# Semiparametric Bespoke Instrumental Variables

Oliver Dukes1, David Richardson2,
Zach Shahn3 and Eric Tchetgen Tchetgen1
1: University of Pennsylvania, Philadelphia, PA, USA
2: University of California Irvine, Irvine, CA, USA
3: CUNY School of Public Health, New York, NY, USA

###### Abstract

Many proposals for the identification of causal effects in the presence of
unmeasured confounding require an instrumental variable or negative control
that satisfies strong, untestable exclusion restrictions. In this paper, we
will instead show how one can identify causal effects for a point exposure by
using a measured confounder as a ‘bespoke instrumental variable’. This
strategy requires an external reference population that does not have access
to the exposure, and a stability condition on the confounder-outcome
association between reference and target populations. Building on recent
identification results of Richardson and Tchetgen Tchetgen, (2021), we develop
the semiparametric efficiency theory for a general bespoke instrumental
variable model, and obtain a multiply robust locally efficient estimator of
the average treatment effect in the treated. The utility of the estimators is
demonstrated in simulation studies and an analysis of the Life Span Study,
concerning atomic bomb survivors in Japan.

##  1 Introduction

In drawing causal inferences from observational data, strong assumptions are
typically made regarding the absence of unmeasured confounding. For example,
in a point exposure study one often assumes that the measured covariates are
sufficient to account for systematic differences between treatment groups at
baseline. This assumption cannot be empirically verified, and may not be
plausible in many contexts. Hence there is now a long tradition in causal
inference of developing identification strategies that do not rely on a ‘no
unmeasured confounding’ (or conditional exchangeability) type assumption.
Arguably the most popular of these is the ‘instrumental variable’ (IV)
approach, long used in economics and recently gaining attention in
epidemiology in the context of Mendelian Randomisation studies (Davey Smith
and Ebrahim,, 2003). For validity, IV-based causal inference relies on having
access to a variable that satisfies the three core IV conditions: IV.1) it is
associated with the exposure (relevance); IV.2) it affects the outcome only
via the exposure (exclusion restriction); and IV.3) it shares no unmeasured
common causes with the outcome (unconfoundedness) (Hernán and Robins,, 2006).
Outside of a double-blinded randomised trial, these conditions are not usually
guaranteed to hold by design. It is known that even under small violations of
one or more of these conditions, IV estimators can exhibit large biases. A
separate strand of work attempts to draw causal inferences based on within-
subject comparisons, when one has access to repeated measurements of an
outcome. Fixed effect models and the difference-in-difference (DiD) strategy
are examples of this approach (Angrist and Pischke,, 2008), with the latter
basing identification on the ‘parallel-trends’ assumption. Synthetic control
approaches allow one to relax this condition to some extent (Abadie et al.,,
2010), if one has access to appropriate ‘donor’ controls which can be used to
approximate a treated unit had it been left untreated. In recent work,
increasing attention has been given to leveraging negative control exposures
and outcomes not just to detect bias, but also to remove it from confounded
estimates (Lipsitch et al.,, 2010; Sofer et al.,, 2016). Negative controls can
be viewed as specific examples of ‘proxy variables’ (Miao et al.,, 2018);
Tchetgen Tchetgen et al., (2020) propose a general nonparametric
identification strategy for leveraging proxies to recover causal effects when
conditional exchangeability fails.

In much of this work, investigators are required to have access to additional
variables beyond the measured confounders, which obey certain untestable
assumptions. However, Richardson and Tchetgen Tchetgen, (2021) take a
difference tact, and use a standard confounder to derive a ‘bespoke’
instrumental variable. For example, in a study attempting to ascertain the
effect of prompt radiation exposure from the atomic bombings of Hiroshima and
Nagasaki on mortality, certain individuals in the database were not present in
these cities at the time of the bombings. The authors show that the
association between a confounder and mortality in this reference population
can be used to predict what would have occurred had exposed individuals in the
target population (those who remained in the cities) been unexposed. The
authors show that the association between a confounder and mortality in this
reference population can be used to anchor the corresponding association among
the individuals in the target population (those who remained in the cities).
In turn, this can be used for identification of the average treatment effect
on the treated (ATT) in a ‘bespoke’ IV analysis. The validity of the approach
relies on having access to a reference population for whom treatment was not
available, as well as the association between the chosen confounder and
treatment-free potential outcome transporting from the reference to the target
population. Importantly however, the bespoke instrument does not have to
satisfy the IV conditions IV.2 and IV.3, besides being predictive of the
exposure, therefore satisfying IV.1.

In this paper, we first review the conditions under which a confounder can be
used as a bespoke instrument to identify causal effects. We draw the links
between this strategy and the IV, DiD and negative control designs and extend
the results of
Richardson and Tchetgen Tchetgen, (2021) for identification of the ATT. We
then develop a rigorous theory of estimation and inference for these causal
estimands, working under a semiparametric model for the observed data
distribution. This yields several novel classes of estimators, which are
distinct from those proposed in Richardson and Tchetgen Tchetgen, (2021). Our
estimators turn out to be quadruply robust; unbiased if one out of four sets
of constraints on the observed data distribution hold. We also characterize
the semiparametric efficiency bound for the parameter indexing the ATT, and
describe a corresponding locally semiparametric efficient estimator in the
special case of a binary treatment and bespoke IV. Finally, we show how the
proposed estimators simplify considerably in the case where a baseline outcome
measurement is used to construct the reference population, in which case a
bespoke IV analysis provides an alternative to standard DiD approach, without
imposing the parallel trend assumption typically invoked to justify DiD.

##  2 Identification

###  2.1 Set-up

Similar to Richardson and Tchetgen Tchetgen, (2021) we will first consider a
setting where we have a point-treatment exposure $A$ and covariate $Z$, which
are measured in a target population ($S=1$) and for the moment both assumed to
be binary. Let $Y$ denote the end-of-study outcome. Suppose furthermore that
data is also available on a reference population ($S=0$), who did not have the
chance to receive the exposure, but for which $Y$ and $Z$ are observed. This
may include pharmacoepidemiologic studies where a reference population may be
sourced from time periods when a new treatment was not yet available.
Alternatively, in environmental health studies, a natural reference population
may be determined based on physical/spatial considerations that would have
prevented access to the exposure.

Let $Y^{a}$ define the potential outcome that would have been observed, had an
individual been assigned to intervention level $A=a$. Then for the moment, we
will consider identification of the conditional ATT in the target population:

$E(Y^{1}-Y^{0}|A=1,Z=z,S=1).$

We will now attempt to link the above expression to the observed data. It
follows from Tchetgen Tchetgen and Vansteelandt, (2013) that one can
parametrise the outcome conditional mean function $E(Y|A=a,Z=z,S=1)$ as

$\displaystyle E(Y|A=a,z,S=1)$
$\displaystyle=E(Y^{a}-Y^{0}|A=a,z,S=1)+E(Y^{0}|A=a,z,S=1)-E(Y^{0}|A=0,z,S=1)$
$\displaystyle\quad-\sum_{a^{\prime}}\left\\{E(Y^{0}|A=a^{\prime},z,S=1)-E(Y^{0}|A=0,z,S=1)\right\\}Pr(A=a^{\prime}|z,S=1)$
$\displaystyle\quad+E(Y^{0}|z,S=1)$
$\displaystyle=\beta(a,z)+\gamma(z)\\{a-f(A=1|z,S=1)\\}+t(z)+c$

where

$\displaystyle\beta(a,z)=E(Y^{a}-Y^{0}|A=a,z,S=1)$
$\displaystyle\gamma(z)=E(Y^{0}|A=1,z,S=1)-E(Y^{0}|A=0,z,S=1)$ $\displaystyle
t(z)=E(Y^{0}|Z=z,S=1)-E(Y^{0}|Z=0,S=1)$ $\displaystyle c=E(Y^{0}|Z=0,S=1)$

where we assume consistency to hold. A quick verification reveals that
$E(Y|A=a,Z=z,S=1)$ has four degrees of freedom: $E(Y|A=1,Z=1,S=1)$,
$E(Y|A=0,Z=1,S=1)$, $E(Y|A=1,Z=0,S=1)$ and $E(Y|A=0,Z=0,S=1)$. However, the
expression $\beta(a,z)+\gamma(z)\\{a-\pi(z)\\}+t(z)+c$ has 6 degrees of
freedom; note that $Pr(A=1|z,S=1)$ can be identified separately based on the
association between $A$ and $Z$ in target population. Thus the identification
task is to reduce this expression by two degrees of freedom.

There are several ways that this can be done; the first is to assume (partial)
conditional exchangeability given $Z$:
$A\rotatebox[origin={c}]{90.0}{$\models$}Y^{0}|Z=z,S=1$, such that
$\gamma(z)=0$ for each $z$. Unfortunately, in many studies, this assumption
may not be plausible, and key confounders may either be missing or measured
with error. Suppose instead that $Z$ satisfies the instrumental variable
conditions IV.1-IV.3. In that case, it follows that $t(z)+c=E(Y^{0})$ (a
constant). However, a further assumption is required to lose an extra degree
of freedom. One option is to restrict the treatment effect, such that
$\beta(a,z)=\beta(a)$; in other words, the effect of $A$ should not be
modified by the instrument (Hernán and Robins,, 2006). It may not be palatable
to place such a restriction on the target of interest, unless there is
existing scientific knowledge to support it. Therefore, Tchetgen Tchetgen and
Vansteelandt, (2013) instead restrict the selection bias term, such that
$\gamma(Z)$ does not depend on $Z$. This leads to alternative results for
identification and inference. However, even if such a restriction is
plausible, this strategy requires having access to a valid instrumental
variable, which may not be possible in many observational studies.

We close this section by describing one last method for identification that is
relevant for developments in later sections. One might lose two degrees of
freedom under an assumption that $\gamma(z)$ can be learnt using the
association between treatment and an additional variable. For example, if one
has access to a negative control outcome $W$, then if

$\displaystyle E(W|A=1,z,S=1)-E(W|A=0,z,S=1)$
$\displaystyle=E(Y^{0}|A=1,z,S=1)-E(Y^{0}|A=0,z,S=1)$ (1)

then $W$ can be harnessed to learn the selection bias. This assumption is know
as additive equi-confounding and essentially requires that $W$ and $Y$ are
measured on the same scale (Sofer et al.,, 2016). A special case arises when
$W$ is a baseline measurement of the outcome, in that case, the above
assumption is equivalent to the parallel-trends assumption, and this
identification strategy boils down to the DiD approach.

###  2.2 Bespoke instrumental variables

Suppose that the measured covariates can be decomposed as
$(Z,\mathbf{C}^{T})^{T}$, where (for the moment) $Z$ is binary. In what
follows, we will convert $Z$, which may be a confounder that fails to satisfy
the three instrumental variable conditions, into a ‘bespoke’ instrument by
leveraging information on the reference population. The key assumptions
underpinning our approach are now given below:

######  Assumption 1.

Consistency: if $A=a$ and $S=1$, then $Y^{a}=Y$.

######  Assumption 2.

Degenerate reference population: we have

$E(Y|z,S=0,\mathbf{c})=E(Y^{0}|z,S=0,\mathbf{c})$

######  Assumption 3.

Partial population exchangeability:

$\displaystyle E(Y^{0}|z,S=1,\mathbf{c})-E(Y^{0}|Z=0,S=1,\mathbf{c})$
$\displaystyle=E(Y^{0}|z,S=0,\mathbf{c})-E(Y^{0}|Z=0,S=0,\mathbf{c})$

######  Assumption 4.

Bespoke IV relevance:

$E(A|z,S=1,\mathbf{c})-E(A|Z=0,S=1,\mathbf{c})\neq 0$

We will discuss each of these assumptions in turn, noting first that
Assumption 1 is standard. Assumption 2 states that in the reference
population, the observed outcome (conditional on $\mathbf{C}$) is equal to the
potential outcome under no treatment. Such a condition can be made plausible
if the reference population is sensibly chosen, such that by an exogenous
intervention which may be a physical or temporal restriction, they were unable
to receive treatment. Note that in those with $S=0$, the usual positivity
assumption is expected to fail, in the sense that $P(A=1|z,S=0,\mathbf{c})=0$;
hence the Bespoke IV design allows one to leverage subgroups of the patient
populations for which positivity is violated. Assumption 3 requires that the
(conditional) association between the potential outcome $Y_{0}$ and $Z$ is
constant between the reference and target populations. This is not a testable
assumption, and facilitates the borrowing of information from the reference
population in order to obtain identification. Note that full population
exchangeability would require that
$E(Y^{0}|z,S=1,\mathbf{c})=E(Y^{0}|z,S=0,\mathbf{c})$, such that this is a
strictly weaker condition. Finally, Assumption 4 demands that $Z$ is
predictive of the exposure, and would be satisfied if $Z$ is a confounder (and
ideally a strong one). This assumption is similar to IV.1; note however, that
$Z$ is not required to satisfy either IV.2 or IV.3.

Similar to the IV set-up, Assumptions 1-4 will be sufficient to test the
causal null hypothesis that $E(Y^{1}-Y^{0}|A=1,\mathbf{z},S=1,\mathbf{c})=0$
but are insufficient to identify the causal effect of interest. Therefore, we
will posit two additional assumptions, of which only one is required to hold
to yield identification:

######  Assumption 5.

No effect modification (NEM):

$E(Y^{1}-Y^{0}|A=1,Z=1,S=1,\mathbf{c})=E(Y^{1}-Y^{0}|A=1,Z=0,S=1,\mathbf{c})=\beta(1,\mathbf{c})$

######  Assumption 6.

No selection modification (NSM):

$E(Y^{0}|A=1,z,S=1,\mathbf{c})-E(Y^{0}|A=0,z,S=1,\mathbf{c})=\gamma(\mathbf{c})$

The first of these requires that the conditional ATT does not depend on $Z$,
whereas the second assumption requires that the selection bias is not a
function of the bespoke IV.

Then we are in a position to give the main theorem for identification of the
conditional ATT.

######  Theorem 2.1.

Suppose that Assumptions 1-3 hold. Then we have that

$\displaystyle t(z,\mathbf{c})$
$\displaystyle=E(Y|Z=z,S=0,\mathbf{c})-E(Y|Z=0,S=0,\mathbf{c})$

Further, under Assumptions 1-5,

$\displaystyle E(Y^{1}-Y^{0}|A=1,Z=z,S=1,\mathbf{C}=\mathbf{c})$
$\displaystyle=\frac{E\\{Y-t(Z,\mathbf{c})|Z=1,S=1,\mathbf{c}\\}-E\\{Y-t(Z,\mathbf{c})|Z=0,S=1,\mathbf{c}\\}}{E(A|Z=1,S=1,\mathbf{c})-E(A|Z=0,S=1,\mathbf{c})}$

and alternatively, under Assumptions 1-4 and 6,

$\displaystyle E(Y^{1}-Y^{0}|A=1,Z=z,S=1,\mathbf{C}=\mathbf{c})$
$\displaystyle=E(Y|A=1,z,S=1,\mathbf{c})-E(Y|A=0,z,S=1,\mathbf{c})$
$\displaystyle\quad+\frac{E\\{Y-t(Z,\mathbf{c})|A=0,Z=1,S=1,\mathbf{c}\\}-E\\{Y-t(Z,\mathbf{c})|A=0,Z=0,S=1,\mathbf{c}\\}}{E(A|Z=1,S=1,\mathbf{c})-E(A|Z=0,S=1,\mathbf{c})}$

The first two results here follow directly from Result 1 in the Appendix of
Richardson and Tchetgen Tchetgen, (2021). The final result can be shown along
the same lines, using the previous parameterisation of the outcome conditional
mean function. To give some intuition for Theorem 2.1 (using this
parametrisation), for a fixed $\mathbf{c}$ we have:

$\displaystyle E(Y|A=a,z,S=1,\mathbf{c})=$
$\displaystyle\beta(a,z,\mathbf{c})+\gamma(z,\mathbf{c})\\{a-f(A=1|z,S=1,\mathbf{c})\\}$
$\displaystyle+t(z,\mathbf{c})+E(Y^{0}|Z=0,S=1,\mathbf{c})$

As before, the right hand side must be reduced by two parameters. The bespoke
IV strategy does this by replacing $t(z,\mathbf{c})$ by a quantity that can be
calculated using a reference population, and then removing one extra parameter
by restricting either the treatment effect or selection bias. A further
implication from Theorem 2.1 is that the marginal ATT $E(Y^{1}-Y^{0}|A=1,S=1)$
can also be identified, by averaging conditional treatment effects over the
distribution of $Z$ and $\mathbf{C}$ in those with $A=1$ and $S=1$. We return
to this in Section 3.4.

One can also harness baseline measurements of the outcome in order to generate
a reference population; for reasons that will become clear, we label this
approach ‘bespoke IV-DiD’. To be specific, let $Y(1)$ now denote the end-of-
study outcome, and $Y(0)$ denote a baseline measurement. Then if we replace
Assumptions 2 and 3 with the conditions that

$\displaystyle E\\{Y(0)|z,\mathbf{c}\\}=E\\{Y^{0}(0)|z,\mathbf{c}\\}$ (2)

and

$\displaystyle
E\\{Y^{0}(1)|z,\mathbf{c}\\}-E\\{Y^{0}(1)|Z=0,\mathbf{c}\\}=E\\{Y^{0}(0)|z,\mathbf{c}\\}-E\\{Y^{0}(0)|Z=0,\mathbf{c}\\}$
(3)

then it follows from Theorem 2.1 that one can identify the effect
$E\\{Y^{1}(1)-Y^{0}(1)|A=1,Z=z,\mathbf{C}=\mathbf{c}\\}$ by leveraging the
(conditional) association between the baseline outcome $Y(0)$ and $Z$.
Specifically, if we assume that the causal effect does not depend on $Z$,

$\displaystyle E\\{Y^{1}(1)-Y^{0}(1)|A=1,Z=z,\mathbf{C}=\mathbf{c}\\}$
$\displaystyle=\frac{E\\{Y(1)-Y(0)|Z=1,\mathbf{c}\\}-E\\{Y(1)-Y(0)|Z=0,\mathbf{c}\\}}{E(A|Z=1,\mathbf{c})-E(A|Z=0,\mathbf{c})}$
(4)

and if we alternatively assume

$\displaystyle
E\\{Y^{0}(1)|A=1,z,\mathbf{c}\\}-E\\{Y^{0}(1)|A=0,z,\mathbf{c}\\}=\gamma(\mathbf{c})$
(5)

e.g. the selection bias does not depend on $Z$ after conditioning on
covariates, then we have

$\displaystyle E\\{Y^{1}(1)-Y^{0}(1)|A=1,Z=z,\mathbf{C}=\mathbf{c}\\}$
$\displaystyle=E\\{Y(1)|A=1,z,\mathbf{c}\\}-E\\{Y(1)|A=0,z,\mathbf{c}\\}$
$\displaystyle\quad+\frac{E\\{Y(1)|A=0,Z=1,\mathbf{c}\\}-E\\{Y(1)|A=0,Z=0,\mathbf{c}\\}}{E(A|Z=1,\mathbf{c})-E(A|Z=0,\mathbf{c})}-\frac{E\\{Y(0)|Z=1,\mathbf{c}\\}-E\\{Y(0)|Z=0,\mathbf{c}\\}}{E(A|Z=1,\mathbf{c})-E(A|Z=0,\mathbf{c})}$

Ye et al., (2020) arrive at an expression similar to (2.2), although they
consider a setting where two measurements of the exposure (in addition to the
outcome) are available. In the denominator of their expression, the contrast
involves the difference in the exposure measurements. In addition, Ye et al.,
(2020) target a different causal estimand and their identification results are
separate from ours, as they require that $Z$ satisfies the first instrumental
condition (unconfoundedness). Their identification functional also appears in
the econometrics literature on ‘fuzzy’ DiD designs (De Chaisemartin and
d’Haultfoeuille,, 2018), where it is usually interpreted (under separate
assumptions) as a local average treatment effect.

The condition (2) are arguably more plausible (in general) than the
corresponding Assumption 2 in the bespoke IV design, from the point of view
that the best reference population is the original sample itself. As we will
see, inference under the bespoke IV-DiD design will turn out to have
additional advantages in terms of robustness and efficiency.

######  Remark.

Instead of (5), one may make the assumption

$\displaystyle
E\\{Y^{0}(1)-Y^{0}(0)|A=1,z,\mathbf{c}\\}-E\\{Y^{0}(1)-Y^{0}(0)|A=0,z,\mathbf{c}\\}=\gamma(\mathbf{c})$
(6)

e.g. the selection bias in a DiD analysis (rather than the comparison of end-
of-study outcomes) does not depend on $Z$ after conditioning on covariates.
This assumption leads to an alternative identification result

$\displaystyle E\\{Y^{1}(1)-Y^{0}(1)|A=1,Z=z,\mathbf{C}=\mathbf{c}\\}$
$\displaystyle=E\\{Y(1)-Y(0)|A=1,z,\mathbf{c}\\}-E\\{Y(1)-Y(0)|A=0,z,\mathbf{c}\\}$
$\displaystyle\quad+\frac{E\\{Y(1)-Y(0)|A=0,Z=1,\mathbf{c}\\}-E\\{Y(1)-Y(0)|A=0,Z=0,\mathbf{c}\\}}{E(A|Z=1,\mathbf{c})-E(A|Z=0,\mathbf{c})}$

which suggests an interesting way of combining the IV and DiD strategies. For
example, assumption (6) weakens the usual parallel trends assumption

$E\\{Y^{0}(1)-Y^{0}(0)|A=1,z,\mathbf{c}\\}-E\\{Y^{0}(1)-Y^{0}(0)|A=0,z,\mathbf{c}\\}=0$

such that the left hand side of the above is no longer required to equal zero
(although it cannot depend on $Z$). A parallel trend-type assumption (3) is
required with respect to the conditional association between $Y(0)$ and $Z$,
although if many covariates are collected, one has at least the freedom to
choose the $Z$ for which (3) is most plausible.

