# De-biased Machine Learning in Instrumental Variable Models for Treatment
Effects

Rahul Singh and Sophie Sun  Rahul Singh
MIT Economics
<EMAIL_ADDRESS>
&Liyang Sun∗
MIT Economics
<EMAIL_ADDRESS>
Equal contribution. Correspondence address: MIT Department of Economics, 50
Memorial Drive, Cambridge, MA 02142. Code:
https://github.com/lsun20/rrr_kappa_replication

(June 10, 2020)

###### Abstract

We introduce a de-biased machine learning (DML) approach to estimating
complier parameters with high-dimensional data. Complier parameters include
local average treatment effect, average complier characteristics, and complier
counterfactual outcome distributions. In our approach, the de-biasing is
itself performed by machine learning, a variant called automatic de-biased
machine learning (Auto-DML). By regularizing the balancing weights, it does
not require ad hoc trimming or censoring. We prove our estimator is
consistent, asymptotically normal, and semi-parametrically efficient. We use
the new approach to estimate the effect of 401(k) participation on the
distribution of net financial assets.

Keywords: Potential outcome, doubly robust, semi-parametric

JEL: C14, C36, C45

##  1 Introduction

Instrumental variable (IV) identification is a strategy in causal statistics
for estimating the counterfactual effect of treatment $D$ on output $Y$
controlling for covariates $\mathbf{X}$ using observational data (Stock and
Trebbi,, 2003). Even in the presence of an unmeasured confounder of $(Y,D)$,
the treatment effect can nonetheless be identified if an instrumental variable
$Z$ is available. Intuitively, $Z$ only influences $Y$ via $D$, identifying
the counterfactual relationship of interest.

This solution comes at a price; the analyst can no longer measure parameters
of the entire population such as average treatment effect (ATE).111Formally,
in the IV setting, ATE is not non-parametrically identified. Angrist and
Fernández-Val, (2013) and Wang and Tchetgen, (2018) present identification
strategies for ATE that require auxiliary assumptions. Instead, the analyst
can only measure parameters defined for the subpopulation of compliers such as
local average treatment effect (LATE). A complier is an individual whose
treatment status $D$ is affected by variation in the instrument $Z$. In public
policy, the instruments take the form of changes in eligibility criteria for
social programs. Compliers are thus of policy interest as they are exactly the
subpopulation to be affected by eligibility changes. In digital platforms, the
instruments take the form of randomized recommendations. Compliers are thus of
business interest as they are exactly the subpopulation to be affected by
different recommendations.222See Appendix A for additional intuition and
recent examples in economic research.

In the present work, we introduce a de-biased machine learning (DML) approach
to estimating complier parameters with high-dimensional data (Chernozhukov et
al., 2018a, ; Foster and Syrgkanis,, 2019). Our instrumental variable
identification assumption is in terms of potential outcomes, and it does not
require any functional form restrictions (Angrist et al.,, 1996). As such, we
allow the possibility of heterogeneous treatment effects, i.e. that the
treatment effect varies with covariate $\mathbf{X}$ as in Frölich, (2007). We
also allow for nonlinear models, which are often appropriate when output $Y$
is binary. Approximating heterogeneity and nonlinearity with a regularized,
black-box machine learning (ML) algorithm can introduce bias into complier
parameter estimation (Chernozhukov et al., 2018a, ). However, by situating the
estimation problem in a DML framework, we correct for this bias. In our
approach, the de-biasing is itself performed by machine learning, a variant
called automatic de-biased machine learning (Auto-DML) (Chernozhukov et al.,
2018c, ; Chernozhukov et al., 2018b, ).

We present a general estimator, then specialize it to the tasks of learning
LATE, average complier characteristics, and complier counterfactual outcome
distributions. Average complier characteristics help to assess the external
validity of any study that uses instrumental variable identification (Angrist
and Evans,, 1998; Angrist and Fernández-Val,, 2013). Counterfactual outcome
distributions are particularly important in welfare analysis of schooling,
subsidized training, union status, minimum wages, and transfer programs
(Abadie,, 2002; Abadie et al.,, 2002). We make the following contributions.

Conceptual. We unify the literature on DML in instrumental variable models for
treatment effects. Rather than considering any specific parameter, our results
apply to the general class of parameters identified by instrumental variables.
To do so, we re-interpret a widely-used algorithm for estimating complier
parameters called $\kappa$-weighting as the Riesz representer in DML; it is in
fact a component of the de-biasing term. As a consequence, we unlock the so-
called forbidden regressions with nonlinear specifications.

Algorithmic. We extend the framework of DML with automatic bias correction
pioneered by Chernozhukov et al., 2018c ; Chernozhukov et al., 2018b . Whereas
Chernozhukov et al., 2018c ; Chernozhukov et al., 2018b  consider parameters
of the full population identified by selection on observables, we consider
parameters of the complier subpopulation identified by instrumental variables.

Statistical. We prove our estimator is consistent, asymptotically normal, and
semi-parametrically efficient. As a consequence of this main result, we
propose a hypothesis test, free of functional form restrictions, to evaluate
whether two different instruments induce the same subpopulation of compliers
and hence identify the same causal parameters. When the parameter of interest
is high-dimensional, we justify simultaneous confidence bands by Gaussian
multiplier bootstrap. To explain the numerical stability conferred by
automatic bias correction, we prove finite sample properties.

Empirical. We show our approach outperforms alternative approaches to
estimating complier parameters and does not require ad hoc pre-processing. We
use the new approach to estimate the effect of 401(k) participation on the
distribution of net financial assets, instrumenting with 401(k) eligibility.

For clarity of exposition, we analyze the setting with discrete or continuous
outcome $Y$, binary treatment $D$, and binary instrument $Z$. This setting has
a simple, intuitive definition of the complier subpopulation. The same
analysis goes through with discrete or continuous treatment $D$ and discrete
instrument $Z$, albeit with more elaborate definitions. For the case of
continuous instrument $Z$, similar results are possible at slower rates by
using our approach to extend results by Kennedy et al., (2017) and Colangelo
and Lee, (2020), who consider the unconfounded setting with continuous
treatment.

##  2 Related Work

Several approaches have been proposed to estimate specific complier parameters
by DML. Ogburn et al., (2015) and Chernozhukov et al., 2018a  present a DML
estimator for LATE. In Belloni et al., (2017), the authors present a DML
estimator for counterfactual outcome distributions. All of these estimators
involve plugging in an estimated propensity score in the denominator, which is
numerically unstable. Unlike previous work, we present a general and unifying
justification that covers any complier parameter. We also present an Auto-DML
variant that eliminates the numerically unstable step of plugging in an
estimated propensity score. As far as we know, ours is the first DML and Auto-
DML estimator of complier characteristics. For a comparison between Auto-DML
and other approaches to semi-parametric estimation that use machine
learning–namely targeted maximum likelihood (Van der Laan and Rose,, 2011),
efficient score (Ning and Liu,, 2017), and approximate residual balancing
(Athey et al.,, 2018)–we recommend Chernozhukov et al., 2018c ; Chernozhukov
et al., 2018b .

Our work also relates to the literature on $\kappa$-weighting, an algorithm
introduced by Abadie, (2003). In $\kappa$-weighting, any complier parameter
can be expressed as a weighted average of the corresponding conditional
population parameter. For example, LATE can be expressed as a weighted average
of conditional ATE across covariate values. Abdulkadiroğlu et al., (2017)
consider a $\kappa$-weighting estimator of LATE. Likewise, Angrist, (2001),
Abadie et al., (2002), and Angrist et al., (2016) propose $\kappa$-weighting
estimators of counterfactual outcome distributions. The weight involves an
estimated propensity score in the denominator, which is numerically unstable.
Theoretically, the literature has not yet justified the use of a regularized,
black-box ML algorithm to learn the propensity score in high dimensional
settings (Abadie,, 2003). By elucidating the relationship between
$\kappa$-weighting and DML, we provide this justification. Moreover, by
introducing the Auto-DML variant, we are able to learn the $\kappa$-weight
directly without estimating its components.

Finally, our paper contributes to the growing literature on instrumental
variables in machine learning. Both Hartford et al., (2017) and Singh et al.,
(2019) consider the problem of nonparametric instrumental variable regression,
where the target parameter is the structural function $h$ that summarizes the
counterfactual relationship $Y=h(D,\mathbf{X})+e$, where $e$ is confounding
noise. Athey et al., (2019) and Syrgkanis et al., (2019) further assume the
function $h$ can be decomposed as
$h(D,\mathbf{X})=\mu(\mathbf{X})+\tau(\mathbf{X})D$, where $\tau(\mathbf{X})$
is interpretable as a heterogeneous treatment effect. Importantly, these works
assume that the confounding noise $e$ is additively separable–a model proposed
by Newey and Powell, (2003). In this setting, Hartford et al., (2017)
introduce nonlinearity with neural networks, Singh et al., (2019) with RKHS
methods, Athey et al., (2019) with random forests, and Syrgkanis et al.,
(2019) with black-box ML. In our setting, we do not assume additive
separability of confounding noise–a model considered by Angrist et al.,
(1996). Our target parameters are functionals of the underlying regression
$\mathbb{E}[\mathbf{V}|Z,\mathbf{X}]$, where $\mathbf{V}$ is a vector of
relevant random variables defined below. Such parameters are called semi-
parametric.333If the instrument $Z$ were continuous, then the target parameter
would be non-parametric. Indeed, the parameters we consider are non-
parametrically identified. We allow black-box ML for nonlinear estimation of
$\mathbb{E}[\mathbf{V}|Z,\mathbf{X}]$.

##  3 Problem Setting and Definitions

Let $\mathbf{W}=(Y,D,Z,\mathbf{X}^{\prime})^{\prime}$ concatenate the random
variables. $Y\in\mathcal{Y}\subset\mathbb{R}$ is the continuous outcome,
$D\in\\{0,1\\}$ is the binary treatment, $Z\in\\{0,1\\}$ is the binary
instrumental variable, and
$\mathbf{X}\in\boldsymbol{\mathcal{X}}\subset{\mathbb{R}^{dim(\mathbf{X})}}$
is the covariate. We observe $n$ i.i.d. observations
$\\{\mathbf{W}_{i}\\}_{i=1}^{n}$. Wherever possible, we suppress index $i$ to
lighten notation.

Instrumental variable identification requires an assumption expressed in terms
of potential outcomes. A potential outcome is a latent random variable
expressing a counterfactual outcome given a hypothetical intervention. We
recommend Imbens and Rubin, (2015), Peters et al., (2017), and Hernan and
Robins, (2019) for a clear introduction to this framework for causal
inference. Following the notation of Angrist et al., (1996), we denote by
$Y^{(z,d)}$ the potential outcome under the intervention $Z=z$ and $D=d$. We
denote by $D^{(z)}$ the potential treatment under the intervention $Z=z$.
Compliers are the subpopulation for whom $D^{(1)}>D^{(0)}$.

We now formalize our causal assumption about the instrument $Z$, quoting
Angrist et al., (1996). This prior knowledge, described informally in the
introduction, allows us to define and recover the counterfactual effect of
treatment $D$ on outcome $Y$ for compliers.

######  Assumption 1 (Identification).

Assume

  1. 1.

Independence:
$\\{Y^{(z,d)}\\}_{z,d\in\\{0,1\\}},\\{D^{(z)}\\}_{d\in\\{0,1\\}}\raisebox{0.50003pt}{\rotatebox[origin={c}]{90.0}{$\models$}}Z|\mathbf{X}$

  2. 2.

Exclusion: $\mathbb{P}(Y^{(1,d)}=Y^{(0,d)}|\mathbf{X})=1$ for $d\in\\{0,1\\}$

  3. 3.

Overlap: $\pi_{0}(\mathbf{X}):=\mathbb{P}(Z=1|\mathbf{X})\in(0,1)$

  4. 4.

Monotonicity: $\mathbb{P}(D^{(1)}\geq D^{(0)}|\mathbf{X})=1$ and
$\mathbb{P}(D^{(1)}>D^{(0)}|\mathbf{X})>0$

Independence states that the instrument $Z$ is as good as randomly assigned
conditional on covariates $\mathbf{X}$. Exclusion imposes that the instrument
$Z$ only affects the outcome $Y$ via the treatment $D$. We can therefore
simplify notation: $Y^{(d)}=Y^{(1,d)}=Y^{(0,d)}$. Overlap ensures that there
are no covariate values for which the instrument is deterministic.
Monotonicity rules out the possibility of defiers: individuals who will always
pursue an opposite treatment status from their assignment.

######  Definition 1 (Complier parameters).

Consider the following examples.

  1. 1.

LATE is $\theta_{0}=\mathbb{E}[Y^{(1)}-Y^{(0)}|D^{(1)}>D^{(0)}]$

  2. 2.

Average complier characteristics are
$\boldsymbol{\theta}_{0}=\mathbb{E}[f(\mathbf{X})|D^{(1)}>D^{(0)}]$ for
characteristics $f(\mathbf{X})$

  3. 3.

Complier counterfactual outcome distributions are
$\boldsymbol{\theta}_{0}=\\{\boldsymbol{\theta}_{0}^{y}\\}_{y\in\mathcal{Y}}$
where

$\boldsymbol{\theta}_{0}^{y}=\begin{bmatrix}\beta_{0}^{y}\\\
\delta_{0}^{y}\end{bmatrix}=\begin{bmatrix}\mathbb{P}(Y^{(0)}\leq
y|D^{(1)}>D^{(0)})\\\ \mathbb{P}(Y^{(1)}\leq y|D^{(1)}>D^{(0)})\end{bmatrix}$

Using the notation of Chernozhukov et al., 2018b , we denote the regression of
a random vector $\mathbf{V}=(V_{1},...,,V_{J})^{\prime}$ conditional on
$(Z,\mathbf{X})$ as

$\gamma_{0}(Z,\mathbf{X})=\begin{bmatrix}\gamma_{0}^{V_{1}}(Z,\mathbf{X})\\\
...\\\ \gamma_{0}^{V_{J}}(Z,\mathbf{X})\\\
\end{bmatrix}=\mathbb{E}[\mathbf{V}|Z,\mathbf{X}]$

where $\gamma_{0}^{V_{j}}(Z,\mathbf{X})=\mathbb{E}[V_{j}|Z,\mathbf{X}]$. The
random vector $\mathbf{V}$ is observable and depends on the complier parameter
of interest; we specify its components for LATE, complier characteristics, and
counterfactual outcome distributions in Theorem 1. We denote the classic
Horvitz-Thompson balancing weight as

$\alpha_{0}(Z,\mathbf{X})=\frac{Z}{\pi_{0}(\mathbf{X})}-\frac{1-Z}{1-\pi_{0}(\mathbf{X})}=\frac{Z-\pi_{0}(\mathbf{X})}{\pi_{0}(\mathbf{X})[1-\pi_{0}(\mathbf{X})]},\quad\pi_{0}(\mathbf{X})=\mathbb{P}(Z=1|\mathbf{X})$

Lastly, we denote by $|\cdot|_{q}$ the $\ell_{q}$ norm of a vector, and we
denote by $\|\cdot\|$ the $\mathcal{L}_{2}$ norm of a random variable, i.e.
$\|V_{j}\|=\sqrt{\mathbb{E}[V_{j}]^{2}}$.

##  4 Learning Problem

DML is a method of moments framework for semi-parametric estimation with de-
biasing and strong statistical guarantees (Chernozhukov et al.,, 2016). We
review the DML learning problem: in stage 1, learn the regression
$\gamma_{0}(Z,\mathbf{X})=\mathbb{E}[\mathbf{V}|Z,\mathbf{X}]$ and the
balancing weight
$\alpha_{0}(Z,\mathbf{X})=\frac{Z}{\pi_{0}(\mathbf{X})}-\frac{1-Z}{1-\pi_{0}(\mathbf{X})}$;
in stage 2, estimate the parameter of interest $\boldsymbol{\theta}_{0}$ using
method of moments with a de-biased moment function and the stage 1 estimates.
We extend DML to estimate complier parameters. Specifically, we demonstrate
how Assumption 1, expressed in terms of potential outcomes, implies a moment
function and a corresponding de-biased moment function for the complier
parameters in Definition 1. Its de-biasing term is precisely the normalized
$\kappa$-weight.

###  4.1 De-biased Learning

Consider a causal parameter $\boldsymbol{\theta}_{0}$ implicitly defined by

$\mathbb{E}[m(\mathbf{W},\gamma_{0},\boldsymbol{\theta})]=0\text{ iff
}\boldsymbol{\theta}=\boldsymbol{\theta}_{0}$

Here $m$ is called the moment function, and it defines the causal parameter
$\boldsymbol{\theta}_{0}$. $\gamma_{0}$ is the regression, a nuisance
parameter that must be estimated in order to estimate
$\boldsymbol{\theta}_{0}$.

The plug-in approach involves estimating $\hat{\gamma}$ in stage 1 by some
black-box ML algorithm, and estimating $\hat{\boldsymbol{\theta}}$ in stage 2
by method of moments with moment function $m$. The plug-in approach is badly
biased when $\hat{\gamma}$ involves regularization (Chernozhukov et al.,
2018a, ).444In the case that the regression $\gamma_{0}$ is linear, an OLS
estimator is unbiased and hence the plug-in approach is unbiased. However, if
the regression $\gamma_{0}$ is nonlinear, an OLS estimator incurs an
approximation error that does not vanish with sample size. By contrast, a
black-box ML algorithm incurs an approximation error that does vanish with
sample size. Its regularization bias is what DML seeks to correct.

The DML approach uses a more sophisticated moment function (Newey,, 1994).

$\displaystyle\mathbb{E}[\psi(\mathbf{W},\gamma_{0},\alpha_{0},\boldsymbol{\theta})]=0\text{
iff }\boldsymbol{\theta}=\boldsymbol{\theta}_{0}$
$\displaystyle\psi(\mathbf{w},\gamma,\alpha,\boldsymbol{\theta})=m(\mathbf{w},\gamma,\boldsymbol{\theta})+\phi(\mathbf{w},\gamma,\alpha,\boldsymbol{\theta})$

$\phi$ is called the de-biasing term. We derive $\phi$ such that $\psi$ is
doubly robust. In particular, we derive $\phi$ such that

$\displaystyle\mathbb{E}[\psi(\mathbf{W},\gamma,\alpha_{0},\boldsymbol{\theta}_{0})]$
$\displaystyle=0\quad\forall\gamma\text{ s.t.
}\mathbb{E}[\gamma_{j}(Z,\mathbf{X})]^{2}<\infty$
$\displaystyle\mathbb{E}[\psi(\mathbf{W},\gamma_{0},\alpha,\boldsymbol{\theta}_{0})]$
$\displaystyle=0\quad\forall\alpha\text{ s.t.
}\mathbb{E}[\alpha(Z,\mathbf{X})]^{2}<\infty$

so stage 2 estimation of $\hat{\boldsymbol{\theta}}$ by method of moments with
moment function $\psi$ is asymptotically invariant to estimation error of
either $\hat{\gamma}$ or $\hat{\alpha}$. In this sense, introducing the
additional term $\phi$ serves to de-bias the original moment function $m$. The
learning problem for the causal parameter $\boldsymbol{\theta}_{0}$ is now
Neyman orthogonal to the learning problem for the nuisance parameters
$(\gamma_{0},\alpha_{0})$.

Importantly, the doubly robust moment function $\psi$ introduces an additional
nuisance parameter $\alpha_{0}$, which must be estimated in stage 1. Whereas
DML involves estimating $\hat{\alpha}$ by estimating its components, we
estimate $\hat{\alpha}$ directly by Auto-DML.

###  4.2 Example: LATE

To fix ideas, we present the example of LATE.

$\displaystyle\theta_{0}$
$\displaystyle=\mathbb{E}[Y^{(1)}-Y^{(0)}|D^{(1)}>D^{(0)}]$
$\displaystyle=\frac{\mathbb{E}[Y|Z=1]-\mathbb{E}[Y|Z=0]}{\mathbb{E}[D|Z=1]-\mathbb{E}[D|Z=0]}$
$\displaystyle=\frac{\mathbb{E}\left\\{\mathbb{E}[Y|Z=1,\mathbf{X}]-\mathbb{E}[Y|Z=0,\mathbf{X}]\right\\}}{\mathbb{E}\left\\{\mathbb{E}[D|Z=1,\mathbf{X}]-\mathbb{E}[D|Z=0,\mathbf{X}]\right\\}}$

where the first expression is from Definition 1, the second expression is the
Wald formula appealing to Assumption 1 and the classic result of Imbens and
Angrist, (1994, Theorem 1), and the third expression we call the expanded Wald
formula appealing to the law of iterated expectations (LIE). Rearranging and
using the notation

$\gamma_{0}(Z,\mathbf{X})=\mathbb{E}[\mathbf{V}|Z,\mathbf{X}],\quad\mathbf{V}=\begin{bmatrix}Y\\\
D\end{bmatrix}$

we arrive at the moment function $m$ formulation of LATE.

$\mathbb{E}\left\\{\begin{bmatrix}1&-\theta\end{bmatrix}[\gamma_{0}(1,\mathbf{X})-\gamma_{0}(0,\mathbf{X})]\right\\}=0\text{
iff }\theta=\theta_{0}$

The plug-in approach involves estimating $\hat{\gamma}$ in stage 1 by some
black-box ML algorithm, and estimating $\hat{\theta}$ in stage 2 by method of
moments with this moment function. Equivalently, the plug-in approach involves
estimating $\mathbb{E}[Y|Z,\mathbf{X}]$ and $\mathbb{E}[D|Z,\mathbf{X}]$ and
plugging these estimates into the expanded Wald formula.

