Extracted main text — title through conclusion, appendix excluded. This is what our citation measures are computed over, published so the extraction can be checked by eye.
151,525 characters · 15 sections · 96 citation commands
Identification of Treatment Effects under Limited Exogenous Variation
\thispagestyle{empty}
Keywords:{ Identification; Conditional nonsingularity; Limited exogenous variation; Treatment effects; Heterogeneous coefficients; Control variable; Discrete instruments; Testability.}
\thispagestyle{empty}
Nonseparable and/or multidimensional heterogeneity is important. It is present in discrete choice models as in McF1973 and HW1978. Multidimensional heterogeneity in demand functions allows price and income elasticities to vary over individuals in unrestricted ways, e.g., HN2016 and KS2018. It allows general variation in production technologies. Treatment effects that vary across individuals require intercept and slope heterogeneity.
Endogeneity is often a problem in these models because we are interested in the effect of an observed choice, or treatment variable on an outcome and the choice or treatment variable is correlated with heterogeneity. Control variables provide an important means of controlling for endogeneity with multidimensional heterogeneity. A control variable is an observed or estimable variable that makes heterogeneity and treatment independent when it is conditioned on. The conditional cumulative distribution function (CDF) of a choice variable given an instrument can serve as a control variable in triangular models (Imbens Newey 2009).
In fully nonparametric and nonseparable models, identification of average or quantile treatment effects requires a full support condition, that the support of the control variable conditional on the treatment variable is equal to the marginal support of the control variable. This restriction is often not satisfied in practice; e.g., see Imbens Newey 2009 for Engel curves. In triangular models the full support condition cannot hold when all instruments are discrete and the treatment variable is continuous.
One approach to this problem is to focus on identified sets for objects of interest, as for quantile effects in Imbens Newey 2009. Another approach is to consider restrictions on the model that allow for point identification. In a triangular model with continuous treatment, Flo Heck Meg Vytl 2008 gave identification results when the outcome equation is a polynomial in the endogenous variable, and MT:2016 when the outcome equation is a linear combination of known transformations of the endogenous variable that are not necessarily polynomials. Torgo 2015 and Fev Hault 2015 gave identification results when there is only scalar heterogeneity in the outcome equation.
In this paper we give identification results when the Control Regression Function (CRF), the regression function of the outcome given the treatment and the controls, is a linear combination of either known functions of the treatment, with unknown coefficients varying with the controls, or known functions of the controls, with unknown coefficients varying with the treatment. We further assume that the Average Structural Function (ASF, Blundell Powell 2003), the outcome structural function when heterogeneity has been integrated out, coincides with the CRF integrated over the controls. A generic sufficient condition for identification of average treatment effects in our models is that the second moment matrix of the known functions given the variable in the varying coefficients is nonsingular with probability one. This framework is a generalization of heterogeneous coefficients formulations where the outcome function is linear in known functions of the treatment, and where the coefficients in this linear combination are mean independant of heterogeneity given the controls.
A main benefit of this modeling framework is that it allows for identification of average treatment effects under limited exogenous variation, i.e., without full support. For important cases of practical relevance, this provides a strong motivation for placing restrictions on the way either the treatment or the control variables affect the outcome, so that point identification is preserved. Leading cases include continuous treatment with conditional support varying across control variable values, continuous treatment in a triangular model with discrete instruments, and multiple treatments without common support or strong overlap. For all these cases, our flexible formulations can deliver point identification. We obtain these results from the assumed varying coefficients structure of the CRF.
We specialize our results to average treatment effects in triangular models with discrete instruments, and we extend the analysis to quantile and distributional treatment effects. For known treatment functions with sufficient variation, we find that a necessary condition for ASF identification is that the number of support points of the discrete instruments is at least as large as the number of coefficients. With known functions of control variables, we find that identification can be achieved with binary or discrete instruments in general nonseparable triangular models when restrictions are placed on how control variables can affect heterogeneity.
These results extend Flo Heck Meg Vytl 2008 in allowing for outcome structural functions that are linear in nonpolynomial functions of the treatment variable, and in allowing for discrete instruments. We also take a different approach to identification, focusing here on conditional nonsingularity of second moment matrices instead of measurable separability. These results also generalize those of MT:2016 to allow for both estimable and observable control variables in the modeling of treatment effects. Our results generalize the heterogeneous coefficients formulations in both papers, to allow for a class of outcome structural functions that are not necessarily linear in known functions of the treatment, and for CRFs that are linear in known functions of the controls given the treatment. We also go beyond average effects by extending the identification analysis to quantile and distributional treatment effects. In addition, our modeling framework generalizes that of NeweyStouli2021 to allow for the CRF to be a linear combination of known functions of only one of either the treatment or the controls, rather than known functions of each. While this literature focuses on sufficient conditions for identification, we study necessity as well as sufficiency, we extend the identification analysis to flexible models of increasing dimension, and we establish testability of our model restrictions.
Second, we give identification conditions that are necessary as well as sufficient for the ASF. Necessity is important in order to demonstrate testability of identification (e.g., Breusch:1986). Conditions that are both necessary and sufficient are important for the determination of minimal conditions for identification. We are thus able to characterize minimal conditions for a very large class of models. In triangular models with discrete instruments, these results allow us to establish that the number of coefficients cannot be larger than the number of support points for the instrument. These results generalize those in NeweyStouli:2022 from discrete to general treatments, from observable to general control variables, and from CRFs that are linear in known functions of the treatment to CRFs that are linear in known functions of either the treatment or the controls, but not both.
Third, we show that average treatment effects are identified in general nonseparable models under limited exogenous variation, when restrictions are placed on how control variables can affect the CRF. For general nonseparable triangular models with binary or discrete instruments, this means that identification can be achieved when restrictions are placed on the relationship between heterogeneity and control variables in the model. This formulation alleviates the full support requirement and provides a novel approach for the modeling of treatment effects in the presence of multidimensional heterogeneity and discrete instruments.
Fourth, we extend the identification analysis to models of increasing dimension, by establishing a connection between semi- and non-parametric identification conditions for treatment effects. When conditional nonsingularity holds for each element of an increasing sequence of suitable approximating functions, we show that the full support condition is satisfied, and hence average treatment effects are nonparametrically identified. Thus our modeling approach provides an encompassing framework for identification of treatment effects, from identification under limited exogenous variation in models of fixed dimension to nonparametric identification under full support.
In Section \ifstrequal{sec:Section2}{ass:Assumption1p}{1(p)}{ \ifstrequal{sec:Section2}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{sec:Section2}}{1}{Assumption 2}{ \ifstrequal{\oldref{sec:Section2}}{2}{Assumption 3}{ \ifstrequal{\oldref{sec:Section2}}{3}{Assumption 4}{ Assumption \oldref{sec:Section2} } } } } } we introduce the modeling framework. Section \ifstrequal{sec:Section3}{ass:Assumption1p}{1(p)}{ \ifstrequal{sec:Section3}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{sec:Section3}}{1}{Assumption 2}{ \ifstrequal{\oldref{sec:Section3}}{2}{Assumption 3}{ \ifstrequal{\oldref{sec:Section3}}{3}{Assumption 4}{ Assumption \oldref{sec:Section3} } } } } } gives the identification analysis and Section \ifstrequal{sec:Section4}{ass:Assumption1p}{1(p)}{ \ifstrequal{sec:Section4}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{sec:Section4}}{1}{Assumption 2}{ \ifstrequal{\oldref{sec:Section4}}{2}{Assumption 3}{ \ifstrequal{\oldref{sec:Section4}}{3}{Assumption 4}{ Assumption \oldref{sec:Section4} } } } } } discusses conditional nonsingularity. In Section \ifstrequal{sec:Section5}{ass:Assumption1p}{1(p)}{ \ifstrequal{sec:Section5}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{sec:Section5}}{1}{Assumption 2}{ \ifstrequal{\oldref{sec:Section5}}{2}{Assumption 3}{ \ifstrequal{\oldref{sec:Section5}}{3}{Assumption 4}{ Assumption \oldref{sec:Section5} } } } } } we specialize our results to triangular models. Section \ifstrequal{sec:Section6}{ass:Assumption1p}{1(p)}{ \ifstrequal{sec:Section6}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{sec:Section6}}{1}{Assumption 2}{ \ifstrequal{\oldref{sec:Section6}}{2}{Assumption 3}{ \ifstrequal{\oldref{sec:Section6}}{3}{Assumption 4}{ Assumption \oldref{sec:Section6} } } } } } gives results on model testability. Section \ifstrequal{sec:Section7}{ass:Assumption1p}{1(p)}{ \ifstrequal{sec:Section7}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{sec:Section7}}{1}{Assumption 2}{ \ifstrequal{\oldref{sec:Section7}}{2}{Assumption 3}{ \ifstrequal{\oldref{sec:Section7}}{3}{Assumption 4}{ Assumption \oldref{sec:Section7} } } } } } discusses estimation, and Section \ifstrequal{sec:Section8}{ass:Assumption1p}{1(p)}{ \ifstrequal{sec:Section8}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{sec:Section8}}{1}{Assumption 2}{ \ifstrequal{\oldref{sec:Section8}}{2}{Assumption 3}{ \ifstrequal{\oldref{sec:Section8}}{3}{Assumption 4}{ Assumption \oldref{sec:Section8} } } } } } concludes. All proofs are given in the Appendix.
Let $X$ denote an endogenous treatment, and $\varepsilon$ a structural disturbance vector of unrestricted dimension, with CDF $F_{\varepsilon}$. A general nonseparable treatment effects model for an outcome variable $Y$ is
where $\varepsilon$ is independent of $X$ conditional on $V$, an observable or estimable control variable with CDF $F_{V}$. For this model, a leading structural object of interest is the ASF, \[ \mu(X) \equiv \int g(X,\varepsilon) dF_{\varepsilon}(\varepsilon). \]
In addition to conditional independence in ( \ifstrequal{eq:g(x,e)}{ass:Assumption1p}{1(p)}{ \ifstrequal{eq:g(x,e)}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{eq:g(x,e)}}{1}{Assumption 2}{ \ifstrequal{\oldref{eq:g(x,e)}}{2}{Assumption 3}{ \ifstrequal{\oldref{eq:g(x,e)}}{3}{Assumption 4}{ Assumption \oldref{eq:g(x,e)} } } } } } ), an assumption required for ASF identification is full support, that the conditional support of $X$ given $V$ is the same as the marginal support of $X$ (Imbens Newey 2009). The ASF can then be expressed in terms of observable or estimable quantities (Blundell Powell 2003):
Full support allows for this integral to be well-defined, and hence for ASF identification as a known functional of the CRF $E[Y | X, V]$. Full support requires the treatment to have unrestricted variation once the control variable is conditioned on.
