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Encompassing Tests for Nonparametric Regressions
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Keywords: Encompassing, Nonparametric Regression, Bootstrap, Bias Correction,
Locally Robust Statistic. \\ JEL Classification: C01, C12, C14
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The encompassing principle was introduced in econometrics by hendry_formulation_1982, gourieroux_testing_1983, and mizon_encompassing_1986, and further developed in gourieroux_testing_1995, florens_encompassing_1996, and dhaene_instrumental_1998 among others. It provides a natural principle for choosing between two competing theories: a new theory must be able to accommodate the results obtained by a concurrent older one. An extensive survey is provided in bontemps_encompassing:_2008.
Our goal is to propose encompassing tests for nonparametric models. Our main steps are (i) to formally define encompassing for nonparametric models, (ii) to develop fully nonparametric encompassing tests, and (iii) to show asymptotic validity of a wild bootstrap method for asymptotic inference. Our first contribution is thus to formally set up a framework to precisely define encompassing for nonparametric regression models. We discuss nonparametric encompassing with respect to previous literature on the comparison of such models, whether nested or non nested, see below for references. We show that encompassing reduces neither to significance of some variables nor to the comparison of the models' theoretical fit. Hence, the null hypothesis of encompassing cannot generally be tested with existing procedures. Our second contribution is to propose fully nonparametric encompassing tests for regression models. Existing encompassing tests rely on parametric functional forms, except for bontemps_parametric_2008 who propose a test aimed at assessing a consequence of encompassing. The new tests we develop directly test the encompassing hypothesis. They are based on an empirical process estimating a continuum of unconditional moments following the Integrated Conditional Moment (ICM) principle introduced by bierens_consistent_1982. Our third contribution is to develop a wild bootstrap method and to show that it provides asymptotically correct inference.
Our fourth contribution is to propose and investigate two approaches to obtain a “small bias property.” Our test statistic depends on an empirical process involving a first-step nonparametric estimator. The small bias property of a semiparametric estimator is that its bias converges to zero faster than the pointwise and integrated bias of the nonparametric estimator on which it is based. A distinguishing feature is that the resulting statistic is $\sqrt{n}$-consistent even when the nonparametric estimator on which it is based converges at the optimal nonparametric rate. Without this property, using a first-step nonparametric estimator necessitates some undersmoothing, and this complicates practical implementation. newey_twicing_2004 have developed a generic technique based on twicing kernels to obtain semiparametric estimators with small bias. We develop here two alternative methods that yield a small bias property. The first one uses a bias-corrected kernel estimator based on the boosting principle di_marzio_boosting_2008, park_l2_2009. The second approach is to make the empirical process of interest {\em locally robust} with respect to the nonparametric regression. This has previously been used successfully in semiparametric estimation, see newey_semiparametric_1990 for an early example, and Chernozhukov2022 for a general approach. We here adapt the two approaches to our empirical process of interest and we show that these yield a small bias property in the asymptotic expansion of our test statistic. This allows for a larger set of smoothing parameters, so the test is expected to be less sensitive to the bandwidth choice.
Our work is related to the extensive literature on consistent specification testing based on empirical processes, see bierens_asymptotic_1997, stinchcombe_consistent_1998, xia_goodness--fit_2004, escanciano_consistent_2006, delgado_distribution-free_2008, lavergne_breaking_2008 to mention just a few. The main features of our tests compared to previous work are that (i) our empirical process contains a nonparametric kernel estimator, and (ii) due to the form of the null hypothesis we cannot use a density-weighted process, and we thus need to control for a random denominator. Our nonparametric encompassing tests are also connected to the comparison of nonparametric regressions in nested and non nested cases, e.g., fan_consistent_1996, LV96, delgado_significance_2001, and lavergne_significance_2015. We show however that the encompassing hypothesis cannot be tested through existing procedures. Our work is also related to the literature on estimation and testing with nonparametric nuisance components, e.g., escanciano_uniform_2014 and mammen_semiparametric_2016. The former authors obtain uniform-in-bandwidth expansions for an empirical process similar to the one we consider. We focus here on obtaining a small bias property, but we do not formally establish uniformity in bandwidth.
Our paper is organized as follows. Section (ref) formalizes the encompassing notion for nonparametric models and compares our framework to the literature on comparison of nonparametric regressions. Section (ref) details the construction of the test statistics and the two approaches used to obtain a small bias property. Section (ref) is devoted to the analysis of the asymptotic behavior of our statistics. Since the asymptotic distribution under the null depends on unknown features of the DGP, we establish in Section (ref) the validity of a wild bootstrap procedure. Section (ref) provides evidence about the small sample performances of our procedures. We check that bootstrapping allows to correctly control size and that our tests have good power. We evaluate the benefits of our bias-reducing approaches, and we investigate thoroughly the influence of the bandwidth as well as of the trimming parameter, which is theoretically necessary. We also provide an empirical illustration. Section (ref) contains the proofs of our main results. A supplementary material contains the proof of a technical lemma.
The definition of encompassing starts with the definition of the {\em binding function}, see gourieroux_testing_1995. In a parametric context, we typically start with two competing parameterized families of densities for $Y$, ${\mathcal{M}_{1}}=\{g_{1}(\cdot,\alpha_{1})\, : \, \alpha_{1}\in A_{1}\}$ and ${\mathcal{M}_{2}}:=\{g_{2}(\cdot,\alpha_{2})\, : \, \alpha_{2}\in A_{2}\}$. The pseudo true value of $\alpha_{i}$, $i=1$ or $2$, is defined as \[ \alpha_{i}^{*}= \arg\min_{\alpha_{i}\in A_{i}} d (f(\cdot) , g_{i}(\cdot,\alpha_{i})) \, , \] where $f(\cdot)$ is the true density of $Y$ and $d$ is some divergence between the two distributions, for instance the Kullback-Leibler divergence $ \operatorname{E}_f \left[ \log \left( f(\cdot) / g(\cdot,\alpha)\right) \right] $, with expectation taken with respect to $f(\cdot)$. The binding function $b(\alpha_{1})$ is a correspondence between an element of model ${\mathcal{M}_{1}}$ and the element of model ${\mathcal{M}}_{2}$ that is closest to it. Specifically, \[ b(\alpha_{1}) = \arg\min_{\alpha_{2}\in A_{2}} d (g_1(\cdot,\alpha_{1}), g_{2}(\cdot,\alpha_{2})) \, . \] We then say that ${\mathcal{M}_{1}}$ encompasses model ${\mathcal{M}_{2}}$ if $ \alpha_{2}^{*}=b(\alpha_{1}^{*}) $. That is, ${\mathcal{M}_{1}}$ encompasses ${\mathcal{M}_{2}}$ if the pseudo-true value of the latter can be obtained from the pseudo-true value of the former.
