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Semiparametric Estimation of Structural Functions in Nonseparable Triangular Models
Keywords: Structural functions, nonseparable models, control function, quantile and distribution regression, semiparametric estimation, uniform inference.
Models with nonadditively separable disturbances provide an important vehicle for incorporating heterogenous effects. However, accounting for endogenous treatments in such a setting can be challenging. One methodology which has been successfully employed in a wide range of models with endogeneity is the use of control functions (see, for surveys, Imbens and Wooldridge 2009, Wooldridge 2015 and Blundell, Newey and Vella 2017). The underlying logic of this approach is to account for the endogeneity by including an appropriate control function in the conditioning variables. This paper proposes some relatively simple control function procedures to estimate objects of interest in a triangular model with nonseparable disturbances. Our approach to circumventing the inherent difficulties in nonparametric estimation associated with the curse of dimensionality is to build our models upon a semiparametric specification. This also alleviates the large support requirement on the exogenous instrument, or exclusion restriction, needed for nonparametric identification. Our goal is thus to provide models and methods that are essentially parametric but still allow for nonseparable disturbances in order to address strong data requirements that come with nonparametric formulations. These models can be interpreted as “baseline" models on which series approximations can be built by adding additional terms.
We consider two kinds of baseline models, quantile regression and distribution regression. These models allow the use of convenient and widely available methods to estimate objects of interest including average, distribution and quantile structural/treatment effects. A main feature of the baseline models is that interaction terms included would not usually be present as leading terms in estimation. These included terms are products of a transformation of the control function with the endogenous treatment. Their presence is meant to allow for heterogeneity in the coefficient of the endogenous variable. Such heterogenous coefficient linear models are of interest in many settings, including demand analysis and estimation of returns to education, and provide a natural starting point for more general models that allow for nonlinear effects of the endogenous treatments.
We use these baseline models to construct estimators of the average, distribution and quantile structural functions based on parametric quantile and distribution regressions. These objects fully characterise the structural relationship between the endogenous treatment and the outcome of interest, and describe the average, distribution and quantiles of the outcome across treatment values, had the treatment been exogenous. We also show how these baseline models can be expanded to include higher order terms, leading to more flexible structural function specifications. The estimation procedure consists of three stages. First, we estimate the control function via quantile regression (QR) or distribution regression (DR) of the endogenous treatment on the exogenous covariates and exclusion restrictions. Second, we estimate the reduced form distribution of the outcome conditional on the treatment, covariates and estimated control function using DR or QR. Third, we construct estimators of the structural functions applying suitable functionals to the reduced form estimator from the second stage. We derive asymptotic theory for the estimators based on DR in all the stages using a trimming device that avoids tail estimation in the construction of the control function. We also establish the validity of the bootstrap for our inference on structural functions, which enables the formulation of convenient inference algorithms which we describe in detail. The modelling framework we propose thus allows us to address three key difficulties that have restricted the use of such models in empirical work -- the curse of dimensionality, the large support condition for identification and the lack of easily implementable inference methods -- while simultaneously retaining important features of the original nonparametric formulation. We give an empirical application based on the estimation of Engel curves which illustrates how our approach leads to complete and flexible estimates of all structural functions and their confidence regions.
Our results for the average structural function in the linear random coefficients model are similar to Garen (1984). Florens, Heckman, Meghir, Vytlacil (2008) give identification and estimation results for a restricted model with random coefficients for powers of the endogenous treatment. Blundell and Powell (2003, 2004) introduce the average structural function, and Imbens and Newey (2009) give general models and results for a variety of objects of interest and control functions, including quantile structural functions, under a large support condition on the exclusion restriction. This work also complements the literature on local identification and estimation in triangular nonseparable models, as in Chesher (2003), Ma and Koenker (2006), and Jun (2009), on global construction of structural functions (Stouli, 2012) and identification in the presence of an exclusion restriction with small support (Fevrier and d'Haultfoeuille, 2015; Torgovitsky, 2015). Chernozhukov, Fernandez-Val and Kowalski (2015) developed a related two-stage quantile regression estimator for triangular nonseparable models . These papers did not consider structural functions defined for nonseparable triangular models with multidimensional unobserved heterogeneity.
