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Nonparametric Instrumental Variable Estimation Under Monotonicity
Despite the pervasive use of linear instrumental variable methods in empirical research, their nonparametric counterparts are far from enjoying similar popularity. Perhaps two of the main reasons for this originate from the observation that point-identification of the regression function in the nonparametric instrumental variable (NPIV) model requires completeness assumptions, which have been argued to be strong (Santos2012) and non-testable (canay2013re), and from the fact that the NPIV model is ill-posed, which may cause regression function estimators in this model to suffer from a very slow, logarithmic rate of convergence (e.g. blundell2007).
In this paper, we explore the possibility of imposing shape restrictions to improve statistical properties of the NPIV estimators and to achieve (partial) identification of the NPIV model in the absence of completeness assumptions. We study the NPIV model
where $Y$ is a dependent variable, $X$ an endogenous regressor, and $W$ an instrumental variable (IV). We are interested in identification and estimation of the nonparametric regression function $g$ based on a random sample of size $n$ from the distribution of $(Y,X,W)$. We impose two monotonicity conditions: (i) monotonicity of the regression function $g$ (we assume that $g$ is increasing\footnote{All results in the paper hold also when $g$ is decreasing. In fact, as we show in Section (ref) the sign of the slope of $g$ is identified under our monotonicity conditions.}) and (ii) monotonicity of the reduced form relationship between the endogenous regressor $X$ and the instrument $W$ in the sense that the conditional distribution of $X$ given $W$ corresponding to higher values of $W$ first-order stochastically dominates the same conditional distribution corresponding to lower values of $W$ (the monotone IV assumption).
We show that these two monotonicity conditions together significantly change the structure of the NPIV model, and weaken its ill-posedness. In particular, we demonstrate that under the second condition, a slightly modified version of the sieve measure of ill-posedness defined in blundell2007 is bounded uniformly over the dimension of the sieve space, when restricted to the set of monotone functions; see Section (ref) for details. As a result, under our two monotonicity conditions, the constrained NPIV estimator that imposes monotonicity of the regression function $g$ possesses a fast rate of convergence in a large but slowly shrinking neighborhood of constant functions.
More specifically, we derive a new non-asymptotic error bound for the constrained estimator. The bound exhibits two regimes. The first regime applies when the function $g$ is not too steep, and the bound in this regime is independent of the sieve measure of ill-posedness, which slows down the convergence rate of the unconstrained estimator. In fact, under some further conditions, the bound in the first regime takes the following form: with high probability, $$ \|\widehat{g}^c - g\|_{2,t} \leq C\Big(\Big(\frac{K\log n}{n}\Big)^{1/2} + K^{-s}\Big) $$ where $\widehat{g}^c$ is the constrained estimator, $\|\cdot\|_{2,t}$ an appropriate $L^2$-norm, $K$ the number of series terms in the estimator $\widehat{g}^c$, $s$ the number of derivatives of the function $g$, and $C$ some constant; see Section (ref) for details. Thus, the constrained estimator $\widehat{g}^c$ has fast rate of convergence in the first regime, and the bound in this regime is of the same order, up to a log-factor, as that for series estimators of conditional mean functions. The second regime applies when the function $g$ is sufficiently steep. In this regime, the bound is similar to that for the unconstrained NPIV estimators. The steepness level separating the two regimes depends on the sample size $n$ and decreases as the sample size $n$ grows large. Therefore, for a given increasing function $g$, if the sample size $n$ is not too large, the bound is in its first regime, where the constrained estimator $\widehat{g}^c$ does not suffer from ill-posedness of the model. As the sample size $n$ grows large, however, the bound eventually switches to the second regime, where ill-posedness of the model undermines the statistical properties of the constrained estimator $\widehat{g}^c$ similarly to the case of the unconstrained estimator.
Intuitively, existence of the second regime of the bound is well expected. Indeed, if the function $g$ is strictly increasing, it lies in the interior of the constraint that $g$ is increasing. Hence, the constraint does not bind asymptotically so that, in sufficiently large samples, the constrained estimator coincides with the unconstrained one and the two estimators share the same convergence rate. In {\em finite samples}, however, the constraint binds with non-negligible probability even if $g$ is strictly increasing. The first regime of our {\em non-asymptotic} bound captures this finite-sample phenomenon, and improvements from imposing the monotonicity constraint on $g$ in this regime can be understood as a boundary effect. Importantly, and perhaps unexpectedly, we show that under the monotone IV assumption, this boundary effect is so strong that ill-posedness of the problem completely disappears in the first regime.\footnote{Even though we have established the result that ill-posedness disappears in the first regime under the monotone IV assumption, currently we do not know whether this assumption is necessary for the result.} In addition, we demonstrate via our analytical results as well as simulations that this boundary effect can be strong even far away from the boundary and/or in large samples.
Our simulation experiments confirm these theoretical findings and demonstrate dramatic finite-sample performance improvements of the constrained relative to the unconstrained NPIV estimator when the monotone IV assumption is satisfied. Imposing the monotonicity constraint on $g$ removes the estimator's non-monotone oscillations due to sampling noise, which in ill-posed inverse problems can be particularly pronounced. Therefore, imposing the monotonicity constraint significantly reduces variance while only slightly increasing bias.
In addition, we show that in the absence of completeness assumptions, that is, when the NPIV model is not point-identified, our monotonicity conditions have non-trivial identification power, and can provide partial identification of the model.
We regard both monotonicity conditions as natural in many economic applications. In fact, both of these conditions often directly follow from economic theory. Consider the following generic example. Suppose an agent chooses input $X$ (e.g. schooling) to produce an outcome $Y$ (e.g. life-time earnings) such that $Y=g(X) + \varepsilon$, where $\varepsilon$ summarizes determinants of outcome other than $X$. The cost of choosing a level $X=x$ is $C(x,W,\eta)$, where $W$ is a cost-shifter (e.g. distance to college) and $\eta$ represents (possibly vector-valued) unobserved heterogeneity in costs (e.g. family background, a family's taste for education, variation in local infrastructure). The agent's optimization problem can then be written as $$ X = \arg\max_x \left\{ g(x)+\varepsilon - c(x,W,\eta) \right\} $$ so that, from the first-order condition of this optimization problem,
if marginal cost are decreasing in $W$ (i.e. $\partial^2 c/\partial X \partial W \leq 0$), marginal cost are increasing in $X$ (i.e. $\partial^2 c/\partial X^2 > 0$), and the production function is concave (i.e. $\partial^2 g/\partial X^2 \leq 0$). As long as $W$ is independent of the pair $(\varepsilon,\eta)$, condition (ref) implies our monotone IV assumption and $g$ increasing corresponds to the assumption of a monotone regression function. Dependence between $\eta$ and $\varepsilon$ generates endogeneity of $X$, and independence of $W$ from $(\varepsilon,\eta)$ implies that $W$ can be used as an instrument for $X$.
Another example is the estimation of Engel curves. In this case, the outcome variable $Y$ is the budget share of a good, the endogenous variable $X$ is total expenditure, and the instrument $W$ is gross income. Our monotonicity conditions are plausible in this example because for normal goods such as food-in, the budget share is decreasing in total expenditure, and total expenditure increases with gross income. Finally, consider the estimation of (Marshallian) demand curves. The outcome variable $Y$ is quantity of a consumed good, the endogenous variable $X$ is the price of the good, and $W$ could be some variable that shifts production cost of the good. For a normal good, the Slutsky inequality predicts $Y$ to be decreasing in price $X$ as long as income effects are not too large. Furthermore, price is increasing in production cost and, thus, increasing in the instrument $W$, and so our monotonicity conditions are plausible in this example as well.