Here we focus on two nonparametric competing models to explain $Y$, where Model $\mathcal{M}_{W}$ uses covariates $W$ and Model $\mathcal{M}_{X}$ uses $X$. A nonparametric model with a specific set of covariates is a function of these variables. To define the pseudo-true value that corresponds to the best explanation of $Y$, we consider the ${L}^{2}$ distance. That is, $\mathcal{M}_{W}$ and $\mathcal{M}_{X}$ are respectively defined as ${L}^{2}(W)$, the space of the square integrable functions of $W$, and ${L}^{2}(X)$. The “pseudo-true functions” are then the regression functions
The binding function is similarly defined as \[ b( g(W) )= \arg\min_{h \in {L}^{2}(X)} \operatorname{E} \left( g(W) - h(X) \right)^{2} = \operatorname{E} \left( g(W) | X\right) \, . \] We thus say that ${\mathcal{M}_{W}}$ encompasses model ${\mathcal{M}_{X}}$ if
That is, the regression function of $Y$ on $X$ can be obtained from the regression function of $Y$ on $W$. Similarly, ${\mathcal{M}_{X}}$ encompasses model ${\mathcal{M}_{W}}$ if $\operatorname{E} \left( \operatorname{E} (Y|X) | W\right) = \operatorname{E} (Y|W)$ almost surely.
In what follows, we explore the implications of the definition of nonparametric encompassing and relate it to previous work on the comparison of regression models. In particular, we show that encompassing reduces neither to significance of some variables nor to the comparison of the models' theoretical fit. Hence, the null hypothesis of encompassing cannot generally be tested with existing procedures.
LV96 proposed comparing nonparametric regression models on the basis of their theoretical fit and considered the hypotheses
Non rejection of the null hypothesis $H_0$ means that both models have the same theoretical fit. Rejection of $H_0$ in favor of either $H_W$ or $H_X$ indicates which model dominates the other. This framework is quite general since it does not make a distinction between nested and non nested situations and treats the two competing models symmetrically. LV96 built a test of $H_{0}$ against $H_W$ and $H_X$ based on the comparison of the empirical analogs of the models' fit.
Comparing nonparametric regressions through their fit seems natural. Does encompassing imply a better fit for the encompassing model? As we detail below, the answer is yes: if $\mathcal{M}_{W}$ encompasses $\mathcal{M}_{X}$, the theoretical fit of $\operatorname{E}(Y|W)$ is at least as good as the one of $\operatorname{E}(Y|X)$, and is strictly better except when $\operatorname{E}(Y|W) = \operatorname{E}(Y|X) \mbox{ a.s.}$
Therefore, except in the case where the two regressions are equal, encompassing implies a strictly better fit for the encompassing model. Conversely, one model can have a better fit than another without encompassing it, see LV96 for some examples where this happens. Hence, the encompassing concept is not tailored for model selection.
The test statistic proposed by LV96 to compare models' fit is asymptotically degenerate under $H_0$ when the two models are “generalized nested regressions,” that is when either $\operatorname{E} ( \operatorname{E}(Y|W) | X) = \operatorname{E}(Y|X) \mbox{ a.s.}$ or $\operatorname{E} ( \operatorname{E}(Y|X) | W) = \operatorname{E}(Y|W) \mbox{ a.s.}$ Hence their test cannot be used when there is encompassing, and the tests we develop below are complementary to theirs. The latter might also be used as preliminary tests to check whether their test can be entertained. This would imply (i) testing whether Model ${\cal M}_X$ encompasses Model ${\cal M}_W$, and (ii) testing whether Model ${\cal M}_W$ encompasses Model ${\cal M}_X$. One rejection among two would imply (we cannot reject) that the encompassing model has a strictly better fit than the encompassed one; no rejection would mean (we cannot reject) that $\operatorname{E}(Y|W) = \operatorname{E}(Y|X) \mbox{ a.s.}$ Only if we rejected twice would we need to entertain the test of LV96 to determine whether one model has a better fit. A more direct alternative to this involved procedure is to apply the test proposed by liao2020nondegenerate, which is universally valid. We leave the study of the respective merits of these competing procedures for future work.
When the two sets of regressors are nested, specifically when $W= (X, Z)$, then it is clear that ((ref)) holds. A more interesting question is whether ${\mathcal{M}_{X}}$ encompasses ${\mathcal{M}_{W}}$, that is whether \[ \operatorname{E} \left( \operatorname{E} (Y | X) | X, Z \right) = \operatorname{E}(Y|X) =\operatorname{E} (Y| X, Z) \mbox{ a.s.} \] In this setup, encompassing is equivalent to whether $Z$ is significant in the regression function of $Y$ on $X$ and $Z$. Significance testing in nonparametric regressions has been a focus of extensive work, see fan_consistent_1996, lavergne2000nonparametric, Abs01, Lav2001, delgado_significance_2001.
Consider now two nonnested sets of regressors, and assume the regressors $X$ are not significant once we control for $W$, that is
By conditioning again with respect to $X$, one obtains that \[ \operatorname{E} (Y|X) = \operatorname{E} [ \operatorname{E} (Y|W) |X] \mbox{ a.s.} \] so that $\mathcal{M}_{W}$ encompasses $\mathcal{M}_{X}$. bontemps_parametric_2008 consider ((ref)) as their hypothesis of interest, since it implies encompassing. The other direction of the implication, however, does not hold: if $\mathcal{M}_{W}$ encompasses $\mathcal{M}_{X}$, then it is not necessarily true that the covariates $X$ are not significant in the nonparametric regression of $Y$ onto $(W,X)$, as shown below.
The previous result highlights that for non nested sets of regressors, the encompassing property does not reduce to the significance of some regressors in the complete regression. As long as $g(W,X)$ is not a.s. equal to zero, $\mathcal{M}_{W}$ encompasses $\mathcal{M}_{X}$, but $X$ is a significant covariate in $\operatorname{E}(Y|W,X)$. The following provides a concrete example.
\paragraph{Example.} Let $W$ be continuous univariate, symmetrically distributed around 0, with density $f_W(\cdot)$. Consider $p(\cdot) \in (0,1)$, $\varphi(\cdot)$ and $\psi(\cdot)$ two univariate densities with mean 0. Define \[ f_{X|W}(x|w) = p(w) \varphi(x) + (1-p(w)) \psi(x) \, , \] so that $X$ has a mixture distribution conditionally on $W$. Then
Let \[ Y = m(W) + h(W) X + \eta\, , \qquad \operatorname{E}(\eta|W,X) = 0 \, . \] Then \[ \operatorname{E}(X|W) = p(W) \int_{}^{}{ x \varphi(x) \, dx} + (1-p(W)) \int_{}^{}{x \psi(x) \, dx} = 0 \, . \] One can select $h(\cdot)$ and $p(\cdot)$ so that $\operatorname{E} [h(W)|X] = 0$. Consider first a case where $h(\cdot)$ is an odd function with mean 0 and $p(\cdot)$ is even. Then by the symmetry of $f_W(\cdot)$, $\operatorname{E} [h(W)|X] = 0$. Another setup used in our simulations below is for $h(\cdot)$ even with mean 0 and $p(-w) = 1-p(w)$. Again, it is easy to show that $\operatorname{E} [h(W)|X] = 0$.