This paper makes four main contributions to the existing literature. First, we establish identification of structural functions in both classes of baseline models, providing conditions that do not impose large support requirements on the exclusion restriction. Second, we derive a functional central limit theorem and a bootstrap functional central limit theorem for the two-stage DR estimators in the second stage. These results are uniform over compact regions of values of the outcome. To the best of our knowledge, this result is new. Chernozhukov, Fernandez-Val and Kowalski (2015) derived similar results for two-stage quantile regression estimators but their results are pointwise over quantile indexes. Our analysis builds on Chernozhukov, Fernandez-Val, and Galichon (2010) and Chernozhukov, Fernandez-Val, and Melly (2013), which established the properties of the DR estimators that we use in the first stage. The theory of the two-stage estimator, however, does not follow from these results using standard techniques due to the dimensionality and entropy properties of the first stage DR estimators. We follow the proof strategy proposed by Chernozhukov, Fernandez-Val and Kowalski (2015) to deal with these issues. Third, we derive functional central limit theorems and bootstrap functional central limit theorems for plug-in estimators of functionals of the distribution of the outcome conditional on the treatment, covariates and control function via functional delta method. These functionals include all the structural functions of interest. We also use a linear functional for the average structural function which had not been previously considered. Fourth, we show that this linear operator that relates the average of a random variable with its distribution is Hadamard differentiable. Our modelling framework and theoretical results are also of interest for the study of nonseparable triangular models in various alternative settings\footnote{See Fernandez-Val et al. (2018) for an application to the analysis of nonseparable sample selection models with censored selection rules.}, and will allow establishing the validity of bootstrap inference for the corresponding estimators.
The rest of the paper is organized as follows. Section (ref) describes the baseline models and objects of interest. Section (ref) presents the estimation and inference methods. Section (ref) gives asymptotic theory. Section (ref) reports the results of an extensive empirical application to Engel curves. Implementation algorithms and proofs of the main result are given in the Appendix. The online Appendix Chernozhukov et al. (2018) contains supplemental material, including results of numerical simulations calibrated to the application.
We begin with a brief review of the triangular nonseparable model and some inherent objects of interest. Let $Y$ denote an outcome variable of interest that can be continuous, discrete or mixed continuous-discrete, $X$ a continuous endogenous treatment, $Z$ a vector of exogenous variables, $ \varepsilon $ a structural disturbance vector of unknown dimension, and $\eta$ a scalar reduced form disturbance\footnote{In our empirical application, we use household level data to study the structural relationship between the share of expenditure on either food or leisure, $Y$, and the log of total expenditure, $X$, with gross earnings of the head of household as the exclusion restriction $Z$. Additional examples and a general economic motivation of nonseparable triangular models are given in Chesher (2003) and Imbens and Newey (2009), for instance.}. The model is
where $\eta\mapsto h(z,\eta)$ is a one-to-one function for each $z$. This model implies that $\varepsilon$ and $X$ are independent conditional on $\eta $ and that $\eta$ is a one-to-one function of $V=F_{X}(X\mid Z)$, the cumulative distribution function (CDF) of $X$ conditional on $Z$ evaluated at the observed variables. Thus, $V$ is a control function.
Objects of interest in this model include the average structural function (ASF), $\mu (x)$, quantile structural function (QSF), $Q(\tau ,x)$, and distribution structural function (DSF), $G(y,x)$, where
and
Here $\mu (\tilde{x})-\mu (\bar{x})$ is like an average treatment effect, $Q(\tau ,\tilde{x})-Q(\tau ,\bar{x})$ is like a quantile treatment effect, and $G(y ,\tilde{x})-G(y ,\bar{x})$ is like a distribution treatment effect from the treatment effects literature. If the support of $V$ conditional on $ X=x$ is the same as the marginal support of $V$ then these objects are nonparametrically identified\footnote{Nonparametric identification thus requires the exclusion restriction $Z$ to have full support conditional on $X=x$; see Imbens and Newey (2009) for a detailed discussion.} by
and
where $G^{\leftarrow }(\tau ,x)$ denotes the left-inverse of $y\mapsto G(y,x)$, i.e. $G^{\leftarrow }(\tau ,x):=\inf \{y\in \mathbb{R}:G(y,x)\geq \tau \}$.