The two NSM conditions (5) and (6) are distinct and one does not imply the
other. We focus for now on analyses under (5) since it is closer in spirit to
the original NSM Assumption 6, where the condition was placed on the selection
bias in the target population. Nevertheless, (6) holds appeal as an
alternative to the standard parallel trends assumption in DiD.

###  2.3 Comparison with existing approaches

####  2.3.1 IV analyses with potentially invalid instruments

As discussed above, $Z$ does not need to satisfy the three IV conditions,
since we can allow for violations of both unconfoundedness and the exclusion
restriction. In the DiD set-up, the partial population exchangeability
assumption (3) can be interpreted as a (conditional) exclusion restriction for
the difference $Y^{0}(1)-Y^{0}(0)$ rather than $Y^{0}(1)$; note that this does
not imply the standard exclusion restriction (and vice versa). Hence the
methods described in this paper can be viewed as part of the growing
literature on IV analyses with invalid instruments (Kang et al.,, 2016;
Tchetgen Tchetgen et al.,, 2021), where there is a focus on robustness to
violations of the exclusion restriction in Mendialian Randomisation studies.
The assumption (3) may in particular be applicable in studies when repeated
measurements of the outcome are collected, and the gene-outcome association
(in the absence of treatment) may be non-null but approximately constant over
time. If $Z$ is in truth a valid instrument (in addition to satisfying
Assumptions 1-4), then the proposed estimators in the following section will
continue to target the causal effects, but may suffer from a loss of
efficiency relative to standard IV estimators by not harnessing the exclusion
restriction.

####  2.3.2 Triple differences

There is also a link with the so-called Difference-in-Difference-in-
Differences or Triple Difference identification strategy (Gruber,, 1994). This
approach typically targets a contrast e.g.
$E\\{Y^{1}(1)-Y^{0}(1)|Z=z,\mathbf{c}\\}$, and allows for violation of the
parallel trend assumption at $Z=z$ so long as the resulting bias in the naïve
DiD analysis is equal to the DiD analysis in those with $Z=z^{\prime}$ for
some $z^{\prime}\neq z$. This essentially requires that the selection bias
operates in the same way at the difference levels of $Z$, similar to (6), but
also that the treatment has no effect at $Z=z^{\prime}$ e.g.
$E\\{Y^{1}(1)-Y^{0}(1)|Z=z^{\prime},\mathbf{c}\\}=0$. This key difference
leads to alternative identification results, as this strategy therefore relies
on a null causal effect rather than stability in the trend $Y^{0}(1)-Y^{0}(0)$
between levels of $Z$.

####  2.3.3 Fuzzy regression discontinuity design

The bespoke IV approach also shares similarities with a certain version of the
Fuzzy Regression Discontinuity Design or Fuzzy RDD (Hahn et al.,, 2001). For
example, Marie, (2015) considered the impact of an early release program on
future criminal behaviour in prisoners in England and Wales. Only prisoners of
a certain age or sentence length were eligible for early release, and around a
quarter of prisoners who were eligible were granted early release. The Fuzzy
RDD approach harnesses discontinuities in the probability of receiving
exposure with respect to a ‘running variable’ (age or sentence length). In
that framework one usually targets the causal effect in individuals whose
exposure status would change if the value of the running variable is shifted
from slightly to the left to slightly to the right of the discontinuity. As
with other complier causal effects, a disadvantage is that this population may
not be well understood, and the resulting estimates may not be more broadly
generalisable.

The bespoke IV methodology provides an alternative approach for leveraging
this type of discontinuity in the chance of being assigned treatment. The
running variable can be used to define the target and reference populations,
so long as there are values at which no individuals could receive treatment.
Then the target population is characterised by the region of the running
variable at which treatment was not degenerate. For identification, one would
leverage an observed covariate that is a priori believed to have a stable
association with the potential outcome under no intervention across different
levels of the running variable.

##  3 Estimation and inference

###  3.1 Semiparametric Theory

In this section, we will first develop a semiparametric efficiency theory for
the conditional treatment effect
$E(Y^{\mathbf{a}}-Y^{\mathbf{0}}|\mathbf{A}=\mathbf{a},\mathbf{Z}=\mathbf{z},S=1,\mathbf{C}=\mathbf{c})=\beta(\mathbf{a,z,c})$
based on a bespoke IV strategy. We allow for $\mathbf{A}$, $\mathbf{Z}$ and
$\mathbf{C}$ to be continuous or discrete and potentially vector-valued; the
previous NEM and NSM assumptions can be generalised as

######  Assumption 7.

No effect modification (NEM):

$\beta(\mathbf{a,z,c})=\beta(\mathbf{a,c})$

######  Assumption 8.

No selection modification (NSM):
The generalised selection bias function

$q(\mathbf{a},\mathbf{Z,C})=E(Y^{\mathbf{0}}|\mathbf{A=a,Z},S=1,\mathbf{C})-E(Y^{\mathbf{0}}|\mathbf{A=0,Z},S=1,\mathbf{C})$

satisfies the restriction $q(\mathbf{a,z,c})=q(\mathbf{a,c})$

Generalised versions of Assumptions 1-4 are given in the Appendix. Also, we
will use the notation

$\displaystyle\epsilon$
$\displaystyle=Y-\beta(\mathbf{A,Z,C})S-[q(\mathbf{A,Z,C})-E\\{q(\mathbf{A,Z,C})|\mathbf{Z},S=1,\mathbf{C}\\}]S$
$\displaystyle\quad-t(\mathbf{Z,C})-b_{1}(\mathbf{C})S-b_{0}(\mathbf{C})$
$\displaystyle\epsilon^{*}$
$\displaystyle=\epsilon+[q(\mathbf{A,Z,C})-E\\{q(\mathbf{A,Z,C})|\mathbf{Z},S=1,\mathbf{C}\\}]S$

where

$\displaystyle
t(\mathbf{z,C})=E(Y^{\mathbf{0}}|\mathbf{Z=z},S=1,\mathbf{C})-E(Y^{\mathbf{0}}|\mathbf{Z=0},S=1,\mathbf{C})$
$\displaystyle
b_{1}(\mathbf{C})=E(Y^{\mathbf{0}}|\mathbf{Z=0},S=1,\mathbf{C})-E(Y^{\mathbf{0}}|\mathbf{Z=0},S=0,\mathbf{C})$
$\displaystyle b_{0}(\mathbf{C})=E(Y^{0}|\mathbf{Z=0},S=0,\mathbf{C}).$

We postulate a semiparametric model

$\displaystyle\beta(\mathbf{a,z,c})=\beta(\mathbf{a,z,c};\psi^{\dagger}),$ (7)

where $\psi^{\dagger}$ is an unknown finite-dimensional parameter vector. We
will let $\mathcal{M}_{NEM}$ denote the model defined by restriction (7) along
with Assumptions 7 and 15-18 (in the Appendix); $\mathcal{M}_{NSM}$
alternatively denotes a model defined by (7), and Assumptions 8 and 15-18.

######  Theorem 3.1.

(Part a - NEM) under the semiparametric model $\mathcal{M}_{NEM}$, the
orthocomplement to the nuisance tangent space is given as

$\displaystyle\bigg{\\{}$
$\displaystyle\phi(\mathbf{Z},S,\mathbf{C})\epsilon^{*}(\psi^{\dagger}):\phi(\mathbf{Z},S,\mathbf{C})\in\Omega\bigg{\\}}\cap
L^{0}_{2}$

where

$\Omega=\\{\phi(\mathbf{Z},S,\mathbf{C}):E\\{\phi(\mathbf{Z},S,\mathbf{C})|\mathbf{Z,C}\\}=E\\{\phi(\mathbf{Z},S,\mathbf{C})|S,\mathbf{C}\\}=0\\}.$

(Part b - NSM) The orthocomplement to the nuisance tangent space under model
$\mathcal{M}_{NSM}$ is given by

$\displaystyle\bigg{\\{}$
$\displaystyle\left\\{S\kappa(\mathbf{A,Z,C})+\phi(\mathbf{Z},S,\mathbf{C})\right\\}\epsilon(\psi^{\dagger})+S\phi(\mathbf{Z},S,\mathbf{C})[q(\mathbf{A,C})-E\\{q(\mathbf{A,C})|\mathbf{Z},S=1,\mathbf{C}\\}]$
$\displaystyle:\kappa(\mathbf{A,Z,C})\in\Gamma,\phi(\mathbf{Z},S,\mathbf{C})\in\Omega\bigg{\\}}\cap
L^{0}_{2}$

where

$\displaystyle\Gamma=\left\\{\kappa(\mathbf{A,Z,C}):E\\{\kappa(\mathbf{A,Z,C})|\mathbf{Z},S=1,\mathbf{C}\\}=E\\{\kappa(\mathbf{A,Z,C})|\mathbf{A},S=1,\mathbf{C}\\}=0\right\\}$

As reviewed e.g. in Bickel et al., (1993), by deriving the orthocomplement of
the nuisance tangent space, we obtain the class of influence functions of all
regular and asymptotically linear (RAL) estimators of $\psi^{\dagger}$. In
turn, knowing this class motivates the construction of RAL estimators e.g. by
solving estimating equations based on a chosen influence function.
Specifically, under Assumption 5, we can estimate $\psi^{\dagger}$ as the
solution to the equations

$\displaystyle
0=\sum^{n}_{i=1}\phi(\mathbf{Z}_{i},S_{i},\mathbf{C}_{i})\epsilon_{i}^{*}(\psi^{\dagger})$

for a chosen $\phi(\mathbf{Z},S,\mathbf{C})$ that satisfies the above
restrictions; likewise, under (8) we solve

$\displaystyle
0=\begin{pmatrix}&\sum^{n}_{i=1}\phi(\mathbf{Z}_{i},S_{i},\mathbf{C}_{i})\epsilon^{*}_{i}(\psi^{\dagger})\\\
&\sum^{n}_{i=1}S_{i}\kappa(\mathbf{A}_{i},\mathbf{Z}_{i},\mathbf{C}_{i})\epsilon_{i}(\psi^{\dagger})\end{pmatrix}$

for a chosen $\phi(\mathbf{Z},S,\mathbf{C})$ and $\kappa(\mathbf{A,Z,C})$.

Converting the above result into an estimation strategy requires a choice of
$\phi(\mathbf{Z},S,\mathbf{C})$ and $\kappa(\mathbf{A,Z,C})$. In order to
provide guidance, we first define a further definition.

######  Definition 1.

Admissible Independence Density (Tchetgen Tchetgen et al.,, 2010). Consider
three potentially vector-valued random variables $\mathbf{X}_{1}$,
$\mathbf{X}_{2}$ and $\mathbf{X}_{3}$. Let
$f^{\ddagger}(\mathbf{X}_{1},\mathbf{X}_{2}|\mathbf{X}_{3})=f^{\ddagger}(\mathbf{X}_{1}|\mathbf{X}_{3})f^{\ddagger}(\mathbf{X}_{2}|\mathbf{X}_{3})$
denote a fixed density that makes $\mathbf{X}_{1}$ and $\mathbf{X}_{2}$
conditionally independent given $\mathbf{X}_{3}$. Then
$f^{\ddagger}(\mathbf{X}_{1},\mathbf{X}_{2}|\mathbf{X}_{3})$ is an admissible
independence density if it is absolutely continuous w.r.t. the true joint law
$f(\mathbf{X}_{1},\mathbf{X}_{2}|\mathbf{X}_{3})$ with probability one.

It follows furthermore from Tchetgen Tchetgen et al., (2010) that, for
admissible independence densities $f^{\ddagger}(\mathbf{Z},S|\mathbf{C})$ and
$f^{\ddagger}(\mathbf{A,Z}|S=1,\mathbf{C})$ we have

$\displaystyle\Omega=$
$\displaystyle\bigg{\\{}\frac{f^{\ddagger}(\mathbf{Z},S|\mathbf{C})}{f(\mathbf{Z},S|\mathbf{C})}[r_{0}(\mathbf{Z},S,\mathbf{C})-E^{\ddagger}\\{r_{0}(\mathbf{Z},S,\mathbf{C})|\mathbf{Z,C}\\}$
$\displaystyle-E^{\ddagger}\\{r_{0}(\mathbf{Z},S,\mathbf{C})|S,\mathbf{C}\\}+E^{\ddagger}\\{r_{0}(\mathbf{Z},S,\mathbf{C})|\mathbf{C}\\}]:\textrm{$r_{0}(\mathbf{Z},S,\mathbf{C})$
unrestricted}\bigg{\\}}$ (8) $\displaystyle\Gamma=$
$\displaystyle\bigg{\\{}\frac{f^{\ddagger}(\mathbf{A,Z}|S=1,\mathbf{C})}{f(\mathbf{A,Z}|S=1,\mathbf{C})}[r_{1}(\mathbf{A,Z,C})-E^{\ddagger}\\{r_{1}(\mathbf{A,Z,C})|\mathbf{Z},S=1,\mathbf{C}\\}$
$\displaystyle-E^{\ddagger}\\{r_{1}(\mathbf{A,Z,C})|\mathbf{A},S=1,\mathbf{C}\\}+E^{\ddagger}\\{r_{1}(\mathbf{A,Z,C})|S=1,\mathbf{C}\\}]:\textrm{$r_{1}(\mathbf{A,Z,C})$
unrestricted}\bigg{\\}}$ (9)

respectively, where $E^{\ddagger}(\cdot|\cdot)$ denotes a (conditional)
expectation taken w.r.t. $f^{\ddagger}(\cdot|\cdot)$. To construct an
estimator, one can choose any $f^{\ddagger}(\mathbf{Z},S|\mathbf{C})$ and
$f^{\ddagger}(\mathbf{A,Z}|S=1,\mathbf{C})$ that satisfy the above definition.
One could choose $f(\mathbf{Z}|\mathbf{C})f(S|\mathbf{C})$ for
$f^{\ddagger}(\mathbf{Z},S|\mathbf{C})$ and
$f(\mathbf{A}|S=1,\mathbf{C})f(\mathbf{Z}|S=1,\mathbf{C})$ for
$f^{\ddagger}(\mathbf{A,Z}|S=1,\mathbf{C})$; it follows from Vansteelandt et
al., (2008) that this may be efficient when $\mathbf{Z}$ and S are
conditionally independent given $\mathbf{C}$, and $\mathbf{A}$ and
$\mathbf{Z}$ are conditionally independent given $\mathbf{C}$ and $S=1$. We
emphasise that there is no requirement to choose the true densities, as the
following example shows.

######  Example 1.

Let $\mathbf{A}$ and $\mathbf{Z}$ both be binary. Suppose one sets
$f^{\ddagger}(Z=1|\mathbf{C})=f^{\ddagger}(S=1|\mathbf{C})=f^{\ddagger}(A=1|S=1,\mathbf{C})=f^{\ddagger}(Z=1|S=1,\mathbf{C})=0.5$
(pretending $A$, $Z$ and $S$ are randomly assigned). Then $\Omega$ and
$\Gamma$ are equivalently characterised respectively as

$\displaystyle\Omega$
$\displaystyle=\left\\{\frac{m(\mathbf{C})(-1)^{Z+S}}{f(Z,S|\mathbf{C})}:\textrm{$m(\mathbf{C})$
unrestricted}\right\\}$ $\displaystyle\Gamma$
$\displaystyle=\left\\{\frac{h(\mathbf{C})(-1)^{A+Z}}{f(A,Z|S=1,\mathbf{C})}:\textrm{$h(\mathbf{C})$
unrestricted}\right\\}$

if we further choose $r_{0}(Z,\mathbf{C},S)=16(A-0.5)(Z-0.5)m(\mathbf{C})$ and
$r_{1}(A,Z,\mathbf{C})=16(A-0.5)(Z-0.5)h(\mathbf{C})$. Then e.g. under NEM,
one can estimate $\psi$ via the unbiased equating function

$\displaystyle\frac{m(\mathbf{C})(-1)^{Z+S}}{f(Z,S|\mathbf{C})}\epsilon^{*}(\psi^{\dagger})$
(10)

An alternative strategy to compute $\phi(\mathbf{Z},S,\mathbf{C})$ and
$\kappa(\mathbf{A,Z,C})$ is through the alternating conditional expectations
(ACE) algorithm (Bickel et al.,, 1993). From an initial choice
$\phi^{(1)}(\mathbf{Z}_{i},S_{i},\mathbf{C}_{i})$, one then computes

$\displaystyle\phi^{(2j)}(\mathbf{Z}_{i},S_{i},\mathbf{C}_{i})=\phi^{(2j-1)}(\mathbf{Z}_{i},S_{i},\mathbf{C}_{i})-E\\{\phi^{(2j-1)}(\mathbf{Z}_{i},S_{i},\mathbf{C}_{i})|\mathbf{Z}_{i},\mathbf{C}_{i}\\}$
$\displaystyle\phi^{(2j+1)}(\mathbf{Z}_{i},S_{i},\mathbf{C}_{i})=\phi^{(2j)}(\mathbf{Z}_{i},S_{i},\mathbf{C}_{i})-E\\{\phi^{(2j)}(\mathbf{Z}_{i},S_{i},\mathbf{C}_{i})|S_{i},\mathbf{C}_{i}\\}$

for $j=1,2,...$ until convergence is reached at
$\lim_{j\to\infty}\phi^{(2j+1)}(\mathbf{Z}_{i},S_{i},\mathbf{C}_{i})$; this
can then be repeated for $\kappa(\mathbf{A,Z,C})$. This may not lead to
closed-form representations of functions of interest, since there is no closed
form expression for $\kappa(\mathbf{A,Z,C})$ when $\mathbf{A}$ and
$\mathbf{Z}$ include continuous components (Vansteelandt et al.,, 2008). It
may nevertheless lead to estimators with improved finite sample performance
since this approach does not involve inverse weighting by a density.

We end this section by extending the previous results to the bespoke IV-DiD
design; for the remainder of this section, let us temporarily define (or
redefine)

$\displaystyle
q(\mathbf{a,Z,C})=E(Y^{\mathbf{0}}(1)|\mathbf{A=a},\mathbf{Z},\mathbf{C})-E(Y^{\mathbf{0}}(1)|\mathbf{A=0},\mathbf{Z},\mathbf{C})$
$\displaystyle
t(\mathbf{z,C})=E(Y^{\mathbf{0}}(1)|\mathbf{Z=z},\mathbf{C})-E(Y^{\mathbf{0}}(1)|\mathbf{Z=0},\mathbf{C})$
$\displaystyle
b^{*}_{1}(\mathbf{C})=E(Y^{\mathbf{0}}(1)|\mathbf{Z=0},\mathbf{C})$
$\displaystyle
b_{0}(\mathbf{C})=E(Y^{\mathbf{0}}(0)|\mathbf{Z=0},\mathbf{C}).$

Under slightly altered versions of Assumptions 16 and 17, $t(\mathbf{z,C})$ is
identified as

$t(\mathbf{z,C})=E(Y(0)|\mathbf{Z=z},\mathbf{C})-E(Y(0)|\mathbf{Z=0},\mathbf{C})$

The following result is a corollary of Theorem 3.1.

######  Corollary 3.1.1.

In the bespoke IV-DiD design, the (conditional) density of $S$ is degenerate
and
$f(\mathbf{Z}|S=1,\mathbf{C})=f(\mathbf{Z}|S=0,\mathbf{C})=f(\mathbf{Z}|\mathbf{C})$,
such that the nuisance tangent space is smaller. Under the semiparametric
model implied by restriction (7) and the NEM Assumption 7, the estimating
function for $\psi^{\dagger}$ implied by Theorem 3.1 simplifies to

$\displaystyle[\phi(\mathbf{Z},\mathbf{C})-E\\{\phi(\mathbf{Z},\mathbf{C})|\mathbf{C}\\}][Y(1)-Y(0)-\beta(\mathbf{A,C};\psi^{\dagger})-\\{b^{*}_{1}(\mathbf{C})-b_{0}(\mathbf{C})\\}]$
(11)

where $\phi(\mathbf{Z,C})$ is arbitrary. Also, if
$q(\mathbf{a,z,c})=q(\mathbf{a,c})$, then the estimating function(s) for
$\psi^{\dagger}$ derived under the NSM Assumption 8 reduce to

$\displaystyle\begin{pmatrix}[\phi(\mathbf{Z},\mathbf{C})-E\\{\phi(\mathbf{Z},\mathbf{C})|\mathbf{C}\\}][Y(1)-Y(0)-\beta(\mathbf{A,Z,C};\psi^{\dagger})-\\{b^{*}_{1}(\mathbf{C})-b_{0}(\mathbf{C})\\}]\\\
\kappa(\mathbf{A,Z,C})\left(Y(1)-\beta(\mathbf{A,Z,C};\psi^{\dagger})-[q(\mathbf{A,C})-E\\{q(\mathbf{A,C})|\mathbf{Z,C}\\}]-t(\mathbf{Z,C})-b^{*}_{1}(\mathbf{C})\right)\end{pmatrix}$
(12)

where $\phi(\mathbf{Z,C})$ is arbitrary and $\kappa(\mathbf{A,Z,C})$ satisfies
$E\\{\kappa(\mathbf{A,Z,C})|\mathbf{A,C}\\}=E\\{\kappa(\mathbf{A,Z,C})|\mathbf{Z,C}=0$.

