Distribution Regression in Duration Analysis:
an Application to Unemployment Spells

Existing semiparametric duration models can be broadly classified into two groups: those based on the conditional hazard and those based on the quantile regression. The former includes the proportional hazard model (Cox, 1972, 1975), the proportional odds model (Clayton, 1976; Bennett, 1983; Murphy et al., 1997), and the accelerated failure time model (Kalbfleisch and Prentice, 1980); see Guo and Zeng (2014) for an overview. In these models, the conditional hazard identifies the conditional cumulative distribution expressed in terms of the error’s marginal distribution of a transformed failure time regression model (see, e.g., Hothorn et al., 2014). Censored quantile regression proposals are relatively more recent, see, e.g., Ying et al. (1995), Honore et al. (2002), Portnoy (2003), Peng and Huang (2008), and Wang and Wang (2009).

Conditional hazard and quantile regression models are alternative modeling strategies with advantages and drawbacks. Classical conditional hazard specifications impose identification conditions that are difficult to justify in some circumstances. For instance, the proportional hazard specification rules out important forms of heterogeneity, see, e.g., Portnoy (2003) and the discussion in Section 2. On the other hand, models based on quantile regression specifications avoid this problem, but impose that the underlying conditional cumulative distribution of duration time is absolutely continuous, which rules out other sampling schemes such as discrete outcomes.

The distribution regression approach was proposed by Foresi and Peracchi (1995) and formalized by Chernozhukov et al. (2013); see also Rothe and Wied (2013, 2019), Chernozhukov et al. (2019) and Chernozhukov et al. (2020) for later developments. One attractive feature of this modeling strategy is that marginal effects associated with distribution regression models are more flexible than those obtained from classical conditional hazard specifications, as they can accommodate richer forms of heterogeneity. In contrast to quantile regression, distribution regression can accommodate discrete, continuous and mixed duration data in a unified manner under fairly weak regularity conditions. Chernozhukov et al. (2013) show that distribution regression encompasses the Cox (1972) model as a special case and represents a useful alternative to quantile regression.

The distribution regression model specifies the distribution of the duration outcome variable by means of a generalized linear model with known link function and nonparametric varying coefficient depending on duration time. When data is uncensored, the varying coefficients at each duration value are consistently estimated by binary regressions. This method is also valid using censored data with known censoring time, which is an unlikely situation in duration studies, e.g., unemployment duration spells. When duration and censoring times are potentially unobserved, the standard binary regression approach lead to inconsistent estimators.

In this article, we propose a weighted binary regression procedure using Kaplan and Meier (1958) weights. The varying coefficients are identified by means of a set of moment restrictions of the duration time and covariates’ joint distribution. These moments are consistently estimated by their corresponding Kaplan-Meier integrals when the duration time is subjected to censoring; if censoring is not an issue, the Kaplan-Meier integrals reduce to the sample analogue of the moment restrictions. The resulting estimator admits a representation as a non-degenerate U-statistic of order three, whose Hájek projection is asymptotically distributed as a normal with an asymptotic variance that can be consistently estimated from the data. The resulting inference procedures on the coefficients do not rely on choosing tuning parameters such as bandwidths, imposing smoothness conditions on the censoring random variable, or using truncation arguments.

We establish the consistency and finite-dimensional distribution convergence of the varying coefficients under weak regularity conditions. Under more restrictive assumptions we justify the convergence in distribution of the estimated coefficient function as a random element of a suitable metric space. We also justify bootstrap-based inference procedures.