Both of our monotonicity assumptions are testable. For example, a test of the monotone IV condition can be found in lee2009fd. In this paper, we extend their results by deriving an {\em adaptive} test of the monotone IV condition, with the value of the involved smoothness parameter chosen in a data-driven fashion. This adaptation procedure allows us to construct a test with desirable power properties when the degree of smoothness of the conditional distribution of $X$ given $W$ is unknown. Regarding our first monotonicity condition, to the best of our knowledge, there are no procedures in the literature that consistently test monotonicity of the function $g$ in the NPIV model ((ref)). We consider such procedures in a separate project and, in this paper, propose a simple test of monotonicity of $g$ given that the monotone IV condition holds.
Matzkin:1994kx advocates the use of shape restrictions in econometrics and argues that economic theory often provides restrictions on functions of interest, such as monotonicity, concavity, and/or Slutsky symmetry. In the context of the NPIV model (ref), Freyberger:2013fk show that, in the absence of point-identification, shape restrictions may yield informative bounds on functionals of $g$ and develop inference procedures when the regressor $X$ and the instrument $W$ are discrete. Blundell:2013fk demonstrate via simulations that imposing Slutsky inequalities in a quantile NPIV model for gasoline demand improves finite-sample properties of the NPIV estimator. grasmair2013tr study the problem of demand estimation imposing various constraints implied by economic theory, such as Slutsky inequalities, and derive the convergence rate of a constrained NPIV estimator under an abstract projected source condition. Our results are different from theirs because we focus on non-asymptotic error bounds, with special emphasis on properties of our estimator in {\em the neighborhood} of the boundary, we derive our results under easily interpretable, low level conditions, and we find that our estimator does not suffer from ill-posedness of the problem in a large but slowly shrinking neighborhood of constant functions.
\paragraph{Other related literature.}
The NPIV model has received substantial attention in the recent econometrics literature. Newey:2003p2167, Hall:2005p2135, blundell2007, and darolles2011 study identification of the NPIV model (ref) and propose estimators of the regression function $g$. See horowitz2011, H14 for recent surveys and further references. In the mildly ill-posed case, Hall:2005p2135 derive the minimax risk lower bound in $L^2$-norm and show that their estimator achieves this lower bound. Under different conditions, chen2011er derive a similar bound for the mildly and the severely ill-posed case and show that the estimator by blundell2007 achieves this bound. Chen:2013fk establish minimax risk bounds in the sup-norm, again both for the mildly and the severely ill-posed case. The optimal convergence rates in the severely ill-posed case were shown to be logarithmic, which means that the slow convergence rate of existing estimators is not a deficiency of those estimators but rather an intrinsic feature of the statistical inverse problem.
There is also large statistics literature on nonparametric estimation of monotone functions when the regressor is exogenous, i.e. $W=X$, so that $g$ is a conditional mean function. This literature can be traced back at least to brunk1955gf. Surveys of this literature and further references can be found in yatchew1998gf, delecroix2000ee, and gijbels2004gf. For the case in which the regression function is both smooth and monotone, many different ways of imposing monotonicity on the estimator have been studied; see, for example, mukerjee1988, Cheng:1981fd, wright1981, friedman1984gf, ramsay1988fd, mammen1991fd, ramsay1998fg, mammen1999rr, hall2001fd, mammen2001ff, and dette2006gg. Importantly, under the mild assumption that the estimators consistently estimate the derivative of the regression function, the standard unconstrained nonparametric regression estimators are known to be monotone with probability approaching one when the regression function is strictly increasing. Therefore, such estimators have the same rate of convergence as the corresponding constrained estimators that impose monotonicity (mammen1991fd). As a consequence, gains from imposing a monotonicity constraint can only be expected when the regression function is close to the boundary of the constraint and/or in finite samples. Zhang:2002dq and Chatterjee:2013eu formalize this intuition by deriving risk bounds of the isotonic (monotone) regression estimators and showing that these bounds imply fast convergence rates when the regression function has flat parts. Our results are different from theirs because we focus on the endogenous case with $W\neq X$ and study the impact of monotonicity constraints on the ill-posedness property of the NPIV model which is absent in the standard regression problem.
\paragraph{Notation.}
For a differentiable function $f:\mathbb{R}\to\mathbb{R}$, we use $Df(x)$ to denote its derivative. When a function $f$ has several arguments, we use $D$ with an index to denote the derivative of $f$ with respect to corresponding argument; for example, $D_w f(w,u)$ denotes the partial derivative of $f$ with respect to $w$. For random variables $A$ and $B$, we denote by $f_{A,B}(a,b)$, $f_{A|B}(a,b)$, and $f_{A}(a)$ the joint, conditional and marginal densities of $(A,B)$, $A$ given $B$, and $A$, respectively. Similarly, we let $F_{A,B}(a,b)$, $F_{A|B}(a,b)$, and $F_{A}(a)$ refer to the corresponding cumulative distribution functions. For an operator $T:L^2[0,1]\to L^2[0,1]$, we let $\|T\|_2$ denote the operator norm defined as $$ \|T\|_2 = \sup_{h\in L^2[0,1]:\,\|h\|_2=1} \|Th\|_2. $$ Finally, by increasing and decreasing we mean that a function is non-decreasing and non-increasing, respectively.
\paragraph{Outline.}
The remainder of the paper is organized as follows. In the next section, we analyze ill-posedness of the model ((ref)) under our monotonicity conditions and derive a useful bound on a restricted measure of ill-posedness for the model ((ref)). Section (ref) discusses the implications of our monotonicity assumptions for estimation of the regression function $g$. In particular, we show that the rate of convergence of our estimator is always not worse than that of unconstrained estimators but may be much faster in a large, but slowly shrinking, neighborhood of constant functions. Section (ref) shows that our monotonicity conditions have non-trivial identification power. Section (ref) provides new tests of our two monotonicity assumptions. In Section (ref), we present results of a Monte Carlo simulation study that demonstrates large gains in performance of the constrained estimator relative to the unconstrained one. Finally, Section (ref) applies the constrained estimator to the problem of estimating gasoline demand functions. All proofs are collected in the appendix.
In this section, we discuss the sense in which the ill-posedness of the NPIV model (ref) is weakened by imposing our monotonicity conditions. In particular, we introduce a restricted measure of ill-posedness for this model (see equation (ref)) and show that, in stark contrast to the existing literature, our measure is bounded (Corollary (ref)) when the monotone IV condition holds.
The NPIV model requires solving the equation ${\mathrm{E}}[Y|W] = {\mathrm{E}}[g(X)|W]$ for the function $g$. Letting $T:L^2[0,1]\to L^2[0,1]$ be the linear operator defined by $(Th)(w):={\mathrm{E}}[h(X) | W=w] f_W(w)$ and denoting $m(w) := {\mathrm{E}}[Y|W=w]f_W(w)$, we can express this equation as
In finite-dimensional regressions, the operator $T$ corresponds to a finite-dimensional matrix whose singular values are typically assumed to be nonzero (rank condition). Therefore, the solution $g$ is continuous in $m$, and consistent estimation of $m$ at a fast convergence rate leads to consistent estimation of $g$ at the same fast convergence rate. In infinite-dimensional models, however, $T$ is an operator that, under weak conditions, possesses infinitely many singular values that tend to zero. Therefore, small perturbations in $m$ may lead to large perturbations in $g$. This discontinuity renders equation ((ref)) ill-posed and introduces challenges in estimation of the NPIV model ((ref)) that are not present in parametric regressions nor in nonparametric regressions with exogenous regressors; see horowitz2011, H14 for a more detailed discussion.