###  3.2 De-biased machine learning and multiply robust estimation

The estimation strategies previously described are not generally feasible,
because the estimating equations for $\psi^{\dagger}$ involve nuisance
parameters that are typically unknown. One approach to deal with this is to
plug-in estimates obtained from nonparametric estimators or flexible
statistical learning methods. It follows from more general results in
semiparametric theory, that if all nuisance parameter estimators converge at a
rate faster than $n^{1/4}$ and sample-splitting/cross-fitting is used, then
the resulting estimators of $\psi^{\dagger}$ are RAL with variance equal to
the variance of its influence function; see e.g. Chernozhukov et al., (2018)
for a review. In the Appendix, we therefore use Theorem 3.1 to construct
cross-fit de-biased machine learning-based estimators of $\psi^{\dagger}$.

The bias of influence function-based estimators is second order, and the
aforementioned $n^{1/4}$-rate condition is sufficient for parametric-rate
inference on the ATT (assuming cross-fitting is used). However, the actual
structure of the bias is context dependent; characterising this structure
gives a more nuanced understanding about the conditions under which the bias
disappears. In certain problems, estimators with a doubly or multiply robust
property may be available. As discussed in Rotnitzky et al., (2021),
understanding whether double or multiple robustness is attainable is important
because it allows one to trade off assumptions on nuisance parameter
estimation (e.g. one can be poorly estimated, if another is estimated well)
and potentially allows one to leverage knowledge on components of the data-
generating mechanism to obtain additional robustness and efficiency, such that
one or more estimator(s) might be allowed to converge at a rate slower that
$n^{1/4}$.

Hence to develop a deeper understanding of the bias properties of the
estimators, we will instead consider a scenario where parametric working
models are used for the conditional expectations/densities that arise in
estimating the causal effect of interest. Let us define the following
parametric models $t(\mathbf{Z,C};\nu^{\dagger})$ for $t(\mathbf{Z,C})$,
$b_{0}(\mathbf{C};\theta^{\dagger}_{0})$ for $b_{0}(\mathbf{C})$ and
$b_{1}(\mathbf{C};\theta^{\dagger}_{1})$ for $b_{1}(\mathbf{C})$ that are
required to model the conditional outcome mean for both the reference and the
target population. Here, $t(\mathbf{Z,C};\nu^{\dagger})$,
$b_{0}(\mathbf{C};\theta^{\dagger}_{0})$ and
$b_{1}(\mathbf{C};\theta^{\dagger}_{1})$ are known functions, which are smooth
in $\nu^{\dagger}$, $\theta^{\dagger}_{0}$ and $\theta^{\dagger}_{1}$
respectively. We will sometimes use the notation
$\theta^{\dagger}=(\theta^{\dagger^{T}}_{1},\theta^{\dagger^{T}}_{0})^{T}$.
For modelling the joint conditional density $f(\mathbf{Z},S|\mathbf{C})$, we
will make use of the following parametrisation based on the generalised odds
ratio function

$OR(\mathbf{Z},S,\mathbf{C})=\frac{f(\mathbf{Z}|S,\mathbf{C})f(\mathbf{Z}=\mathbf{z_{0}}|S={s_{0}},\mathbf{C})}{f(\mathbf{Z}=\mathbf{z_{0}}|S,\mathbf{C})f(\mathbf{Z}|S={s_{0}},\mathbf{C})}$

(Tchetgen Tchetgen et al.,, 2010); one can generate
$f(\mathbf{Z},S|\mathbf{C})$, $f(\mathbf{Z}|S,\mathbf{C})$ and
$f(S|\mathbf{Z},\mathbf{C})$ based on specification of
$f(\mathbf{Z}|S=s,\mathbf{C})$, $f(S|\mathbf{Z}=\mathbf{z},\mathbf{C})$ and
$OR(\mathbf{Z},S,\mathbf{C})$ (Chen,, 2007, 2010). The reference values of
$\mathbf{z_{0}}$ and $s_{0}$ are user-specified; in what fallows, we will use
$\mathbf{z_{0}}=\mathbf{0}$ and $s_{0}=0$ as a generic notation. We will
postulate smooth parametric models
$f(\mathbf{Z}|S=0,\mathbf{C};\tau^{\dagger})$,
$f(S|\mathbf{Z}=\mathbf{0},\mathbf{C};\alpha^{\dagger})$ and
$OR(\mathbf{Z},S,\mathbf{C};\rho)$ for $f(\mathbf{Z}|S=0,\mathbf{C})$,
$f(S|\mathbf{Z}=\mathbf{0},\mathbf{C})$ and $OR(\mathbf{Z},S,\mathbf{C})$
respectively. We note that $\nu^{\dagger}$, $\theta^{\dagger}$,
$\tau^{\dagger}$, $\alpha^{\dagger}$ and $\rho^{\dagger}$ are all finite
dimensional parameters.

If we first assume the NEM restriction holds, then consider the following sets
of restrictions on the observed data distribution:

  * •

$\mathcal{M}_{1}$: $t(\mathbf{Z,C})=t(\mathbf{Z,C};\nu^{\dagger})$,
$b_{0}(\mathbf{C})=b_{0}(\mathbf{C};\theta^{\dagger}_{0})$ and
$b_{1}(\mathbf{C})=b_{1}(\mathbf{C};\theta^{\dagger}_{1})$.

  * •

$\mathcal{M}_{2}$:
$f(\mathbf{Z}|S=0,\mathbf{C})=f(\mathbf{Z}|S=0,\mathbf{C};\tau^{\dagger})$,
$OR(\mathbf{Z},S,\mathbf{C})=OR(\mathbf{Z},S,\mathbf{C};\rho^{\dagger})$
and $t(\mathbf{Z,C})=t(\mathbf{Z,C};\nu^{\dagger})$.

  * •

$\mathcal{M}_{3}$:
$f(S|\mathbf{Z}=0,\mathbf{C})=f(S|\mathbf{Z}=0,\mathbf{C};\alpha^{\dagger})$,
$OR(\mathbf{Z},S,\mathbf{C})=OR(\mathbf{Z},S,\mathbf{C};\rho^{\dagger})$, and
$b_{1}(\mathbf{C})=b_{1}(\mathbf{C};\theta^{\dagger}_{1})$.

  * •

$\mathcal{M}_{4}$:
$f(\mathbf{Z}|S=0,\mathbf{C})=f(\mathbf{Z}|S=0,\mathbf{C};\tau^{\dagger})$,
$f(S|\mathbf{Z}=0,\mathbf{C})=f(S|\mathbf{Z}=0,\mathbf{C};\alpha^{\dagger})$
and $OR(\mathbf{Z},S,\mathbf{C})=OR(\mathbf{Z},S,\mathbf{C};\rho^{\dagger})$.

Our goal is to construct an estimator that is quadruply robust, that is,
unbiased if one of these four restrictions on the observed data (in addition
to model $\mathcal{M}_{NEM}$) holds. In order to achieve this, care needs to
be taken regarding the estimation of nuisance parameters. We propose the
following strategy:

  1. 1.

Estimate $\tau^{\dagger}$ and $\alpha^{\dagger}$ as $\hat{\tau}$ and
$\hat{\alpha}$ using maximum likelihood or $m$-estimation.

  2. 2.

Estimate $\rho^{\dagger}$ as $\hat{\rho}$ by solving the equations

$\displaystyle 0=\sum^{n}_{i=1}$
$\displaystyle\left[e_{1}(\mathbf{Z}_{i},\mathbf{C}_{i})-\frac{E\\{e_{1}(\mathbf{Z}_{i},\mathbf{C}_{i})f(S_{i}=1|\mathbf{Z}_{i},\mathbf{C}_{i};\hat{\alpha},\rho^{\dagger})f(S_{i}=0|\mathbf{Z}_{i},\mathbf{C}_{i};\hat{\alpha},\rho^{\dagger})|\mathbf{C}_{i};\hat{\alpha},\hat{\tau},\rho^{\dagger}\\}}{E\\{f(S_{i}=1|\mathbf{Z}_{i},\mathbf{C}_{i};\hat{\alpha},\rho^{\dagger})f(S_{i}=0|\mathbf{Z}_{i},\mathbf{C}_{i};\hat{\alpha},\rho^{\dagger})|\mathbf{C}_{i};\hat{\alpha},\hat{\tau},\rho^{\dagger}\\}}\right]$
$\displaystyle\times\\{S_{i}-f(S_{i}=1|\mathbf{Z}_{i},\mathbf{C}_{i};\hat{\alpha},\rho^{\dagger})\\}$

for $\rho^{\dagger}$ (Tchetgen Tchetgen et al.,, 2010), where
$e_{1}(\mathbf{C})$ is an arbitrary function of the same dimension of
$\rho^{\dagger}$.

  3. 3.

Estimate $\nu^{\dagger}$ and $\theta_{0}^{\dagger}$ as $\tilde{\nu}$ and
$\hat{\theta}_{0}$ by solving the equations

$\displaystyle 0=\sum^{n}_{i=1}$
$\displaystyle(1-S_{i})\begin{pmatrix}e_{2}(\mathbf{Z}_{i},\mathbf{C}_{i})\\\
e_{3}(\mathbf{C}_{i})\end{pmatrix}\epsilon^{*}_{i}(\psi^{\dagger},\nu^{\dagger},\theta^{\dagger})$

for $\nu^{\dagger}$ and $\theta_{0}^{\dagger}$, where
$e_{2}(\mathbf{Z},\mathbf{C})$ and $e_{3}(\mathbf{C})$ are of the same
dimension as $\nu^{\dagger}$ and $\theta^{\dagger}_{0}$ respectively. Re-
estimate $\nu^{\dagger}$ as $\hat{\nu}$ by solving the equations

$\displaystyle
0=\sum^{n}_{i=1}(1-S_{i})[e_{2}(\mathbf{Z}_{i},\mathbf{C}_{i})-E\\{e_{2}(\mathbf{Z}_{i},\mathbf{C}_{i})|S_{i}=0,\mathbf{C}_{i};\hat{\tau}\\}]\epsilon^{*}_{i}(\psi^{\dagger},\nu^{\dagger},\hat{\theta}_{0},\theta_{1}^{\dagger})$

for $\nu^{\dagger}$ (Robins et al.,, 1992).

  4. 4.

Estimate $\theta_{1}^{\dagger}$ and $\psi^{\dagger}$ as $\hat{\theta}_{1}$ and
$\tilde{\psi}$ respectively, by solving the equations

$\displaystyle 0=\sum^{n}_{i=1}$
$\displaystyle\begin{pmatrix}e_{4}(\mathbf{C}_{i})\\{S_{i}-E\\{S_{i}|\mathbf{Z}_{i},\mathbf{C}_{i};\hat{\alpha},\hat{\rho}\\}\\\
e_{5}(\mathbf{Z}_{i},\mathbf{C}_{i})\\{S_{i}-E\\{S_{i}|\mathbf{Z}_{i},\mathbf{C}_{i};\hat{\alpha},\hat{\rho}\\}\end{pmatrix}\epsilon^{*}_{i}(\psi^{\dagger},\hat{\nu},\hat{\theta}_{0},\theta_{1}^{\dagger})$

for $\theta_{1}^{\dagger}$ and $\psi^{\dagger}$, where $e_{4}(\mathbf{C}_{i})$
and $e_{5}(\mathbf{Z}_{i},\mathbf{C}_{i})$ are of the same dimension as
$\theta_{1}^{\dagger}$ and $\psi^{\dagger}$ respectively.

  5. 5.

Re-estimate $\psi^{\dagger}$ as $\hat{\psi}_{MR-NEM}$ by solving the equations

$0=\sum^{n}_{i=1}\phi(\mathbf{Z}_{i},S_{i},\mathbf{C}_{i};\hat{\tau},\hat{\alpha},\hat{\rho})\epsilon^{*}(\psi^{\dagger},\hat{\nu},\hat{\theta})$

for $\psi^{\dagger}$.

What is crucial here is that $\hat{\rho}$, $\hat{\nu}$, $\hat{\theta}_{1}$ are
themselves doubly robust estimators (see the proof of the following theorem
for details); we can obtain an unbiased estimator of $t(\mathbf{Z,C})$ even
when $b_{1}(\mathbf{C};\theta^{\dagger}_{1})$ is misspecified (and vice
versa). Likewise, we can obtain an unbiased estimate of $f(\mathbf{Z}|S,C)$
even when $f(S|\mathbf{Z}=\mathbf{0},C)$ is poorly modelled.

######  Theorem 3.2.

Under the union model
$\mathcal{M}_{NEM}\cap(\mathcal{M}_{1}\cup\mathcal{M}_{2}\cup\mathcal{M}_{3}\cup\mathcal{M}_{4})$
and assuming standard regularity conditions hold, $\hat{\psi}_{MR-NEM}$ is a
consistent and asymptotically normal (CAN) estimator of $\psi^{\dagger}$.

A nonparametric estimator of the standard error for $\hat{\psi}_{MR-NEM}$ can
be obtained either using a sandwich estimator or the nonparametric bootstrap.
Quadruple rather than triple robustness here hinges on the chosen
parameterisation of $f(\mathbf{Z},S|\mathbf{C})$. If one were instead to use
the factorisation e.g.
$f(\mathbf{Z},S|\mathbf{C})=f(\mathbf{Z}|S,\mathbf{C})f(S|\mathbf{C})$ and
posit separate models for $f(\mathbf{Z}|S,\mathbf{C})$ and $f(S|\mathbf{C})$,
then the resulting estimator would be triply robust, as it would not generally
be unbiased under correct models for $f(S|\mathbf{C})$, $b_{0}(\mathbf{C})$
and $b_{1}(\mathbf{C})$ alone. Additional benefits in terms of robustness (and
precision) may be obtained when one has additional knowledge on
$f(\mathbf{Z},S|\mathbf{C})$ e.g. in Mendelian Randomisation studies where it
is sometimes plausible that the joint density factorises as
$f(\mathbf{Z})f(S|\mathbf{C})$.

Under NSM, we will utilise an additional parametric model
$q(\mathbf{A,C};\omega^{\dagger})$ for the selection bias function
$q(\mathbf{A,C})$, where $q(\mathbf{A,C};\omega^{\dagger})$ is a known
function smooth in a finite dimensional parameter $\omega^{\dagger}$. A smooth
model
$f(\mathbf{A}|\mathbf{Z},S=1,\mathbf{C};\pi^{\dagger})f(\mathbf{Z}|\mathbf{C},S=1,\mathbf{C};\tau^{\dagger},\rho^{\dagger})$
is postulated for $f(\mathbf{A,Z}|S=1,\mathbf{C})$; the event that model
$f(A|\mathbf{Z},S,\mathbf{C};\pi^{\dagger})$ is correctly specified is denoted
by $\mathcal{M}_{a}$. We will also consider alternative versions of the
previous restrictions $\mathcal{M}_{1q}$ and $\mathcal{M}_{3q}$ on the data:

  * •

$\mathcal{M}_{1q}$: $t(\mathbf{Z,C})=t(\mathbf{Z,C};\nu^{\dagger})$,
$b_{0}(\mathbf{C})=b_{0}(\mathbf{C};\theta^{\dagger}_{0})$,
$b_{1}(\mathbf{C})=b_{1}(\mathbf{C};\theta^{\dagger}_{1})$ and
$q(\mathbf{A,C})=q(\mathbf{A,C};\omega^{\dagger})$.

  * •

$\mathcal{M}_{3q}$:
$f(S|\mathbf{Z}=0,\mathbf{C})=f(S|\mathbf{Z}=0,\mathbf{C};\alpha^{\dagger})$,
$OR(\mathbf{Z},S,\mathbf{C})=OR(\mathbf{Z},S,\mathbf{C};\rho^{\dagger})$,
$b_{1}(\mathbf{C})=b_{1}(\mathbf{C};\theta^{\dagger}_{1})$, and
$q(\mathbf{A,C})=q(\mathbf{A,C};\omega^{\dagger})$.

In order to estimate nuisance parameters under NSM, one can straightforwardly
update the previous strategy. For example, $\pi^{\dagger}$ can be estimated as
$\hat{\pi}$ via maximum likelihood; one can also obtain an estimator
$\hat{\omega}$ by solving the equations

$\displaystyle 0=\sum^{n}_{i=1}$ $\displaystyle
e_{6}(A_{i},\mathbf{C}_{i})\epsilon_{i}(\psi^{\dagger},\nu^{\dagger},\theta^{\dagger},\omega^{\dagger},\hat{\pi})$

for $\omega^{\dagger}$. Finally, $\hat{\psi}_{MR-NSM}$ is obtained by solving
the equations

$\displaystyle 0=\sum^{n}_{i=1}$
$\displaystyle\begin{pmatrix}\phi(\mathbf{Z}_{i},S_{i},\mathbf{C}_{i};\hat{\tau},\hat{\alpha},\hat{\rho})\epsilon_{i}^{*}(\psi^{\dagger},\hat{\nu},\hat{\theta})\\\
S\kappa(\mathbf{A,Z,C};\hat{\pi},\hat{\tau})\epsilon_{i}(\psi^{\dagger},\hat{\nu},\hat{\theta},\hat{\omega},\hat{\pi})\end{pmatrix}$

for $\psi^{\dagger}$.

######  Theorem 3.3.

Under the union
$\mathcal{M}_{NSM}\cap\mathcal{M}_{a}\cap(\mathcal{M}_{1q}\cup\mathcal{M}_{2}\cup\mathcal{M}_{3q}\cup\mathcal{M}_{4})$
and standard regularity conditions, $\hat{\psi}_{MR-NSM}$ is a CAN estimator
of $\psi^{\dagger}$.

Hence, we can obtain a similar quadruple robustness result under the NSM
assumption, although we now require a correctly specified propensity score
model. Note that a similar parametrisation of the joint density
$f(\mathbf{A,Z}|S=1,\mathbf{C})$ based on the generalised odds ratio function
could be used here. Nevertheless, since one needs to model
$f(\mathbf{A}|\mathbf{Z},S=1,\mathbf{C})$ correctly in order to obtain an
unbiased estimator, adopting such a parametrisation will not generally lead to
greater robustness. Finally, since estimation under NSM requires consistent
estimation of the propensity score, one approach would be to use flexible
nonparametric/machine learning methods of
$f(\mathbf{A}|\mathbf{Z},S=1,\mathbf{C})$ combined with cross-fitting (see the
general algorithm described in the Appendix), even if one were to then use
parametric models for the other nuisance parameters.

######  Remark.

In the bespoke IV-DiD design, note that estimators of $\psi^{\dagger}$ based
on the estimating functions given in Corollary 3.1.1 are doubly robust.
Specifically, an estimator based on equation (11) is unbiased if either i)
$f(\mathbf{Z}|\mathbf{C})$ or ii) $b^{*}_{1}(\mathbf{C})$ and
$b_{0}(\mathbf{C})$ (or their difference) are consistently estimated.
Similarly, an estimator based on equation (12) is unbiased if, in addition to
$f(\mathbf{A}|\mathbf{Z,C})$, either i) $f(\mathbf{Z}|\mathbf{C})$ or ii)
$q(\mathbf{A,C})$, $t(\mathbf{Z,C})$ $b^{*}_{1}(\mathbf{C})$ and
$b_{0}(\mathbf{C})$ are consistently estimated. Hence, estimators in the
bespoke IV-DiD model are expected to be additionally robust and efficient
relative to estimators in the general bespoke IV setting.

###  3.3 Local semiparametric efficiency

Once we have obtained the class of regular and asymptotically linear
estimators of $\psi^{\dagger}$ under a semiparametric model, it remains to
identify the optimal estimator within a given class. Under NEM, we are able to
compute a closed form representation of the efficient score. Before doing
this; we will define some additional notation. Let us define

$\mu(\mathbf{Z},S,\mathbf{C})=E\left\\{\frac{\partial\beta(\mathbf{A,Z,C};\psi^{\dagger})S}{\partial\psi}\bigg{|}_{\psi=\psi^{\dagger}}\bigg{|}\mathbf{Z},S,\mathbf{C}\right\\},$
$P_{\sigma}(\mathbf{Z,C})=\frac{E\\{S\sigma^{-2}(\mathbf{Z},S,\mathbf{C})|\mathbf{Z,C}\\}}{E\\{\sigma^{-2}(\mathbf{Z},S,\mathbf{C})|\mathbf{Z,C}\\}}$

and
$\delta\mu(\mathbf{Z,C})=\mu(\mathbf{Z},1,\mathbf{C})-\mu(\mathbf{Z},0,\mathbf{C})$,
where
$\sigma^{2}(\mathbf{Z},S,\mathbf{C})=E(\epsilon^{*2}(\psi^{\dagger})|\mathbf{Z},S,\mathbf{C})$.