In order to justify the asymptotic inferences based on our proposed method, we impose restrictions on the joint distribution of the duration time, censoring time and covariates. Stute (1993, 1996) provided strong consistency and a central limit theorem for Kaplan-Meier integrals that estimate moments of (partially observed) durations time and covariates under the following identification conditions: (a) the duration and censoring times are independent, and (b) the event of being censored is conditionally independent of the covariates given the duration time. Stute shows that the two conditions suffice to identify the joint distribution of duration time and covariates from the observed censored data, and, hence, to identify corresponding moments of these conditional distributions. Of course, the identification of alternative models may require alternative regularity conditions. For instance, identification of the Cox (1972)’s hazard model, Aalen (1980)’s additive hazard model, or censored quantile regressions models (Portnoy, 2003) only requires that duration and censoring times are independent given covariates. However, as we mention in Section 2, these models usually exclude important types of heterogeneity, or restrict their attention to absolutely continuous duration data. Alternatively, one can potentially estimate the moment equations using inverse probability weighting (IPW) methods, as suggested by Robins and Rotnitzky (1992, 1995) and later extended by Wooldridge (2007) to general missing data problems. Although this path allows one to only assume that duration and censoring times are independent given covariates, it also requires that the conditional distribution of the censoring times given covariates is (uniformly) bounded away from one, which rules out discrete distributions and the most popular distribution functions used in survival analysis such as exponential, Gamma, log-normal and Weibull, among others. If one is not comfortable with these additional restrictions, truncation/trimming arguments need to be carefully introduced, and the choice of tuning parameters becomes a first-order concern. By adopting the Kaplan-Meier integrals approach, we avoid these alternative restrictions, but rely on conditions (a) and (b) above.

We illustrate the relevance of our proposal by assessing how changes in unemployment insurance benefits affect, on average, the distribution of unemployment duration, using data from the Survey of Income and Program Participation for the period 1985-2000. We find that, by allowing the distributional marginal effects to vary over the unemployment spell, our proposed method can reveal interesting insights when compared to traditional hazard models as those used by Chetty (2008). For instance, our results suggest a non-monotone marginal effect of an increase in unemployment insurance on the unemployment duration distribution. Such a finding is in sharp contrast with those obtained using classical proportional hazard models. On the other hand, our results agree with Chetty (2008) in that increases in unemployment insurance have larger effects on liquidity-constrained workers, suggesting that unemployment insurance affects unemployment duration not only through a moral hazard channel but also through a liquidity effect channel.

The rest of the article is organized as follows. Section 2 introduces the basic notation and motivates distribution regression models using duration data as an alternative to classical conditional hazard modeling. Section 3 describes our estimation procedure, whereas Section 4 introduces regularity conditions needed to justify inferences on the distribution regression varying coefficients. All the results are proved in the Supplementary Appendix. Application of the results to different contexts are placed in Section 5. Section 6 briefly summarizes the results of the Monte Carlo simulations – detailed discussion is presented in the Supplementary Appendix¹¹1The Supplementary Appendix is available at https://pedrohcgs.github.io/files/Delgado_GarciaSuaza_SantAnna_2021_EJ-Appendix.pdf. Finally, we apply the proposed techniques to investigate the effect of unemployment benefits on unemployment duration in Section 7. All our replication files are available at https://github.com/pedrohcgs/KMDR-replication.

Consider the $\mathbb{R}^{1+k}-valued$ random vector $\left(T,X\right)$ defined on $\left(\Omega,\mathcal{F},\mathbb{P}\right),$ where $T$ is the duration outcome of interest and $X$ is a $k-$dimensional vector of time-invariant covariates, with supports $\mathcal{T}$ and $\mathcal{X},$ respectively. Henceforth, let $\mathbf{X}=(1,X^{\prime})^{\prime}$ and $\mathbf{x}=(1,x^{\prime})^{\prime}$. We assume that the conditional cumulative distribution function (CDF) of $T$ given $X$ follows the distribution regression (DR) model,

where $\theta_{0}\left(\cdot\right)=\left(\varphi\left(\cdot\right),\beta_{0}\left(\cdot\right)^{\prime}\right)^{\prime}\mapsto\Theta\subseteq\mathbb{R}^{k+1}$ is a vector of nonparametric functions, and $\Lambda$ is a known link function.