In this section, we show that, under our monotonicity conditions, there exists a finite constant $\bar{C}$ such that for any monotone function $g'$ and any constant function $g''$, with $m'=T g'$ and $m''=T g''$, we have $$ \|g'-g''\|_{2,t} \leq \bar{C} \|m'-m''\|_2,$$ where $\|\cdot\|_{2,t}$ is a truncated $L^2$-norm defined below. This result plays a central role in our derivation of the upper bound on the restricted measure of ill-posedness, of identification bounds, and of fast convergence rates of a constrained NPIV estimator that imposes monotonicity of $g$ in a large but slowly shrinking neighborhood of constant functions.
We now introduce our assumptions. Let $0\leq x_1<\widetilde{x}_1<\widetilde{x}_2<x_2\leq 1$ and $0\leq w_1<w_2\leq 1$ be some constants. We implicitly assume that $x_1$, $\widetilde{x}_1$, and $w_1$ are close to $0$ whereas $x_2$, $\widetilde{x}_2$, and $w_2$ are close to $1$. Our first assumption is the monotone IV condition that requires a monotone relationship between the endogenous regressor $X$ and the instrument $W$.
Assumption (ref) is crucial for our analysis. The first part, condition (ref), requires first-order stochastic dominance of the conditional distribution of the endogenous regressor $X$ given the instrument $W$ as we increase the value of the instrument $W$. This condition (ref) is testable; see, for example, lee2009fd. In Section (ref) below, we extend the results of lee2009fd by providing an {\em adaptive} test of the first-order stochastic dominance condition ((ref)).
The second and third parts of Assumption (ref), conditions (ref) and (ref), strengthen the stochastic dominance condition (ref) in the sense that the conditional distribution is required to “shift to the right” by a {\em strictly} positive amount at least between two values of the instrument, $w_1$ and $w_2$, so that the instrument is not redundant. Conditions (ref) and (ref) are rather weak as they require such a shift only in some intervals $(0,x_2)$ and $(x_1,1)$, respectively.
Condition (ref) can be equivalently stated in terms of monotonicity with respect to the instrument $W$ of the reduced form first stage function. Indeed, by the Skorohod representation, it is always possible to construct a random variable $U$ distributed uniformly on $[0,1]$ such that $U$ is independent of $W$, and equation $X=r(W,U)$ holds for the reduced form first stage function $r(w,u):=F^{-1}_{X|W}(u|w):=\inf\{x: F_{X|W}(x|w)\geq u\}$. Therefore, condition ((ref)) is equivalent to the assumption that the function $w\mapsto r(w,u)$ is increasing for all $u\in[0,1]$. Notice, however, that our condition (ref) allows for general unobserved heterogeneity of dimension larger than one, for instance as in Example (ref) below.
Condition (ref) is related to a corresponding condition in Kasy2014tr who assumes that the (structural) first stage has the form $X=\widetilde{r}(W,\widetilde{U})$ where $\widetilde{U}$, representing (potentially multidimensional) unobserved heterogeneity, is independent of $W$, and the function $w\mapsto \widetilde{r}(w,\widetilde{u})$ is increasing for all values $\widetilde{u}$. Kasy employs his condition for identification of (nonseparable) triangular systems with multidimensional unobserved heterogeneity whereas we use our condition (ref) to derive a useful bound on the restricted measure of ill-posedness and to obtain a fast rate of convergence of a monotone NPIV estimator of $g$ in the (separable) model (ref). Condition (ref) is not related to the monotone IV assumption in the influential work by manski2000 which requires the function $w\mapsto {\mathrm{E}}[\varepsilon|W=w]$ to be increasing. Instead, we maintain the mean independence condition ${\mathrm{E}}[\varepsilon|W]=0$.
This is a mild regularity assumption. The first part of the assumption implies that the operator $T$ is compact. The second and the third parts of the assumption require the conditional distribution of $X$ given $W=w_1$ or $w_2$ and the marginal distribution of $W$ to be bounded away from zero over some intervals. Recall that we have $0\leq x_1<x_2\leq 1$ and $0\leq w_1<w_2\leq 1$. We could simply set $[x_1,x_2]=[w_1,w_2]=[0,1]$ in the second part of the assumption but having $0<x_1<x_2<1$ and $0<w_1<w_2<1$ is required to allow for densities such as the normal, which, even after a transformation to the interval $[0,1]$, may not yield a conditional density $f_{X|W}(x|w)$ bounded away from zero; see Example (ref) below. Therefore, we allow for the general case $0\leq x_1<x_2\leq 1$ and $0\leq w_1<w_2\leq 1$. The restriction $f_W(w)\leq C_W$ for all $w\in[0,1]$ imposed in Assumption (ref) is not actually required for the results in this section, but rather those of Section (ref).
We now provide two examples of distributions of $(X,W)$ that satisfy Assumptions (ref) and (ref), and show two possible ways in which the instrument $W$ can shift the conditional distribution of $X$ given $W$. Figure (ref) displays the corresponding conditional distributions.
Figure (ref) shows that, in Example (ref), the conditional distribution at two different values of the instrument is shifted to the right at every value of $X$, whereas, in Example (ref), the conditional support of $X$ given $W=w$ changes with $w$, but the positive shift in the cdf of $X | W=w$ occurs only for values of $X$ in a subinterval of $[0,1]$.
Before stating our results in this section, we introduce some additional notation. Define the truncated $L^2$-norm $\|\cdot\|_{2,t}$ by $$ \|h\|_{2,t} := \left( \int_{\widetilde{x}_1}^{\widetilde{x}_2} h(x)^2 d x \right)^{1/2},\quad h\in L^2[0,1]. $$ Also, let $\mathcal{M}$ denote the set of all monotone functions in $L^2[0,1]$. Finally, define $\zeta:=(c_f, c_W, C_F, C_T, w_1, w_2, x_1, x_2, \widetilde{x}_1, \widetilde{x}_2)$. Below is our first main result in this section.
To prove this theorem, we take a function $h\in\mathcal{M}$ with $\|h\|_{2,t}=1$ and show that $\|Th\|_2$ is bounded away from zero. A key observation that allows us to establish this bound is that, under monotone IV Assumption (ref), the function $w\mapsto {\mathrm{E}}[h(X)|W=w]$ is monotone whenever $h$ is. Together with non-redundancy of the instrument $W$ implied by conditions ((ref)) and ((ref)) of Assumption (ref), this allows us to show that ${\mathrm{E}}[h(X)|W=w_1]$ and ${\mathrm{E}}[h(X)|W=w_2]$ cannot both be close to zero so that $\|{\mathrm{E}}[h(X)|W=\cdot]\|_2$ is bounded from below by a strictly positive constant from the values of ${\mathrm{E}}[h(X)|W=w]$ in the neighborhood of either $w_1$ or $w_2$. By Assumption (ref), $\|T h\|_2$ must then also be bounded away from zero.
