# Nonparametric Regression with Common Shocks

^{1}

^{1}

## Abstract

**:**

## 1. Introduction

## 2. Regression Model and Conditional Densities

#### Data Generation

**Assumption 1**

#### Existence of Conditional Densities

**Assumption 2**

**Lemma**

**1.**

**Example**

**1.**

## 3. Regression Estimator

**Condition**

**1.**

**Condition**

**2.**

**Condition**

**3.**

**Condition**

**4.**

**Condition**

**5.**

**Condition**

**6.**

**Condition**

**7.**

**Condition**

**8.**

**Remark**

**1.**

**Proposition**

**1.**

- 1.
- $\widehat{m}\left(x\right)\stackrel{p}{\u27f6}m\left(x,C\right)$ as $n\to \infty .$
- 2.
- $\widehat{m}\left(x\right)-m\left(x,C\right)={O}_{p}\left({n}^{-\frac{2}{4+k}}\right).$
- 3.
- Suppose also that $\int {\left|K\left(u\right)\right|}^{2+\delta}du<\infty $ and $E\left[{\left|{\epsilon}_{i}\right|}^{2+\delta}\right]<\infty $, for some $\delta >0$. Define ${\sigma}^{2}\left(x,C\right)=E\left({\epsilon}_{i}^{2}|{X}_{i}=x,C\right)$. Then, (i) as $n\to \infty $:$$\sqrt{n{h}_{n}^{k}}\left(\widehat{m}\left(x\right)-E\left[\widehat{m}\left(x\right)|X={\left\{{x}_{i}\right\}}_{i=1}^{n},C\right]\right)\stackrel{d}{\u27f6}\left(\frac{{\sigma}^{2}\left(x,C\right)}{f\left(x|C\right)}\int {K}^{2}\left(u\right)du\right)N\left(0,1\right)$$$$\sqrt{n{h}_{n}^{k}}\left(\widehat{m}\left(x\right)-m\left(x,C\right)\right)\stackrel{d}{\u27f6}\left(\frac{{\sigma}^{2}\left(x,C\right)}{f\left(x|C\right)}\int {K}^{2}\left(u\right)du\right)N\left(0,1\right)$$

**Remark**

**2.**

**Remark**

**3.**

**Remark**

**4.**

**Remark**

**5.**

## 4. Conclusions

## Supplementary Materials

Supplementary File 1## Acknowledgments

## Conflicts of Interest

## Appendix A. Disintegration Theory

**Definition**

**1.**

- 1.
- ${P}_{c}\left\{C\ne c\right\}=0$, for Q-almost all $c\in \mathcal{C}$;and for each nonnegative measurable function f on Ω:
- 2.
- the map $c\mapsto \int f\left(\omega \right)d{P}_{c}\left(\omega \right)$ is $\mathcal{B}$-measurable; and
- 3.
- the equality $\int f\left(\omega \right)dP=\int \left[\int f\left(\omega \right)d{P}_{c}\left(\omega \right)\right]dQ\left(c\right)$ holds.

**Definition**

**2.**

**Definition**

**3.**

- 1.
- ${\lambda}_{c}$ is a sigma-finite measure on $\mathcal{F}$ concentrated on $\left\{C=c\right\}$, that is ${\lambda}_{c}\left\{C\ne c\right\}=0$ for μ-almost all c; and for each nonnegative measurable function f on Ω:
- 2.
- the map $c\mapsto \int f\left(\omega \right)d{\lambda}_{c}\left(\omega \right)$ is $\mathcal{B}$-measurable; and
- 3.
- the equality $\int f\left(\omega \right)d\lambda =\int \left[\int f\left(\omega \right)d{\lambda}_{c}\left(\omega \right)\right]d\mu \left(c\right)$ holds.

