Extracted main text — title through conclusion, appendix excluded. This is what our citation measures are computed over, published so the extraction can be checked by eye.
96,033 characters · 6 sections · 71 citation commands
Rate-Optimal Estimation of the Intercept in a Semiparametric Sample-Selection Model
\newtheorem{definition}{Definition}\newtheorem{hypothesis}{Hypothesis}\newtheorem{lemma}{Lemma}\newtheorem{proposition}{Proposition}\newtheorem{theorem}{Theorem}\newtheorem{corollary}{Corollary} \newtheorem{assumption}{Assumption}\newtheorem{remark}{Remark}\newtheorem{example}{Example}\newtheorem{condition}{Condition}
JEL {\it Classification:} C14, C31, C34
{\it Keywords:} sample selection, rate optimality, minimax efficiency, nearest neighbours
This paper considers a sample-selection model Heckman79 given by
where $[
]$ is an observed random vector, where $[
]$ is an unobserved random vector such that $E\left[U^2\right]<\infty$ and $E[U|\bm{X}]=0$ and $E[U|\bm{Z}]=E[U]$ almost surely. Equations~(\ref{outcome}) and (\ref{selection}) are typically referred to as the {\it outcome} and {\it selection} equations, respectively. Variants of the model given in (\ref{outcome})--(\ref{observation}) have been considered at least since the contributions of \citet{Gronau74,Heckman74} and \citet{Lewis74}. These authors were primarily concerned with the issue of selectivity bias in empirical analyses of individual labour-force participation decisions, particularly for women. This bias, which is present in least-squares estimates of the parameter $[
]$ appearing above in (\ref{outcome}), arises from the assumption that the observed wages of workers are affected by the self selection of those workers into the workforce. In particular, one can only observe a wage that exceeds the reservation wage of the individual in question. In terms of the model given in (\ref{outcome})--(\ref{observation}), the wage offer is represented by the variable $Y^*$, while the observed wage is denoted by $Y$. The employment status of the individual in question is denoted by the binary variable $D$, which takes a value of one if the unobserved difference $\bm{Z}^{\top}\bm{\gamma}_0-V$ between the wage offer and the reservation wage is positive; $D$ is otherwise equal to zero. Observed variables influencing individual participation decisions are collected in the vector $\bm{Z}$, while observed determinants of individual wage offers are collected in $\bm{X}$. Variants of the sample-selection model appearing in ((ref))--((ref)) have been found useful for a wide variety of applied problems apart from the analysis of indidivual labour supply decisions Vella98. Relatively recent economic applications include those of HelpmanMelitzRubinstein08, MulliganRubinstein08 and JimenezOngenaPeydroSaurina14.
This paper focuses on statistical inference regarding the intercept $\theta_0$ appearing above in ((ref)). The intercept in the outcome equation is often of inherent interest in various applications of the model given by ((ref))--((ref)). For example, suppose that ((ref)) accurately describes the selection of individuals into some treatment group. In this case, the difference between the intercepts in the outcome equations for treated and non-treated individuals may be interpreted as the causal effect of treatment when selection to treatment is mean independent of the unobservable $U$ in the outcome equation AndrewsSchafgans98. The intercept in the outcome equation is similarly crucial in computing the average wage difference in problems where the sample-selection model given above is applied to the analysis of wage differences between workers in two different socioeconomic groups, or between unionized and non-unionized workers Oaxaca73,Lewis86,Heckman90. Finally, the intercept in the outcome equation permits the evaluation of the net benefits of a social program in terms of the differences between the observed outcomes of participants and their counterfactual expected outcomes had they chosen not to participate HeckmanRobb85.
Early applied work generally proceeded from the assumption that the unobservables $[
]$ appearing above in (\ref{outcome})--(\ref{selection}) are bivariate normal mean-zero with an unknown covariance matrix and independent of $[
]$. This assumption in turn allowed for the estimation of the parameters appearing in (\ref{outcome})--(\ref{selection}) via the method of maximum likelihood or the related two-step procedure of \citet{Heckman74,Heckman76}. These estimates, however, are generally inconsistent under departures from the assumed bivariate normality of $[
]$ \citep[e.g.,][]{ArabmazarSchmidt82,Goldberger83,Schafgans04}. The desirability of not imposing a parametric specification on the joint distribution of the unobservables in the outcome and selection equations has led in turn to the development of distribution-free methods of estimating the parameters appearing above in (\ref{outcome})--(\ref{selection}). Distribution-free methods of estimating the intercept $\theta_0$ in ((ref)) include the proposals of GallantNychka87,Heckman90,AndrewsSchafgans98 and Lewbel07.
Estimators of the intercept of the outcome equation implemented by distribution-free procedures have large-sample behaviours that vary depending on the extent of endogeneity in the selection mechanism, i.e., on the nature and extent of any dependence between the random variables $U$ and $V$ appearing above in ((ref)) and ((ref)), respectively. Given that these features of the joint distribution of $[
]$ are typically unknown in empirical practice, the dependence of the asymptotic behaviour of intercept estimators on these features potentially complicates statistical inference regarding $\theta_0$. This issue is easily and starkly illustrated in the case of ordinary least squares (OLS). In particular, suppose that selection in the model is based strictly on observables, which is equivalent to assuming that the unobservable $U$ appearing in (\ref{outcome}) and the selection indicator $D$ are conditionally mean independent given $\bm{X}$ and $\bm{Z}$, i.e., $P\left[E[U|D=1,\bm{X},\bm{Z}]=E[U|D=0,\bm{X},\bm{Z}]\right]=1$. In this case $\theta_0$ can be consistently estimated at a parametric rate with no additional assumptions imposed on the joint distribution of $[
]$ by applying OLS to the outcome equation using only those observations for which $D=1$. On the other hand, the OLS estimate of $\theta_0$ obtained in this way is inconsistent if the difference $1-P\left[E[U|D=1,\bm{X},\bm{Z}]=E[U|D=0,\bm{X},\bm{Z}]\right]$ is positive, even if arbitrarily small. It follows that OLS generates inferences regarding $\theta_0$ that vary drastically with respect to the degree to which $E[U|D=1,\bm{X},\bm{Z}]$ may differ from $E[U|D=0,\bm{X},\bm{Z}]$.
