Estimation of Average Derivatives of Latent Regressors: With an Application to Inference on Buffer-Stock Saving

Estimation of Average Derivatives of Latent Regressors: With an Application to Inference on Buffer-Stock Saving

Introduction

Rational forward-looking agents should have a unit marginal propensity to consume out of permanent shocks in income (MPCP). Carroll (2009) demonstrates that the MPCP is strictly less than one in the context of the buffer-stock model where inpatient consumers have a standard precautionary saving motive, and further shows through simulations across a wide range of structural assumptions that the MPCP ranges from 0.75 to 0.92. Thus, differentiating between the standard model and Carroll's buffer-stock model can be accomplished by accessing whether the MPCP is strictly less than one.

When we take this testable implication of the buffer-stock model to statistical inference based on empirical data, we encounter a fundamental issue. Namely, we do not observe the permanent income in data. If we could observe the permanent income shock $X^{\ast}$, as well as the consumption growth $Y$, then a statistical inference about the theory of Carroll boils down to inference about the nonparametric regression function $g(\cdot)=E[Y|X^{\ast}=\cdot]$. Specifically, rejecting the null hypothesis that a (weighted) average derivative of $g$ is greater than or equal to one against the alternative that it is strictly less than one will provide a statistical support for the theory of Carroll. Under the unobservability of the permanent income shock $X^{\ast}$ in data, however, the existing econometric methods of estimation and inference for (weighted) average derivatives do not apply. To overcome this, we develop a novel method and theory of estimation and inference for weighted average derivatives when the latent regressor $X^{\ast}$ is unobserved, but two noisy measures of $X^{\ast}$ are available in data, as is the case with the standard permanent-transitory models of earnings and income dynamics.

This paper, in terms of its technical aspects, belongs to the vast literature on measurement error models and deconvolution. See books by Carroll, Ruppert, Stefanski and Crainiceanu (2006), Meister (2009) and Horowitz (2009) and surveys by Chen, Hong and Nekipelov (2011), Schennach (2016) and Schennach (2021) for reviews. The literature on deconvolution started out with the deconvolution kernel density methods under known error distributions (Carroll and Hall, 1988; Stefanski and Carroll, 1990; Fan, 1991a,b; Bissantz, D�mbgen, Holzmann and Munk, 2007; Bissantz and Holzmann, 2008; van Es and Gugushvili, 2008; Lounici and Nickl, 2011; Schmidt-Hieber, Munk and D�mbgen, 2013), followed by those under unknown error distributions (Diggle and Hall, 1993; Horowitz and Markatou, 1996; Neumann and H�ssjer, 1997; Efromovich, 1997; Li and Vuong, 1998; Delaigle, Hall and Meister, 2008; Johannes, 2009; Comte and Lacour, 2011; Kato and Sasaki, 2018; Kato, Sasaki and Ura, 2021). Among the latter set of papers, Horowitz and Markatou (1996) and Delaigle, Hall and Meister (2008) use repeated measurements with symmetrically and identically distributed errors, while Li and Vuong (1998) propose an alternative estimator based on Kotlarski's lemma (Kotlarski, 1967) that does not require known error distribution -- also see Bohnomme and Robin (2010) and Comte and Kappus (2015). Also related is Adusumilli, Kurisu, Otsu and Whang (2020) who studies distribution function instead of density function.

These deconvolution kernel density methods extend to methods for nonparametric errors-in-variables regression. Fan and Truong (1993) and Fan and Masry (1992) study Nadaraya-Watson estimator under known error distribution, followed by extensions by Delaigle and Meister (2007), Delaigle, Fan and Carroll (2009) and Delaigle, Hall and Jamshidi (2015). Delaigle, Hall and Meister (2008), Adusumilli and Otsu (2018) and Kato and Sasaki (2019) consider cases of unknown error distribution with symmetrically and identically distributed errors, while Li (2002), Schennach (2004), Schennach, White and Chalak (2012), Schennach and Hu (2013) and Hu and Sasaki (2015) consider cases of unknown error distribution with repeated measurements. Fan (1995) and Dong, Otsu and Taylor (2021) study average derivatives of nonparametric errors-in-variables regression under known error distribution and unknown symmetric error distribution, respectively. The current paper is closely related to the last two references in that we are also interested in $\sqrt{n}$-consistent estimation and inference for average derivatives for the purpose of the aforementioned statistical inference about the hypothesis of buffer-stock saving. However, unlike Fan (1995) or Dong, Otsu and Taylor (2021), we allow for the unknown error distribution to be non-symmetric, in light of the recent empirical reports that components of earnings and income have skewed distributions (e.g., Bonhomme and Robin, 2010; Guvenen, Ozkan and Song, 2014; Hu, Moffitt and Sasaki, 2019; Guvenen, Karahan, Ozkan and Song, 2021).