######  Theorem 3.4.

Under the semiparametric model $\mathcal{M}_{NEM}$ and with admissible
independence densities

$f^{{\ddagger}}\left(\mathbf{Z},S|\mathbf{C}\right)=\left(\frac{1}{2}\right)^{S}\left(\frac{1}{2}\right)^{1-S}f(\mathbf{Z}|\mathbf{C})$

the optimal index function $r^{opt}_{0}(\mathbf{Z},S,\mathbf{C})$ in (8) is

$r_{0}^{opt}(\mathbf{Z},S,\mathbf{C})=\left\\{\delta\mu(\mathbf{Z,C})-\frac{E\left\\{\delta\mu(\mathbf{Z},\mathbf{C})P_{\sigma}(\mathbf{Z,C})|\mathbf{C}\right\\}}{E\left\\{P_{\sigma}(\mathbf{Z,C})|\mathbf{C}\right\\}}\right\\}2(-1)^{1-S}P_{\sigma}(\mathbf{Z,C}).$

Furthermore, the efficient score is an unbiased estimating function in the
union model
$\mathcal{M}_{NEM}\cap(\mathcal{M}_{1}\cup\mathcal{M}_{2}\cup\mathcal{M}_{3}\cup\mathcal{M}_{4})$
that is locally efficient at the intersection submodel
$\mathcal{M}_{NEM}\cap\mathcal{M}_{1}\cap\mathcal{M}_{2}\cap\mathcal{M}_{3}\cap\mathcal{M}_{4}$.

Similar to the standard IV set-up, efficient estimation generally requires
modelling the conditional mean of $\mathbf{A}$ given $\mathbf{Z}$,
$\mathbf{S}$ and $\mathbf{C}$. However, even if this model is misspecified,
the resulting estimating of $\psi^{\dagger}$ will remain consistent (although
it will no longer generally be efficient). In order to further simplify the
above result, we revisit Example 1.

######  Corollary 3.4.1.

If $\mathbf{Z}$ is binary, then the optimal choice $m_{opt}(\mathbf{C})$ of
$m(\mathbf{C})$ in the estimating function (10) is

$\displaystyle
m_{{opt}}(\mathbf{C})=E\left[\left\\{\frac{(-1)^{Z+S}}{f(Z,S|\mathbf{C})}\right\\}^{2}\sigma^{2}(\mathbf{Z},S,\mathbf{C})\bigg{|}\mathbf{C}\right]^{-1}\\{\mu(Z=1,S=1,\mathbf{C})-\mu(Z=0,S=1,\mathbf{C})\\}.$

If $\sigma^{2}(\mathbf{Z},S,\mathbf{C})=\sigma^{2}$, then
$m_{{opt}}(\mathbf{C})$ reduces to
$w_{0}(\mathbf{C})^{-1}\sigma^{-2}\\{\mu(Z=1,S=1,\mathbf{C})-\mu(Z=0,S=1,\mathbf{C})\\}$,
where

$w_{0}(\mathbf{C})=\left\\{\frac{1}{f(Z=1,S=1|\mathbf{C})}+\frac{1}{f(Z=1,S=0|\mathbf{C})}+\frac{1}{f(Z=0,S=1|\mathbf{C})}+\frac{1}{f(Z=0,S=0|\mathbf{C})}\right\\}$

Note that the weight function $w_{0}(\mathbf{C})^{-1}$ will automatically give
high weight given in the estimating equations to strata of the data where
there is most overlap in terms of the distribution $f(Z,S|\mathbf{C})$. In
studies where the inverse weights $1/f(Z,S|\mathbf{C})$ are extreme, we might
expect an efficient estimator to thus perform considerably better than other
multiply robust estimators both in terms of precision as well as stability and
finite sample bias.

Under NSM, there appears to exist no closed-form expression for the efficient
score for arbitrary $\mathbf{A}$ and $\mathbf{Z}$. In the Appendix, the
efficient score is given in closed form when $\mathbf{A}$ is binary; below, we
obtain a result for binary exposure and instrument.

######  Theorem 3.5.

Suppose that $A$ and $Z$ are binary, and let

$\displaystyle Q_{1}$
$\displaystyle=\frac{S(-1)^{A+Z}}{f(A,Z|S=1,\mathbf{C})}\epsilon(\psi^{\dagger})$
$\displaystyle Q_{2}$
$\displaystyle=\frac{(-1)^{Z+S}}{f(Z,S|\mathbf{C})}\epsilon^{*}(\psi^{\dagger})$
$\displaystyle Q_{3}$
$\displaystyle=Q_{2}-\frac{E(Q_{1}Q_{2}|\mathbf{C})}{E(Q^{2}_{1}|\mathbf{C})}Q_{1}$

where $U_{\psi^{\dagger}}$ is the score for $\psi^{\dagger}$. Then under the
model $\mathcal{M}_{NSM}$,

$\displaystyle m_{opt}(\mathbf{C})$
$\displaystyle=\frac{E(U_{\psi^{\dagger}}Q_{3}|\mathbf{C})}{E(Q^{2}_{3}|\mathbf{C})}$
$\displaystyle h_{opt}(\mathbf{C})$
$\displaystyle=\frac{E(U_{\psi^{\dagger}}Q_{1}|\mathbf{C})}{E(Q^{2}_{1}|\mathbf{C})}-\frac{E(U_{\psi^{\dagger}}Q_{3}|\mathbf{C})}{E(Q^{2}_{3}|\mathbf{C})}\frac{E(Q_{1}Q_{2}|\mathbf{C})}{E(Q^{2}_{1}|\mathbf{C})}$

Furthermore, if $\sigma^{2}(\mathbf{Z},S,\mathbf{C})=\sigma^{2}$ and
$E(\epsilon^{2}|A,Z,S,\mathbf{C})=\bar{\sigma}^{2}$, then

$\displaystyle
m_{{opt}}(\mathbf{C})=w_{0}(\mathbf{C})^{-1}\sigma^{2}\\{\mu(Z=1,S=1,\mathbf{C})-\mu(Z=0,S=1,\mathbf{C})\\}$
$\displaystyle
h_{{opt}}(\mathbf{C})=w_{1}(\mathbf{C})^{-1}\bar{\sigma}^{2}E\left\\{\frac{(-1)^{A+Z}}{f(A,Z|S=1,\mathbf{C})}\frac{\partial\beta(A,Z,\mathbf{C};\psi^{\dagger})}{\partial\psi}\bigg{|}_{\psi=\psi^{\dagger}}\bigg{|}S=1,\mathbf{C}\right\\}$

where

$\displaystyle w_{1}(\mathbf{C})=$
$\displaystyle\bigg{\\{}\frac{1}{f(A=1,Z=1|S=1,\mathbf{C})}+\frac{1}{f(A=0,Z=1|S=1,\mathbf{C})}$
$\displaystyle+\frac{1}{f(A=1,Z=0|S=1,\mathbf{C})}+\frac{1}{f(A=0,Z=0|S=1,\mathbf{C})}\bigg{\\}}$

Estimators in a model defined by restriction (6) require a consistently
estimated propensity score model, as opposed to working under the restriction
(5). Along the lines of Tchetgen Tchetgen and Vansteelandt, (2013), if one is
willing to entertain that a parametric model for the propensity is correctly
specified, then this can be exploited in order to make additional efficiency
gains. Below, we will derive the efficient score in the submodel
$\mathcal{M}_{NSM}\cap\mathcal{M}_{a}\cap(\mathcal{M}_{1q}\cup\mathcal{M}_{2}\cup\mathcal{M}_{3q}\cup\mathcal{M}_{4})$
under the assumption that a parametric model
$f(\mathbf{A}|\mathbf{Z},S=1,\mathbf{C};\pi^{\dagger})$ for
$f(\mathbf{A}|\mathbf{Z},S=1,\mathbf{C})$ is correctly specified and one of
the three sets of restrictions in the second part of Theorem (3.2) holds.

######  Theorem 3.6.

The set of influence functions of $\psi^{\dagger},\pi^{\dagger}$ in the
semiparametric model $\mathcal{M}_{NSM}\cap\mathcal{M}_{a}$ consists of
elements proportional to

$\displaystyle\bigg{\\{}$ $\displaystyle
S\left\\{\kappa(\mathbf{A,Z,C})+\phi(\mathbf{Z},S,\mathbf{C})\right\\}\epsilon_{1}(\psi^{\dagger})+S\phi(\mathbf{Z},S,\mathbf{C})[q(\mathbf{A,C})-E\\{q(\mathbf{A,C})|\mathbf{Z},S=1,\mathbf{C}\\}]$
$\displaystyle+S[g(\mathbf{A,Z,C})-E\\{g(\mathbf{A,Z,C})|\mathbf{Z},S=1,\mathbf{C}\\}]:$
$\displaystyle\kappa(\mathbf{A,Z,C})\in\Gamma,\phi(\mathbf{Z},S,\mathbf{C})\in\Omega,\textrm{$g(\mathbf{A,Z})$
unrestricted}\bigg{\\}}\cap L^{0}_{2}$

If $\mathbf{A}$ is binary, then functions
$S[g(\mathbf{A,Z,C})-E\\{g(\mathbf{A,Z,C})|\mathbf{Z},S=1,\mathbf{C}\\}]$ can
be equivalently expressed as
$Sg(\mathbf{Z,C})\\{A-E(A|\mathbf{Z},S=1,\mathbf{C})\\}$. Further, if
$\mathbf{A}$ and $\mathbf{Z}$ are binary and we redefine $Q_{1}$ as

$\displaystyle Q_{1}$
$\displaystyle=\frac{S(-1)^{A+Z}}{f(A,Z|S=1,\mathbf{C};\pi^{\dagger})}\epsilon_{1}(\psi^{\dagger})$

and define $Q_{4}=\left(Q_{1},Q_{2}\right)^{T}$, then the efficient choices
$h_{opt}(\mathbf{C})$, $m_{opt}(\mathbf{C})$, $g_{opt}(\mathbf{Z,C})$ of
$h(\mathbf{C})$, $m(\mathbf{C})$, $g(\mathbf{Z,C})$ are given by

$\displaystyle\left(h_{opt}(\mathbf{C}),m_{opt}(\mathbf{C})\right)^{T}$
$\displaystyle=E(U_{\psi^{\dagger},\pi^{\dagger}}Q_{4}|\mathbf{C})E^{-1}(Q_{4}Q^{T}_{4}|\mathbf{C})Q_{4}$
$\displaystyle g_{opt}(\mathbf{Z,C})$
$\displaystyle=\frac{E[U_{\psi^{\dagger},\pi^{\dagger}}\\{A-f(A=1|\mathbf{Z},S=1,\mathbf{C};\pi^{\dagger})\\}|\mathbf{Z},S=1,\mathbf{C}]}{E[\\{A-f(A=1|\mathbf{Z},S=1,\mathbf{C};\pi^{\dagger})\\}^{2}|\mathbf{Z},S=1,\mathbf{C}]}$

where
$U_{\psi^{\dagger},\pi^{\dagger}}=(U^{T}_{\psi^{\dagger}},U^{T}_{\pi^{\dagger}})^{T}$
is the score for $(\psi^{\dagger},\pi^{\dagger})$. Therefore efficient
estimators can be constructed by inputting $h_{opt}(\mathbf{C})$,
$m_{opt}(\mathbf{C})$ and $g_{opt}(\mathbf{Z,C})$ in the scores given above.

###  3.4 Nonparametric inference for the marginal ATT

It may be that interest is in the marginal rather than the conditional ATT.
Suppose that $\mathbf{A}$ and $\mathbf{Z}$ are binary; then in the case that
$\beta(Z)=\beta$, previous methods can be used due to the collapsibility of
the linear link function, but one may not wish to invoke this restriction. It
follows from the identification results in Section 2 that under the NEM
Assumption 5,

$\psi^{*}=E(Y^{1}-Y^{0}|A=1,S=1)=E\\{E(Y^{1}-Y^{0}|A=1,Z=z,S=1)|A=1,S=1\\}$

is identified as

$\displaystyle\psi^{*}=\int\beta(\mathbf{c})dF(\mathbf{c}|A=1,S=1)$

where

$\displaystyle\beta(\mathbf{C})$
$\displaystyle=\frac{\delta^{Y}(\mathbf{C})-t(1,\mathbf{C})}{\delta^{A}(\mathbf{C})}$
$\displaystyle\delta^{Y}(\mathbf{C})$
$\displaystyle=E(Y|Z=1,S=1,\mathbf{C})-E(Y|Z=0,S=1,\mathbf{C})$
$\displaystyle\delta^{A}(\mathbf{C})$
$\displaystyle=f(A=1|Z=1,S=1,\mathbf{C})-f(A=1|Z=0,S=1,\mathbf{C})$

and alternatively, under the NSM Assumption 6, we have

$\displaystyle\psi^{*}=$ $\displaystyle E(Y|A=1,S=1)-\int
E(Y|A=0,z,S=1,\mathbf{c})dF(z,\mathbf{c}|A=1,S=1)$
$\displaystyle-\int\gamma(\mathbf{c})dF(\mathbf{c}|A=1,S=1)$

where

$\displaystyle\gamma(\mathbf{C})$
$\displaystyle=-\frac{\delta_{0}^{Y}(\mathbf{C})-t(1,\mathbf{C})}{\delta^{A}(\mathbf{C})}$
$\displaystyle\delta_{0}^{Y}(\mathbf{C})$
$\displaystyle=E(Y|A=0,Z=1,S=1,\mathbf{C})-E(Y|A=0,Z=0,S=1,\mathbf{C})$

In either case, we have a closed form representation of $\psi^{*}$ as a
functional of the observed data distribution, Since $\psi^{*}$ is a pathwise
differentiable parameter, we can then obtain nonparametric inference on
$\psi$, by obtaining the efficient influence function under a nonparametric
model. The following results may be of independent interest with respect to
the literature of nonparametric inference in the IV model (Wang and Tchetgen
Tchetgen,, 2018).

######  Theorem 3.7.

(Part a) In the nonparametric model defined by Assumptions 1-5, the efficient
influence function is equal to:

$\displaystyle EIF_{1}(\psi^{*})=$
$\displaystyle\frac{f(A=1,S=1|\mathbf{C})(-1)^{Z+S}}{f(A=1,S=1)\delta^{A}(\mathbf{C})f(Z,S|\mathbf{C})}$
$\displaystyle\times[Y-\beta(\mathbf{C})\\{A-f(A=1|Z=0,S=1,\mathbf{C})\\}S-t(1,\mathbf{C})Z-E(Y|Z=0,S,\mathbf{C})]$
$\displaystyle+\frac{1}{f(A=1,S=1)}AS\left\\{\beta(\mathbf{C})-\psi^{*}\right\\}$

(Part b) In the nonparametric model defined by Assumptions 1-4 and 6, the
efficient influence function is given by:

$\displaystyle EIF_{2}(\psi^{*})$
$\displaystyle=\frac{f(A=1,S=1|\mathbf{C})(2Z-1)}{f(A=1,S=1)\delta^{A}(\mathbf{C})f(Z,S|\mathbf{C})}$
$\displaystyle\quad\times\bigg{[}\frac{(1-A)S}{f(A=0|Z,S=1,\mathbf{C})}\\{Y+\gamma(\mathbf{C})\delta^{A}(\mathbf{C})Z-t(1,\mathbf{C})Z-E(Y|A=0,Z=0,S=1,\mathbf{C})\\}$
$\displaystyle\quad+S\\{A-f(A=1|Z,S=1,C)\\}\gamma(\mathbf{C})-(1-S)\\{Y-t(1,\mathbf{C})Z-E(Y|Z=0,S=0,\mathbf{C})\\}\bigg{]}$
$\displaystyle\quad-\frac{(1-A)Sf(A=1|Z,S=1,\mathbf{C})}{f(A=1,S=1)f(A=0|Z,S=1,\mathbf{C})}\\{Y+\gamma(\mathbf{C})\delta^{A}(\mathbf{C})Z-t(1,\mathbf{C})Z-E(Y|A=0,Z=0,S=1,\mathbf{C})\\}$
$\displaystyle\quad+\frac{AS}{f(A=1,S=1)}\left\\{Y+\gamma(\mathbf{C})\delta^{A}(\mathbf{C})Z-t(1,\mathbf{C})Z-E(Y|A=0,Z=0,S=1,\mathbf{C})-\gamma(\mathbf{C})-\psi^{*}\right\\}$

One can use these results to create estimators of $\psi^{*}$, after obtaining
estimates of the (many) nuisance parameters involved. Then the following
theorem describes the robustness properties of our estimators, when parametric
working models for the nuisance parameters are employed.

######  Theorem 3.8.

Consider the following five restrictions on the observed data distribution.

  * •

$\mathcal{M}^{*}_{1}$: Parametric working models for $\beta(\mathbf{C})$,
$f(A=1|Z,S=1,\mathbf{C})$, $E(Y|Z=0,S,\mathbf{C})$ and $t(1,\mathbf{C})$ are
correct.

  * •

$\mathcal{M}^{*}_{2}$: Models for $f(Z|S,\mathbf{C})$, $t(1,\mathbf{C})$ and
$f(A=1|Z,S=1,\mathbf{C})$ are correct.

  * •

$\mathcal{M}^{*}_{3}$: Models for $\beta(\mathbf{C})$, $f(Z|S,\mathbf{C})$ and
$t(1,\mathbf{C})$ are correct.

  * •

$\mathcal{M}^{*}_{4}$: Models for $f(Z|S,\mathbf{C})$, $f(S|\mathbf{C})$ and
$f(A=1|Z,S=1,\mathbf{C})$ are correct.

  * •

$\mathcal{M}^{*}_{5}$: Models for $\beta(\mathbf{C})$, $f(Z|S,\mathbf{C})$ and
$f(S|\mathbf{C})$ are correct.

Under Assumptions 1-5, $EIF_{1}$ (evaluated at the values of the parameters
indexing the working models) has mean zero under the union model
$\mathcal{M}^{*}_{1}\cup\mathcal{M}^{*}_{2}\cup\mathcal{M}^{*}_{3}\cup\mathcal{M}^{*}_{4}\cup\mathcal{M}^{*}_{5}$.

Consider now the following alternative restrictions.

  * •

$\mathcal{M}^{*}_{1\gamma}$: Models for $\gamma(\mathbf{C})$,
$E(Y|A=0,Z=0,S=1,\mathbf{C})$, $E(Y|Z=0,S,\mathbf{C})$ and $t(1,\mathbf{C})$
are correct.

  * •

$\mathcal{M}^{*}_{2\gamma}$: Models for $f(Z|S,\mathbf{C})$ and
$t(1,\mathbf{C})$ are correct.

  * •

$\mathcal{M}^{*}_{3\gamma}$: Models for $f(Z|S,\mathbf{C})$ and
$f(S|\mathbf{C})$ are correct.

Under Assumptions 1-4 and 6, $EIF_{2}$ (evaluated at the values of the
parameters indexing the working models) has mean zero under the union model
$\mathcal{M}_{a}\cap(\mathcal{M}^{*}_{1\gamma}\cup\mathcal{M}^{*}_{2\gamma}\cup\mathcal{M}^{*}_{3\gamma})$.

Hence under NEM, the resulting estimators are quintuply robust; they are
unbiased if at least one (out of five) combinations of nuisance parameters are
consistently estimated. Note that unlike in previous sections, it is possible
to obtain valid inference on the target causal parameter even when a model for
the conditional treatment effect is misspecified.

##  4 Longitudinal studies

We will consider now inference on treatment effects in longitudinal studies
where repeated measurements of outcomes, exposures and covariates are
collected at times $j=0,...,K$. Specifically, suppose that we observe the data
$\bar{\mathbf{O}}=\bar{\mathbf{O}}(K)=\\{\bar{\mathbf{Y}}(K),\bar{\mathbf{A}}(K),\bar{\mathbf{Z}}(K),\bar{\mathbf{C}}(K)\\}$
where we use the notation
$\bar{\mathbf{X}}(j)=\\{\mathbf{X}(1),...,\mathbf{X}(j)\\}$. Further,
$\bar{\mathbf{H}}(K)=\\{\bar{\mathbf{A}}(K),\bar{\mathbf{Z}}(K),\bar{\mathbf{C}}(K)\\}$;
if $j=0$, then $\bar{\mathbf{H}}(j-1)$ is empty. We are interested in
identification of the following causal effect on outcome $Y(k)$:

$\beta_{j,k}\\{\bar{\mathbf{h}}(j)\\}=E\\{Y^{\bar{\mathbf{a}}(j),\mathbf{0}}(k)-Y^{\bar{\mathbf{a}}(j-1),\mathbf{0}}(k)|\bar{\mathbf{H}}(j)=\bar{\mathbf{h}}(j)\\}$

for $0\leq j<k\leq K$, where $Y^{\bar{\mathbf{a}}(j),\mathbf{0}}(k)$ denotes
the potential outcome if regime $\bar{\mathbf{a}}(j)$ was followed up to time
$t$, and then no further treatment was given up to time $k$. The above causal
contrast then expresses the effect of one final portion of treatment at time
$t$ on a future outcome $Y(k)$, conditional on treatment and covariate
history; $\beta_{j,k}\\{\bar{\mathbf{a}}(j),\mathbf{z}(j),\mathbf{c}(j)\\}$
encodes a structural mean model for this effect (Robins, 1994).