This paper develops a distribution-free estimator of the intercept $\theta_0$ in the outcome equation that is consistent and asymptotically normal with a rate of convergence that is the same regardless of the joint distribution of $[
]$. I show that there exists an implementation of the proposed estimator of $\theta_0$ that converges uniformly at the rate $n^{-p/(2p+1)}$, where $n$ denotes the sample size and where $p\ge 2$ is an integer that indexes the strength of certain smoothness assumptions described below. The uniformity of this convergence involves uniformity over a class of joint distributions of $[
]$ satisfying necessary conditions for the identification of $\theta_0$. In other words, the estimator developed below is {\it adaptive} to the nature of selection in the model.
This paper also shows that the uniform $n^{-p/(2p+1)}$-rate attainable by the proposed estimator is in fact the optimal rate of convergence of an estimator of $\theta_0$ in terms of a minimax criterion. It follows that the proposed estimator may be implemented in such a way as to converge in probability to $\theta_0$ at the fastest possible minimax rate.
The estimator developed below differs from earlier proposals of Heckman90 and AndrewsSchafgans98 that involve the {\it rate-adaptive} estimation of the intercept in the outcome equation. These proposals also involve estimators that are consistent and asymptotically normal regardless of the extent to which selection is endogenous, but converge to the limiting normal distribution at unknown rates; in particular, see SchafgansZindeWalsh02 for the estimator of Heckman90 and AndrewsSchafgans98.
The estimator developed below also differs from estimators of $\theta_0$ that take the form of averages weighted by the reciprocal of an estimate of the density of the selection index $\bm{Z}^{\top}\bm{\gamma}_0$ appearing above in ((ref)). Such estimators Lewbel07, in common with the estimators of Heckman90 and AndrewsSchafgans98, are known to converge generically at unknown rates. In addition, estimators taking the form of inverse density-weighted averages may have sampling distributions that are not even asymptotically normal KhanTamer10,KhanNekipelov13,ChaudhuriHill16. In general, the estimators of Heckman90 and AndrewsSchafgans98 and of estimators in the class of inverse density-weighted averages converge at rates that depend critically on conditions involving the relative tail thicknesses of the distributions of the selection index $\bm{Z}^{\top}\bm{\gamma}_0$ and of the latent selection variable $V$ appearing in ((ref)) KhanTamer10. These conditions may be difficult to verify in applications. The estimator developed below converges by contrast at a known rate under conditions implied by the identification of $\theta_0$ to a normal distribution uniformly over the underlying parameter space regardless of the relative tail behaviours of $\bm{Z}^{\top}\bm{\gamma}_0$ and $V$. This facilitates statistical inference regarding $\theta_0$.
The remainder of this paper proceeds as follows. The following section discusses identification of the intercept parameter in the outcome equation and presents the new estimator along with its first-order asymptotic properties. Section (ref) derives the minimax rate optimality of the new estimator. Section (ref) presents the results of simulation experiments that investigate the behaviour in finite samples of the proposed estimator in relation to other methods. Section (ref) considers an application of the new estimator to the analysis of gender wage gaps in Malaysia. Section (ref) concludes. Proofs of all theoretical results are collected in the appendix.
This section presents the new estimator of the intercept $\theta_0$ appearing in ((ref)) and describes its asymptotic behaviour to first order. Let $\hat{\bm{\beta}}_n$ and $\hat{\bm{\gamma}}_n$ denote $\sqrt{n}$-consistent estimators of the parameters $\bm{\beta}_0$ and $\bm{\gamma}_0$ appearing above in ((ref)) and ((ref)), respectively, where $\bm{\gamma}_0$ is assumed to be identified up to a location and scale normalization. The existence of such estimators has long been established; see e.g., the proposals of Han87,Robinson88,PowellStockStoker89,Andrews91,IchimuraLee91,Ichimura93,KleinSpady93,Powell01 or Newey09. In addition, suppose that $\left\{[
]:\, i=1,\ldots,n\right\}$ are iid copies of the random vector $[
]$. Let $\bm{z}\in\mathbb{R}^{l}$ be an arbitrary vector, and define
Let
for each $i\in\{1,\ldots,n\}$. This paper proposes to estimate $\theta_0$ via a locally linear smoother of the form
where $\bm{e}_1=[
]^{\top}$ and where
and
for $i=1,\ldots,n$. The quantity $h_n$ appearing in each $K_i$ denotes a bandwidth $h_n>0$ such that $h_n\to 0$ with $nh_n^{3}\to\infty$ as $n\to\infty$, while for some $p\ge 2$, $K(\cdot)$ denotes a smoothing kernel of order $p$, i.e., one where $\int_{-\infty}^{\infty} K(u) du=1$, $\int_{-\infty}^{\infty} u^r K(u)=0$ for all $r\in \{1,\ldots,p-1\}$ and $\int_{-\infty}^{\infty} u^{p} K(u)<\infty$.
