Testing exogeneity in the functional linear regression model

In functional linear regression models, goodness-of-fit tests are much more complicated to construct then e.g. in the multiple linear setting. This, among others stems to the fact that in functional linear regression models the $L_{2}$-distance of the slope function estimator to the true function has no proper limiting distribution. This was shown in [Cardot et al. (2006)] and [Ruymgaart et al. (2000)] for two estimators in the classical functional linear regression model under exogeneity. It turns out, that the lack of a proper limiting distribution also applies for other estimators using different model assumptions. This phenomenon inherent to functional data setups is probably one of the reasons why goodness-of-fit testing is generally not that widely developed for functional regression models yet. In particular, desirable counterparts of standard tests that are well-established in the multiple linear model are still missing in the functional linear setting.

In functional data settings, existing goodness-of-fit tests are described in [Müller and Stadtmüller (2005)], who use a suitable scalar product to transform the functions to a different space using the autocovariance operator to obtain a test statistic having a proper limiting distribution. Further approaches are given in [Cuesta-Albertos et al. (2019)] and [García-Portugués et al. (2014), García-Portugués et al. (2020)], who use random projections together with empirical process techniques.

In practice, one important model assumption is the exogeneity of the regressor. Especially in economics, this assumption is often violated such that the regressors are correlated with the error terms which leads to endogeneity issues. Estimating in such a model is an inverse problem. Neglecting endogeneity generally results in inconsistent estimators. Hence, it is important to test the data for exogeneity first. If the null hypothesis of exogeneity is rejected, different estimators such as e.g. instrumental variable (IV) estimators are required to achieve consistent estimation. See [Johannes (2016)], [Florens et al. (2011)] or [Florens and Van Bellegem (2014)], who consider such IV estimators in functional regression setups and derive asymptotic theory. While in the multiple linear regression model the Hausman test (see [Hausman (1978)] und [Wu (1973)]) is a standard and natural approach for testing exogeneity, this method cannot be transferred directly to the functional linear model since it is based on the $L_{2}$-distance of two slope function estimators due to the following proposition which transfers the results in [Cardot et al. (2006)] and [Ruymgaart et al. (2000)] to the present setting.

In the following, let $\hat{\beta}$ denote the estimator of the slope function in the exogeneous model described in [Johannes (2013)] and [Joh13add], which is consistent under exogeneity, but inconsistent under endogeneity, and by $\hat{\beta}_{IV}$ the IV estimator in the endogeneous functional linear model given in [Johannes (2016)], which is consistent in both cases. Then, we get the following result.

The proof of this result mainly goes along the lines of the one in [Cardot et al. (2006)], see also [Dorn (2021)] for further details. This is why we just state the result here and use it as motivation for a different approach in the following. Motivated by the fact, that in contrast to the $L_{2}$-distance, the projection error typically has an asymptotic distribution (see e.g. [Müller and Stadtmüller (2005)] and [Florens and Van Bellegem (2014)]), we propose to use the sum of the squared difference of projections of the two estimators as test statistic.

The rest of the paper is organized as follows. In Section 2, we state the model assumptions, construct the test statistic and derive its asymptotic distribution. As the limiting distribution turns out to depend on unknown functional nuisance parameters, which are difficult to estimate, we propose residual-based bootstrap methods in Section 3 and prove their consistency. The finite sample performance of all discussed tests is investigated in Section 5. All longer proofs are deferred to the Appendix and additional auxiliary results to a supplement.

We consider the functional linear regression model

$$Y=\int_{[0,1]}X(t)\beta(t)dt+U=\left\langle\beta,X\right\rangle+U,$$

(2.1)

where $Y$ is a real-valued random variable, $U$ is a real-valued error term with $E[U]=0$ and $E[U^{2}]=\sigma^{2}{\in(0,\infty)}$, $X$ is a functional random variable with values in $L_{2}([0,1])$ such that $\int_{0}^{1}E|X(t)|^{2}\,dt<\infty$. In this setup, the error variance $\sigma^{2}$ is unknown, and $\beta$ is an unknown function from the Sobolev space of periodically extendable square integrable functions denoted by

$$\mathcal{W}_{\nu}:=\Big{\{}f\in L^{2}[0,1]:\;\|f\|_{\nu}^{2}:=\sum_{k\in\mathbb{Z}}\gamma_{k}^{\nu}|\langle f,\phi_{k}\rangle|^{2}<\infty\Big{\}},$$