Chernozhukov et al. (2013) point out that the DR model is a flexible alternative to classical duration models as it allows coefficients to vary with duration time. In particular, they show that it nests the Cox (1972) proportional hazard (PH) model. Model (2.1) also generalizes other traditional models commonly adopted in duration analysis such as Kalbfleisch and Prentice (1980) accelerated failure time (AFT) model, and Clayton (1976) proportional odds (PO) model.

We note that all the aforementioned classical duration models are special cases of the linear transformation model

where $\beta_{0}\in\mathbb{R}^{k}$ is a vector of unknown parameters, $\varphi$ is a (potentially unknown) monotonically increasing transformation function, $\Lambda^{-1}$ is the $\Lambda^{\prime}s$ quantile function, and $U$ is uniformly distributed in $\left[0,1\right]$ independently of $X$. For instance, $(a)$ the PH model corresponds to model (2.2) with $\Lambda$ the complementary log-log (cloglog) link function, $\Lambda\left(u\right)=1-\exp\left(-\exp\left(u\right)\right)$; $(b)$ the AFT model corresponds to (2.2) with $\Lambda$ commonly assumed to be the link function of a log-logistic or of a Weibull distribution; and $(c)$ the PO model corresponds to (2.2) with $\Lambda$ the logistic link function, $\Lambda\left(u\right)=\exp\left(u\right)\left/\left(1+\exp\left(u\right)\right)\right.$; see, e.g. Doksum and Gasko (1990), Cheng et al. (1995) and Hothorn et al. (2014). Indeed, the CDF associated with (2.2) is given by

Therefore, (2.1) is a natural generalization of (2.3) that allows all the slope coefficients varying with $t$. Other duration models with varying slope coefficients are also special cases of (2.1). For instance, Aalen (1980) semi-parametric additive hazard model reduces to (2.1) with $\Lambda\left(u\right)=1-\exp\left(-u\right)$. The non-proportional odds model considered by McCullagh (1980) and Armstrong and Sloan (1989) can also be expressed in terms of the DR specification (2.1) when $\Lambda$ is the logistic link function.

We conclude this section by emphasizing that allowing for varying slope coefficients is particularly important to capture richer heterogeneous effects of covariates across the distribution of the duration outcome. Indeed, traditional duration specifications, such as the PH, PO or AFT models, implicitly impose that all covariates affect the CDF of $T$ in a proportional, and monotonic form. If $\Lambda$ is differentiable with Lebesgue density $\lambda$, partial effects for these models have the form

Thus, the sign of the marginal effect of any $X$’s component is not allowed to vary with $t$, which may be too restrictive in some applications. For instance, in the context of the empirical application in Section 7, (2.3) does not allow a non-monotonic effect, e.g. U-shaped, ruling out non-stationary search models in the spirit of van den Berg (1990). The DR model (2.1) bypass such limitations since

which sign is allowed to vary with $t$. We view this added flexibility as an attractive feature of the DR modeling approach.

The main challenge when dealing with duration data is that the outcome of interest $T$ is subject to censoring according to a variable $C$. As so, inferences must be based on observations $\left\{Y_{i},X_{i},\delta_{i}\right\}_{i=1}^{n}$, with $Y_{i}=\min\left(T_{i},C_{i}\right)$ the realized (potentially censored) outcome, $\delta_{i}=1_{\left\{T_{i}\leq C_{i}\right\}}$ indicates whether or not $T$ is observed, and $\left\{T_{i},C_{i},X_{i}\right\}_{i=1}^{n}$ are independent and identically distributed ($iid$) as $\left(T,C,X\right)$. Henceforth, $1_{\left\{A\right\}}$ is the indicator function of the event $A$.