Assume that the disturbance $U$ in the outcome equation ((ref)) satisfies $E[|U|]<\infty$ with $P\left[E[U|\bm{Z}]=0\right]=1$. In addition, let the selection index $\bm{Z}^{\top}\bm{\gamma}_0$ be distributed with distribution $F_0$, assumed to be absolutely continuous. The estimator of $\theta_0$ given in ((ref)) exploits the fact that identification of $\theta_0$ occurs “at infinity” Chamberlain86, or in any case depends crucially on the selection index $\bm{Z}^{\top}\bm{\gamma}_0$ being able to take values sufficiently large so that the corresponding conditional probabilities of selection take values close to one. In particular, $\theta_0$ is characterized by the equalities
The proposed estimator of $\theta_0$ exploits the representation of the estimand in ((ref)), which suggests the estimation of $\theta_0$ by direct estimation of the quantity $E\left[\left.D\left(Y-\bm{X}^{\top}\bm{\beta}_0\right)\right|F_0\left(\bm{Z}^{\top}\bm{\gamma}_0\right)=1\right]$.
One can view the estimator given in ((ref)) as an extension of the Yang--Stute symmetrized nearest-neighbours (SNN) estimator of a conditional mean Yang81,Stute84 to the problem of estimating the intercept in the outcome equation ((ref)). SNN estimators are characterized by asymptotic behaviours that are asymptotically “design adaptive” in the sense that their asymptotic normality can generally be established without technical conditions on the probability of the design variable taking values in regions of low density Stute84. In the present context, it is shown under certain conditions that the estimator $\hat{\theta}_n$ in ((ref)) is asymptotically normal with a rate of convergence that depends neither on the extent to which the unobservables $U$ and $V$ are dependent, nor on the relationship between the behaviours of the selection index $\bm{Z}^{\top}\bm{\gamma}_0$ and the unobservable $V$ in the right tails of their respective marginal distributions. This is essentially accomplished by transforming the estimated indices $\bm{Z}^{\top}_i\hat{\bm{\gamma}}_n$ into random variables $\hat{\eta}_n(\bm{Z}_i)$ that are approximately uniformly distributed on $[0,1]$.
It is worth noting that estimators of $\theta_0$ that take the form of inverse density-weighted averages Lewbel07 have rates of convergence that generally vary with the extent to which $U$ and $V$ are dependent, as well as with the relative right-tail behaviours of $\bm{Z}^{\top}\bm{\gamma}_0$ and $V$ KhanTamer10. These estimators rely on an alternative representation of ((ref)) and are consistent under additional regularity conditions. In particular, suppose $m_0\left(\bm{z}^{\top}\bm{\gamma}_0\right)\equiv E\left[\left. D\left(Y-\bm{X}^{\top}\bm{\beta}_0\right)\right|\bm{Z}^{\top}\bm{\gamma}_0=\bm{z}^{\top}\bm{\gamma}_0\right]$ is everywhere differentiable in $\bm{z}^{\top}\bm{\gamma}_0$ with derivative given by $m_0^{(1)}\left(\bm{z}^{\top}\bm{\gamma}_0\right)\equiv \partial m_0\left(\bm{z}^{\top}\bm{\gamma}_0\right)/\partial\bm{z}^{\top}\bm{\gamma}_0$. Suppose in addition that the distribution of $\bm{Z}^{\top}\bm{\gamma}_0$ is absolutely continuous with density $f_0(\cdot)$ such that $E\left[\left|m_0^{(1)}\left(\bm{Z}^{\top}\bm{\gamma}_0\right)/f_0\left(\bm{Z}^{\top}\bm{\gamma}_0\right)\right|\right]<\infty$. One can then write
which suggests estimating $\theta_0$ via its approximate sample analogue in which $m_0^{(1)}(\cdot)$ and $f_0(\cdot)$ are replaced by suitable preliminary estimates, and in which the systematic trimming of observations corresponding to small values of $f_0\left(\bm{Z}^{\top}_i\bm{\gamma}_0\right)$ may be required. This approach to estimating $\theta_0$ follows that proposed by Lewbel97 for estimating a binary choice model arising from a latent linear model in which a mean restriction is imposed on the latent error term, and is the approach to estimating $\theta_0$ considered in more recent work by Lewbel07,KhanTamer10 and KhanNekipelov13. Consistent estimators of $\theta_0$ that exploit ((ref)) in this way naturally depend critically on the assumed finiteness of $E\left[\left|m_0^{(1)}\left(\bm{Z}^{\top}\bm{\gamma}_0\right)/f_0\left(\bm{Z}^{\top}\bm{\gamma}_0\right)\right|\right]$, an assumption that in turn leads the rates at which they converge to their limiting distributions to depend on the relative tail behaviours of the variables in the selection equation or the extent to which selection is endogenous KhanTamer10. The non-uniformity in the rate of convergence as one varies the relative tail behaviours of the determinants of selection or the dependence between the disturbance terms $U$ and $V$ is a feature that is also shared by estimators of $\theta_0$ that involve locally constant or polynomial regressions of $\hat{W}_i$ on the untransformed estimated selection indices $\bm{Z}^{\top}_i\hat{\bm{\gamma}}_n$ Heckman90,AndrewsSchafgans98. Results of this nature significantly complicate the task of statistical inference regarding $\theta_0$. By way of contrast, the transformations to $\hat{\eta}_n(\bm{Z}_i)$ of the selection indices used in the locally linear SNN estimator given in ((ref)) permit the locally linear SNN estimator to enjoy asymptotic normality with a rate of convergence that varies neither with the endogeneity of selection nor with the relative tail behaviours of the variables appearing in the selection equation.