**Theorem**

**1.**

**Theorem**

**2.**

- 1.
- The image measure $Q=P\left({C}^{-1}\right)$ (i.e., the probability distribution of C induced by P) is absolutely continuous with respect to μ, with density $q\left(c\right)\equiv \int f\left(\omega \right)d{\lambda}_{c}\left(\omega \right)$.
- 2.
- The set $\left\{\left(\omega ,c\right)\in \mathrm{\Omega}\times \mathcal{C}:q\left(c\right)=\infty \phantom{\rule{4.pt}{0ex}}\mathrm{or}\phantom{\rule{4.pt}{0ex}}q\left(c\right)=0\right\}$ has zero $\mu \otimes \mathrm{\Lambda}$ measure.
- 3.
- The probability measure P has conditional distribution $\left\{{P}_{c}:c\in \mathcal{C}\right\}$ given C, where ${P}_{c}$ is defined by having density:$$f\left(\omega |c\right)\equiv \frac{f\left(\omega \right)}{q\left(c\right)}\left\{0<q\left(c\right)<\infty \right\}$$

^{1.}See, for example, Arbia [1], the proceedings of the 2008 Cowles Summer Conference [2], the special issue of the Journal of Econometrics (“Analysis of Spatially Dependent Data,” 2007, 140(1), edited by Baltagi, Kelejian and Prucha), and the special issue of Econometrics (“Spatial Econometrics,” 2015, edited by Arbia and Lee). For recent surveys, see [3,4,5].^{3.}Formally, the probability limit of the kernel estimator for a nonstationary process can be obtained using the concept of local time, as in Wang and Phillips [26]. However, the probability limit of the kernel regression estimator may not be measurable with respect to the conditioning variables, including the common shocks. This is a particularly important problem when we extend the results to panel data models, as in Souza-Rodrigues [27].^{4.}Although separability is not a necessary condition, it seems difficult to avoid it if we are to obtain the existence of conditional densities; see the discussion about the role of separability in the Appendix. Note that several separable metric spaces satisfying the sufficient conditions are available, but careful interpretation is needed in particular cases. For instance, suppose that an infinite-dimensional common shock can be well-approximated by a finite dimensional object. Because the sigma-field generated by the common shock may be different from the sigma-field generated by the approximating object, the conditional expectations given the common shock and given the finite-dimensional object are different. Ignoring this difference leads to problems such as the Borel paradox.^{5.}The motivation for this application is that numerous studies have documented an inverse relationship between hospital volumes of operations and mortality rates (see [32]). This suggests that thousands of deaths per year could have been prevented if hospitals with inadequate experience (i.e., with low volume of operations) had performed fewer surgical procedures. The evidence, however, is weak for most operations. Furthermore, existing papers have estimated parametric models that may be misspecified and have not considered the potential correlation between hospital volume of operations and hospital unobserved quality.^{6.}The second step runs a nonparametric instrumental variable regression across groups (hospitals) of the predicted outcome obtained in the first step on the group-level observables. It separates the impacts of group-level observables (hospital volume of surgeries) and unobservables (hospital unobserved quality).^{8.