It is possible to directly derive the de-biasing term $\phi$ and hence the
doubly robust moment function $\psi$ for LATE by standard arguments (Tan,,
2006). Using the notation

$\alpha_{0}(Z,\mathbf{X})=\frac{Z}{\pi_{0}(\mathbf{X})}-\frac{1-Z}{1-\pi_{0}(\mathbf{X})},\quad\pi_{0}(\mathbf{X})=\mathbb{P}(Z=1|\mathbf{X})$

the doubly robust moment function $\psi$ formulation of LATE is

$\mathbb{E}\left\\{\begin{bmatrix}1&-\theta\end{bmatrix}[\gamma_{0}(1,\mathbf{X})-\gamma_{0}(0,\mathbf{X})]+\alpha_{0}(Z,\mathbf{X})\begin{bmatrix}1&-\theta\end{bmatrix}[\mathbf{V}-\gamma_{0}(Z,\mathbf{X})]\right\\}=0\text{
iff }\theta=\theta_{0}$

It turns out that a single, unifying argument–given in Theorem 1 below–can
derive the doubly robust moment function for not only LATE but also any
complier parameter, including average complier characteristics and complier
counterfactual outcome distributions. In Appendix B, we compare this de-biased
LATE characterization with the so-called forbidden regression discussed in
Angrist and Pischke, (2008, Section 4.6.1). The de-biased framework allows for
consistent estimation even for forbidden regressions, i.e. nonlinear
$\mathbb{E}[D|Z,\mathbf{X}]$ specifications.

###  4.3 Complier Parameters

In our first theorem, we derive the doubly robust moment functions for the
complier parameters in Definition 1. We show that these moment functions share
a common structure.

######  Theorem 1 (Doubly robust moment functions).

Under Assumption 1, the doubly robust moment functions for LATE, average
complier characteristics, and complier counterfactual outcome distributions
are of the form

$\displaystyle\psi(\mathbf{w},\gamma,\alpha,\boldsymbol{\theta})$
$\displaystyle=m(\mathbf{w},\gamma,\boldsymbol{\theta})+\phi(\mathbf{w},\gamma,\alpha,\boldsymbol{\theta})$
$\displaystyle m(\mathbf{w},\gamma,\boldsymbol{\theta})$
$\displaystyle=A(\boldsymbol{\theta})[\gamma(1,\mathbf{x})-\gamma(0,\mathbf{x})]$
$\displaystyle\phi(\mathbf{w},\gamma,\alpha,\boldsymbol{\theta})$
$\displaystyle=\alpha(z,\mathbf{x})A(\boldsymbol{\theta})[\mathbf{v}-\gamma(z,\mathbf{x})]$

where

  1. 1.

For LATE, $\mathbf{V}=(Y,D)^{\prime}$ and
$A(\theta)=\begin{bmatrix}1&-\theta\end{bmatrix}$

  2. 2.

For complier characteristics,
$\mathbf{V}=(Df(\mathbf{X})^{\prime},D)^{\prime}$ and
$A(\boldsymbol{\theta})=\begin{bmatrix}1&-\boldsymbol{\theta}\end{bmatrix}$

  3. 3.

For complier counterfactual distributions, $\mathbf{V}^{y}=((D-1)1_{Y\leq
y},D1_{Y\leq y},D)^{\prime}$ and
$A(\boldsymbol{\theta}^{y})=\begin{bmatrix}1&0&-\beta^{y}\\\
0&1&-\delta^{y}\end{bmatrix}$

Formally, $\alpha_{0}(z,\mathbf{x})A(\mathbf{\boldsymbol{\theta}}_{0})$ is the
Riesz representer to the continuous linear functional
$\gamma\mapsto\mathbb{E}[A(\boldsymbol{\theta}_{0})[\gamma(1,\mathbf{x})-\gamma(0,\mathbf{x})]]$,
i.e.

$\mathbb{E}[A(\boldsymbol{\theta}_{0})[\gamma(1,\mathbf{X})-\gamma(0,\mathbf{X})]]=\mathbb{E}[\alpha_{0}(Z,\mathbf{X})A(\boldsymbol{\theta}_{0})\gamma(Z,\mathbf{X})],\quad\forall\gamma\text{
s.t. }\mathbb{E}[\gamma_{j}(Z,\mathbf{X})]^{2}<\infty$

Indeed, this fact directly follows from the classic Horvitz-Thompson
derivation that the balancing weight $\alpha_{0}(z,\mathbf{x})$ is the Riesz
representer to the continuous linear functional
$\gamma\mapsto\mathbb{E}[\gamma(1,\mathbf{X})-\gamma(0,\mathbf{X})]$, i.e.

$\mathbb{E}[\gamma(1,\mathbf{X})-\gamma(0,\mathbf{X})]=\mathbb{E}[\alpha_{0}(Z,\mathbf{X})\gamma(Z,\mathbf{X})],\quad\forall\gamma\text{
s.t. }\mathbb{E}[\gamma_{j}(Z,\mathbf{X})]^{2}<\infty$

In Appendix C, we review the classic Horvitz-Thompson derivation. We also
prove a more general version of Theorem 1 for the entire class of complier
parameters. In doing so, we demonstrate that the $\kappa$-weight is a
reparametrization of the Riesz representer
$\alpha_{0}(z,\mathbf{x})A(\boldsymbol{\theta}_{0})$.

##  5 Algorithm

We present the algorithm and then discuss its finite sample properties.
Specifically, we show that Auto-DML automatically attenuates the influence of
outliers, which is a central issue in applied economic research. We also show
that Auto-DML is more robust to mis-specification than DML, conferring
numerical stability.

###  5.1 Automatic De-biased Machine Learning

In Chernozhukov et al., (2016), the authors show it is data-efficient and
theoretically elegant to use sample splitting in DML (Bickel,, 1982; Schick,,
1986). We extend the Auto-DML algorithm of Chernozhukov et al., 2018c ;
Chernozhukov et al., 2018b  as follows.

######  Algorithm 1 (De-biased machine learning).

Partition the sample into subsets $\\{I_{\ell}\\}_{\ell=1:L}$.

  1. 1.

For each $\ell$, estimate $\hat{\gamma}_{-\ell}$ and $\hat{\alpha}_{-\ell}$
from observations not in $I_{\ell}$

  2. 2.

Estimate $\hat{\boldsymbol{\theta}}$ as the solution to

$\frac{1}{n}\sum_{\ell=1}^{L}\sum_{i\in
I_{\ell}}\psi(\mathbf{W}_{i},\hat{\gamma}_{-\ell},\hat{\alpha}_{-\ell},\boldsymbol{\theta})\bigg{|}_{\boldsymbol{\theta}=\hat{\boldsymbol{\theta}}}=0$

Our theoretical guarantees apply to Dantzig selector or Lasso estimators of
the balancing weight $\hat{\alpha}_{-\ell}$, originally presented in
Chernozhukov et al., 2018c  and Chernozhukov et al., 2018b , respectively. In
what follows, we restrict attention to Lasso.

Consider the projection of the balancing weight $\alpha_{0}(z,\mathbf{x})$
onto the $p$-dimensional dictionary $b(z,\mathbf{x})$. A high-dimensional
dictionary $b$ allows for flexible approximation of $\alpha_{0}$ (Belloni et
al.,, 2014). With $\ell_{1}$-regularization, the objective becomes

$\boldsymbol{\rho}_{L}=\operatorname*{\arg\\!\min}_{\boldsymbol{\rho}}\mathbb{E}[\alpha_{0}(Z,\mathbf{X})-\boldsymbol{\rho}^{\prime}b(Z,\mathbf{X})]^{2}+2\lambda_{L}|\boldsymbol{\rho}|_{1}$

where $\lambda_{L}=\sqrt{\frac{\ln p}{n}}$ is a theoretical regularization
level. Next, extend the Riesz representer result component-wise.

$\mathbb{E}[b(1,\mathbf{X})-b(0,\mathbf{X})]=\mathbb{E}[\alpha_{0}(Z,\mathbf{X})b(Z,\mathbf{X})],\quad\forall
b\text{ s.t. }\mathbb{E}[b_{j}(Z,\mathbf{X})]^{2}<\infty$

Expanding the square, ignoring terms without $\boldsymbol{\rho}$, and using
this Riesz representer result

$\boldsymbol{\rho}_{L}=\operatorname*{\arg\\!\min}_{\boldsymbol{\rho}}-2\boldsymbol{\rho}^{\prime}\mathbb{E}[b(1,\mathbf{X})-b(0,\mathbf{X})]+\boldsymbol{\rho}^{\prime}\mathbb{E}[b(Z,\mathbf{X})b(Z,\mathbf{X})^{\prime}]\boldsymbol{\rho}+2\lambda_{L}|\boldsymbol{\rho}|_{1}$

The empirical analogue to the above expression yields an estimator of
$\hat{\boldsymbol{\rho}}$. In this paper, we consider
$\hat{\alpha}(z,\mathbf{x})=\hat{\boldsymbol{\rho}}^{\prime}b(z,\mathbf{x})$
as in Chernozhukov et al., 2018b .

######  Algorithm 2 (Regularized balancing weight).

For observations in $I_{-\ell}=\\{1,...,n\\}\setminus I_{\ell}$

  1. 1.

Calculate $p\times p$ matrix
$\hat{\mathbf{G}}_{-\ell}=\frac{1}{n-n_{\ell}}\sum_{i\in
I_{-\ell}}b(Z_{i},\mathbf{X}_{i})b(Z_{i},\mathbf{X}_{i})^{\prime}$

  2. 2.

Calculate $p\times 1$ vector
$\hat{\mathbf{M}}_{-\ell}=\frac{1}{n-n_{\ell}}\sum_{i\in
I_{-\ell}}b(1,\mathbf{X}_{i})-b(0,\mathbf{X}_{i})$

  3. 3.

Set
$\hat{\alpha}_{-\ell}(z,\mathbf{x})=b(z,\mathbf{x})^{\prime}\hat{\boldsymbol{\rho}}_{-\ell}$
where
$\hat{\boldsymbol{\rho}}_{-\ell}=\operatorname*{\arg\\!\min}_{\boldsymbol{\rho}}\boldsymbol{\rho}^{\prime}\hat{\mathbf{G}}_{-\ell}\boldsymbol{\rho}-2\boldsymbol{\rho}^{\prime}\hat{\mathbf{M}}_{-\ell}+2\lambda_{n}|\boldsymbol{\rho}|_{1}$

In Appendix D, we provide and justify an iterative tuning procedure for data-
driven regularization parameter $\lambda_{n}$.

Likewise, we can project the regression $\gamma_{0}(z,\mathbf{x})$ onto the
$p$-dimensional dictionary $b(z,\mathbf{x})$ using the functional
$b\mapsto\mathbb{E}[b(Z,\mathbf{X})\mathbf{V}^{\prime}]$. Our theoretical
results are agnostic about the choice of estimator $\hat{\gamma}$; it may be
this estimator or any other black-box ML algorithm satisfying the rate
condition specified in Assumption 6.

Suppose we wish to test the null hypothesis that two different instruments
$Z_{1}$ and $Z_{2}$ induce the same complier subpopulation and hence identify
the same causal parameters. Denote by $\hat{\boldsymbol{\theta}}_{1}$ and
$\hat{\boldsymbol{\theta}}_{2}$ the Auto-DML estimators for complier
characteristics using the different instruments $Z_{1}$ and $Z_{2}$,
respectively. The following algorithm allows us to test this hypothesis from
some estimator $\hat{\mathbf{C}}$ for the asymptotic variance $\mathbf{C}$ of
$\hat{\boldsymbol{\theta}}=\left(\hat{\boldsymbol{\theta}}^{\prime}_{1},\hat{\boldsymbol{\theta}}^{\prime}_{2}\right)^{\prime}$.

######  Algorithm 3 (Hypothesis test for difference of complier
subpopulations).

Given $\hat{\mathbf{C}}$,

  1. 1.

Calculate the Wald statistic
$W=[\hat{\boldsymbol{\theta}}_{1}-\hat{\boldsymbol{\theta}}_{2}]^{\prime}\left\\{\mathbf{R}\hat{\mathbf{C}}\mathbf{R}^{\prime}\right\\}^{-1}[\hat{\boldsymbol{\theta}}_{1}-\hat{\boldsymbol{\theta}}_{2}]$
where $\mathbf{R}=\begin{bmatrix}\mathbf{I}&-\mathbf{I}\end{bmatrix}$

  2. 2.

Compute the value $c_{\alpha}$ as the $(1-\alpha)$-quantile of $\chi^{2}(1)$

  3. 3.

Reject the null hypothesis if $W>c_{\alpha}$

Suppose we wish to form a simultaneous confidence band for the components of
$\hat{\boldsymbol{\theta}}$, particularly relevant for the estimation of
counterfactual outcome distributions based on a grid
$\mathcal{U}\subseteq\mathcal{Y}$. The following algorithm allows us to do so
from some estimator $\hat{\mathbf{C}}$ for the asymptotic variance
$\mathbf{C}$ of $\hat{\boldsymbol{\theta}}$. Let
$\hat{\mathbf{S}}=diag(\hat{\mathbf{C}})$ and $\mathbf{S}=diag(\mathbf{C})$
collect the diagonal elements of these matrices.

######  Algorithm 4 (Simultaneous confidence band).

Given $\hat{\mathbf{C}}$,

  1. 1.

Calculate
$\hat{\boldsymbol{\Sigma}}=\hat{\mathbf{S}}^{-1/2}\hat{\mathbf{C}}\hat{\mathbf{S}}^{-1/2}$

  2. 2.

Sample
$\mathbf{Q}\overset{i.i.d.}{\sim}\mathcal{N}(0,\hat{\boldsymbol{\Sigma}})$ and
compute the value $c$ as the $(1-\alpha)$-quantile of sampled
$|\mathbf{Q}|_{\infty}$

  3. 3.

Form the confidence band
$[l_{j},u_{j}]=\left[\hat{\theta}_{j}-c\sqrt{\frac{\hat{C}_{jj}}{n}},\
\hat{\theta}_{j}+c\sqrt{\frac{\hat{C}_{jj}}{n}}\right]$ where $\hat{C}_{jj}$
is the diagonal entry of $\hat{\mathbf{C}}$ corresponding to $j$-th element
$\hat{\theta}_{j}$ of $\boldsymbol{\theta}$

###  5.2 Finite Sample Balance

Next, we discuss how Algorithm 2 automatically attenuates the influence of
outliers, which is a central issue in applied statistical research. Recall
that in Algorithm 1, the asymptotic influence
$\psi(\mathbf{W}_{i},\gamma_{0},\alpha_{0},\boldsymbol{\theta}_{0})$ of
observation $\mathbf{W}_{i}$ in fold $\ell$ is estimated by empirical
influence
$\psi(\mathbf{W}_{i},\hat{\gamma}_{-\ell},\hat{\alpha}_{-\ell},\hat{\boldsymbol{\theta}})$.

In original DML, the propensity score $\hat{\pi}_{-\ell}$ is explicitly
estimated to serve as a component of the balancing weight
$\hat{\alpha}^{\text{original}}_{-\ell}(Z_{i},\mathbf{X}_{i})=\frac{Z_{i}}{\hat{\pi}_{-\ell}(\mathbf{X}_{i})}-\frac{1-Z_{i}}{1-\hat{\pi}_{-\ell}(\mathbf{X}_{i})}$.
In the finite sample, $\hat{\pi}_{-\ell}(\mathbf{X}_{i})$ can be close to 0 or
1, causing the empirical influence in original DML
$\psi(\mathbf{W}_{i},\hat{\gamma}_{-\ell},\hat{\alpha}^{\text{original}}_{-\ell},\hat{\boldsymbol{\theta}})$
to diverge. This scenario would arise if, for example, there exists an
imbalanced stratum $\mathbf{x}$: there is an outlier $\mathbf{W}_{i}$ with
$Z_{i}=1$ and $\mathbf{X}_{i}=\mathbf{x}$ but no other observations
$\mathbf{W}_{j}$ with $Z_{j}=0$ and $\mathbf{X}_{j}$ close to $\mathbf{x}$.
This issue may cause an analyst to worry about the choice of covariates
$\mathbf{X}$, or to introduce ad hoc trimming or censoring of propensity
scores. It is a general concern in propensity score-based methods, including
matching, $\kappa$-weighting, and original DML.

Auto-DML automatically addresses outliers and finite-sample imbalance in three
ways. First, Auto-DML considers $\alpha_{0}$ rather than $\pi_{0}$ to be a
nuisance parameter. $\hat{\alpha}_{-\ell}$, estimated by Algorithm 2, enters
$\psi(\mathbf{W}_{i},\hat{\gamma}_{-\ell},\hat{\alpha}_{-\ell},\hat{\boldsymbol{\theta}})$
additively, whereas $\hat{\pi}_{-\ell}$ enters
$\psi(\mathbf{W}_{i},\hat{\gamma}_{-\ell},\hat{\alpha}^{\text{original}}_{-\ell},\hat{\boldsymbol{\theta}})$
inversely. Second, Auto-DML confers a finite sample guarantee of balance on
average. Consider the choice of dictionary $b$ and corresponding partition of
parameter $\boldsymbol{\rho}$ to be

$b(z,\mathbf{x})=\begin{bmatrix}zq(\mathbf{x})\\\
(1-z)q(\mathbf{x})\end{bmatrix},\quad\boldsymbol{\rho}=\begin{bmatrix}\boldsymbol{\rho}^{(z=1)}\\\
\boldsymbol{\rho}^{(z=0)}\end{bmatrix}$

######  Proposition 1 (Balance).

$\displaystyle\left|\frac{1}{n-n_{\ell}}\sum_{i\in
I_{-\ell}}q(\mathbf{X}_{i})-\frac{1}{n-n_{\ell}}\sum_{i\in
I_{-\ell}}q(\mathbf{X}_{i})Z_{i}\cdot\hat{\omega}_{-\ell,i}^{(z=1)}\right|_{\infty}\leq\lambda_{n}$
$\displaystyle\left|\frac{1}{n-n_{\ell}}\sum_{i\in
I_{-\ell}}q(\mathbf{X}_{i})-\frac{1}{n-n_{\ell}}\sum_{i\in
I_{-\ell}}q(\mathbf{X}_{i})(1-Z_{i})\cdot\hat{\omega}_{-\ell,i}^{(z=0)}\right|_{\infty}\leq\lambda_{n}$

where
$\hat{\omega}_{-\ell,i}^{(z=1)}=q(\mathbf{X}_{i})^{\prime}\hat{\boldsymbol{\rho}}^{(z=1)}_{-\ell}$
and
$\hat{\omega}_{-\ell,i}^{(z=0)}=q(\mathbf{X}_{i})^{\prime}\hat{\boldsymbol{\rho}}^{(z=0)}_{-\ell}$

Proposition 1 shows that the weights
$\\{\hat{\omega}_{\ell,i}^{(z=1)},\hat{\omega}_{\ell,i}^{(z=0)}\\}$ serve to
approximately balance the overall sample average with the sample average of
the group having $Z=1$ and the sample average of the group having $Z=0$,
across each element of dictionary $q$. The result is similar to the balancing
conditions of Zubizarreta, (2015) and Athey et al., (2018). Auto-DML
automatically calculates these weights.

Third, Auto-DML is a Lasso-type estimator that delivers a sparse estimate
$\hat{\boldsymbol{\rho}}_{-\ell}$. As such, it automatically determines which
elements of dictionary $q$ to use in the calculation of influence
$\psi(\mathbf{W}_{i},\hat{\gamma}_{-\ell},\hat{\alpha}_{-\ell},\hat{\boldsymbol{\theta}})$.
It discerns that, in the finite sample, some elements are best ignored, and
those elements may be different for the group with $Z=1$ and the group with
$Z=0$. If there exists an imbalanced stratum $\mathbf{X}=\mathbf{x}$, it may
not necessarily correspond to an imbalanced stratum
$q(\mathbf{x})=\mathbf{q}^{\prime}$. If $\mathbf{q}^{\prime}$ is an imbalanced
stratum, then Auto-DML can zero out specific components of $q(\mathbf{x})$
where $\mathbf{q}^{\prime}$ is imbalanced; geometrically, it can collapse
those dimensions in the space of basis functions. In this sense, Auto-DML
learns which transformations of covariates to match on.

###  5.3 Finite Sample Orthogonality

By construction, the doubly robust moment function $\psi$ ensures asymptotic
invariance of $\hat{\boldsymbol{\theta}}$ to estimation error of either
$\hat{\gamma}$ or $\hat{\alpha}$. Any consistent estimators
$(\hat{\alpha},\hat{\gamma})$ of the nuisance parameters that satisfy the
well-known DML product condition
$\|\hat{\alpha}-\alpha_{0}\|\cdot\|\hat{\gamma}-\gamma_{0}\|=o_{p}(1)$ will
deliver consistent and asymptotically normal estimators
$\hat{\boldsymbol{\theta}}$ via Algorithm 1; many approaches are
asymptotically equivalent. We present a finite sample property of Algorithm 2,
as a theoretical explanation of its numerical stability relative to previous
approaches.