It should also be noted that the result given above in ((ref)), in which the estimand is identified as $\theta_0=E\left[\left.D\left(Y-\bm{X}^{\top}\bm{\beta}_0\right)\right|F_0(\bm{Z}^{\top}\bm{\gamma}_0)=1\right]$, motivates the formulation of the estimator in ((ref)) as the intercept in a locally linear regression. One could as easily in this context use ((ref)) to motivate an estimator of $\theta_0$ as the corresponding variant of a Nadaraya--Watson (i.e., locally constant regression) estimator; see in particular the approach taken in StuteZhu05. The focus on a locally linear regression estimator of $\theta_0$ is purely to improve the rate at which the bias of the proposed estimator vanishes in large samples, given that locally linear regression estimators have biases that converge at the same rate regardless of whether the conditioning variable is evaluated at an interior or at a limit point of its support FanGijbels92. Nadaraya--Watson estimators, on the other hand, have biases that converge relatively slowly when the conditioning variable is evaluated at a limit point of its support.
Assumptions underlying the first-order asymptotic behaviour of the estimator given by $\hat{\theta}_n$ in ((ref)) are given as follows.
The conditions of Assumption (ref) are largely standard and notably suffice for the selection parameter $\bm{\gamma}_0$ to be identified up to the particular location and scale normalization imposed by Assumption (ref).(ref). Assumption (ref) also does not restrict the components $V$ and $\bm{Z}^{\top}\bm{\gamma}_0$ of the selection equation to be independent.
Assumption (ref) plays a crucial role in controlling the asymptotic bias of the proposed estimator $\hat{\theta}_n$. In particular, identification of $\bm{\gamma}_0$ subject to the conditions of Assumption (ref), along with the differentiability conditions of Assumption (ref).(ref)--(ref).(ref), imply a smoothness restriction on the conditional mean function
This smoothness restriction takes the form of differentiability of $m_{F_0}(q)$ for $q\in (0,1)$ up to order no less than $p$, where $p\ge 2$ is the constant specified in Assumption (ref).(ref), along with finiteness of the left-hand limit of $\left(d^p/dq^p\right) E\left[D\left(Y-\bm{X}^{\top}\bm{\beta}_0\right)\left|F_0\left(\bm{Z}^{\top}\bm{\gamma}_0\right)=q\right.\right]$ at $q=1$. This smoothness restriction, in other words, corresponds to a standard assumption in the literature on kernel estimation of conditional mean functions. On the other hand, the differentiability to $p^*$-order in the second argument of $g_{U,V|\bm{x},\bm{z}}(\cdot,\cdot)$ is slightly stronger than the usual assumption of differentiability to order $p$. This slight strengthening of the standard differentiability condition is used in the rate optimality arguments developed below in Section (ref). Details are contained in the proof of Theorem (ref) below.
The smoothness restriction on $m_{F_0}(q)$ given in ((ref)) can also be seen to be implied by the identification of $\bm{\gamma}_0$ up to a location and scale normalization and by the smoothness conditions imposed in Assumption (ref).(ref) on the conditional densities $g_{U,V|\bm{x},\bm{z}}(\cdot,\cdot)$ and $g_{V|\bm{x},\bm{z}}(\cdot,\cdot)$ for any $[
]$ in the support of $[
]$. In particular, one can show that under the conditions of Assumption~\ref{a1} and \ref{a2}.\ref{a2d}, $U$ has a conditional distribution given $F_0(V)=q$ for any $q\in [0,1]$ that is absolutely continuous with density
where $g_{UV}(\cdot,\cdot)$ and $g_V(\cdot)$ are respectively the joint density of $[
]$ and the marginal density of $V$. The conditional density $r_{U|Q}(u|q)$ is, given the absolute continuity of $F_0$ and the differentiability conditions on $g_{UV}$ and $g_V$ implied by Assumption~\ref{a2}.\ref{a2d}, $(p+1)$-times differentiable in $q$ on $(0,1)$ for any $u\in\mathbb{R}$. The $(p+1)$-times differentiability of $r_{U|Q}(u|q)$ in $q$ on $(0,1)$ in turn implies the finiteness of $\left.\left(\partial^p/\partial q^p\right) r_{U|Q}(u|q)\right|_{q=1}$ for any $u\in\mathbb{R}$. It is the finiteness of $\left.\left(\partial^p/\partial q^p\right) r_{U|Q}(u|q)\right|_{q=1}$ that implies the smoothness restrictions on $m_{F_0}(q)$ mentioned above. Further details are supplied below in Appendix (ref).
Let $\sigma^2_{U|F_0(\bm{Z}^{\top}\bm{\gamma}_0)}(q)\equiv E\left[U^2\left|F_0\left(\bm{Z}^{\top}\bm{\gamma}_0\right)=q\right.\right]$, where $U$ is the disturbance in the outcome equation ((ref)). The following result summarizes the large-sample behaviour to first order of the proposed estimator:
It follows from Theorem (ref) that the rate of convergence of $\hat{\theta}_n$ to its limiting normal distribution is unaffected by the dependence, if any, between the disturbance terms $U$ and $V$ in ((ref)) and ((ref)), respectively. The rate of convergence of $\hat{\theta}_n$ is also unaffected by the relative upper tail thicknesses of the distributions of $V$ and of the selection index $\bm{Z}^{\top}\bm{\gamma}_0$.
We also have from the statement of Theorem (ref) that a necessary condition for the consistency of $\hat{\theta}_n$ is the finiteness of the derivative $m^{(p)}_{F_0}(1)$. The finiteness of $m^{(p)}_{F_0}(1)$, as discussed above, is implied by the identification of $\bm{\gamma}_0$ up to a location and scale normalization as well as by the differentiability conditions specified in Assumptions (ref).(ref)--(ref).(ref). From this it follows that the consistency of the proposed estimator is implied by natural restrictions on the joint distribution of $[
]$. These distributional restrictions correspond collectively to a standard smoothness restriction in the literature on kernel estimation of conditional mean functions.
