: §3.4.1 · p. 17

3.4.2 WH(θ)-Distance

  • The proposed WH(θ) distance optimally distinguishes distinct measures using a Hilbert-space unit ball and metrizes the weak topology on probability measures.The paper proposes estimating W(θ) with an asymptotically unbiased empirical estimate based on this new metric.Source: §3.4.2 · pp. 18–20
  • WH(θ) is comparable with W(θ) in the probability metric using d_k, under C10 and C11.C11 requires bounded kernel diagonals and a separable metric space (Q, ρ).Source: §3.4.2 · p. 20
  • A transition measure γ captures connections between Hθ(x, ε) and Pθ(x, ε), providing the optimal criterion function for their minimum-distance problem.The criterion uses a general Wasserstein formulation over two probability measures.Source: §3.4.2 · pp. 20–21
  • Kantorovich duality and kernel representation embed the infinite-dimensional criterion in a tractable space as a kernel-based distance.The resulting distance is equivalent to total variation, so the modified estimation problem maintains finer topological details.Source: §3.4.2 · p. 21

4 Estimation and Inference

The section develops a kernel-based distance estimator and inference statistic, establishes convergence of sample counterparts, and relates the optimal measure to efficiency. It also describes the shared form of the WH(θ)- and SH(θ)-distance estimators and presents general inference results that do not require densities.

4.1 Identification

  • Under the stated kernel conditions, Hθ ≠ Pθ implies WH(θ) ≠ 0, so the distance separates the true model from alternatives.The section also states that WH and W are equivalent to total variation distance when the primal distance uses a Hamming metric.Source: §4.1 · p. 22

4.2 Convergence of Pnθ and Hnθ

  • On compact metric spaces with finite covering numbers, sample counterparts of Pθ and Hθ converge by a uniform law of large numbers.Source: §4.2 · p. 22

n-Biased and Unbiased Estimates

  • The baseline cWH criterion has asymptotic bias and may be m-inconsistent, motivating a squared-distance criterion that eliminates root-n asymptotic bias.The paper proposes the square of the WH(θ)-distance as an asymptotically unbiased criterion.Source: n-Biased and Unbiased Estimates · p. 23, §4.4 · p. 25
  • Under Theorem 7’s conditions, the minimum-distance estimator θ∗ is consistent.The result assumes bounded kernels and, for each θ ≠ θ0, an open set Bθ containing θ with h(θ|Bθ) > 0.Source: n-Biased and Unbiased Estimates · p. 24

4.4 Asymptotic Distribution of the Unbiased Statistic

  • The quadratic unbiased statistic has a standard Gaussian limit at the root-n rate, supporting hypothesis testing.The paper presents it as a nonparametric test statistic whose asymptotic properties may extend beyond the current setting.Source: §4.4 · p. 25

4.5 Optimal Information

  • The section states that the proposed estimator is more efficient than Brown and Wegkamp’s and links the minimum optimal measure to the most efficient information.The discussion emphasizes the measure’s endogenous effect in simultaneous equations.Source: §4.5 · pp. 26–27
  • Estimators based on WH(θ)- or SH(θ)-distance share the same form, with the optimum attained at wθ* = k.The covariance discussion notes that the densities hθ(q) and pθ(q) may not exist in the general inference results.Source: §4.5 · p. 28, The covariance matrix · p. 28

The covariance matrix

  • To distinguish the optimal measure from BW’s underlying measure µ, the covariance formulation uses wθ(q)dq instead of dµ and modifies C14′ to C14.C14′ is a condition on probability measures Hθ and Pθ; C14 is a condition on their densities.Source: The covariance matrix · p. 28

5 Computation and Numerical Experiment

The estimator is computed by solving a dual linear program for the empirical transport problem, with the estimated parameter and optimal measure obtained together. The numerical experiment illustrates sparse transport assignments and irregular optimal marginal densities, and connects the transport measure to matching.