(2.2)

where $(\phi_{k})_{k\in\mathbb{Z}}$ is the Fourier basis of $L^{2}([0,1])$, $\nu\in\mathbb{R}$ and $\gamma_{k}=1+|2\pi k|$, $k\in\mathbb{Z}$, see e.g. [Neuba, Neubb], [MR96] or [Tsybakov (2004)]. In the setup of (2.1), we will speak of exogeneity (and call $X$ an exogenous regressor), if

$$H_{0}:\ \ E[X(t)U]=0\mbox{ for all }t\in[0,1].$$

(2.3)

Otherwise, we will speak of endogeneity (and call $X$ an endogeneous regressor), if

$$H_{1}:\ \ E[X(t)U]\neq 0\mbox{ for at least one }t\in[0,1].$$

(2.4)

For consistent estimation in the endogeneous case, we assume to additionally have a functional instrumental variable $W$ with values in $L_{2}([0,1])$ such that $\int_{0}^{1}E|W(t)|^{2}\,dt<\infty$ and $E[UW(t)]=0$ for all $t\in[0,1]$. For the sake of simplicity, it is often assumed in the literature, that $E[X(t)]=E[W(t)]=0$ holds for all $t\in[0,1]$. However, the general case can be handled along the same lines by centering with the sample mean in a first step and our results are stated for the general case. For estimating the cross-covariance operator, we also assume that $(X,W)$ is second-order stationary, see [Johannes (2016)].

By imposing continuity of $c_{X}$, we have that whenever (2.4) holds for one $t\in[0,1]$, this immediately implies $E[X(t)U]\neq 0$ on some set with positive Lebesgue measure. This condition ensures, that the test statistic proposed in the following can be used to consistently test for the null hypothesis (2.3) against alternatives (2.4).

Note, that $c_{X}$ and $c_{W}$ define the kernels of the covariance operators $\Gamma_{X}$ of $X$ and $\Gamma_{W}$ of $W$, respectively, and $c_{WX}$ is the kernel of the cross covariance operator $\Gamma_{WX}$ of $X$ and $W$. The (joint) weak stationarity of $(X,W)$ ensures, that both covariance operators as well as the cross covariance operator have the same exponential system of eigenfunctions, which we denote by $(\phi_{k})_{k\in\mathbb{N}}$. Hence, let $(x_{k},\phi_{k})_{k\in\mathbb{N}}$ be the eigensystem of $\Gamma_{X}$, $(w_{k},\phi_{k})_{k\in\mathbb{N}}$ the eigensystem of $\Gamma_{W}$ and $(c_{k},\phi_{k})_{k\in\mathbb{N}}$ the eigensystem of $\Gamma_{WX}$. Furthermore, denote $\lambda_{k}=\frac{|c_{k}|^{2}}{w_{k}}$.

The last assumptions ensures, that the linear prediction of $X$ with respect to $W$ is well defined.

In principle, if they were available, IV estimation would be based on the optimal instrument $\tilde{W}$ defined by

$\displaystyle\tilde{W}=\Gamma_{WX}\Gamma_{W}^{-1}W=\sum_{k\in\mathbb{Z}}\frac{\overline{c_{k}}}{w_{k}}\langle W,\phi_{k}\rangle\phi_{k}$

and the eigenvalues $(\lambda_{k})_{k\in\mathbb{N}}$ of the corresponding cross covariance operator $\Gamma_{\tilde{W}X}$. However, this is usuall not the case and the optimal instrument respectively the corresponding eigenvalues of the cross covariance operator have to be estimated. Note, that $\tilde{W}$ could be exactly computed from $X$ and $W$, if the (cross) covariance operators were known and remark that $\lambda_{k}=\frac{|c_{k}|^{2}}{w_{k}}\leq x_{k}$ for all $k\in\mathbb{Z}$.