}In a panel data setting, one typically allows for time-varying regressors ${X}_{it}$, but restricts ${S}_{i}$, so that it does not vary over time, and the common shock C, so that it does not vary across individuals. Fixed-effect panel data models let ${X}_{it}$ and ${S}_{i}$ be correlated.^{9.}Note that this approach does not require known economic distances, but can readily accommodate them by taking ${U}_{i}={\sigma}_{i}\left({X}_{i}\right){\sum}_{j=1}^{\infty}{b}_{ij}{e}_{i}$, $e=\left({e}_{1},...,{e}_{n}\right)$ and by making some assumptions regarding how ${b}_{ij}$ depends on the distance $\left|i-j\right|$.^{10.}When Andrews [8] specializes to factor structure models, he imposes more restrictions on the common shocks, which makes his approach more similar to ours.^{11.}A regular conditional probability, $Pr\left(Y|X=x\right)$, is a family of probability distribution, such that (i) for a fixed x, $Pr\left(\text{\xb7}|X=x\right)$ is a probability measure and (ii) for a fixed measurable set A, $Pr\left(A|X=x\right)$ is a measurable function mapping x to $\left[0,1\right]$.^{12.}The measure λ is Radon if (i) $\lambda \left(K\right)<\infty $ for each compact K and $\lambda \left(B\right)=sup\left\{\lambda \left(K\right):B\supseteq K,\phantom{\rule{4.pt}{0ex}}K\phantom{\rule{4.pt}{0ex}}\mathrm{compact}\right\}.$^{13.}It is possible to characterize all of the objects in Assumption 2 when $\mathcal{W}=\mathcal{Z}\times \mathcal{C}$. First, we have that (i) $\mathcal{W}$ is a separable metric space provided that $\mathcal{C}$ is a separable metric space, as well, and (ii) the Borel σ-field $\mathcal{A}$ on $\mathcal{W}$ equals the product Borel σ-field ${\mathcal{A}}_{Z}\otimes \mathcal{B}$, where we denote ${\mathcal{A}}_{Z}$ the Borel σ-field on $\mathcal{Z}$ (see [46], Proposition 1.5). Second, let ${\pi}_{c}$ be the projection of $\mathcal{W}$ onto the coordinate space $\mathcal{C}$, i.e., ${\pi}_{c}:\mathcal{W}\to \mathcal{C}$. Then, (i) the sub-sigma field ${\pi}_{c}^{-1}\left(\mathcal{B}\right)$ is contained in $\mathcal{A}$ and (ii) because $C\left(w\right)={\pi}_{c}\left(w\right)$, for all $w\in \mathcal{W}$; the sigma-field generated by C is $\sigma \left(C\right)={\pi}_{c}^{-1}\left(\mathcal{B}\right)\subset \mathcal{A}$. Furthermore, if we define the sigma-finite Radon λ on $\left(\mathcal{W},\mathcal{A}\right)$ to be the product measure $\lambda =\nu \otimes \mu $, where $\nu \phantom{\rule{0.166667em}{0ex}}\phantom{\rule{4pt}{0ex}}$ is defined on $\left(\mathcal{Z},{\mathcal{A}}_{Z}\right)$ and μ on $\left(\mathcal{C},\mathcal{B}\right)$, then the measure $\lambda \left({C}^{-1}\right)$ induced by C and λ on $\left(\mathcal{C},\mathcal{B}\right)$ equals μ, and so, $\lambda \left({C}^{-1}\right)$ is (trivially) absolutely continuous with respect to μ. Finally, we have to assume both ν and μ are sigma-finite Radon, so that λ is sigma-finite Radon on $\mathcal{A}$, as well.^{14.}Note that we can manipulate the conditional density (6) on $\mathcal{Z}\otimes \mathcal{C}$ as is usually done. Fix $C=c$ and think of $\mathcal{Z}\otimes \left\{c\right\}$ as a copy of $\mathcal{Z}$ embedded into the product space. For a fixed $c\in \mathcal{C}$, take the measure ${\lambda}_{c}$ living on $\mathcal{Z}\otimes \left\{c\right\}$ to coincide with the Lebesgue measure on $\mathcal{Z}$. If $r($·) is a vector-valued function with $E\u2225r\left(Z\right)\u2225\phantom{\rule{0.166667em}{0ex}}<\infty $, then:$$E\left[r\left(Z\right)|C=c\right]=\int r\left(\tilde{z}\right)d{P}_{c}\left(\tilde{z}\right)=\int r\left(\tilde{z}\right)f\left(\tilde{z}|c\right)d{\lambda}_{c}\left(\tilde{z}\right)=\int r\left(\tilde{z}\right)f\left(\tilde{z}|c\right)d\tilde{z}.