Suppose for the moment that $\delta_{i}=1,$ i.e., $T_{i}=Y_{i},$ for all $i=1,\dots,n$. In this case, the joint $\left(T,X^{\prime}\right)^{\prime}$ CDF, $F\left(t,x\right)$, is consistently estimated by its sample version $\tilde{F}_{n}\left(t,x\right)=n^{-1}\sum_{i=1}^{n}1_{\left\{Y_{i}\leq y,X_{i}\leq x\right\}}$, where inequalities are coordinate-wise. Therefore, following Foresi and Peracchi (1995) and Chernozhukov et al. (2013), $\theta_{0}\left(t\right)$ can be consistently estimated by $\tilde{\theta}_{n}(t)$, the maximizer of the conditional likelihood function of $\left\{1_{\left\{Y_{i}\leq t\right\}},X_{i}\right\}_{i=1}^{n}$

with

However, when $T$ is subject to right-censoring, i.e., $T\neq Y$, $\tilde{F}_{n}$ is no longer a consistent estimator of $F$ and, hence, neither is $\tilde{\theta}_{n}\left(t\right)$.

Since $T$ is not always observed, $F$ and $\theta(t)$ must be identified from the joint distribution of the observed data on $\left(Y,X,\delta\right)$. However, this is not always possible without additional information. Henceforth, for any random variable $\xi$, which can be $T,~C$ or $Y$, define $F_{\xi}(t)\equiv\mathbb{P}\left(\xi\leq t\right)$ and $\tau_{\xi}\equiv\inf\left(t:\mathbb{P}\left(\xi\leq t\right)=1\right)$, and let $A$ denote the (possibly empty) set of $F_{T}$ jumps. Also, for any generic function $g$, $g(t-)\equiv\lim_{s\uparrow t}g(s)$. If no information about $T$ beyond $\tau_{Y}$ is available from the data, identification of

is the best that one can hope for; see, e.g., Tsiatis (1975, 1981), and Stute (1993). In order to identify $F_{\ast}$ we impose the following assumptions.

Assumption 3.1 is introduced to identify $F_{\ast}\left(\cdot,\infty,...,\infty\right)$ (see Tsiatis, 1975, 1981 for discussion), whereas Assumption 3.2 is introduced by Stute (1993) and allows us to incorporate covariates into the analysis. Taken together, these two assumptions imply that covariates should have no effect on the probability of being censored once $T$ is known. Of course, Assumptions 3.1 and 3.2 hold if $C$ is independent of $\left(T,X\right)$, but can also hold under more general circumstances. See Stute (1993, 1996, 1999) for a discussion on Assumptions 3.1 and and 3.2, and the identification of $F_{*}$ and its moments. Identification can be achieved under alternative conditions, in the context of alternative conditional survival models, or alternative estimation procedures. See Introduction for further comments.

In order to identify $\theta_{0}(t)$, we ensure that $F=F_{\ast}$ by imposing the following condition; see Stute (1999) for a similar assumption in the case of nonlinear regression with randomly censored outcomes.

Thus, $\theta_{0}\left(t\right)$ is identified as the parameter value that maximizes $Q\left(\theta,t\right)$ under standard conditions in binary regression, e.g., Amemiya (1985, Section 9.2.2), or more recently, van der Vaart (1998, Example 5.40).

Henceforth, $Y_{1:n}\leq...\leq Y_{n:n}$ are the ordered $Y-values$, where ties within outcomes $T$ or within censoring times $C$ are ordered arbitrarily and ties among $T$ and $C$ are treated as if the former precedes the latter. For observations $\left\{\xi_{i}\right\}_{i=1}^{n}$ of a random variable $\xi$, which may be $\delta$ or $X$, $\xi_{\left[i:n\right]}$ is the $i-th$ $\xi-$concomitant of the order statistics $\left\{Y_{i:n}\right\}_{i=1}^{n},$ i.e., $\xi_{\left[i:n\right]}=\xi_{j}$ if $Y_{i:n}=Y_{j}$.