In the following, let $\{(X_{i},W_{i},Y_{i})\}_{i=1,\ldots,n}$ be independent and identically distributed (i.i.d.) copies of $(X,W,Y)$ and suppose (2.3) is valid. Then, we can consistently estimate the unknown slope function $\beta$ due to [Johannes (2013)] and [Johannes (2016)] in two different ways. For this purpose, let $(\alpha_{n})_{n\in\mathbb{N}}$ be a sequence of regularization parameters such that $\alpha_{n}>0$ for all $n\in\mathbb{N}$ and $\lim_{n\to\infty}\alpha_{n}=0$. To simplify notation, we will write $\alpha$ for the regularization keeping in mind that it still depends on $n$. Since the covariance operators and therefore the corresponding eigenvalues are unknown, they have to be estimated in a first step. Further, let $\Gamma_{WX,n},\Gamma_{X,n},\Gamma_{W,n}:\,L_{2}([0,1])\to L_{2}([0,1])$ denote the empirical versions of $\Gamma_{WX},\Gamma_{X}$ and $\Gamma_{W}$, respectively, defined by

$\displaystyle\Gamma_{WX,n}f:=\frac{1}{n}\sum_{i=1}^{n}\langle W_{i},f\rangle X_{i},\quad\Gamma_{X,n}f$

$\displaystyle:=\frac{1}{n}\sum_{i=1}^{n}\langle X_{i},f\rangle X_{i},\quad\text{and}\quad\Gamma_{W,n}f$

$\displaystyle:=\frac{1}{n}\sum_{i=1}^{n}\langle W_{i},f\rangle W_{i}$

for $f\in L_{2}([0,1])$. These estimators as well as the deduced estimators

	$\displaystyle\hat{w}_{k}:=\frac{1}{n}\sum_{i=1}^{n}|\langle W_{i},\phi_{k}\rangle|^{2},\,$	$\displaystyle\hat{x}_{k}:=\frac{1}{n}\sum_{i=1}^{n}|\langle X_{i},\phi_{k}\rangle|^{2},$
	$\displaystyle\hat{c}_{k}:=\frac{1}{n}\sum_{i=1}^{n}\langle\phi_{k},X_{i}\rangle\langle W_{i},\phi_{k}\rangle,\,$	$\displaystyle\hat{\lambda}_{k}:=\frac{|\hat{c}_{k}|^{2}}{\hat{w}_{k}}I\{\hat{w}_{k}\geq\alpha\}$

for the eigenvalues $w_{k}$, $x_{k}$, $c_{k}$ and $\lambda_{k}$, respectively, are consistent for all $k\in\mathbb{Z}$. Hence, observations of the optimal linear instrument $\widetilde{W}$ can be estimated by

$\displaystyle\widetilde{W}_{n,i}:=\sum_{k\in\mathbb{Z}}\frac{\overline{\hat{c}_{k}}}{\hat{w}_{k}}I\{\hat{w}_{k}\geq\alpha\}\langle W_{i},\phi_{k}\rangle\phi_{k},\quad i=1,\ldots,n,$

and the corresponding cross covariance operator by

$\displaystyle\widetilde{\Gamma}_{n}:=\frac{1}{n}\sum_{i=1}^{n}\langle\widetilde{W}_{n,i},\cdot\rangle X_{i}=\frac{1}{n}\sum_{k\in\mathbb{Z}}\frac{\overline{\hat{c}_{k}}}{\hat{w}_{k}}I\{\hat{w}_{k}\geq\alpha\}\sum_{i=1}^{n}\langle\cdot,X_{i}\rangle\langle W_{i},\phi_{k}\rangle\phi_{k}.$

(2.6)

This allows to construct the IV-based estimator $\hat{\beta}_{IV}$ of the slope function $\beta$ defined by

$$\hat{\beta}_{IV}:=\sum_{k\in\mathbb{Z}}\frac{\hat{g}_{k}}{\hat{\lambda}_{k}}I\{\hat{\lambda}_{k}\geq\gamma_{k}^{\nu}\alpha\}\phi_{k}=\sum_{k\in\mathbb{Z}}\frac{\frac{1}{n}\sum_{i=1}^{n}\langle W_{i},\phi_{k}\rangle Y_{i}}{\hat{c}_{k}}I\{\hat{\lambda}_{k}\geq\gamma_{k}^{\nu}\alpha\}I\{\hat{w}_{k}\geq\alpha\}\phi_{k},$$

(2.7)

where

$\displaystyle\hat{g}_{k}$

$\displaystyle=\frac{1}{n}\sum_{i=1}^{n}Y_{i}\langle\tilde{W}_{n,i},\phi_{k}\rangle.$

As shown in [Johannes (2016)], under Assumptions 1 and 2 the estimator $\hat{\beta}_{IV}$ is consistent under the exogeneity assumption (2.3) as well as under endogeneity of (2.4). In contrast, again under Assumptions 1 and 2, the estimator

$$\hat{\beta}=\sum_{k\in\mathbb{Z}}\frac{\frac{1}{n}\sum_{i=1}^{n}\langle X_{i},\phi_{k}\rangle Y_{i}}{\hat{x}_{k}}I\{\hat{\lambda}_{k}\geq\alpha\gamma_{k}^{\nu}\}\phi_{k}$$