A common occurrence in practice is that the full support assumption does not hold. This means that, with positive probability, the support of $X$ given $V$ is only a strict subset of the marginal support of $X$, i.e., the treatment has limited exogenous variation. To accommodate this feature, the CRF in ( \ifstrequal{eq:integrated_CRF}{ass:Assumption1p}{1(p)}{ \ifstrequal{eq:integrated_CRF}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{eq:integrated_CRF}}{1}{Assumption 2}{ \ifstrequal{\oldref{eq:integrated_CRF}}{2}{Assumption 3}{ \ifstrequal{\oldref{eq:integrated_CRF}}{3}{Assumption 4}{ Assumption \oldref{eq:integrated_CRF} } } } } } ) constitutes a natural modeling target, on which restrictions can be placed to alleviate the full support requirement.
In this paper, we introduce a modeling framework that allows for two types of restrictions on the CRF, each corresponding to a distinct class of flexible models, and each allowing for identification of treatment effects under limited exogenous variation. Specifically, we place restrictions on how either $X$ or $V$ affects the CRF, which we specify as either a linear combination of known functions of $X$, or a linear combination of known functions of $V$, but not both.
In the first class of models we consider, the CRF is a linear combination of known functions $p(X)$, of finite and known dimension $J$, with varying coefficients that are unknown functions of $V$.
Under Assumption \ifstrequal{ass:Assumption1p}{ass:Assumption1p}{1(p)}{ \ifstrequal{ass:Assumption1p}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{ass:Assumption1p}}{1}{Assumption 2}{ \ifstrequal{\oldref{ass:Assumption1p}}{2}{Assumption 3}{ \ifstrequal{\oldref{ass:Assumption1p}}{3}{Assumption 4}{ Assumption \oldref{ass:Assumption1p} } } } } } , while the functional relationship between the treatment and the CRF is effectively restricted to belong to a known class of functions, the way the treatment affects the CRF is unrestricted within that class across control variable values. For example, in the simplest case $p(X)=(1,X)'$, the implied CRF specification $E[Y|X,V]=q_{01}(V)+q_{02}(V)X$ imposes linearity of the CRF in $X$, while allowing the intercept and slope coefficients to vary freely across values of $V$.
A leading example of a model that satisfies Assumption \ifstrequal{ass:Assumption1p}{ass:Assumption1p}{1(p)}{ \ifstrequal{ass:Assumption1p}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{ass:Assumption1p}}{1}{Assumption 2}{ \ifstrequal{\oldref{ass:Assumption1p}}{2}{Assumption 3}{ \ifstrequal{\oldref{ass:Assumption1p}}{3}{Assumption 4}{ Assumption \oldref{ass:Assumption1p} } } } } } is a heterogeneous coefficients formulation of $g(X,\varepsilon)$ in ( \ifstrequal{eq:g(x,e)}{ass:Assumption1p}{1(p)}{ \ifstrequal{eq:g(x,e)}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{eq:g(x,e)}}{1}{Assumption 2}{ \ifstrequal{\oldref{eq:g(x,e)}}{2}{Assumption 3}{ \ifstrequal{\oldref{eq:g(x,e)}}{3}{Assumption 4}{ Assumption \oldref{eq:g(x,e)} } } } } } ), of the form
where the coefficients vector $\beta(\varepsilon)$ is mean independent of the endogenous variable $X$ conditional on $V$:
This class of models ( \ifstrequal{eq:hcoefs_model_p}{ass:Assumption1p}{1(p)}{ \ifstrequal{eq:hcoefs_model_p}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{eq:hcoefs_model_p}}{1}{Assumption 2}{ \ifstrequal{\oldref{eq:hcoefs_model_p}}{2}{Assumption 3}{ \ifstrequal{\oldref{eq:hcoefs_model_p}}{3}{Assumption 4}{ Assumption \oldref{eq:hcoefs_model_p} } } } } } )-( \ifstrequal{eq:CMI}{ass:Assumption1p}{1(p)}{ \ifstrequal{eq:CMI}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{eq:CMI}}{1}{Assumption 2}{ \ifstrequal{\oldref{eq:CMI}}{2}{Assumption 3}{ \ifstrequal{\oldref{eq:CMI}}{3}{Assumption 4}{ Assumption \oldref{eq:CMI} } } } } } ) is one where $X$ is known to affect the CRF only through a vector of known functions $p(X)$. Conditional mean independence property ( \ifstrequal{eq:CMI}{ass:Assumption1p}{1(p)}{ \ifstrequal{eq:CMI}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{eq:CMI}}{1}{Assumption 2}{ \ifstrequal{\oldref{eq:CMI}}{2}{Assumption 3}{ \ifstrequal{\oldref{eq:CMI}}{3}{Assumption 4}{ Assumption \oldref{eq:CMI} } } } } } ) and the form of the outcome function $p(X)'\beta(\varepsilon)$ in ( \ifstrequal{eq:hcoefs_model_p}{ass:Assumption1p}{1(p)}{ \ifstrequal{eq:hcoefs_model_p}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{eq:hcoefs_model_p}}{1}{Assumption 2}{ \ifstrequal{\oldref{eq:hcoefs_model_p}}{2}{Assumption 3}{ \ifstrequal{\oldref{eq:hcoefs_model_p}}{3}{Assumption 4}{ Assumption \oldref{eq:hcoefs_model_p} } } } } } ) together imply that:
with unknown coefficients $q_{0}(V)$. This specification places no restrictions on $E[\beta(\varepsilon)|V]$, and hence does not restrict how control variables can affect the coefficients. This is a generalization of Flo Heck Meg Vytl 2008 to allow $p(X)$ to be any functions of $X$ rather than just powers of $X$. This restricted nonparametric regression is of the varying coefficients type considered by Cai:2006.
An example for $p(X)$ known naturally arises with discrete treatment. A general formulation is to let $X$ be a vector of dummy variables $X(t)$, $t\in\{1,\ldots,T\}$, taking value one if treatment regime $t$ occurs and zero otherwise, and setting
This formulation generalizes the RR:1983 binary treatment effects model to multivalued and/or multiple treatments, and is not restrictive when using mutually exclusive treatment regimes in the definition of $X$. This formulation also relaxes the conditional independence assumption commonly imposed for identification of treatment effects, an assumption that is not necessary for identification and can be replaced by the weaker conditional mean independence property ( \ifstrequal{eq:CMI}{ass:Assumption1p}{1(p)}{ \ifstrequal{eq:CMI}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{eq:CMI}}{1}{Assumption 2}{ \ifstrequal{\oldref{eq:CMI}}{2}{Assumption 3}{ \ifstrequal{\oldref{eq:CMI}}{3}{Assumption 4}{ Assumption \oldref{eq:CMI} } } } } } ).\footnote{See Section 4 in NeweyStouli:2022 for a detailed analysis and an equivalent formulation of the conditional mean independence assumption in terms of potential outcomes.}
In the second class of models we consider, the CRF is a linear combination of known functions $q(V)$, of finite and known dimension $K$, with varying coefficients that are unknown functions of $X$.
Under Assumption \ifstrequal{ass:Assumption1q}{ass:Assumption1p}{1(p)}{ \ifstrequal{ass:Assumption1q}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{ass:Assumption1q}}{1}{Assumption 2}{ \ifstrequal{\oldref{ass:Assumption1q}}{2}{Assumption 3}{ \ifstrequal{\oldref{ass:Assumption1q}}{3}{Assumption 4}{ Assumption \oldref{ass:Assumption1q} } } } } } , while the functional relationship between the control variables and the CRF is effectively restricted to belong to a known class of functions, the way control variables affect the CRF is unrestricted within that class across treatment values. For example, in the simplest case $q(V)=(1,V)'$, the implied CRF specification $E[Y|X,V]=p_{01}(X)+p_{02}(X)V$ imposes linearity of the CRF in $V$, while allowing the intercept and slope coefficients to vary freely across values of $X$.
