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Valid GMM Inference with Non-parametrically Generated Discrete Type

Taegyu Yang

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Abstract

The recent progress in machine learning and classification methods has enabled researchers to predict economic discrete attributes that are not observed in standard datasets. These range from missing demographics, such as ethnicity and poverty status, to economic categories, such as innovative firms, metropolitan areas, occupational groups, binary treatment status, and counterfactual status. Although the plug-in procedure is widely used in empirical applications, nonparametric first-stage estimation recovering the labels can introduce regularization bias because of its slow convergence rate. Even if the double/debiased machine learning approach provides a way to correct the regularization bias, the according bias-correction term is not well defined in our problem, because even a small first-stage error can change the type discontinuously. To address this problem, we smooth the discontinuous type indicators using a smooth kernel function and a shrinking tuning parameter bb. We then construct a well-defined Neyman-orthogonal score and define the appropriate debiased estimator. We show that this debiased estimator satisfies Nb\sqrt{Nb}-consistency and asymptotic normality under regularity conditions. The slower Nb\sqrt{Nb} convergence rate is the cost of correcting the regularization bias when the nonparametric first-stage learner enters the moment function through discontinuous functions.

Introduction

We study valid inference in generalized method of moments (GMM) models in which discrete latent types are non-parametrically generated regressors. A researcher specifies a moment function m(⋅)m(\cdot) to learn the parameter of interest θ\theta, and its truth θ0\theta_0 satisfies the following moment condition:

E[m(θ,γ0,i,Wi)]=0⇔θ=θ0,\mathbb{E}\big[ m(\theta, \pmb{\gamma} _{0,i}, W_{i}) \big] = 0 {\quad\Leftrightarrow\quad} \theta = \theta_0,

where WiW_i denotes the data and γ0,i∈{0,1}\pmb{\gamma} _{0,i}\in\{0,1\} the unobserved discrete type. To recover the latent type, the researcher specifies a classification algorithm, consisting of a loss function identifying a nuisance function δ(Zi)\delta(Z_i) representing group-probability or distance to the group-centroid, and a classifier 1{δ(Zi)≥cutoff}\mathbf{1}\{ \delta(Z_i) \geq \textnormal{cutoff} \}.

The recent advances in machine learning (ML) and classification methods have made it possible to predict discrete attributes that are not usually measured in conventional datasets. Applications range from socioeconomic demographics including ethnicity [58, 60, 59], gender [61, 62], and poverty status [57, 54, 55, 67, 56], to economically defined types, such as innovative firms [64, 63], metropolitan areas [53, 50], product-substitution or occupational groups [52, 51, 49, 48, 70]. Other applications include binary treatment or discrete counterfactual status [69, 65, 66, 68], as well as complier groups in instrumental variable settings [1].

A natural and widely used approach in these empirical practices is the naïve plug-in procedure: replacing the label γ0,i\pmb{\gamma} _{0,i} with the estimated classifier 1{δ^(Zi)≥cutoff}\mathbf{1}\{ \widehat{\delta}(Z_i) \geq \textnormal{cutoff} \}, and then minimizing the following GMM criterion:

θ~plug-in=argmin θ∥E^[m(θ,1{δ^(Zi)≥cutoff},Wi)]∥2.{\widetilde{\theta}_{\textnormal{plug-in}} } = \underset{\theta}{\textnormal{argmin\,}}{ \big\| \widehat{\mathbb{E}}[ m(\theta, \mathbf{1}\{\widehat{\delta}(Z_i)\geq \textnormal{cutoff}\}, W_i) ] \big\|^2 } .

However, nonparametric first-stage estimators usually results in regularization bias and contaminates downstream inference, unless the first-stage estimator converges to the truth sufficiently fast [8] or unless the machine learner exactly recovers all true types with high probability [43]. Since the seminal paper of [7], double/debiased machine learning (DML) approach has provided a framework to achieve valid inference when the nonparametric or ML-first-stage estimators resulting the regularization bias. A key device is to adapt the specified moment function so that the modified one obeys Neyman orthogonality [15, 14, 8]: small perturbations of the first-stage estimate have no first-order effect on the expectation of the moment function.

In our problem, however, the standard DML approach is not directly applicable, because the bias-correction term is not well defined, even if the expectation of the moment function involving the latent type is smooths in the nuisance function δ(Zi)\delta(Z_i). Formally, the bias-correction term involves an L2\mathcal{L}^2-Riesz representer that solves a certain functional equation, but this equation has no solution in L2\mathcal{L}^2. The reason is that observations, whose discrete type changes under a small perturbation of the first-stage nuisance, are concentrated near the classification boundary and therefore even a small error can change the type discontinuously. Taking expectations smooths this discontinuity, but the resulting effect depends on the density at the boundary.