Whenever $\operatorname{E} [h(W)|X] = 0$, then $\operatorname{E} [ h(W) X|W] = \operatorname{E}[ h(W) X|X]=0$, and $\mathcal{M}_{W}$ encompasses $\mathcal{M}_{X}$, but $X$ is significant in $\operatorname{E}(Y|W,X)$. Furthermore, $\mathcal{M}_{X}$ does not encompass $\mathcal{M}_{W}$ in general. If this holds, then $m(W)=\operatorname{E}(Y|X)$ a.s. \ from Proposition (ref). Integrating both sides with respect to the marginal density $f_X(\cdot)$ implies that $m(W)$ and thus $\operatorname{E}(Y|X)$ are both constant almost surely.
We want to test \[ {H}_{0} \, : \, \operatorname{E} \left( \operatorname{E}(Y|W) | X\right) = \operatorname{E}(Y|X) \mbox{ a.s.} \Leftrightarrow \operatorname{E} \left( Y- \operatorname{E}(Y|W) | X\right) = 0 \mbox{ a.s.} \] against its logical complement ${H}_{1} = {H}_{0}^{c}$. While $H_0$ is a conditional moment restriction, we can consider instead an equivalent continuum of unconditional moments. Assume $X \in \mathbb{R}^d$ has bounded support, which is without loss of generality, as we can always transform $X$ by a one-to-one function that maps it to a compact set. Then the null hypothesis is equivalent to
where ${\cal S}$ is a (arbitrary) neighborhood of the origin in $\mathbb{R}^d$ and $\varphi(\cdot)$ is a well-chosen function, see Assumption (ref) below for precise conditions. Some convenient choices for $\varphi(\cdot)$ are as follows. bierens_consistent_1982 shows the previous equivalence for the complex exponential $\varphi(u) =\exp (i u) $, bierens_consistent_1990 considers the exponential $\varphi(u) =\exp (u) $, bierens_asymptotic_1997 the logistic c.d.f. $\varphi(u) = 1/(1+\exp(c-u))$, see also stinchcombe_consistent_1998. Other types of functions could be used such as indicator functions escanciano_consistent_2006, delgado_significance_2001.
If we observe a random sample $(Y_i,W_i,X_i)$, $i=1, \ldots n$, from $(Y,W,X)$, and if we know the precise form of $\operatorname{E}(Y|W)$, then Bierens' Integrated Conditional Moment (ICM) statistic for testing $H_0$ is
where $\mu$ is some probability measure on ${\cal S}$, such as the uniform distribution on ${\cal S}$. Alternatively, a Kolmogorov-Smirnov type statistic could be considered, but the Cramer-von-Mises form appears to be easier to deal with in practice, see below.
In practice, we use a kernel nonparametric estimator of the conditional expectation $\operatorname{E} (Y|W)$. Let $K(\cdot)$ be a kernel on $\mathbb{R}^{p}$, $h\,(=h_n)$ a bandwidth, and $K_{{h}} (u) = K \left( u_{1}/h, \ldots u_{p}/h \right)$.\footnote{For notational simplicity we are assuming the same bandwidth across regressors, but our proofs would carry over when each regressor has a specific bandwidth.} Define
With $e = (1, \ldots, 1)'$, let $\widehat{f} (w) = \overline{e} (w)$, and $\widehat{m} (w) = \overline{Y}(w)/\widehat{f} (w)$.
To control for the random denominator in the kernel estimator, we introduce a trimming factor $\widehat{t}(w) = \mathbf{1} \left( \widehat{f}(w) \geq \tau_{n} \right)$, where $\tau_n$ converges to zero. Let $\widehat{\varepsilon}_{i} = Y_i - \widehat{m} (W_i) $, $\phi_s(\cdot) = \varphi(s'\cdot)$, and $\mathbb{P}_{n}(g) = n^{-1} \sum_{i=1}^{n}g(Z_{i})$ denote the empirical mean process based on $g(\cdot)$. An ICM test statistic can be built upon the process $\mathbb{P}_{n}\left( \widehat{\varepsilon} \phi_{s} \widehat{t} \right)$ as \[ S_n = n \, \int_{{\cal S}} | \mathbb{P}_{n} \left( \widehat{\varepsilon} \phi_{s} \widehat{t} \right) |^{2} \, d\mu(s) \, . \] In our practical implementation, we chose $\varphi(\cdot)$ as the complex exponential and $\mu$ to be symmetric around the origin. Define \[ a(z) = \int_{{\cal S}} \exp(i s'z) \, d\mu(s) = \int_{{\cal T}} \cos(s'z) \, d\mu(s) \, , \] due to the symmetry of $\mu$, and let $\widehat{\bm{ \varepsilon}} = \left( \widehat{\varepsilon}_{j}, j=1, \ldots n \right)'$ and $\widehat{t}_{i} = \widehat{t} (W_i)$. Then the statistic becomes
where $\bm{A}$ is a matrix with generic element $n^{-1} a\left(X_{j}-X_{m}\right) \widehat{t}_j \widehat{t}_m$. In practice the function $a(\cdot)$ is the Fourier transform of $\mu$, so we can choose the latter so that the former has an analytic expression, and computation of the matrix $A$ is fast. To achieve scale invariance, we recommend, as in bierens_consistent_1982, to scale each component of $X$ by a measure of dispersion, such as the empirical standard deviation.
The behavior of $\sqrt{n} \mathbb{P}_{n}\left( \widehat{\varepsilon} \phi_{s} \widehat{t} \right)$ is studied in detail by escanciano_uniform_2014, who derived a uniform expansion. Hence, the properties of the ICM test based on $S_n$ can be derived from their results. However, these impose undersmoothing in kernel estimation, which ensures that the bias disappears fast enough, but makes the practical bandwidth choice tricky. In what follows, we develop two approaches to avoid undersmoothing, and in particular to allow for an optimal nonparametric bandwidth. This is convenient because there are well-known methods for approximating such bandwidths.
newey_twicing_2004 obtained a small bias property for density weighted average semiparametric estimators. Here we instead rely on an idea developed in the boosting literature for kernel regressions di_marzio_boosting_2008, park_l2_2009. xia_goodness--fit_2004 used a similar bias correction in a specification test for a single-index model.