For this discussion, we use Algorithm 2 to estimate both the balancing weight
$\hat{\alpha}(z,\mathbf{x})=\hat{\boldsymbol{\rho}}^{\prime}b(z,\mathbf{x})$
and the regression
$\hat{\gamma}(z,\mathbf{x})=\hat{\boldsymbol{\beta}}^{\prime}b(z,\mathbf{x})$.
For simplicity, consider LATE so that $\theta_{0}$ is scalar. As a
consequence, the notation simplifies:
$\psi(\mathbf{W}_{i},\hat{\gamma}_{-\ell},\hat{\alpha}_{-\ell},\boldsymbol{\theta})=\psi(\mathbf{W}_{i},\hat{\boldsymbol{\beta}}_{-\ell},\hat{\boldsymbol{\rho}}_{-\ell},\theta)$.
We have the following result.

######  Proposition 2 (Orthogonality).

$\displaystyle\left|\frac{1}{n-n_{\ell}}\sum_{i\in
I_{-\ell}}\frac{\partial}{\partial\boldsymbol{\beta}}\psi(\mathbf{W}_{i},\hat{\boldsymbol{\beta}}_{-\ell},\hat{\boldsymbol{\rho}}_{-\ell},\theta_{0})\right|_{\infty}\leq\\{1+|\theta_{0}|\\}\lambda_{n}\rightarrow
0$ $\displaystyle\left|\frac{1}{n-n_{\ell}}\sum_{i\in
I_{-\ell}}\frac{\partial}{\partial\boldsymbol{\rho}}\psi(\mathbf{W}_{i},\hat{\boldsymbol{\beta}}_{-\ell},\hat{\boldsymbol{\rho}}_{-\ell},\theta_{0})\right|_{\infty}\leq\\{1+|\theta_{0}|\\}\lambda_{n}\rightarrow
0$

For any fixed sample size, Algorithm 2 delivers estimators
$(\hat{\alpha},\hat{\gamma})$ parametrized by
$(\hat{\boldsymbol{\rho}},\hat{\boldsymbol{\beta}})$ such that estimation of
the target $\theta_{0}$ is approximately invariant to each nuisance even under
mis-specification of both nuisances $(\alpha_{0},\gamma_{0})$.

This property differs from Neyman orthogonality in three important respects.
First, it is a finite sample property rather than a population property. As
such, it can distinguish our estimator from asymptotically equivalent
estimators. Second, the invariance with respect to one nuisance does not
require plugging in the correct value of the other. In other words,
orthogonality holds even if both nuisances $(\alpha_{0},\gamma_{0})$ are mis-
specified. Third, the property holds approximately, with tolerance that
vanishes at the rate of $\lambda_{n}$.

Finally, Proposition 2 connects our work with the literature on bias-reduced
doubly robust estimation, which seeks to improve the stability of balancing
weights. Vermeulen and Vansteelandt, (2015) propose parametric estimators of
nuisances $(\alpha_{0},\gamma_{0})$ based on an exact version of the finite
sample orthogonality property. The authors prove that their estimators locally
minimize the squared first-order asymptotic bias; the same is true about our
estimators, up to tolerance $\lambda_{n}$. Unlike Vermeulen and Vansteelandt,
(2015), we consider semi-parametric estimation in a high-dimensional setting.

##  6 Consistency and Asymptotic Normality

In this section, we present asymptotic statistical guarantees for our
approach. We generalize the main results of Chernozhukov et al., 2018b  to the
instrumental variable setting.

###  6.1 Stage 1

We analyze our stage 1 estimators $(\hat{\gamma},\hat{\alpha})$ of the
nuisance parameters $(\gamma_{0},\alpha_{0})$. First, we place a weak
assumption on the dictionary $b$.

######  Assumption 2 (Bounded dictionary).

There exists $C>0$ s.t. $\max_{j}|b_{j}(Z,\mathbf{X})|\leq C$ almost surely

Alternatively, it is possible to allow the bound on the dictionary to be a
sequence $B_{n}^{b}$ that increases in $n$; the core analysis remains the same
with additional notation.

Next, we articulate assumptions required for convergence of $\hat{\alpha}$
under two regimes: the regime in which $\alpha_{0}$ is dense and the regime in
which $\alpha_{0}$ is sparse.

######  Assumption 3 (Dense balancing weight).

Assume there exist some $\boldsymbol{\rho}_{n}\in\mathbb{R}^{p}$ and
$C<\infty$ s.t. $|\boldsymbol{\rho}_{n}|_{1}\leq C$ and
$\|\alpha_{0}-b^{\prime}\boldsymbol{\rho}_{n}\|^{2}=O\left(\sqrt{\frac{\ln
p}{n}}\right)$

Assumption 3 is a statement about the quality of approximation of the
balancing weight $\alpha_{0}$ by dictionary $b$. It is satisfied if, for
example, $\alpha_{0}$ is a linear combination of $b$. It is possible to allow
the bound on $|\boldsymbol{\rho}_{n}|_{1}$ to be a sequence $B_{n}$ that
increases in $n$; the core analysis remains the same with additional notation.

######  Assumption 4 (Sparse balancing weight).

Assume

  1. 1.

There exist $C>1,\xi>0$ s.t. for all $\bar{s}\leq C\left(\frac{\ln
p}{n}\right)^{-\frac{1}{1+2\xi}}$, there exists some
$\bar{\boldsymbol{\rho}}\in\mathbb{R}^{p}$ with
$|\bar{\boldsymbol{\rho}}|_{1}\leq C$ and $\bar{s}$ nonzero elements s.t.
$\|\alpha_{0}-b^{\prime}\bar{\boldsymbol{\rho}}\|^{2}\leq C(\bar{s})^{-\xi}$

  2. 2.

$\mathbf{G}=\mathbb{E}[b(Z,\mathbf{X})b(Z,\mathbf{X})^{\prime}]$ is
nonsingular with largest eigenvalue uniformly bounded in $n$

  3. 3.

Denote $\mathcal{J}_{\boldsymbol{\rho}}=support(\boldsymbol{\rho})$. There
exists $k>3$ s.t. for
$\boldsymbol{\rho}\in\\{\boldsymbol{\rho}_{L},\bar{\boldsymbol{\rho}}\\}$

$RE(k)=\inf_{\\{\boldsymbol{\delta}:\boldsymbol{\delta}\neq
0,\sum_{j\in\mathcal{J}^{c}_{\boldsymbol{\rho}}}|\delta_{j}|\leq
k\sum_{j\in\mathcal{J}_{\boldsymbol{\rho}}}|\delta_{j}|\\}}\frac{\boldsymbol{\delta}^{\prime}\mathbf{G}\boldsymbol{\delta}}{\sum_{j\in\mathcal{J}_{\boldsymbol{\rho}}}\delta_{j}^{2}}>0$

  4. 4.

$\ln p=O(\ln n)$

Assumption 4 is a statement about the quality of approximation of the
balancing weight $\alpha_{0}$ by a subset of dictionary $b$. It is satisfied
if, for example, $\alpha_{0}$ is sparse or approximately sparse (Chernozhukov
et al., 2018b, ). It is possible to allow the bound on
$|\bar{\boldsymbol{\rho}}|_{1}$ to be a sequence $B_{n}$ that increases in
$n$; the core analysis remains the same with additional notation. The uniform
bound on the largest eigenvalue of $\mathbf{G}$ rules out the possibility that
$\mathbf{G}$ is an equal-correlation matrix. $RE$ is the population version of
the restricted eigenvalue condition of Bickel et al., (2009). The final
condition $\ln p=O(\ln n)$ rules out the possibility that $p=\exp(n)$;
dimension cannot grow too much faster than sample size.

We adapt stage 1 convergence guarantees from Chernozhukov et al., 2018b  for
the balancing weight estimator $\hat{\alpha}$ in Algorithm 2. We obtain a slow
rate for dense $\alpha_{0}$ and a fast rate for sparse $\alpha_{0}$. In both
cases, we require the data-driven regularization parameter $\lambda_{n}$ to
approach $0$ slightly slower than $\sqrt{\frac{\ln p}{n}}$.

######  Assumption 5 (Regularization).

$\lambda_{n}=a_{n}\sqrt{\frac{\ln p}{n}}$ for some $a_{n}\rightarrow\infty$

For example, one could set $a_{n}=\ln(\ln(n))$ (Chatterjee and Jafarov,,
2015). In Appendix D, we provide and justify an iterative tuning procedure to
determine data-driven regularization parameter $\lambda_{n}$. The guarantees
are as follows.

######  Lemma 1 (Dense balancing weight rate).

Under Assumptions 1, 2, 3, and 5,

$\|\hat{\alpha}-\alpha_{0}\|^{2}=O_{p}\left(a_{n}\sqrt{\frac{\ln
p}{n}}\right),\quad|\hat{\boldsymbol{\rho}}|_{1}=O_{p}(1)$

######  Lemma 2 (Sparse balancing weight rate).

Under Assumptions 1, 2, 4, and 5,

$\|\hat{\alpha}-\alpha_{0}\|^{2}=O_{p}\left(a_{n}^{2}\left(\frac{\ln
p}{n}\right)^{\frac{2\xi}{1+2\xi}}\right),\quad|\hat{\boldsymbol{\rho}}|_{1}=O_{p}(1)$

Whereas Lemma 1 does not require an explicit sparsity condition, Lemma 2 does.
When $\xi>\frac{1}{2}$, the rate in Lemma 2 is faster than the rate in Lemma 1
for $a_{n}$ growing slowly enough. Interpreting the rate in Lemma 2,
$n^{-\frac{2\xi}{1+2\xi}}$ is the well-known rate of convergence if the
identity of the nonzero components of $\bar{\boldsymbol{\rho}}$ were known.
The fact that their identity is unknown introduces a cost of $(\ln
p)^{\frac{2\xi}{1+2\xi}}$. The cost $a_{n}^{2}$ can be made arbitrarily small.

We place a rate assumption on black-box estimator $\hat{\gamma}$. It is a weak
condition that allows $\hat{\gamma}$ to converge at a rate slower than
$n^{-\frac{1}{2}}$. Importantly, it allows the analyst a broad variety of
choices of ML algorithms to estimate $\gamma_{0}$. In our empirical
application, we choose Lasso and neural networks.

######  Assumption 6 (Regression rate).

$\|\hat{\gamma}-\gamma_{0}\|=O_{p}(n^{-d_{\gamma}})$ where

  1. 1.

in the dense balancing weight regime,
$d_{\gamma}\in\left(\frac{1}{4},\frac{1}{2}\right)$

  2. 2.

in the sparse balancing weight regime,
$d_{\gamma}\in\left(\frac{1}{2}-\frac{\xi}{1+2\xi},\frac{1}{2}\right)$

These regime-specific lower bounds on $d_{\gamma}$ are sufficient conditions
for the DML product condition.

######  Corollary 1.

Suppose the conditions of Lemma 1 or Lemma 2 hold as well as Assumption 6.
Then
$\|\hat{\alpha}-\alpha_{0}\|\cdot\|\hat{\gamma}-\gamma_{0}\|=o_{p}(n^{-\frac{1}{2}})$

The product condition in Corollary 1 formalizes the trade-off in estimation
error permitted in learning the stage 1 nuisance parameters
$(\gamma_{0},\alpha_{0})$. In particular, faster convergence of $\hat{\alpha}$
permits slower convergence of $\hat{\gamma}$. Prior information about the
balancing weight $\alpha_{0}$ used to estimate $\hat{\alpha}$, encoded by
sparsity or perhaps by additional moment restrictions, can be helpful in this
way. We will appeal to this product condition while proving learning
guarantees for stage 2 causal parameter $\boldsymbol{\theta}_{0}$.555Faster
convergence of $\hat{\alpha}$ does not imply faster convergence of
$\hat{\boldsymbol{\theta}}$, which already occurs at the parametric rate. Nor
does it imply efficiency gains in the asymptotic variance of
$\hat{\boldsymbol{\theta}}$, which is already the semi-parametric lower bound.

###  6.2 Stage 2

We now present our second theorem. We prove our Auto-DML estimator for
complier parameters is consistent, asymptotically normal, and semi-
parametrically efficient. We build on the theory in Chernozhukov et al.,
(2016) to generalize the main result in Chernozhukov et al., 2018b . To do so,
we place weak regularity conditions on the propensity score $\pi_{0}$,
conditional variance $var(\mathbf{V}|Z,\mathbf{X})$, Jacobian
$\mathbf{\mathbf{J}}$, and target parameter space $\boldsymbol{\Theta}$.

######  Assumption 7 (Regularity).

Assume

  1. 1.

$\pi_{0}(\mathbf{x})\in(\bar{c},1-\bar{c})$ for some $\bar{c}>0$

  2. 2.

$var(\mathbf{V}|Z,\mathbf{X})$ is bounded

  3. 3.

Jacobian
$\mathbf{\mathbf{J}}=\mathbb{E}\left[\frac{\partial\psi(\mathbf{W},\gamma_{0},\alpha_{0},\boldsymbol{\theta})}{\partial\boldsymbol{\theta}}\bigg{|}_{\boldsymbol{\theta}=\boldsymbol{\theta}_{0}}\right]$
is nonsingular

  4. 4.

$\boldsymbol{\theta}_{0},\hat{\boldsymbol{\theta}}\in\boldsymbol{\Theta}$, a
compact parameter space

######  Theorem 2 (Auto-DML consistency and asymptotic normality).

Suppose the conditions of Corollary 1 hold as well as Assumption 7. Then
$\hat{\boldsymbol{\theta}}\overset{p}{\rightarrow}\boldsymbol{\theta}_{0}$,
$\sqrt{n}(\hat{\boldsymbol{\theta}}-\boldsymbol{\theta}_{0})\overset{d}{\rightarrow}\mathcal{N}(0,\mathbf{C})$,
and $\hat{\mathbf{C}}\overset{p}{\rightarrow}\mathbf{C}$ where

$\displaystyle\mathbf{\mathbf{J}}=\mathbb{E}\left[\frac{\partial\psi_{0}(\mathbf{W})}{\partial\boldsymbol{\theta}}\right],\quad\hat{\mathbf{\mathbf{J}}}=\frac{1}{n}\sum_{\ell=1}^{L}\sum_{i\in
I_{\ell}}\frac{\partial\hat{\psi}_{i}(\hat{\boldsymbol{\theta}})}{\partial\boldsymbol{\theta}}$
$\displaystyle\boldsymbol{\Omega}=\mathbb{E}[\psi_{0}(\mathbf{W})\psi_{0}(\mathbf{W})^{\prime}],\quad\hat{\boldsymbol{\Omega}}=\frac{1}{n}\sum_{\ell=1}^{L}\sum_{i\in
I_{\ell}}\hat{\psi}_{i}(\hat{\boldsymbol{\theta}})\hat{\psi}_{i}(\hat{\boldsymbol{\theta}})^{\prime}$
$\displaystyle\mathbf{C}=\mathbf{J}^{-1}\boldsymbol{\Omega}\mathbf{J}^{-1},\quad\hat{\mathbf{C}}=\hat{\mathbf{J}}^{-1}\hat{\boldsymbol{\Omega}}\hat{\mathbf{J}}^{-1}$
$\displaystyle\psi_{0}(\mathbf{w})=\psi(\mathbf{w},\gamma_{0},\alpha_{0},\boldsymbol{\theta}_{0}),\quad\hat{\psi}_{i}(\boldsymbol{\theta})=\psi(\mathbf{W}_{i},\hat{\gamma}_{-\ell},\hat{\alpha}_{-\ell},\boldsymbol{\theta})$

It follows that $\hat{\boldsymbol{\theta}}$ is semi-parametrically efficient
(Chernozhukov et al.,, 2016). See Ackerberg et al., (2014) for a discussion of
semi-parametric efficiency. Moreover, it follows that the proposed hypothesis
test is valid.

######  Corollary 2 (Hypothesis test for difference of complier
subpopulations).

Under the conditions of Theorem 2, the hypothesis test in Algorithm 3 falsely
rejects the null hypothesis with probability approaching the nominal level,
i.e. $\lim_{n\rightarrow\infty}\mathbb{P}(W>c_{\alpha}|H_{0})=1-\alpha.$

###  6.3 Simultaneous Confidence Band

Finally, we present the third theorem of this paper: validity of the bootstrap
procedure presented in Algorithm 4 for simultaneous inference on the
counterfactual distributions $\boldsymbol{\theta}_{0}$. We assume the
following sufficient conditions.

######  Assumption 8 (Grid size).

There are positive constants $(c^{\prime},C^{\prime})$ s.t.
$\left(\ln\left(dn\right)\right)^{7}/n\leq C^{\prime}n^{-c^{\prime}}$ where
$d=dim(\mathcal{U})$ is the dimension of the grid on which the counterfactual
distributions $\boldsymbol{\theta}_{0}$ are evaluated.

######  Assumption 9 (Tail bounds).

$c^{\prime}$ and $C^{\prime}$ also satisfy

  1. 1.

$\mathbb{P}\left(\sqrt{\ln
d}\Delta_{1}>C^{\prime}n^{-c^{\prime}}\right)<C^{\prime}n^{-c^{\prime}}$

  2. 2.

$\mathbb{P}\left(\left(\ln\left(dn\right)\right)^{2}\Delta_{2}>C^{\prime}n^{-c^{\prime}}\right)<C^{\prime}n^{-c^{\prime}}$

where

$\displaystyle\Delta_{1}$
$\displaystyle=\left|\frac{1}{\sqrt{n}}\sum_{\ell=1}^{L}\sum_{i\in
I_{\ell}}\left(\hat{\mathbf{S}}^{-1/2}\hat{\mathbf{J}}^{-1}\hat{\psi}_{i}(\boldsymbol{\theta}_{0})-\mathbf{S}^{-1/2}\mathbf{J}^{-1}\psi_{0}(\mathbf{W}_{i})\right)\right|_{\infty}$
$\displaystyle\Delta_{2}$
$\displaystyle=\left|\frac{1}{n}\sum_{\ell=1}^{L}\sum_{i\in
I_{\ell}}\left(\hat{\mathbf{S}}^{-1/2}\hat{\mathbf{J}}^{-1}\hat{\psi}_{i}(\boldsymbol{\theta}_{0})-\mathbf{S}^{-1/2}\mathbf{J}^{-1}\psi_{0}(\mathbf{W}_{i})\right)^{2}\right|_{\infty}$

and the square is taken element-wise.

Belloni et al., (2018) give examples of primitive conditions under which
Assumption 9 holds.

######  Theorem 3 (Simultaneous confidence band).

Suppose the conditions of Theorem 2 hold as well as Assumptions 8 and 9. Then
the confidence band in Algorithm 4 jointly covers the true counterfactual
distributions $\boldsymbol{\theta}_{0}$ at all grid points $y\in\mathcal{U}$
with probability approaching the nominal level, i.e.
$\lim_{n\rightarrow\infty}\mathbb{P}((\boldsymbol{\theta}_{0})_{j}\in[l_{j},u_{j}]\;\forall
j)=1-\alpha.$

##  7 Experiments

We compare the performance of our Auto-DML estimator with original DML
(Chernozhukov et al., 2018a, ) and $\kappa$-weighting (Abadie,, 2003) in
simulations. We focus on counterfactual distributions as our choice of
complier parameter $\boldsymbol{\theta}_{0}$. We then apply our Auto-DML
estimator to real-world data to estimate the counterfactual distributions of
employee net financial assets with and without 401(k) participation.

###  7.1 Simulation

We consider a simulation design detailed in Appendix F. Each simulation
consists of $n=1000$ observations, and we use a dictionary $b$ with dimension
$p=10$. We apply Auto-DML, DML, and $\kappa$-weighting to estimate a
counterfactual outcome distribution at each value in the grid $\mathcal{U}$
specified on the horizontal axis of Figure 1.

For each algorithm, we implement 500 simulations and visualize the median as
well as the 10% and 90% quantiles for each value in the grid $\mathcal{U}$.
Figure 1 summarizes results: Auto-DML outperforms DML by a large margin due to
numerical stability. Auto-DML modestly outperforms $\kappa$-weighting, perhaps
because the former uses regularized ML to estimate nuisance parameters while
the latter does not.

(a) $\beta_{0}^{y}=\mathbb{P}(Y^{(0)}\leq y\mid D^{(1)}>D^{(0)})$

(b) $\delta_{0}^{y}=\mathbb{P}(Y^{(1)}\leq y\mid D^{(1)}>D^{(0)})$

Figure 1:  Counterfactual distribution simulation

In Appendix F.4, we consider whether DML and $\kappa$-weighting can be
improved by addressing the numerical instability that results from inverting
$\hat{\pi}$. Specifically, we try trimming and censoring. Trimming means
excluding observations for which $\hat{\pi}$ is extreme (Belloni et al.,,
2017). Censoring means imposing bounds on $\hat{\pi}$ for such observations.
We find that trimming improves DML. There is no noticeable improvement for
$\kappa$-weighting with either trimming or censoring. Auto-DML without
trimming or censoring outperforms DML and $\kappa$-weighting even with
trimming or censoring. This property is convenient, since ad hoc trimming and
censoring have limited theoretical justification (Crump et al.,, 2009).

###  7.2 Effect of 401(k) on Assets

Next, we use Auto-DML to investigate the effect of 401(k) participation on the
distribution of net financial assets, using 401(k) eligibility as the
instrument. We follow the identification strategy of Poterba and Venti, (1994)
and Poterba et al., (1995). The authors assume that when 401(k) was
introduced, workers ignored whether a given job offered 401(k) and instead
made employment decisions based on income and other observable job
characteristics; after conditioning on income and job characteristics, 401(k)
eligibility was as good as randomly assigned at the time. The independence and
exclusion conditions of Assumption 1 are thus satisfied. Since ineligibilty
implies no participation, the monotonicity condition of Assumption 1 is
satisfied by construction.