The presence of $m^{(p)}_{F_0}(1)$ in the bias term appearing in Theorem (ref), however, indicates that the approximate large-sample bias of $\hat{\theta}_n$ does depend on the extent to which $U$ is mean dependent on $V$. In particular, the conditional mean derivative $m^{(p)}_{F_0}(1)$ depends on the smoothness of the conditional mean $E\left[U\left|F_0(V)=q\right.\right]$ as a function of $q$ for values of $q$ near one; Appendix (ref) below contains further discussion. It is worth noting in this connection that $m^{(p)}_{F_0}(1)=0$ when the selection mechanism is exogenous to the extent that $U$ is mean independent of $V$, i.e., when $P\left[E[U|V]= E[U]\right]=1$.
The dependence of the approximate large-sample bias of $\hat{\theta}_n$ on the joint distribution of $[
]$ through the conditional mean derivative $m^{(p)}_{F_0}(1)$ can be ameliorated in practice by a judicious choice of variable bandwidth; see Corollary~\ref{cormain} below and the corresponding discussion and simulation evidence presented in Section~\ref{mc}. Theorem~\ref{mainthm} in any case indicates that the approximate large-sample bias, but not the variance, of the proposed estimator depends on the joint distribution of $[
]$. Theorem~\ref{mainthm} as such distinguishes the asymptotic behaviour of $\hat{\theta}_n$ from those of existing estimators of $\theta_0$ \citep[e.g.,][]{Heckman90,Lewbel07} whose biases and variances both depend on the joint distribution of $[
]$.
The following corollary is immediate from Theorem (ref):
Continue to let $p\ge 2$ be as specified above in Assumption (ref) in the previous section, and $n$ the sample size. This section shows that under the conditions of Assumptions (ref) and (ref), the rate $n^{-p/(2p+1)}$ is the fastest achievable, or {\it optimal}, rate of convergence of an estimator of the intercept $\theta_0$ in ((ref)). The optimality in question is relative to the convergence rates of all other estimators of $\theta_0$ and excludes by definition those estimators that are asymptotically superefficient at particular points in the underlying parameter space. The exclusion of superefficient estimators in this context notably rules out estimators that converge at the parametric rate of $n^{-1/2}$ under conditions stronger than those given above in Assumptions (ref) and (ref). For example, the OLS estimator of $\theta_0$ based on observations for which $D_i=1$ is superefficient for specifications of ((ref))--((ref)) over submodels in which the disturbance $U$ and the selection indicator $D$ are conditionally mean independent given $\bm{X}$ and $\bm{Z}$, i.e., models where $P\left[E[U|D=1,\bm{X},\bm{Z}]=E[U|D=0,\bm{X},\bm{Z}]=0\right]=1$. As noted in the Introduction, the OLS estimator of $\theta_0$ converges at the standard rate of $n^{-1/2}$ when $U$ and $D$ are conditionally mean independent given $\bm{X}$ and $\bm{Z}$ but is otherwise inconsistent.
The approach to optimality taken here follows that of Horowitz93, which was in turn based on the approach of Stone80. In particular, let $\left\{\Psi_n:\,n=1,2,3,\ldots\right\}$ denote a sequence of sets of the form
where $\bm{\psi}_1=(\theta,\bm{\beta}^{\top},\bm{\gamma}^{\top})^{\top}\in\mathbb{R}^{1+k+l}$ and where $g$ denotes the joint conditional density given $\bm{X}$ and $\bm{Z}$ of the disturbances $U$ and $V$ appearing above in ((ref)) and ((ref)), respectively. The quantity $\psi$ may depend generically on $n$.
Consider the observable random variables $D$, $Y$, $\bm{X}$ and $\bm{Z}$ appearing above in ((ref))--((ref)). Suppose that for each $n$, the joint conditional distribution of the vector $(D,Y)$ given $\bm{X}$ and $\bm{Z}$ is indexed by some $\psi\in\Psi_n$. Let $P_{\psi}[\cdot]\equiv P_{(\bm{\psi}_1,g)}[\cdot]$ denote the corresponding probability measure. Following Stone80, one may in this context define a constant $\rho>0$ to be an {\it upper bound} on the rate of convergence of estimators of the intercept parameter $\theta_0$ if for every estimator sequence $\left\{\theta_n\right\}$,
for all $s>0$, and if
where $\theta$ as it appears in ((ref)) and ((ref)) refers to the first component of the finite-dimensional component $\bm{\psi}_1$ of $\psi$.
In addition, define $\rho>0$ to be an {\it achievable} rate of convergence for the intercept parameter if there exists an estimator sequence $\left\{\theta_n\right\}$ such that
One calls $\rho>0$ the {\it optimal} rate of convergence for estimation of the intercept parameter if it is both an upper bound on the rate of convergence and achievable. In what follows, I first show that for large $n$, $p/(2p+1)$ is an upper bound on the rate of convergence. I then show that there exists an implementation of the estimator given in ((ref)) that attains the $n^{-p/(2p+1)}$-rate of convergence uniformly over $\Psi_n$ as $n\to\infty$.
The approach taken first involves the specification for each $n$ of a subset $\Psi^*_n$ of the parameter set $\Psi_n$ in which the finite-dimensional component $\bm{\psi}_1\equiv\bm{\psi}_{1n}$ lies in a shrinking neighbourhood $\Psi^*_{1n}$ of some point $[
]^{\top}\in\mathbb{R}^{1+k+l}$. In addition, the infinite-dimensional component $g$ is embedded in a curve (i.e., parametrization) indexed by a scalar $\psi_{2n}$ on a shrinking neighbourhood $\Psi^*_{2n}$ of a bivariate density function $g_0$ satisfying all relevant conditions of Assumptions~\ref{a1} and \ref{a2} for a conditional density of $U$ and $V$ given $\bm{X}$ and $\bm{Z}$.