In the absence of covariates, $F_{\ast T}(t)=F_{\ast}\left(t,\infty,...,\infty\right)$ is consistently estimated by the Kaplan and Meier (1958) product-limit estimator,

By noticing that the jumps of $\hat{F}_{Tn}$ at $Y_{i:n}$ are

we can express $\hat{F}_{Tn}$ in an additive form as

When covariates are present, the natural $F_{\ast}\left(t,x\right)$ estimator is

see, Stute (1993, 1996). Stute (1993) showed that, under Assumptions 3.1-3.3,

Thus, (3.3) is indeed a natural candidate to replace $\tilde{F}_{n}$ in (3.1) under censoring. This suggests estimating $Q_{\ast}\left(t,\theta\right)$ by the Kaplan-Meier integral,

which is a weighted version of $\tilde{Q}_{n}\left(\theta,t\right)$.

The Kaplan-Meier distribution regression (KMDR) estimator of $\theta_{0}(t)$ is then given by

Notice that, to compute the KMDR estimators, one simply needs to run a weighted binary regression, where the weights are the (random) Kaplan-Meier weights $W_{in}$. In the absence of censoring, i.e., $Y_{i}=T_{i},$ $\delta_{i}=1$, $i=1,..,n$, we have that the Kaplan-Meier weights $W_{in}=n^{-1}$, and $\hat{\theta}_{n}\left(t\right)$ reduces to $\tilde{\theta}_{n}\left(t\right).$

In this section, we present the asymptotic properties of our proposed KMDR estimator $\hat{\theta}_{n}(t)$ for $\theta_{0}\left(t\right)$.

In order to establish consistency, we impose the following fairly weak assumption which is compatible with discrete, continuous and mixed duration outcomes.

Next theorem, like any other result in the paper, is proved in the Supplemental Appendix. The proof applies the arguments in Example 5.40 in van der Vaart (1998) to show that

In order to provide the finite dimensional asymptotic distribution of $\hat{\theta}_{n}(t),$ we need the following assumption.

The restriction on $\Lambda$ in Assumption 4.5 is a classical assumption, which can be found, for instance, in Amemiya (1985) Assumption 9.2.1. The assumption is satisfied by the normal and logistic distribution (probit and logit), but also for many other distributions. In the general case, Assumption 4.5 essentially rules out extremes $t$, since, in these cases, $\Lambda$ may be near zero or one. This assumption can be relaxed at the cost of more involved proofs.

The score function is given by

where

The conditional Fisher information that the binary variable $1_{\left\{T\leq t\right\}}$ contains about $\theta_{0}(t)$ given $X$ is $\mathcal{I}_{0}\left(t\right)=\mathcal{I}\left(\theta_{0}\left(t\right),t\right)$, where

Notice that, under our assumptions, $\theta\mapsto\mathcal{I}\left(\theta,\cdot\right)$ is continuously differentiable.

We first pay attention to the asymptotic distribution of the score $\hat{\Psi}_{n}^{0}\left(t\right)=\hat{\Psi}_{n}\left(\theta_{0}(t),t\right).$ Applying Stute (1995, 1996) lemmata, $\hat{\Psi}_{n}\left(\theta,t\right)$ can be expressed as an $U-$statistic of order three with Hájek projection $\hat{U}_{n}^{0}(t)=\hat{U}_{n}(\theta_{0}(t),t)$, where

and for any function $\left(y,x\right)\mapsto\varphi\left(y,x\right)$,

where $F_{Y}^{0}(t)=\mathbb{P}\left(Y\leq t,\delta=0\right)\text{ and }F^{1}(t,x)=\mathbb{P}\left(Y\leq t,X\leq x,\delta=1\right)$.

To derive the asymptotic normality of the score, we need the following extra moment conditions.

Assumption 4.6$\left(a\right)$ guarantees that the variance of the leading term in $\zeta_{\psi_{\theta_{0}\left(t\right)t}}$ is bounded, and implies that $\mathbb{E}\left\|\zeta_{\psi_{\theta t}}(Y,X,\delta)\right\|^{2}$ is finite. The bias of the Kaplan-Meier integral is not necessarily $o\left(n^{-1/2}\right)$ for any integrand function, and may decrease to zero at a polynomial rate depending on the degree of censoring, which is characterized by the function $S$. Assumption 4.6$\left(b\right)$ on $\psi_{\theta_{0}(t)t}$ guarantees that the $\hat{\Psi}_{n}^{0}\left(t\right)$ bias is of order $o\left(n^{-1/2}\right)$; see Stute (1994), Chen and Ying (1996), and Chen and Lo (1997).