Theorem (ref) has an important consequence. Indeed, consider the linear equation (ref). By Assumption (ref)(i), the operator $T$ is compact, and so
Property (ref) means that $\|T h\|_2$ being small does not necessarily imply that $\|h\|_{2}$ is small and, therefore, the inverse of the operator $T: L^2[0,1]\to L^2[0,1]$, when it exists, cannot be continuous. Therefore, (ref) is ill-posed in Hadamard's sense\footnote{Well- and ill-posedness in Hadamard's sense are defined as follows. Let $A:D\to R$ be a continuous mapping between metric spaces $(D,\rho_D)$ and $(R,\rho_R)$. Then, for $d\in D$ and $r\in R$, the equation $Ad=r$ is called “well-posed” on $D$ in Hadamard's sense (see Hadamard:1923ty) if (i) $A$ is bijective and (ii) $A^{-1}:R\to D$ is continuous, so that for each $r\in R$ there exists a unique $d=A^{-1}r\in D$ satisfying $Ad=r$, and, moreover, the solution $d=A^{-1}r$ is continous in “the data” $r$. Otherwise, the equation is called “ill-posed” in Hadamard's sense.}, if no other conditions are imposed. This is the main reason why standard NPIV estimators have (potentially very) slow rate of convergence. Theorem (ref), on the other hand, implies that, under Assumptions (ref) and (ref), (ref) is not possible if $h_k$ belongs to the set $\mathcal{M}$ of monotone functions in $L^2[0,1]$ for all $k\geq 1$ and we replace the $L^2$-norm $\|\cdot\|_2$ in the numerator of the left-hand side of (ref) by the truncated $L^2$-norm $\|\cdot\|_{2,t}$, indicating that shape restrictions may be helpful to improve statistical properties of the NPIV estimators. Also, in Remark (ref), we show that replacing the norm in the numerator is not a significant modification in the sense that for most ill-posed problems, and in particular for all severely ill-posed problems, ((ref)) holds even if we replace $L^2$-norm $\|\cdot\|_2$ in the numerator of the left-hand side of ((ref)) by the truncated $L^2$-norm $\|\cdot\|_{2,t}$.
Next, we derive an implication of Theorem (ref) for the (quantitative) measure of ill-posedness of the model ((ref)). We first define the restricted measure of ill-posedness. For $a\in \mathbb{R}$, let $$ \mathcal{H}(a) := \left\{ h\in L^2[0,1] :\, \inf_{0\leq x'<x''\leq 1}\frac{h(x'')-h(x')}{x''-x'} \geq -a \right\} $$ be the space containing all functions in $L^2[0,1]$ with lower derivative bounded from below by $-a$ uniformly over the interval $[0,1]$. Note that $\mathcal{H}(a')\subset \mathcal{H}(a'')$ whenever $a'\leq a''$ and that $\mathcal{H}(0)$ is the set of increasing functions in $L^2[0,1]$. For continuously differentiable functions, $h\in L^2[0,1]$ belongs to $\mathcal{H}(a)$ if and only if $\inf_{x\in[0,1]}Dh(x)\geq -a$. Further, define the {\em restricted measure of ill-posedness}:
As we discussed above, under our Assumptions (ref) and (ref), $\tau(\infty)=\infty$ if we use the $L^2$-norm instead of the truncated $L^2$-norm in the numerator in ((ref)). We show in Remark (ref) below, that $\tau(\infty)=\infty$ for many ill-posed and, in particular, for all severely ill-posed problems even with the truncated $L^2$-norm as defined in (ref). However, Theorem (ref) implies that $\tau(0)$ is bounded from above by $\bar{C}$ and, by definition, $\tau(a)$ is increasing in $a$, i.e. $\tau(a')\leq \tau(a'')$ for $a'\leq a''$. It turns out that $\tau(a)$ is bounded from above even for some positive values of $a$:
This is our second main result in this section. It is exactly this corollary of Theorem (ref) that allows us to obtain a fast convergence rate of our constrained NPIV estimator $\widehat{g}^c$ not only when the regression function $g$ is constant but, more generally, when $g$ belongs to a large but slowly shrinking neighborhood of constant functions.
The rate at which unconstrained NPIV estimators converge to $g$ depends crucially on the so-called sieve measure of ill-posedness, which, unlike $\tau(a)$, does not measure ill-posedness over the space $\mathcal{H}(a)$, but rather over the space $\mathcal{H}_n(\infty)$, a finite-dimensional (sieve) approximation to $\mathcal{H}(\infty)$. In particular, the convergence rate is slower the faster the sieve measure of ill-posedness grows with the dimensionality of the sieve space $\mathcal{H}_n(\infty)$. The convergence rates can be as slow as logarithmic in the severely ill-posed case. Since by Corollary (ref), our monotonicity assumptions imply boundedness of $\tau(a)$ for some range of finite values $a$, we expect these assumptions to translate into favorable performance of a constrained estimator that imposes monotonicity of $g$. This intuition is confirmed by the novel non-asymptotic error bounds we derive in this section (Theorem (ref)).
Let $(Y_i,X_i,W_i)$, $i=1,\dots,n$, be an i.i.d. sample from the distribution of $(Y,X,W)$. To define our estimator, we first introduce some notation. Let $\{p_k(x),k\geq 1\}$ and $\{q_k(w),k\geq 1\}$ be two orthonormal bases in $L^2[0,1]$. For $K=K_n\geq 1$ and $J=J_n\geq K_n$, denote
Let $\mathbf{P} := (p(X_1),\ldots, p(X_n))'$ and $\mathbf{Q} := (q(W_1),\ldots, q(W_n))'$. Similarly, stack all observations on $Y$ in $\mathbf{Y}:= (Y_1,\ldots,Y_n)'$. Let $\mathcal{H}_n(a)$ be a sequence of finite-dimensional spaces defined by $$\mathcal{H}_n(a) := \left\{ h\in \mathcal{H}(a) :\, \exists b_1,\ldots, b_{K_n}\in\mathbb{R} \text{ with } h=\sum_{j=1}^{K_n} b_j p_j \right\}$$ which become dense in $\mathcal{H}(a)$ as $n\to\infty$. Throughout the paper, we assume that $\|g\|_2\leq C_b$ where $C_b$ is a large but finite constant known by the researcher. We define two estimators of $g$: the unconstrained estimator $\widehat{g}^u(x) := p(x)'\widehat{\beta}^u$ with
which is similar to the estimator defined in horowitz2012 and a special case of the estimator considered in blundell2007, and the constrained estimator $\widehat{g}^c(x):=p(x)'\widehat{\beta}^c$ with
which imposes the monotonicity of $g$ through the constraint $p(\cdot)'b \in \mathcal{H}_{n}(0)$.
To study properties of the two estimators we introduce a finite-dimensional, or sieve, counterpart of the restricted measure of ill-posedness $\tau(a)$ defined in (ref) and also recall the definition of the (unrestricted) sieve measure of ill-posedness. Specifically, define the {\em restricted} and {\em unrestricted} sieve measures of ill-posedness $\tau_{n,t}(a)$ and $\tau_n$ as $$\tau_{n,t}(a) := \sup_{h\in \mathcal{H}_n(a)\atop \|h\|_{2,t}=1} \frac{\|h\|_{2,t}}{\|Th\|_2}\quad\text{ and }\quad \tau_n:=\sup_{h\in\mathcal{H}_n(\infty)}\frac{\|h\|_{2}}{\|T h\|_2}.$$ The sieve measure of ill-posedness defined in blundell2007 and also used, for example, in horowitz2012 is $\tau_n$. blundell2007 show that $\tau_{n}$ is related to the singular values of $T$.\footnote{In fact, blundell2007 talk about the eigenvalues of $T^*T$, where $T^*$ is the adjoint of $T$ but there is a one-to-one relationship between eigenvalues of $T^*T$ and singular values of $T$.} If the singular values converge to zero at the rate $K^{-r}$ as $K\to\infty$, then, under certain conditions, $\tau_n$ diverges at a polynomial rate, that is $\tau_{n} = O(K_n^r)$. This case is typically referred to as “mildly ill-posed”. On the other hand, when the singular values decrease at a fast exponential rate, then $\tau_{n} = O(e^{cK_n})$, for some constant $c>0$. This case is typically referred to as “severely ill-posed”.