With $X$ continuous, a leading example of a model that satisfies Assumption \ifstrequal{ass:Assumption1q}{ass:Assumption1p}{1(p)}{ \ifstrequal{ass:Assumption1q}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{ass:Assumption1q}}{1}{Assumption 2}{ \ifstrequal{\oldref{ass:Assumption1q}}{2}{Assumption 3}{ \ifstrequal{\oldref{ass:Assumption1q}}{3}{Assumption 4}{ Assumption \oldref{ass:Assumption1q} } } } } } is a heterogeneous coefficients representation of ( \ifstrequal{eq:g(x,e)}{ass:Assumption1p}{1(p)}{ \ifstrequal{eq:g(x,e)}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{eq:g(x,e)}}{1}{Assumption 2}{ \ifstrequal{\oldref{eq:g(x,e)}}{2}{Assumption 3}{ \ifstrequal{\oldref{eq:g(x,e)}}{3}{Assumption 4}{ Assumption \oldref{eq:g(x,e)} } } } } } ), of the form
where $\varepsilon \!\perp\!\!\!\perp X | V$ and $p^{*}(X)$ is a vector of unknown functions of arbitrarily large dimension, and where $E[\beta(\varepsilon)|V]$ is a vector of linear combinations of known functions
for an unknown $K\times J$ matrix $\Omega$. When $p^{*}(X)$ is a vector of approximating functions such as splines or wavelets, this model can be viewed as an approximation to the general nonseparable model ( \ifstrequal{eq:g(x,e)}{ass:Assumption1p}{1(p)}{ \ifstrequal{eq:g(x,e)}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{eq:g(x,e)}}{1}{Assumption 2}{ \ifstrequal{\oldref{eq:g(x,e)}}{2}{Assumption 3}{ \ifstrequal{\oldref{eq:g(x,e)}}{3}{Assumption 4}{ Assumption \oldref{eq:g(x,e)} } } } } } ) where $\beta(\varepsilon)$ are varying coefficients in an expansion of $g(X,\varepsilon)$ in $p^{*}(X)$, as in HN2016. With $p^{*}(X)$ a vector of unknown functions of arbitrarily large dimension, the outcome function $g(X,\varepsilon)$ is effectively unrestricted under ( \ifstrequal{eq:hcoefs_model_p*}{ass:Assumption1p}{1(p)}{ \ifstrequal{eq:hcoefs_model_p*}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{eq:hcoefs_model_p*}}{1}{Assumption 2}{ \ifstrequal{\oldref{eq:hcoefs_model_p*}}{2}{Assumption 3}{ \ifstrequal{\oldref{eq:hcoefs_model_p*}}{3}{Assumption 4}{ Assumption \oldref{eq:hcoefs_model_p*} } } } } } ).\footnote{For $x\mapsto g(x,\varepsilon)$ a smooth function uniformly in $\varepsilon$ and with $p^{*}(X)$ chosen from a suitable class of approximating functions, there is $\beta(\varepsilon)$ that depends on $J$ such that $E[\{g(X,\varepsilon)-p^{*}(X)'\beta(\varepsilon)\}^{2}]\rightarrow0$ as $J\rightarrow\infty$. This is because the (uniform) approximation error for $x\mapsto g(x,\varepsilon)$ is bounded for each value of $\varepsilon$ (e.g., Powell:1981), and hence uniformly over $\varepsilon$ if $x\mapsto g(x,\varepsilon)$ is smooth uniformly in $\varepsilon$. The case where $Y$ is binary or discrete can be accommodated if there is $\rho$ such that $Y=g(X,\varepsilon)+\rho$ and $E[\rho|X,V]=0$ and $x\mapsto g(x,\varepsilon)$ is smooth uniformly in $\varepsilon$. This formulation extends representations proposed in CDGHN:2025 to the control variable case. } In this example, restrictions are thus placed on the way control variables affect $\beta(\varepsilon)$, rather than on $g(X,\varepsilon)$.
This class of models ( \ifstrequal{eq:hcoefs_model_p*}{ass:Assumption1p}{1(p)}{ \ifstrequal{eq:hcoefs_model_p*}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{eq:hcoefs_model_p*}}{1}{Assumption 2}{ \ifstrequal{\oldref{eq:hcoefs_model_p*}}{2}{Assumption 3}{ \ifstrequal{\oldref{eq:hcoefs_model_p*}}{3}{Assumption 4}{ Assumption \oldref{eq:hcoefs_model_p*} } } } } } )-( \ifstrequal{eq:cond mean het coefs}{ass:Assumption1p}{1(p)}{ \ifstrequal{eq:cond mean het coefs}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{eq:cond mean het coefs}}{1}{Assumption 2}{ \ifstrequal{\oldref{eq:cond mean het coefs}}{2}{Assumption 3}{ \ifstrequal{\oldref{eq:cond mean het coefs}}{3}{Assumption 4}{ Assumption \oldref{eq:cond mean het coefs} } } } } } ) is one where $V$ is known to affect the CRF only through a vector of known functions $q(V)$. Conditional independence and the form of the structural function $p^{*}(X)'\beta(\varepsilon)$ in ( \ifstrequal{eq:hcoefs_model_p*}{ass:Assumption1p}{1(p)}{ \ifstrequal{eq:hcoefs_model_p*}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{eq:hcoefs_model_p*}}{1}{Assumption 2}{ \ifstrequal{\oldref{eq:hcoefs_model_p*}}{2}{Assumption 3}{ \ifstrequal{\oldref{eq:hcoefs_model_p*}}{3}{Assumption 4}{ Assumption \oldref{eq:hcoefs_model_p*} } } } } } ) together imply that:
with unknown coefficients $p_{0}(X)$. Compared to ( \ifstrequal{eq:hcoefs_model_p}{ass:Assumption1p}{1(p)}{ \ifstrequal{eq:hcoefs_model_p}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{eq:hcoefs_model_p}}{1}{Assumption 2}{ \ifstrequal{\oldref{eq:hcoefs_model_p}}{2}{Assumption 3}{ \ifstrequal{\oldref{eq:hcoefs_model_p}}{3}{Assumption 4}{ Assumption \oldref{eq:hcoefs_model_p} } } } } } )-( \ifstrequal{eq:CMI}{ass:Assumption1p}{1(p)}{ \ifstrequal{eq:CMI}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{eq:CMI}}{1}{Assumption 2}{ \ifstrequal{\oldref{eq:CMI}}{2}{Assumption 3}{ \ifstrequal{\oldref{eq:CMI}}{3}{Assumption 4}{ Assumption \oldref{eq:CMI} } } } } } ), specification ( \ifstrequal{eq:hcoefs_model_p*}{ass:Assumption1p}{1(p)}{ \ifstrequal{eq:hcoefs_model_p*}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{eq:hcoefs_model_p*}}{1}{Assumption 2}{ \ifstrequal{\oldref{eq:hcoefs_model_p*}}{2}{Assumption 3}{ \ifstrequal{\oldref{eq:hcoefs_model_p*}}{3}{Assumption 4}{ Assumption \oldref{eq:hcoefs_model_p*} } } } } } )-( \ifstrequal{eq:cond mean het coefs}{ass:Assumption1p}{1(p)}{ \ifstrequal{eq:cond mean het coefs}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{eq:cond mean het coefs}}{1}{Assumption 2}{ \ifstrequal{\oldref{eq:cond mean het coefs}}{2}{Assumption 3}{ \ifstrequal{\oldref{eq:cond mean het coefs}}{3}{Assumption 4}{ Assumption \oldref{eq:cond mean het coefs} } } } } } ) generalizes the outcome function to allow for general nonseparable models, but restricts the way control variables can affect the CRF, thereby defining an alternative flexible varying coefficients structure for the CRF.
An example for $q(V)$ known naturally arises with discrete control variables $V$. A general formulation is to let $V$ be a vector of dummy variables $V(s)$, $s\in\{1,\ldots,S\}$, taking value one if control value $s$ occurs and zero otherwise, and setting \[ \Omega\in\mathbb{R}^{(S+1)\times J},\quad q(V)=(1,V(1),\ldots,V(S))'. \] This formulation allows for unrestricted modeling of $E[\beta(\varepsilon)|V]$ when using mutually exclusive control regimes in the definition of $V$.
For both classes of models in Assumptions \ifstrequal{ass:Assumption1p}{ass:Assumption1p}{1(p)}{ \ifstrequal{ass:Assumption1p}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{ass:Assumption1p}}{1}{Assumption 2}{ \ifstrequal{\oldref{ass:Assumption1p}}{2}{Assumption 3}{ \ifstrequal{\oldref{ass:Assumption1p}}{3}{Assumption 4}{ Assumption \oldref{ass:Assumption1p} } } } } } and \ifstrequal{ass:Assumption1q}{ass:Assumption1p}{1(p)}{ \ifstrequal{ass:Assumption1q}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{ass:Assumption1q}}{1}{Assumption 2}{ \ifstrequal{\oldref{ass:Assumption1q}}{2}{Assumption 3}{ \ifstrequal{\oldref{ass:Assumption1q}}{3}{Assumption 4}{ Assumption \oldref{ass:Assumption1q} } } } } } , the integrated CRF $\int E[Y |X,V=v] dF_{V}(v)$ is linear in the treatment functions and, in general, need not coincide with the ASF $\int g(X,\varepsilon) dF_{\varepsilon}(\varepsilon)$. We thus restrict outcome functions to those with implied ASF that does coincide with the integrated CRF. In this way, both the CRF and the ASF are linear in the treatment functions, and the ASF is expressed as a known functional of observable or estimable quantities only. This relation will form the basis of our ASF identification strategy.
Assumption \ifstrequal{ass:DGP}{ass:Assumption1p}{1(p)}{ \ifstrequal{ass:DGP}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{ass:DGP}}{1}{Assumption 2}{ \ifstrequal{\oldref{ass:DGP}}{2}{Assumption 3}{ \ifstrequal{\oldref{ass:DGP}}{3}{Assumption 4}{ Assumption \oldref{ass:DGP} } } } } } encapsulates two types of restrictions. First, a functional form restriction on $g(X,\varepsilon)$, with implied ASF required to be in linear form. The ASF derivative is also in linear form, such that $\partial\mu(X)/\partial x=\{\partial \varphi(X)/\partial x\}^{\prime}E[\chi(V)]$, a derivative version of the average treatment effect. Assumption \ifstrequal{ass:DGP}{ass:Assumption1p}{1(p)}{ \ifstrequal{ass:DGP}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{ass:DGP}}{1}{Assumption 2}{ \ifstrequal{\oldref{ass:DGP}}{2}{Assumption 3}{ \ifstrequal{\oldref{ass:DGP}}{3}{Assumption 4}{ Assumption \oldref{ass:DGP} } } } } } is a generalization of the heterogeneous coefficients functional form ( \ifstrequal{eq:hcoefs_model_p}{ass:Assumption1p}{1(p)}{ \ifstrequal{eq:hcoefs_model_p}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{eq:hcoefs_model_p}}{1}{Assumption 2}{ \ifstrequal{\oldref{eq:hcoefs_model_p}}{2}{Assumption 3}{ \ifstrequal{\oldref{eq:hcoefs_model_p}}{3}{Assumption 4}{ Assumption \oldref{eq:hcoefs_model_p} } } } } } ), and hence of the models in Flo Heck Meg Vytl 2008 and MT:2016.\footnote{For an example of outcome function not linear in known functions $p(X)$ while the implied ASF is, consider $g(X,\varepsilon) = p(X)'\beta(\varepsilon) + \xi(X,\varepsilon^{*})$, with $\varepsilon^{*}\mapsto \xi(X,\varepsilon^{*})$ an odd function, and $\varepsilon^{*}$ a component of $\varepsilon$ with distribution symmetric about zero. Then $\int \xi(X,\varepsilon^{*}) dF_{\varepsilon^{*}}(\varepsilon^{*})=0$, and hence: $\int g(X,\varepsilon) dF_{\varepsilon}(\varepsilon) =\int \{ p(X)'\beta(\varepsilon) + \xi(X,\varepsilon^{*}) \} dF_{\varepsilon}(\varepsilon) =p(X)'E[\beta(\varepsilon)] + \int \xi(X,\varepsilon^{*}) dF_{\varepsilon^{*}}(\varepsilon^{*}) =p(X)'E[\beta(\varepsilon)]$.} Assumption \ifstrequal{ass:DGP}{ass:Assumption1p}{1(p)}{ \ifstrequal{ass:DGP}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{ass:DGP}}{1}{Assumption 2}{ \ifstrequal{\oldref{ass:DGP}}{2}{Assumption 3}{ \ifstrequal{\oldref{ass:DGP}}{3}{Assumption 4}{ Assumption \oldref{ass:DGP} } } } } } also generalizes these models by allowing for discrete or continuous treatment with an observable control variable. Moreover, under restrictions on how $V$ affects the CRF in Assumption \ifstrequal{ass:Assumption1q}{ass:Assumption1p}{1(p)}{ \ifstrequal{ass:Assumption1q}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{ass:Assumption1q}}{1}{Assumption 2}{ \ifstrequal{\oldref{ass:Assumption1q}}{2}{Assumption 3}{ \ifstrequal{\oldref{ass:Assumption1q}}{3}{Assumption 4}{ Assumption \oldref{ass:Assumption1q} } } } } } and regularity conditions on $g(X,\varepsilon)$ in footnote \ifstrequal{fn:approximation}{ass:Assumption1p}{1(p)}{ \ifstrequal{fn:approximation}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{fn:approximation}}{1}{Assumption 2}{ \ifstrequal{\oldref{fn:approximation}}{2}{Assumption 3}{ \ifstrequal{\oldref{fn:approximation}}{3}{Assumption 4}{ Assumption \oldref{fn:approximation} } } } } } , our formulation allows for unrestricted $g(X,\varepsilon)$.