It is straightforward to extend this approach to allow for covariates in the model by further conditioning on or integrating over them. Suppose that $Z_{1}\subset Z$ is included in the structural equation, which is now $g(X,Z_{1},\varepsilon ).$ Under the assumption that $\varepsilon $ and $V$ are jointly independent of $Z$, then $\varepsilon $ will be independent of $X$ and $Z_{1}$ conditional on $V$. Conditional on covariates and unconditional average structural functions are identified by
and
Similarly, conditional on covariates and unconditional quantile and distribution structural functions are identified by
and
respectively.
Without functional form restrictions, the curse of dimensionality makes it difficult to estimate the control function $V=F_{X}(X \mid Z)$, the conditional mean $E[Y \mid X,Z_{1},V],$ and the conditional CDF $F_{Y}(Y \mid X,Z_{1},V)$, and the full support condition makes it difficult to achieve point identification of the structural functions. These difficulties motivate our specification of baseline parametric models in what follows. These baseline models provide good starting points for nonparametric estimation and may be of interest in their own right.
We start with a simplified specification with one endogenous treatment $X$, one exclusion restriction $Z$, and a continuous outcome $Y$. We show below how additional excluded variables and covariates can be included.
The baseline first stage is the QR model
Note that $v\mapsto \pi _{1}(v)$ and $v\mapsto \pi _{2}(v)$ are infinite dimensional parameters (functions). We can recover the control function $V$ from $V=F_{X}(X\mid Z)=Q_{X}^{-1}(X\mid Z)$ or equivalently from
This generalized inverse representation of the CDF is convenient for estimation because it does not require the conditional quantile function to be strictly increasing to be well-defined.
The baseline second stage has a reduced form:
where $\Phi ^{-1}$ is the standard normal inverse CDF. This transformation is included to expand the support of $V$ and to encompass the normal system of equations as a special case. An example of a structural model with this reduced form is the random coefficient model
with the restrictions
These restrictions include the control function assumption $\varepsilon _{j}\perp \!\!\!\perp X\mid V$ and a joint functional form restriction, where the unobservable $U$ is the same for $\varepsilon _{1}$ and $ \varepsilon _{2}$. Substituting in the second stage equation,
which has the form of (ref)-(ref).
The specification (ref)-(ref) is a baseline, or starting point, for a more general series approximation to the quantiles of $Y$ conditional on $X$ and $V$ based on including additional functions of $X$ and $\Phi ^{-1}(V)$. The baseline is unusual as it includes the interaction term $\Phi ^{-1}(V)X;$ it is more usual to take the starting point to be $(1,\Phi ^{-1}(V),X),$ which is linear in the regressors $X$ and $\Phi ^{-1}(V).$ The inclusion of the interaction term is motivated by allowing the coefficient of $X$ to vary with individuals, so that $\Phi ^{-1}(V)$ then interacts $X$ in the conditional distribution of $\varepsilon _{2}$ given the control functions. All the parameters of model (ref)-(ref) can be estimated using the QR estimator (Koenker and Bassett, 1978).
The ASF of the baseline specification is:
where the second equality follows by $\int_{0}^{1}\Phi ^{-1}(v)dv=0$ and
with $\beta _{j}:=\int_{0}^{1}\beta _{j}(u)du$, $j\in \{1,\ldots ,4\}$. The QSF does not appear to have a closed form expression. It is the solution to
A special case of the QR baseline is a heteroskedastic normal system of equations. We use this specification in the numerical simulations of Section (ref).