We use data from the 1991 US Survey of Income and Program Participation,
studied in Abadie, (2003); Chernozhukov and Hansen, (2004, 2005); Ogburn et
al., (2015); Belloni et al., (2017). We use sample selection and variable
construction as in Chernozhukov et al., 2018a . The outcome $Y$ is net
financial assets (NFA) defined as the sum of IRA balances, 401(k) balances,
checking accounts, US saving bonds, other interest-earning accounts, stocks,
mutual funds, and other interest-earning assets minus non-mortgage debt. The
treatment $D$ is participation in a 401(k) plan. The instrument $Z$ is
eligibility to enroll in a 401(k) plan. The covariates $\mathbf{X}$ are age,
income, years of education, family size, marital status, two-earner status,
benefit pension status, IRA participation, and home-ownership.

The data include $n=9875$ observations after we numerically impose the overlap
condition of Assumption 1, following Farrell, (2015) and Chernozhukov et al.,
2018a . In Appendix G, we show that our results remain unchanged without this
pre-processing step. We follow Belloni et al., (2017) in the choice of grid
points $\mathcal{U}$ and Chernozhukov et al., 2018a  in the choice of
dictionary $b$. We take $\mathcal{U}$ as the $5^{th}$ through $95^{th}$
percentiles of $\mathcal{Y}$, a total of 91 different values of $y$. We
consider a high-dimensional dictionary with $p=277$. See Appendix G for
further details on the dictionary and Auto-DML implementation.

Figure 2 visualizes point estimates and simultaneous 95% confidence bands. We
find that 401(k) participation significantly shifts out the distribution of
NFA, consistent with results reported in Belloni et al., (2017). For
compliers, the distribution of potential NFA under participation first order
stochastic dominates the distribution of potential NFA under non-
participation. Moreover, the Auto-DML algorithm is robust in the high
dimensional setting, yielding similar results using Lasso or a neural network
to estimate $\hat{\gamma}$. Our counterfactual distribution estimates are non-
decreasing. Generically, this property may not hold in the finite sample
(Imbens and Rubin,, 1997); in such case, monotone rearrangement is possible
(Chernozhukov et al.,, 2009).

(a) $\hat{\gamma}$ estimated by Lasso

(b) $\hat{\gamma}$ estimated by neural network

Figure 2: Effect of 401(k) on net financial assets for compliers

##  8 Conclusion

We extend de-biased machine learning with automatic bias correction to the
task of learning causal parameters from confounded, high-dimensional data. The
procedure is easily implemented and semi-parametrically efficient. As a
contribution to the instrumental variable literature, we reinterpret the
$\kappa$-weight as the Riesz representer in the problem of learning complier
parameters and we allow for high-dimensional covariates. As a contribution to
the de-biased machine learning literature, we generalize the theory of Auto-
DML and provide a unified framework for estimating causal parameters
identified by instrumental variables. In simulations, Auto-DML outperforms DML
and $\kappa$-weighting and eliminates the ad hoc step of trimming or
censoring, suggesting Auto-DML may be an effective paradigm in high-
dimensional causal inference.

### Acknowledgements

We are grateful to Alberto Abadie, Anish Agarwal, Isaiah Andrews, Joshua
Angrist, Abhijit Banerjee, Victor Chernozhukov, Joseph Kibe, Anna Mikusheva,
Whitney Newey, Garima Sharma, and Suhas Vijaykumar for helpful comments. We
are also grateful for support from the Jerry A. Hausman Graduate Dissertation
Fellowship.

## References

  * Abadie, (2002) Abadie, A. (2002).  Bootstrap tests for distributional treatment effects in instrumental variable models.  Journal of the American Statistical Association, 97(457):284–292.
  * Abadie, (2003) Abadie, A. (2003).  Semiparametric instrumental variable estimation of treatment response models.  Journal of Econometrics, 113(2):231–263.
  * Abadie et al., (2002) Abadie, A., Angrist, J. D., and Imbens, G. W. (2002).  Instrumental variables estimates of the effect of subsidized training on the quantiles of trainee earnings.  Econometrica, 70(1):91–117.
  * Abdulkadiroğlu et al., (2017) Abdulkadiroğlu, A., Angrist, J. D., Narita, Y., and Pathak, P. A. (2017).  Research design meets market design: Using centralized assignment for impact evaluation.  Econometrica, 85(5):1373–1432.
  * Ackerberg et al., (2014) Ackerberg, D., Chen, X., Hahn, J., and Liao, Z. (2014).  Asymptotic efficiency of semiparametric two-step GMM.  Review of Economic Studies, 81(3):919–943.
  * Angrist, (2001) Angrist, J. D. (2001).  Estimation of limited dependent variable models with dummy endogenous regressors: Simple strategies for empirical practice.  Journal of Business & Economic Statistics, 19(1):2–28.
  * Angrist et al., (2016) Angrist, J. D., Cohodes, S. R., Dynarski, S. M., Pathak, P. A., and Walters, C. R. (2016).  Stand and deliver: Effects of Boston’s charter high schools on college preparation, entry, and choice.  Journal of Labor Economics, 34(2):275–318.
  * Angrist et al., (2010) Angrist, J. D., Dynarski, S. M., Kane, T. J., Pathak, P. A., and Walters, C. R. (2010).  Inputs and impacts in charter schools: KIPP Lynn.  American Economic Review, 100(2):239–43.
  * Angrist and Evans, (1998) Angrist, J. D. and Evans, W. N. (1998).  Children and their parents’ labor supply: Evidence from exogenous variation in family size.  American Economic Review, 88(3):450–477.
  * Angrist and Fernández-Val, (2013) Angrist, J. D. and Fernández-Val, I. (2013).  ExtrapoLATE-ing: External validity and overidentification in the LATE framework.  In Advances in Economics and Econometrics, pages 401–434.
  * Angrist et al., (1996) Angrist, J. D., Imbens, G. W., and Rubin, D. B. (1996).  Identification of causal effects using instrumental variables.  Journal of the American Statistical Association, 91(434):444–455.
  * Angrist et al., (2013) Angrist, J. D., Pathak, P. A., and Walters, C. R. (2013).  Explaining charter school effectiveness.  American Economic Journal: Applied Economics, 5(4):1–27.
  * Angrist and Pischke, (2008) Angrist, J. D. and Pischke, J.-S. (2008).  Mostly Harmless Econometrics: An Empiricist’s Companion.  Princeton University Press.
  * Athey et al., (2018) Athey, S., Imbens, G. W., and Wager, S. (2018).  Approximate residual balancing: Debiased inference of average treatment effects in high dimensions.  Journal of the Royal Statistical Society: Series B (Statistical Methodology), 80(4):597–623.
  * Athey et al., (2019) Athey, S., Tibshirani, J., and Wager, S. (2019).  Generalized random forests.  The Annals of Statistics, 47(2):1148–1178.
  * Belloni et al., (2012) Belloni, A., Chen, D., Chernozhukov, V., and Hansen, C. B. (2012).  Sparse models and methods for optimal instruments with an application to eminent domain.  Econometrica, 80(6):2369–2429.
  * Belloni and Chernozhukov, (2013) Belloni, A. and Chernozhukov, V. (2013).  Least squares after model selection in high-dimensional sparse models.  Bernoulli, 19(2):521–547.
  * Belloni et al., (2018) Belloni, A., Chernozhukov, V., Chetverikov, D., Hansen, C., and Kato, K. (2018).  High-dimensional econometrics and regularized GMM.  arXiv:1806.01888.
  * Belloni et al., (2017) Belloni, A., Chernozhukov, V., Fernández-Val, I., and Hansen, C. B. (2017).  Program evaluation and causal inference with high-dimensional data.  Econometrica, 85(1):233–298.
  * Belloni et al., (2014) Belloni, A., Chernozhukov, V., and Hansen, C. B. (2014).  Inference on treatment effects after selection among high-dimensional controls.  The Review of Economic Studies, 81(2):608–650.
  * Bickel, (1982) Bickel, P. J. (1982).  On adaptive estimation.  The Annals of Statistics, pages 647–671.
  * Bickel et al., (2009) Bickel, P. J., Ritov, Y., and Tsybakov, A. B. (2009).  Simultaneous analysis of Lasso and Dantzig selector.  The Annals of Statistics, 37(4):1705–1732.
  * Chatterjee and Jafarov, (2015) Chatterjee, S. and Jafarov, J. (2015).  Prediction error of cross-validated lasso.  arXiv:1502.06291.
  * (24) Chernozhukov, V., Chetverikov, D., Demirer, M., Duflo, E., Hansen, C. B., Newey, W. K., and Robins, J. M. (2018a).  Double/debiased machine learning for treatment and structural parameters.  The Econometrics Journal, 21(1):C1–C68.
  * Chernozhukov et al., (2013) Chernozhukov, V., Chetverikov, D., and Kato, K. (2013).  Gaussian approximations and multiplier bootstrap for maxima of sums of high-dimensional random vectors.  The Annals of Statistics, 41(6):2786–2819.
  * Chernozhukov et al., (2016) Chernozhukov, V., Escanciano, J. C., Ichimura, H., Newey, W. K., and Robins, J. M. (2016).  Locally robust semiparametric estimation.  arXiv:1608.00033.
  * Chernozhukov et al., (2009) Chernozhukov, V., Fernández-Val, I., and Galichon, A. (2009).  Improving point and interval estimators of monotone functions by rearrangement.  Biometrika, 96(3):559–575.
  * Chernozhukov and Hansen, (2004) Chernozhukov, V. and Hansen, C. B. (2004).  The effects of 401(k) participation on the wealth distribution: An instrumental quantile regression analysis.  Review of Economics and Statistics, 86(3):735–751.
  * Chernozhukov and Hansen, (2005) Chernozhukov, V. and Hansen, C. B. (2005).  An IV model of quantile treatment effects.  Econometrica, 73(1):245–261.
  * (30) Chernozhukov, V., Newey, W. K., and Singh, R. (2018b).  Automatic debiased machine learning of causal and structural effects.  arXiv:1809.05224.
  * (31) Chernozhukov, V., Newey, W. K., and Singh, R. (2018c).  De-biased machine learning of global and local parameters using regularized Riesz representers.  arXiv:1802.08667.
  * Colangelo and Lee, (2020) Colangelo, K. and Lee, Y.-Y. (2020).  Double debiased machine learning nonparametric inference with continuous treatments.  arXiv:2004.03036.
  * Crump et al., (2009) Crump, R. K., Hotz, V. J., Imbens, G. W., and Mitnik, O. A. (2009).  Dealing with limited overlap in estimation of average treatment effects.  Biometrika, 96(1):187–199.
  * Farrell, (2015) Farrell, M. H. (2015).  Robust inference on average treatment effects with possibly more covariates than observations.  Journal of Econometrics, 189(1):1–23.
  * Foster and Syrgkanis, (2019) Foster, D. J. and Syrgkanis, V. (2019).  Orthogonal statistical learning.  arXiv:1901.09036.
  * Frölich, (2007) Frölich, M. (2007).  Nonparametric IV estimation of local average treatment effects with covariates.  Journal of Econometrics, 139(1):35–75.
  * Hartford et al., (2017) Hartford, J., Lewis, G., Leyton-Brown, K., and Taddy, M. (2017).  Deep IV: A flexible approach for counterfactual prediction.  In International Conference on Machine Learning, pages 1414–1423.
  * Hernan and Robins, (2019) Hernan, M. A. and Robins, J. M. (2019).  Causal Inference.  CRC.
  * Imbens and Angrist, (1994) Imbens, G. W. and Angrist, J. D. (1994).  Identification and estimation of local average treatment effects.  Econometrica, 62(2):467–475.
  * Imbens and Rubin, (1997) Imbens, G. W. and Rubin, D. B. (1997).  Estimating outcome distributions for compliers in instrumental variables models.  The Review of Economic Studies, 64(4):555–574.
  * Imbens and Rubin, (2015) Imbens, G. W. and Rubin, D. B. (2015).  Causal Inference in Statistics, Social, and Biomedical Sciences.  Cambridge University Press.
  * Kennedy et al., (2017) Kennedy, E. H., Ma, Z., McHugh, M. D., and Small, D. S. (2017).  Nonparametric methods for doubly robust estimation of continuous treatment effects.  Journal of the Royal Statistical Society. Series B: Statistical Methodology, 79(4):1229.
  * Newey, (1994) Newey, W. K. (1994).  The asymptotic variance of semiparametric estimators.  Econometrica, pages 1349–1382.
  * Newey and McFadden, (1994) Newey, W. K. and McFadden, D. (1994).  Large sample estimation and hypothesis testing.  Handbook of Econometrics, 4:2111–2245.
  * Newey and Powell, (2003) Newey, W. K. and Powell, J. L. (2003).  Instrumental variable estimation of nonparametric models.  Econometrica, 71(5):1565–1578.
  * Ning and Liu, (2017) Ning, Y. and Liu, H. (2017).  A general theory of hypothesis tests and confidence regions for sparse high dimensional models.  The Annals of Statistics, 45(1):158–195.
  * Ogburn et al., (2015) Ogburn, E. L., Rotnitzky, A., and Robins, J. M. (2015).  Doubly robust estimation of the local average treatment effect curve.  Journal of the Royal Statistical Society: Series B (Statistical Methodology), 77(2):373–396.
  * Peters et al., (2017) Peters, J., Janzing, D., and Schölkopf, B. (2017).  Elements of Causal Inference: Foundations and Learning Algorithms.  MIT press.
  * Poterba and Venti, (1994) Poterba, J. M. and Venti, S. F. (1994).  401(k) plans and tax-deferred saving.  In Studies in the Economics of Aging, pages 105–142. University of Chicago Press.
  * Poterba et al., (1995) Poterba, J. M., Venti, S. F., and Wise, D. A. (1995).  Do 401(k) contributions crowd out other personal saving?  Journal of Public Economics, 58(1):1–32.
  * Schick, (1986) Schick, A. (1986).  On asymptotically efficient estimation in semiparametric models.  The Annals of Statistics, 14(3):1139–1151.
  * Singh et al., (2019) Singh, R., Sahani, M., and Gretton, A. (2019).  Kernel instrumental variable regression.  In Advances in Neural Information Processing Systems, pages 4595–4607.
  * Stock and Trebbi, (2003) Stock, J. H. and Trebbi, F. (2003).  Retrospectives: Who invented instrumental variable regression?  Journal of Economic Perspectives, 17(3):177–194.
  * Syrgkanis et al., (2019) Syrgkanis, V., Lei, V., Oprescu, M., Hei, M., Battocchi, K., and Lewis, G. (2019).  Machine learning estimation of heterogeneous treatment effects with instruments.  In Advances in Neural Information Processing Systems, pages 15167–15176.
  * Tan, (2006) Tan, Z. (2006).  Regression and weighting methods for causal inference using instrumental variables.  Journal of the American Statistical Association, 101(476):1607–1618.
  * Van der Laan and Rose, (2011) Van der Laan, M. J. and Rose, S. (2011).  Targeted Learning: Causal Inference for Observational and Experimental Data.  Springer Science & Business Media.
  * Vermeulen and Vansteelandt, (2015) Vermeulen, K. and Vansteelandt, S. (2015).  Bias-reduced doubly robust estimation.  Journal of the American Statistical Association, 110(511):1024–1036.
  * Wang and Tchetgen, (2018) Wang, L. and Tchetgen, E. T. (2018).  Bounded, efficient and multiply robust estimation of average treatment effects using instrumental variables.  Journal of the Royal Statistical Society: Series B (Statistical Methodology), 80(3):531.
  * Zubizarreta, (2015) Zubizarreta, J. R. (2015).  Stable weights that balance covariates for estimation with incomplete outcome data.  Journal of the American Statistical Association, 110(511):910–922.

##  Appendix A Intuition and Notation

###  A.1 Examples of Instrumental Variables

To fix ideas, we provide examples with continuous outcome $Y$, binary
treatment $D$, and binary instrument $Z$.

Charter school admission by lottery ($Z$) only influences student test scores
($Y$) via actually attending the charter school ($D$), identifying the
counterfactual effect of the charter school on test scores even if there is
selection bias in which students choose to accept an offer of admission
(Angrist et al.,, 2010, 2013). However, the analyst can only learn the
treatment effect on the subpopulation of complier students: those who would
attend the charter school if they won the lottery and who would not attend the
charter school if they lost the lottery.

As another example, the randomized recommendation to enroll on a digital
platform ($Z$) only influences platform engagement ($Y$) via actual enrollment
($D$), identifying the counterfactual effect of enrollment on engagement even
in the scenario of imperfect compliance (Syrgkanis et al.,, 2019). However,
the analyst can only learn the treatment effect on the subpopulation of
complier consumers: those who would enroll if recommended and who would not
enroll if not recommended.

###  A.2 Notation Glossary

Let $\mathbf{W}=(Y,D,Z,\mathbf{X}^{\prime})^{\prime}$ concatenate the random
variables. $Y\in\mathcal{Y}\subset\mathbb{R}$ is the continuous outcome,
$D\in\\{0,1\\}$ is the binary treatment, $Z\in\\{0,1\\}$ is the binary
instrumental variable, and
$\mathbf{X}\in\boldsymbol{\mathcal{X}}\subset{\mathbb{R}^{dim(\mathbf{X})}}$
is the covariate. We observe $n$ i.i.d. observations
$\\{\mathbf{W}_{i}\\}_{i=1}^{n}$. Where possible, we suppress index $i$ to
lighten notation.

Following the notation of Angrist et al., (1996), we denote by $Y^{(z,d)}$ the
potential outcome under the intervention $Z=z$ and $D=d$. Due to Assumption 1,
we can simplify notation: $Y^{(d)}=Y^{(1,d)}=Y^{(0,d)}$. We denote by
$D^{(z)}$ the potential treatment under the intervention $Z=z$. Compliers are
the subpopulation for whom $D^{(1)}>D^{(0)}$.

Using the notation of Chernozhukov et al., 2018b , we denote the regression of
random vector $\mathbf{V}=(V_{1},...,,V_{J})^{\prime}$ conditional on
$(Z,\mathbf{X})$ as

$\gamma_{0}(Z,\mathbf{X})=\begin{bmatrix}\gamma_{0}^{V_{1}}(Z,\mathbf{X})\\\
...\\\ \gamma_{0}^{V_{J}}(Z,\mathbf{X})\\\
\end{bmatrix}=\mathbb{E}[\mathbf{V}|Z,\mathbf{X}]$

where $\gamma_{0}^{V_{j}}(Z,\mathbf{X})=\mathbb{E}[V_{j}|Z,\mathbf{X}]$. The
random vector $\mathbf{V}$ is observable and depends on the complier parameter
of interest; we specify its components for LATE, complier characteristics, and
counterfactual outcome distributions in Theorem 1.

We denote the propensity score
$\pi_{0}(\mathbf{x})=\mathbb{P}(Z=1|\mathbf{X}=\mathbf{x})$. We denote the
classic Horvitz-Thompson weight with

$\alpha_{0}(z,\mathbf{x})=\frac{z}{\pi_{0}(\mathbf{x})}-\frac{1-z}{1-\pi_{0}(\mathbf{x})}=\frac{z-\pi_{0}(\mathbf{x})}{\pi_{0}(\mathbf{x})[1-\pi_{0}(\mathbf{x})]}$

We denote by $|\cdot|_{q}$ the $\ell_{q}$ norm of a vector. We denote by
$\|\cdot\|$ the $\mathcal{L}_{2}$ norm of a random variable $V_{j}$, i.e.
$\|V_{j}\|=\sqrt{\mathbb{E}[V_{j}]^{2}}$. For random vector
$\mathbf{V}=(V_{1},...,V_{J})^{\prime}$, we slightly abuse notation by writing

$\|\mathbf{V}\|=\begin{bmatrix}\|V_{1}\|\\\ \|V_{2}\|\\\ \vdots\\\
\|V_{J}\|\end{bmatrix},\quad\mathbb{E}[\mathbf{V}]^{2}=\|\mathbf{V}\|^{2}=\begin{bmatrix}\|V_{1}\|^{2}\\\
\|V_{2}\|^{2}\\\ \vdots\\\ \|V_{J}\|^{2}\end{bmatrix}$

Likewise, we write the element-wise absolute value as

$|\mathbf{V}|=\begin{bmatrix}|V_{1}|\\\ |V_{2}|\\\ \vdots\\\
|V_{J}|\end{bmatrix}$

Consider a causal parameter $\boldsymbol{\theta}_{0}\in\boldsymbol{\Theta}$,
where $\boldsymbol{\Theta}$ is some compact parameter space. It is implicitly
defined by moment function $m$.

$\mathbb{E}[m(\mathbf{W},\gamma_{0},\boldsymbol{\theta})]=0\text{ iff
}\boldsymbol{\theta}=\boldsymbol{\theta}_{0}$

We denote the doubly robust moment function for $\boldsymbol{\theta}_{0}$ by

$\displaystyle\mathbb{E}[\psi(\mathbf{W},\gamma_{0},\alpha_{0},\boldsymbol{\theta})]=0\text{
iff }\boldsymbol{\theta}=\boldsymbol{\theta}_{0}$
$\displaystyle\psi(\mathbf{w},\gamma,\alpha,\boldsymbol{\theta})=m(\mathbf{w},\gamma,\boldsymbol{\theta})+\phi(\mathbf{w},\gamma,\alpha,\boldsymbol{\theta})$

$\phi$ is called the de-biasing term.