As discussed previously, repeated measures of an outcome form a natural
reference population in longitudinal studies. In what follows, we will
leverage the association between $\mathbf{Z}(j)$ and the concurrent $Y(j)$ in
order to deal with bias due to unmeasured confounding. In order to do this, we
must first restate the key assumptions for identification to allow for time
varying treatments and confounders (both measured and unmeasured).

######  Assumption 9.

(Consistency) If $\bar{\mathbf{A}}(j)=\bar{\mathbf{a}}(j)$, then
$Y^{\bar{\mathbf{a}}(j)}(k)=Y(k)$, $0\leq j<k\leq K$.

######  Assumption 10.

(Degenerate reference population) We have

$E\\{Y(j)|\bar{\mathbf{A}}(j-1)=\bar{\mathbf{a}}(j-1),\bar{\mathbf{z}}(j),\bar{\mathbf{c}}(j)\\}=E\\{Y^{\bar{\mathbf{a}}(j-1),\mathbf{0}}(j)|\bar{\mathbf{A}}(j-1)=\bar{\mathbf{a}}(j-1),\bar{\mathbf{z}}(j),\bar{\mathbf{c}}(j)\\},\quad
j=0,...,K.$

######  Assumption 11.

(Partial population exchangeability)

$\displaystyle
E\\{Y^{\bar{\mathbf{a}}(j-1),\mathbf{0}}(k)|\bar{\mathbf{A}}(j-1)=\bar{\mathbf{a}}(j-1),\bar{\mathbf{z}}(j),\bar{\mathbf{c}}(j)\\}$
$\displaystyle-E\\{Y^{\bar{\mathbf{a}}(j-1),\mathbf{0}}(k)|\bar{\mathbf{A}}(j-1)=\bar{\mathbf{a}}(j-1),\mathbf{Z}(j)=0,\bar{\mathbf{z}}(j-1),\bar{\mathbf{c}}(j)\\}$
$\displaystyle=E\\{Y^{\bar{\mathbf{a}}(j-1),\mathbf{0}}(j)|\bar{\mathbf{A}}(j-1)=\bar{\mathbf{a}}(j-1),\bar{\mathbf{z}}(j),\bar{\mathbf{c}}(j)\\}$
$\displaystyle\quad-E(Y^{\bar{\mathbf{a}}(j-1),\mathbf{0}}(j)|\bar{\mathbf{A}}(j-1)=\bar{\mathbf{a}}(j-1),\mathbf{Z}(j)=0,\bar{\mathbf{z}}(j-1),\bar{\mathbf{c}}(j)\\},\quad
0\leq j<k\leq K.$

######  Assumption 12.

(Bespoke IV relevance)
$\mathbf{A}(k)\centernot{\rotatebox[origin={c}]{90.0}{$\models$}}\mathbf{Z}(j)|\bar{\mathbf{A}}(j-1),\bar{\mathbf{Z}}(j-1),\bar{\mathbf{C}}(j)$,
$0\leq j\leq k\leq K$.

######  Assumption 13.

(Partial effect homogeneity)

$\displaystyle\beta_{j,k}\\{\bar{\mathbf{h}}(j)\\}=\beta_{j,k}\\{\bar{\mathbf{a}}(j),\bar{\mathbf{z}}(j-1),\bar{\mathbf{c}}(j)\\},\quad
0\leq j<k\leq K.$

######  Assumption 14.

(Partial selection bias homogeneity) The general selection bias function

$\displaystyle q_{j,k}\\{\bar{\mathbf{h}}(j)\\}=$ $\displaystyle
E\\{Y^{\bar{\mathbf{a}}(j-1),\mathbf{0}}(k)-Y^{\bar{\mathbf{a}}(j-1),\mathbf{0}}(j)|\bar{\mathbf{A}}(j)=\bar{\mathbf{a}}(j),\bar{\mathbf{z}}(j),\bar{\mathbf{c}}(j)\\}$
$\displaystyle-E\\{Y^{\bar{\mathbf{a}}(j-1),\mathbf{0}}(k)-Y^{\bar{\mathbf{a}}(j-1),\mathbf{0}}(j)|\mathbf{A}(j)=0,\bar{\mathbf{A}}(j-1)=\bar{\mathbf{a}}(j-1),\bar{\mathbf{z}}(j),\bar{\mathbf{c}}(j)\\}$

follows the restriction

$\displaystyle
q_{j,k}\\{\bar{\mathbf{h}}(j)\\}=q_{j,k}\\{\bar{\mathbf{a}}(j),\bar{\mathbf{z}}(j-1),\bar{\mathbf{c}}(j)\\},\quad
0\leq j<k\leq K.$

Assumptions 9 and 10 carry over straightforwardly from the point treatment
setting. Assumption 11 extends the (conditional) parallel trends-type
assumption (3) for studies with time-varying confounding. To give some further
intuition, consider a two time point study (i.e. $K=2$). The short term
effects of $\mathbf{A}(1)$ on $Y(2)$ and $\mathbf{A}(0)$ on $Y(1)$ can be
identified using an assumption akin to (3). However to identify

$E\\{Y^{\mathbf{a}(0),\mathbf{0}}(2)-Y^{\mathbf{0}}(2)|\mathbf{h}(0)\\}$

then we require an assumption

$\displaystyle
E\\{Y^{\mathbf{0}}(2)|\mathbf{z}(0),\mathbf{c}(0)\\}-E\\{Y^{\mathbf{0}}(2)|\mathbf{Z}(0)=\mathbf{0},\mathbf{c}(0)\\}$
$\displaystyle=E\\{Y^{\mathbf{0}}(0)|\mathbf{z}(0),\mathbf{c}(0)\\}-E(Y^{\mathbf{0}}(0)|\mathbf{Z}(0)=\mathbf{0},\mathbf{c}(0)\\}\neq
0$

which is made to respect to a ‘blipped-down’ potential outcome
$Y^{\mathbf{0}}(2)$ e.g an outcome where the impact of treatment at time 1 has
been removed. Conditions closely related to Assumptions 13 and 14 have already
been discussed in the context of time-varying confounding in Robins, (1994)
and Tchetgen Tchetgen and Vansteelandt, (2013) respectively.

######  Proposition 4.0.1.

Under Assumptions 9-12 we have the following re-parametrisation of the
conditional mean function:

$\displaystyle E\\{Y(k)|\bar{\mathbf{h}}(k-1)\\}$
$\displaystyle=E\\{Y^{\mathbf{0}}(0)\\}+\sum^{k-1}_{j=0}\beta_{j,k}\\{\bar{\mathbf{h}}(j)\\}+q_{j,k}\\{\bar{\mathbf{h}}(j)\\}-E[q_{j,k}\\{\bar{\mathbf{h}}(j)\\}|\bar{\mathbf{a}}(j-1),\bar{\mathbf{z}}(j),\bar{\mathbf{c}}(j)]$
$\displaystyle+b_{j,k}\\{\bar{\mathbf{h}}(j-1),\mathbf{c}(j)\\}-E[b_{j,k}\\{\bar{\mathbf{h}}(j-1),\mathbf{C}(j)\\}|\bar{\mathbf{h}}(j-1)]$
$\displaystyle+E\\{Y(j)|\bar{\mathbf{h}}(j)\\}-E\\{Y^{\bar{\mathbf{a}}(j-1),\mathbf{0}}(j)|\bar{\mathbf{h}}(j-1)\\}$

where

$\displaystyle b_{j,k}\\{\bar{\mathbf{h}}(j-1),\mathbf{c}(j)\\}=$
$\displaystyle
E\\{Y^{\bar{\mathbf{a}}(j-1),\mathbf{0}}(k)-Y^{\bar{\mathbf{a}}(j-1),\mathbf{0}}(j)|\bar{\mathbf{h}}(j-1),\mathbf{c}(j)\\}$
$\displaystyle-E\\{Y^{\bar{\mathbf{a}}(j-1),\mathbf{0}}(k)-Y^{\bar{\mathbf{a}}(j-1),\mathbf{0}}(j)|\bar{\mathbf{h}}(j-1),\mathbf{C}(j)=0\\}$

$\forall j<k\leq K$, and $E\\{b_{0,k}()\\}=0$.

It follows from this that

$\displaystyle E\\{Y(k)-Y(k-1)|\bar{\mathbf{h}}(k-1)\\}$ $\displaystyle=$
$\displaystyle\beta_{k-1,k}\\{\bar{\mathbf{h}}(k-1)\\}+q_{k-1,k}\\{\bar{\mathbf{h}}(k-1)\\}-E[q_{k-1,k}\\{\bar{\mathbf{h}}(k-1)\\}|\bar{\mathbf{a}}(k-2),\bar{\mathbf{z}}(k-1),\bar{\mathbf{c}}(k-1)]$
$\displaystyle+b_{k-1,k}\\{\bar{\mathbf{h}}(k-2),\mathbf{c}(k-1)\\}$

and also

$\displaystyle
E\\{Y^{\bar{\mathbf{a}}(j),\mathbf{0}}(k)-Y(j)|\bar{\mathbf{h}}(j)\\}=$
$\displaystyle\beta_{j,k}\\{\bar{\mathbf{h}}(j)\\}+q_{j,k}\\{\bar{\mathbf{h}}(j)\\}-E[q_{j,k}\\{\bar{\mathbf{h}}(j)\\}|\bar{\mathbf{a}}(j-1),\bar{\mathbf{z}}(j),\bar{\mathbf{c}}(j)]$
$\displaystyle+b_{j,k}\\{\bar{\mathbf{h}}(j-1),\mathbf{c}(j)\\}$

for $0\leq j<k\leq K$.

The above parametrisation therefore essentially reduces to Robins’ re-
parametrisation of the conditional mean function for longitudinal data (Robins
et al.,, 2000) after differencing the (blipped-down) outcome, which implies
that $\beta_{j,k}\\{\bar{\mathbf{h}}(j)\\}$ can be identified along the lines
of previous results for structural nested means models with instrumental
variables after transforming the outcome (Robins,, 1994). For example, in the
special case where the exposure and bespoke instrument are binary and
Assumption 13 holds, we have the following identification functional

$\displaystyle\beta_{j,k}\\{\bar{\mathbf{h}}(j)\\}$
$\displaystyle=\frac{E\\{Y(k)-\sum^{k-1}_{j^{\prime}=j+1}\beta_{j^{\prime},k}\\{\bar{\mathbf{A}}(j^{\prime}),\bar{\mathbf{Z}}(j^{\prime}-1),\bar{\mathbf{C}}(j^{\prime})\\}-Y(j)|Z(j)=1,\mathbf{c}(j),\bar{\mathbf{h}}(j-1)\\}}{E\\{A(j)|Z(j)=1,\mathbf{c}(j),\bar{\mathbf{h}}(j-1)\\}-E\\{A(j)|Z(j)=0,\mathbf{c}(j),\bar{\mathbf{h}}(j-1)\\}}$
$\displaystyle\quad-\frac{E\\{Y(k)-\sum^{k-1}_{j^{\prime}=j+1}\beta_{j^{\prime},k}\\{\bar{\mathbf{A}}(j^{\prime}),\bar{\mathbf{Z}}(j^{\prime}-1),\bar{\mathbf{C}}(j^{\prime})\\}-Y(j)|Z(j)=0,\mathbf{c}(j),\bar{\mathbf{h}}(j-1)\\}}{E\\{A(j)|Z(j)=1,\mathbf{c}(j),\bar{\mathbf{h}}(j-1)\\}-E\\{A(j)|Z(j)=0,\mathbf{c}(j),\bar{\mathbf{h}}(j-1)\\}}$

which generalises the conditional Wald-type estimand (2.2) to the longitudinal
setting with time-varying confounding.

Let

$\displaystyle\epsilon_{j,k}=$ $\displaystyle
Y(k)-\left[\sum^{k-1}_{j^{\prime}=j}\beta_{j^{\prime},k}\\{\bar{\mathbf{H}}(j^{\prime})\\}\right]-\left(q_{j,k}\\{\bar{\mathbf{H}}(j)\\}-E[q_{j,k}\\{\bar{\mathbf{H}}(j)\\}|\bar{\mathbf{A}}(j-1),\bar{\mathbf{Z}}(j),\bar{\mathbf{C}}(j)]\right)$
$\displaystyle-b_{j,k}\\{\bar{\mathbf{H}}(j-1),\mathbf{C}(j)\\}-Y(j)$
$\displaystyle\epsilon^{*}_{j,k}=$
$\displaystyle\epsilon_{j,k}+\left(q_{j,k}\\{\bar{\mathbf{H}}(j)\\}-E[q_{j,k}\\{\bar{\mathbf{H}}(j)\\}|\bar{\mathbf{A}}(j-1),\bar{\mathbf{Z}}(j),\bar{\mathbf{C}}(j)]\right)$

We can again postulate a semiparametric model

$\displaystyle\beta_{j,k}\\{\bar{\mathbf{h}}(j)\\}=\beta_{j,k}\\{\bar{\mathbf{h}}(j);\psi^{\dagger}_{j,k}\\},$
(13)

where $\psi_{j,k}^{\dagger}$ is an unknown finite-dimensional parameter
vector. Then one can construct doubly robust estimators of the parameter
$\psi^{\dagger}_{j,k}$ indexing the structural nested model. Specifically, it
follows from results in Robins, (1994) that the NEM estimating equation

$\left(m_{j,k}\\{\bar{\mathbf{A}}(j-1),\bar{\mathbf{Z}}(j),\bar{\mathbf{C}}(j)\\}-E[m_{j,k}\\{\bar{\mathbf{A}}(j-1),\bar{\mathbf{Z}}(j),\bar{\mathbf{C}}(j)\\}|\bar{\mathbf{H}}(j-1),\bar{\mathbf{C}}(j)]\right)\epsilon^{*}_{j,k}$

where $m_{j,k}$ is unrestricted, has mean zero if parametric working models
are substituted for
$f(\bar{\mathbf{Z}}(j)|\bar{\mathbf{H}}(j-1),\bar{\mathbf{C}}(j))$ or
$b_{j,k}\\{\bar{\mathbf{H}}(j-1),\bar{\mathbf{C}}(j)\\}$, so long as at least
one out of the two is correctly specified. Similarly, it follows from Tchetgen
Tchetgen and Vansteelandt, (2013) that NSM estimating equation

$\displaystyle\left(m_{j,k}\\{\bar{\mathbf{A}}(j-1),\bar{\mathbf{Z}}(j),\bar{\mathbf{C}}(j)\\}-E[m_{j,k}\\{\bar{\mathbf{A}}(j-1),\bar{\mathbf{Z}}(j),\bar{\mathbf{C}}(j)\\}|\bar{\mathbf{H}}(j-1),\mathbf{C}(j)]\right)\epsilon^{*}_{j,k}(\psi^{\dagger}_{j,k})$
$\displaystyle+\kappa_{j,k}\\{\bar{\mathbf{H}}(j)\\}\epsilon_{j,k}(\psi^{\dagger}_{j,k})$

where $m_{j,k}$ is unrestricted, $\kappa_{j,k}\in\Gamma_{j,k}$, and

$\displaystyle\Gamma_{j,k}=\bigg{\\{}$
$\displaystyle\kappa_{j,k}\\{\bar{\mathbf{H}}(j)\\}:E[\kappa_{j,k}\\{\bar{\mathbf{H}}(j)\\}|\bar{\mathbf{A}}(j-1),\bar{\mathbf{Z}}(j),\bar{\mathbf{C}}(j)]=E[\kappa_{j,k}\\{\bar{\mathbf{H}}(j)\\}|\bar{\mathbf{A}}(j),\bar{\mathbf{Z}}(j-1),\bar{\mathbf{C}}(j)]=0\bigg{\\}}$

has mean zero if parametric working models are substituted for
$f(\mathbf{Z}(j)|\bar{\mathbf{H}}(j-1),\bar{\mathbf{C}}(j))$ or
$b_{j,k}\\{\bar{\mathbf{H}}(j-1),\mathbf{C}(j)\\}$, if at least one out of the
two is correctly specified and one can correctly model
$f(\mathbf{A}(j)|\bar{\mathbf{A}(j-1)},\bar{\mathbf{Z}}(j),\bar{\mathbf{C}}(j))$.
See the aforementioned papers for a discussion of efficient choices of
$m_{j,k}\\{\bar{\mathbf{A}}(j-1),\bar{\mathbf{Z}}(j),\bar{\mathbf{C}}(j)\\}$
and $\kappa_{j,k}\\{\bar{\mathbf{H}}(j)\\}$.

##  5 Simulations

We conducted a series of simulation studies to first evaluate the robustness
properties of the proposed estimators from Section 3.2. Specifically, we
generated covariates $C_{1}$ and $C_{2}$ from a a Bernoulli distribution with
expectation 0.5 and a uniform distribution on an open $(-1,1)$ interval
respectively; let $\mathbf{C}=(C_{1},C_{2})$. An unmeasured confounder $U$ was
also a random binary variable. Then the sample selection probability was
$f(S=1|\mathbf{C})=expit(-0.5+C_{1}+0.6C_{2}+0.5C_{1}C_{2})$, and the bespoke
instrument $Z$ was generated from a Bernoulli distribution where
$f(Z=1|\mathbf{C})=expit(0.25C_{1}-0.25C_{2}+0.5C_{1}C_{2})$. Generating $S$
in this manner led to around 60% of individuals in a simulated data set being
members of the target population ($S$=1). The exposure was simulated from a
Bernoulli distribution with
$f(A=1|Z,S=1,\mathbf{C},U)=expit(1-1.5Z-0.75C_{1}-0.3C_{2}-0.5C_{1}C_{2}+U)$
for individuals in the population $S=1$ and was fixed at zero for all others.
Finally, we generated the outcome from the distribution

$\mathcal{N}(1+A+U+0.5C_{1}+0.5C_{2}-0.5C_{1}C_{2}+Z(1-0.4C_{1}-0.4C_{2}+0.5C_{1}C_{2})+S(0.5C_{1}+0.5C_{2}+0.5C_{1}C_{2}),1).$

This data generating mechanism is compatible with the identification
Assumptions 1-5, but not with conditional exchangeability if one does not have
access to $U$. Nor would it be compatible with the IV conditions, since $Z$
has a direct effect on $Y$. Note that $Z$ is the only valid bespoke IV out of
the three generated covariates, since the effects of the others in
$\mathbf{C}$ on the outcome depend on $S$, which violates Assumption 3.

In our simulations, we considered a suite of estimators of the ATT in the
target population. Specifically, we looked at

  * •

$\hat{\psi}_{TSLS}$: a two-stage least squares estimator described in
Richardson and Tchetgen Tchetgen, (2021) that is unbiased under the model
$\mathcal{M}_{NEM}\cap\mathcal{M}_{1}$.

  * •

$\hat{\psi}_{g-Z}$: a $g$-estimator described in Richardson and Tchetgen
Tchetgen, (2021) that is unbiased under the model
$\mathcal{M}_{NEM}\cap\mathcal{M}_{2}$.

  * •

$\hat{\psi}_{g-S}$: an alternative $g$-estimator that is unbiased under the
model $\mathcal{M}_{NEM}\cap\mathcal{M}_{3}$.

  * •

$\hat{\psi}_{IPW}$: an inverse probability weighted estimator that is unbiased
under the model $\mathcal{M}_{NEM}\cap\mathcal{M}_{4}$.

  * •

$\hat{\psi}_{MR}$: the multiply-robust estimator described in Section 3.2 with
$h(\mathbf{C})=m(\mathbf{C})=1$, which is unbiased under the model
$\mathcal{M}_{NEM}\cap(\mathcal{M}_{1}\cup\mathcal{M}_{2}\cup\mathcal{M}_{3}\cup\mathcal{M}_{4})$.

  * •

$\hat{\psi}_{MR-eff}$: the multiply-robust estimator with $h(\mathbf{C})$ and
$m(\mathbf{C})$ set to the optimal choices described in Section 3.3, which is
unbiased under the model
$\mathcal{M}_{NEM}\cap(\mathcal{M}_{1}\cup\mathcal{M}_{2}\cup\mathcal{M}_{3}\cup\mathcal{M}_{4})$.

Details of the first four estimators are given in the Appendix. Nuisance
parameters were estimated using maximum likelihood, and a sandwich estimator
was obtained for each of the estimators given above. For comparison, we also
considered a doubly robust $g$-estimator that is valid under a conditional
exchangeability assumption (Robins et al.,, 1992), where a linear model was
fittted for the outcome, and a logistic model for the exposure; both models
were adjusted for $Z$, $C_{1}$, $C_{2}$ and a $C_{1}C_{2}$ interaction term.

We considered five settings:

  1. 1.

All nuisance models are correctly specified: the intersection model
$\mathcal{M}_{NEM}\cap\mathcal{M}_{1}\cap\mathcal{M}_{2}\cap\mathcal{M}_{3}\cap\mathcal{M}_{4}$
holds.