We start again with a simplified specification with one endogenous treatment $X$ and one excluded $Z$, but now the outcome $Y$ can be continuous, discrete or mixed.
Let $\Gamma$ denote a strictly increasing continuous CDF such as the standard normal or logistic CDF. The first stage equation is the distribution regression model
which corresponds to the specification of the control variable $V$ as \
For the DR baseline, the QSF is the solution to
Compared to the QR baseline model, the ASF cannot be obtained as a linear projection but it can be conveniently expressed as a linear functional of $ G(y,x)$. Let $\mathcal{Y}$ denote the support of $Y$, $\mathcal{Y}^{+}= \mathcal{Y}\cap \lbrack 0,\infty )$ and $\mathcal{Y}^{-}=\mathcal{Y}\cap (-\infty ,0)$. The ASF can be characterized as
where $\nu $ is either the counting measure when $\mathcal{Y}$ is countable or the Lebesgue measure otherwise, and we exploit the linear relationship between the expected value and the distribution of a random variable. This characterization simplifies both the computation and theoretical treatment of the DR-based estimator for the ASF. It also applies to the QR specification upon using the corresponding expression for $G(y,x)$.
Section (ref) provides an example of a special case of the DR model.
The structural functions of the baseline models involve quantile and distribution regressions on the same set of regressors. A sufficient condition for identification of the coefficients of these regressions is that the second moment matrix of those regressors is nonsingular. The regressors have a Kronecker product form $p(X) \otimes r_{1}(Z_{1}) \otimes q(V)$. The second moment matrix for these regressors will be nonsingular if the joint distribution dominates a distribution where $X$, $Z_{1}$ and $V$ are independent and the second moment matrices of $X$, $Z_{1}$ and $V$ are positive definite\footnote{This condition is sufficient for identification and is in principle testable. However, in practice it will be easier to check directly if the sample second moment matrix for the regressors is of full rank.}. Define the product probability measure $\varsigma(z_{1}):=\times_{l=1}^{d_{z_{1}}} \varsigma_{l}(z_{1l})$.
Assumptions (ref)-(ref) are sufficient conditions for the map $y\mapsto F_{Y}(y\mid x,z_{1},v)$ to be well-defined for all $(x,z_{1},v)$, and therefore for identification of the structural functions.
Given the semiparametric specifications in Assumption (ref), identification of structural functions does not restrict the support of $Z$ to be continuous, and the full support assumption of Imbens and Newey (2009) need not be satisfied. When $q(V)=(1,\Phi ^{-1}(V))^{\prime }$, for the second moment matrix of regressors to be nonsingular only requires the control function to have strictly positive variance across the support of $X$, which can be satisfied even if the support of $Z$ is binary or discrete. This is in sharp contrast with nonparametric identification which requires full support of the control variable at each value of $X$. Theorem (ref) thus illustrates the identifying power of semiparametric restrictions and the trade-off between these restrictions and the full support condition for identification of structural functions.
The QR and DR baselines of the previous section lead to three-stage analog estimation and inference methods for the DSF, QSF and ASF. The first stage estimates the control function $V=F_{X}(X \mid Z)$. The second stage estimates the conditional distribution function $F_{Y}(y \mid X,Z_{1},V)$, replacing $V$ by the estimator from the first stage. The third stage obtains estimators of the structural functions, which are functionals of the first and second stages building blocks. We provide a detailed description of the implementation of each step for both QR and DR methods. We also describe a weighted bootstrap procedure to perform uniform inference on all structural functions considered. Detailed implementation algorithms are given in Appendix (ref).
We assume that we observe a sample of $n$ independent and identically distributed realizations $\{(Y_{i},X_{i},Z_{i})\}_{i=1}^{n}$ of the random vector $(Y,X,Z)$, and that $\textrm{dim}(X)=1$. Calligraphic letters such as $\mathcal{Y}$ and $\mathcal{X}$ denote the supports of $Y$ and $X$; and $\mathcal{YX}$ denotes the joint support of $(Y,X)$. The description of all the stages includes individual weights $e_{i}$ which are set to $1$ for the estimators, or drawn from a distribution that satisfies Assumption (ref) in Section (ref) for the weighted bootstrap version of the estimators.