Second, a restriction on the conditional distribution of $\varepsilon$ given $X$ and $V$, imposing a form of conditional independence between $\varepsilon$ and $X$ given $V$, with the ASF required to coincide with the integrated CRF. This is made apparent by writing ( \ifstrequal{eq:model}{ass:Assumption1p}{1(p)}{ \ifstrequal{eq:model}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{eq:model}}{1}{Assumption 2}{ \ifstrequal{\oldref{eq:model}}{2}{Assumption 3}{ \ifstrequal{\oldref{eq:model}}{3}{Assumption 4}{ Assumption \oldref{eq:model} } } } } } ) as
by $\varphi(X)' E[\chi(V)] = \int \{\varphi(X)'\chi(v)\}dF_V(v) = \int E[Y|X,V=v]dF_{V}(v)$ under Assumption \ifstrequal{ass:Assumption1p}{ass:Assumption1p}{1(p)}{ \ifstrequal{ass:Assumption1p}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{ass:Assumption1p}}{1}{Assumption 2}{ \ifstrequal{\oldref{ass:Assumption1p}}{2}{Assumption 3}{ \ifstrequal{\oldref{ass:Assumption1p}}{3}{Assumption 4}{ Assumption \oldref{ass:Assumption1p} } } } } } or \ifstrequal{ass:Assumption1q}{ass:Assumption1p}{1(p)}{ \ifstrequal{ass:Assumption1q}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{ass:Assumption1q}}{1}{Assumption 2}{ \ifstrequal{\oldref{ass:Assumption1q}}{2}{Assumption 3}{ \ifstrequal{\oldref{ass:Assumption1q}}{3}{Assumption 4}{ Assumption \oldref{ass:Assumption1q} } } } } } . Clearly, for general $g(X,\varepsilon)$ independence of $\varepsilon$ and $X$ given $V$ is sufficient for ( \ifstrequal{eq:implicit_independence}{ass:Assumption1p}{1(p)}{ \ifstrequal{eq:implicit_independence}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{eq:implicit_independence}}{1}{Assumption 2}{ \ifstrequal{\oldref{eq:implicit_independence}}{2}{Assumption 3}{ \ifstrequal{\oldref{eq:implicit_independence}}{3}{Assumption 4}{ Assumption \oldref{eq:implicit_independence} } } } } } ), and hence also for ( \ifstrequal{eq:model}{ass:Assumption1p}{1(p)}{ \ifstrequal{eq:model}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{eq:model}}{1}{Assumption 2}{ \ifstrequal{\oldref{eq:model}}{2}{Assumption 3}{ \ifstrequal{\oldref{eq:model}}{3}{Assumption 4}{ Assumption \oldref{eq:model} } } } } } ) under Assumption \ifstrequal{ass:Assumption1p}{ass:Assumption1p}{1(p)}{ \ifstrequal{ass:Assumption1p}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{ass:Assumption1p}}{1}{Assumption 2}{ \ifstrequal{\oldref{ass:Assumption1p}}{2}{Assumption 3}{ \ifstrequal{\oldref{ass:Assumption1p}}{3}{Assumption 4}{ Assumption \oldref{ass:Assumption1p} } } } } } or \ifstrequal{ass:Assumption1q}{ass:Assumption1p}{1(p)}{ \ifstrequal{ass:Assumption1q}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{ass:Assumption1q}}{1}{Assumption 2}{ \ifstrequal{\oldref{ass:Assumption1q}}{2}{Assumption 3}{ \ifstrequal{\oldref{ass:Assumption1q}}{3}{Assumption 4}{ Assumption \oldref{ass:Assumption1q} } } } } } .
For restricted $g(X,\varepsilon)$, ( \ifstrequal{eq:model}{ass:Assumption1p}{1(p)}{ \ifstrequal{eq:model}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{eq:model}}{1}{Assumption 2}{ \ifstrequal{\oldref{eq:model}}{2}{Assumption 3}{ \ifstrequal{\oldref{eq:model}}{3}{Assumption 4}{ Assumption \oldref{eq:model} } } } } } ) can hold under weaker forms of conditional independence. In the leading examples ( \ifstrequal{eq:hcoefs_model_p}{ass:Assumption1p}{1(p)}{ \ifstrequal{eq:hcoefs_model_p}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{eq:hcoefs_model_p}}{1}{Assumption 2}{ \ifstrequal{\oldref{eq:hcoefs_model_p}}{2}{Assumption 3}{ \ifstrequal{\oldref{eq:hcoefs_model_p}}{3}{Assumption 4}{ Assumption \oldref{eq:hcoefs_model_p} } } } } } ) and ( \ifstrequal{eq:hcoefs_model_p*}{ass:Assumption1p}{1(p)}{ \ifstrequal{eq:hcoefs_model_p*}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{eq:hcoefs_model_p*}}{1}{Assumption 2}{ \ifstrequal{\oldref{eq:hcoefs_model_p*}}{2}{Assumption 3}{ \ifstrequal{\oldref{eq:hcoefs_model_p*}}{3}{Assumption 4}{ Assumption \oldref{eq:hcoefs_model_p*} } } } } } )-( \ifstrequal{eq:cond mean het coefs}{ass:Assumption1p}{1(p)}{ \ifstrequal{eq:cond mean het coefs}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{eq:cond mean het coefs}}{1}{Assumption 2}{ \ifstrequal{\oldref{eq:cond mean het coefs}}{2}{Assumption 3}{ \ifstrequal{\oldref{eq:cond mean het coefs}}{3}{Assumption 4}{ Assumption \oldref{eq:cond mean het coefs} } } } } } ), relation ( \ifstrequal{eq:model}{ass:Assumption1p}{1(p)}{ \ifstrequal{eq:model}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{eq:model}}{1}{Assumption 2}{ \ifstrequal{\oldref{eq:model}}{2}{Assumption 3}{ \ifstrequal{\oldref{eq:model}}{3}{Assumption 4}{ Assumption \oldref{eq:model} } } } } } ) holds under mean independence of $\beta(\varepsilon)$ and $X$ given $V$. In these examples, the ASF is $\mu(X)= p(X)^{\prime}E[\beta(\varepsilon)]$ and $\mu(X)= p^{*}(X)^{\prime}E[\beta(\varepsilon)]$, respectively; see Cham:1984 and Wool:2005. Relation ( \ifstrequal{eq:model}{ass:Assumption1p}{1(p)}{ \ifstrequal{eq:model}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{eq:model}}{1}{Assumption 2}{ \ifstrequal{\oldref{eq:model}}{2}{Assumption 3}{ \ifstrequal{\oldref{eq:model}}{3}{Assumption 4}{ Assumption \oldref{eq:model} } } } } } ) is then verified by expressing the ASF as a linear combination of $E[q_{0}(V)]$ and $p_{0}(X)=\Omega p^{*}(X)$, respectively. For ( \ifstrequal{eq:hcoefs_model_p}{ass:Assumption1p}{1(p)}{ \ifstrequal{eq:hcoefs_model_p}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{eq:hcoefs_model_p}}{1}{Assumption 2}{ \ifstrequal{\oldref{eq:hcoefs_model_p}}{2}{Assumption 3}{ \ifstrequal{\oldref{eq:hcoefs_model_p}}{3}{Assumption 4}{ Assumption \oldref{eq:hcoefs_model_p} } } } } } ), by iterated expectations and $E[\beta(\varepsilon)| V]=q_{0}(V)$,
Similarly, for ( \ifstrequal{eq:hcoefs_model_p*}{ass:Assumption1p}{1(p)}{ \ifstrequal{eq:hcoefs_model_p*}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{eq:hcoefs_model_p*}}{1}{Assumption 2}{ \ifstrequal{\oldref{eq:hcoefs_model_p*}}{2}{Assumption 3}{ \ifstrequal{\oldref{eq:hcoefs_model_p*}}{3}{Assumption 4}{ Assumption \oldref{eq:hcoefs_model_p*} } } } } } )-( \ifstrequal{eq:cond mean het coefs}{ass:Assumption1p}{1(p)}{ \ifstrequal{eq:cond mean het coefs}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{eq:cond mean het coefs}}{1}{Assumption 2}{ \ifstrequal{\oldref{eq:cond mean het coefs}}{2}{Assumption 3}{ \ifstrequal{\oldref{eq:cond mean het coefs}}{3}{Assumption 4}{ Assumption \oldref{eq:cond mean het coefs} } } } } } ), by $E[\beta(\varepsilon)|V]=\Omega'q(V)$ and $\Omega p^{*}(X) = p_{0}(X)$,
This discussion shows that Assumption \ifstrequal{ass:DGP}{ass:Assumption1p}{1(p)}{ \ifstrequal{ass:DGP}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{ass:DGP}}{1}{Assumption 2}{ \ifstrequal{\oldref{ass:DGP}}{2}{Assumption 3}{ \ifstrequal{\oldref{ass:DGP}}{3}{Assumption 4}{ Assumption \oldref{ass:DGP} } } } } } allows for a wide class of models, beyond heterogeneous coefficients models that are linear in known functions of the treatment.