Despite the extensive econometric and statistical literature on deconvolution as summarized in the prior two paragraphs, none of the existing papers to the best of our knowledge can conduct statistical inference for the MPCP under an unknown and possibly asymmetric error distribution, even though the skewness has been reported to be very likely by recent empirical studies on earnings and income dynamics. Motivated by the aforementioned economic question concerning the MPCP, therefore, this paper fills this important void in the deconvolution literature by proposing novel methods of estimation and inference for average derivatives of latent regressors whose error distribution can be both unknown and non-symmetric.

Regarding the application to the inference on buffer-stock saving motivated by Carroll (2009), there are a couple of related existing papers. In particular, Blundell, Pistaferri and Preston (2008) and Arellano, Blundell and Bonhomme (2017) investigate MPCP using empirical data. Blundell, Pistaferri and Preston (2008) use a linear parametric consumption model and conduct inference on the parameter that represents a constant MPCP. In contrast, we use a nonparametric consumption model with a possibly non-constant MPCP. Arellano, Blundell and Bonhomme (2017) use a nonlinear model of income dynamics, although their identification and estimation approach per se does not lead to statistical inference on MPCP. In contrast, at the cost of linear model of income dynamics, our proposed approach allows for statistical inference on MPCP under a nonparametric consumption model, and thus nonparametrically enables hypothesis testing about buffer-stock saving. In this way, the novel method proposed in this paper complements the existing literature on the empirical analysis of MPCP.

Methodology

Motivation and overview

Consider the permanent-transitory model of income dynamics

where $\iota_{jt}$, $\pi_{jt}$, $\tau_{jt}$ and $\eta_{jt}$ denote observed log income, latent log permanent income, latent log transitory income and latent permanent income shock, respectively, of individual $j$ in year $t$. Under this setup,

hold, where $(X_{j},W_{j})$ is observed and $(X_{j}^{\ast},\epsilon_{j},\nu_{j})$ is unobserved by a researcher. Although not necessary, if we assume that the shocks, $\eta_{jt-1}$, $\eta_{jt}$, $\eta_{jt+1}$, $\tau_{jt-2}$, $\tau_{jt-1}$, $\tau_{jt}$, and $\tau_{jt+1}$, are mutually independent, then the latent components $X_{j}^{\ast}$, $\epsilon_{j}$, and $\nu_{j}$ are also mutually independent.\footnote{\doublespacing Even with more periods of data, if we wish errors like $\epsilon_{j}$ and $\nu_{j}$ to keep sharing no common component, due to the structure of the permanent-transitory model of income dynamics, we can have no more than two noisy measures of $X_{j}^{*}$. Hence, the repeated measurements setting here is fundamental.}

Let $Y_{j}$ denote consumption growth $C_{jt}-C_{jt-1}$ from year $t-1$ to year $t$. Consider the nonparametric regression function $g(\cdot)=E[Y_{j}|X_{j}^{\ast}=\cdot]$. The derivative $g^{\prime}$ of this function quantifies the MPCP introduced in Section (ref). The theory of Carroll (2009) is that the buffer-stock model that arises under inpatient consumers with a standard precautionary saving motive implies $g^{\prime}<1$ rather than $g^{\prime}=1$. This testable implication leads to the null and alternative hypotheses

where $\theta_{1}=E[\{g^{\prime}(X^{\ast})-1\}f(X^{\ast})]$ and $f$ is the density of $X^{*}$. Rejection of $H_{0}$ in favor of $H_{1}$ implies that there is at least some location $x^{\ast}$ in the support of $X^{*}$ such that $g^{\prime}(x^{\ast})<1$, thus providing statistical support for the theory of Carroll (2009).