The first stage estimates the $n$ target values of the control function, $ V_{i}=F_{X}(X_{i}\mid Z_{i})$, $i=1,\ldots ,n$. We estimate the conditional distribution of $X$ in a trimmed support $\overline{\mathcal{X}}$ that excludes extreme values. The purpose of the trimming is to avoid the far tails. We consider a fixed trimming rule, which greatly simplifies the derivation of the asymptotic properties. In our numerical and empirical examples we find that the results are not sensitive to the trimming rule and the choice of $\overline{\mathcal{X}}$ as the observed support of $X,$ i.e. no trimming, works well. We use bars to denote trimmed supports with respect to $X$, e.g., $\overline{\mathcal{X}\mathcal{Z}}=\{(x,z)\in \mathcal{X} \mathcal{Z}:x\in \overline{\mathcal{X}}\}$. A subscript in a set denotes a finite grid covering the set, where the subscript is the number of grid points. Unless otherwise specified, the points of the grid are sample quantiles of the corresponding variable at equidistant probabilities in $ [0,1]$. For example, $\mathcal{X}_{5}$ denotes a grid of $5$ points covering $\mathcal{X}$ located at the $0$, $1/4$, $1/2$, $3/4$ and $1$ sample quantiles of $X$.
Denoting the usual check function by $\rho _{v}(z)=(v-1(z<0))z$, the first stage in the QR baseline is
for some small constant $\epsilon >0$. The adjustment in the limits of the integral in ((ref)) avoids tail estimation of quantiles.\footnote{ Chernozhukov, Fernandez-Val and Melly (2013) provide conditions under which this adjustment does not introduce bias.} The first stage in the DR baseline is,
When $e_{i}=1$ for all $i=1,\dots,n$, expressions ((ref))-((ref)) and ((ref))-((ref)) define $\widehat{F}_{X}$, the QR and DR estimators of $F_{X}$. For $(X_{i},Z_{i})\in \overline{\mathcal{XZ}}$, the estimator and weighted bootstrap version of the control function are then $\widehat{V}_{i}=\widehat{ F}_{X}(X_{i}\mid Z_{i})$ and $\widehat{V}_{i}^{e}=\widehat{F} _{X}^{e}(X_{i}\mid Z_{i})$, respectively, and we set $\widehat{V}_{i}= \widehat{V}_{i}^{e}=0$ otherwise.
With the estimated control function in hand, the second building block required for the estimation of structural functions is an estimate of the reduced form CDF of $Y$ given $(X,Z_{1},V)$. The baseline models provide direct estimation procedures based on QR and DR.
Let $T := 1(X \in \overline{\mathcal{X}})$ be a trimming indicator, which is formally defined in Assumption (ref) of Section (ref). The estimator of $F_{Y}$ in the QR baseline is
As for the first stage, the adjustment in the limits of the integral in ((ref)) avoids tail estimation of quantiles. The estimator of $F_{Y}$ in the DR baseline is
When $e_{i}=1$ for all $i=1,\dots,n$, expressions ((ref))-((ref)) and ((ref))-((ref)) define $\widehat{F}_{Y}$, the quantile and distribution regression estimators of $F_{Y}$, respectively.
The weighted bootstrap versions of the DSF, QSF and ASF estimators are obtained by rerunning the estimation procedure introduced in Section (ref) with sampling weights drawn from a distribution that satisfies Assumption (ref) in Section (ref); see Algorithm (ref) in Appendix (ref) for details. They can then be used to perform uniform inference over the region of interest.