In order to construct a well-defined Neyman-orthogonal score, and thereby ensuring the existence of an L2\mathcal{L}^2-Riesz representer, particularly, we approximate the indicator 1 ⁣{δ≥cutoff}\mathbf{1}_{}\!\left\{\delta \geq \textnormal{cutoff}\right\} by k((δ−cutoff)/b)k\big( (\delta - \textnormal{cutoff})/b\big), where k(⋅)k(\cdot) is a distribution kernel function and b>0b>0 is a shrinking tuning parameter. We then define the debiased estimator using the smoothed types and show that it is Nb\sqrt{Nb}-consistent and asymptotically normal. We emphasize that smoothing is essential to this construction: without it, the Riesz representer is not defined. Thus, the slower Nb\sqrt{Nb} convergence rate is the cost of correcting the bias when the nonparametric first-stage learner enters the moment function through discontinuous functions.

We contribute to two strands of literature. First, we provide an inference procedure for economic models including latent discrete types. Finite-mixture models [24, 22, 29, 25, 27, 23, 26, 28, 30] and grouped-fixed effects models [19, 18, 21, 20] study related problems. A distinguishing feature of our approach is its separate first stage, which permits nonparametric identification of the groups. Finite-mixture models typically impose parametric assumptions on the distributions of outcomes and treatments conditional on type, which may be restrictive in many applications. In grouped-fixed effects models, the target parameters and group labels jointly minimize a common and known criterion. This joint structure is natural when addressing incidental parameter problems arising from fixed effects in linear panel models. However, if specifying a full model for the structural parameters, latent types, and observed variables is restrictive or even infeasible because the underlying system is too complicated, a separate first-stage approach may be attractive.

Second, we contribute to the DML literature [7, 9, 8] and the growing literature on AI-/ML-generated regressors [16, 17]. We study a nonstandard problem in which the L2\mathcal{L}^2-Riesz representer required for the bias-correction term does not exist. We construct a bias-correction term through smoothing and provide sufficient conditions for valid asymptotic inference. [1] also discuss how estimation error from a first-stage classifier can affect the plug-in estimator. Their result, like ours, shows why plug-in inference may be invalid. Though they establish N\sqrt{N}-consistency and asymptotic normality of an estimator under specific rate conditions on the first-stage error. In contrast, we show how to orthogonalize the effect of first-stage estimation and obtain a normal approximation, and thereby our approach accommodates flexible first-stage estimator.

We note that the idea of smoothing indicators is not new. For example, [31] proposed the smoothed maximum score estimator to address the nonregularity of the maximum score estimator [32]. [11] and [12] consider a related two-stage inference framework in which a first-stage estimator enters the moment function through the max⁡{δ(Z),0}\max\{\delta(Z), 0\}, which they approximate using a logistic distribution function or a softmax function. Their setting does not face the same problem of existence of an L2\mathcal{L}^2-Riesz representer, because the max function max⁡{δ(Z),0}\max\{\delta(Z),0\} is continuous in δ(Z)\delta(Z). On the other hand, we consider the case under which δ(Z)\delta(Z) changes the moment function m(θ,1{δ(Z)≥cutoff},Wi)m(\theta, \textbf{1}\{\delta(Z)\geq \textnormal{cutoff}\}, W_i) in discontinuous way.

This paper is organized as follows. In § 2, we use a simple model to show why plug-in inference is invalid, why an L2\mathcal{L}^2-Riesz representer does not exist without modification, and how smoothing the indicator addresses this problem. In § 3, we introduce the econometric framework, formalize the problem, and state the basic assumptions. In § 4, we impose restrictions on the smoothing kernel and the rate at which the tuning parameter bb shrinks, derive the Neyman-orthogonal score, and define the debiased estimator. We then establish its Nb\sqrt{Nb}-consistency and asymptotic normality. Finally, in § 5, we report Monte Carlo simulation results and show the debiased estimator achieves the nominal coverage probability.

A simple example

In this section, we provide a simple example to briefly illustrate what this paper does: how the plug-in procedure may lead to invalid inference; why the standard way to construct the Neyman-orthogonal score may not work; and how to modify the problem to make the DML approach applicable.

Setup

Let a binary latent type and an outcome of interest be denoted by γ0,i∈{0,1}\pmb{\gamma} _{0,i} \in \{0,1\} and YiY_i, respectively, with π0≡E[γ0,i]∈(0,1)\pi_0 \equiv \mathbb{E}\big[ \pmb{\gamma} _{0,i}\big] \in (0,1). We assume that the pairs (Yi,γ0,i)(Y_i, \pmb{\gamma} _{0,i}) are independent across i=1,...,Ni=1,...,N.