(2.8)

is only consistent under the exogeneity assumption (2.3) (see [Johannes (2013)]) and inconsistent under endogeneity of (2.4). Note, that in comparison to the original definition in [Johannes (2013)], for $\hat{\beta}$, we use the same indicator function $I\{\hat{\lambda}_{k}\geq\alpha\gamma_{k}^{\nu}\}$ as in $\hat{\beta}_{IV}$. It turned out, that the tests perform better if the same regularization is used in both estimators although it might not be the best choice for estimating $\beta$ by $\hat{\beta}$ under assumption (2.3).

Based on the two estimators (2.7) and (2.8), we construct the test statistic as

$\displaystyle T_{n}$

$\displaystyle=\frac{1}{n}\sum_{i=1}^{n}\left|\left\langle\hat{\beta}_{IV}-\hat{\beta},X_{i}\right\rangle\right|^{2}=\left\langle\hat{\beta}_{IV}-\hat{\beta},\Gamma_{X,n}\left(\hat{\beta}_{IV}-\hat{\beta}\right)\right\rangle.$

(2.9)

The last representation above corresponds to the idea used in [Müller and Stadtmüller (2005)] to construct a goodness-of-fit test. The equivalence of both approaches can be seen by using the singular value decomposition for the estimators and for the covariance operator.

For the next results, different moment conditions for $X$, $W$ and $U$ are required. To simplify the notation, we introduce the following sets. In doing so, we assume, that all conditions on $X$ and $W$ mentioned above are fulfilled and define

	$\displaystyle\mathcal{F}_{\eta}^{m}$	$\displaystyle:=\Big{\{}(X,W)\Big{|}\sup_{k\in\mathbb{Z}}\text{E}\left|\frac{\langle X,\phi_{k}\rangle}{\sqrt{x_{k}}}\right|^{m}\leq\eta\text{ and }\sup_{k\in\mathbb{Z}}\text{E}\left|\frac{\langle W,\phi_{k}\rangle}{\sqrt{w_{k}}}\right|^{m}\leq\eta\Big{\}},$		(2.10)
	$\displaystyle\mathcal{G}_{\eta}^{m}$	$\displaystyle:=\Big{\{}X\Big{|}\Gamma_{X}>0\text{ and }\sup_{k\in\mathbb{Z}}\text{E}\left|\frac{\langle X,\phi_{k}\rangle}{\sqrt{x_{k}}}\right|^{m}\leq\eta\Big{\}}.$		(2.11)

In the following, for an operator $\Delta$, we denote by $\Delta^{\dagger}$ the regularized inverse of the operator, that is

$$\Delta^{\dagger}=\sum_{k\in\mathbb{Z}}\frac{1}{\delta_{k}}I\{|\delta_{k}|>\alpha\gamma_{k}^{\nu}\langle\cdot,\phi_{k}\rangle\phi_{k}\}$$

and we define

$$t_{n}^{2}:=\|(\tilde{\Gamma}_{X,n}^{\dagger}-\Gamma_{X}^{\dagger})\Gamma_{X}\|_{HS}^{2}=\sum_{k\in\mathcal{K}_{n}}\left(\frac{x_{k}w_{k}}{|c_{k}|^{2}}-1\right)^{2},$$

(2.12)

where $\|\cdot\|_{HS}$ denotes the Hilbert-Schmidt norm and we set

$$\mathcal{K}_{n}:=\{k\in\mathbb{Z}\mid\lambda_{k}\geq\alpha\gamma_{k}^{\nu}\}.$$

Now, we are in a position to state an asymptotic result for the test statistic.

To apply the above result for testing, the bias and variance term have to be estimated. To this end, note that $\sigma^{2}$ can be consistently estimated by

$${\hat{\sigma}}_{n}^{2}=\frac{1}{n}\sum_{i=1}^{n}(Y_{i}-\langle\hat{\beta}_{IV},X_{i}\rangle)^{2}$$

(2.15)

due to the law of large numbers and since $\frac{1}{n}\sum_{i=1}^{n}\langle\beta-\hat{\beta}_{IV},X_{i}\rangle^{2}=o_{P}(1)$ by similar calculations as in the derivation of the asymptotic distribution of $T_{n}$.