For instance, a $(1-\alpha )$-confidence band for the DSF over the region $ \mathcal{I}_G$ can be constructed as
where $\widehat{\sigma}_{G}(y,x)$ is an estimator of $\sigma _{G}(y,x),$ the asymptotic standard deviation of $\widehat{G}(y,x)$, such as the rescaled weighted bootstrap interquartile range\footnote{An alternative is to use the bootstrap standard deviation, but its validity requires convergence of bootstrap moments in addition to convergence of the bootstrap distribution; cf. Remark 3.2 in Chernozhukov et al. (2013) for a discussion.}
and $\widehat{k}_{G}(1-\alpha )$ denote a consistent estimator of the $ (1-\alpha )$-quantile of the maximal $t$-statistic
such as the $(1-\alpha )$-quantile of the bootstrap draw of the maximal $t$ -statistic
Confidence bands for the ASF can be constructed by a similar procedure, using the bootstrap draws of the ASF estimator. For the QSF, we can either use the same procedure based on the bootstrap draws of the QSF, or invert the confidence bands for the DSF following the generic method of Chernozhukov et al (2016). The first possibility works only when $Y $ is continuous, whereas the second method is more generally applicable. We provide algorithms for the construction of the bands in Appendix (ref).
We derive asymptotic theory for the estimators of the ASF, DSF and QSF where both the first and second stages are based on DR. The theory for the estimators based on QR can be derived using similar arguments.
The second assumption is about the first stage where we estimate the control function $(x,z) \mapsto \vartheta_{0}(x,z)$ defined as
with trimmed support $\overline{\mathcal{V}}=\{\vartheta_{0}(x,z):(x,z)\in \overline{\mathcal{X}\mathcal{Z}}\}$. We assume a logistic DR model for the conditional distribution of $X$ in the trimmed support $\overline{\mathcal{X}}$.
For $x\in\overline{\mathcal{X}}$, let
and set
if $(x,r)\in\overline{\mathcal{X}\mathcal{R}},$ and $\vartheta_{0}(x,r)= \widehat{\vartheta}^{e}(x,r)=0$ otherwise.
Theorem 4 of Chernozhukov, Fernandez-Val and Kowalski (2015) established the asymptotic properties of the DR estimator of the control function. We repeat the result here as a lemma for completeness and to introduce notation that will be used in the results below. Let $T(x):=1(x\in\overline{\mathcal{X}})$ , $\|f\|_{T,\infty}:=\sup_{a\in\mathcal{A}}|T(x)f(a)|$ for any function $f: \mathcal{A}\mapsto\mathbb{R}$, $\lambda=\Lambda(1-\Lambda)$, the density of the logistic distribution.
The next assumptions are about the second stage. We assume a logistic DR model for the conditional distribution of $Y$ given $(X,Z_{1},V)$, impose compactness and smoothness conditions, and provide sufficient conditions for identification of the parameters. Compactness is imposed over the trimmed supports and can be relaxed at the cost of more complicated and cumbersome proofs. The smoothness conditions are fairly tight. The assumptions on $ \mathcal{Y}$ cover continuous, discrete and mixed outcomes in the second stage. We denote partial derivatives as $\partial_x f(x,y) := \partial f(x,y)/\partial x.$
For $y\in\mathcal{Y}$, let
where
and $\widehat{\vartheta}$ is the estimator of the control function in the unweighted sample; and
where $\widehat{\vartheta}^{e}$ is the estimator of the control function in the weighted sample.
The following lemma establishes a functional central limit theorem and a functional central limit theorem for the bootstrap for the estimator of the DR coefficients in the second stage. Let $d_{w}:=\dim(W)$, and $ \ell^{\infty}(\mathcal{Y})$ be the set of all uniformly bounded real functions on $\mathcal{Y}$, and define the matrix $J(y) := {\mathrm{E}}_{P}[\lambda(W'\beta_{0}(y))WW' T]$ for $y\in \mathcal{Y}$. We use $\rightsquigarrow_{\mathbb{P}}$ to denote bootstrap consistency, i.e. weak convergence conditional on the data in probability, which is formally defined in Appendix (ref).