Our restricted sieve measure of ill-posedness $\tau_{n,t}(a)$ is smaller than the unrestricted sieve measure of ill-posedness $\tau_n$ because we replace the $L^2$-norm in the numerator by the truncated $L^2$-norm and the space $\mathcal{H}_n(\infty)$ by $\mathcal{H}_n(a)$. As explained in Remark (ref), replacing the $L^2$-norm by the truncated $L^2$-norm does not make a crucial difference but, as follows from Corollary (ref), replacing $\mathcal{H}_n(\infty)$ by $\mathcal{H}_n(a)$ does. In particular, since $\tau(a)\leq C_\tau$ for all $a\leq c_\tau$ by Corollary (ref), we also have $\tau_{n,t}(a)\leq C_\tau$ for all $a\leq c_\tau$ because $\tau_{n,t}(a)\leq \tau(a)$. Thus, for all values of $a$ that are not too large, $\tau_{n,t}(a)$ remains bounded uniformly over all $n$, no matter how fast the singular values of $T$ converge to zero.
We now specify conditions that we need to derive non-asymptotic error bounds for the constrained estimator $\widehat{g}^c(x)$.
Assumption (ref), along with Assumption (ref), is our main monotonicity condition. Assumption (ref) is a mild moment condition. Assumption (ref) requires that the dimension of the vector $q(w)$ is not much larger than the dimension of the vector $p(x)$. Let $s>0$ be some constant.
The first part of this condition requires the approximating function $g_n$ to be increasing. The second part requires a particular bound on the approximation error in the $L^2$-norm. devore1977gg,devore1977ok show that the assumption $\|g-g_n\|_2\leq C_g K^{-s}$ holds when the approximating basis $p_1,\dots,p_K$ consists of polynomial or spline functions and $g$ belongs to a H\"{o}lder class with smoothness level $s$. Therefore, approximation by monotone functions is similar to approximation by all functions. The third part of this condition is similar to Assumption 6 in blundell2007.
This condition is similar to Assumption 3(iii) in horowitz2012. Also, define the operator $T_n: L^2[0,1]\to L^2[0,1]$ by $$ (T_n h)(w):=q(w)'{\mathrm{E}}[q(W)p(X)']{\mathrm{E}}[p(U) h(U)],\qquad w\in[0,1] $$ where $U\sim U[0,1]$.
This condition is similar to Assumption 5 in horowitz2012. Finally, let
We start our analysis in this section with a simple observation that, if the function $g$ is strictly increasing and the sample size $n$ is sufficiently large, then the constrained estimator $\widehat{g}^c$ coincides with the unconstrained estimator $\widehat{g}^u$, and the two estimators share the same rate of convergence.
The result in Lemma (ref) is similar to that in Theorem 1 of mammen1991fd, which shows equivalence (in the sense of (ref)) of the constrained and unconstrained estimators of conditional mean functions. Lemma (ref) implies that imposing monotonicity of $g$ cannot lead to improvements in the rate of convergence of the estimator if $g$ is strictly increasing. However, the result in Lemma (ref) is asymptotic and only applies to the interior of the monotonicity constraint. It does not rule out faster convergence rates on or near the boundary of the monotonicity constraint nor does it rule out significant performance gains in finite samples. In fact, our Monte Carlo simulation study in Section (ref) shows significant finite-sample performance improvements from imposing monotonicity even if $g$ is strictly increasing and relatively far from the boundary of the constraint. Therefore, we next derive a {\em non-asymptotic} estimation error bound for the constrained estimator $\widehat{g}^c$ and study the impact of the monotonicity constraint on this bound.
This is the main result of this section. An important feature of this result is that since the constant $C$ depends only on the constants appearing in Assumptions (ref)-(ref), the bounds (ref) and (ref) hold uniformly over all data-generating processes that satisfy those assumptions with the same constants. In particular, for any two data-generating processes in this set, the same finite-sample bounds (ref) and (ref) hold with the same constant $C$, even though the unrestricted sieve measure of ill-posedness $\tau_n$ may be of different order of magnitude for these two data-generating processes.
Another important feature of the bound ((ref)) is that it depends on the restricted sieve measure of ill-posedness that we know to be smaller than the unrestricted sieve measure of ill-posedness, appearing in the analysis of the unconstrained estimator. In particular, we know from Section (ref) that $\tau_{n,t}(a)\leq \tau(a)$ and that, by Corollary (ref), $\tau(a)$ is uniformly bounded if $a$ is not too large. Employing this result, we obtain the bound ((ref)) of Theorem (ref).\footnote{Ideally, it would be of great interest to have a tight bound on the restricted sieve measure of ill-posedness $\tau_{n,t}(a)$ for all $a\geq 0$, so that it would be possible to optimize ((ref)) over $\delta$. Results of this form, however, are not yet available in the literature, and so the optimization is not possible.}
The bound (ref) has two regimes depending on whether the following inequality
holds. The most interesting feature of this bound is that in the first regime, when the inequality (ref) is satisfied, the bound is independent of the (unrestricted) sieve measure of ill-posedness $\tau_n$, and can be small if the function $g$ is not too steep, regardless of whether the original NPIV model ((ref)) is mildly or severely ill-posed. This is the regime in which the bound relies upon the monotonicity constraint imposed on the estimator $\widehat{g}^c$. For a given sample size $n$, this regime is active if the function $g$ is not too steep.
As the sample size $n$ grows large, the right-hand side of inequality (ref) decreases (if $K=K_n$ grows slowly enough) and eventually becomes smaller than the left-hand side, and the bound (ref) switches to its second regime, in which it depends on the (unrestricted) sieve measure of ill-posedness $\tau_n$. This is the regime in which the bound does not employ the monotonicity constraint imposed on $\widehat{g}^c$. However, since $\tau_n\to \infty$, potentially at a very fast rate, even for relatively large sample sizes $n$ and/or relatively steep functions $g$, the bound may be in its first regime, where the monotonicity constraint is important. The presence of the first regime and the observation that it is active in a (potentially very) large set of data generated processes provides a theoretical justification for the importance of imposing the monotonicity constraint on the estimators of the function $g$ in the NPIV model ((ref)) when the monotone IV Assumption (ref) is satisfied.
A corollary of the existence of the first regime in the bound (ref) is that the constrained estimator $\widehat{g}^c$ possesses a very fast rate of convergence in a large but slowly shrinking neighborhood of constant functions, independent of the (unrestricted) sieve measure of ill-posedness $\tau_n$:
The local-to-constant asymptotics considered in this corollary captures the finite sample situation in which the regression function is not too steep relative to the sample size. The convergence rate in this corollary is the standard polynomial rate of nonparametric conditional mean regression estimators up to a $(\log n)^{1/2}$ factor, regardless of whether the original NPIV problem without our monotonicity assumptions is mildly or severely ill-posed. One way to interpret this result is that the constrained estimator $\widehat{g}^c$ is able to recover regression functions in the shrinking neighborhood of constant functions at a fast polynomial rate. Notice that the neighborhood of functions $g$ that satisfy $\sup_{x\in[0,1]}Dg(x)=O((K\log n/n)^{1/2})$ is shrinking at a slow rate because $K\to\infty$, in particular the rate is much slower than $n^{-1/2}$. Therefore, in finite samples, we expect the estimator to perform well for a wide range of (non-constant) regression functions $g$ as long as the maximum slope of $g$ is not too large relative to the sample size.
In conclusion, in general, the convergence rate of the constrained estimator is the same as the standard minimax optimal rate, which depends on the degree of ill-posedness and may, in the worst-case, be logarithmic. This case occurs in the interior of the monotonicity constraint when $g$ is strictly monotone. On the other hand, under the monotone IV assumption, the constrained estimator converges at a very fast rate, independently of the degree of ill-posedness, in a large but slowly shrinking neighborhood of constant functions, a part of the boundary of the monotonicity constraint. In finite samples, we expect to experience cases between the two extremes, and the bounds (ref) and (ref) provide information on what the performace of the constrained estimator depends in that general case. Since the first regime of bound ((ref)) is active in a large set of data generating processes and sample size combinations, and since the fast convergence rate in Corollary (ref) is obtained in a large but slowly shrinking neighborhood of constant functions, we expect the boundary effect due to the monotonicity constraint to be strong even far away from the boundary and for relatively large sample sizes.