A researcher is interested in the effect of the γ0,i\pmb{\gamma} _{0,i} on the outcome YiY_i. For the sake of simplicity, we consider the simplest linear model without an intercept and any covariates, as follow:

Yi=θ0γ0,i+ϵiwhereE[ϵi ∣ γ0,i]=0 , E[ϵi2 ∣ γ0,i]=1almost surely.Y_i = \theta_0 \pmb{\gamma} _{0,i} + \epsilon_i\quad\textnormal{where}\quad \mathbb{E}\big[ \epsilon_i \,\big|\, \pmb{\gamma} _{0,i} \big] = 0\,,\, \mathbb{E}\big[ \epsilon_i^2 \,\big|\, \pmb{\gamma} _{0,i} \big] = 1\quad\textnormal{almost surely.}

Because the type is not observed in the data, we have to predict or estimate the latent type. To do so, we assume that the latent type can be expressed with a discrete classifier 1 ⁣{δ0(Zi)≥0}\mathbf{1}_{}\!\left\{\delta_0(Z_i) \geq 0\right\} with high probability, where δ0(⋅)\delta_0(\cdot) indicates a normalized discriminant function. We assume that the signal ZZ is a 1-dimensional continuous random variable, whose Lebesgue density is denoted by fZ(⋅)f_{Z}(\cdot) with fZ(0)>0f_Z(0) > 0.

We assume that the true value δ0(Z)\delta_0(Z) is a solution to:

E[ζ(Z)(T−δ0(Z))]=0for any ζ(⋅)∈D,\begin{equation} \mathbb{E}\big[ \zeta(Z)(T - \delta_0(Z)) \big] = 0 \quad\textnormal{for any }\zeta(\cdot)\in \mathcal{D} , \tag*{(2.1)} \end{equation}

where the random variable TT is an observed proxy variable and D\mathcal{D} denotes a nuisance space. To simplify our discussion, we further assume δ0(Z)=Z\delta_0(Z) = Z in this example. A nonparametric estimator for δ0(⋅)\delta_0(\cdot), denoted by δ^(⋅)\widehat{\delta}(\cdot), is a solution to the sample analogue of the problem (2.1). We simply assume the estimated type γ^i≡1{δ^(Zi)≥0}\widehat{ \pmb{\gamma} }_i \equiv \mathbf{1}\{\widehat{\delta}(Z_i) \geq 0\} satisfies the following condition: for each i=1,...,Ni = 1,...,N,

γ^i−γ0,i={1 with probability ρN+−1 with probability ρN−0 with probability 1−ρN+−ρN−,\begin{equation} \widehat{ \pmb{\gamma} }_i - \pmb{\gamma} _{0,i} = \begin{cases} 1 & \textnormal{ with probability }\rho^{+}_{N} \\ -1 & \textnormal{ with probability }\rho^{-}_{N} \\ 0 & \textnormal{ with probability }1 - \rho^{+}_{N} - \rho^{-}_{N}, \end{cases} \tag*{(2.2)} \end{equation}

with ρN+↓0\rho^{+}_{N} \downarrow 0 and ρN−↓0\rho^{-}_{N} \downarrow 0. In this example, we rule out the miraculous case that ρN+=ρN−\rho^{+}_{N} = \rho^{-}_{N} holds for every N∈Z+N\in{ \mathbb{Z}_{+} }, implying that the direction of misclassification error is symmetrically distributed.

For a moment contribution mi(θ,δ)≡Yi−θ1 ⁣{δ(Zi)>0}m_i(\theta,\delta) \equiv Y_i - \theta\mathbf{1}_{}\!\left\{ \delta(Z_i) > 0 \right\}, we have a moment restriction:

E[mi(θ,δ0)]=0⇔θ=θ0.\mathbb{E}\big[ m_{i}(\theta, \delta_0) \big] = 0 {\quad\Leftrightarrow\quad} \theta = \theta_0.

For a given type profile Γ≡[γ1,...,γN]\pmb{\Gamma} \equiv [ \pmb{\gamma} _{1}, ..., \pmb{\gamma} _{N}], an estimator Γ↦θ~(Γ)\pmb{\Gamma} \mapsto \widetilde{\theta}( \pmb{\Gamma} ) is defined as follow:

θ~(Γ)=(∑i=1Nγi)−1∑i=1NYi.\widetilde{\theta}( \pmb{\Gamma} ) = \left( \sum_{i=1}^{N} \pmb{\gamma} _{i} \right)^{-1}\sum_{i=1}^{N}Y_i.

An oracle estimator and the plug-in estimator are defined by θ~oracle≡θ~(Γ0){\widetilde{\theta}_{\textnormal{oracle}} } \equiv \widetilde{\theta}( { \pmb{\Gamma} }_0 ) and θ~plug-in≡θ~(Γ^){\widetilde{\theta}_{\textnormal{plug-in}} } \equiv \widetilde{\theta}(\widehat{ \pmb{\Gamma} }), respectively.

The plug-in procedure may not be valid

Our first interest is the asymptotic behavior of a statistic N(θ~plug-in−θ0)\sqrt{N}\big( {\widetilde{\theta}_{\textnormal{plug-in}} } - \theta_0 \big), in particular, under what conditions this statistic converges in distribution to a normal distribution with consistently estimable mean and variance. By adding and subtracting θ~oracle{\widetilde{\theta}_{\textnormal{oracle}} }, the statistic is expressed as:

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