In particular, consider a parametrization of the conditional joint density $g$ of $(U,V)$ given $\bm{X}$ and $\bm{Z}$ given by $g_{\psi_{2n}}$ for $\psi_2\in\Psi^*_{2n}$, where for some $\psi_{2n0}\in\Psi^*_{2n}$, we have $g_{\psi_{2n0}}(u,v|\bm{x},\bm{z})=g_0(u,v|\bm{x},\bm{z})$ for each $[
]\in\mathbb{R}^{2+k+l}$; i.e., the curve on $\Psi^*_{2n}$ given by $\psi_{2n}\to g_{\psi_{2n}}$ passes through the true conditional joint density $g_0$ at some point $\psi_{2n0}\in\Psi^*_{2n}$.
Now let $\Psi^*_n\equiv \Psi^*_{1n}\times\Psi^*_{2n}$. Let $s>0$ be arbitrary, and let $\left\{\theta_n\right\}$ denote an arbitrary sequence of estimators of $\theta_0$. Consider that if
then ((ref)) holds with $\rho=p/(2p+1)$. This is because the set $\Psi_n$ in ((ref)) contains the set over which the supremum is taken in ((ref)). Similarly, if
holds, then so does ((ref)).
It follows that proving ((ref)) and ((ref)) suffices to show that $p/(2p+1)$ is an upper bound on the rate of convergence; the key step in the proof is the specification of a suitable parametrization $\psi_{2n}\to g_{\psi_{2n}}$ for $\psi_{2n}\in\Psi^*_{2n}$. This is in fact the approach taken in Appendix (ref), which contains a proof of the following result:
Theorem (ref) implies that $p/(2p+1)$ is an upper bound on the rate of convergence of an estimator sequence $\left\{\theta_n\right\}$ in the minimax sense of ((ref)) and ((ref)) above.
Next, it is shown that $p/(2p+1)$ is an achievable rate of convergence in the sense of ((ref)) by exhibiting an estimator sequence $\left\{\theta_n\right\}$ such that ((ref)) holds with $\rho=p/(2p+1)$. In this connection, let $\hat{\theta}_n^*$ denote the proposed estimator given above in ((ref)) implemented with a bandwidth $h_n^*=cn^{-1/(2p+1)}$ for some constant $c>0$. In this case, ((ref)) is satisfied with $\theta_n=\hat{\theta}^*_n$ and $\rho=p/(2p+1)$:
Theorems (ref) and (ref) jointly imply that $p/(2p+1)$ is the optimal rate of convergence for estimation of $\theta_0$.
This section reports the results of simulation experiments that compare the finite-sample behaviour of the estimator in ((ref)) to the behaviours of alternative estimators. The simulations involved:
Each simulation experiment involved 1000 replicated samples of sizes $n\in\{100,400\}$ from the model given above in ((ref))--((ref)), where the parameter of interest was fixed at $\theta_0=1$, the variance of the unobservable term in the selection equation was fixed at $Var[V]=1$ and where for some constant $\rho\in [-1,1]$, the unobservable term in the outcome equation was specified as $U=\rho V+E$ for a random variable $E$ independent of $V$, where $E\sim N\left(0,1-\rho^2\right)$. The parameter $\rho$ in this case is by construction the correlation coefficient between $U$ and $V$. The simulations considered the settings $\rho\in\{0,.25,.50,.75,.95\}$.
In addition, the vector $\bm{Z}$ of observable predictors of selection was taken to be $l$-variate with $\bm{Z}=[
]^{\top}$, while the vector $\bm{X}$ of outcome predictors was specified to be $k$-variate with $k<l$ and $\bm{X}=[
]^{\top}$. The coefficient vector attached to $\bm{X}$ was set to $\bm{\beta}_0=\bm{\iota}_k$, i.e., the $k$-dimensional unit vector. The simulations imposed the settings $l=7$, $k=4$, which were intentionally set to equal the dimensions of the corresponding vectors appearing in the empirical example used below in Section (ref).
The selection index $\bm{Z}^{\top}\bm{\gamma}_0$ and the selection-equation disturbance term were simulated from two data-generating processes (DGPs), considered in turn. The distributions of $\bm{Z}^{\top}\bm{\gamma}_0$ under both DGPs, as required by Assumption (ref), are absolutely continuous. In addition, the parameter $\alpha>0$ used in the specification of both models is defined so as to index the degree to which the parameter of interest $\theta_0$ is identified. In particular, $\alpha\ge 1$ can be seen in this context to be a necessary condition for the identification of $\theta_0$, with $\alpha\ge 1$ corresponding to the case where the transformation $F_0(V)$ has an absolutely continuous distribution with density given by the ratio $r_Q(q)$ defined in ((ref)) below. Values of $\alpha<1$, on the other hand, correspond to the non-identifiability of $\theta_0$, with $\alpha\in (0,1)$ in the context of either of the following two DGPs implying failure of the condition that $F_0(V)$ have an absolutely continuous distribution supported on $[0,1]$. In particular, $\alpha\in (0,1)$ in the following two DGPs implies that the quantity $r_Q(q)$ given in ((ref)) below has the property that $r_Q(1)=\infty$:
The simulations under both DGPs considered the settings $\alpha\in\{2.00,1.50,1.25,1.00\}$. The effect of variation in the correlation coefficient $\rho$ and the parameter $\alpha$ on estimation of the intercept was a primary focus of these simulations. The outcome-equation nuisance parameter $\bm{\beta}_0$ and the selection parameter $\bm{\gamma}_0$ were accordingly fixed at their true values in these simulations in order to provide a clearer picture of the effects of variation in $\rho$ and $\alpha$ on the behaviour of the various intercept estimators considered.