$$^{15.}Any infinite-dimensional separable Hilbert space, say $\mathcal{H}$, is isometrically isomorphic to a suitable ${\ell}_{2}\left(I\right)$, where the cardinality of the set I is the cardinality of an arbitrary Hilbertian basis for $\mathcal{H}$, i.e., there exists a linear operator $L:\mathcal{H}\to {\ell}_{2}\left(I\right)$, such that ${\u2225Lh\u2225}_{2}={\u2225h\u2225}_{\mathcal{H}}$, where $h\in \mathcal{H}$, ${\u2225\text{\xb7}\u2225}_{\mathcal{H}}$ is the norm on $\mathcal{H}$ and ${\u2225\text{\xb7}\u2225}_{2}$ is the ${\ell}_{2}$-norm.^{16.}Conditioning on the event $\left\{\left({C}_{1},{C}_{2},...\right)=\left({c}_{1},{c}_{2},...\right)\right\}$ is only one possibility. For some $a\in \mathbb{R}$, we could condition either on the event $\left\{C\left({S}_{i}\right)=a\right\}=\left\{{\sum}_{j=1}^{\infty}{C}_{j}{\varphi}_{j}\left({S}_{i}\right)=a\right\}$, or on the event $\left\{c\left({S}_{i}\right)=a\right\}=\left\{{\sum}_{j=1}^{\infty}{c}_{j}{\varphi}_{j}\left({S}_{i}\right)=a\right\}$, where the randomness of the event comes from ${S}_{i}$, or on $\left\{C\left(s\right)=a\right\}=\left\{{\sum}_{j=1}^{\infty}{C}_{j}{\varphi}_{j}\left(s\right)=a\right\}$, where the randomness comes from $\left({C}_{1},{C}_{2},...\right)$.^{17.}It should be clear that it is not possible to separately identify ${m}_{1}\left(X\right)$ from $C\left(X\right)$ in this example.^{18.}Recall that the nonparametric version of the factor model takes ${Y}_{i}={m}_{1}\left({X}_{i}\right)+{U}_{i}$, with ${U}_{i}={\sum}_{j=1}^{J}{S}_{ij}{C}_{j}+{\epsilon}_{i}$. The parametric model imposes ${m}_{1}\left(x\right)=\alpha +{x}^{\prime}\beta $.^{19.}Note that if we were able to estimate the conditional expectation $m\left(x\right)$ instead of $m\left(x,C\right)$, it would be impossible to separate ${x}^{\prime}\beta $ from ${\sum}_{j=1}^{J}{b}_{ij}E\left[{C}_{j}|X=x\right]$, and so, we would not be able to identify β.^{20.}In the Supplemental Material, we provide conditions under which the kernel density estimator is consistent: $\widehat{f}\left(x\right)\stackrel{p}{\u27f6}f\left(x|C\right)$. For the variance ${\sigma}^{2}\left(x,c\right)=E\left({Y}_{i}^{2}|{X}_{i}=x,C=c\right)-{\left[m\left(x,c\right)\right]}^{2}$, we can take ${\widehat{\sigma}}^{2}\left(x\right)$ to be:$${\widehat{\sigma}}^{2}\left(x\right)=\left[\frac{{\mathrm{\Sigma}}_{i=1}^{n}{Y}_{i}^{2}K\left(\frac{{X}_{i}-x}{h}\right)}{{\mathrm{\Sigma}}_{i=1}^{n}K\left(\frac{{X}_{i}-x}{h}\right)}\right]-{\left[\widehat{m}\left(x\right)\right]}^{2}.$$$${T}_{n}=\sqrt{n{h}_{n}^{k}}\frac{\left(\widehat{m}\left(x\right)-m\left(x,C\right)\right)}{{\left(\frac{{\widehat{\sigma}}^{2}\left(x\right)}{\widehat{f}\left(x\right)}\int {K}^{2}\left(u\right)du\right)}^{1/2}}+\sqrt{n{h}_{n}^{k}}\frac{\left(m\left(x,C\right)-{m}_{0}\left(x\right)\right)}{{\left(\frac{{\widehat{\sigma}}^{2}\left(x\right)}{\widehat{f}\left(x\right)}\int {K}^{2}\left(u\right)du\right)}^{1/2}}.$$^{22.}Formally, the necessary condition is that λ must be approximated by a compact paving that is closed under countable unions.

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Souza-Rodrigues, E.A. Nonparametric Regression with Common Shocks. *Econometrics* **2016**, *4*, 36.
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Souza-Rodrigues, Eduardo A. 2016. "Nonparametric Regression with Common Shocks" *Econometrics* 4, no. 3: 36.
https://doi.org/10.3390/econometrics4030036