An important kind of control variable arises in a triangular model where an instrumental variable $Z$ is excluded from the outcome equation $Y=g(X,\varepsilon)$ and where $X$ is a scalar with
with $\eta\mapsto h(Z,\eta)$ strictly monotonic. If $(\varepsilon,\eta)$ is jointly independent of $Z$, then $\varepsilon$ is independent of $X$ given $V$ for $V=F_{X|Z}(X|Z)$, the CDF of $X$ conditional on $Z$ (Imbens Newey 2009).\footnote{When strict monotonicity does not hold, for instance with discrete $X$, the control function is set-valued and structural objects of interest such as average and quantile treatment effects are partially identified. This case is considered in Chesher 2005 and HK2024.} Alternatively, $V=F_{X|Z}(X|Z)$ is a control variable in the leading examples ( \ifstrequal{eq:hcoefs_model_p}{ass:Assumption1p}{1(p)}{ \ifstrequal{eq:hcoefs_model_p}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{eq:hcoefs_model_p}}{1}{Assumption 2}{ \ifstrequal{\oldref{eq:hcoefs_model_p}}{2}{Assumption 3}{ \ifstrequal{\oldref{eq:hcoefs_model_p}}{3}{Assumption 4}{ Assumption \oldref{eq:hcoefs_model_p} } } } } } ) and ( \ifstrequal{eq:hcoefs_model_p*}{ass:Assumption1p}{1(p)}{ \ifstrequal{eq:hcoefs_model_p*}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{eq:hcoefs_model_p*}}{1}{Assumption 2}{ \ifstrequal{\oldref{eq:hcoefs_model_p*}}{2}{Assumption 3}{ \ifstrequal{\oldref{eq:hcoefs_model_p*}}{3}{Assumption 4}{ Assumption \oldref{eq:hcoefs_model_p*} } } } } } ) where the outcome functions are in heterogeneous coefficients form, under the weaker conditions that $\eta$ is independent from $Z$ and that $\beta(\varepsilon)$ is mean independent of $Z$ conditional on $\eta$.
Let $x \mapsto h^{-1}(Z,x)$ denote the inverse function of $\eta \mapsto h(Z,\eta)$. Since $\eta=h^{-1}(Z,X)$ and $V=F_{X|Z}(X|Z)=F_\eta(h^{-1}(Z,X))$, the two leading examples in triangular form, i.e., ( \ifstrequal{eq:hcoefs_model_p}{ass:Assumption1p}{1(p)}{ \ifstrequal{eq:hcoefs_model_p}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{eq:hcoefs_model_p}}{1}{Assumption 2}{ \ifstrequal{\oldref{eq:hcoefs_model_p}}{2}{Assumption 3}{ \ifstrequal{\oldref{eq:hcoefs_model_p}}{3}{Assumption 4}{ Assumption \oldref{eq:hcoefs_model_p} } } } } } ) and ( \ifstrequal{eq:hcoefs_model_p*}{ass:Assumption1p}{1(p)}{ \ifstrequal{eq:hcoefs_model_p*}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{eq:hcoefs_model_p*}}{1}{Assumption 2}{ \ifstrequal{\oldref{eq:hcoefs_model_p*}}{2}{Assumption 3}{ \ifstrequal{\oldref{eq:hcoefs_model_p*}}{3}{Assumption 4}{ Assumption \oldref{eq:hcoefs_model_p*} } } } } } ) augmented with ( \ifstrequal{eq:h(z,eta)}{ass:Assumption1p}{1(p)}{ \ifstrequal{eq:h(z,eta)}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{eq:h(z,eta)}}{1}{Assumption 2}{ \ifstrequal{\oldref{eq:h(z,eta)}}{2}{Assumption 3}{ \ifstrequal{\oldref{eq:h(z,eta)}}{3}{Assumption 4}{ Assumption \oldref{eq:h(z,eta)} } } } } } ), are particular cases of the class of nonseparable triangular models of the form: \[ Y=g(X,\varepsilon), \quad X=h(Z,\eta), \quad \eta\mid Z \sim F_{\eta}, \quad \int g(X,\varepsilon) dF_{\varepsilon}(\varepsilon) = \varphi(X)'E[\chi(F_\eta(\eta))], \] with $F_\eta$ the CDF of $\eta$, $E[Y|X,\eta]=\varphi(X)'\chi(F_\eta(\eta))$, and where either $\varphi(\cdot)$ or $\chi(\cdot)$ is known, but not both. This formulation characterizes the set of nonseparable triangular models to which the results in this paper will apply.
One main contribution of this paper is to highlight and show that in our modeling framework the ASF is identified under nonsingularity of the second moment matrix of the known functions of either the treatment given the controls, or the controls given the treatment.
Assumption \ifstrequal{ass:cond_nonsingularity}{ass:Assumption1}{ \ifstrequal{p}{p}{\hyperlink{ass:Assumption1p}{ \ifstrequal{ass:cond_nonsingularity}{ass:Assumption1p}{1(p)}{ \ifstrequal{ass:cond_nonsingularity}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{ass:cond_nonsingularity}}{1}{Assumption 2}{ \ifstrequal{\oldref{ass:cond_nonsingularity}}{2}{Assumption 3}{ \ifstrequal{\oldref{ass:cond_nonsingularity}}{3}{Assumption 4}{ Assumption \oldref{ass:cond_nonsingularity} } } } } } (p)}}{ \ifstrequal{p}{q}{\hyperlink{ass:Assumption1q}{ \ifstrequal{ass:cond_nonsingularity}{ass:Assumption1p}{1(p)}{ \ifstrequal{ass:cond_nonsingularity}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{ass:cond_nonsingularity}}{1}{Assumption 2}{ \ifstrequal{\oldref{ass:cond_nonsingularity}}{2}{Assumption 3}{ \ifstrequal{\oldref{ass:cond_nonsingularity}}{3}{Assumption 4}{ Assumption \oldref{ass:cond_nonsingularity} } } } } } (q)}}{
\ifstrequal{ass:cond_nonsingularity}{ass:Assumption1p}{1(p)}{ \ifstrequal{ass:cond_nonsingularity}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{ass:cond_nonsingularity}}{1}{Assumption 2}{ \ifstrequal{\oldref{ass:cond_nonsingularity}}{2}{Assumption 3}{ \ifstrequal{\oldref{ass:cond_nonsingularity}}{3}{Assumption 4}{ Assumption \oldref{ass:cond_nonsingularity} } } } } } (p) } } }{ \ifstrequal{ass:cond_nonsingularity}{ass:cond_nonsingularity}{ \ifstrequal{p}{p}{\hyperlink{ass:cond_nonsingularity_p}{ \ifstrequal{ass:cond_nonsingularity}{ass:Assumption1p}{1(p)}{ \ifstrequal{ass:cond_nonsingularity}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{ass:cond_nonsingularity}}{1}{Assumption 2}{ \ifstrequal{\oldref{ass:cond_nonsingularity}}{2}{Assumption 3}{ \ifstrequal{\oldref{ass:cond_nonsingularity}}{3}{Assumption 4}{ Assumption \oldref{ass:cond_nonsingularity} } } } } } (p)}}{ \ifstrequal{p}{q}{\hyperlink{ass:cond_nonsingularity_q}{ \ifstrequal{ass:cond_nonsingularity}{ass:Assumption1p}{1(p)}{ \ifstrequal{ass:cond_nonsingularity}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{ass:cond_nonsingularity}}{1}{Assumption 2}{ \ifstrequal{\oldref{ass:cond_nonsingularity}}{2}{Assumption 3}{ \ifstrequal{\oldref{ass:cond_nonsingularity}}{3}{Assumption 4}{ Assumption \oldref{ass:cond_nonsingularity} } } } } } (q)}}{
\ifstrequal{ass:cond_nonsingularity}{ass:Assumption1p}{1(p)}{ \ifstrequal{ass:cond_nonsingularity}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{ass:cond_nonsingularity}}{1}{Assumption 2}{ \ifstrequal{\oldref{ass:cond_nonsingularity}}{2}{Assumption 3}{ \ifstrequal{\oldref{ass:cond_nonsingularity}}{3}{Assumption 4}{ Assumption \oldref{ass:cond_nonsingularity} } } } } } (p) } } }{
\ifstrequal{ass:cond_nonsingularity}{ass:Assumption1p}{1(p)}{ \ifstrequal{ass:cond_nonsingularity}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{ass:cond_nonsingularity}}{1}{Assumption 2}{ \ifstrequal{\oldref{ass:cond_nonsingularity}}{2}{Assumption 3}{ \ifstrequal{\oldref{ass:cond_nonsingularity}}{3}{Assumption 4}{ Assumption \oldref{ass:cond_nonsingularity} } } } } } (p) } }
is sufficient for identification of the unknown coefficients $q_{0}(V)$ in CRF models with known $p(X)$, and Assumption \ifstrequal{ass:cond_nonsingularity}{ass:Assumption1}{ \ifstrequal{q}{p}{\hyperlink{ass:Assumption1p}{ \ifstrequal{ass:cond_nonsingularity}{ass:Assumption1p}{1(p)}{ \ifstrequal{ass:cond_nonsingularity}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{ass:cond_nonsingularity}}{1}{Assumption 2}{ \ifstrequal{\oldref{ass:cond_nonsingularity}}{2}{Assumption 3}{ \ifstrequal{\oldref{ass:cond_nonsingularity}}{3}{Assumption 4}{ Assumption \oldref{ass:cond_nonsingularity} } } } } } (p)}}{ \ifstrequal{q}{q}{\hyperlink{ass:Assumption1q}{ \ifstrequal{ass:cond_nonsingularity}{ass:Assumption1p}{1(p)}{ \ifstrequal{ass:cond_nonsingularity}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{ass:cond_nonsingularity}}{1}{Assumption 2}{ \ifstrequal{\oldref{ass:cond_nonsingularity}}{2}{Assumption 3}{ \ifstrequal{\oldref{ass:cond_nonsingularity}}{3}{Assumption 4}{ Assumption \oldref{ass:cond_nonsingularity} } } } } } (q)}}{