In the previous section, we derived non-asymptotic error bounds on the constrained estimator in the NPIV model (ref) assuming that $g$ is point-identified, or equivalently, that the linear operator $T$ is invertible. Newey:2003p2167 linked point-identification of $g$ to completeness of the conditional distribution of $X$ given $W$, but this completeness condition has been argued to be strong (Santos2012) and non-testable (canay2013re). In this section, we therefore discard the completeness condition and explore the identification power of our monotonicity conditions, which appear natural in many economic applications. Specifically, we derive informative bounds on the identified set of functions $g$ satisfying (ref). This means that, under our two monotonicity assumptions, the identified set is a proper subset of all monotone functions $g\in \mathcal{M}$.
By a slight abuse of notation, we define the sign of the slope of a differentiable, monotone function $f\in\mathcal{M}$ by $$sign(Df) := \left\{
\right. $$ and the sign of a scalar $b$ by $sign(b) := 1\{b>0 \} - 1\{b<0 \}$. We first show that if the function $g$ is monotone, the sign of its slope is identified under our monotone IV assumption (and some other technical conditions):
This theorem shows that, under certain regularity conditions, the monotone IV assumption and monotonicity of the regression function $g$ imply identification of the sign of the regression function's slope, even though the regression function itself is, in general, not point-identified. This result is useful because in many empirical applications it is natural to assume a monotone relationship between outcome variable $Y$ and the endogenous regressor $X$, given by the function $g$, but the main question of interest concerns not the exact shape of $g$ itself, but whether the effect of $X$ on $Y$, given by the slope of $g$, is positive, zero, or negative; see, for example, the discussion in abrevaya2010re).
It turns out that our two monotonicity assumptions possess identifying power even beyond the slope of the regression function.
The following theorem provides necessary conditions for observational equivalence.
Under Assumption (ref) that $g$ is increasing, Theorem (ref) suggests the construction of a set $\Theta'$ that includes the identified set $\Theta$ by $ \Theta' := \mathcal{M}_+\backslash \Delta,$ where $\mathcal{M}_+ := \mathcal{H}(0)$ denotes all increasing functions in $\mathcal{M}$ and
We emphasize that $\Delta$ is not empty, which means that our Assumptions (ref)--(ref) possess identifying power leading to nontrivial bounds on $g$. Notice that the constant $\bar{C}$ depends only on the observable quantities $c_w$, $c_f$, and $C_F$ from Assumptions (ref)--(ref), and on the known constants $\widetilde{x}_1$, $\widetilde{x}_2$, $x_1$, $x_2$, $w_1$, and $w_2$. Therefore, the set $\Theta'$ could, in principle, be estimated, but we leave estimation and inference on this set to future research.
In this section, we propose tests of our two monotonicity assumptions based on an i.i.d. sample $(X_i,W_i)$, $i=1,\dots,n$, from the distribution of $(X,W)$. First, we discuss an adaptive procedure for testing the stochastic dominance condition (ref) in our monotone IV Assumption (ref). The null and alternative hypotheses are
respectively. The null hypothesis, $H_0$, is equivalent to stochastic monotonicity of the conditional distribution function $F_{X|W}(x|w)$. Although there exist several good tests of $H_0$ in the literature (see lee2009fd, Delgado:2012fk and Lee:2014zl, for example), to the best of our knowledge there does not exist any procedure that adapts to the unknown smoothness level of $F_{X|W}(x|w)$. We provide a test that is adaptive in this sense, a feature that is not only theoretically attractive, but also important in practice: it delivers a data-driven choice of the smoothing parameter $h_n$ (bandwidth value) of the test whereas nonadaptive tests are usually based on the assumption that $h_n\to 0$ with some rate in a range of prespecified rates, leaving the problem of the selection of an appropriate value of $h_n$ in a given data set to the researcher (see, for example, lee2009fd and Lee:2014zl). We develop the critical value for the test that takes into account the data dependence induced by the data-driven choice of the smoothing parameter. Our construction leads to a test that controls size, and is asymptotically non-conservative.
Our test is based on the ideas in Chetverikov:2012fk who in turn builds on the methods for adaptive specification testing in horowitz2001ww and on the theoretical results on high dimensional distributional approximations in CCK1 (CCK). Note that $F_{X|W}(x|w)={\mathrm{E}}[1\{X\leq x\}|W=w]$, so that for a fixed $x\in(0,1)$, the hypothesis that $F_{X|W}(x|w')\geq F_{X|W}(x,w'')$ for all $0\leq w'\leq w''\leq 1$ is equivalent to the hypothesis that the regression function $w\mapsto {\mathrm{E}}[1\{X\leq x\}|W=w]$ is decreasing. An adaptive test of this hypothesis was developed in Chetverikov:2012fk. In our case, $H_0$ requires the regression function $w\mapsto {\mathrm{E}}[1\{X\leq x\}|W=w]$ to be decreasing not only for a particular value $x\in(0,1)$ but for all $x\in(0,1)$, and so we need to extend the results obtained in Chetverikov:2012fk.
Let $K:\mathbb{R}\to \mathbb{R}$ be a kernel function satisfying the following conditions:
We assume that the kernel function $K(w)$ has bounded support, is continuous, and is strictly positive on the support. The last condition excludes higher-order kernels. For a bandwidth value $h>0$, define $$ K_h(w):=h^{-1}K(w/h),\quad w\in\mathbb{R}. $$ Suppose $H_0$ is satisfied. Then, by the law of iterated expectations,
for all $x,w\in(0,1)$ and $i,j=1,\dots,n$. Denoting $$ K_{i j,h}(w):=\text{sign}(W_i-W_j)K_h(W_i-w)K_h(W_j-w), $$ taking the sum of the left-hand side in ((ref)) over $i,j=1,\dots,n$, and rearranging give $$ {\mathrm{E}}\left[\sum_{i=1}^n 1\{X_i\leq x\}\sum_{j=1}^n(K_{i j,h}(w)-K_{j i,h}(w))\right]\leq 0, $$ or, equivalently,
where $$ k_{i,h}(w):=\sum_{j=1}^n(K_{i j,h}(w)-K_{j i,h}(w)). $$
To define the test statistic $T$, let $\mathcal{B}_n$ be a collection of bandwidth values satisfying the following conditions:
The collection of bandwidth values $\mathcal{B}_n$ is a geometric progression with the coefficient $u\in(0,1)$, the largest value $1/2$, and the smallest value converging to zero not too fast. As the sample size $n$ increases, the collection of bandwidth values $\mathcal{B}_n$ expands.
Let $\mathcal{W}_n:=\{W_1,\dots,W_n\}$, and $\mathcal{X}_n:=\{\epsilon+l(1-2\epsilon)/n:l=0,1,\dots,n\}$ for some small $\epsilon>0$. We define our test statistic by
The statistic $T$ is most closely related to that in lee2009fd. The main difference is that we take the maximum with respect to the set of bandwidth values $h\in\mathcal{B}_n$ to achieve adaptiveness of the test.