$L^2$ boosting starts with an initial nonparametric estimator and builds a bias-corrected updated estimator. The bias correction is roughly based on nonparametric residuals. The method can be iterated, but we will restrict to a single boosting step, which is sufficient for our purpose. In our context, the bias correction to be applied to $\widehat{m} (w)$ is \[ \widehat{B} (w) = \frac{\overline{\widehat{m}}(w)}{\widehat{f} (w)} - \widehat{m}(w) \, , \] where
and the trimming controls for the random denominator of $\widehat{m}$. The bias-corrected estimator thus is
We then consider $\widetilde{\varepsilon}_{i} = Y_{i} - \widetilde{m} (W_i)$ and the bias-corrected ICM statistic \[ S_{n}^{BC}= n\, \int_{{\cal S}} | \mathbb{P}_{n} \left( \widetilde{\varepsilon} \phi_{s} \widehat{t} \right) |^{2} \, d\mu(s) \, . \] The form ((ref)) of our bias corrected estimator is similar to the one discussed in newey_twicing_2004 who considered density-weighted nonparametric estimators. To use their technique in our setup would necessitate applying their correction to both the numerator and the denominator of the regression estimator. We feel more natural and practically more convenient to correct for the bias of the regression estimator as described above.
Locally robust semiparametric estimation has been considered by several authors, see newey_semiparametric_1990 for an early example. Chernozhukov2022 consider a general GMM estimation problem, where the moments depend on a first-step nonparametric estimator. Locally robust semiparametric GMM estimators are built from moment conditions that have zero derivatives with respect to the first-step estimator so that the latter does not affect the asymptotic variance of the parameters of interest. They have smaller bias and better small sample properties than standard GMM estimators.
In our case, we have a continuum of moment conditions. Our goal is to modify these moment conditions so that (i) we can still test for our null hypothesis of interest using these modified moments, and (ii) estimation becomes “adaptive” with respect to the nonparametric regression. Hence, we consider the moments \[ \operatorname{E} \left[ \left( Y - \operatorname{E}(Y|W) \right) \left( \varphi(s'X) - \operatorname{E} (\varphi(s'X) | W) \right) \right] = 0 \ \quad \forall s \in {\cal S} \, . \] Because for any $s$, $\operatorname{E} \left[ \left( Y - \operatorname{E}(Y|W) \right) \operatorname{E} (\varphi(s'X) | W) \right] = 0$, the above moment conditions are equivalent to ((ref)). We show below that for the empirical equivalent of these moments, estimation of nonparametric components has no first order effect.
Let $\iota_{s}(W) = \operatorname{E} (\phi_{s}(X)|W)$ and its estimator \[ \widehat{\iota}_s (w) = \overline{\phi_{s}}(w)/\widehat{f} (w) \, , \qquad \overline{\phi_{s}}(w) = \left(n h^p \right)^{-1} \sum_{i=1}^n \phi_s(X_i) K_h \left( {W_i - w} \right) \, . \] Our locally robust approach thus delivers the statistic \[ S_{n}^{LR}= n \, \int_{{\cal S}} | \mathbb{P}_{n} \left( \widehat{\varepsilon} ( \phi_{s} - \widehat{\iota}_s) \widehat{t} \right) |^{2} \, d\mu(s) \, . \] newey_twicing_2004 note that their bias correction based on twicing kernels is equivalent to a locally robust density weighted average. By contrast, our bias correction detailed in the previous section does not yield the same statistic as the one based on the locally robust process. In practice, there is no need to compute $\widehat{\iota}_{s}(W)$ for each $s \in {\cal S}$ and to integrate. Simple algebra reveals that, if $\bm{K}$ denotes the $n\times n$ matrix of generic element $K_h(W_i-W_j)/[nh^p\widehat{f}(W_j)]$, then \[ S_n^{LR} = \widehat{\bm{\varepsilon}}' \bm{A} \widehat{\bm{\varepsilon}} - 2\widehat{\bm{\varepsilon}}' \bm{A} \bm{K} \widehat{\bm{\varepsilon}} + \widehat{\bm{\varepsilon}}' \bm{K}' \bm{A} \bm{K}\widehat{\bm{\varepsilon}} \, , \] so the locally robust version of the statistic is practically straightforward to compute.
We here focus on the asymptotic expansion of the empirical processes on which our test statistics $S_{n}^{BC}$ and $S_{n}^{LR}$ are based. We do not formally consider the empirical process entering $S_{n}$: its properties would be similar but would necessitate assuming some undersmoothing.
We first introduce some definitions. Define the differential operator \[ \partial^{l}g(w) = \frac{\partial^{|l|}}{\partial^{l_{1}}w_{1} \ldots \partial^{l_{p}}w_{p}} g(w) \qquad l=(l_{1},..,l_{p})', \ |l| = l_{1}+..+l_{p} \, . \]
Define the uniform convergence rate of the kernel density estimator as
Assumption (ref) ensures that the encompassing hypothesis can be written as the continuum of moment conditions ((ref)). In particular, bierens_econometric_2017 builds on previous results by bierens_consistent_1982 and stinchcombe_consistent_1998, and shows that (ref)-(iii) is sufficient. Intuitively, if $\varphi(\cdot)$ is analytic non-polynomial, Equation ((ref)) implies that $Y - \operatorname{E}(Y|W)$ is uncorrelated with any polynomial in $s'X$, and thus yields the required equivalence. This allows for different functions $\varphi(\cdot)$ as previously detailed.
Assumption (ref) imposes conditions that are commonly found in the literature on nonparametric estimation. In particular, they impose that the density of $W$ is differentiable over $\cal W$ and its derivatives go smoothly to zero as we approach to the boundaries of the support. Part (ii) is necessary only for the study of the locally robust empirical process, which involves a nonparametric estimator of $\iota_s (\cdot)$.
Assumption (ref) sets the main conditions on the bandwidths. Together with Assumption (ref), it implies that the kernel estimators asymptotically belong to a class of sufficiently smooth functions with limited entropy/complexity. This is needed for the asymptotic stochastic equicontinuity of the empirical processes at the basis of our test statistics. Assumptions (ref) and (ref) impose conditions on the bandwidths in connection with the trimming parameter. We need the effect of trimming to vanish quickly enough to avoid large biases. Abstracting from the appearance of the trimming, Condition (ii) in Assumption (ref) ensures that the kernel estimators are $n^{-1/4}$-consistent. It requires in particular that $n h_{n}^{4r}=o(1)$. Without the bias correction or locally robust approach employed in the construction of our statistics, this condition would become $n h_{n}^{2r}=o(1)$ to ensure that the bias of the nonparametric estimator is negligible compared to the variability of the empirical process, see delgado_significance_2001 or escanciano_uniform_2014. This means that our empirical processes have a small bias property (in their stochastic expansion). Our conditions allow the bandwidth to be optimal for nonparametric estimation purposes and avoid the need for undersmoothing. The main practical advantage is that one can use for instance a simple rule-of-thumb.\footnote{We note however that the optimal nonparametric bandwidth is likely not optimal for estimation of $\operatorname{E} \left[ \left( Y -\operatorname{E}(Y|W) \right) \varphi(s'X) \right]$.} We could extend our theoretical results to stochastic bandwidths to allow for the use of data-driven methods, as was done in other contexts andrews_nonparametric_1995, Mammen1992, Lav2001. The details would be more involved here because of the stochastic trimming, so we do not pursue this issue further.