\ifstrequal{ass:cond_nonsingularity}{ass:Assumption1p}{1(p)}{ \ifstrequal{ass:cond_nonsingularity}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{ass:cond_nonsingularity}}{1}{Assumption 2}{ \ifstrequal{\oldref{ass:cond_nonsingularity}}{2}{Assumption 3}{ \ifstrequal{\oldref{ass:cond_nonsingularity}}{3}{Assumption 4}{ Assumption \oldref{ass:cond_nonsingularity} } } } } } (q) } } }{ \ifstrequal{ass:cond_nonsingularity}{ass:cond_nonsingularity}{ \ifstrequal{q}{p}{\hyperlink{ass:cond_nonsingularity_p}{ \ifstrequal{ass:cond_nonsingularity}{ass:Assumption1p}{1(p)}{ \ifstrequal{ass:cond_nonsingularity}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{ass:cond_nonsingularity}}{1}{Assumption 2}{ \ifstrequal{\oldref{ass:cond_nonsingularity}}{2}{Assumption 3}{ \ifstrequal{\oldref{ass:cond_nonsingularity}}{3}{Assumption 4}{ Assumption \oldref{ass:cond_nonsingularity} } } } } } (p)}}{ \ifstrequal{q}{q}{\hyperlink{ass:cond_nonsingularity_q}{ \ifstrequal{ass:cond_nonsingularity}{ass:Assumption1p}{1(p)}{ \ifstrequal{ass:cond_nonsingularity}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{ass:cond_nonsingularity}}{1}{Assumption 2}{ \ifstrequal{\oldref{ass:cond_nonsingularity}}{2}{Assumption 3}{ \ifstrequal{\oldref{ass:cond_nonsingularity}}{3}{Assumption 4}{ Assumption \oldref{ass:cond_nonsingularity} } } } } } (q)}}{
\ifstrequal{ass:cond_nonsingularity}{ass:Assumption1p}{1(p)}{ \ifstrequal{ass:cond_nonsingularity}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{ass:cond_nonsingularity}}{1}{Assumption 2}{ \ifstrequal{\oldref{ass:cond_nonsingularity}}{2}{Assumption 3}{ \ifstrequal{\oldref{ass:cond_nonsingularity}}{3}{Assumption 4}{ Assumption \oldref{ass:cond_nonsingularity} } } } } } (q) } } }{
\ifstrequal{ass:cond_nonsingularity}{ass:Assumption1p}{1(p)}{ \ifstrequal{ass:cond_nonsingularity}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{ass:cond_nonsingularity}}{1}{Assumption 2}{ \ifstrequal{\oldref{ass:cond_nonsingularity}}{2}{Assumption 3}{ \ifstrequal{\oldref{ass:cond_nonsingularity}}{3}{Assumption 4}{ Assumption \oldref{ass:cond_nonsingularity} } } } } } (q) } }
is sufficient for identification of the unknown coefficients $p_{0}(X)$ in CRF models with known $q(V)$.
In Section \ifstrequal{sec:Section5}{ass:Assumption1p}{1(p)}{ \ifstrequal{sec:Section5}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{sec:Section5}}{1}{Assumption 2}{ \ifstrequal{\oldref{sec:Section5}}{2}{Assumption 3}{ \ifstrequal{\oldref{sec:Section5}}{3}{Assumption 4}{ Assumption \oldref{sec:Section5} } } } } } we discuss conditions under which $E[p(X)p(X)'|V]$ and $E[q(V)q(V)'|X]$ are nonsingular in triangular models. All those conditions are sufficient for identification of $q_{0}(V)$ and of $p_{0}(X)$, respectively, including those that allow for discrete valued instrumental variables. We also note that identification of $q_{0}(V)$ and of $p_{0}(X)$ means uniqueness on sets of $V$ and of $X$ having probability one, respectively. Thus, under Assumption \ifstrequal{ass:DGP}{ass:Assumption1p}{1(p)}{ \ifstrequal{ass:DGP}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{ass:DGP}}{1}{Assumption 2}{ \ifstrequal{\oldref{ass:DGP}}{2}{Assumption 3}{ \ifstrequal{\oldref{ass:DGP}}{3}{Assumption 4}{ Assumption \oldref{ass:DGP} } } } } } with Assumption \ifstrequal{ass:Assumption1p}{ass:Assumption1p}{1(p)}{ \ifstrequal{ass:Assumption1p}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{ass:Assumption1p}}{1}{Assumption 2}{ \ifstrequal{\oldref{ass:Assumption1p}}{2}{Assumption 3}{ \ifstrequal{\oldref{ass:Assumption1p}}{3}{Assumption 4}{ Assumption \oldref{ass:Assumption1p} } } } } } , the ASF will be identified as \[ \mu(X)=p(X)'E[q_{0}(V)], \] and, under Assumption \ifstrequal{ass:DGP}{ass:Assumption1p}{1(p)}{ \ifstrequal{ass:DGP}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{ass:DGP}}{1}{Assumption 2}{ \ifstrequal{\oldref{ass:DGP}}{2}{Assumption 3}{ \ifstrequal{\oldref{ass:DGP}}{3}{Assumption 4}{ Assumption \oldref{ass:DGP} } } } } } with Assumption \ifstrequal{ass:Assumption1q}{ass:Assumption1p}{1(p)}{ \ifstrequal{ass:Assumption1q}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{ass:Assumption1q}}{1}{Assumption 2}{ \ifstrequal{\oldref{ass:Assumption1q}}{2}{Assumption 3}{ \ifstrequal{\oldref{ass:Assumption1q}}{3}{Assumption 4}{ Assumption \oldref{ass:Assumption1q} } } } } } , as \[ \mu(X)=p_{0}(X)'E[q(V)]. \] In other words, the ASF is identified under Assumption \ifstrequal{ass:Assumption1p}{ass:Assumption1p}{1(p)}{ \ifstrequal{ass:Assumption1p}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{ass:Assumption1p}}{1}{Assumption 2}{ \ifstrequal{\oldref{ass:Assumption1p}}{2}{Assumption 3}{ \ifstrequal{\oldref{ass:Assumption1p}}{3}{Assumption 4}{ Assumption \oldref{ass:Assumption1p} } } } } }
because $p\left(X\right)$ is a known function, and $q_{0}(V)$ is identified, and hence $E[q_{0}(V)]$ also is; the ASF is identified under Assumption \ifstrequal{ass:Assumption1q}{ass:Assumption1p}{1(p)}{ \ifstrequal{ass:Assumption1q}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{ass:Assumption1q}}{1}{Assumption 2}{ \ifstrequal{\oldref{ass:Assumption1q}}{2}{Assumption 3}{ \ifstrequal{\oldref{ass:Assumption1q}}{3}{Assumption 4}{ Assumption \oldref{ass:Assumption1q} } } } } } because $p_{0}(X)$ is identified, and $q(V)$ is a known function, and hence $E[q(V)]$ also is.
In CRF models with known $q(V)$, the ASF $p_{0}(X)'E[q(V)]$ is now a linear combination of unknown functions $p_{0}(X)$ with known coefficients $E[q(V)]$. In general, nonsingularity of $E[q(V)q(V)'|X]$ with probability one is not necessary for ASF identification in this case. A condition that is both necessary and sufficient is that $E[q(V)]$ belongs to the range $\mathcal{R}(E[q(V)q(V)'|X])$ of $E[q(V)q(V)'|X]$ with probability one.
When $E[q(V)q(V)'|X]$ is nonsingular with probability one, its range is $\mathbb{R}^{K}$ and hence the range condition is automatically satisfied. When conditional nonsingularity only holds with positive probability, then the range condition allows for identification of the ASF, but is in general not sufficient for point identification of $p_{0}(X)$. Like nonsingularity, the range condition depends on observable or estimable quantities only.
For CRF models with known $p(X)$, nonsingularity of $E[p(X)p(X)^{\prime}|V]$ allows for the conditional support of $X$ given $V$ to be a strict subset of the marginal support of $X$ with positive probability, for instance when the conditional support of $X$ given $V$ is discrete whereas both $X$ and $V$ are continuous. Under conditional nonsingularity, $p(X)$ known and identification of $q_{0}(V)$ together imply uniqueness of the CRF $p(X)'q_0(V)$ on a set of $(X,V)$ having probability one. Therefore the integral in characterization ( \ifstrequal{eq:integrated_CRF}{ass:Assumption1p}{1(p)}{ \ifstrequal{eq:integrated_CRF}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{eq:integrated_CRF}}{1}{Assumption 2}{ \ifstrequal{\oldref{eq:integrated_CRF}}{2}{Assumption 3}{ \ifstrequal{\oldref{eq:integrated_CRF}}{3}{Assumption 4}{ Assumption \oldref{eq:integrated_CRF} } } } } } ) of the ASF is well-defined for $E[Y|X,V]=p(X)'q_0(V)$ because integration then occurs over a range of $v$ values conditional on $X$ where the CRF is identified. Stronger sufficient conditions for identification are full support (Imbens Newey 2009) and measurable separability (Flo Heck Meg Vytl 2008), that any function of $X$ equal to a function of $V$ with probability one must be equal to a constant with probability one. Both conditions require $X$ to have continuous support conditional on $V$.
Conditional nonsingularity for identification in models of fixed dimension and full support for nonparametric identification are related under regularity conditions. For $p^{J}(X)$ denoting an increasing sequence of mean-square spanning approximating functions, nonsingularity for each element of this sequence implies full support, and hence also nonparametric identification. To state this result formally, denote the smallest and largest eigenvalue of a matrix $A$ by $\lambda_{\min}(A)$ and $\lambda_{\max}(A)$, respectively, and let $p^{J}(X)$ be mean-square spanning if, for any $f(X)$ such that $E[f(X)^{2}]<\infty$, there is $\gamma^{J}$ such that $E[\{f(X)-p^{J}(X)'\gamma^{J}\}^{2}]\rightarrow0$ as $J\rightarrow\infty$.