Using Corollary 2.2, it is possible to construct a test for the null hypothesis

$$H_{0}:\ \mbox{E}[X(t)U]=0\mbox{ for all }t\in[0,1]$$

(2.16)

against

$$H_{1}:\ \mbox{E}[X(t)U]\neq 0\mbox{ for at least one }t\in[0,1].$$

(2.17)

For given size $\gamma\in(0,1)$, we can reject $H_{0}$ if

$$\frac{n}{\hat{t}_{n}}\frac{T_{n}-\hat{\mathfrak{B}}_{n}-\hat{\mathfrak{R}}_{n}}{\sqrt{\hat{\mathfrak{V}}_{n}}}>u_{1-\gamma}$$

(2.18)

where $u_{1-\gamma}$ denotes the $(1-\gamma)$-quantile of the standard normal distribution. That is, we get a one-sided test for $H_{0}$ against $H_{1}$. In the special case $\mu_{X}=0$ we can neglect the additional bias term which avoids the use of its plug-in estimator such that the test has the simpler structure $I\{n(T_{n}-\hat{\mathfrak{R}}_{n})/{{\hat{t}_{n}}\sqrt{\hat{\mathfrak{V}}_{n}}}>u_{1-\gamma}\}$.

In practice, we do not know, if $\mu_{X}=0$ such that a naive application of the asymptotic test without estimating ${\mathfrak{B}}_{n}$ could result in wrong decisions. In addition, asymptotic tests based on plug-in methods as above usually exhibit a smaller power compared to other methods. This is due to the additional estimation step. The bootstrap version of the test discussed in the next section is expected to have better finite sample behavior, since it is not required to estimate the unknown bias and variance. This has additionally the effect, that we need not distinguish between the cases $\mu_{X}=0$ and $\mu_{X}\neq 0$ which is a clear advantage of the bootstrap test.

In this section, we use residual-based bootstrap procedures to estimate the distribution of

$$\frac{n}{t_{n}}\left(T_{n}-\mathfrak{B}_{n}-\mathfrak{R}_{n}\right)$$

under the null of exogeneity . To this end, we first estimate the residuals from the original data set and define

$\displaystyle\hat{U}_{i}=Y_{i}-\langle\hat{\beta}_{IV},X_{i}\rangle,\quad i=1,\ldots,n,$

where we use the IV-based estimator, because it is consistent under the null hypothesis as well as under the alternative. However, using the classical estimator $\hat{\beta}$ would also result in a proper bootstrap scheme to approximate the distribution of the test statistics under the null of exogeneity, since the independence of error and regressor in the bootstrap sample is achieved by the (fixed-design) bootstrap procedure itself. However, to get bootstrap data that mimics the true distribution under the null hypothesis of exogeneity given the original sample as close as possible, the IV-based estimator turns out to be more natural and performs better in simulations. In the sequel, different versions of residual-based bootstraps are considered. All bootstrap methods will follow these steps

Step 1.)

Given i.i.d. observations $(X_{i},W_{i},Y_{i})$, $i=1,\ldots,n$, we generate a bootstrap sample $(X_{i},W_{i},Y_{i}^{*})$, $i=1,\ldots,n$, by

$\displaystyle Y_{i}^{*}=\langle\hat{\beta}_{IV},X_{i}\rangle+U_{i}^{*},$

where the bootstrap errors $U_{i}^{*}$ are generated from the residuals $\hat{U}_{1},\ldots,\hat{U}_{n}$ in such a way that the conditional independence of $U_{i}^{*}$ and $(X_{i},W_{i})$ is ensured. A thorough discussion, which types of bootstrap are appropriate in this sense follows in the next subsection.

Step 2.)

From $(X_{i},W_{i},Y_{i}^{*})$, $i=1,\ldots,n$, a bootstrap test statistic $T_{n}^{*}$ is calculated.

Step 3.)

Repeat Steps 1.) and 2.) $B$ times, where $B$ is large, to get bootstrap realizations $T_{n}^{*,1},\ldots,T_{n}^{*,B}$ of the test statistic and denote by $q_{1-\gamma}^{*}=T_{n}^{*,(\lfloor B(1-\gamma)\rfloor)}$ the corresponding empirical $(1-\gamma)$-quantile.