Assumption (ref) as a whole is used to ensure that trimming has a negligible effect overall, see e.g. LV96 or escanciano_uniform_2014 for similar assumptions. It may be difficult to check, but our simulations seem to indicate that trimming is actually not crucial: when a data-driven bandwidth is used, results with no trimming are comparable, and mostly better, than results with trimming.
The following proposition establishes the influence function representations of the empirical processes used in our statistics $S_{n}^{BC}$ and $S_{n}^{LR}$.
From the above result, under $H_0$ the limiting distribution, say $\mathbb{G}_s$, of both empirical processes is the limiting one of $\sqrt{n}\mathbb{P}_{n}(\varepsilon (\phi_{s} - \iota_s))$. This $\mathbb{G}_s$ is a zero-mean tight Gaussian process valued in $L^\infty(\cal S)$, the space of uniformly bounded functionals over $\cal S$, and characterized by the collection of covariances $\{\operatorname{E} (Y-m(W))^2 (\phi_s(X)-\iota_s(X)) (\phi_t(X)-\iota_t(W)) :s,t \in\mathcal{S}\}$. Our asymptotic expansions directly yield \[ S_{n}^{BC} \quad \mbox{or } \quad S_{n}^{LR} \ \mbox{$\stackrel{d}{\longrightarrow}\,$} \int_{}^{}{\left|\mathbb{G}_s \right|^2 \, d\mu(s)} \, . \] However, the result is not useful in practice since the covariance function of $\mathbb{G}_s$ depends on the unknown data generating process. In what follows, we develop a bootstrap procedure for obtaining critical and p-values.
We use a wild bootstrap procedure that imposes the null hypothesis $H_{0}$ when resampling the observations. The bootstrap data generating process (DGP) writes as
where $\{\xi_{i}, i=1, \ldots n\}$ is a sequence of independent bootstrap weights with $\operatorname{E}\xi=0$ and $\operatorname{E}\xi^{2}=1$. From each bootstrap sample $\{\left(Y_{i}^*, W_i, X_i\right)\text{ : }i=1,..,n\}$, we proceed as above to obtain $\widehat{\varepsilon}^{*} = Y_{i}^{*}- \widehat{m}^{*}(W_{i})$ and $\widetilde{\varepsilon}_{i}^{*} = Y_{i}^{*}-\widetilde{m}^{*}(W_{i})$, where
The kernel smoothers $\overline{Y^*}$ and $\overline{\widehat{m}^*}$ are constructed in the same way as in Equation ((ref)) to control for the random denominators in $Y^*$ and $\widehat{m}^*$. The bias correction could be the one used in the original statistic, as done by xia_goodness--fit_2004, however we noted in simulations that recomputing the bias correction with bootstrap data yields a better behavior. We thus consider the bootstrap statistics
In practice, one can compute many bootstrap statistics and obtain their $1-\alpha$ quantiles, denoted as ${q}_{1-\alpha}^{BC}$ and ${q}_{1-\alpha}^{LR}$. The bootstrap test rejects $H_0$ whenever $S_{n}^{j} > {q}_{1-\alpha}^{j}$, for $j=BC$ or $LR$.
We used a DGP in line with our example in Section (ref). Specifically, \[ Y = m(W) + h(W) X + \gamma \delta(X) + \eta\, , \qquad \operatorname{E}(\eta|W,X) = 0 \, , \] where $\eta \sim N(0,1)$ is independent of $(W,X)$, $W \sim N(0,\sigma^{2} = (1/2)^2)$, \[ f_{X|W}(x|w) = p(w) \varphi(x; 1/4) + (1-p(w)) \varphi(x; 3/4) \, , \] with $\varphi(\cdot; s)$ the density of a $N(0, \sigma^2 =s^2)$ and $p(\cdot)$ the cumulative distribution function of $W$. We set $m(w) = w^3 - 2 w$, $h(w) = w^2 - (1/2)^2$, $\delta(x) = x^4 - 3 x^2$. When $\gamma = 0$, ${\cal M}_W$ encompasses ${\cal M}_X$, while it does not when $\gamma \neq 0$.
To study the behavior of our tests based on $S_n^{BC}$ and $S_n^{LR}$ and compare them to the test based on the uncorrected ICM statistic $S_n$, we ran 10000 simulations for different values of $\gamma$ and for sample sizes $n=$200 and 400. For our implementation, the weighting function was $a(x) = \operatorname{sinc}(\pi x)$, which corresponds to a uniform $\mu$ with complex exponential function $\varphi(\cdot)$. The observations on $X$ were transformed by the logistic c.d.f. then standardized before being passed as arguments to $a(\cdot)$. As there is no clear way to choose the amount of trimming, we trimmed the 2% more extreme observations in a first step, and we subsequently investigated the influence of trimming. For bootstrapping, we used the two-point distribution defined through $ \Pr ( \xi = \frac{3-\sqrt{5}}{2}) = \frac{5 + \sqrt{5}}{10}$ and $\Pr( \xi = \frac{3+\sqrt{5}}{2}) = \frac{5 - \sqrt{5}}{10}$. We chose this simple distribution with third central moment equal to one in the hope to better approximate the distribution of the statistic, as is the case in simpler setups Mammen1992. To speed up computations, we used the warp-speed method proposed by davidson_improving_2007 and studied by giacomini_warp-speed_2013. Specifically, we drew one bootstrap sample for each simulated data, and we used the whole set of bootstrap statistics to compute the bootstrap p-values associated with each original statistic.