The proposed intercept estimator given above in ((ref)) was implemented with a standard (i.e., second-order) Epanechnikov kernel. The bandwidth used to implement $\hat{\theta}_n$ was initially set to the sample analogue of the theoretical asymptotic MSE-optimal bandwidth $h^*_n$ specified above in Corollary (ref). In particular, the simulations involved the bandwidth $\hat{h}^*_n$, where $\hat{h}^*_n$ was taken to be the sample analogue of $h^*_n$. As such, $\hat{h}^*_n$ was set to decay at the MSE-optimal rate of $n^{-1/5}$ corresponding to the order of kernel employed (i.e., $p=2$), while its leading constant was specified as the sample analogue of the leading constant appearing in $h^*_n$. The unknown parameters appearing in the leading constant of $h^*_n$ were estimated via auxiliary locally cubic regressions as described in FanGijbels96. The sensitivity of the proposed estimator's sampling behaviour to the choice of bandwidth was also assessed by considering implementations using the bandwidth settings $h_n=(2/3)\hat{h}_{n}^*$ and $h_n=(3/2)\hat{h}_{n}^*$.
Comparisons of the corresponding sampling behaviours in terms of squared bias, standard deviation and root mean-squared error (RMSE) over 1000 Monte Carlo replications for values of $(\rho,\alpha)\in\{0,.25,.50,.75,.95\}\times\{2.00,1.50,1.25,1.00\}$ are presented below for samples of size $n=100$ in Tables (ref) and (ref) for DGP1 and DGP2, respectively. The corresponding results for samples of size $n=400$ appear in Tables (ref) and (ref). The RMSE figures displayed in these tables are multiplied by $\sqrt{n}$ in order to provide a clearer indication of the rate of convergence of the proposed estimator. The increases in $\sqrt{n}\times$ RMSE as one moves from simulated samples of size $n=100$ to those of $n=400$ indicate the slower-than-parametric rate of convergence of the proposed estimator regardless of the precise setting of $(\rho,\alpha)$. It is also clear that the rate of convergence of the proposed estimator is slower for settings of $(\rho,\alpha)$ with one or both of $\rho$ and $\alpha$ close to one. In addition, and as predicted by Theorem (ref) above, one can see that the effect of variation in $(\rho,\alpha)$ on the squared bias of the proposed estimator is more pronounced than the corresponding effect on the variance; indeed the standard deviation of the proposed estimator tends to be relatively stable over the various settings of $(\rho,\alpha)$ used in the simulations, particularly for values of $(\rho,\alpha)\in\{.25,.50,.75\}\times\{2.00,1.50,1.25\}$. Finally, the tabulated results indicate that the sampling performance of $\hat{\theta}_n$ is not sensitive to moderate variations in bandwidth.
\FloatBarrier
\FloatBarrier
I next consider the simulated performances over 1000 Monte Carlo replications across DGPs, sample sizes and settings of $(\rho,\alpha)$ of several alternative estimators of the intercept $\theta_0$. The results for samples of size $n=100$ are summarized below in Tables (ref) and (ref) for DGPs 1 and 2, respectively. The corresponding results for samples of size $n=400$ appear in Tables (ref) and (ref). The standard Heckman 2-step estimator is found under DGP1 to have a performance in terms of RMSE that is comparable to that of the proposed estimator in ((ref)). The proposed estimator under DGP2, on the other hand, is found to dominate in terms of RMSE the performance of the following alternative estimators under all combinations of $(\rho,\alpha)$ considered:
\FloatBarrier
In summary, the simulations presented here show the potential of the proposed estimator to exhibit good performance in terms of RMSE across two different parametric families of data-generating process, across variation in the degree to which the errors $U$ and $V$ are dependent and across variation in the extent to which the parameter of interest is identified. This assessment is unaffected by moderate variation in the estimated MSE-optimal bandwidth used to implement the proposed estimator. Tables (ref) and (ref) also support the conclusion of Theorem (ref) in indicating the sensitivity of the bias of the proposed estimator to variation in the parameter $(\rho,\alpha)$ under both DGP1 and DGP2. These results also show that the estimated MSE-optimal bandwidth used to implement the proposed estimator was effective in limiting the extent to which the RMSE of the proposed estimator was sensitive to variation in $(\rho,\alpha)$. In particular, the RMSE of the proposed estimator was found under DGP2 to dominate those of the other estimators considered.
This section reconsiders individual labour-market data from Malaysia that were originally analyzed by Schafgans00. The estimator developed above is applied to the problem of estimating the extent of plausible gender wage discrimination in Malaysia using data from the Second Malaysian Family Life Survey (MFLS2) conducted between August 1988 and January 1989. Inferences available from the proposed estimator are compared with those obtained via the same alternative estimators considered in Section (ref). The proposed estimator was found to generate inferences that differ significantly from those obtainable via the alternative estimators used in the simulation experiments described in Section (ref). All estimators applied to the MFLS2 data considered in this section were implemented in precisely the same way in which they were implemented in the simulation experiments presented earlier.
I consider a decomposition of the female--male log-wage difference for ethnic Malay workers. The consideration of gender wage gaps for Malaysian workers of the same ethnicity is potentially important because of the differential treatment of Malays in the labour force after 1970 Schafgans00. I follow Schafgans00 by analyzing gender wage gaps in the MFLS2 using the basic decomposition technique of Oaxaca73. In particular, suppose that the generic model given above in ((ref))--((ref)) holds for both men and women, i.e.,
where $Y^*_j$ is the natural logarithm of the offered average hourly wage, and where the index $j\in\{0,1\}$ denotes a given gender. For $j\in\{0,1\}$ let $\bar{Y}_j\equiv E\left[\left.Y_j\right|D_j=1\right]$, and let $\bar{\bm{X}}_j$ denote the average “endowments” of wage-determining attributes for workers of gender $j$. The observed log-wage gap $\bar{Y}_1-\bar{Y}_0$ between the two genders can then be decomposed as
where $A$ is that part of the gap due to differences in wage structures between genders; $B$ is due to observable differences between men and women in wage-determining characteristics and $C$ is the contribution of differential self-selection into the labour force. Following Schafgans00 the quantity $\bar{Y}_1-\bar{Y}_0-C=A+B$ is referred to as the {\it selection-corrected log-wage gap}.