We consider now the estimators of the main quantities of interest -- the structural functions. Let $W_{x}:=w(x,Z_{1},V)$, $\widehat{W}_{x}:=w(x,Z_{1}, \widehat{V})$, and $\widehat{W}_{x}^{e}:=w(x,Z_{1},\widehat{V}^{e})$. The DR estimator and bootstrap draw of the DSF in the trimmed support, $G_T(y,x)={ \mathrm{E}}_{P}\{\Lambda[\beta_{0}(y)^{\prime }W_{x}] \mid T = 1\}$, are $ \widehat{G}(y,x)=\sum_{i=1}^{n}\Lambda[\widehat{\beta}(y)^{\prime }\widehat{W }_{xi}]T_{i}/n_T$, and $\widehat{G}^{e}(y,x)=\sum_{i=1}^{n}e_{i}\Lambda[ \widehat{\beta}^{e}(y)^{\prime }\widehat{W}_{xi}^{e}]T_{i}/n_T^e$. Let $p_T := P(T=1)$. The next result gives large sample theory for these estimators.
When $Y$ is continuous and $y\mapsto G_T(y,x)$ is strictly increasing, we can also characterize the asymptotic distribution of $\widehat{Q}(\tau,x)$, the estimator of the QSF in the trimmed support. Let $g_T(y,x)$ be the density of $y\mapsto G_T(y,x)$ , $\overline{\mathcal{T}}:=\{\tau\in(0,1):Q(\tau,x)\in \mathcal{Y}, g_T(Q(\tau,x),x)>\epsilon, x \in \overline{\mathcal{X}} \}$ for fixed $ \epsilon>0$, and $Q_{T}(\tau,x)$ the QSF in the trimmed support $\overline{ \mathcal{TX}}$ defined as
The estimator and its bootstrap draw given in (ref)- (ref) follow the functional central limit theorem:
Finally, we consider the ASF in the trimmed support
The estimator and its bootstrap draw given in (ref)- (ref) follow the functional central limit theorem:
In this section we apply our methods to the estimation of a semiparametric nonseparable triangular model for Engel curves. We focus on the structural relationship between household's total expenditure and household's demand for two goods: food and leisure. We take the outcome $Y$ to be the expenditure share on either food or leisure, and $X$ the logarithm of total expenditure. Following Blundell, Chen and Kristensen (2007) we use as an exclusion restriction the logarithm of gross earnings of the head of household. We also include an additional binary covariate $Z_{1}$ accounting for the presence of children in the household.
There is an extensive literature on Engel curve estimation (e.g., see Lewbel (2006) for a review), and the use of nonseparable triangular models for the identification and estimation of Engel curves has been considered in the recent literature. Blundell, Chen and Kristensen (2007) estimate semi-nonparametrically Engel curves for several categories of expenditure, Imbens and Newey (2009) estimate the QSF nonparametrically for food and leisure, and Chernozhukov, Fernandez-Val and Kowalski (2015) estimate Engel curves for alcohol accounting for censoring. For comparison purposes we use the same dataset as these papers, the 1995 U.K. Family Expenditure Survey. We restrict the sample to 1,655 married or cohabiting couples with two or fewer children, in which the head of the household is employed and between the ages of 20 and 55 years. For this sample we estimate the DSF, QSF and ASF for both goods. Unlike Imbens and Newey (2009) we also account for the presence of children in the household and we impose semiparametric restrictions through our baseline models. In contrast to Chernozhukov, Fernandez-Val and Kowalski (2015), we do not impose separability between the control function and other regressors, and we estimate the structural functions.