We employed Gaussian kernels of order 2 for nonparametric estimation. In a first step, we used the bandwidth rule $h=C\widehat{\sigma}_W n^{-1/5}$, where $\widehat{\sigma}_{W}$ is the estimated standard deviation of $W$. To check the performance of our tests under different bandwidth choices, we let the constant $C$ vary. We report actual rejection probabilities for 5% and 10% nominal sizes in Table (ref). For $C=.5$ or $C=1$, the three tests have equal size control and the empirical size becomes closer to the nominal level when increasing the sample size. When $C$ increases to 1.5 then 2, size control deteriorates for the uncorrected ICM test, while it improves for our two tests. We also report in Figure (ref) errors in rejection probability (ERP), that is the difference between the empirical rejection proportion and the nominal size under the null hypothesis $H_0$. A perfect test would exhibit an ERP of zero for any nominal size. This gives us a visual way to evaluate whether the null distribution of the test statistic is well approximated by its bootstrap approximation. These graphs clearly show the high sensitivity of the ICM test to the bandwidth choice, and the relative robustness of our two procedures. We complemented our study with an investigation of the trimming influence. While we do not report details, the most striking result we obtained was that the absence of trimming adversely affected the behavior of the locally robust test for large bandwidths.
Following the suggestion of a referee, we investigated the issue of bandwidth selection. Since there is no role for the bandwidth in the first order expansion of our test statistics, we decided to implement a data-driven bandwidth that should be asymptotically optimal for nonparametric estimation. We used an improved version of the Akaike Information Criterion proposed by hurvich1998smoothing, which has been found to have excellent small sample performances by Li-cross-2004. The criterion is
where $H$ is the “hat matrix” such that $\left( \widehat m (W_1),\ldots,\widehat m(W_n)\right)'= H \,\left(Y_1,\ldots,Y_n\right)'$. As seen from Table (ref), with this data-driven bandwidth, the three tests have good size control. We also report errors in rejection probability in Figure (ref), which shows the excellent adequation of the bootstrap approximation. We then repeated our experiment without any trimming and observed that ERP are closer to zero for our tests compared to the case where trimming is implemented.
Finally, for a data-driven bandwidth and no trimming, we report in Figure (ref) the power of the three tests when $\gamma$ varies. Our two tests basically have the same power, and power increases with sample size, while the uncorrected ICM test is significantly less powerful.
We apply our encompassing tests to competing models of consumption behavior, in the spirit of gaver1974discriminating and greene2003econometric. The first model assumes that consumption only depends on income, and relates consumption to current and past income. The second assumes the presence of habits in consumption behavior, and relates consumption to current income and past consumption. Formally, the first model is
where $C_{i}$ denotes (log of) consumption of household $i$, $I_{i}$ denotes (log of) income of household $i$, and $PI_i$ denotes past (log of) income of household $i$. The second model is
where $PC_i$ denotes (log of) past consumption of household $i$.
We used data compiled by arellano2017earnings from the Panel Study of Income Dynamics, which concerns $n=792$ households. We focus on the first two time periods, namely the years $1999$ and $2001$, so that $C_i$ and $I_i$ correspond to variables in 2001, while $PC_i$ and $PI_i$ correspond to 1999. We first tested if Model ((ref)) encompasses Model ((ref)), i.e.
We then tested the reversed hypothesis that Model ((ref)) encompasses Model ((ref)), i.e.,
We applied our encompassing tests with the bandwidth selected by the $AIC_c$ criterion and 999 bootstrap samples drawn to obtain critical values. Table (ref) reports the values of our test statistics, together with their 95 percent bootstrap quantiles and bootstrap p-values. The first encompassing hypothesis $H_0$ is clearly rejected, hence Model ((ref)) does not encompass Model ((ref)). The encompassing hypothesis $H_0'$ however cannot be rejected by any of our tests, even at a 10% nominal level. This suggests that consumption may be adequately modeled as a function of income and past consumption, in a way coherent with the consumption habit formation theory.
We have studied two general approaches to obtain a small bias property for our encompassing test statistic. These approaches could potentially be used in other estimation and testing problems involving a first-step nonparametric estimation. Our simulation experiment seems to indicate that when coupled with a data-driven bandwidth these two approaches are robust to trimming choices and deliver good performances.
For any real or complex valued function $g(\cdot)$, we denote with $||g||_\infty$ the supremum norm taken over the support of its argument and $P g = \int g(z) \, dP(z)$. We define the population counterpart of $\widehat{t}$ as $t(w)=\mathbf{1}(f(w)\geq \tau_n)$. By convention, $(\overline{Y}/\widehat{f})(w)$ will be set to 0 whenever $\widehat{f}(w)=0$. The same notation will hold for $\overline{\widehat{m}}/\widehat{f}$, $\overline{\phi}_s/\widehat{f}$, $\overline{Y^*}/\widehat{f}$, and $\overline{\widehat{m}^*}/\widehat{f}$. $C$ denotes a generic constant that may vary from line to line.
(a) We study the bias corrected empirical process