5.1 Computation of the Estimator

  • The computation minimizes the empirical distance over θ; grid search is feasible in one dimension but computationally difficult for simultaneous equations with at least two variables.The proposed routine solves the dual problem and computes the distance using a kernel representation.Source: §5.1 · pp. 29–30
  • The dual of the empirical transport problem provides a linear-programming route to the estimator, with solutions to the primal and dual corresponding under the stated condition.The paper describes solving the dual with the simplex method and presents the kernel-based Kantorovich formulation.Source: §5.1 · pp. 29–30

5.2 Numerical Experiment

  • In the experiment, the optimal transport matrix is sparse: most transitions between observations have zero probability.The example uses n = 140 and m = 150, so the transport matrix is 150×140.Source: §5.2 · p. 31
  • The optimal measure is not uniformly distributed, and the experiment’s smoothed marginal densities have irregular, mixture-like shapes.The authors note that the optimal measure may therefore fall outside regularly specified choices such as uniform or normal distributions.Source: §5.2 · p. 32
  • The transport measure is presented as both an inference mechanism and an optimal matching mechanism, with the representation applicable to general matching problems.The experiment illustrates the measure by connecting initial points to posterior points.Source: §5.2 · p. 32

6 Related Likelihood-based Inference

The paper relates its distance-based approach to likelihood inference by seeking a map that pushes the joint distribution Hθ to Pθ. It notes that selecting this map is infeasible without restrictions, while a Gaussian restriction makes it attainable in a linear additive example.

6 Related Likelihood-based Inference

  • The likelihood-based formulation seeks a map that pushes Hθ(x, ε) forward to Pθ(x, ε), with the identity map sufficient when the independence condition holds.At θ = θ0, the transformed density is stated to equal hθ(q).Source: §6 · pp. 33–34
  • Without further restrictions on the transportation function’s class, selecting the optimal map is infeasible because the function is unknown.The passage contrasts this criterion’s simpler representation with the tractability of W(θ) in practice.Source: §6 · p. 34
  • In the linear additive example, a Gaussian restriction makes the push-forward function attainable; with Gaussian q, the translated variable q′ remains Gaussian.The example assumes q′ = A(θ) + q and describes q as zero-mean Gaussian and independent of the model parameters.Source: §6 · p. 34

7 Conclusion

The conclusion presents a minimum integrated distance estimator that endogenizes and nonparametrically estimates its weighting measure. It reports standard asymptotic results and broader applicability to compact metric spaces without requiring densities.

7 Conclusion

  • The estimator generalizes BW via Kantorovich optimal transport, endogenizing the weighting measure and estimating it nonparametrically alongside parameters.It does not impose an a priori structure on the weighting measure.Source: §7 · p. 35
  • The approach applies to probability measures on compact metric spaces without assuming densities or requiring knowledge of the reduced form.The conclusion also describes the assumptions on the measure as minimal.Source: §7 · p. 35
  • The authors report standard convergence results for empirical estimates across different statistical analyses and greater efficiency than competitors.The estimator's asymptotic properties are established.Source: §7 · p. 35

A Mathematical Preliminary

  • A kernel is associated with a feature map into a Hilbert feature space, and neither the map nor the space is unique.The preliminary section also identifies a canonical feature space for a given kernel as a reproducing kernel Hilbert space.Source: A Mathematical Preliminary · p. 39

B Probabilistic Preliminary

  • The probabilistic preliminary restricts attention to pre-compact spaces of probability measures with Hilbertian-type metrics.For complete spaces, compactness is equivalent to pre-compactness.Source: B Probabilistic Preliminary · p. 40

C Proof of Main Theorems

The proof section uses Hilbert-space representation, transportation-distance bounds, and empirical-process arguments to establish results for the estimator.