  2. 2.

$f(Z|S=0,\mathbf{C};\tau^{\dagger})$ and
$f(S|Z=0,\mathbf{C};\alpha^{\dagger})$ were misspecified:
$\mathcal{M}_{NEM}\cap\mathcal{M}_{1}$ holds.

  3. 3.

$f(S|Z=0,\mathbf{C};\alpha^{\dagger})$,
$b_{0}(\mathbf{C};\theta_{0}^{\dagger})$ and
$b_{1}(\mathbf{C};\theta_{1}^{\dagger})$, were misspecified:
$\mathcal{M}_{NEM}\cap\mathcal{M}_{2}$ holds.

  4. 4.

$f(Z|S=0,\mathbf{C};\tau^{\dagger})$ ,
$b_{0}(\mathbf{C};\theta_{0}^{\dagger})$ and $t(Z,\mathbf{C};\nu^{\dagger})$,
were misspecified: $\mathcal{M}_{NEM}\cap\mathcal{M}_{3}$ holds.

  5. 5.

$t(Z,\mathbf{C};\nu^{\dagger})$, $b_{0}(\mathbf{C};\theta_{0}^{\dagger})$ and
$t(Z,\mathbf{C};\nu^{\dagger})$, were misspecified:
$\mathcal{M}_{NEM}\cap\mathcal{M}_{4}$ holds.

Misspecification was induced via omission of interaction terms. For each
setting, we simulated 2,000 data sets, with a sample size of 5,000.

Table 1: Empirical bias (Bias), empirical standard error (SE), mean estimated standard errors (ESE) and coverage probability (Cov) Model | Estimator | Bias | SE | ESE | Cov
---|---|---|---|---|---
All correct | $\hat{\psi}_{TSLS}$ | 0.01 | 0.26 | 0.26 | 0.95
| $\hat{\psi}_{g-Z}$ | 0 | 0.41 | 0.41 | 0.95
| $\hat{\psi}_{g-S}$ | 0.01 | 0.23 | 0.23 | 0.96
| $\hat{\psi}_{IPW}$ | 0.01 | 0.33 | 0.32 | 0.95
| $\hat{\psi}_{MR}$ | 0 | 0.25 | 0.25 | 0.95
| $\hat{\psi}_{MR-eff}$ | 0.01 | 0.23 | 0.23 | 0.95
$\mathcal{M}_{NSM}\cap\mathcal{M}_{1}$ | $\hat{\psi}_{TSLS}$ | 0 | 0.27 | 0.26 | 0.95
| $\hat{\psi}_{g-Z}$ | 0.08 | 0.39 | 0.38 | 0.94
| $\hat{\psi}_{g-S}$ | 0.18 | 0.26 | 0.25 | 0.89
| $\hat{\psi}_{IPW}$ | -0.54 | 0.28 | 0.27 | 0.48
| $\hat{\psi}_{MR}$ | -0.01 | 0.25 | 0.25 | 0.95
| $\hat{\psi}_{MR-eff}$ | -0.01 | 0.24 | 0.23 | 0.94
$\mathcal{M}_{NSM}\cap\mathcal{M}_{2}$ | $\hat{\psi}_{TSLS}$ | -0.59 | 0.25 | 0.25 | 0.35
| $\hat{\psi}_{g-Z}$ | -0.02 | 0.41 | 0.41 | 0.95
| $\hat{\psi}_{g-S}$ | -0.3 | 0.22 | 0.23 | 0.73
| $\hat{\psi}_{IPW}$ | -0.36 | 0.28 | 0.28 | 0.74
| $\hat{\psi}_{MR}$ | -0.01 | 0.26 | 0.26 | 0.95
| $\hat{\psi}_{MR-eff}$ | -0.01 | 0.23 | 0.23 | 0.95
$\mathcal{M}_{NSM}\cap\mathcal{M}_{3}$ | $\hat{\psi}_{TSLS}$ | -0.51 | 0.26 | 0.25 | 0.48
| $\hat{\psi}_{g-Z}$ | -1.13 | 0.36 | 0.36 | 0.1
| $\hat{\psi}_{g-S}$ | 0 | 0.24 | 0.23 | 0.95
| $\hat{\psi}_{IPW}$ | -0.25 | 0.32 | 0.31 | 0.85
| $\hat{\psi}_{MR}$ | -0.01 | 0.26 | 0.25 | 0.95
| $\hat{\psi}_{MR-eff}$ | 0 | 0.23 | 0.23 | 0.95
$\mathcal{M}_{NSM}\cap\mathcal{M}_{4}$ | $\hat{\psi}_{TSLS}$ | -0.58 | 0.25 | 0.25 | 0.34
| $\hat{\psi}_{g-Z}$ | -0.42 | 0.36 | 0.36 | 0.8
| $\hat{\psi}_{g-S}$ | -0.14 | 0.22 | 0.22 | 0.92
| $\hat{\psi}_{IPW}$ | 0 | 0.33 | 0.32 | 0.95
| $\hat{\psi}_{MR}$ | 0 | 0.26 | 0.25 | 0.96
| $\hat{\psi}_{MR-eff}$ | 0 | 0.23 | 0.23 | 0.95

The results of the experiments can be seen in Table 1. As a comparison, the
benchmark doubly robust $g$-estimator has a bias in all experiments of 0.24,
with coverage probability $<$0.01. We see that the bias and coverage
properties of the estimators reflect the theory; the multiply robust
estimators are the only estimators that maintain good coverage properties
across all experiments. The $g$-estimator $\hat{\psi}_{g-Z}$ exhibited wide
confidence intervals across settings and sometimes large bias when the models
were misspecified; improvements could potentially be made via modelling
$b_{0}(\mathbf{C})$ and making the estimator doubly robust e.g. unbiased under
model $\mathcal{M}_{NEM}\cap(\mathcal{M}_{1}\cup\mathcal{M}_{2})$, as
described in Richardson and Tchetgen Tchetgen, (2021). The efficient multiply
robust estimator outperformed the two-stage least squares estimator both in
terms of bias in the presence of misspecification, but also precision when
models were correct. The latter aspect is perhaps not totally surprising,
given that two-stage least squares estimators are not necessarily efficient
even when the conditional outcome mean is correctly specified (Vansteelandt
and Didelez,, 2018).

##  6 Data analysis

To illustrate the estimators described in the paper, we analysed data from the
Life Span Study of atomic bomb survivors in Japan. We used the a sample of
8,463 survivors who 45-49 years old at the time of the bombings and who were
followed up to December 31, 2000. These individuals were residents of
Hiroshima or Nagasaki; out of the this sample, 1,787 people were away from the
cities at the time that the atomic bomb was dropped. These individuals were
used to form a reference population. A measure of high vs. low exposure to
prompt radiation was based on weighted DS02 colon dose estimates, expressed as
the weighted dose in gray (Gy); those in the reference population were assumed
to have dose estimates equal to 0 Gy. For those who were present in the
cities, an individual was considered to have had a high level of exposure if
their dose estimates were above the median; otherwise they were considered to
have low exposure. The outcome of interest was age at death, measured in
years. The dataset included two covariates: city of residence and sex.

In a previous analysis of this dataset, Richardson and Tchetgen Tchetgen,
(2021) found that a bespoke IV approach revealed a more pronounced harmful
effect of high exposure to radiation compared to an analysis assuming
conditional exchangeability. They only considered a two stage least squares-
type estimator, that relies on correct modelling of the conditional mean of
outcome; we re-analysed the data using the estimators described in the
previous section that depend on alternative modelling assumptions, as well as
multiply robust approaches. We chose city of residence as the bespoke
instrument, and adjusted for sex as a covariate; since both variables were
binary in this case, it was possible to carry out a fully nonparametric
analysis. Nevertheless, as in Richardson and Tchetgen Tchetgen, (2021), a non-
saturated model for the treatment effect under a NEM assumption was chosen,
which excluded an interaction between $A$ and $C$. Further, to evaluate
sensitivity to model misspecification, we considered five different model
specifications for the nuisance parameters:

  1. 1.

All working models for the nuisance parameters were saturated.

  2. 2.

The models $OR(Z,S,C;\rho^{\dagger})$ and $t(Z,C;\nu^{\dagger})$ excluded
interactions between $Z$ and $C$.

  3. 3.

The models $OR(Z,S,C;\rho^{\dagger})$, $b_{1}(C;\theta^{\dagger}_{1})$
excluded interactions between $Z$ and $C$.

  4. 4.

The model $t(Z,C;\nu^{\dagger})$, $b_{1}(C;\theta^{\dagger}_{1})$ excluded an
interaction between $Z$ and $C$.

  5. 5.

All working models for the nuisance parameters excluded interactions between
$Z$ and $C$.

Varying the fitted models in this way enabled us to check the robustness of
the estimators $\hat{\psi}_{TSLS}$, $\hat{\psi}_{g-Z}$, $\hat{\psi}_{g-S}$,
$\hat{\psi}_{IPW}$, $\hat{\psi}_{MR}$ and $\hat{\psi}_{MR-eff}$ to departures
from a nonparametric model. We note that if all true underlying models for the
nuisance parameters include interactions, then in theory one can only
construct estimators that are unbiased under the first and fourth
specifications. To check sensitivity, we calculated the change in each
estimator from estimates based on the first specification (saturated models),
then scaling by the latter estimates.

The results of the analysis can be found in Table 2. As might be expected,
when all working models are saturated (except the exposure effect model), the
estimates are in reasonable agreement. Fitting a linear model adjusted for
radiation exposure, city of residence, sex and an interaction between the
covariates yielded an estimate of the exposure effect -0.32 (95% -0.84,0.20).
Hence the estimates from the bespoke IV analysis tended to be considerably
more marked (and arguably more plausible) than an analysis under conditional
exchangeability, albeit with much wider confidence intervals. As one varies
the model specifications, one can see the the multiply robust estimators
produce stable estimates; in contrast, $\hat{\psi}_{g-Z}$, $\hat{\psi}_{g-S}$,
$\hat{\psi}_{IPW}$ all appear to be highly sensitive to departures from
fitting saturated models, both in terms of the point estimates as well as
confidence interval width.

Table 2: Results from the analysis of the Life Span study under different model choices. 95% CI: 95% confidence interval; % Change: proportion of change from estimate based on saturated nuisance models. Specification | Estimator | Estimate |  $95\%$ CI |  $\%$ Change
---|---|---|---|---
All models saturated | $\hat{\psi}_{TSLS}$ | -1.06 | (-4.52,2.39) | 0
| $\hat{\psi}_{g-Z}$ | -1.06 | (-4.52,2.39) | 0
| $\hat{\psi}_{g-S}$ | -1.19 | (-4.64,2.27) | 0
| $\hat{\psi}_{IPW}$ | -1.09 | (-4.54,2.36) | 0
| $\hat{\psi}_{MR}$ | -1.09 | (-4.54,2.36) | 0
| $\hat{\psi}_{MR-eff}$ | -1.17 | (-4.62,2.28) | 0
$OR(\mathbf{Z},S,\mathbf{C};\rho^{\dagger})$, $t(\mathbf{Z,C};\nu^{\dagger})$ restricted | $\hat{\psi}_{TSLS}$ | -1.18 | (-4.62,2.26) | 0.11
| $\hat{\psi}_{g-Z}$ | -1.7 | (-5.21,1.8) | 0.6
| $\hat{\psi}_{g-S}$ | -1.69 | (-5.2,1.81) | 0.43
| $\hat{\psi}_{IPW}$ | -3.34 | (-7.72,1.05) | 2.06
| $\hat{\psi}_{MR}$ | -1.12 | (-4.56,2.32) | 0.03
| $\hat{\psi}_{MR-eff}$ | -1.17 | (-4.61,2.28) | 0
$OR(\mathbf{Z},S,\mathbf{C};\rho^{\dagger})$, $b_{1}(\mathbf{C};\theta^{\dagger}_{1})$ restricted | $\hat{\psi}_{TSLS}$ | -1.08 | (-4.53,2.38) | 0.01
| $\hat{\psi}_{g-Z}$ | -5.53 | (-13.49,2.43) | 4.2
| $\hat{\psi}_{g-S}$ | -1.7 | (-5.21,1.81) | 0.43
| $\hat{\psi}_{IPW}$ | -3.34 | (-7.72,1.05) | 2.06
| $\hat{\psi}_{MR}$ | -1.1 | (-4.55,2.35) | 0.01
| $\hat{\psi}_{MR-eff}$ | -1.14 | (-4.59,2.3) | -0.02
$t(\mathbf{Z,C};\nu^{\dagger})$, $b_{1}(\mathbf{C};\theta^{\dagger}_{1})$ restricted | $\hat{\psi}_{TSLS}$ | -1.19 | (-4.64,2.25) | 0.12
| $\hat{\psi}_{g-Z}$ | -1.18 | (-4.62,2.26) | 0.11
| $\hat{\psi}_{g-S}$ | -1.11 | (-4.57,2.34) | -0.06
| $\hat{\psi}_{IPW}$ | -1.09 | (-4.54,2.36) | 0
| $\hat{\psi}_{MR}$ | -1.09 | (-4.54,2.36) | 0
| $\hat{\psi}_{MR-eff}$ | -1.17 | (-4.62,2.28) | 0
All models restricted | $\hat{\psi}_{TSLS}$ | -1.19 | (-4.64,2.25) | 0.12
| $\hat{\psi}_{g-Z}$ | -1.7 | (-5.21,1.8) | 0.6
| $\hat{\psi}_{g-S}$ | -1.7 | (-5.21,1.81) | 0.43
| $\hat{\psi}_{IPW}$ | -3.34 | (-7.72,1.05) | 2.06
| $\hat{\psi}_{MR}$ | -1.14 | (-4.58,2.3) | 0.05
| $\hat{\psi}_{MR-eff}$ | -1.19 | (-4.63,2.26) | 0.01

##  7 Discussion

In this article we have proposed strategies for identifying causal effects in
studies prone to unmeasured confounding by leveraging bespoke instruments.
These strategies are applicable to studies where there is access to a
reference population as well as longitudinal studies with repeated
measurements of the outcome. In certain cases, the proposed estimators are
unbiased so long as one out of four restrictions on the observed data hold,
and hence have two additional layers of robustness than the doubly robust
estimators described in Richardson and Tchetgen Tchetgen, (2021). Potential
areas for future work include developing the proposal for binary, count and
survival outcomes, by postulating a multiplicative model for the causal effect
and (in the binary case) harnessing the nuisance model parametrisation of
Richardson et al., (2017). However, results in Vansteelandt et al., (2008)
suggest that only triple rather than quadruple robustness may be attainable in
the multiplicative model.

In this identification strategy, one must restrict either the conditional ATT
or the selection bias not to depend on the bespoke IV. Which restriction one
chooses will depend to some extent on a priori information on the bespoke
IV(s) of choice, but will also be a matter of taste. Restrictions on the
treatment effect are perhaps less appealing that in the standard IV set-up,
where the IV-outcome relationship is already constrained via the exclusion
restriction. On the other hand, under NEM, the semiparametric model
$\mathcal{M}_{NEM}$ is at least guaranteed to be correctly specified under the
null hypothesis.

Finally, one may wish to supplement our proposals with a sensitivity analysis.
Attention may be given in particular to the crucial assumption of partial
population exchangeability; one could then proceed by parametrising deviations
from this condition. With multiple plausible bespoke IVs, the ATT may become
over identified, and specification tests (like the Sargen-Hansen test) can
then be employed to assess the validity of the identifying assumptions
described here. How to optimally combine or synthesise evidence from multiple
bespoke IVs is an important area for future work.

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##  Appendix A Additional information

###  A.1 Assumptions for general bespoke IV model

######  Assumption 15.

Consistency: $Y=Y^{\mathbf{A}}$ almost surely in those with $S=1$.

######  Assumption 16.

Degenerate reference population:

$E(Y|\mathbf{Z},S=0,\mathbf{C})=E(Y^{\mathbf{0}}|\mathbf{Z},S=0,\mathbf{C})$

######  Assumption 17.

Partial population exchangeability:

$\displaystyle
E(Y^{\mathbf{0}}|\mathbf{Z=z},S=1,\mathbf{C})-E(Y^{\mathbf{0}}|\mathbf{Z=0},S=1,\mathbf{C})$
$\displaystyle=E(Y^{\mathbf{0}}|\mathbf{Z=z},S=0,\mathbf{C})-E(Y^{\mathbf{0}}|\mathbf{Z=0},S=0,\mathbf{C})\neq
0$

######  Assumption 18.

Bespoke IV relevance:
$\mathbf{A}\centernot{\rotatebox[origin={c}]{90.0}{$\models$}}\mathbf{Z}|\mathbf{C},S=1$.

###  A.2 Algorithm for constructing cross-fit de-biased machine learning
estimators

In what follows, let $\mathbf{O}=(Y,\mathbf{A},\mathbf{Z},S,\mathbf{C})$ and
$\eta(\mathbf{O})$ refer to the nuisance parameters (at the truth).

  1. 1.

Split the sample into parts $I_{k}$ (that are each are of size $n_{k}=n/K$).
Here, $K$ is an integer (we shall assume $n$ is a multiple of $K$). For each
$I_{k}$, $I^{c}_{k}$ denotes the indices that are not in $I_{k}$.

  2. 2.

For each $k=1,...,k$, using $I^{c}_{k}$ only, estimate $\eta$ as
$\hat{\eta}^{c}_{k}=\hat{\eta}((\mathbf{O}_{i})_{i\in I^{c}_{k}})$.

  3. 3.

Construct $K$ estimators $\hat{\psi}_{k},k=1,...,k$ of $\psi^{\dagger}$: under
NEM, solve the equations

$\displaystyle 0=\sum_{i\in
I_{k}}\phi(\mathbf{Z}_{i},S_{i},\mathbf{C}_{i};\hat{\eta}^{c}_{k})\epsilon_{i}^{*}(\psi^{\dagger},\hat{\eta}_{k})$

for $\psi^{\dagger}$, for each $k=1,...,k$. Under NSM, solve the equations

$\displaystyle 0=\begin{pmatrix}&\sum_{i\in
I_{k}}\phi(\mathbf{Z}_{i},S_{i},\mathbf{C}_{i};\hat{\eta}^{c}_{k})\epsilon^{*}_{i}(\psi^{\dagger},\hat{\eta}^{c}_{k})\\\
&\sum_{i\in
I_{k}}S_{i}\kappa(\mathbf{A}_{i},\mathbf{Z}_{i},\mathbf{C}_{i};\hat{\eta}^{c}_{k})\epsilon_{i}(\psi^{\dagger},\hat{\eta}^{c}_{k})\end{pmatrix}$

for $\psi^{\dagger}$, for each $k=1,...,K$.

  4. 4.

Take the average of the $K$ estimators of $\psi^{\dagger}$ to obtain
$\hat{\psi}_{CF}$.

  5. 5.

Estimate the standard error for the cross-fit estimator of $\hat{\psi}_{CF}$
using a sandwich estimator, ignoring the data-splitting; e.g. under NEM,
estimate the standard error as $\sqrt{\hat{V}/n}$, where

$\hat{V}=\frac{1}{n}\sum^{n}_{i=1}\left\\{\phi(\mathbf{Z}_{i},S_{i},\mathbf{C}_{i};\hat{\eta}^{c}_{k(i)})\epsilon^{*}_{i}(\hat{\psi}_{CF},\hat{\eta}^{c}_{k(i)})\right\\}^{2}$

where $k(i)=\\{k=1,..,K:i\in I_{k}\\}$.

This follows the ‘DML1’ proposal described in Chernozhukov et al., (2018),
along with the proposal for estimating standard errors described in
Chernozhukov et al., (2017). Justification can be found in those papers.

As in other semiparametric regression problems, a complication is how to
perform machine-learning based estimation of the nuisance parameters, given
that they are not all conditional mean functions that can be estimated
directly; see Section 4.2 of Chernozhukov et al., (2018) for further
discussion. One option is to restrict to estimation methods that can more
easily respect the structure of the semiparametric model (generalised additive
models, Lasso, deep neural networks etc); the other is to adopt the ‘localised
de-biased machine learning’ procedure described in Kallus et al., (2019),
where e.g. an initial (biased) estimator of $\psi^{\dagger}$ could be used to
construct estimators of other nuisance parameters.

##  Appendix B Proofs

###  B.1 Proof of Theorem 3.1

####  B.1.1 Proof of part (a)

###### Proof.

We introduce some additional notation:

$\displaystyle\epsilon_{1}$
$\displaystyle=Y-\beta(\mathbf{A,Z,C})-[q(\mathbf{A,Z,C})-E\\{q(\mathbf{A,Z,C})|\mathbf{Z},S=1,\mathbf{C}\\}]-b^{*}_{1}(\mathbf{C})-t(\mathbf{Z,C})$
$\displaystyle\epsilon^{*}_{1}$
$\displaystyle=\epsilon_{1}+[q(\mathbf{A,Z,C})-E\\{q(\mathbf{A,Z,C})|\mathbf{Z},S=1,\mathbf{C}\\}]$
$\displaystyle\epsilon_{0}$
$\displaystyle=Y-b_{0}(\mathbf{C})-t(\mathbf{Z,C})$

where $b^{*}_{1}(\mathbf{C})=b_{1}(\mathbf{C})+b_{0}(\mathbf{C})$. Note that
in order to obtain identification, we enforce that
$\beta(\mathbf{A,Z,C})=\beta(\mathbf{A,C})$.