Theorem \ifstrequal{thm:Theorem6}{ass:Assumption1p}{1(p)}{ \ifstrequal{thm:Theorem6}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{thm:Theorem6}}{1}{Assumption 2}{ \ifstrequal{\oldref{thm:Theorem6}}{2}{Assumption 3}{ \ifstrequal{\oldref{thm:Theorem6}}{3}{Assumption 4}{ Assumption \oldref{thm:Theorem6} } } } } } is a novel kind of result that relates semi- and non-parametric identification conditions, embedding nonparametric identification as a particular case in a general class of identification results for flexible outcome models with ASF and CRF in linear form, under Assumption \ifstrequal{ass:DGP}{ass:Assumption1p}{1(p)}{ \ifstrequal{ass:DGP}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{ass:DGP}}{1}{Assumption 2}{ \ifstrequal{\oldref{ass:DGP}}{2}{Assumption 3}{ \ifstrequal{\oldref{ass:DGP}}{3}{Assumption 4}{ Assumption \oldref{ass:DGP} } } } } } with Assumption \ifstrequal{ass:Assumption1p}{ass:Assumption1p}{1(p)}{ \ifstrequal{ass:Assumption1p}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{ass:Assumption1p}}{1}{Assumption 2}{ \ifstrequal{\oldref{ass:Assumption1p}}{2}{Assumption 3}{ \ifstrequal{\oldref{ass:Assumption1p}}{3}{Assumption 4}{ Assumption \oldref{ass:Assumption1p} } } } } } . Denoting by $q_{0}^{J}(V)$ the coefficients in CRFs with known functions $p^{J}(X)$, this result reveals that a unified treatment of semi- and non-parametric identification is achieved under conditional nonsingularity: by $p^{J}(X)$ known, nonsingularity for given $J$ implies identification of $p^{J}(X)'q_0^{J}(V)$ viewed as a correct CRF model, and nonsingularity for all $J$ implies uniqueness on a set of $(X,V)$ with probability one of each $p^{J}(X)'q_0^{J}(V)$ in a sequence of CRF approximating models, and hence is sufficient for nonparametric identification. This embedding of semi- and non-parametric identification into a common framework also applies to the other control regressions in Remark \ifstrequal{rem:control CQF and CDF}{ass:Assumption1p}{1(p)}{ \ifstrequal{rem:control CQF and CDF}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{rem:control CQF and CDF}}{1}{Assumption 2}{ \ifstrequal{\oldref{rem:control CQF and CDF}}{2}{Assumption 3}{ \ifstrequal{\oldref{rem:control CQF and CDF}}{3}{Assumption 4}{ Assumption \oldref{rem:control CQF and CDF} } } } } } , with nonsingularity for all approximating models implying full support, and hence also nonparametric identification of distributional and quantile treatment effects (Imbens Newey 2009).
For the class of models defined by Assumption \ifstrequal{ass:DGP}{ass:Assumption1p}{1(p)}{ \ifstrequal{ass:DGP}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{ass:DGP}}{1}{Assumption 2}{ \ifstrequal{\oldref{ass:DGP}}{2}{Assumption 3}{ \ifstrequal{\oldref{ass:DGP}}{3}{Assumption 4}{ Assumption \oldref{ass:DGP} } } } } } with Assumption \ifstrequal{ass:Assumption1p}{ass:Assumption1p}{1(p)}{ \ifstrequal{ass:Assumption1p}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{ass:Assumption1p}}{1}{Assumption 2}{ \ifstrequal{\oldref{ass:Assumption1p}}{2}{Assumption 3}{ \ifstrequal{\oldref{ass:Assumption1p}}{3}{Assumption 4}{ Assumption \oldref{ass:Assumption1p} } } } } } , i.e., with $p(X)$ known, a converse result is that full support implies conditional nonsingularity with probability one, under the additional maintained assumption that $E[p(X)p(X)']$ be nonsingular. This is a corollary of Theorem \ifstrequal{thm:Theorem4}{ass:Assumption1p}{1(p)}{ \ifstrequal{thm:Theorem4}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{thm:Theorem4}}{1}{Assumption 2}{ \ifstrequal{\oldref{thm:Theorem4}}{2}{Assumption 3}{ \ifstrequal{\oldref{thm:Theorem4}}{3}{Assumption 4}{ Assumption \oldref{thm:Theorem4} } } } } } .
The relaxation of full support afforded by our framework is important in practice. We discuss below the leading case of triangular models with discrete instruments, where full support cannot hold. More generally, within the class of models we consider, treatment effects can be identified for continuous $X$ when the joint support of $X$ and $V$ is not rectangular. For discrete $X$, treatment effects can be identified when the conditional support of $X$ given $V$ has fewer points than the marginal support of $X$ with positive probability, with $p(X)$ of dimension smaller than the number of treatment regimes.
For CRF models with known $p(X)$, a condition weaker than conditional nonsingularity with probability one is nonsingularity of $E[p(X)p(X)'|V]$ with positive probability. This condition has been used by MT:2016 for identification in the triangular model formed by ( \ifstrequal{eq:hcoefs_model_p}{ass:Assumption1p}{1(p)}{ \ifstrequal{eq:hcoefs_model_p}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{eq:hcoefs_model_p}}{1}{Assumption 2}{ \ifstrequal{\oldref{eq:hcoefs_model_p}}{2}{Assumption 3}{ \ifstrequal{\oldref{eq:hcoefs_model_p}}{3}{Assumption 4}{ Assumption \oldref{eq:hcoefs_model_p} } } } } } ) and ( \ifstrequal{eq:h(z,eta)}{ass:Assumption1p}{1(p)}{ \ifstrequal{eq:h(z,eta)}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{eq:h(z,eta)}}{1}{Assumption 2}{ \ifstrequal{\oldref{eq:h(z,eta)}}{2}{Assumption 3}{ \ifstrequal{\oldref{eq:h(z,eta)}}{3}{Assumption 4}{ Assumption \oldref{eq:h(z,eta)} } } } } } ). Our identification results that are based on the control variable $V=F_{X|Z}(X|Z)$ are thus related to their approach. Suppose nonsingularity of $E[p(X)p(X)'|V]$ holds on a set with positive probability, and $\overline{q}(v)\neq q_{0}(v)$ for a value $v$ in that set. Then, with $\lambda(v)\equiv\overline{q}(v)-q_{0}(v)$,
If $E[\beta(\varepsilon)]$ and $E[p(X)p(X)']$ exist then the expectation in ( \ifstrequal{eq:id_v}{ass:Assumption1p}{1(p)}{ \ifstrequal{eq:id_v}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{eq:id_v}}{1}{Assumption 2}{ \ifstrequal{\oldref{eq:id_v}}{2}{Assumption 3}{ \ifstrequal{\oldref{eq:id_v}}{3}{Assumption 4}{ Assumption \oldref{eq:id_v} } } } } } ) exists, and hence $q_{0}(V)$ is identified from $E[Y|X,V=v]$, by definition ( \ifstrequal{eq:controlregression_1}{ass:Assumption1p}{1(p)}{ \ifstrequal{eq:controlregression_1}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{eq:controlregression_1}}{1}{Assumption 2}{ \ifstrequal{\oldref{eq:controlregression_1}}{2}{Assumption 3}{ \ifstrequal{\oldref{eq:controlregression_1}}{3}{Assumption 4}{ Assumption \oldref{eq:controlregression_1} } } } } } ). MT:2016 showed this, and noted that $E[q_{0}(V)]$ is identified if the set of $v$ with $E[p(X)p(X)'|V=v]$ nonsingular has probability one. Their approach is local (pointwise in $v$) and constructive for $q_{0}(v)$. In contrast, directly considering uniqueness with probability one of $q_{0}(V)$ is useful for our analysis, which focuses on average treatment effects and hence requires identification of $E[q_{0}(V)]$, and is constructive for $q_{0}(V)$.
Nonsingularity of $E[p(X)p(X)'|V]$ and of $E[q(V)q(V)'|X]$ are generic conditions for identification in model specifications above, for any observable or estimable $V$. These conditions, however, do not use the specific structure of triangular models. Explicitly accounting for their specific features leads to the formulation of primitive conditions that can be considerably easier to interpret and verify. Thus we specialize our identification analysis to triangular models with implied CRF of the form ( \ifstrequal{eq:controlregression_1}{ass:Assumption1p}{1(p)}{ \ifstrequal{eq:controlregression_1}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{eq:controlregression_1}}{1}{Assumption 2}{ \ifstrequal{\oldref{eq:controlregression_1}}{2}{Assumption 3}{ \ifstrequal{\oldref{eq:controlregression_1}}{3}{Assumption 4}{ Assumption \oldref{eq:controlregression_1} } } } } } ), with known functions of $X$. Results with known functions of $V$ can be derived using analogous arguments and are summarized in Remark
\ifstrequal{rem:model2}{ass:Assumption1p}{1(p)}{ \ifstrequal{rem:model2}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{rem:model2}}{1}{Assumption 2}{ \ifstrequal{\oldref{rem:model2}}{2}{Assumption 3}{ \ifstrequal{\oldref{rem:model2}}{3}{Assumption 4}{ Assumption \oldref{rem:model2} } } } } } below.
In triangular models with control variable $V=F_{X|Z}(X|Z)$, the identification conditions can equivalently be stated in terms of the first-stage representation $X=Q_{X|Z}(V|Z)$ and the instrument $Z$. By independence of $V$ from $Z$, the identification condition with ( \ifstrequal{eq:controlregression_1}{ass:Assumption1p}{1(p)}{ \ifstrequal{eq:controlregression_1}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{eq:controlregression_1}}{1}{Assumption 2}{ \ifstrequal{\oldref{eq:controlregression_1}}{2}{Assumption 3}{ \ifstrequal{\oldref{eq:controlregression_1}}{3}{Assumption 4}{ Assumption \oldref{eq:controlregression_1} } } } } } ) is that \[ E[p(Q_{X\mid Z}(v\mid Z))p(Q_{X\mid Z}(v\mid Z))'] \] be nonsingular for almost every (a.e.) $v$ in the support $\mathcal{V}$ of $V$.