As the bootstrap errors are generated such that conditional independence of $U_{i}^{*}$ and $(X_{i},W_{i})$ is ensured, the bootstrap automatically adopts the exogeneity assumption. For the naive (Efron-type)residual bootstrap, this is trivially the case, because the bootstrap errors are drawn independently with replacement from the residuals, and for the wild bootstrap, since suitable bootstrap multiplier variables $V_{i}$ will also be drawn independently from $X_{i}$ and $W_{i}$.

Based on this result, we can again construct a one-sided test for the hypotheses (2.17) which rejects the null hypothesis if $T_{n}>q_{1-\gamma}^{*}$ from Step 3 since $T_{n},T_{n}^{\ast}\geq 0$ and both asymptotically have the same bias and variance.

While the above results are stated for the spectral-cut-off estimators as proposed in [Johannes (2013)] and [Johannes (2016)], it is also possible to derive analogue results for other types of estimators like cut-off as in [Müller and Stadtmüller (2005)] or ones based on Tikhonov or ridge-type regularization. A quite general approach is given in [Cardot et al. (2006)] with a sequence of regularization functions $f_{n}:[c_{n},\infty)\to\mathbb{R}_{0}^{+}$ such that $f_{n}$ is decreasing on $[c_{n},2z_{1}-z_{2}]$ where $(z_{j})_{j\in\mathbb{Z}}$ are the eigenvalues of the relevant covariance operator and $(c_{n})_{n\in\mathbb{N}}$ is a decreasing sequence of positive values with $c_{n}<z_{1}$. Furthermore $\lim_{n\to\infty}\sup_{z\geq c_{n}}|zf_{n}(z)-1|=o(1/\sqrt{n})$ and $f_{n}$ is differentiable on $[c_{n},\infty)$ which replaces Assumption 3. While the estimator $\hat{\beta}$ from (2.8) above does not completely fit this situation it is not neccessary to consider this modification if one is only interested in testing goodness-of-fit in an exogeneous model (2.1). For the sake of shorter notation we assume here $\mu_{X}\equiv 0$. If $\tilde{\beta}$ denotes the estimator proposed in [Cardot et al. (2006)] we obtain under Assumption 1 and the moment conditions in Theorem 2.1 the following result

$$\frac{n}{s_{n}}\left(\frac{1}{n}\sum_{i=1}^{n}\left|\left\langle\tilde{\beta}-\beta,X_{i}\right\rangle\right|^{2}-\mathfrak{R}_{n}\right)\stackrel{{\scriptstyle\mathcal{D}}}{{\to}}\mathcal{N}(0,\mathfrak{V})$$

with $s_{n}=\sum_{k\in\mathbb{Z}}x_{k}^{4}f_{n}^{4}(x_{k})$, $\mathfrak{V}$ as in Theorem 2.1 and

$$\mathfrak{R}_{n}=\frac{1}{n}\left(\sigma^{2}+\sum_{m\in\mathbb{Z}}|\langle\beta,\phi_{m}\rangle|^{2}x_{m}\right)\sum_{k\in\mathbb{Z}}f_{n}(x_{k})=O\left(\frac{1}{\sqrt{n}}\right).$$

If is straight forward to also generalize the instrumental variable estimator to other regularization schemes. We get an estimator

$$\tilde{\beta}_{IV}=\sum_{k\in\mathbb{Z}}\hat{g}_{k}f_{n}(\hat{\lambda}_{k})$$

and, if we are willing to assume exogeneity here, derive by the same arguments as above

$$\frac{n}{s_{n,IV}}\left(\frac{1}{n}\sum_{i=1}^{n}\left|\left\langle\tilde{\beta}_{IV}-\beta,X_{i}\right\rangle\right|^{2}-\mathfrak{R}_{n}\right)\stackrel{{\scriptstyle\mathcal{D}}}{{\to}}\mathcal{N}(0,\mathfrak{V})$$

with $s_{n,IV}=\sum_{k\in\mathbb{Z}}x_{k}^{2}\lambda_{k}^{2}f_{n}^{4}(\lambda_{k})$, $\mathfrak{V}$ as in Theorem 2.1 and

$$\mathfrak{R}_{n}=\frac{1}{n}\mathfrak{V}^{1/2}\sum_{k\in\mathbb{Z}}x_{k}\lambda_{k}f_{n}(\lambda_{k})=O\left(\frac{1}{\sqrt{n}}\right).$$

The assumption of exegeneity is in this case not realistic because one would only use the instrumental variable estimator under endogeneity. Proving an analogue result under endogeneity is in principle possible but the proof differs in several points from the one presented here.