All structural functions are estimated by both QR and DR methods, following exactly the description of the implementation presented in Section (ref) with the specifications $r(Z) = (1,Z)^{\prime}$, $r_{1}(Z_{1})=(1,Z_{1})^{\prime}$, $p(X) = (1,X)^{\prime}$, and $q(V) = (1,\Phi^{-1}(V))^{\prime}$. We set $M=599$ and $ \epsilon=0.01$ in Algorithm (ref), approximate the integrals using $S=599$ points, and run $B=199$ bootstrap replications in Algorithm (ref) for both methods. The regions of interest are $\widetilde{\mathcal{X}}=[\widehat Q_{X}(0.1),\widehat Q_{X}(0.9)]$ and $\widetilde{\mathcal{Y}}=[\widehat Q_{Y}(0.1),\widehat Q_{Y}(0.9)]$, where $\widehat Q_{X}(u)$ and $\widehat Q_{Y}(u)$ are the sample $u$-quantiles of $X$ and $Y$. We approximate $ \widetilde{\mathcal{X}}$ by a grid $\widetilde{\mathcal{X}}_{K}$ with $K = 3,5$, and $\widetilde{\mathcal{Y}}$ by a grid $\widetilde{\mathcal{Y}}_{15}$ . We estimate the structural functions and perform uniform inference over the following regions:
We implement the DR estimator using the logit link function. Since the estimated DSF may be non-monotonic in $y$, we apply rearrangement to $ y\mapsto \widehat{G}(y,x)$ at each value of $x$ in $\mathcal{I}_G$. None of the methods uses trimming, that is we set $T=1$ a.s.
Figures (ref)-(ref) show the QSF, ASF and DSF for both goods \footnote{ For graphical representation the QSF and ASF are interpolated by splines over $\overline{\mathcal{X}}$ and the DSF over $\overline{\mathcal{Y}}$.}. For each structural function, we report weighted bootstrap 90%-confidence bands that are uniform over the corresponding region specified above. Our empirical results illustrate that QR and DR specifications are able to capture different features of structural functions, and are therefore complementary. For food, both estimation methods deliver very similar QSF estimates, close to being linear, although linearity is not imposed in the estimation procedure. For leisure, the QSF and ASF estimated by DR are able to capture some nonlinearity which is absent from those obtained by QR. For QR, this reflects the specified linear structure of the ASF which also constrains the shape of the QSF. In addition, some degree of heteroskedasticity appears to be a feature of the structural model for both goods, although much more markedly for leisure, so our methods are well-suited for this problem. Increased dispersion across quantile levels in Figure (ref) is reflected by the increasing spread across probability levels between the two extreme DSF estimates in Figure (ref). Finally, our semiparametric specifications are able to capture the asymmetry across leisure expenditure shares, an important feature of the data highlighted in Imbens and Newey (2009).
Our baseline models naturally allow for the inclusion of transformations of covariates - for instance spline transformations - in order to account for potential nonlinearities in data. In practice, these augmented specifications are useful to verify the robustness of the baseline specifications empirical findings. In order to illustrate nonlinear implementations of our approach and robustness of our baseline estimates, the QSF for food and leisure obtained by taking cubic B-splines transformations with 4 knots of log-total expenditure are shown in Figure (ref), for both DR and QR methods. A complete description of the structural stochastic relationship between total expenditure and food and leisure shares is then obtained, and confirms the essentially linear form of the QSF for food, as well as the nonlinearity already detected by DR for leisure in the empirical application - without the inclusion of nonlinear transformations of log-total expenditure.
Compared to existing studies of this dataset, the empirical results presented for the DSF are new. Our semiparametric estimates of the ASF and QSF capture the main features displayed by the nonparametric estimates of Imbens and Newey (2009), or those we obtain with more flexible specifications in Figure (ref). Moreover, our results and methods further make it possible to construct uniform confidence regions for structural functions, thereby providing applied researchers with useful inferential tools. These empirical results thus illustrate that our parsimonious models are able to capture complex features of the data, such as asymmetric distributions and nonlinear structural relationships, while leading to relatively easy-to-implement estimators and inferential methods that can be augmented straightforwardly for robustness checks and additional flexibility. This is demonstrated further in the Supplementary Material where we perform a thorough sensitivity analysis which further shows that our empirical results are robust to the modelling, estimation and integration choices.