When $Z$ has discrete support $\mathcal{Z}=\left\{ z:\Pr(Z=z)\geq\delta>0\right\} $ of finite cardinality $|\mathcal{Z}|$, a necessary condition for nonsingularity is that the set $\mathcal{Q}(V)$ of distinct values of $z\mapsto Q_{X\mid Z}(V|z)$ has cardinality $|\mathcal{Q}(V)|$ greater than or equal to $J=\dim(p(X))$ with probability one.\footnote{Formally, for $v\in(0,1)$, we define $\mathcal{Q}(v)=\left\{ Q_{X\mid Z}(v\mid z_{m})\right\} _{m\in\mathcal{M}(v)}$, where \[ \mathcal{M}(v)=\left\{ m\in\{1,\ldots,|\mathcal{Z}|\}:Q_{X\mid Z}(v\mid z_{m})\neq Q_{X\mid Z}(v\mid z_{m'})\,\textrm{for all }m'\in\{1,\ldots,|\mathcal{Z}|\}\backslash\{m\}\right\} . \] } Thus, for a sufficiently rich set of $p(X)$ values, Theorem \ifstrequal{thm:Theorem4}{ass:Assumption1p}{1(p)}{ \ifstrequal{thm:Theorem4}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{thm:Theorem4}}{1}{Assumption 2}{ \ifstrequal{\oldref{thm:Theorem4}}{2}{Assumption 3}{ \ifstrequal{\oldref{thm:Theorem4}}{3}{Assumption 4}{ Assumption \oldref{thm:Theorem4} } } } } } implies that the ASF cannot be identified if $|\mathcal{Z}|<J$.
Theorem \ifstrequal{thm:Theorem9}{ass:Assumption1p}{1(p)}{ \ifstrequal{thm:Theorem9}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{thm:Theorem9}}{1}{Assumption 2}{ \ifstrequal{\oldref{thm:Theorem9}}{2}{Assumption 3}{ \ifstrequal{\oldref{thm:Theorem9}}{3}{Assumption 4}{ Assumption \oldref{thm:Theorem9} } } } } } formalizes the intuitive notion that the complexity of the model, as measured by the dimension of its known component $p(X)$, is restricted by the cardinality of the set of instrumental values: the ASF can only be identified when the number of support points in $\mathcal{Z}$ is not smaller than $J$, the number of treatment functions. Thus only when $p(X)$ is two-dimensional can identification be achieved in the presence of a binary instrument $Z\in\{0,1\}$. A more primitive condition for identification in this case is that a change in the value of the instrument shifts the value of the conditional quantile function $z\mapsto Q_{X\mid Z}(V|z)$ with probability one, the condition stated in MT:2016 that $\text{Var}(Q_{X\mid Z}(v|Z))>0$ for a.e. $v\in\mathcal{V}$. Here we further show that this condition is also necessary when $E[p(X)p(X)']$ is nonsingular. L:2024 gives a related analysis in a panel random coefficient model.
We present two examples that illustrate implications of our identification analysis for quantile and distributional treatment effects in alternative control quantile and distribution regression model specifications. To the best of our knowledge, identification with discrete instruments has not been previously established in these models.
This paper uses functional formal restrictions to achieve identification without full support. In this section we establish testability of the model specifications implied by these restrictions, and we characterize all testable implications.
We first consider testability of the CRF model ( \ifstrequal{eq:controlregression_1}{ass:Assumption1p}{1(p)}{ \ifstrequal{eq:controlregression_1}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{eq:controlregression_1}}{1}{Assumption 2}{ \ifstrequal{\oldref{eq:controlregression_1}}{2}{Assumption 3}{ \ifstrequal{\oldref{eq:controlregression_1}}{3}{Assumption 4}{ Assumption \oldref{eq:controlregression_1} } } } } } ) with known functions of $X$. Results for CRFs with known functions of $V$ are stated in Remark \ifstrequal{rem:Testability}{ass:Assumption1p}{1(p)}{ \ifstrequal{rem:Testability}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{rem:Testability}}{1}{Assumption 2}{ \ifstrequal{\oldref{rem:Testability}}{2}{Assumption 3}{ \ifstrequal{\oldref{rem:Testability}}{3}{Assumption 4}{ Assumption \oldref{rem:Testability} } } } } } below.
Theorem \ifstrequal{thm:Theorem10}{ass:Assumption1p}{1(p)}{ \ifstrequal{thm:Theorem10}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{thm:Theorem10}}{1}{Assumption 2}{ \ifstrequal{\oldref{thm:Theorem10}}{2}{Assumption 3}{ \ifstrequal{\oldref{thm:Theorem10}}{3}{Assumption 4}{ Assumption \oldref{thm:Theorem10} } } } } } characterizes the complete set of orthogonality conditions implied by CRF specification ( \ifstrequal{eq:controlregression_1}{ass:Assumption1p}{1(p)}{ \ifstrequal{eq:controlregression_1}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{eq:controlregression_1}}{1}{Assumption 2}{ \ifstrequal{\oldref{eq:controlregression_1}}{2}{Assumption 3}{ \ifstrequal{\oldref{eq:controlregression_1}}{3}{Assumption 4}{ Assumption \oldref{eq:controlregression_1} } } } } } ), in terms of known functions and observable or estimable random variables only. In particular $q^{*}(V)$ is a vector of conditional least-squares projections of $Y$ on $p(X)$ given $V$, where $Y$, $X$ and $V$ are each observable or estimable.
There are several possible choices of test functions $a(X,V)$. One natural approach to exploit this characterization of testable implications of the model is to specify test functions $a(X,V)$ to be a vector of $L$ power transformations of the CRF:
Orthogonality conditions of the form \[ E[\{Y-p(X)'q^{*}(V)\}a(X,V)]=0 \] with ( \ifstrequal{eq:Overid3}{ass:Assumption1p}{1(p)}{ \ifstrequal{eq:Overid3}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{eq:Overid3}}{1}{Assumption 2}{ \ifstrequal{\oldref{eq:Overid3}}{2}{Assumption 3}{ \ifstrequal{\oldref{eq:Overid3}}{3}{Assumption 4}{ Assumption \oldref{eq:Overid3} } } } } } ) extend the classical approach of Ramsay:1969 for specification testing of mean regression functions to the control regression setting. Alternative choices of $a(X,V)$ are revealing functions of Bierens 1982 and SW:1998.
A key implication of Theorem \ifstrequal{thm:Theorem10}{ass:Assumption1p}{1(p)}{ \ifstrequal{thm:Theorem10}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{thm:Theorem10}}{1}{Assumption 2}{ \ifstrequal{\oldref{thm:Theorem10}}{2}{Assumption 3}{ \ifstrequal{\oldref{thm:Theorem10}}{3}{Assumption 4}{ Assumption \oldref{thm:Theorem10} } } } } } is that if $E[Y|X,V]$ misspecified, i.e., is not of the specified form $p(X)'q_{0}(V)$, then $E[\{Y-p(X)'q^{*}(V)\}a(X,V)]\neq0$ for some test function $a(X,V)$ with $E[a(X,V)^{2}]<\infty$. This provides the basis of a test that can detect violations of the model specification. Empirical likelihood-based testing procedures for unconditional orthogonality conditions have been developed in a parametric setting, with $q^{*}(V)=q(V;\theta^{*})$ (DIN 2003). It is beyond the scope of this paper to extend these approaches to the case with infinite-dimensional parameters.
Our identification analysis leads to direct estimation methods for the heterogeneous coefficients models we consider. One approach to making estimation feasible is through approximation of the nonparametric components $q_{0}(V)$ or $p_{0}(X)$ by approximating functions such as splines or wavelets. Here we focus on $q_{0}(V)$ in CRF model ( \ifstrequal{eq:controlregression_1}{ass:Assumption1p}{1(p)}{ \ifstrequal{eq:controlregression_1}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{eq:controlregression_1}}{1}{Assumption 2}{ \ifstrequal{\oldref{eq:controlregression_1}}{2}{Assumption 3}{ \ifstrequal{\oldref{eq:controlregression_1}}{3}{Assumption 4}{ Assumption \oldref{eq:controlregression_1} } } } } } ), and an estimator for $p_{0}(X)$ in CRF model ( \ifstrequal{eq:controlregression_2}{ass:Assumption1p}{1(p)}{ \ifstrequal{eq:controlregression_2}{ass:Assumption1q}{1(q)}{ \ifstrequal{\oldref{eq:controlregression_2}}{1}{Assumption 2}{ \ifstrequal{\oldref{eq:controlregression_2}}{2}{Assumption 3}{ \ifstrequal{\oldref{eq:controlregression_2}}{3}{Assumption 4}{ Assumption \oldref{eq:controlregression_2} } } } } } ) can be constructed analogously.
For the specification $E[Y|X,V]=p(X)'q_{0}(V)$, we approximate each component $q_{0j}(V)$, $j\in\{1,\ldots,J\}$, of the unknown functional coefficient vector $q_{0}(V)$ by a linear combination of $K$ basis functions $\psi^{K}=(\psi_{1}^{K},\ldots,\psi_{K}^{K})'$,
where $b_{j}=(b_{j1},\ldots,b_{jK})'$, which yields an approximation of the form \[ E\left[Y\mid X,V\right]=p(X)'q_{0}(V)\approx\sum_{j=1}^{J}\left\{ b_{j}'\psi^{K}(V)\right\} p_{j}(X)=b'[p(X)\otimes\psi^{K}(V)], \] where $b=(b_{1}',\ldots,b_{J}')'$. Such an approximation is well-defined under our conditions with $b=b_{\textrm{LS}}^{K}$, the coefficient vector of a least squares regression of $Y$ on $p(X)\otimes\psi^{K}(V)$,
The proposed approximation is valid for the CRF $E[Y|X,V]$ if the specified basis functions satisfy the following condition.
This paper introduces a unified modeling framework for treatment effects under minimal identification conditions. This framework is general enough to encompass a wide range of models of interest to applied researchers, and we provide a comprehensive treatment of identification, model testability and estimation for all classes of models considered. For flexible models of increasing dimension, we elucidate and characterize the connection between conditional nonsingularity for identification in these models, and the full support condition for nonparametric identification of treatment effects. In the presence of multidimensional heterogeneity and discrete instruments, we give conditions for identification of average treatment effects in general nonseparable triangular models, and our results demonstrate testability of both identification and model specification. These results extend to other types of control regressions, and our models can be conveniently estimated by series-based least squares estimators with well-understood properties.