The likelihood for a given observation can be written as

$f(\mathbf{O})=f(\epsilon_{0}|\mathbf{Z},S=0,\mathbf{C})^{(1-S)}\left\\{f(\epsilon_{1}|\mathbf{A},\mathbf{Z},S=1,\mathbf{C})f(\mathbf{A}|\mathbf{Z},S=1,\mathbf{C})\right\\}^{S}f(\mathbf{Z},S|\mathbf{C})f(\mathbf{C})$

Then we will consider the parametric submodel:

$f_{r}(\mathbf{O})=f_{r}(\epsilon_{0_{r}}|\mathbf{Z},S=0,\mathbf{C})^{(1-S)}\left\\{f_{r}(\epsilon_{1_{r}}|\mathbf{A,Z},S=1,\mathbf{C})f_{r}(A|\mathbf{Z},S=1,\mathbf{C})\right\\}^{S}f_{r}(\mathbf{Z},S|\mathbf{C})f_{r}(S,\mathbf{C})$

which varies in the direction of $q_{r}(A,\mathbf{Z})$,
$b_{0_{r}}(\mathbf{C})$, $b^{*}_{1_{r}}(\mathbf{C})$, $t_{r}(\mathbf{Z})$,
$f_{r}(\mathbf{A}|\mathbf{Z},S=1,\mathbf{C})$,
$f_{r}(\mathbf{Z},S|\mathbf{C})$ and $f_{r}(\mathbf{C})$. Here,

$\displaystyle\epsilon_{1_{r}}$
$\displaystyle=Y-\beta(\mathbf{A,Z,C})-[q_{r}(\mathbf{A,Z,C})-E_{r}\\{q_{r}(\mathbf{A,Z,C})|\mathbf{Z},S=1,\mathbf{C}\\}]-b^{*}_{1_{r}}(\mathbf{C})-t_{r}(\mathbf{Z,C})$
$\displaystyle\epsilon_{0_{r}}$
$\displaystyle=Y-b_{0_{r}}(\mathbf{C})-t_{r}(\mathbf{Z,C})$

The nuisance tangent space $\lambda_{nuis}$ under $\mathcal{M}_{NEM}$ can be
characterised as

$\lambda_{nuis}=\lambda_{nuis_{1}}\oplus\lambda_{nuis_{2}}\oplus\lambda_{nuis_{3}}\oplus\lambda_{nuis_{4}}\oplus\lambda_{nuis_{5}}\oplus\lambda_{nuis_{6}}\oplus\lambda_{nuis_{7}}\oplus\lambda_{nuis_{8}}\oplus\lambda_{nuis_{9}}$

where

$\displaystyle\Lambda_{nuis_{1}}=\bigg{\\{}$ $\displaystyle
Sd_{1}(\epsilon_{1},\mathbf{A,Z,C}):$ $\displaystyle
E\\{d_{1}(\epsilon_{1},\mathbf{A,Z,C})|\mathbf{A,Z},S=1,\mathbf{C}\\}=E\\{\epsilon_{1}d_{1}(\epsilon_{1},\mathbf{A,Z,C})|\mathbf{A,Z},S=1,\mathbf{C}\\}=0\bigg{\\}}\cap
L_{2}^{0}$ $\displaystyle\Lambda_{nuis_{2}}=\bigg{\\{}$
$\displaystyle(1-S)d_{2}(\epsilon_{0},\mathbf{Z,C}):E\\{d_{2}(\epsilon_{0},\mathbf{Z,C})|\mathbf{Z},S=0,\mathbf{C}\\}=E\\{\epsilon_{0}d_{2}(\epsilon_{1},\mathbf{Z})|\mathbf{Z},S=0,\mathbf{C}\\}=0\bigg{\\}}\cap
L_{2}^{0}$ $\displaystyle\Lambda_{nuis_{3}}=\bigg{\\{}$ $\displaystyle
d_{3}(\mathbf{Z},S,\mathbf{C}):E\\{d_{3}(\mathbf{Z},S,\mathbf{C})|\mathbf{C}\\}=0\bigg{\\}}\cap
L_{2}^{0}$ $\displaystyle\Lambda_{nuis_{4}}=\bigg{\\{}$ $\displaystyle
d_{4}(\mathbf{C}):E\\{d_{4}(\mathbf{C})\\}=0\bigg{\\}}\cap L_{2}^{0}$
$\displaystyle\Lambda_{nuis_{5}}=\bigg{\\{}$ $\displaystyle
S[d_{5}(\mathbf{A,Z,C})-E\\{d_{5}(\mathbf{A,Z,C})|\mathbf{Z},S=1,\mathbf{C}\\}]f^{\prime}_{\epsilon_{1}}:\textrm{$d_{5}(\mathbf{A,Z,C})$
unrestricted}\bigg{\\}}\cap L_{2}^{0}$
$\displaystyle\Lambda_{nuis_{6}}=\bigg{\\{}$ $\displaystyle
Sd_{6}(A,\mathbf{Z,C})+Sf^{\prime}_{\epsilon_{1}}\int
d_{6}(A^{*},\mathbf{Z,C})q(A^{*},\mathbf{Z,C})dF(A^{*}|\mathbf{Z},S=1,\mathbf{C}):$
$\displaystyle
E\\{d_{6}(A,\mathbf{Z,C})|\mathbf{Z},S=1,\mathbf{C}\\}=0\bigg{\\}}\cap
L_{2}^{0}$ $\displaystyle\Lambda_{nuis_{7}}=\bigg{\\{}$ $\displaystyle
Sd_{7}(\mathbf{C})f^{\prime}_{\epsilon_{1}}:\textrm{$d_{7}(\mathbf{C})$
unrestricted}\bigg{\\}}\cap L_{2}^{0}$
$\displaystyle\Lambda_{nuis_{8}}=\bigg{\\{}$
$\displaystyle(1-S)d_{8}(\mathbf{C})f^{\prime}_{\epsilon_{0}}:\textrm{$d_{8}(\mathbf{C})$
unrestricted}\bigg{\\}}\cap L_{2}^{0}$
$\displaystyle\Lambda_{nuis_{9}}=\bigg{\\{}$ $\displaystyle
Sd_{9}(\mathbf{Z,C})f^{\prime}_{\epsilon_{1}}+(1-S)d_{9}(\mathbf{Z,C})f^{\prime}_{\epsilon_{0}}:\textrm{$d_{9}(\mathbf{C})$
unrestricted}\bigg{\\}}\cap L_{2}^{0}$

where $f^{\prime}_{\epsilon_{1}}$ is the derivative of
$f(\epsilon_{1}|A,\mathbf{Z},S=1,\mathbf{C})$ w.r.t. $\epsilon_{1}$ and
likewise $f^{\prime}_{\epsilon_{0}}$ is the derivative of
$f(\epsilon_{0}|\mathbf{Z},S=0,\mathbf{C})$ w.r.t. $\epsilon_{0}$.

Using standard results on the restricted mean model e.g. Tsiatis, (2007), we
have that

$\displaystyle\Lambda^{\perp}_{nuis_{1}}\cap\Lambda^{\perp}_{nuis_{3}}\cap\Lambda^{\perp}_{nuis_{4}}=\bigg{\\{}$
$\displaystyle
S\epsilon_{1}c_{1}(\mathbf{A,Z,C})+Sc_{2}(\mathbf{A,Z},\mathbf{C}):$
$\displaystyle\textit{$c_{1}(\mathbf{A,Z,C})$
unrestricted},E\\{c_{2}(\mathbf{A,Z},\mathbf{C})|\mathbf{Z},S=1,\mathbf{C}\\}=0\bigg{\\}}\cap
L_{2}^{0}$

Then to find elements in this space that are orthogonal to
$\Lambda_{nuis_{5}}$, we must find the elements in
$\Lambda^{\perp}_{nuis_{1}}\cap\Lambda^{\perp}_{nuis_{3}}\cap\Lambda^{\perp}_{nuis_{4}}$
that satisfy:

$\displaystyle 0$
$\displaystyle=E\left(S\\{\epsilon_{1}c_{1}(\mathbf{A,Z,C})+c_{2}(\mathbf{A,Z},\mathbf{C})\\}[d_{5}(\mathbf{A,Z,C})-E\\{d_{5}(\mathbf{A,Z,C})|\mathbf{Z},S=1,\mathbf{C}\\}]f^{\prime}_{\epsilon_{1}}\right)$
$\displaystyle=E\left(Sc_{1}(\mathbf{A,Z,C})[d_{5}(\mathbf{A,Z,C})-E\\{d_{5}(\mathbf{A,Z,C})|\mathbf{Z},S=1,\mathbf{C}\\}]\right)$

It follows that elements of $Sc_{1}(\mathbf{A,Z,C})$ that will satisfy the
equality are of the form $S\phi_{1}(\mathbf{Z,C})$, where
$\phi_{1}(\mathbf{Z,C})$ are unrestricted. One can also show that the elements

$S\epsilon_{1}[\phi_{1}(\mathbf{Z,C})-E\\{\phi_{1}(\mathbf{Z,C})|S=1,\mathbf{C}\\}]+Sc_{2}(\mathbf{A,Z},\mathbf{C})$

are orthogonal to $\Lambda_{nuis_{7}}$.

Next, in considering $\Lambda_{nuis_{6}}$, we need elements that satisfy:

$\displaystyle 0=E\bigg{\\{}$ $\displaystyle
S\left(\epsilon_{1}[\phi_{1}(\mathbf{Z,C})-E\\{\phi_{1}(\mathbf{Z,C})|S=1,\mathbf{C}\\}]+c_{2}(\mathbf{A,Z},\mathbf{C})\right)$
$\displaystyle\times[d_{6}(A,\mathbf{Z,C})-E\\{Sd_{6}(A,\mathbf{Z,C})q(\mathbf{A,Z,C})|\mathbf{Z},S=1,\mathbf{C}\\}]f^{\prime}_{\epsilon_{1}}\bigg{\\}}$

Then note that

$\displaystyle
E\bigg{\\{}S\epsilon_{1}[\phi_{1}(\mathbf{Z,C})-E\\{\phi_{1}(\mathbf{Z,C})|S=1,\mathbf{C}\\}]d_{6}(\mathbf{A,Z,C})\bigg{\\}}=0$
$\displaystyle
E\bigg{\\{}-Sc_{2}(\mathbf{A,Z},\mathbf{C})E\\{d_{6}(A,\mathbf{Z,C})q(\mathbf{A,Z,C})|\mathbf{Z},S=1,\mathbf{C}\\}f^{\prime}_{\epsilon_{1}}\bigg{\\}}=0$

We are left with the restriction that

$\displaystyle 0=E\bigg{(}$ $\displaystyle
Sc_{2}(\mathbf{A,Z},\mathbf{C})d_{6}(\mathbf{A,Z,C})$
$\displaystyle-S\epsilon_{1}[\phi_{1}(\mathbf{Z,C})-E\\{\phi_{1}(\mathbf{Z,C})|S=1,\mathbf{C}\\}]E\\{d_{6}(A,\mathbf{Z,C})q(\mathbf{A,Z,C})|\mathbf{Z},S=1,\mathbf{C}\\}f^{\prime}_{\epsilon_{1}}\bigg{)}$
$\displaystyle=E\bigg{(}$ $\displaystyle Sd_{6}(\mathbf{A,Z,C})$
$\displaystyle\times\left(c_{2}(\mathbf{A,Z},\mathbf{C})-[\phi_{1}(\mathbf{Z,C})-E\\{\phi_{1}(\mathbf{Z,C})|S=1,\mathbf{C}\\}][q(\mathbf{A,Z,C})-E\\{q(\mathbf{A,Z,C})|\mathbf{Z},S=1,\mathbf{C}\\}]\right)\bigg{)}$

and so

$c_{2}(\mathbf{A,Z},\mathbf{C})=[\phi_{1}(\mathbf{Z,C})-E\\{\phi_{1}(\mathbf{Z,C})|S=1,\mathbf{C}\\}][q(\mathbf{A,Z,C})-E\\{q(\mathbf{A,Z,C})|\mathbf{Z},S=1,\mathbf{C}\\}]$

Hence, so far we have shown that

$\displaystyle\Lambda^{\perp}_{nuis_{1}}\cap\Lambda^{\perp}_{nuis_{3}}\cap\Lambda^{\perp}_{nuis_{4}}\cap\Lambda^{\perp}_{nuis_{5}}\cap\Lambda^{\perp}_{nuis_{6}}\cap\Lambda^{\perp}_{nuis_{7}}$
$\displaystyle=\bigg{\\{}S\epsilon^{*}_{1}[\phi_{1}(\mathbf{Z,C})-E\\{\phi_{1}(\mathbf{Z,C})|S=1,\mathbf{C}\\}]:\textit{$\phi_{1}(\mathbf{Z,C})$
unrestricted}\bigg{\\}}\cap L_{2}^{0}$

Considering elements in
$\Lambda^{\perp}_{nuis_{2}}\cap\Lambda^{\perp}_{nuis_{8}}$, then using
previous reasoning, we have that

$\Lambda^{\perp}_{nuis_{1}}\cap\Lambda^{\perp}_{nuis_{7}}=\left\\{(1-S)\epsilon_{0}[\phi_{2}(\mathbf{Z,C})-E\\{\phi_{2}(\mathbf{Z,C})|\mathbf{C},S=0\\}];\textit{$\phi_{2}(\mathbf{Z,C})$
unrestricted}\right\\}$

It is also straightforward to show that elements in the space
$\Lambda^{\perp}_{nuis_{2}}\cap\Lambda^{\perp}_{nuis_{8}}$ are orthogonal to
those in the space
$\Lambda^{\perp}_{nuis_{1}}\cap\Lambda^{\perp}_{nuis_{3}}\cap\Lambda^{\perp}_{nuis_{4}}\cap\Lambda^{\perp}_{nuis_{5}}\cap\Lambda^{\perp}_{nuis_{6}}\cap\Lambda^{\perp}_{nuis_{7}}$,
and therefore

$\displaystyle\Lambda^{\perp}_{nuis_{1}}\cap\Lambda^{\perp}_{nuis_{2}}\cap\Lambda^{\perp}_{nuis_{3}}\cap\Lambda^{\perp}_{nuis_{4}}\cap\Lambda^{\perp}_{nuis_{5}}\cap\Lambda^{\perp}_{nuis_{6}}\cap\Lambda^{\perp}_{nuis_{7}}\cap\Lambda^{\perp}_{nuis_{8}}$
$\displaystyle=\bigg{\\{}S\epsilon^{*}_{1}[\phi_{1}(\mathbf{Z,C})-E\\{\phi_{1}(\mathbf{Z,C})|S=1,\mathbf{C}\\}]$
$\displaystyle\quad+(1-S)\epsilon_{0}[\phi_{2}(\mathbf{Z,C})-E\\{\phi_{2}(\mathbf{Z,C})|\mathbf{C},S=0\\}]:\textit{$\phi_{1}(\mathbf{Z,C}),\phi_{2}(\mathbf{Z,C})$
unrestricted}\bigg{\\}}$

It remains to find elements in this space that are orthogonal to
$\Lambda^{\perp}_{nuis_{9}}$; these must satisfy

$\displaystyle 0=$ $\displaystyle
E\bigg{\\{}\bigg{(}S\epsilon^{*}_{1}[\phi_{1}(\mathbf{Z,C})-E\\{\phi_{1}(\mathbf{Z,C})|S=1,\mathbf{C}\\}]+(1-S)\epsilon_{0}[\phi_{2}(\mathbf{Z,C})-E\\{\phi_{2}(\mathbf{Z,C})|S=0,\mathbf{C}\\}]\bigg{)}$
$\displaystyle\times\\{Sd_{9}(\mathbf{Z,C})f^{\prime}_{\epsilon_{1}}+(1-S)d_{9}(\mathbf{Z,C})f^{\prime}_{\epsilon_{0}}\\}\bigg{)}$
$\displaystyle=$ $\displaystyle
E\bigg{\\{}\bigg{(}S[\phi_{1}(\mathbf{Z,C})-E\\{\phi_{1}(\mathbf{Z,C})|S=1,\mathbf{C}\\}]+(1-S)[\phi_{2}(\mathbf{Z,C})-E\\{\phi_{2}(\mathbf{Z,C})|S=0,\mathbf{C}\\}]\bigg{)}d_{9}(\mathbf{Z,C})\bigg{\\}}$

It follows from Tchetgen Tchetgen et al., (2010) that the space of elements

$S[\phi_{1}(\mathbf{Z,C})-E\\{\phi_{1}(\mathbf{Z,C})|S=1,\mathbf{C}\\}]+(1-S)[\phi_{2}(\mathbf{Z,C})-E\\{\phi_{2}(\mathbf{Z,C})|\mathbf{C},S=0\\}]$

that satisfy this equality can be represented as

$\Omega=\\{\phi(\mathbf{Z},S,\mathbf{C}):E\\{\phi(\mathbf{Z},S,\mathbf{C})|\mathbf{Z,C}\\}=E\\{\phi(\mathbf{Z},S,\mathbf{C})|S,\mathbf{C}\\}=0\\}.$

Then the main result follows. ∎

####  B.1.2 Proof of part (b)

###### Proof.

Many of the steps in in this proof follow along the lines of the proof of part
(a) and can therefore be omitted for brevity. We note however that NSM
Assumption (8) imposes different restrictions on the nuisance tangent space,
such that we redefine $\Lambda_{nuis_{5}}$ as

$\displaystyle\Lambda_{nuis_{5}}=\bigg{\\{}$ $\displaystyle
S[d_{5}(\mathbf{A,C})-E\\{d_{5}(\mathbf{A,C})|\mathbf{Z},S=1,\mathbf{C}\\}]f^{\prime}_{\epsilon_{1}}:d_{5}(\mathbf{A,C})\textrm{
unrestricted}\bigg{\\}}\cap L_{2}^{0}.$

Along the lines of the proof of Theorem 4 in Tchetgen Tchetgen and
Vansteelandt, (2013), one can show that

$\displaystyle\Lambda^{\perp}_{nuis_{1}}\cap\Lambda^{\perp}_{nuis_{3}}\cap\Lambda^{\perp}_{nuis_{4}}\cap\Lambda^{\perp}_{nuis_{5}}\cap\Lambda^{\perp}_{nuis_{6}}\cap\Lambda^{\perp}_{nuis_{7}}$
$\displaystyle=\bigg{\\{}S\epsilon_{1}[\kappa(\mathbf{A,Z,C})+\phi_{1}(\mathbf{Z,C})-E\\{\phi_{1}(\mathbf{Z,C})|S=1,\mathbf{C}\\}]$
$\displaystyle\quad\quad+S[q(\mathbf{A,Z,C})-E\\{q(\mathbf{A,Z,C})|\mathbf{Z},S=1,\mathbf{C}\\}][\phi_{1}(\mathbf{Z,C})-E\\{\phi_{1}(\mathbf{Z,C})|S=1,\mathbf{C}\\}]:$
$\displaystyle\quad\quad\textit{$\kappa(\mathbf{A,Z,C})\in\Gamma$,
$\phi_{1}(\mathbf{Z,C})$ unrestricted}\bigg{\\}}\cap L_{2}^{0}$

and as in the previous proof,

$\Lambda^{\perp}_{nuis_{2}}\cap\Lambda^{\perp}_{nuis_{8}}=\left\\{(1-S)\epsilon_{0}[\phi_{2}(\mathbf{Z,C})-E\\{\phi_{2}(\mathbf{Z,C})|\mathbf{C},S=0\\}]:\textit{$\phi_{2}(\mathbf{Z,C})$
unrestricted}\right\\}$

Then one can find the elements

$\displaystyle
S\epsilon_{1}[\kappa(\mathbf{A,Z,C})+\phi_{1}(\mathbf{Z,C})-E\\{\phi_{1}(\mathbf{Z,C})|S=1,\mathbf{C}\\}]$
$\displaystyle+[q(\mathbf{A,Z,C})-E\\{q(\mathbf{A,Z,C})|\mathbf{Z},S=1,\mathbf{C}\\}][\phi_{1}(\mathbf{Z,C})-E\\{\phi_{1}(\mathbf{Z,C})|S=1,\mathbf{C}\\}]$
$\displaystyle+(1-S)\epsilon_{0}[\phi_{2}(\mathbf{Z,C})-E\\{\phi_{2}(\mathbf{Z,C})|\mathbf{C},S=0\\}]$

that are orthogonal to $\lambda_{nuis_{9}}$ as in the previous proof, and the
main result follows.