Let $\left\{Z\left(t\right)\right\}_{t=t_{1}}^{t_{m}}$ be a $k+1$ Gaussian random vector with zero mean and covariances

with $j,m=1,...,k+1$.

With Assumption 4.6, we show that $\hat{\Psi}_{n}^{0}\left(t\right)$ is asymptotically equivalent to a $U-statistic$ of order 3 with Hájek projection $\hat{U}_{n}^{0}\left(t\right)$. Thus, $\sqrt{n}\hat{U}_{n}^{0}$ is asymptotically normal with finite dimensional covariance $\Omega_{0}\left(j,m\right)$. Theorem 4.2 below follows by applying Theorem 5.21 in van der Vaart (1998).

In order to conduct inferences on $\theta_{0}(\cdot)$, $\Omega_{0}\left(j,m\right)$ is estimated by

where

and $\mathcal{I}_{0}\left(t\right)$ is estimated by

where $\hat{\psi}_{i:n}(t)=\psi_{\hat{\theta}_{n}(t)t}\left(Y_{i:n},X_{[i:n]}\right)$²²2Alternatively, one can estimate $-\mathcal{I}_{0}\left(t\right)$ using the Hessian of $\hat{Q}_{n}(\hat{\theta}_{n},t)$. We follow this path in our simulations..

We can also justify inferences with the assistance of a multiplier bootstrap technique in the spirit of Theorem 2.3 in Stute et al. (2000) and Theorem 4 in Sant’Anna (2020), using an external resample of $\left\{\hat{\zeta}_{i}(t)\right\}_{i=1}^{n}.$ Once we generate $iid$ random numbers $\left\{V_{i}\right\}_{i=1}^{n}$ independently of the data with mean $0$, variance $1$, and finite third moment, the “resampled” $\left\{\hat{\zeta}_{i}^{\ast}(t)\right\}_{i=1}^{n}$, with $\hat{\zeta}_{i}^{\ast}(t)=\hat{\zeta}_{i}(t)V_{i}$, forms a basis to compute the bootstrap analog of $\hat{\theta}_{n}(t)$ based on the asymptotic linearization (4.7),

where

is the bootstrap version of $\hat{U}_{n}^{0}\left(t\right)$.

The theorem follows by checking the conditional version of the Linderberg-Levy conditions to show that, with probability 1, $\hat{U}_{n}^{\ast}\left(t\right)$ converges in distribution to $Z$ under the bootstrap law, i.e., conditional on the data, with probability 1.

Next, we strengthen the pointwise results in Theorems 4.2 and 4.3 to hold uniformly in $t$. Towards this end, let the space $\ell^{\infty}\left(\mathcal{G}\right)^{q}$ be the set of all $q\times 1$ vectors of bounded functions on $\mathcal{G}$, that is, all the functions’ vectors $g:u\mapsto\mathbb{R}$ such that $\sup_{u\in\mathcal{G}}\left\|g\left(u\right)\right\|<\infty.$ $\mathcal{G}$ can be either an Euclidean or a functional space. We interpret the multivariate empirical process indexed by functions in $\mathcal{G}$ as a random element in the metric space $\ell^{\infty}\left(\mathcal{G}\right)^{q}$ endowed with the sup-norm. The empirical process $\hat{\Psi}_{n}$ is indexed by $\left(\theta,t\right)\in\Theta\times\mathcal{T}$; in this case $\mathcal{G}=\Theta\times\mathcal{T}_{0},$ where $\mathcal{T}_{0}$ is the compact interval of $\mathcal{T}$ of interest. But $\hat{\Psi}_{n}$ can also be interpreted as an empirical process indexed by the class $\mathcal{F}$ of functions $\psi_{\theta t}:\mathcal{X\times}\mathcal{T\times}\left[0,1\right]\mapsto\mathbb{R}^{1+k};$ in this case $\mathcal{G}=\mathcal{F}$. The random process $\hat{\Psi}_{n}$ is viewed as a random element of the metric space $\ell^{\infty}\left(\Theta\times\mathcal{T}_{0}\right)^{k+1}$ or $\ell^{\infty}\left(\mathcal{F}\right)^{k+1},$ where $\mathcal{T}_{0}\in\mathcal{T}$ is the compact subset of $\mathcal{T}$ we are interested in.