In sample splitting, we partition the sample into $L$ folds
$\\{I_{\ell}\\}_{\ell=1:L}$, each with $n_{\ell}=n/L$ observations. We denote
by $(\hat{\gamma}_{-\ell},\hat{\alpha}_{-\ell})$ the estimates from
observations not in $I_{\ell}$. We denote by $b(z,\mathbf{x})$ a
$p$-dimensional dictionary of basis functions.

The population regularized balancing weight parameter $\boldsymbol{\rho}_{L}$
is the solution to

$\boldsymbol{\rho}_{L}=\operatorname*{\arg\\!\min}_{\boldsymbol{\rho}}\mathbb{E}[\alpha_{0}(Z,\mathbf{X})-\boldsymbol{\rho}^{\prime}b(Z,\mathbf{X})]^{2}+2\lambda_{L}|\boldsymbol{\rho}|_{1}$

where $\lambda_{L}=\sqrt{\frac{\ln p}{n}}$ is the theoretical regularization
parameter.

The sample balancing weight parameter $\hat{\boldsymbol{\rho}}_{-\ell}$
estimated from $I_{-\ell}$ is the solution to

$\hat{\boldsymbol{\rho}}_{-\ell}=\operatorname*{\arg\\!\min}_{\boldsymbol{\rho}}\boldsymbol{\rho}^{\prime}\hat{\mathbf{G}}_{-\ell}\boldsymbol{\rho}-2\boldsymbol{\rho}^{\prime}\hat{\mathbf{M}}_{-\ell}+2\lambda_{n}|\boldsymbol{\rho}|_{1}$

where

$\hat{\mathbf{G}}_{-\ell}=\frac{1}{n-n_{\ell}}\sum_{i\in
I_{-\ell}}b(Z_{i},\mathbf{X}_{i})b(Z_{i},\mathbf{X}_{i})^{\prime},\quad\hat{\mathbf{M}}_{-\ell}=\frac{1}{n-n_{\ell}}\sum_{i\in
I_{-\ell}}b(1,\mathbf{X}_{i})-b(0,\mathbf{X}_{i})$

and $\lambda_{n}$ is a data-driven regularization parameter. We denote
$\hat{\alpha}_{-\ell}(z,\mathbf{x})=b(z,\mathbf{x})^{\prime}\hat{\boldsymbol{\rho}}_{-\ell}$.

In estimating a simultaneous confidence band, we denote the $d$-dimensional
grid $\mathcal{U}\subset\mathcal{Y}$. $\hat{\mathbf{C}}$ is the estimator of
the asymptotic variance $\mathbf{C}$ of $\hat{\boldsymbol{\theta}}$. Let
$\hat{\mathbf{S}}=diag(\hat{\mathbf{C}})$ and $\mathbf{S}=diag(\mathbf{C})$
collect the diagonal elements of these matrices.

The remaining symbols are concisely defined in the assumptions and theorems of
Section 6

##  Appendix B Forbidden Regression and Parametric Estimation

We compare our semi-parametric approach with standard parametric approaches to
estimation using instrumental variables. Specifically, we relate Auto-DML to
the so-called forbidden regression discussed in Angrist and Pischke, (2008,
Section 4.6.1). We focus our discussion on LATE as the target parameter
$\theta_{0}$. We show that specifications which may be inconsistent in the
parametric framework are indeed consistent in the semi-parametric framework.
By considering an Auto-DML approach to LATE, we may unlock the forbidden
regressions.

###  B.1 2SLS

To simplify the exposition, we consider a model without covariates
$\mathbf{X}$. The linear (parametric) specification of Angrist and Pischke,
(2008, Section 4.6.1) is

$\displaystyle D$
$\displaystyle=\pi_{0}Z+\epsilon_{1},\quad\mathbb{E}[Z\epsilon_{1}]=0,\quad\mathbb{E}[\epsilon_{1}]=0$
$\displaystyle Y$
$\displaystyle=\theta_{0}D+\epsilon_{2},\quad\mathbb{E}[Z\epsilon_{2}]=0,\quad\mathbb{E}[\epsilon_{2}]=0$

Economists refer to the former equation as the first-stage and the latter as
the second-stage. Substituting the former into the latter yields an equation
called the reduced-form. Note that the first-stage and second-stage of 2SLS
are different than stage 1 and stage 2 of DML; we maintain the semantic
distinction throughout. $\theta_{0}$ is LATE. By standard projection geometry

$\displaystyle\bar{D}$ $\displaystyle:=P_{Z}D=\pi_{0}Z$ $\displaystyle\bar{Y}$
$\displaystyle:=P_{Z}Y=P_{Z}(\theta_{0}D)=\theta_{0}\bar{D}$

where $P_{Z}$ denotes the projection onto instrument $Z$. It follows that
$\theta_{0}$ can be obtained by projecting $\bar{Y}$ on $\bar{D}$. Since
$Y=\bar{Y}+\bar{Y}^{\perp}$ where $\bar{Y},\bar{D}\in span\\{Z\\}$ and
$Y^{\perp}\in span\\{Z\\}^{\perp}$, $\theta_{0}$ can also be obtained by
projecting $Y$ on $\bar{D}$. This procedure is the widely-used two-stage least
squares algorithm (2SLS).

2SLS is robust to functional form mis-specification in two senses. First, if
the model is saturated–i.e. if $(D,Z,\mathbf{X})$ have discrete support–then
the linear specification is w.l.o.g. Second, even if the first-stage
relationship is actually nonlinear, projection geometry ensures consistent
estimation of a well-defined best linear approximation of LATE.666In the
absence of covariates $\mathbf{X}$ or under the assumption of homogeneous
treatment effects, the estimand of 2SLS is LATE. In the presence of covariates
and heterogeneous treatment effects, the estimand of 2SLS is not LATE but
rather a variance-reweighting of the expanded Wald formula (Angrist and
Pischke,, 2008, Theorem 4.5.1). By contrast, even in the presence of
heterogeneous treatment effects, the estimand of Auto-DML is LATE.

###  B.2 Forbidden Regression

A forbidden regression is a variant of 2SLS in which the analyst attempts to
guess the functional form of the nonlinear first-stage relationship. For
example, for binary treatment $D$ one may consider the probit specification

$\displaystyle D$
$\displaystyle=\Phi(\tilde{\pi}_{0}Z)+\tilde{\epsilon}_{1},\quad\mathbb{E}[\tilde{\epsilon}_{1}|Z]=0$
$\displaystyle Y$
$\displaystyle=\theta_{0}D+\tilde{\epsilon}_{2},\quad\mathbb{E}[\tilde{\epsilon}_{2}|Z]=0$

Note that the moment restrictions are stronger than in 2SLS: conditional
moment restrictions implying correct specification of the conditional means,
rather than unconditional moment restrictions implying orthogonal noise.
Substituting the former into the latter yields a reduced-form.

Under correct specification, $\theta_{0}$ may be consistently estimated using
the first-stage and reduced-form, similar to 2SLS. However, this approach is
not robust to functional form mis-specification. Under mis-specification,
projection geometry no longer applies and the resulting estimator for
$\theta_{0}$ is inconsistent.

The concept of a forbidden regression highlights the drawbacks of using a
parametric approach to LATE, i.e. a finite-dimensional nuisance parameter
($\pi_{0}$ or $\tilde{\pi}_{0}$) and finite-dimensional causal parameter
$\theta_{0}$. Any nonlinear parametric approach is highly sensitive to mis-
specification. Indeed, to guarantee consistency under mis-specification, the
analyst must revert back to the linear specification and projection geometry
of 2SLS. Forfeiting data fit entails forfeiting statistical precision. Another
important drawback is that 2SLS consistently estimates an approximation to
LATE in a highly restricted class.

###  B.3 Auto-DML

By contrast, the Auto-DML approach is semi-parametric, with infinite-
dimensional nuisance parameters $(\gamma_{0},\alpha_{0})$ and finite-
dimensional causal parameter $\theta_{0}$. In Theorem 1, we show that the
Auto-DML estimator for LATE accommodates black-box machine learning of

$\gamma_{0}(Z,\mathbf{X}):=\mathbb{E}[\mathbf{V}|Z,\mathbf{X}]=\mathbb{E}\left[\begin{bmatrix}Y\\\
D\\\ \end{bmatrix}\bigg{|}Z,\mathbf{X}\right]$

These conditional expectations correspond to the reduced-form and first-stage
estimating equations of 2SLS, respectively. Equivalently, they correspond to
the numerator and denominator of the expanded Wald formula in Section 4.2. In
particular, the Auto-DML estimator accommodates the forbidden regression: an
analyst may specify probit first-stage, linear second-stage, and the resulting
reduced-form.

By using a semi-parametric approach to LATE, any nonlinear approach to
estimating the first-stage and the reduced-form is permitted. Mis-
specification is no longer a concern; rather, the approximation of
$\gamma_{0}$ must be of sufficiently high quality as stated in Hypothesis 6.
By Theorem 2, such an estimator is consistent, asymptotically normal, and
semi-parametrically efficient. Auto-DML consistently estimates an
approximation to LATE in a much broader class, namely the class in which
$\mathbb{E}[\gamma_{0}(Z,\mathbf{X})]^{2}<\infty$.

##  Appendix C Identification

We review the derivation of the classic Horvitz-Thompson weight, relate Auto-
DML to $\kappa$-weighting, and prove a general identification result. We then
specialize this result to LATE, complier characteristics, and counterfactual
outcome distributions.

######  Proposition 3.

$\alpha_{0}(z,\mathbf{x})$ is the Riesz representer to the continuous linear
functional
$\gamma\mapsto\mathbb{E}[\gamma(1,\mathbf{X})-\gamma(0,\mathbf{X})]$, i.e.

$\mathbb{E}[\gamma(1,\mathbf{X})-\gamma(0,\mathbf{X})]=\mathbb{E}[\alpha_{0}(Z,\mathbf{X})\gamma(Z,\mathbf{X})],\quad\forall\gamma\text{
s.t. }\mathbb{E}[\gamma(Z,\mathbf{X})]^{2}<\infty$

###### Proof.

Observe that

$\displaystyle\mathbb{E}\left[\gamma(Z,\mathbf{X})\frac{Z}{\pi_{0}(\mathbf{X})}\bigg{|}\mathbf{X}\right]$
$\displaystyle=\mathbb{E}\left[\gamma(Z,\mathbf{X})\frac{1}{\pi_{0}(\mathbf{X})}\bigg{|}Z=1,\mathbf{X}\right]\mathbb{P}(Z=1|\mathbf{X})$
$\displaystyle=\mathbb{E}\left[\gamma(Z,\mathbf{X})\frac{1}{\pi_{0}(\mathbf{X})}\bigg{|}Z=1,\mathbf{X}\right]\pi_{0}(\mathbf{X})$
$\displaystyle=\gamma(1,\mathbf{X})$

and likewise

$\mathbb{E}\left[\gamma(Z,\mathbf{X})\frac{1-Z}{1-\pi_{0}(\mathbf{X})}\bigg{|}\mathbf{X}\right]=\gamma(0,\mathbf{X})$

In summary, we can write

$\displaystyle\mathbb{E}[\gamma(1,\mathbf{X})-\gamma(0,\mathbf{X})]$
$\displaystyle=\int\\{\gamma(1,\mathbf{x})-\gamma(0,\mathbf{x})\\}d\mathbb{P}(\mathbf{x})$
$\displaystyle=\int\bigg{\\{}\mathbb{E}\left[\gamma(Z,\mathbf{X})\frac{Z}{\pi_{0}(\mathbf{X})}\bigg{|}\mathbf{X}=\mathbf{x}\right]$
$\displaystyle\quad-\mathbb{E}\left[\gamma(Z,x)\frac{1-Z}{1-\pi_{0}(\mathbf{X})}\bigg{|}\mathbf{X}=\mathbf{x}\right]\bigg{\\}}d\mathbb{P}(\mathbf{x})$
$\displaystyle=\mathbb{E}\left[\gamma(Z,\mathbf{X})\frac{Z}{\pi_{0}(\mathbf{X})}\right]-\mathbb{E}\left[\gamma(Z,\mathbf{X})\frac{1-Z}{1-\pi_{0}(\mathbf{X})}\right]$

∎

######  Definition 2.

Define

$\displaystyle\kappa^{(0)}(\mathbf{w})$
$\displaystyle=(1-d)\frac{(1-z)-(1-\pi_{0}(\mathbf{x}))}{[1-\pi_{0}(\mathbf{x})]\pi_{0}(\mathbf{x})}$
$\displaystyle\kappa^{(1)}(\mathbf{w})$
$\displaystyle=d\frac{z-\pi_{0}(\mathbf{x})}{[1-\pi_{0}(\mathbf{x})]\pi_{0}(\mathbf{x})}$
$\displaystyle\kappa(\mathbf{w})$
$\displaystyle=[1-\pi_{0}(\mathbf{x})]\kappa^{(0)}(\mathbf{w})+\pi_{0}(\mathbf{x})\kappa^{(1)}(\mathbf{w})$

These are the $\kappa$-weights introduced in Abadie, (2003).

######  Proposition 4.

The $\kappa$-weights can be rewritten as

$\displaystyle\kappa^{(0)}(\mathbf{w})$
$\displaystyle=\alpha_{0}(z,\mathbf{x})[d-1]$
$\displaystyle\kappa^{(1)}(\mathbf{w})$
$\displaystyle=\alpha_{0}(z,\mathbf{x})d$ $\displaystyle\kappa(\mathbf{w})$
$\displaystyle=\alpha_{0}(z,\mathbf{x})[d-1+\pi_{0}(\mathbf{x})]$

###### Proof.

$\alpha_{0}(z,\mathbf{x})=\frac{z}{\pi_{0}(\mathbf{x})}-\frac{1-z}{1-\pi_{0}(\mathbf{x})}=\frac{z-\pi_{0}(\mathbf{x})}{\pi_{0}(\mathbf{x})[1-\pi_{0}(\mathbf{x})]}$
∎

######  Theorem 4.

Suppose Assumption 1 holds. Let $g(y,d,\mathbf{x},\boldsymbol{\theta})$ be a
measurable, real-valued function s.t.
$\mathbb{E}[g(Y,D,\mathbf{X},\boldsymbol{\theta})]^{2}<\infty$ for all
$\boldsymbol{\theta}\in\boldsymbol{\Theta}$.

  1. 1.

If $\boldsymbol{\theta}_{0}$ is defined by the moment condition
$\mathbb{E}[g(Y^{(0)},\mathbf{X},\boldsymbol{\theta}_{0}))|D^{(1)}>D^{(0)}]=0$,
let

$v(\mathbf{w},\boldsymbol{\theta})=[d-1]g(y,\mathbf{x},\boldsymbol{\theta})$

  2. 2.

If $\boldsymbol{\theta}_{0}$ is defined by the moment condition
$\mathbb{E}[g(Y^{(1)},\mathbf{X},\boldsymbol{\theta}_{0}))|D^{(1)}>D^{(0)}]=0$,
let

$v(\mathbf{w},\boldsymbol{\theta})=d\cdot g(y,\mathbf{x},\boldsymbol{\theta})$

  3. 3.

If $\boldsymbol{\theta}_{0}$ is defined by the moment condition
$\mathbb{E}[g(Y,D,\mathbf{X},\boldsymbol{\theta}_{0}))|D^{(1)}>D^{(0)}]=0$,
let

$v(\mathbf{w},\boldsymbol{\theta})=[d-1+\pi_{0}(\mathbf{x})]g(y,d,\mathbf{x},\boldsymbol{\theta})$

Then the doubly robust moment function for $\boldsymbol{\theta}_{0}$ is of the
form

$\displaystyle\psi(\mathbf{w},\gamma,\alpha,\boldsymbol{\theta})$
$\displaystyle=\tilde{m}(\mathbf{w},\gamma,\boldsymbol{\theta})+\phi(\mathbf{w},\gamma,\alpha,\boldsymbol{\theta})$
$\displaystyle\tilde{m}(\mathbf{w},\gamma,\boldsymbol{\theta})$
$\displaystyle=\gamma(1,\mathbf{x},\boldsymbol{\theta})-\gamma(0,\mathbf{x},\boldsymbol{\theta})$
$\displaystyle\phi(\mathbf{w},\gamma,\alpha,\boldsymbol{\theta})$
$\displaystyle=\alpha(z,\mathbf{x})[v(\mathbf{w},\boldsymbol{\theta})-\gamma(z,\mathbf{x})]$

where
$\gamma_{0}(z,\mathbf{x},\boldsymbol{\theta}):=\mathbb{E}[v(\mathbf{W},\boldsymbol{\theta})|z,\mathbf{x}]$

###### Proof.

Consider the first case. Under Assumption 1, we can appeal to Abadie, (2003,
Theorem 3.1).

$0=\mathbb{E}[g(Y^{(0)},\mathbf{X},\boldsymbol{\theta}_{0}))|D^{(1)}>D^{(0)}]=\dfrac{\mathbb{E}[\kappa^{(0)}(\mathbf{W})g(Y^{(0)},\mathbf{X},\boldsymbol{\theta}_{0}))]}{\mathbb{P}(D^{(1)}>D^{(0)})}$

Hence

$\displaystyle 0$
$\displaystyle=\mathbb{E}[\kappa^{(0)}(\mathbf{W})g(Y^{(0)},\mathbf{X},\boldsymbol{\theta}_{0}))]$
$\displaystyle=\mathbb{E}[\alpha_{0}(Z,\mathbf{X})[D-1]g(Y^{(0)},\mathbf{X},\boldsymbol{\theta}_{0}))]$
$\displaystyle=\mathbb{E}[\alpha_{0}(Z,\mathbf{X})v(\mathbf{W},\boldsymbol{\theta}_{0})]$
$\displaystyle=\mathbb{E}[\alpha_{0}(Z,\mathbf{X})\gamma_{0}(Z,\mathbf{X},\boldsymbol{\theta}_{0})]$
$\displaystyle=\mathbb{E}[\gamma_{0}(1,\mathbf{X},\boldsymbol{\theta}_{0})-\gamma_{0}(0,\mathbf{X},\boldsymbol{\theta}_{0})]$

appealing to Assumption 1, Proposition 4, and the fact that $\alpha_{0}$ is
the Riesz representer for
$\gamma\mapsto\mathbb{E}[\gamma(1,\mathbf{X},\boldsymbol{\theta}_{0})-\gamma(0,\mathbf{X},\boldsymbol{\theta}_{0})]$.
Likewise for the second and third cases. ∎

###### Proof of Theorem 1.

Suppose we can decompose
$v(\mathbf{w},\boldsymbol{\theta})=h(\mathbf{w},\boldsymbol{\theta})+a(\boldsymbol{\theta})$
for some function $a(\cdot)$ that does not depend on data. Then we can replace
$v(\mathbf{w},\boldsymbol{\theta})$ with $h(\mathbf{w},\boldsymbol{\theta})$
without changing $\tilde{m}$ and $\phi$. This is because
$\gamma^{v}(z,\mathbf{x},\boldsymbol{\theta})=\gamma^{h}(z,\mathbf{x},\boldsymbol{\theta})+a(\boldsymbol{\theta})$
and hence
$v(\mathbf{w},\boldsymbol{\theta})-\gamma^{v}(z,\mathbf{x})=h(\mathbf{w},\boldsymbol{\theta})-\gamma^{h}(z,\mathbf{x})$.
Whenever we use this reasoning, we write
$v(\mathbf{w},\boldsymbol{\theta})\propto h(\mathbf{w},\boldsymbol{\theta})$.

  1. 1.

For LATE we can write $\theta_{0}=\delta_{0}-\beta_{0}$, where $\delta_{0}$ is
defined by the moment condition
$\mathbb{E}[Y^{(1)}-\delta_{0}|D^{(1)}>D^{(0)}]=0$ and $\beta_{0}$ is defined
by the moment condition $\mathbb{E}[Y^{(0)}-\beta_{0}|D^{(1)}>D^{(0)}]=0$.
Applying Case 2 of Theorem 4 to $\delta_{0}$, we have
$v(\mathbf{w},\delta)=d\cdot(y-\delta)$. Applying Case 1 of Theorem 4 to
$\beta_{0}$, we have $v(\mathbf{w},\beta)=(d-1)\cdot(y-\beta)\propto(d-1)\cdot
y-d\cdot\beta$. Writing $\theta=\delta-\beta$, the moment function for
$\theta_{0}$ can thus be derived with
$v(\mathbf{w},\theta)=v(\mathbf{w},\delta)-v(\mathbf{w},\beta)=y-d\cdot\theta$.
Note that this expression decomposes into $\mathbf{V}=(Y,D)^{\prime}$ and
$A(\theta)=\begin{bmatrix}1&-\boldsymbol{\theta}\end{bmatrix}$ in Theorem 1.

  2. 2.

For average complier characteristics, $\boldsymbol{\theta}_{0}$ is defined by
the moment condition
$\mathbb{E}[f(\mathbf{X})-\boldsymbol{\theta}_{0}|D^{(1)}>D^{(0)}]=0$.
Applying Case 2 of Theorem 4 setting
$g(Y^{(1)},\mathbf{X},\boldsymbol{\theta}_{0})=f(\mathbf{X})-\boldsymbol{\theta}_{0}$,
we have
$v(\mathbf{w},\boldsymbol{\theta})=d\cdot(f(\mathbf{x})-\boldsymbol{\theta}_{0})$.
This expression decomposes into
$\mathbf{V}=(Df(\mathbf{X})^{\prime},D)^{\prime}$ and
$A(\boldsymbol{\theta})=\begin{bmatrix}1&-\boldsymbol{\theta}\end{bmatrix}$ in
Theorem 1.