C.1 Proof of Theorem 3

  • Theorem 3's proof represents a bounded linear functional on a Hilbert space as an inner product with a unique element.It applies the Riesz representation theorem for each probability measure.Source: §C.1 · p. 41

C.2 Proof of Theorem 4

  • Theorem 4's proof establishes the bound W_H(θ) ≤ W(θ) by taking an infimum over couplings.The proof also gives an upper bound on W(θ).Source: §C.2 · pp. 42–43

C.3 Proof of Theorem 6

  • Theorem 6's proof bounds the difference between an empirical quantity Δ and its expectation using symmetrization and a ghost sample.The ghost sample is a second independent set of observations from the same distribution.Source: §C.3 · p. 43

C.4 Proof of Theorem 7

The proof of Theorem 7 partitions the parameter space around θ0 and considers convergence on the resulting sets.

C.4 Proof of Theorem 7

  • For each set in the partition, h(q_i, q_j) converges almost surely to c_s, and the proof uses positive separation away from θ0.When Θ is not compact, the proof covers the complement of a neighborhood of θ0 with finitely many sets.Source: §C.4 · p. 45

C.5 Proof of Theorem 8

The proof of Theorem 8 develops an eigenfunction expansion for a centered covariance kernel and uses the independence condition in its argument.

C.5 Proof of Theorem 8

  • The proof represents the centered kernel through eigenvalues and basis functions of an eigenvalue equation.It identifies the covariance function of zero-mean Gaussian vectors as characterizing the space B_H.Source: §C.5 · pp. 45–46
  • The kernel induces a compact self-adjoint operator with unique countable eigenvalues, which the proof orders in decreasing sequence.The passage states that the eigenvalues converge to zero.Source: §C.5 · pp. 46–47
  • Under the independence condition at θ0, the proof uses zero covariance between ψ_i(x) and ψ_j(ε) for all i and j.The proof also states that the basis functions have zero expectation for finite indices with nonzero eigenvalues.Source: §C.5 · pp. 47–48

C.6 Proof of Theorem 10

The proofs establish optimal-information and convergence properties for the estimator’s minimizer and the kernel-induced metric.

C.6 Proof of Theorem 10

  • Using Hellinger-distance properties, the proof shows terms (I) and (II) are ignorable when SH(θ*)→0.This yields the stated representation.Source: §C.6 · pp. 48, 51
  • The proof concludes that (θ*, k) attains optimal information for every f∈F.Source: §C.6 · p. 52

D.1 Proof of Corollary 1

  • Corollary 1 equates weak convergence Pn⇝P with convergence in the kernel-induced H-norm, using bounded continuity of k and density in BH.Source: §D.1 · p. 52

D.2 Proof of Lemma 1

The proofs characterize zero distance through signed-measure decomposition and derive a dual formulation using transport constraints.

D.2 Proof of Lemma 1

  • Jordan decomposition gives mutually singular positive and negative measures with equal mass α, whose normalization defines distinct probability measures Hθ and Pθ.Source: §D.2 · p. 52
  • Conversely, distinct Hθ and Pθ with WH(θ)=0 yield a nonzero finite signed Borel measure with total mass zero.Source: §D.2 · p. 53

D.3 Proof of Corollary 2

  • The dual formulation associates γi,j(θ) with the constraints ui+vj≥Ci,j through Lagrange multipliers.Source: §D.3 · p. 53

E Auxiliary Results

The auxiliary section states concentration and symmetrization results and illustrates the optimal transport measure and its representation.

E.1 Large Deviation Bounds

  • Theorem 13 gives a concentration bound for functions on Q^m whose change when coordinate i is replaced is bounded by C_i.Source: §E.1 · p. 54
  • Theorem 14 states Rademacher symmetrization for the unit ball BH on compact Q, with a bounded kernel and an i.i.d. sample from Hθ.Source: §E.1 · p. 54
  • Figures 1–5 depict point clouds, movement toward target clouds, the γ matrix, density plots, and γ as an optimal action.Source: Figure 1 · p. 55, Figure 2 · p. 56, Figure 3 · p. 57
Loading 1412.2143v1…