Wage discrimination in favor of members of gender $j=1$ is empirically plausible if the overall log-wage gap $\bar{Y}_1-\bar{Y}_0$ cannot be entirely explained by differential self-selection into paid work, differences in observed endowments or by differing returns to those endowments. Moreover, given the definitions of the quantities $A$ and $B$ appearing above in ((ref)), the extent of plausible wage discrimination favoring gender 1 may be equated with the difference in intercepts $\theta_1-\theta_0$.
The analysis that follows considers a subset of the sample taken from the MFLS2 of 1988--89 that was analyzed by Schafgans00. This particular dataset is publicly available from the {\it Journal of Applied Econometrics} Data Archive at \url{http://qed.econ.queensu.ca/jae/1998-v13.5/schafgans/}. Each observation in this sample corresponds to a member of the labour force. I specifically consider ethnic Malays residing in non-urban settings who were observed to have some level of unearned household income in terms of dividends, interests or rents, and who were also observed to have passed the highest level of schooling (i.e., primary on the one hand, or secondary or above) corresponding to the number of years of schooling observed. This subset of the MFLS2 consisted of 965 women and 878 men.
I also use the same variable specifications used by Schafgans00. In particular, the outcome variable $Y^*_j$ is LWAGE, the log hourly real wage in the local currency deflated using the 1985 consumer price index. The selection variable $D_j$ is the indicator PAIDWORK for whether the individual in question is in fact a wage worker. The exogenous variables appearing in the selection equations for each gender include UNEARN, a measure of household unearned income in terms of dividends, interest and rents; HOUSEH, the value of household real estate owned, computed as the product of an indicator variable for house ownership and the cost of the house owned; and AMTLAND, the extent of household landholding in hundreds of acres. In addition, selection into wage work is also assumed to be determined by AGE, in years; AGESQ, the square of AGE divided by 100; YPRIM, years of primary schooling and YSEC, years of schooling at the secondary level or above. The variables appearing on the right-hand side of the outcome equations for each gender or ethnic group are AGE, AGESQ, YPRIM and YSEC. Schafgans00 contains further details regarding variable definitions.
For each gender $j\in\{\mbox{female, male}\}$, the proposed estimator and that of Heckman90 and AndrewsSchafgans98 rely on the preliminary procedure described in Schafgans98 to estimate the nuisance parameters $\bm{\beta}_j$ and $\bm{\gamma}_j$ appearing in ((ref)) and ((ref)), respectively. This involves estimating the selection equation for each group via the method of KleinSpady93 and estimating the slope parameters in each outcome equation using the method of Robinson88. This is followed by estimation of the intercept parameter in each outcome equation via the proposed estimator. Standard errors are calculated by bootstrapping with replacement with $B=200$ replications.
Estimates of the outcome-equation parameters obtained via the proposed estimator are given in which the proposed estimator is implemented using the same kernel and estimated MSE-optimal bandwidth $\hat{h}^*_n$ used in the simulations reported in Section (ref). In common with the results given earlier in Section (ref), I also considered implementations of the proposed estimator in which the bandwidth was set to $h_n=(2/3)\hat{h}^*_n$ and $h_n=(3/2)\hat{h}^*_n$.
The decomposition of the observed gender log-wage gaps for ethnic Malay workers is presented in Table (ref). In keeping with the theory developed above, the focus is on the extent of plausible gender wage discrimination, which is identified with the difference between the estimated intercepts. A striking result is the evidence provided by the proposed estimator of positive wage discrimination in favor of women. In particular, Table (ref) indicates that all three implementations of the proposed estimator imply a large, positive and significant difference in intercepts, while the OLS and 2-step procedures generated estimated intercept differences that were both insignificant. The implementation of the H90 procedure with $b_n$ set to the .90-quantile of the estimated selection index generated a similarly insignificant estimate of the difference in intercepts. The other implementation of the H90 procedure, along with the AS98 procedure, proved to be numerically unstable. Table (ref) indicates that there exists a clear difference in inferences regarding the extent of gender wage discrimination amongst ethnic Malay workers between estimates generated by the proposed estimator and those generated by established procedures.
\FloatBarrier
This paper has developed a new estimator of the intercept of a sample-selection model in which the joint distribution of the unobservables and the selection index is unspecified. It has been shown that the new estimator can be made under mild conditions to converge in probability at an $n^{-p/(2p+1)}$-rate, where $p\ge 2$ is an integer that indexes the strength of certain smoothness assumptions as given above in Assumption (ref).(ref). This rate of convergence is shown to be the optimal rate of convergence for estimation of the intercept parameter in terms of a minimax criterion. The new estimator is under mild conditions consistent and asymptotically normal with a rate of convergence that is the same regardless of the joint distribution of the unobservables and the selection index. This differs from other proposals in the literature and is convenient in practice, as the extent to which selection is endogenous is typically unknown in applications. In addition, the rate of convergence of the new estimator, unlike those of better known estimators, does not depend on assumptions regarding the relative tail behaviours of the determinants of selection beyond those necessary for the identification of the estimand. This similarly facilitates statistical inference regarding the intercept. Simulations presented above show the potential accuracy of the proposed estimator relative to that of established procedures across different model specifications. An empirical example using individual labour-market data from Malaysia shows the potential of the proposed estimator to generate inferences regarding the extent of plausible gender wage discrimination that differ from those available from better known estimators.