  3. 3.

For complier distribution of $Y^{(0)}$, $\beta^{\bar{y}}_{0}$ is defined by
the moment condition
$\mathbb{E}[1_{Y^{(0)}\leq\bar{y}}-\beta^{\bar{y}}_{0}|D^{(1)}>D^{(0)}]=0$.
Applying Case 1 of Theorem 4 to $\beta^{\bar{y}}_{0}$, we have
$v(\mathbf{w},\beta^{\bar{y}})=(d-1)\cdot(1_{y\leq{\bar{y}}}-\beta^{\bar{y}})\propto(d-1)\cdot
1_{y\leq{\bar{y}}}-d\cdot\beta^{\bar{y}}$. For complier distribution of
$Y^{(1)}$, $\delta^{\bar{y}}_{0}$ is defined by the moment condition
$\mathbb{E}[1_{Y^{(1)}\leq{\bar{y}}}-\delta^{\bar{y}}_{0}|D^{(1)}>D^{(0)}]=0$.
Applying Case 2 of Theorem 4 to $\delta_{0}$, we have
$v(\mathbf{w},\delta^{\bar{y}})=d\cdot(1_{y\leq{\bar{y}}}-\delta^{\bar{y}})$.
Concatenating $v(\mathbf{w},\beta^{\bar{y}})$ and
$v(\mathbf{w},\delta^{\bar{y}})$, we arrive at the decomposition in Theorem 1.

∎

##  Appendix D Tuning

Algorithm 2 takes as given the value of regularization parameter
$\lambda_{n}$. For practical use, we provide an iterative tuning procedure to
empirically determine $\lambda_{n}$. This is precisely the tuning procedure of
Chernozhukov et al., 2018b , adapted from Chernozhukov et al., 2018c . Due to
its iterative nature, the tuning procedure is most clearly stated as a
replacement for Algorithm 2.

Recall that the inputs to Algorithm 2 are observations in $I_{-\ell}$, i.e.
excluding fold $\ell$. The analyst must also specify the $p$-dimensional
dictionary $b$. For notational convenience, we assume $b$ includes the
intercept in its first component: $b_{1}(z,\mathbf{x})=1$. In this tuning
procedure, the analyst must further specify a low-dimensional sub-dictionary
$b^{\text{low}}$ of $b$. As in Algorithm 2, the output of the tuning procedure
is $\hat{\alpha}_{-\ell}$, an estimator of the balancing weight trained only
on observations in $I_{-\ell}$.

The tuning procedure is as follows.

######  Algorithm 5 (Regularized balancing weight with tuning).

For observations in $I_{-\ell}$

  1. 1.

Initialize $\hat{\boldsymbol{\rho}}_{-\ell}$ using $b^{\text{low}}$

$\displaystyle\hat{\mathbf{G}}^{\text{low}}_{-\ell}$
$\displaystyle=\frac{1}{n-n_{\ell}}\sum_{i\in
I_{-\ell}}b^{\text{low}}(Z_{i},\mathbf{X}_{i})b^{\text{low}}(Z_{i},\mathbf{X}_{i})^{\prime}$
$\displaystyle\hat{\mathbf{M}}^{\text{low}}_{-\ell}$
$\displaystyle=\frac{1}{n-n_{\ell}}\sum_{i\in
I_{-\ell}}b^{\text{low}}(1,\mathbf{X}_{i})-b^{\text{low}}(0,\mathbf{X}_{i})$
$\displaystyle\hat{\boldsymbol{\rho}}_{-\ell}$
$\displaystyle=\begin{bmatrix}\left(\hat{\mathbf{G}}^{\text{low}}_{-\ell}\right)^{-1}\hat{\mathbf{M}}^{\text{low}}_{-\ell}\\\
0\end{bmatrix}$

  2. 2.

Calculate moments

$\displaystyle\hat{\mathbf{G}}_{-\ell}$
$\displaystyle=\frac{1}{n-n_{\ell}}\sum_{i\in
I_{-\ell}}b(Z_{i},\mathbf{X}_{i})b(Z_{i},\mathbf{X}_{i})^{\prime}$
$\displaystyle\hat{\mathbf{M}}_{-\ell}$
$\displaystyle=\frac{1}{n-n_{\ell}}\sum_{i\in
I_{-\ell}}b(1,\mathbf{X}_{i})-b(0,\mathbf{X}_{i})$

  3. 3.

While $\hat{\boldsymbol{\rho}}_{-\ell}$ has not converged

    1. (a)

Update normalization

$\hat{\mathbf{D}}_{-\ell}=\sqrt{\frac{1}{n-n_{\ell}}\sum_{i\in
I_{-\ell}}[b(Z_{i},\mathbf{X}_{i})b(Z_{i},\mathbf{X}_{i})^{\prime}\hat{\boldsymbol{\rho}}_{-\ell}-(b(1,\mathbf{X}_{i})-b(0,\mathbf{X}_{i}))]^{2}}$

    2. (b)

Update $(\lambda_{n},\hat{\boldsymbol{\rho}}_{-\ell})$

$\displaystyle\lambda_{n}$
$\displaystyle=\frac{c_{1}}{\sqrt{n-n_{\ell}}}\Phi^{-1}\left(1-\frac{c_{2}}{2p}\right)$
$\displaystyle\hat{\boldsymbol{\rho}}_{-\ell}$
$\displaystyle=\operatorname*{\arg\\!\min}_{\boldsymbol{\rho}}\boldsymbol{\rho}^{\prime}\hat{\mathbf{G}}_{-\ell}\boldsymbol{\rho}-2\boldsymbol{\rho}^{\prime}\hat{\mathbf{M}}_{-\ell}+2\lambda_{n}c_{3}|\hat{D}_{-\ell,11}\cdot\rho_{1}|+2\lambda_{n}\sum_{j=2}^{p}|\hat{D}_{-\ell,jj}\cdot\rho_{j}|$

where $\rho_{j}$ is the $j$-th coordinate of $\boldsymbol{\rho}$ and
$\hat{D}_{-\ell,jj}$ is the $j$-th diagonal entry of
$\hat{\mathbf{D}}_{-\ell}$

  4. 4.

Set
$\hat{\alpha}_{-\ell}(z,\mathbf{x})=b(z,\mathbf{x})^{\prime}\hat{\boldsymbol{\rho}}_{-\ell}$

In step 1, $b^{low}$ is sufficiently low-dimensional that
$\hat{\mathbf{G}}_{-\ell}^{\text{low}}$ is invertible. In practice, we take
$dim(b^{low})=dim(b)/40$.

In step 3, $(c_{1},c_{2},c_{3})$ are hyper-parameters taken as $(1,0.1,0.1)$
in practice. We implement the optimization via generalized coordinate descent
with soft-thresholding. See Chernozhukov et al., 2018b  for a detailed
derivation of this soft-thresholding routine. In the optimization, we
initialize at the previous value of $\hat{\boldsymbol{\rho}}_{-\ell}$. For
numerical stability, we use $\hat{\mathbf{D}}_{-\ell}+0.2\mathbf{I}$ instead
of $\hat{\mathbf{D}}_{-\ell}$, and we cap the maximum number of iterations at
10.

We justify Algorithm 5 in the same manner as Chernozhukov et al., 2018c (,
Section 5.1). Specifically, we appeal to Belloni and Chernozhukov, (2013,
Theorem 8) for the homoscedastic case and Belloni et al., (2012, Theorem 1)
for the heteroscedastic case.

##  Appendix E Finite Sample and Asymptotic Properties

###  E.1 Balancing and Orthogonality

###### Proof of Proposition 1.

The Lasso FOC gives
$|\hat{\mathbf{M}}_{-\ell}-\hat{\mathbf{G}}_{-\ell}\hat{\boldsymbol{\rho}}_{-\ell}|_{\infty}\leq\lambda_{n}$.
∎

###### Proof of Proposition 2.

The derivatives are

$\displaystyle\frac{\partial}{\partial\boldsymbol{\beta}}\psi(\mathbf{W}_{i},\boldsymbol{\beta},\boldsymbol{\rho},\theta_{0})$
$\displaystyle=\\{b(1,\mathbf{X}_{i})-b(0,\mathbf{X}_{i})-\boldsymbol{\rho}^{\prime}b(Z_{i},\mathbf{X}_{i})b(Z_{i},\mathbf{X}_{i})\\}A(\theta_{0})$
$\displaystyle\frac{\partial}{\partial\boldsymbol{\rho}}\psi(\mathbf{W}_{i},\boldsymbol{\beta},\boldsymbol{\rho},\theta_{0})$
$\displaystyle=b(Z_{i},\mathbf{X}_{i})A(\theta_{0})\\{\mathbf{V}_{i}-\boldsymbol{\beta}^{\prime}b(Z_{i},\mathbf{X}_{i})\\}$

We analyze the former expression; the argument for the latter is analogous,
replacing $b\mapsto b(1,\mathbf{x})-b(0,\mathbf{x})$ with $b\mapsto
b(z,\mathbf{x})\mathbf{v}^{\prime}$. In the former,

$\frac{1}{n-n_{\ell}}\sum_{i\in
I_{-\ell}}\frac{\partial}{\partial\boldsymbol{\beta}}\psi(\mathbf{W}_{i},\hat{\boldsymbol{\beta}}_{-\ell},\hat{\boldsymbol{\rho}}_{-\ell},\theta_{0})=\\{\hat{\mathbf{M}}_{-\ell}-\hat{\mathbf{G}}_{-\ell}\hat{\rho}_{-\ell}\\}A(\theta_{0})$

Finally appeal to the Lasso FOC. ∎

###  E.2 Lemmas

######  Definition 3.

$\displaystyle\mathbf{G}$
$\displaystyle=\mathbb{E}[b(Z,\mathbf{X})b(Z,\mathbf{X})^{\prime}]$
$\displaystyle\mathbf{M}$
$\displaystyle=\mathbb{E}[m(\mathbf{W},b,\boldsymbol{\theta}_{0})]$

######  Proposition 5.

Under Assumption 2,
$|\hat{\mathbf{G}}-\mathbf{G}|_{\infty}=O_{p}\left(\sqrt{\frac{\ln
p}{n}}\right)$

###### Proof.

Chernozhukov et al., 2018b (, Lemma C1) ∎

######  Proposition 6.

Under Assumptions 1 and 2,
$|\hat{\mathbf{M}}-\mathbf{M}|_{\infty}=O_{p}\left(\sqrt{\frac{\ln
p}{n}}\right)$

###### Proof.

Chernozhukov et al., 2018b (, Lemma 4) ∎

Denote
$\tilde{m}(\mathbf{w},\gamma):=\gamma(1,\mathbf{x})-\gamma(0,\mathbf{x})$.

######  Proposition 7.

Under Assumptions 1 and 7

  1. 1.

$\mathbb{E}[\tilde{m}(\mathbf{W},\gamma_{0})^{2}]<\infty$

  2. 2.

$\mathbb{E}[\tilde{m}(\mathbf{W},\gamma)-\tilde{m}(\mathbf{W},\gamma_{0})]^{2}$
is continuous at $\gamma_{0}$ w.r.t. $\|\gamma-\gamma_{0}\|$

  3. 3.

$\max_{j}|\tilde{m}(\mathbf{W},b_{j})-\tilde{m}(\mathbf{W},0)|\leq C$

###### Proof.

Chernozhukov et al., 2018b (, Theorem 6) ∎

######  Proposition 8.

Under Assumption 7

  1. 1.

$\mathbb{E}[\gamma_{0}(z,\mathbf{X})]^{2}\leq
C\mathbb{E}[\gamma_{0}(Z,\mathbf{X})]^{2}$ for $z\in\\{0,1\\}$

  2. 2.

$\mathbb{E}[\gamma(z,\mathbf{X})-\gamma_{0}(z,\mathbf{X})]^{2}\leq
C\|\gamma-\gamma_{0}\|^{2}$ for $d\in\\{0,1\\}$

###### Proof.

Chernozhukov et al., 2018b (, Theorem 6) ∎

######  Proposition 9.

Consider the estimator
$\hat{\boldsymbol{\theta}}=\operatorname*{\arg\\!\min}_{\boldsymbol{\theta}\in\boldsymbol{\Theta}}\hat{Q}(\boldsymbol{\theta})$,
where $\hat{Q}:\boldsymbol{\Theta}\rightarrow\mathbb{R}$ estimates
$Q_{0}:\boldsymbol{\Theta}\rightarrow\mathbb{R}$. If

  1. 1.

$\boldsymbol{\Theta}$ is compact

  2. 2.

$Q_{0}$ is continuous in $\boldsymbol{\theta}\in\boldsymbol{\Theta}$

  3. 3.

$Q_{0}$ is uniquely maximized at $\boldsymbol{\theta}_{0}$

  4. 4.

$\sup_{\boldsymbol{\theta}\in\boldsymbol{\Theta}}|\hat{Q}(\boldsymbol{\theta})-Q_{0}(\boldsymbol{\theta})|\overset{p}{\rightarrow}0$

then
$\hat{\boldsymbol{\theta}}\overset{p}{\rightarrow}\boldsymbol{\theta}_{0}$

###### Proof.

Newey and McFadden, (1994, Theorem 2.1) ∎

###  E.3 Stage 1

###### Proof of Lemma 1.

Proposition 5, Proposition 6, and Chernozhukov et al., 2018b (, Theorem 1) ∎

###### Proof of Lemma 2.

Proposition 5, Proposition 6, and Chernozhukov et al., 2018b (, Theorem 3).
The argument that $|\hat{\boldsymbol{\rho}}|_{1}=O_{p}(1)$ is analogous to
Chernozhukov et al., 2018b (, Lemmas 2 and 3). ∎

###  E.4 Stage 2

######  Proposition 10.

Under Assumptions 1, 2, either 3 or 4, 5, 6, and 7

$\frac{1}{\sqrt{n}}\sum_{\ell=1}^{L}\sum_{i\in
I_{\ell}}\psi(\mathbf{W}_{i},\hat{\gamma}_{-\ell},\hat{\alpha}_{-\ell},\hat{\boldsymbol{\theta}})\overset{p}{\rightarrow}\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\psi_{0}(\mathbf{W}_{i})$

###### Proof.

Chernozhukov et al., 2018b (, Theorem 5) ∎

######  Proposition 11.

Under Assumptions 1, 2, either 3 or 4, 5, 6, and 7, for each $\ell$

  1. 1.

$\mathbb{E}[m(\mathbf{W},\hat{\gamma}_{-\ell},\boldsymbol{\theta}_{0})-m(\mathbf{W},\gamma_{0},\boldsymbol{\theta}_{0})]^{2}\overset{p}{\rightarrow}0$

  2. 2.

$\mathbb{E}[\phi(\mathbf{W},\hat{\gamma}_{-\ell},\alpha_{0},\boldsymbol{\theta}_{0})-\phi(\mathbf{W},\gamma_{0},\alpha_{0},\boldsymbol{\theta}_{0})]^{2}\overset{p}{\rightarrow}0$

  3. 3.

$\mathbb{E}[\phi(\mathbf{W},\gamma_{0},\hat{\alpha}_{-\ell},\boldsymbol{\theta}_{0})-\phi(\mathbf{W},\gamma_{0},\alpha_{0},\boldsymbol{\theta}_{0})]^{2}\overset{p}{\rightarrow}0$

###### Proof.

First note that

$\displaystyle\phi(\mathbf{W},\hat{\gamma}_{-\ell},\alpha_{0},\boldsymbol{\theta}_{0})-\phi(\mathbf{W},\gamma_{0},\alpha_{0},\boldsymbol{\theta}_{0})$
$\displaystyle=\alpha_{0}(z,\mathbf{x})A(\boldsymbol{\theta}_{0})[\gamma_{0}(z,\mathbf{x})-\hat{\gamma}_{-\ell}(z,\mathbf{x})]$
$\displaystyle\phi(\mathbf{W},\gamma_{0},\hat{\alpha}_{-\ell},\boldsymbol{\theta}_{0})-\phi(\mathbf{W},\gamma_{0},\alpha_{0},\boldsymbol{\theta}_{0})$
$\displaystyle=[\hat{\alpha}_{-\ell}(z,\mathbf{x})-\alpha_{0}(z,\mathbf{x})]A(\boldsymbol{\theta}_{0})[\mathbf{v}-\gamma_{0}(z,\mathbf{x})]$

  1. 1.

Assumption 6 then Proposition 7

  2. 2.

By Assumption 6 and Assumption 7

$\|\alpha_{0}A(\boldsymbol{\theta}_{0})[\gamma_{0}-\hat{\gamma}_{-\ell}]\|\leq
CA(\boldsymbol{\theta}_{0})\|\gamma_{0}-\hat{\gamma}_{-\ell}\|\overset{p}{\rightarrow}0$

  3. 3.

By Lemma 1 or Lemma 2, Assumption 7, and LIE w.r.t.
$\mathbf{W}_{-\ell}:=\\{\mathbf{W}_{i}\\}_{i\notin I_{\ell}}$

$\|[\hat{\alpha}_{-\ell}-\alpha_{0}]A(\boldsymbol{\theta}_{0})[\mathbf{v}-\gamma_{0}(z,\mathbf{x})]\|\leq\|\hat{\alpha}_{-\ell}-\alpha_{0}\|A(\boldsymbol{\theta}_{0})C\cdot\vec{1}\overset{p}{\rightarrow}0$

∎

######  Proposition 12.

Under Assumptions 1, 2, either 3 or 4, 5, 6, and 7,

$\displaystyle\frac{1}{\sqrt{n}}\sum_{\ell=1}^{L}\sum_{i\in
I_{\ell}}[\phi(\mathbf{W}_{i},\hat{\gamma}_{-\ell},\hat{\alpha}_{-\ell},\boldsymbol{\theta}_{0})-\phi(\mathbf{W}_{i},\hat{\gamma}_{-\ell},\alpha_{0},\boldsymbol{\theta}_{0})$
$\displaystyle\quad-\phi(\mathbf{W}_{i},\gamma_{0},\hat{\alpha}_{-\ell},\boldsymbol{\theta}_{0})+\phi(\mathbf{W}_{i},\gamma_{0},\alpha_{0},\boldsymbol{\theta}_{0})]\overset{p}{\rightarrow}0$

###### Proof.

Note that

$\displaystyle\phi(\mathbf{w},\hat{\gamma}_{-\ell},\hat{\alpha}_{-\ell},\boldsymbol{\theta}_{0})-\phi(\mathbf{w},\hat{\gamma}_{-\ell},\alpha_{0},\boldsymbol{\theta}_{0})-\phi(\mathbf{w},\gamma_{0},\hat{\alpha}_{-\ell},\boldsymbol{\theta}_{0})+\phi(\mathbf{w},\gamma_{0},\alpha_{0},\boldsymbol{\theta}_{0})$
$\displaystyle=-[\hat{\alpha}_{-\ell}(z,\mathbf{x})-\alpha_{0}(z,\mathbf{x})]A(\boldsymbol{\theta}_{0})[\hat{\gamma}_{-\ell}(z,\mathbf{x})-\gamma_{0}(z,\mathbf{x})]$

Because convergence in first mean implies convergence in probability, it
suffices to analyze

$\displaystyle\mathbb{E}\left[\left|\frac{1}{\sqrt{n}}\sum_{\ell=1}^{L}\sum_{i\in
I_{\ell}}-[\hat{\alpha}_{-\ell}(Z_{i},\mathbf{X}_{i})-\alpha_{0}(Z_{i},\mathbf{X}_{i})]A(\boldsymbol{\theta}_{0})[\hat{\gamma}_{-\ell}(Z_{i},\mathbf{X}_{i})-\gamma_{0}(Z_{i},\mathbf{X}_{i})]\right|\right]$
$\displaystyle\leq\sum_{\ell=1}^{L}\mathbb{E}\left[\sqrt{n}\frac{1}{n}\sum_{i\in
I_{\ell}}\left|-[\hat{\alpha}_{-\ell}(Z_{i},\mathbf{X}_{i})-\alpha_{0}(Z_{i},\mathbf{X}_{i})]A(\boldsymbol{\theta}_{0})[\hat{\gamma}_{-\ell}(Z_{i},\mathbf{X}_{i})-\gamma_{0}(Z_{i},\mathbf{X}_{i})]\right|\right]$
$\displaystyle=\sum_{\ell=1}^{L}\mathbb{E}\mathbb{E}\left[\sqrt{n}\frac{1}{n}\sum_{i\in
I_{\ell}}\left|[\hat{\alpha}_{-\ell}(Z_{i},\mathbf{X}_{i})-\alpha_{0}(Z_{i},\mathbf{X}_{i})]A(\boldsymbol{\theta}_{0})[\hat{\gamma}_{-\ell}(Z_{i},\mathbf{X}_{i})-\gamma_{0}(Z_{i},\mathbf{X}_{i})]\right|\bigg{|}\mathbf{W}_{-\ell}\right]$
$\displaystyle=\sum_{\ell=1}^{L}\mathbb{E}\mathbb{E}\left[\left|\sqrt{n}\frac{n_{\ell}}{n}[\hat{\alpha}_{-\ell}(Z_{i},\mathbf{X}_{i})-\alpha_{0}(Z_{i},\mathbf{X}_{i})]A(\boldsymbol{\theta}_{0})[\hat{\gamma}_{-\ell}(Z_{i},\mathbf{X}_{i})-\gamma_{0}(Z_{i},\mathbf{X}_{i})]\right|\bigg{|}\mathbf{W}_{-\ell}\right]$

By element-wise Hölder and Proposition 1 we have convergence for each summand
as

$\displaystyle\mathbb{E}\left[|\sqrt{n}\frac{n_{\ell}}{n}[\hat{\alpha}_{-\ell}(Z_{i},\mathbf{X}_{i})-\alpha_{0}(Z_{i},\mathbf{X}_{i})]A(\boldsymbol{\theta}_{0})[\hat{\gamma}_{-\ell}(Z_{i},\mathbf{X}_{i})-\gamma_{0}(Z_{i},\mathbf{X}_{i})]|\bigg{|}\mathbf{W}_{-\ell}\right]$
$\displaystyle\leq\mathbb{E}\left[|\sqrt{n}[\hat{\alpha}_{-\ell}(Z_{i},\mathbf{X}_{i})-\alpha_{0}(Z_{i},\mathbf{X}_{i})]A(\boldsymbol{\theta}_{0})[\hat{\gamma}_{-\ell}(Z_{i},\mathbf{X}_{i})-\gamma_{0}(Z_{i},\mathbf{X}_{i})]|\bigg{|}\mathbf{W}_{-\ell}\right]$
$\displaystyle\leq\sqrt{n}\|\hat{\alpha}_{-\ell}-\alpha_{0}\|A(\boldsymbol{\theta}_{0})\|\hat{\gamma}_{-\ell}-\gamma_{0}\|$
$\displaystyle\overset{p}{\rightarrow}0$

∎

######  Proposition 13.

Under Assumption 1, for each $\ell$

  1. 1.