Since $|\varepsilon|$ is bounded and $|\phi_{s}(\cdot)|$ is uniformly bounded, \[ \sup_s \left| \sqrt{n}\mathbb{P}_{n} (\varepsilon \phi_{s} (\widehat{t} - 1) ) \right| \leq C \sqrt{n}\mathbb{P}_{n} |\widehat{t} - 1| = o_p(1) \] by Lemma (ref)-(i). Hence, $\sqrt{n}\mathbb{P}_{n} (\varepsilon \phi_{s} \widehat{t}) =\sqrt{n}\mathbb{P}_{n} (\varepsilon \phi_{s} ) + o_{p}^{\mathcal{S}}(1)$, where $o_{p}^{\mathcal{S}}(1)$ denotes a uniform in $s \in {\cal S}$ $o_p(1)$. For the second term, as $(\widehat{t}-t)= \widehat{t}^{2}-t^{2}=(\widehat{t}+t)(\widehat{t}-t)$, and $ \| (m - \widehat{m}) (\widehat{t} +t) \|_{\infty} = o_{p} (n^{-1/4})$ by Lemma (ref)-(ii), \[ \sup_s \Big| \sqrt{n} \mathbb{P}_{n} (m - \widehat{m}) \phi_{s} (\widehat{t}-t) \Big| \leq C \| (m - \widehat{m}) (\widehat{t} + t) \|_{\infty} \sqrt{n} \mathbb{P}_{n} |\widehat{t}-t| = o_p(1) \, . \]
We now show that the third term is such that \[ \sqrt{n}\mathbb{P}_{n} ( \widehat{B} \phi_{s} \widehat{t}) = - \sqrt{n}\mathbb{P}_{n} (\varepsilon \iota_s ) - \sqrt{n}\mathbb{P}_{n} \left( (m - \widehat{m}) \iota_s {t} \right) +o_{p}^{\mathcal{S}}(1) \, . \] First, use $\|\widehat{B} (\widehat{t} + t) \|_{\infty} = o_{p}(n^{-1/4})$, see Lemma (ref)-(iii), and similar arguments as above to show that $\sqrt{n}\mathbb{P}_{n} ( \widehat{B} \phi_{s} \widehat{t}) = \sqrt{n}\mathbb{P}_{n} ( \widehat{B} \phi_{s} {t}) +o_{p}^{\mathcal{S}}(1)$. Lemma (ref)-(i)-(ii) ensures in addition that $\what{B}\in \mathcal{G}_{l}(\mathcal{W}_{n})$ with probability approaching one, where $l =\lceil (p+1)/2 \rceil$ and $\mathcal{W}_n := \left\{ w : f(w) \geq \tau_{n}/2 \right\}$. So, Lemma (ref)-(i) yields
Since $\| [\widehat{B}-(\overline{\what{m}}-\overline{Y})/ f ] t \|_{\infty} = o_p(n^{-1/2})$ from Lemma (ref)-(iv) and $\iota_s$ is uniformly bounded,
where the second equality follows from a change of variable. Let us deal with each term separately. Since $\varepsilon$ is bounded and $|t(w+uh)-1|=\mathbf{1}\{f(w+uh)<\tau_n\}$,
By a mean-value expansion, for all $(u,w)\in\text{Supp}(K) \times \mathbb{R}^p$ we have $f(w+hu) - f(w) = \partial^T f(\widetilde{w}) u h$, for some $\widetilde{w}$. By Assumptions (ref)-(ref), $ \left| \partial^T f(w) uh\right| \leq C h \quad \text{for all }(u,w) \in \text{Supp}(K) \times\mathbb{R}^p $. Since $h\tau_n^{-1}=o(1)$, for $n$ large enough $ \mathbf{1}\{f(w+uh)<\tau_n\} \leq \mathbf{1}\{ f(w)\leq3\tau_{n}/2\} $. Now use $\mathbb{P}_{n} \mathbf{1}\Big\{ f(W)\leq3\tau_{n}/2\Big\} = O_{P}(p_{n}) = o_P(n^{-1/2})$, from Assumption (ref)(i), to obtain \[ \sqrt{n}\mathbb{P}_{n} \left[\varepsilon \int K(u) (t \iota_{s})(W+uh)\, du \right] = \sqrt{n}\mathbb{P}_{n} \left[\varepsilon \int K(u) \iota_s(W+uh)\, du \right] + o_p^{\mathcal{S}}(1) \, . \] The same reasoning allows the same replacement in the second and third term of the decomposition, using that $\|(\what{m}-m)\what{t}\|_\infty $ is $o_p(1)$ by Lemma (ref)-(ii) and that $m(\widehat{t}-1)$ is bounded.
Then
Lemma (ref)(ii) yields \[ \sqrt{n}\mathbb{P}_{n}\left( \varepsilon \int K(u) \iota_s(W+uh)\, du \right) = \sqrt{n}\mathbb{P}_{n} (\varepsilon \iota_s) + o_p^{\mathcal{S}}(1) \, , \] as $P \left(\varepsilon \int K(u) \iota_s(W+uh)\, du \right) = P \left(\varepsilon \iota_s \right) = 0 $. A Taylor expansion of order $r$ guarantees that $\int K(u)\iota_s(w+uh)du=\iota_s(w)+O(h^r)$ uniformly in $\mathcal{S}\times \mathcal{W}$ as $\iota_s(\cdot)$ has uniformly bounded derivatives of order $r$. Since $\|(\what{m}-m)\what{t}\|_{\infty} = o_p(n^{-1/4})$ and $nh^{4r}=o(1)$,
Now use similar arguments as before to replace $\widehat{t}$ by $t$. The last term in ((ref)) is negligible. Indeed, the argument of $\mathbb{P}_n$ is uniformly bounded, and $\sqrt{n} \mathbb{P}_{n}|t-1|= o_P(1)$ from Lemma (ref)-(i). Hence,
Gathering results, \[ \sqrt{n}\mathbb{P}_{n} (\widetilde{\varepsilon}\phi_{s} \widehat{t}) = \sqrt{n}\mathbb{P}_{n} (\varepsilon (\phi_{s} - \iota_s) ) + \sqrt{n}\mathbb{P}_{n} \left( (m - \widehat{m}) (\phi_{s} - \iota_s) {t} \right) +o_p^{\mathcal{S}}(1) \, . \] From arguments similar to those used for ((ref)), since $\| (m - \widehat{m}) t\|_\infty = o_{p}(n^{-1/4})$ and $\what{m}\in \mathcal{G}_l(\mathcal{W}_{n})$ with probability approaching one, \[ \sqrt{n}\mathbb{P}_{n} \left( (m - \widehat{m}) (\phi_{s} - \iota_s) {t} \right) = \sqrt{n}P \left( (m - \widehat{m}) (\phi_{s} - \iota_s) {t} \right) +o_p^{\mathcal{S}}(1) \, . \] But $\sqrt{n}P \left( (m - \widehat{m}) (\phi_{s} - \iota_s) {t} \right) = 0$ as $\operatorname{E} (\phi_s(X)|W) = \iota_s(W)$. Finally, we obtain $ \sqrt{n}\mathbb{P}_{n} (\widetilde{\varepsilon}\phi_{s} \widehat{t}) = \sqrt{n}\mathbb{P}_{n} (\varepsilon (\phi_{s} - \iota_s) ) +o_p^{\mathcal{S}}(1) $ as expected.
(b) We now study the locally robust empirical process
A similar reasoning as in Part (a) allows replacing $\widehat{t}$ by $t$ and yields
based on the boundedness of $\varepsilon$, $\sqrt{n} \mathbb{P}_{n} |\widehat{t} - t| = o_{p}(1)$, and the boundedness in probability of $\sup_s \| (\phi_s - \what{\iota}_s) {t}\|_\infty$, $\sup_s \| (\phi_s - \what{\iota}_s) \widehat{t}\|_\infty$, $\|(\what{m}-m)t\|_\infty$, and $\|(\what{m}-m)\what{t}\|_\infty$, see Lemma (ref). From Lemma (ref), $\sup_s \|(\what{\iota}_s - \iota_s)t\|_\infty = o_{p}(n^{-1/4})$ and $\|(m-\what{m})t\|_\infty=o_P(n^{-1/4})$, hence \[ \sqrt{n}\mathbb{P}_{n} \left( (m - \widehat{m}) (\phi_{s} - \what{\iota}_s) {t} \right) = \sqrt{n}\mathbb{P}_{n} \left( (m - \widehat{m}) (\phi_{s} - {\iota}_s) {t} \right) +o_p^{\mathcal{S}}(1) = o_p^{\mathcal{S}}(1) \, , \] where the last equality is established in Part (a). Now, \[ \mathbb{P}_n \varepsilon (\phi_s-\widehat{\iota}_s) t = \mathbb{P}_n \varepsilon (\phi_s - \iota_s) t - \mathbb{P}_n \varepsilon (\widehat{\iota}_s - \iota_s) t \, , \] and $ \sup_s \left|\sqrt{n}\mathbb{P}_{n} (\varepsilon (\phi_{s} - \iota_s) (t - 1) ) \right| \leq C \sqrt{n}\mathbb{P}_{n} |t -1| = o_P(1)$ by Lemma (ref)-(i). Last, $ \sqrt{n}\mathbb{P}_n \varepsilon (\widehat{\iota}_s - \iota_s) t = o_{p}^\mathcal{S}(1) $ by Lemma (ref)-(i). Gathering results, $ \sqrt{n}\mathbb{P}_{n} (\widehat{\varepsilon} (\phi_{s} - \what{\iota}_s) \widehat{t}) = \sqrt{n}\mathbb{P}_{n} ({\varepsilon} (\phi_{s} - {\iota}_s) ) + o_p^{\mathcal{S}}(1) $.