We now discuss the construction of a critical value for the test. Suppose that we would like to have a test of level (approximately) $\alpha$. As succinctly demonstrated by lee2009fd, the derivation of the asymptotic distribution of $T$ is complicated even when $\mathcal{B}_n$ is a singleton. Moreover, when $\mathcal{B}_n$ is not a singleton, it is generally unknown whether $T$ converges to some nondegenerate asymptotic distribution after an appropriate normalization. We avoid these complications by employing the non-asymptotic approach developed in CCK and using a multiplier bootstrap critical value for the test. Let $e_1,\dots,e_n$ be an i.i.d. sequence of $N(0,1)$ random variables that are independent of the data. Also, let $\widehat{F}_{X|W}(x|w)$ be an estimator of $F_{X|W}(x|w)$ satisfying the following conditions:
This is a mild assumption implying uniform consistency of an estimator $\widehat{F}_{X|W}(x|w)$ of $F_{X|W}(x|w)$ over $(x,w)\in\mathcal{X}_n\times\mathcal{W}_n$. Define a bootstrap test statistic by $$ T^b:=\max_{(x,w,h)\in\mathcal{X}_n\times\mathcal{W}_n\times \mathcal{B}_n}\frac{\sum_{i=1}^ne_i\left(k_{i,h}(w)(1\{X_i\leq x\}-\widehat{F}_{X|W}(x|W_i))\right)}{\left(\sum_{i=1}^nk_{i,h}(w)^2\right)^{1/2}}. $$ Then we define the critical value\footnote{In the terminology of the moment inequalities literature, $c(\alpha)$ can be considered a “one-step” or “plug-in” critical value. Following Chetverikov:2012fk, we could also consider two-step or even multi-step (stepdown) critical values. For brevity of the paper, however, we do not consider these options here.} $c(\alpha)$ for the test as $$ c(\alpha):=(1-\alpha)\text{ conditional quantile of }T^b\text{ given the data}. $$
We reject $H_0$ if and only if $T>c(\alpha)$. To prove validity of this test, we assume that the conditional distribution function $F_{X|W}(x|w)$ satisfies the following condition:
The first theorem in this section shows that our test controls size asymptotically and is not conservative:
Our second result in this section concerns the ability of our test to detect models in the alternative $H_a$. Let $\epsilon>0$ be the constant appearing in the definition of $T$ via the set $\mathcal{X}_n$.
This theorem shows that our test is consistent against any model in $H_a$ (with smooth $F_{X|W}(x|w)$) whose deviation from $H_0$ is not on the boundary, so that the deviation $D_wF_{X|W}(x|w)>0$ occurs for $x\in(\epsilon,1-\epsilon)$. It is also possible to extend our results to show that Theorems (ref) and (ref) hold with $\epsilon=0$ at the expense of additional technicalities. Further, using the same arguments as those in Chetverikov:2012fk, it is possible to show that the test suggested here has minimax optimal rate of consistency against the alternatives belonging to certain H\"{o}lder classes for a reasonably large range of smoothness levels. We do not derive these results here for the sake of brevity of presentation.
We conclude this section by proposing a simple test of our second monotonicity assumption, that is, monotonicity of the regression function $g$. The null and alternative hypotheses are
respectively. The discussion in Remark (ref) reveals that, under Assumptions (ref) and (ref), monotonicity of $g(x)$ implies monotonicity of $w\mapsto {\mathrm{E}}[Y|W=w]$. Therefore, under Assumptions (ref) and (ref), we can test $H_0$ by testing monotonicity of the conditional expectation $w\mapsto {\mathrm{E}}[Y|W=w]$ using existing tests such as Chetverikov:2012fk and Lee:2014zl, among others. This procedure tests an implication of $H_0$ instead of $H_0$ itself and therefore may have low power against some alternatives. On the other hand, it does not require solving the model for $g(x)$ and therefore avoids the ill-posedness of the problem.
In this section, we study the finite-sample behavior of our constrained estimator that imposes monotonicity and compare its performance to that of the unconstrained estimator. We consider the NPIV model $Y=g(X) + \varepsilon$, ${\mathrm{E}}[\varepsilon | W]=0$, for two different regression functions, one that is strictly increasing and a weakly increasing one that is constant over part of its domain:
where $\varepsilon=\kappa\sigma_{\varepsilon}\bar{\varepsilon}$ and $\bar{\varepsilon} = \eta \epsilon + \sqrt{1-\eta^2}\nu$. The regressor and instrument are generated by $X=\Phi(\xi)$ and $W=\Phi(\zeta)$, respectively, where $\Phi$ is the standard normal cdf and $\xi = \rho \zeta + \sqrt{1-\rho^2}\epsilon$. The errors are generated by $(\nu,\zeta,\epsilon)\sim N(0,I)$.
We vary the parameter $\kappa$ in $\{1,0.5,0.1\}$ to study how the constrained and unconstrained estimators' performance compares depending on the maximum slope of the regression function. $\eta$ governs the dependence of $X$ on the regression error $\varepsilon$ and $\rho$ the strength of the first stage. All results are based on $1,000$ MC samples and the normalized B-spline basis for $p(x)$ and $q(w)$ of degree $3$ and $4$, respectively.
Tables (ref)--(ref) report the Monte Carlo approximations to the squared bias, variance, and mean squared error (“MSE”) of the two estimators, each averaged over a grid on the interval $[0,1]$. We also show the ratio of the constrained estimator's MSE divided by the unconstrained estimator's MSE. $k_X$ and $k_W$ denote, respectively, the number of knots used for the basis $p(x)$ and $q(w)$. The first two tables vary the number of knots, and the latter two the dependence parameters $\rho$ and $\eta$. Different sample sizes and different values for $\rho$, $\eta$, and $\sigma_{\varepsilon}$ yield qualitatively similar results. Figures (ref) and (ref) show the two estimators for a particular combination of the simulation parameters. The dashed lines represent confidence bands, computed as two times the (pointwise) empirical standard deviation of the estimators across simulation samples. Both, the constrained and the unconstrained, estimators are computed by ignoring the bound $\|b\|\leq C_b$ in their respective definitions. Horowitz2012as and horowitz2012 also ignore the constraint $\|b\|\leq C_b$ and state that it does not affect the qualitative results of their simulation experiment.
The MSE of the constrained estimator (and, interestingly, also of the unconstrained estimator) decreases as the regression function becomes flatter. This observation is consistent with the error bound in Theorem (ref) depending positively on the maximum slope of $g$.
Because of the joint normality of $(X,W)$, the simulation design is severely ill-posed and we expect high variability of both estimators. In all simulation scenarios, we do in fact observe a very large variance relative to bias. However, the magnitude of the variance differs significantly across the two estimators: in all scenarios, even in the design with a strictly increasing regression function, imposing the monotonicity constraint significantly reduces the variance of the NPIV estimator. The MSE of the constrained estimator is therefore much smaller than that of the unconstrained estimator, from about a factor of two smaller when $g$ is strictly increasing and the noise level is low ($\sigma_{\varepsilon}=0.1$), to around 20 times smaller when $g$ contains a flat part and the noise level is high ($\sigma_{\varepsilon}=0.7$). Generally, the gains in MSE from imposing monotonicity are larger the higher the noise level $\sigma_{\varepsilon}$ in the regression equation and the higher the first-stage correlation $\rho$.\footnote{Since Tables (ref) and (ref) report results for the lower level of $\rho$, and Tables (ref) and (ref) results for the lower noise level $\sigma_{\varepsilon}$, we consider the selection of results as, if at all, favoring the unconstrained estimator.}
In this section, we revisit the problem of estimating demand functions for gasoline in the United States. Because of the dramatic changes in the oil price over the last few decades, understanding the elasticity of gasoline demand is fundamental to evaluating tax policies. Consider the following partially linear specification of the demand function: $$Y = g(X,Z_1) + \gamma'Z_2 + \varepsilon,\qquad {\mathrm{E}}[\varepsilon | W,Z_1,Z_2]=0,$$ where $Y$ denotes annual log-gasoline consumption of a household, $X$ log-price of gasoline (average local price), $Z_1$ log-household income, $Z_2$ are control variables (such as population density, urbanization, and demographics), and $W$ distance to major oil platform. We allow for price $X$ to be endogenous, but assume that $(Z_1,Z_2)$ is exogenous. $W$ serves as an instrument for price by capturing transport cost and, therefore, shifting the cost of gasoline production. We use the same sample of size $4,812$ from the 2001 National Household Travel Survey and the same control variables $Z_2$ as blundell2012fg. More details can be found in their paper.