$\sqrt{n}\mathbb{E}[\psi(\mathbf{W},\hat{\gamma}_{-\ell},\alpha_{0},\boldsymbol{\theta}_{0})]\overset{p}{\rightarrow}0$

  2. 2.

$\sqrt{n}\mathbb{E}[\phi(\mathbf{W},\gamma_{0},\hat{\alpha}_{-\ell},\boldsymbol{\theta}_{0})]\overset{p}{\rightarrow}0$

###### Proof.

Note that

$\displaystyle\mathbb{E}[\psi(\mathbf{W},\hat{\gamma}_{-\ell},\alpha_{0},\boldsymbol{\theta}_{0})]$
$\displaystyle=\mathbb{E}[A(\boldsymbol{\theta}_{0})[\hat{\gamma}_{-\ell}(1,\mathbf{X})-\hat{\gamma}_{-\ell}(0,\mathbf{X})]+\alpha_{0}(Z,\mathbf{X})A(\boldsymbol{\theta}_{0})[\mathbf{V}-\hat{\gamma}_{-\ell}(Z,\mathbf{X})]]$
$\displaystyle\mathbb{E}[\phi(\mathbf{W},\gamma_{0},\hat{\alpha}_{-\ell},\boldsymbol{\theta}_{0})]$
$\displaystyle=\mathbb{E}[\hat{\alpha}_{-\ell}(Z,\mathbf{X})A(\boldsymbol{\theta}_{0})[\mathbf{V}-\gamma_{0}(Z,\mathbf{X})]]$

  1. 1.

The LHS is exactly 0 by Theorem 1 as well as LIE w.r.t. $\mathbf{W}_{-\ell}$

  2. 2.

The LHS is exactly 0 by Theorem 1 as well as LIE w.r.t. $\mathbf{W}_{-\ell}$

∎

######  Proposition 14.

Under Assumptions 1, 2, either 3 or 4, 5, 6, and 7,

  1. 1.

$\mathbf{\mathbf{J}}$ exists

  2. 2.

$\exists$ neighborhood $\mathcal{N}$ of $\boldsymbol{\theta}_{0}$ w.r.t.
$|\cdot|_{2}$ s.t.

    1. (a)

$\|\hat{\gamma}_{-\ell}-\gamma_{0}\|\overset{p}{\rightarrow}0$

    2. (b)

$\|\hat{\alpha}_{-\ell}-\alpha_{0}\|\overset{p}{\rightarrow}0$

    3. (c)

for $\|\gamma-\gamma_{0}\|$ and $\|\alpha-\alpha_{0}\|$ small enough,
$\psi(\mathbf{W}_{i},\gamma,\alpha,\boldsymbol{\theta})$ is diff. in
$\boldsymbol{\theta}$ w.p.a. 1

    4. (d)

$\exists\zeta>0$ and $d(\mathbf{W})$ s.t $\mathbb{E}[d(\mathbf{W})]<\infty$
and for $\|\gamma-\gamma_{0}\|$ small enough,

$\left|\frac{\partial\psi(\mathbf{w},\gamma,\alpha,\boldsymbol{\theta})}{\partial\boldsymbol{\theta}}-\frac{\partial\psi(\mathbf{w},\gamma,\alpha,\boldsymbol{\theta}_{0})}{\partial\boldsymbol{\theta}}\right|_{2}\leq
d(\mathbf{w})|\boldsymbol{\theta}-\boldsymbol{\theta}_{0}|_{2}^{\zeta}$

  3. 3.

$\mathbb{E}\left|\frac{\partial\psi_{j}(\mathbf{W},\hat{\gamma}_{-\ell},\hat{\alpha}_{-\ell},\boldsymbol{\theta}_{0})}{\partial\boldsymbol{\theta}_{k}}-\frac{\partial\psi_{j}(\mathbf{W},\gamma_{0},\alpha_{0},\boldsymbol{\theta}_{0})}{\partial\boldsymbol{\theta}_{k}}\right|\overset{p}{\rightarrow}0$,
$\forall\ell,j,k$

###### Proof.

Note that

$\frac{\partial\psi(\mathbf{w},\gamma,\alpha,\boldsymbol{\theta})}{\partial\boldsymbol{\theta}}=\frac{\partial
A(\boldsymbol{\theta})}{\partial\boldsymbol{\theta}}[\gamma(1,\mathbf{x})-\gamma(0,\mathbf{x})]+\alpha(z,\mathbf{x})\frac{\partial
A(\boldsymbol{\theta})}{\partial\boldsymbol{\theta}}[\mathbf{v}-\gamma(z,\mathbf{x})]$

where $\frac{\partial A(\boldsymbol{\theta})}{\partial\boldsymbol{\theta}}$ is
a tensor consisting of 1s and 0s.

  1. 1.

It suffices to show finite second moment. By triangle inequality and
Assumption 7

$\displaystyle\left\|\frac{\partial
A(\boldsymbol{\theta}_{0})}{\partial\boldsymbol{\theta}}[\gamma_{0}(1,\mathbf{x})-\gamma_{0}(0,\mathbf{x})]+\alpha_{0}(z,\mathbf{x})\frac{\partial
A(\boldsymbol{\theta})}{\partial\boldsymbol{\theta}}[\mathbf{v}-\gamma_{0}(z,\mathbf{x})]\right\|$
$\displaystyle\quad\leq\frac{\partial
A(\boldsymbol{\theta}_{0})}{\partial\boldsymbol{\theta}}\left\\{\|\gamma_{0}(1,\mathbf{x})-\gamma_{0}(0,\mathbf{x})\|+CC^{\prime}\right\\}$

To bound the RHS, appeal to Proposition 8.

$\|\gamma_{0}(1,\mathbf{x})-\gamma_{0}(0,\mathbf{x})\|\leq\|\gamma_{0}(1,\mathbf{x})\|+\|\gamma_{0}(0,\mathbf{x})\|\leq
C\|\gamma_{0}\|<\infty$

  2. 2.

    1. (a)

Assumption 6

    2. (b)

Lemma 1 or Lemma 2

    3. (c)

$\frac{\partial A(\boldsymbol{\theta})}{\partial\boldsymbol{\theta}}$ is a
tensor consisting of 1s and 0s

    4. (d)

the LHS is exactly $\vec{0}$

  3. 3.

It suffices to analyze the difference

$\displaystyle\xi$
$\displaystyle=\hat{\gamma}_{-\ell}(1,\mathbf{x})-\hat{\gamma}_{-\ell}(0,\mathbf{x})+\hat{\alpha}_{-\ell}(z,\mathbf{x})[\mathbf{v}-\hat{\gamma}_{-\ell}(z,\mathbf{x})]$
$\displaystyle\quad-\left\\{\gamma_{0}(1,\mathbf{x})-\gamma_{0}(0,\mathbf{x})+\alpha_{0}(z,\mathbf{x})[\mathbf{v}-\gamma_{0}(z,\mathbf{x})]\right\\}$
$\displaystyle=\hat{\gamma}_{-\ell}(1,\mathbf{x})-\gamma_{0}(1,\mathbf{x})$
$\displaystyle\quad-\hat{\gamma}_{-\ell}(0,\mathbf{x})+\gamma_{0}(0,\mathbf{x})$
$\displaystyle\quad+\hat{\alpha}_{-\ell}(z,\mathbf{x})[\mathbf{v}-\hat{\gamma}_{-\ell}(z,\mathbf{x})]-\alpha_{0}(z,\mathbf{x})[\mathbf{v}-\hat{\gamma}_{-\ell}(z,\mathbf{x})]$
$\displaystyle\quad+\alpha_{0}(z,\mathbf{x})[\mathbf{v}-\hat{\gamma}_{-\ell}(z,\mathbf{x})]-\alpha_{0}(z,\mathbf{x})[\mathbf{v}-\gamma_{0}(z,\mathbf{x})]$
$\displaystyle=\hat{\gamma}_{-\ell}(1,\mathbf{x})-\gamma_{0}(1,\mathbf{x})$
$\displaystyle\quad-\hat{\gamma}_{-\ell}(0,\mathbf{x})+\gamma_{0}(0,\mathbf{x})$
$\displaystyle\quad+[\hat{\alpha}_{-\ell}(z,\mathbf{x})-\alpha_{0}(z,\mathbf{x})][\mathbf{v}-\gamma_{0}(z,\mathbf{x})]$
$\displaystyle\quad+[\hat{\alpha}_{-\ell}(z,\mathbf{x})-\alpha_{0}(z,\mathbf{x})][\gamma_{0}(z,\mathbf{x})-\hat{\gamma}_{-\ell}(z,\mathbf{x})]$
$\displaystyle\quad+\alpha_{0}(z,\mathbf{x})[\gamma_{0}(z,\mathbf{x})-\hat{\gamma}_{-\ell}(z,\mathbf{x})]$

where we use the decomposition

$\displaystyle\hat{\alpha}_{-\ell}(z,\mathbf{x})[\mathbf{v}-\hat{\gamma}_{-\ell}(z,\mathbf{x})]-\alpha_{0}(z,\mathbf{x})[\mathbf{v}-\hat{\gamma}_{-\ell}(z,\mathbf{x})]$
$\displaystyle=[\hat{\alpha}_{-\ell}(z,\mathbf{x})-\alpha_{0}(z,\mathbf{x})][\mathbf{v}-\gamma_{0}(z,\mathbf{x})+\gamma_{0}(z,\mathbf{x})-\hat{\gamma}_{-\ell}(z,\mathbf{x})]$

Hence

$\displaystyle\mathbb{E}|\xi|$
$\displaystyle\leq\mathbb{E}|\hat{\gamma}_{-\ell}(1,\mathbf{X})-\gamma_{0}(1,\mathbf{X})|$
$\displaystyle\quad+\mathbb{E}|\hat{\gamma}_{-\ell}(0,\mathbf{X})-\gamma_{0}(0,\mathbf{X})|$
$\displaystyle\quad+\mathbb{E}|[\hat{\alpha}_{-\ell}(Z,\mathbf{X})-\alpha_{0}(Z,\mathbf{X})][\mathbf{V}-\gamma_{0}(Z,\mathbf{X})]|$
$\displaystyle\quad+\mathbb{E}|[\hat{\alpha}_{-\ell}(Z,\mathbf{X})-\alpha_{0}(Z,\mathbf{X})][\gamma_{0}(Z,\mathbf{X})-\hat{\gamma}_{-\ell}(Z,\mathbf{X})]|$
$\displaystyle\quad+\mathbb{E}|\alpha_{0}(Z,\mathbf{X})[\gamma_{0}(Z,\mathbf{X})-\hat{\gamma}_{-\ell}(Z,\mathbf{X})]|$

Consider the first two terms. By Jensen, Proposition 8, and Assumption 6

$\displaystyle\mathbb{E}|\hat{\gamma}_{-\ell}(1,\mathbf{X})-\gamma_{0}(1,\mathbf{X})|$
$\displaystyle\leq\|\hat{\gamma}_{-\ell}(1,\mathbf{x})-\gamma_{0}(1,\mathbf{x})\|\leq
C\|\hat{\gamma}_{-\ell}-\gamma_{0}\|\overset{p}{\rightarrow}0$
$\displaystyle\mathbb{E}|\hat{\gamma}_{-\ell}(0,\mathbf{X})-\gamma_{0}(0,\mathbf{X})|$
$\displaystyle\leq\|\hat{\gamma}_{-\ell}(0,\mathbf{x})-\gamma_{0}(0,\mathbf{x})\|\leq
C\|\hat{\gamma}_{-\ell}-\gamma_{0}\|\overset{p}{\rightarrow}0$

Consider the third term. By Hölder, Lemma 1 or Lemma 2, Assumption 7, and LIE
w.r.t. $\mathbf{W}_{-\ell}$

$\mathbb{E}|[\hat{\alpha}_{-\ell}(Z,\mathbf{X})-\alpha_{0}(Z,\mathbf{X})][\mathbf{V}-\gamma_{0}(Z,\mathbf{X})]|\leq\|\hat{\alpha}_{-\ell}-\alpha_{0}\|\|\mathbf{v}-\gamma_{0}(z,\mathbf{x})\|\leq
C\|\hat{\alpha}_{-\ell}-\alpha_{0}\|\overset{p}{\rightarrow}0$

Consider the fourth term. By Hölder and Proposition 1

$\mathbb{E}|[\hat{\alpha}_{-\ell}(Z,\mathbf{X})-\alpha_{0}(Z,\mathbf{X})][\gamma_{0}(Z,\mathbf{X})-\hat{\gamma}_{-\ell}(Z,\mathbf{X})]|\leq\|\hat{\alpha}_{-\ell}-\alpha_{0}\|\|\gamma_{0}-\hat{\gamma}_{-\ell}\|\overset{p}{\rightarrow}0$

Consider the fifth term. By Assumption 7, Jensen, and Assumption 6

$\mathbb{E}|\alpha_{0}(Z,\mathbf{X})[\gamma_{0}(Z,\mathbf{X})-\hat{\gamma}_{-\ell}(Z,\mathbf{X})]|\leq
C\mathbb{E}|\gamma_{0}(Z,\mathbf{X})-\hat{\gamma}_{-\ell}(Z,\mathbf{X})|\leq
C\|\gamma_{0}-\hat{\gamma}_{-\ell}\|\overset{p}{\rightarrow}0$

∎

######  Proposition 15.

Suppose Assumptions 1, 2, either 3 or 4, 5, 6, and 7, hold. Then
$\hat{\boldsymbol{\theta}}\overset{p}{\rightarrow}\boldsymbol{\theta}_{0}$

###### Proof.

We verify the conditions of Proposition 9 with

$\displaystyle Q_{0}(\boldsymbol{\theta})$
$\displaystyle=\mathbb{E}[\psi_{0}(\boldsymbol{\theta})]^{\prime}\mathbb{E}[\psi_{0}(\boldsymbol{\theta})]$
$\displaystyle\hat{Q}(\boldsymbol{\theta})$
$\displaystyle=\left[\frac{1}{n}\sum_{\ell=1}^{L}\sum_{i\in
I_{\ell}}\hat{\psi}_{i}(\boldsymbol{\theta})\right]^{\prime}\frac{1}{n}\sum_{\ell=1}^{L}\sum_{i\in
I_{\ell}}\hat{\psi}_{i}(\boldsymbol{\theta})$
$\displaystyle\psi_{0}(\boldsymbol{\theta})$
$\displaystyle=\psi(\mathbf{W},\gamma_{0},\alpha_{0},\boldsymbol{\theta})$
$\displaystyle\hat{\psi}_{i}(\boldsymbol{\theta})$
$\displaystyle=\psi(\mathbf{W}_{i},\hat{\gamma}_{-\ell},\hat{\alpha}_{-\ell},\boldsymbol{\theta})$

  1. 1.

Assumption 7,

  2. 2.

Theorem 1

  3. 3.

Theorem 1

  4. 4.

Define

$\displaystyle\eta_{0}(\mathbf{w})$
$\displaystyle=\gamma_{0}(1,\mathbf{x})-\gamma_{0}(0,\mathbf{x})+\alpha_{0}(z,\mathbf{x})[\mathbf{v}-\gamma_{0}(z,\mathbf{x})]$
$\displaystyle\hat{\eta}_{-\ell}(\mathbf{w})$
$\displaystyle=\hat{\gamma}_{-\ell}(1,\mathbf{x})-\hat{\gamma}_{-\ell}(0,\mathbf{x})+\hat{\alpha}_{-\ell}(z,\mathbf{x})[\mathbf{v}-\hat{\gamma}_{-\ell}(z,\mathbf{x})]$

It follows that

$\displaystyle\psi_{0}(\boldsymbol{\theta})$
$\displaystyle=A(\boldsymbol{\theta})\eta_{0}(\mathbf{W}),\quad\mathbb{E}\psi_{0}(\boldsymbol{\theta})=A(\boldsymbol{\theta})\mathbb{E}\eta_{0}(\mathbf{W})$
$\displaystyle\hat{\psi}_{i}(\boldsymbol{\theta})$
$\displaystyle=A(\boldsymbol{\theta})\hat{\eta}_{-\ell}(\mathbf{W}_{i}),\quad\frac{1}{n}\sum_{\ell=1}^{L}\sum_{i\in
I_{\ell}}\hat{\psi}_{i}(\boldsymbol{\theta})=A(\boldsymbol{\theta})\frac{1}{n}\sum_{\ell=1}^{L}\sum_{i\in
I_{\ell}}\hat{\eta}_{-\ell}(\mathbf{W}_{i})$

It suffices to show $\frac{1}{n}\sum_{\ell=1}^{L}\sum_{i\in
I_{\ell}}\hat{\eta}_{-\ell}(\mathbf{W}_{i})\overset{p}{\rightarrow}\mathbb{E}\eta_{0}(\mathbf{W})$
since by continuous mapping theorem this implies that
$\forall\boldsymbol{\theta}\in\boldsymbol{\Theta}$,
$\frac{1}{n}\sum_{\ell=1}^{L}\sum_{i\in
I_{\ell}}\hat{\psi}_{i}(\boldsymbol{\theta})\overset{p}{\rightarrow}\mathbb{E}\psi_{0}(\boldsymbol{\theta})$
and hence
$\hat{Q}(\boldsymbol{\theta})\overset{p}{\rightarrow}Q_{0}(\boldsymbol{\theta})$
uniformly.

We therefore turn to arguing the sufficient condition. Write

$\displaystyle\frac{1}{n}\sum_{\ell=1}^{L}\sum_{i\in
I_{\ell}}\hat{\eta}_{-\ell}(\mathbf{W}_{i})-\mathbb{E}\eta_{0}(\mathbf{W})$
$\displaystyle=\quad\frac{1}{n}\sum_{\ell=1}^{L}\sum_{i\in
I_{\ell}}[\hat{\eta}_{-\ell}(\mathbf{W}_{i})-\eta_{0}(\mathbf{W}_{i})]+\frac{1}{n}\sum_{\ell=1}^{L}\sum_{i\in
I_{\ell}}\eta_{0}(\mathbf{W}_{i})-\mathbb{E}\eta_{0}(\mathbf{W})$

Consider the first and second term. Denote
$\xi_{i}=\hat{\eta}_{-\ell}(\mathbf{W}_{i})-\eta_{0}(\mathbf{W}_{i})$ as in
Proposition 14.3. Since convergence in mean implies convergence in
probability, it suffices to analyze

$\displaystyle\mathbb{E}\left|\frac{1}{n}\sum_{\ell=1}^{L}\sum_{i\in
I_{\ell}}\xi_{i}\right|$
$\displaystyle\leq\sum_{\ell=1}^{L}\mathbb{E}\frac{1}{n}\sum_{i\in
I_{\ell}}|\xi_{i}|$
$\displaystyle=\sum_{\ell=1}^{L}\mathbb{E}\mathbb{E}\left[\frac{1}{n}\sum_{i\in
I_{\ell}}|\xi_{i}|\bigg{|}\mathbf{W}_{-\ell}\right]$
$\displaystyle=\sum_{\ell=1}^{L}\mathbb{E}\frac{n_{\ell}}{n}\mathbb{E}[|\xi_{i}||\mathbf{W}_{-\ell}]$
$\displaystyle\leq\sum_{\ell=1}^{L}\mathbb{E}\mathbb{E}[|\xi_{i}||\mathbf{W}_{-\ell}]$
$\displaystyle\rightarrow 0$

by triangle inequality, LIE, and the proof of Proposition 14.3.

Consider the third and fourth term.

$\frac{1}{n}\sum_{\ell=1}^{L}\sum_{i\in
I_{\ell}}\eta_{0}(\mathbf{W}_{i})-\mathbb{E}\eta_{0}(\mathbf{W})=\frac{1}{n}\sum_{i=1}^{n}\eta_{0}(\mathbf{W}_{i})-\mathbb{E}\eta_{0}(\mathbf{W})\overset{p}{\rightarrow}0$

by WLLN and the fact that $\mathbb{E}[\eta_{0}(\mathbf{W})]^{2}<\infty$,
guaranteed by the argument in Proposition 16.3 below.