In order to establish the weak convergence of $\sqrt{n}\left(\hat{\theta}_{n}-\theta_{0}\right)$ as a random element of the metric space $\ell^{\infty}\left(\mathcal{T}_{0}\right)^{k+1}$, we impose the following additional assumption, which is also assumed in Chernozhukov et al. (2013).

Notice that $f_{\left.T\right|X}\left(\left.t\right|X\right)=\dot{\theta}_{0}^{\prime}\left(t\right)\cdot\boldsymbol{X\cdot}\lambda\left(\theta_{0}^{\prime}\left(t\right)\boldsymbol{X}\right)$ a.s., where $\dot{\theta}_{0}\left(t\right)=d\theta_{0}\left(t\right)/dt.$ Therefore, assuming that $f_{\left.T\right|X}$ is a proper probability density may not be innocuous. In practice, $\mathcal{T}_{0}$ should be bounded by the minimum and maximum value of the observed uncensored duration.

Let $\mathbb{Z}$ be a centered tight Gaussian element of $\ell^{\infty}\left(\mathcal{T}_{0}\right)^{k+1}$ with matrix of variance and covariance functions

with $\Omega_{0}\left(t_{1},t_{2}\right)$ defined in (4.6).

This functional CLT is proved extending Chernozhukov et al. (2013) (CFM henceforth) strategy to our setup. To this end, we first show, using Stute (1995, 1996) lemmata, that the score function $\hat{\Psi}_{n}$ and $U_{n}$ are asymptotically equivalent in the metric space $\ell^{\infty}(\Theta,\mathcal{T}_{0})^{k+1}$, i.e.,

where $U_{n}$ is a $U-process$ with Hájek projection $\hat{U}_{n}$. Then, we show that

applying asymptotic results for U-process in Arcones and Giné (1993, 1995). Since the class $\mathcal{E}$ of functions $\left(y,x,d\right)\mapsto\zeta_{\psi_{\theta t}}\left(y,x,d\right)$ is Donsker,

converges weakly in $\ell^{\infty}\left(\Theta\times\mathcal{T}_{0}\right)^{k+1}$. Then, noticing that Assumptions 4.4 - 4.7 imply Assumption DR in CFM, the functional $\theta\mapsto\Psi_{0}\left(\theta,t\right)$, with $\Psi_{0}\left(\theta,t\right)=\mathbb{E}\left[\zeta_{\psi_{\theta t}}\left(Y,X,\delta\right)\right]$, is continuously differentiable for each $t\in\mathcal{T}_{0},$ with $\left.\left.d\Psi_{0}\left(\theta,t\right)\right/d\theta^{\prime}\right\rfloor_{\theta=\theta_{0}\left(t\right)}=-\mathcal{I}_{0}\left(t\right),$ which is invertible. Then, according to CFM’s Lemma E.3,

with $\sup_{t\in\mathcal{T}_{0}}\left\|\hat{r}_{n}\left(t\right)\right\|=o_{p}\left(1\right)$. This result and $\mathcal{E}$’s Donskerness prove the theorem.

A bootstrap version of this theorem follows by applying the multiplier CLT (see sections 2.6 and 3.6 in van der Vaart and Wellner, 1996).