Moving away from constant price and income elasticities is likely very important as individuals' responses to price changes vary greatly with price and income level. Since economic theory does not provide guidance on the functional form of $g$, finding an appropriate parametrization is difficult. hausman1995 and blundell2012fg, for example, demonstrate the importance of employing flexible estimators of $g$ that do not suffer from misspecification bias due to arbitrary restrictions in the model. Blundell:2013fk argue that prices at the local market level vary for several reasons and that they may reflect preferences of the consumers in the local market. Therefore, one would expect prices $X$ to depend on unobserved factors in $\varepsilon$ that determine consumption, rendering price an endogenous variable. Furthermore, the theory of the consumer requires downward-sloping compensated demand curves. Assuming a positive income derivative\footnote{blundell2012fg estimate this income derivative and do, in fact, find it to be positive over the price range of interest.} $\partial g/\partial z_1$, the Slutsky condition implies that the uncompensated (Marshallian) demand curves are also downward-sloping, i.e. $g(\cdot,z_1)$ should be monotone for any $z_1$, as long as income effects do not completely offset price effects. Finally, we expect the cost shifter $W$ to monotonically increase cost of producing gasoline and thus satisfy our monotone IV condition. In conclusion, our constrained NPIV estimator appears to be an attractive estimator of demand functions in this setting.
We consider three benchmark estimators. First, we compute the unconstrained nonparametric (“uncon. NP”) series estimator of the regression of $Y$ on $X$ and $Z_1$, treating price as exogenous. As in blundell2012fg, we accommodate the high-dimensional vector of additional, exogenous covariates $Z_2$ by (i) estimating $\gamma$ by 198807's procedure, (ii) then removing these covariates from the outcome, and (iii) estimating $g$ by regressing the adjusted outcomes on $X$ and $Z_1$. The second benchmark estimator (“con. NP”) repeats the same steps (i)--(iii) except that it imposes monotonicity (in price) of $g$ in steps (i) and (iii). The third benchmark estimator is the unconstrained NPIV estimator (“uncon. NPIV”) that accounts for the covariates $Z_2$ in similar fashion as the first, unconstrained nonparametric estimator, except that (i) and (iii) employ NPIV estimators that impose additive separability and linearity in $Z_2$.
The fourth estimator we consider is the constrained NPIV estimator (“con. NPIV”) that we compare to the three benchmark estimators. We allow for the presence of the covariates $Z_2$ in the same fashion as the unconstrained NPIV estimator except that, in steps (i) and (iii), we impose monotonicity in price.
We report results for the following choice of bases. All estimators employ a quadratic B-spline basis with $3$ knots for price $X$ and a cubic B-spline with $10$ knots for the instrument $W$. Denote these two bases by $\mathbf{P}$ and $\mathbf{Q}$, using the same notation as in Section (ref). In step (i), the NPIV estimators include the additional exogenous covariates $(Z_1,Z_2)$ in the respective bases for $X$ and $W$, so they use the estimator defined in Section (ref) except that the bases $\mathbf{P}$ and $\mathbf{Q}$ are replaced by $\widetilde{\mathbf{P}} := [\mathbf{P}, \mathbf{P}\times \mathbf{Z}_1, \mathbf{Z}_2]$ and $\widetilde{\mathbf{Q}} := [\mathbf{Q}, \mathbf{Q}\times (\mathbf{Z}_1, \mathbf{Z}_2)]$, respectively, where $\mathbf{Z}_k := (Z_{k,1},\ldots,Z_{k,n})'$, $k=1,2$, stacks the observations $i=1,\ldots,n$ and $\mathbf{P}\times \mathbf{Z}_1$ denotes the tensor product of the columns of the two matrices. Since, in the basis $\widetilde{\mathbf{P}}$, we include interactions of $\mathbf{P}$ with $\mathbf{Z}_1$, but not with $\mathbf{Z}_2$, the resulting estimator allows for a nonlinear, nonseparable dependence of $Y$ on $X$ and $Z_1$, but imposes additive separability in $Z_2$. The conditional expectation of $Y$ given $W$, $Z_1$, and $Z_2$ does not have to be additively separable in $Z_2$, so that, in the basis $\widetilde{\mathbf{Q}}$, we include interactions of $\mathbf{Q}$ with both $\mathbf{Z}_1$ and $\mathbf{Z}_2$.\footnote{Notice that $\mathbf{P}$ and $\mathbf{Q}$ include constant terms so it is not necessary to separately include $\mathbf{Z}_k$ in addition to its interactions with $\mathbf{P}$ and $\mathbf{Q}$, respectively.}
We estimated the demand functions for many different combinations of the order of B-spline for $W$, the number of knots in both bases, and even with various penalization terms (as discussed in Remark (ref)). While the shape of the unconstrained NPIV estimate varied slightly across these different choices of tuning parameters (mostly near the boundary of the support of $X$), the constrained NPIV estimator did not exhibit any visible changes at all.
Figure (ref) shows a nonparametric kernel estimate of the conditional distribution of the price $X$ given the instrument $W$. Overall the graph indicates an increasing relationship between the two variables as required by our stochastic dominance condition (ref). We formally test this monotone IV assumption by applying our new test proposed in Section (ref). We find a test statistic value of $0.139$ and $95\%$-critical value of $1.720$.\footnote{The critical value is computed from $1,000$ bootstrap samples, using the bandwidth set $\mathcal{B}_n = \{2,1,0.5,0.25,0.125,0.0625\}$, and a kernel estimator for $\widehat{F}_{X|W}$ with bandwidth $0.3$ which produces the estimate in Figure (ref).} Therefore, we fail to reject the monotone IV assumption.
Figure (ref) shows the estimates of the demand function at three income levels, at the lower quartile ($\$ 42,500$), the median ($\$ 57,500$), and the upper quartile ($\$ 72,500$). The area shaded in grey represents the 90% uniform confidence bands around the unconstrained NPIV estimator as proposed in Horowitz2012as.\footnote{Critical values are computed from $1,000$ bootstrap samples and the bands are computed on a grid of $100$ equally-spaced points in the support of the data for $X$.} The black lines correspond to the estimators assuming exogeneity of price and the red lines to the NPIV estimators that allow for endogeneity of price. The dashed black line shows the kernel estimate of blundell2012fg and the solid black line the corresponding series estimator that imposes monotonicity. The dashed and solid red lines similarly depict the unconstrained and constrained NPIV estimators, respectively.
All estimates show an overall decreasing pattern of the demand curves, but the two unconstrained estimators are both increasing over some parts of the price domain. We view these implausible increasing parts as finite-sample phenomena that arise because the unconstrained nonparametric estimators are too imprecise. The wide confidence bands of the unconstrained NPIV estimator are consistent with this view. hausman1995 and Horowitz2012as find similar anomalies in their nonparametric estimates, assuming exogenous prices. Unlike the unconstrained estimates, our constrained NPIV estimates are downward-sloping everywhere and smoother. They lie within the $90\%$ uniform confidence bands of the unconstrained estimator so that the monotonicity constraint appears compatible with the data.
The two constrained estimates are very similar, indicating that endogeneity of prices may not be important in this problem, but they are both significantly flatter than the unconstrained estimates across all three income groups, which implies that households appear to be less sensitive to price changes than the unconstrained estimates suggest. The small maximum slope of the constrained NPIV estimator also suggests that the error bound in Theorem (ref) may be small and therefore we expect the constrained NPIV estimate to be precise for this data set.