∎

###  B.2 Proof of Theorems 3.2

###### Proof.

We will first show that

$E\\{\phi(\mathbf{Z},S,\mathbf{C};\tau^{\dagger},\alpha^{\dagger},\rho^{\dagger})\epsilon^{*}(\psi^{\dagger},\nu^{\dagger},\theta^{\dagger})\\}$

evaluated at the limiting (rather than estimated) values is an unbiased
estimating function under the union model. Under model
$\mathcal{M}_{NEM}\cap\mathcal{M}_{1}$, by the law of iterated expectation,

$\displaystyle
E\\{\phi(\mathbf{Z},S,\mathbf{C};\tau^{\dagger},\alpha^{\dagger},\rho^{\dagger})\epsilon^{*}(\psi^{\dagger},\nu^{\dagger},\theta^{\dagger})\\}$
$\displaystyle=E\bigg{(}\phi(\mathbf{Z},S,\mathbf{C};\tau^{\dagger},\alpha^{\dagger},\rho^{\dagger})[b_{0}(\mathbf{C})+b_{1}(\mathbf{C})S+t(\mathbf{Z,C})-b_{0}(\mathbf{C};\theta_{0}^{\dagger})-b_{1}(\mathbf{C};\theta_{1}^{\dagger})S-t(\mathbf{Z,C};\nu^{\dagger})$
$\displaystyle\quad+q(\mathbf{A,Z,C})-E\\{q(\mathbf{A,Z,C})|\mathbf{Z},S=1,\mathbf{C}\\}]\bigg{)}$
$\displaystyle=E\left[\phi(\mathbf{Z},S,\mathbf{C};\tau^{\dagger},\alpha^{\dagger},\rho^{\dagger})\\{b_{0}(\mathbf{C})+b_{1}(\mathbf{C})S+t(\mathbf{Z,C})-b_{0}(\mathbf{C};\theta_{0}^{\dagger})-b_{1}(\mathbf{C};\theta_{1}^{\dagger})S-t(\mathbf{Z,C};\nu^{\dagger})\\}\right]=0$

Under model $\mathcal{M}_{NEM}\cap\mathcal{M}_{2}$, looking back to the final
line of the previous expression,

$\displaystyle
E\left[\phi(\mathbf{Z},S,\mathbf{C};\tau^{\dagger},\alpha^{\dagger},\rho^{\dagger})\\{b_{0}(\mathbf{C})+b_{1}(\mathbf{C})S+t(\mathbf{Z,C})-b_{0}(\mathbf{C};\theta_{0}^{\dagger})-b_{1}(\mathbf{C};\theta_{1}^{\dagger})S-t(\mathbf{Z,C};\nu^{\dagger})\\}\right]$
$\displaystyle=E\left[\phi(\mathbf{Z},S,\mathbf{C};\tau^{\dagger},\alpha^{\dagger},\rho^{\dagger})\\{b_{0}(\mathbf{C})+b_{1}(\mathbf{C})S-b_{0}(\mathbf{C};\theta_{0}^{\dagger})-b_{1}(\mathbf{C};\theta_{1}^{\dagger})S\\}\right]$
$\displaystyle=E\left[E\\{\phi(\mathbf{Z},S,\mathbf{C};\tau^{\dagger},\alpha^{\dagger},\rho^{\dagger})|S,\mathbf{C}\\}\\{b_{0}(\mathbf{C})+b_{1}(\mathbf{C})S-b_{0}(\mathbf{C};\theta_{0}^{\dagger})-b_{1}(\mathbf{C};\theta_{1}^{\dagger})S\\}\right]=0$

where the final equality follows since
$f(\mathbf{Z}|S,\mathbf{C})=f(\mathbf{Z}|S,\mathbf{C};\nu^{\dagger},\rho^{\dagger})$.

Under model $\mathcal{M}_{NEM}\cap\mathcal{M}_{3}$,

$\displaystyle
E\left[\phi(\mathbf{Z},S,\mathbf{C};\tau^{\dagger},\alpha^{\dagger},\rho^{\dagger})\\{b_{0}(\mathbf{C})+b_{1}(\mathbf{C})S+t(\mathbf{Z,C})-b_{0}(\mathbf{C};\theta_{0}^{\dagger})-b_{1}(\mathbf{C};\theta_{1}^{\dagger})S-t(\mathbf{Z,C};\nu^{\dagger})\\}\right]$
$\displaystyle=E\left[E\\{\phi(\mathbf{Z},S,\mathbf{C};\tau^{\dagger},\alpha^{\dagger},\rho^{\dagger})|\mathbf{Z,C}\\}\\{b_{0}(\mathbf{C})+t(\mathbf{Z,C})-b_{0}(\mathbf{C};\theta_{0}^{\dagger})-t(\mathbf{Z,C};\nu^{\dagger})\\}\right]=0$

where the final equality follows since
$f(S|S,\mathbf{Z,C})=f(S|\mathbf{Z,C};\alpha^{\dagger},\rho^{\dagger})$.

Finally, under model $\mathcal{M}_{NEM}\cap\mathcal{M}_{4}$,

$\displaystyle
E\left[\phi(\mathbf{Z},S,\mathbf{C};\tau^{\dagger},\alpha^{\dagger},\rho^{\dagger})\\{b_{0}(\mathbf{C})+b_{1}(\mathbf{C})S+t(\mathbf{Z,C})-b_{0}(\mathbf{C};\theta_{0}^{\dagger})-b_{1}(\mathbf{C};\theta_{1}^{\dagger})S-t(\mathbf{Z,C};\nu^{\dagger})\\}\right]$
$\displaystyle=E\left[E\\{\phi(\mathbf{Z},S,\mathbf{C};\tau^{\dagger},\alpha^{\dagger},\rho^{\dagger})|S,\mathbf{C}\\}\\{b_{1}(\mathbf{C})S-b_{1}(\mathbf{C};\theta_{1}^{\dagger})S\\}\right]$
$\displaystyle\quad+E\left[E\\{\phi(\mathbf{Z},S,\mathbf{C};\tau^{\dagger},\alpha^{\dagger},\rho^{\dagger})|\mathbf{Z,C}\\}\\{t(\mathbf{Z,C})-t(\mathbf{Z,C};\nu^{\dagger})\\}\right]$
$\displaystyle\quad+E\left\\{\phi(\mathbf{Z},S,\mathbf{C};\tau^{\dagger},\alpha^{\dagger},\rho^{\dagger})\\{b_{0}(\mathbf{C})-b_{0}(\mathbf{C};\theta_{0}^{\dagger})\\}\right\\}=0$

Note furthermore that

  * •

$f(\mathbf{Z}|S,\mathbf{C};\hat{\tau},\hat{\rho})$ is a CAN estimator of
$f(\mathbf{Z}|S,\mathbf{C})$ in model $\mathcal{M}_{2}\cup\mathcal{M}_{4}$.

  * •

$f(S|\mathbf{Z,C};\hat{\alpha},\hat{\rho})$ is a CAN estimator of
$f(S|\mathbf{Z,C})$ in model $\mathcal{M}_{3}\cup\mathcal{M}_{4}$.

  * •

$t(\mathbf{Z,C};\hat{\nu})$ is a CAN estimator of $t(\mathbf{Z,C})$ in model
$\mathcal{M}_{NEM}\cap(\mathcal{M}_{1}\cup\mathcal{M}_{2})$.

  * •

$b_{1}(\mathbf{C};\hat{\theta}_{1})$ is a CAN estimator of $b_{1}(\mathbf{C})$
in model $\mathcal{M}_{NEM}\cap(\mathcal{M}_{1}\cup\mathcal{M}_{3})$.

  * •

$b_{0}(\mathbf{C};\hat{\theta}_{0})$ is a CAN estimator of $b_{0}(\mathbf{C})$
in model
$\mathcal{M}_{NEM}\cap(\mathcal{M}_{1}\cup\mathcal{M}_{2}\cup\mathcal{M}_{3})$.

These results follow from Robins et al., (1992) and Tchetgen Tchetgen et al.,
(2010). This completes the proof. ∎

###  B.3 Proof of Theorem 3.3

###### Proof.

If we consider the estimating function

$S\kappa(\mathbf{A,Z,C};\pi^{\dagger},\tau^{\dagger},\rho^{\dagger})\epsilon(\psi^{\dagger},\nu^{\dagger},\theta^{\dagger},\omega^{\dagger},\pi^{\dagger})+\phi(\mathbf{Z},S,\mathbf{C};\tau^{\dagger},\alpha^{\dagger},\rho^{\dagger})\epsilon^{*}(\psi^{\dagger},\nu^{\dagger},\theta^{\dagger})$

then it follows from the previous proof that

$E\\{\phi(\mathbf{Z},S,\mathbf{C};\tau^{\dagger},\alpha^{\dagger},\rho^{\dagger})\epsilon^{*}(\psi^{\dagger},\nu^{\dagger},\theta^{\dagger})\\}=0$

under the union model.

Under model $\mathcal{M}_{NSM}\cap\mathcal{M}_{a}\cap\mathcal{M}_{1q}$,

$\displaystyle
E\\{S\kappa(\mathbf{A,Z,C};\pi^{\dagger},\tau^{\dagger},\rho^{\dagger})\epsilon(\psi^{\dagger},\nu^{\dagger},\theta^{\dagger},\omega^{\dagger},\pi^{\dagger})\\}$
$\displaystyle=E\bigg{(}S\kappa(\mathbf{A,Z,C};\pi^{\dagger},\tau^{\dagger},\rho^{\dagger})[b_{0}(\mathbf{C})+b_{1}(\mathbf{C})S+t(\mathbf{Z,C})-b_{0}(\mathbf{C};\theta_{0}^{\dagger})-b_{1}(\mathbf{C};\theta_{1}^{\dagger})S-t(\mathbf{Z,C};\nu^{\dagger})$
$\displaystyle\quad+q(\mathbf{A,C})-E\\{q(\mathbf{A,C})|\mathbf{Z},S=1,\mathbf{C}\\}-q(\mathbf{A,C};\omega^{\dagger})+E\\{q(\mathbf{A,C};\omega^{\dagger})|\mathbf{Z},S=1,\mathbf{C};\pi^{\dagger}\\}]\bigg{)}=0$

Under model $\mathcal{M}_{NSM}\cap\mathcal{M}_{a}\cap\mathcal{M}_{2}$,

$\displaystyle
E\bigg{(}S\kappa(\mathbf{A,Z,C};\pi^{\dagger},\tau^{\dagger},\rho^{\dagger})[b_{0}(\mathbf{C})+b_{1}(\mathbf{C})S+t(\mathbf{Z,C})-b_{0}(\mathbf{C};\theta_{0}^{\dagger})-b_{1}(\mathbf{C};\theta_{1}^{\dagger})S-t(\mathbf{Z,C};\nu^{\dagger})$
$\displaystyle\quad+q(\mathbf{A,C})-E\\{q(\mathbf{A,C})|\mathbf{Z},S=1,\mathbf{C}\\}-q(\mathbf{A,C};\omega^{\dagger})+E\\{q(\mathbf{A,C};\omega^{\dagger})|\mathbf{Z},S=1,\mathbf{C};\pi^{\dagger}\\}]\bigg{)}$
$\displaystyle=E\bigg{(}S\kappa(\mathbf{A,Z,C};\pi^{\dagger},\tau^{\dagger},\rho^{\dagger})[b_{0}(\mathbf{C})+b_{1}(\mathbf{C})S-b_{0}(\mathbf{C};\theta_{0}^{\dagger})-b_{1}(\mathbf{C};\theta_{1}^{\dagger})S-$
$\displaystyle\quad+q(\mathbf{A,C})-E\\{q(\mathbf{A,C})|\mathbf{Z},S=1,\mathbf{C}\\}-q(\mathbf{A,C};\omega^{\dagger})+E\\{q(\mathbf{A,C};\omega^{\dagger})|\mathbf{Z},S=1,\mathbf{C};\pi^{\dagger}\\}]\bigg{)}$
$\displaystyle=E\left[SE\\{\kappa(\mathbf{A,Z,C};\pi^{\dagger},\tau^{\dagger},\rho^{\dagger})|S=1,\mathbf{C}\\}\\{b_{0}(\mathbf{C})+b_{1}(\mathbf{C})S-b_{0}(\mathbf{C};\theta_{0}^{\dagger})-b_{1}(\mathbf{C};\theta_{1}^{\dagger})S\\}\right]$
$\displaystyle\quad+E\left[SE\\{\kappa(\mathbf{A,Z,C};\pi^{\dagger},\tau^{\dagger},\rho^{\dagger})|\mathbf{A},S=1,\mathbf{C}\\}\\{q(\mathbf{A,C})-q(\mathbf{A,C};\omega^{\dagger})\\}\right]$
$\displaystyle\quad-E\bigg{(}SE\\{\kappa(\mathbf{A,Z,C};\pi^{\dagger},\tau^{\dagger},\rho^{\dagger})|\mathbf{Z},S=1,\mathbf{C}\\}[E\\{q(\mathbf{A,C})|\mathbf{Z},S=1,\mathbf{C}\\}$
$\displaystyle\quad-E\\{q(\mathbf{A,C};\omega^{\dagger})|\mathbf{Z},S=1,\mathbf{C};\pi^{\dagger}\\}]\bigg{)}$
$\displaystyle=0$

where the final equality follows since
$f(\mathbf{A,Z}|S=1,\mathbf{C})=f(\mathbf{A,Z}|S=1,\mathbf{C};\pi^{\dagger},\tau^{\dagger},\rho^{\dagger})$.

Under model $\mathcal{M}_{NSM}\cap\mathcal{M}_{a}\cap\mathcal{M}_{3q}$,

$\displaystyle
E\bigg{(}S\kappa(\mathbf{A,Z,C};\pi^{\dagger},\tau^{\dagger},\rho^{\dagger})[b_{0}(\mathbf{C})+b_{1}(\mathbf{C})S+t(\mathbf{Z,C})-b_{0}(\mathbf{C};\theta_{0}^{\dagger})-b_{1}(\mathbf{C};\theta_{1}^{\dagger})S-t(\mathbf{Z,C};\nu^{\dagger})$
$\displaystyle\quad+q(\mathbf{A,C})-E\\{q(\mathbf{A,C})|\mathbf{Z},S=1,\mathbf{C}\\}-q(\mathbf{A,C};\omega^{\dagger})+E\\{q(\mathbf{A,C};\omega^{\dagger})|\mathbf{Z},S=1,\mathbf{C};\pi^{\dagger}\\}]\bigg{)}$
$\displaystyle=E\left[S\kappa(\mathbf{A,Z,C};\pi^{\dagger},\tau^{\dagger},\rho^{\dagger})\\{b_{0}(\mathbf{C})+t(\mathbf{Z,C})-b_{0}(\mathbf{C};\theta_{0}^{\dagger})-b_{1}(\mathbf{C};\theta_{1}^{\dagger})S-t(\mathbf{Z,C};\nu^{\dagger})\\}\right]$
$\displaystyle=E\left[SE\\{\kappa(\mathbf{A,Z,C};\pi^{\dagger},\tau^{\dagger},\rho^{\dagger})|\mathbf{Z},S=1,\mathbf{C}\\}\\{b_{0}(\mathbf{C})+t(\mathbf{Z,C})-b_{0}(\mathbf{C};\theta_{0}^{\dagger})-b_{1}(\mathbf{C};\theta_{1}^{\dagger})S-t(\mathbf{Z,C};\nu^{\dagger})\\}\right]$
$\displaystyle=0$

where the final equality follows since
$f(\mathbf{A}|,\mathbf{Z},S=1,\mathbf{C})=f(\mathbf{A}|\mathbf{Z},S=1,\mathbf{C};\pi^{\dagger})$.

Under model $\mathcal{M}_{NSM}\cap\mathcal{M}_{a}\cap\mathcal{M}_{4}$, it
follows from previous arguments that

$\displaystyle
E\bigg{(}S\kappa(\mathbf{A,Z,C};\pi^{\dagger},\tau^{\dagger},\rho^{\dagger})[b_{0}(\mathbf{C})+b_{1}(\mathbf{C})S+t(\mathbf{Z,C})-b_{0}(\mathbf{C};\theta_{0}^{\dagger})-b_{1}(\mathbf{C};\theta_{1}^{\dagger})S-t(\mathbf{Z,C};\nu^{\dagger})$
$\displaystyle\quad+q(\mathbf{A,C})-E\\{q(\mathbf{A,C})|\mathbf{Z},S=1,\mathbf{C}\\}-q(\mathbf{A,C};\omega^{\dagger})+E\\{q(\mathbf{A,C};\omega^{\dagger})|\mathbf{Z},S=1,\mathbf{C};\pi^{\dagger}\\}]\bigg{)}=0$

since
$f(\mathbf{A,Z}|S=1,\mathbf{C})=f(\mathbf{A,Z}|S=1,\mathbf{C};\pi^{\dagger},\tau^{\dagger})$.

∎

###  B.4 Proof of Theorem 3.4

###### Proof.

Under a restricted moment model where $t(\mathbf{Z,C})$, $b_{0}(\mathbf{C})$,
$b_{1}(\mathbf{C})$ and $q(\mathbf{A,Z,C})$ are known, it follows e.g. from
Robins, (1994) that the efficient score for $\psi^{\dagger}$ is equal to

$U_{RM-
eff}=\mu(\mathbf{Z},S,\mathbf{C})\sigma^{-2}(\mathbf{Z},S,\mathbf{C})\epsilon^{*}(\psi^{\dagger})$

It also follows from the proofs of Theorem 1 and 4 in Vansteelandt et al.,
(2008) that the nuisance tangent space under $\mathcal{M}_{NEM}$ obtained in
Theorem 3.1 can be equivalently represented as

$\bigg{\\{}\\{e_{0}(\mathbf{C})+e_{1}(S,\mathbf{C})+e_{2}(\mathbf{Z,C})\\}\sigma^{-2}(\mathbf{Z},S,\mathbf{C})\epsilon^{*}(\psi^{\dagger}):e_{0}\in\mathcal{C}_{0},e_{1}\in\mathcal{C}_{1},e_{2}\in\mathcal{C}_{2}\bigg{\\}}\cap
L_{2}^{0}$

where $\mathcal{C}_{0}=\\{e_{0}(\mathbf{C})\\}\cap\mathcal{H}$,
$\mathcal{C}_{1}=\\{e_{1}(S,\mathbf{C})\\}\cap\mathcal{H}$,
$\mathcal{C}_{2}=\\{e_{2}(\mathbf{Z,C})\\}\cap\mathcal{H}$ and $\mathcal{H}$
is the Hilbert space of functions of $\mathbf{Z},S,\mathbf{C}$ with inner
product given by
$E\\{\sigma^{-2}(\mathbf{Z},S,\mathbf{C})e_{1}(\mathbf{Z},S,\mathbf{C})e_{2}(\mathbf{Z},S,\mathbf{C})\\}$
for all $e_{1},e_{2}$.

Then the efficient score under model $\mathcal{M}_{NEM}$ is equal to the
following projection

$\displaystyle\Pi_{L_{2}^{0}}\left(U_{RM-
eff}|[\\{e_{0}(\mathbf{C})+e_{1}(S,\mathbf{C})+e_{2}(\mathbf{Z,C})\\}\sigma^{-2}(\mathbf{Z},S,\mathbf{C})\epsilon^{*}(\psi^{\dagger})]^{\perp}:e_{0}\in\mathcal{C}_{0},e_{1}\in\mathcal{C}_{1},e_{2}\in\mathcal{C}_{2}\right)$
$\displaystyle\Pi_{\mathcal{H}}\left(\mu(\mathbf{Z},S,\mathbf{C})|\\{e_{0}(\mathbf{C})+e_{1}(S,\mathbf{C})+e_{2}(\mathbf{Z,C})\\}^{\perp}:e_{0}\in\mathcal{C}_{0},e_{1}\in\mathcal{C}_{1},e_{2}\in\mathcal{C}_{2}\right)\sigma^{-2}(\mathbf{Z},S,\mathbf{C})\epsilon^{*}(\psi^{\dagger})$

where $\Pi_{L_{2}^{0}}(\cdot|\cdot)$ is the orthogonal projection operator in
$L_{2}^{0}$, and $\Pi_{H}(\cdot|\cdot)$ is similarly defined w.r.t.
$\mathcal{H}$. Vansteelandt et al., (2008) obtain a closed form representation
of the projection in $\mathcal{H}$ of any function $D\in\mathcal{H}$ onto
$(\mathcal{C}_{0}\cup\mathcal{C}_{1}\cup\mathcal{C}_{2})^{\perp}$ as

$\displaystyle
J\left(D\right)-\frac{E\\{D\sigma^{-2}(\mathbf{Z},S,\mathbf{C})\tilde{S}|\mathbf{C}\\}}{E\\{\sigma^{-2}(\mathbf{Z},S,\mathbf{C})\tilde{S}^{2}|\mathbf{C}\\}}\tilde{S}-J\left(\frac{E\\{D\sigma^{-2}(\mathbf{Z},S,\mathbf{C})|\mathbf{Z,C}\\}}{E\\{\sigma^{-2}(\mathbf{Z},S,\mathbf{C})|\mathbf{Z,C}\\}}\right)$

where we define the $J(\cdot)$ operator on function $D^{*}$ as

$J(D^{*})=D^{*}-\frac{E\\{D^{*}\sigma^{-2}(\mathbf{Z},S,\mathbf{C})|\mathbf{C}\\}}{E\\{\sigma^{-2}(\mathbf{Z},S,\mathbf{C})|\mathbf{C}\\}}$

and

$\tilde{S}=S-P_{\sigma}(\mathbf{Z,C}).$

Applying this result, we obtain the efficient score as