∎

######  Proposition 16.

Under Assumptions 1, 2, either 3 or 4, 5, 6, and 7,

  1. 1.

$\hat{\boldsymbol{\theta}}\overset{p}{\rightarrow}\boldsymbol{\theta}_{0}$

  2. 2.

$\mathbf{J}^{\prime}\mathbf{J}$ is nonsingular

  3. 3.

$\mathbb{E}[\psi_{0}(\mathbf{W})]^{2}<\infty$

  4. 4.

$\mathbb{E}[\phi(\mathbf{W},\hat{\gamma}_{-\ell},\hat{\alpha}_{-\ell},\boldsymbol{\theta}_{0})-\phi(\mathbf{W},\hat{\gamma}_{-\ell},\alpha_{0},\boldsymbol{\theta}_{0})-\phi(\mathbf{W},\gamma_{0},\hat{\alpha}_{-\ell},\boldsymbol{\theta}_{0})+\phi(\mathbf{W},\gamma_{0},\alpha_{0},\boldsymbol{\theta}_{0})]^{2}\overset{p}{\rightarrow}0$

###### Proof.

As before

$\displaystyle\phi(\mathbf{w},\hat{\gamma}_{-\ell},\hat{\alpha}_{-\ell},\boldsymbol{\theta}_{0})-\phi(\mathbf{w},\hat{\gamma}_{-\ell},\alpha_{0},\boldsymbol{\theta}_{0})-\phi(\mathbf{w},\gamma_{0},\hat{\alpha}_{-\ell},\boldsymbol{\theta}_{0})+\phi(\mathbf{w},\gamma_{0},\alpha_{0},\boldsymbol{\theta}_{0})$
$\displaystyle=-[\hat{\alpha}_{-\ell}(z,\mathbf{x})-\alpha_{0}(z,\mathbf{x})]A(\boldsymbol{\theta}_{0})[\hat{\gamma}_{-\ell}(z,\mathbf{x})-\gamma_{0}(z,\mathbf{x})]$

  1. 1.

Proposition 15

  2. 2.

Assumption 7

  3. 3.

By triangle inequality, Assumption 7, and Proposition 8

$\|\gamma_{0}(1,\mathbf{x})-\gamma_{0}(0,\mathbf{x})+\alpha_{0}(z,\mathbf{x})[\mathbf{v}-\gamma_{0}(z,\mathbf{x})]\|\leq\|\gamma_{0}(1,\mathbf{x})-\gamma_{0}(0,\mathbf{x})\|+CC^{\prime}$

To bound the RHS, appeal to Proposition 8.

$\|\gamma_{0}(1,\mathbf{x})-\gamma_{0}(0,\mathbf{x})\|\leq\|\gamma_{0}(1,\mathbf{x})\|+\|\gamma_{0}(0,\mathbf{x})\|\leq
C\|\gamma_{0}\|<\infty$

  4. 4.

It suffices to analyze

$\displaystyle\|-[\hat{\alpha}_{-\ell}-\alpha_{0}]A(\boldsymbol{\theta}_{0})[\hat{\gamma}_{-\ell}-\gamma_{0}]\|$
$\displaystyle\leq\|\hat{\alpha}_{-\ell}A(\boldsymbol{\theta}_{0})[\hat{\gamma}_{-\ell}-\gamma_{0}]\|+\|\alpha_{0}A(\boldsymbol{\theta}_{0})[\hat{\gamma}_{-\ell}-\gamma_{0}]\|$

Consider the first term. By Hölder, Assumption 2, and either Lemma 1 or Lemma
2

$|\hat{\alpha}_{-\ell}(z,\mathbf{x})|=|\hat{\boldsymbol{\rho}}_{-\ell}^{\prime}b(z,\mathbf{x})|\leq|\hat{\boldsymbol{\rho}}_{-\ell}|_{1}|b(z,\mathbf{x})|_{\infty}=O_{p}(1)$

It follows by Assumption 6

$\|\hat{\alpha}_{-\ell}A(\boldsymbol{\theta}_{0})[\hat{\gamma}_{-\ell}-\gamma_{0}]\|=O_{p}(1)\|\hat{\gamma}_{-\ell}-\gamma_{0}\|=O_{p}(n^{-d_{\gamma}})\overset{p}{\rightarrow}0$

Consider the second term. By Assumption 6 and Assumption 7

$\|\alpha_{0}A(\boldsymbol{\theta}_{0})[\hat{\gamma}_{-\ell}-\gamma_{0}]\|\leq
CA(\boldsymbol{\theta}_{0})\|\hat{\gamma}_{-\ell}-\gamma_{0}\|\overset{p}{\rightarrow}0$

∎

###### Proof of Theorem 2.

Chernozhukov et al., (2016, Theorems 16 and 17). In particular, Proposition 11
verifies Chernozhukov et al., (2016, Assumption 4), Proposition 12 verifies
Chernozhukov et al., (2016, Assumption 5), Proposition 13 verifies
Chernozhukov et al., (2016, Assumption 6), Proposition 14 verifies
Chernozhukov et al., (2016, Assumption 7), and Proposition 16 verifies the
additional conditions in Chernozhukov et al., (2016, Theorems 16 and 17).
Finally, note that parameter $\boldsymbol{\theta}_{0}$ is exactly identified;
$\mathbf{\mathbf{J}}$ is a square matrix, the GMM weighting can be taken as
the identity, and the covariance formulas simplify. ∎

###### Proof of Corollary 2.

Immediate from Theorem 2 and (Newey and McFadden,, 1994, Section 9). ∎

###  E.5 Simultaneous Confidence Band

###### Proof of Theorem 3.

Let $c_{0}$ be the $(1-\alpha)$-quantile of
$|\mathcal{N}(0,\boldsymbol{\Sigma})|_{\infty}$ where
$\boldsymbol{\Sigma}=\mathbf{S}^{-1/2}\mathbf{C}\mathbf{S}^{-1/2}$ and
$\mathbf{S}=diag(\mathbf{C})$. We first show that this critical value ensures
correct (asymptotic) simultaneous coverage of confidence bands in the form of
the rectangle

$[(l_{0})_{j},(u_{0})_{j}]=\left[\hat{\theta}_{j}-c_{0}\sqrt{\frac{{C}_{jj}}{n}},\
\hat{\theta}_{j}+c_{0}\sqrt{\frac{{C}_{jj}}{n}}\right]$

where ${C}_{jj}$ is the diagonal entry of ${\mathbf{C}}$ corresponding to
$j$-th element $\hat{\theta}_{j}$ of $\boldsymbol{\theta}$.

The argument is as follows. Denote
$[l_{0},u_{0}]=\times_{j=1:2d}[(l_{0})_{j},(u_{0})_{j}]$ and
$d=dim(\mathcal{U})$. Then the simultaneous coverage probability is

$\displaystyle\mathbb{P}(\boldsymbol{\theta}_{0}\in[l_{0},u_{0}])$
$\displaystyle=\mathbb{P}(\sqrt{n}(\hat{\boldsymbol{\theta}}-\boldsymbol{\theta}_{0})\in\mathbf{S}^{1/2}[-c_{0},c_{0}]^{2d})$
$\displaystyle=\mathbb{P}(\mathcal{N}(0,\mathbf{C})\in\mathbf{S}^{1/2}[-c_{0},c_{0}]^{2d})+o(1)$
$\displaystyle=\mathbb{P}(\mathbf{S}^{-1/2}\mathcal{N}(0,\mathbf{C})\in[-c_{0},c_{0}]^{2d})+o(1)$
$\displaystyle=\mathbb{P}(|\mathcal{N}(0,\boldsymbol{\Sigma})|_{\infty}\leq
c_{0})+o(1)$ $\displaystyle=1-\alpha+o(1)$

Next, we prove the validity of Gaussian multiplier bootstrap for approximating
this $c_{0}$, appealing to Chernozhukov et al., (2013, Theorem 5.1). Gaussian
multiplier bootstrap is operationally equivalent to approximating $c_{0}$ with
$c$ calculated in Algorithm 4. We match symbols with Chernozhukov et al.,
(2013) to formalize the equivalence, designating notation from Chernozhukov et
al., (2013) with tildes.

$\tilde{T}=\left|\hat{\mathbf{S}}^{-1/2}\sqrt{n}(\hat{\boldsymbol{\theta}}-\boldsymbol{\theta}_{0})\right|_{\infty},\
\tilde{T}_{0}=\left|\mathbf{S}^{-1/2}\sqrt{n}\frac{1}{n}\sum_{i=1}^{n}\mathbf{J}^{-1}\psi_{0}(\mathbf{W}_{i})\right|_{\infty}$

and

$\displaystyle\tilde{W}$
$\displaystyle=\left|\hat{\mathbf{S}}^{-1/2}\sqrt{n}\frac{1}{n}\sum_{\ell=1}^{L}\sum_{i\in
I_{\ell}}\hat{\mathbf{J}}^{-1}\hat{\psi}_{i}(\hat{\boldsymbol{\theta}})e_{i}\right|_{\infty}=\left|\hat{\mathbf{S}}^{-1/2}\mathcal{N}(0,\hat{\mathbf{C}})\right|_{\infty}$
$\displaystyle\tilde{W}_{0}$
$\displaystyle=\left|\mathbf{S}^{-1/2}\sqrt{n}\frac{1}{n}\sum_{i=1}^{n}\mathbf{J}^{-1}\psi_{0}(\mathbf{W}_{i})e_{i}\right|_{\infty}=\left|\mathbf{S}^{-1/2}\mathcal{N}(0,\mathbf{J}^{-1}\left\\{\frac{1}{n}\sum_{i=1}^{n}\psi_{0}(\mathbf{W}_{i})\psi_{0}(\mathbf{W}_{i})^{\prime}\right\\}\mathbf{J}^{-1})\right|_{\infty}$

where $e_{i}\sim\mathcal{N}(0,1)$.

By linearity of $\hat{\psi}_{i}(\boldsymbol{\theta})$, we can write

$\frac{1}{n}\sum_{\ell=1}^{L}\sum_{i\in
I_{\ell}}\hat{\psi}_{i}(\hat{\boldsymbol{\theta}})-\frac{1}{n}\sum_{\ell=1}^{L}\sum_{i\in
I_{\ell}}\hat{\psi}_{i}(\boldsymbol{\theta}_{0})=\hat{\mathbf{J}}(\hat{\boldsymbol{\theta}}-\boldsymbol{\theta}_{0})$

Furthermore, since $\boldsymbol{\theta}_{0}$ is exactly identified, the first
term on the LHS is $\frac{1}{n}\sum_{\ell=1}^{L}\sum_{i\in
I_{\ell}}\hat{\psi}_{i}(\hat{\boldsymbol{\theta}})=0$. Premultiplying the
above equation by $\hat{\mathbf{S}}^{-1/2}\hat{\mathbf{J}}^{-1}$, we observe
that
$\hat{\mathbf{S}}^{-1/2}\sqrt{n}(\hat{\boldsymbol{\theta}}-\boldsymbol{\theta}_{0})$
can be represented as an approximate mean: it is equal to a scaled average of
$\psi_{0}(\mathbf{W}_{i})$ plus a remainder term $\mathbf{r}_{n}$

$\displaystyle\hat{\mathbf{S}}^{-1/2}\sqrt{n}(\hat{\boldsymbol{\theta}}-\boldsymbol{\theta}_{0})$
$\displaystyle=-\frac{1}{\sqrt{n}}\hat{\mathbf{S}}^{-1/2}\sum_{\ell=1}^{L}\sum_{i\in
I_{\ell}}\hat{\mathbf{J}}^{-1}\hat{\psi}_{i}(\boldsymbol{\theta}_{0})=-\frac{1}{\sqrt{n}}\mathbf{S}^{-1/2}\sum_{i=1}^{n}\mathbf{J}^{-1}\psi_{0}(\mathbf{W}_{i})+\mathbf{r}_{n}$

where

$\mathbf{r}_{n}:=\frac{1}{\sqrt{n}}\mathbf{S}^{-1/2}\sum_{i=1}^{n}\mathbf{J}^{-1}\psi_{0}(\mathbf{W}_{i})-\frac{1}{\sqrt{n}}\hat{\mathbf{S}}^{-1/2}\sum_{\ell=1}^{L}\sum_{i\in
I_{\ell}}\hat{\mathbf{J}}^{-1}\hat{\psi}_{i}(\boldsymbol{\theta}_{0})$

After matching symbols, we verify Chernozhukov et al., (2013, Condition M),
which is a sufficient condition for Chernozhukov et al., (2013, Theorem 5.1).
Assumption 9 verifies Chernozhukov et al., (2013, Conditions M.i, M.ii). As
argued in Chernozhukov et al., (2013, Comment 2.2), a sufficient condition for
the first clause of Chernozhukov et al., (2013, Condition M.iii) is that
$\mathbf{S}^{-1/2}\mathbf{J}^{-1}\psi_{0}(\mathbf{W}_{i})$ is bounded.
Assumption 8 verifies the second clause of Chernozhukov et al., (2013,
Condition M.iii).

Finally, we prove that
$\mathbf{S}^{-1/2}\mathbf{J}^{-1}\psi_{0}(\mathbf{W}_{i})$ is indeed bounded.
Observe that

$\|\mathbf{J}^{-1}\|_{op}=\lambda_{\max}(\mathbf{J}^{-1})=\frac{1}{\lambda_{\min}(\mathbf{J})}<\infty$

where $\|\cdot\|_{op}$ is operator norm and the inequality appeals to
Hypothesis 7.3. Recall that

$\psi_{0}(\mathbf{W}_{i})=A(\boldsymbol{\theta}_{0})[\gamma_{0}(1,\mathbf{X}_{i})-\gamma_{0}(0,\mathbf{X}_{i})]+\alpha_{0}(Z_{i},\mathbf{X}_{i})A(\boldsymbol{\theta}_{0})[\mathbf{V}_{i}-\gamma_{0}(Z_{i},\mathbf{X}_{i})]$

By Hypothesis 7.1, $|\alpha_{0}(Z_{i},\mathbf{X}_{i})|<\infty$. For complier
counterfactual distributions,
$\mathbf{V}_{i}^{y}\in\\{-1,0\\}\times\\{0,1\\}\times\\{0,1\\}$ and hence
$\gamma_{0}^{\mathbf{V}^{y}}(Z_{i},\mathbf{X}_{i})\in[-1,0]\times[0,1]\times[0,1]$
for all $y\in\mathcal{U}$. ∎

##  Appendix F Simulation

###  F.1 Design

We implement 500 simulations. One simulation consists of a sample of $n=1000$
observations. A given observation is generated from the following IV model:

$\displaystyle X\stackrel{{\scriptstyle i.i.d.}}{{\sim}}Unif[0,1],\quad$
$\displaystyle Z\mid X=x\stackrel{{\scriptstyle
i.i.d.}}{{\sim}}Bern(\pi_{0}(x)),$ $\displaystyle D\mid
Z=z,X=x\stackrel{{\scriptstyle
i.i.d.}}{{\sim}}Bern(\gamma_{0}^{D}(z,x)),\quad$ $\displaystyle Y\mid
Z=z,X=x\stackrel{{\scriptstyle
i.i.d.}}{{\sim}}\mathcal{N}(\gamma_{0}^{Y}(z,x),1)$

where $Y$ is the continuous outcome, $D$ is the binary treatment, $Z$ is the
binary instrumental variable, and $X$ is the covariate. $Y$ and $D$ are drawn
independently given $Z$ and $X$. In particular,

$\pi_{0}(x)=0.05\cdot 1_{x\leq 0.5}+0.95\cdot 1_{x>0.5},\
\gamma_{0}^{Y}(z,x)=2zx^{2},\ \gamma_{0}^{D}(z,x)=zx$

From observations of $\mathbf{W}=(Y,D,Z,X)^{\prime}$, we estimate complier
counterfactual outcome distributions
$\hat{\boldsymbol{\theta}}=(\hat{\boldsymbol{\beta}}^{\prime},\hat{\boldsymbol{\delta}}^{\prime})^{\prime}$
at a few grid points. For
$\boldsymbol{\beta}_{0}=\\{\beta_{0}^{y}\\}=\\{\mathbb{P}(Y^{(0)}\leq
y|D^{(1)}>D^{(0)})\\}$, we set $y\in\\{-3,-2,\dots,3,4\\}$. For
$\boldsymbol{\delta}_{0}=\\{\delta_{0}^{y}\\}=\\{\mathbb{P}(Y^{(1)}\leq
y|D^{(1)}>D^{(0)})\\}$, we set $y\in\\{-2,-1,\dots,4,5\\}$. The true parameter
values are

$\displaystyle\beta_{0}^{y}$
$\displaystyle=\frac{\int_{0}^{1}[\Phi(y-2x^{2})(x-1)+\Phi(y)]dx}{\int_{0}^{1}xdx}$
$\displaystyle\delta_{0}^{y}$
$\displaystyle=\frac{\int_{0}^{1}[\Phi(y-2x^{2})x]dx}{\int_{0}^{1}xdx}$

###  F.2 Algorithms

We estimate $\hat{\gamma}$ and $\hat{\alpha}$ with a dictionary $b(Z,X)$
consisting of fourth-order polynomials of $X$ and interactions between $Z$ and
the polynomials. We estimate $\hat{\pi}$ with a dictionary $b(X)$ consisting
of fourth-order polynomials of $X$. We set $L=5$ and $n_{\ell}=n/L$ for sample
splitting.

Auto-DML. We estimate $\hat{\alpha}_{-\ell}$ by Lasso according to Algorithm
5. We estimate $\hat{\gamma}^{V}_{-\ell}$ by Lasso according to Algorithm 5
and by neural network using the settings in Chernozhukov et al., 2018a .
Subsequently, we estimate $\hat{\boldsymbol{\theta}}$ by Algorithm 1.

DML. The only difference from Auto-DML is how we estimate
$\hat{\alpha}_{-\ell}$. For DML, we first estimate $\hat{\pi}_{-\ell}$ by
$\ell_{1}$-regularized logistic regression, and then form
$\hat{\alpha}_{-\ell}$ by plugging $\hat{\pi}_{-\ell}$ into its formula.

kappa-weight. We estimate $\hat{\pi}$ by logistic regression, then plug
$\hat{\pi}$ into Definition 2 to obtain $\hat{\kappa}^{(0)}(\mathbf{w})$,
$\hat{\kappa}^{(1)}(\mathbf{w})$, or $\hat{\kappa}(\mathbf{w})$. The
$\kappa$-weighted estimator is then the sample analog of formula given in
Abadie, (2003). For example, for
$\boldsymbol{\theta}_{0}=\mathbb{E}[g(Y^{(0)},\mathbf{X})|D^{(1)}>D^{(0)}]$,
the $\kappa$-weighted estimator is

$\hat{\boldsymbol{\theta}}^{\kappa}=\frac{\frac{1}{n}\sum_{i=1}^{n}\hat{\kappa}^{(0)}(\mathbf{W}_{i})\cdot
g(Y_{i},\mathbf{X}_{i})}{\frac{1}{n}\sum_{i=1}^{n}\hat{\kappa}^{(0)}(\mathbf{W}_{i})}$

MATLAB code for replicating the simulations is available in the supplementary
materials.

###  F.3 Sensitivity

To investigate the sensitivity of Auto-DML to hyperparameters, we revisit our
counterfactual distribution simulation exercise. We first consider the
sensitivity of Auto-DML to the number of folds $L$ used in sample splitting.
In Figure 3, we report simulation results for $L\in\\{2,5,10\\}$. Across $L$,
we estimate $\hat{\gamma}$ by Lasso and tune the regularization parameter
$\lambda$ according to Algorithm 5. We find that Auto-DML is insensitive to
$L$.

(a) $\beta_{0}^{y}=\mathbb{P}(Y^{(0)}\leq y\mid D^{(1)}>D^{(0)})$

(b) $\delta_{0}^{y}=\mathbb{P}(Y^{(1)}\leq y\mid D^{(1)}>D^{(0)})$

Figure 3:  Sensitivity to $L$

Next, we consider the sensitivity of Auto-DML to the regularization parameter
$\lambda$. We consider
$\lambda\in\\{0.5\lambda_{n},\lambda_{n},2\lambda_{n}\\}$ where $\lambda_{n}$
is the value obtained by Algorithm 5. Across $\lambda$, we estimate
$\hat{\gamma}$ by Lasso and we fix $L=5$. The simulation results in Figure 4
demonstrate that Auto-DML is insensitive to the value of $\lambda$.

(a) $\beta_{0}^{y}=\mathbb{P}(Y^{(0)}\leq y\mid D^{(1)}>D^{(0)})$

(b) $\delta_{0}^{y}=\mathbb{P}(Y^{(1)}\leq y\mid D^{(1)}>D^{(0)})$

Figure 4:  Sensitivity to $\lambda$

###  F.4 Trimming and Censoring

We assess the effectiveness of trimming and censoring in the simulation
considered in Section 7.1. We impose trimming according to Belloni et al.,
(2017), dropping observations with $\hat{\pi}\not\in[10^{-12},1-10^{-12}]$. We
impose censoring by setting $\hat{\pi}<10^{-12}$ to be $10^{-12}$ and
$\hat{\pi}>1-10^{-12}$ to be $1-10^{-12}$. Trimming and censoring improve