We here consider statements relative to $P^\xi\otimes P$, the joint probability measure of both the bootstrap weights and the sample data.
(a) We study the bootstrap version of the bias corrected empirical process
From here we only stress the differences with the proof of Proposition 1. Proceeding as in the latter proof and using results in Lemma (ref),
For the last term, use $\|\widehat{B}^{*} t \|_{\infty} = o_{P}(n^{-1/4})$, Lemma (ref)-(ii)-(iv)-(v), and Lemma (ref)-(i) to show that
From the law of iterated expectations, $ P (\widehat{B}^{*} t \phi_s) = P (\widehat{B}^{*} t \iota_s)$. Since $\| [\widehat{B}^{*}-(\overline{\what{m}^{*}}-\overline{Y^{*}})/ f ] t \|_{\infty} = o_P(n^{-1/2})$ from Lemma (ref)-(viii),
where the second equality follows from a change of variable. Replace $(Y^{*}-\what{m}^{*})\what{t}$ by $\xi\varepsilon {t} + \xi\varepsilon(\what{t}-t)+\xi(m-\what{m})\what{t}+(\what{m}-\what{m}^*)\what{t}$ and proceed as in Proposition 1 to obtain
Gathering results,
A reasoning similar to Proposition 1 then yields $ \sqrt{n}\mathbb{P}_{n} (\xi \varepsilon (\phi_{s} - \iota_s) )t = \sqrt{n}\mathbb{P}_{n} (\xi \varepsilon (\phi_{s} - \iota_s) ) +o_P^{\mathcal{S}}(1) $, $\sqrt{n}\mathbb{P}_{n} \left(\xi (m - \widehat{m}) (\phi_{s} - \iota_s) {t} \right) = o_P^{\mathcal{S}}(1)$, and $\sqrt{n}\mathbb{P}_n((\what{m}-\what{m}^*)t(\phi_s-\iota_s))=o_P^\mathcal{S}(1)$. Hence $$\sqrt{n}\mathbb{P}_n \widetilde{\varepsilon}^*\what{t}\phi_s=\sqrt{n}\mathbb{P}_n\xi\varepsilon(\phi_s-\iota_s)+o_P^\mathcal{S}(1).$$
For the locally robust empirical process,
The proof proceeds along similar lines to show that the first term equals $\sqrt{n}\mathbb{P}_{n} (\xi \varepsilon (\phi_{s} - {\iota}_s) ) + o_P^{\mathcal{S}}(1)$ and the other terms are both $o_P^{\mathcal{S}}(1)$.
{(b)} Since the class $\{(y,w,x)\mapsto(y-m(w))(\phi_s(x)-\iota_s(w)):s\in \mathcal{S}\}$ is Donsker, $\sqrt{n}\mathbb{P}_n\varepsilon(\phi_s-\iota_s)$ converges weakly to a tight zero-mean Gaussian process $\mathbb{G}_s$ under $H_0$, see the main text. By the continuity of the Cramer-Von Mises functional and Proposition 1, $S_n^{BC}$ and $S_n^{LR}$ weakly converge to $\int|\mathbb{G}_s|^2\mu(ds)$. From bentkus_asymptotic_1993, this distribution is continuous, so pointwise convergence implies uniform convergence.
Using van_der_vaart_weak_1996, we get weak convergence of $\sqrt{n}\mathbb{P}_n \xi \varepsilon(\phi_s-\iota_s)$ in probability conditionally upon the initial sample to a tight zero-mean Gaussian process $\mathbb{G}'_s$ with the same covariance function as $\mathbb{G}_s$. From (a), the bootstrap statistics ((ref)) weakly converge to $\int|\mathbb{G}'_s|^2\mu(ds)$ in probability conditionally upon the initial sample. The desired result then follows.
{(c)} From (a) and a Glivenko-Cantelli property of the class $\{(y,w,x)\mapsto(y-m(w))(\phi_s(x)-\iota_s(w)):s\in \mathcal{S}\}$, $\mathbb{P}_n\widetilde{\varepsilon}\what{t}\phi_s= P \varepsilon (\phi_s - \iota_s) +o_P^\mathcal{S}(1)$ and similarly for $\mathbb{P}_n\widehat{\varepsilon}(\phi_s-\what{\iota})\what{t}$. Hence, $S_n^{LR}/n$ and $S_n^{BC}/n \mbox{$\stackrel{p}{\longrightarrow}\,$} \int|\operatorname{E}\varepsilon (\phi_s - \iota_s) |^2d\mu(s)$. From bierens_econometric_2017, under $H_1$, $\operatorname{E}\varepsilon (\phi_s - \iota_s) \neq 0 $ for almost all $s$ and $\int \left|\operatorname{E}\varepsilon (\phi_s - \iota_s) \right|^2 \, d\mu(s)>0$. From (b), it holds that the bootstrap statistics are bounded in probability, and the result follows.
The following lemma provides the regularity features needed to apply stochastic equicontinuity results. Similar results can be found in AndrewsHB and andrews_nonparametric_1995. The differences with respect to these works are the presence of the bias correction components $\overline{\what{m}}/\what{f}$ and the different assumptions on the bandwidths. Recall that $$\mathcal{W}_n:=\left\{ w\text{ : }f_W(w)\geq \tau_n/2\right\}\text{ . }$$
The following lemma provides a stochastic equicontinuity result. Similar results can be found in AndrewsHB or andrews_nonparametric_1995. Let us introduce some notation that will be used in the proof. For a generic space of functions $\mathcal{F}$ endowed with the $L_2(P)$ metric $||\cdot||_{2,P}$, we denote by $N(\epsilon,\mathcal{F},||\cdot||_{2,P})$ the $\epsilon$ covering number and with $N_{[\cdot]}(\epsilon,\mathcal{F},||\cdot||_{2,P})$ the $\epsilon$ bracketing number, see vaart_asymptotic_1998 and van_der_vaart_weak_1996.
\paragraph{Acknowledgements} We thank the Editor Michael Jansson and two referees for their comments that helped to improve the paper.