As we formally present in Section (ref) ahead, we propose to estimate $\theta_{1}$ by

where $\hat{\mathbb{K}}(u)=\frac{1}{2\pi}\int e^{-\mathrm{i}tu}\frac{K^{\mathrm{ft}}(t)}{\hat{f}_{\epsilon}^{\mathrm{ft}}(t/b_{n})}dt$, $\mathrm{i}=\sqrt{-1}$, $K^{\mathrm{ft}}$ is the Fourier transform of a kernel function $K$, $b_{n}$ is a bandwidth parameter, and $\hat{f}_{\epsilon}^{{\rm ft}}(t)=\frac{\hat{f}_{X}^{{\rm ft}}(t)}{\hat{f}^{{\rm ft}}(t)}$ with $\hat{f}_{X}^{{\rm ft}}(t)=\frac{1}{n}\sum_{j=1}^{n}e^{{\rm i}tX_{j}}$ and $\hat{f}^{{\rm ft}}(t)=\exp\left(\int_{0}^{t}\frac{{\rm i}\sum_{j=1}^{n}X_{j}e^{{\rm i}sW_{j}}}{\sum_{j=1}^{n}e^{{\rm i}sW_{j}}}ds\right)$. In Section (ref), we further show that $\sqrt{n}(\hat{\theta}_{1}-\theta_{1})$ converges to a normal distribution, which facilitates statistical inference for $\theta_{1}$. Given the estimator $\hat{\theta}_{1}$ along with its estimated standard error, we may conduct a formal statistical test of the implication ((ref)) of the theory of Carroll (2009).

Average derivative estimator

To understand $\hat{\theta}_{1}$ defined in ((ref)) in a general framework, we consider the estimation of $\theta_{c}=E[\{g^{\prime}(X^{*})-c\}f(X^{*})]$ for a constant $c$. We set $c=1$ for the test of ((ref)), whereas one can set $c=0$ if inference for the average derivative is the objective per se. Suppose that $f$ is continuously differentiable and $\{g(x)-cx\}f^{2}(x)\to0$ as $|x|\to\infty$. By integration by parts, $\theta_{c}$ can be expressed by \[ \theta_{c}=-2E[(Y-cX^{*})f^{\prime}(X^{*})]. \]

If $X^{*}$ were directly observed, $\theta_{c}$ could be estimated by $\tilde{\theta}_{c}=-\frac{2}{n}\sum_{j=1}^{n}(Y_{j}-cX_{j}^{*})\tilde{f}^{\prime}(X_{j}^{*})$ , where $\tilde{f}(x)$ denotes the kernel density estimator of $f(x)$, and $\tilde{\theta}_{c}$ could be understood as the density-weighted average derivative estimator by Powell, Stock and Stoker (1989) with dependent variable $Y-cX^{*}$. In the case when $X^{*}$ is unobserved, however, $\tilde{\theta}_{c}$ is infeasible.

Motivated by (ref)--(ref), suppose that we can observe two noisy measurements of $X^{*}$, denoted by $X$ and $W$, which are generated by

where $\epsilon$ and $\nu$ are measurement errors associated with $X$ and $W$, respectively. In particular, $\epsilon$ and $\nu$ have zero mean and are classical; that is, $\epsilon$ and $\nu$ are independent of $X^{*}$.\footnote{\doublespacing We will invoke a weaker assumption than the full statistical independence formally in Section (ref).} To construct an estimator of $\theta_{c}$ in this case, note that

where $h_{c}(x)=\{g(x)-cx\}f(x)$. Let $f_{A}$ denote the density of a random variable $A$, $a^{{\rm ft}}(t)=\int e^{{\rm i}tx}a(x)dx$ denote the Fourier transform of a function $a$, and $\{Y_{j},X_{j},W_{j}\}_{j=1}^{n}$ be an i.i.d. sample of $(Y,X,W)$.

If $f_{\epsilon}$ were known, $h_{c}$ and $f$ could be estimated using the deconvolution techniques by

\[ \check{f}(x)=\frac{1}{nb_{n}}\sum_{j=1}^{n}\mathbb{K}\left(\frac{x-X_{j}}{b_{n}}\right), \] where $\mathbb{K}(u)=\frac{1}{2\pi}\int e^{-\mathrm{i}tu}\frac{K^{\mathrm{ft}}(t)}{f_{\epsilon}^{\mathrm{ft}}(t/b_{n})}dt$ is the deconvolution kernel function based on the characteristic function $f_{\epsilon}^{{\rm ft}}$ of the true measurement error $\epsilon$. Hence, it would be natural to estimate $\theta_{c}$ by the following plug-in estimator

Now, suppose that $f_{\epsilon}$ is unknown. Assume that $f^{{\rm ft}}$, $f_{\epsilon}^{{\rm ft}}$ and $f_{\nu}^{{\rm ft}}$ do not vanish anywhere, according to Kotlarski's (1967) identity, $f^{{\rm ft}}(t)=\exp\left(\int_{0}^{t}\frac{{\rm i}E[Xe^{{\rm i}sW}]}{E[e^{{\rm i}sW}]}ds\right)$, which together with $f_{\epsilon}^{{\rm ft}}(t)=\frac{E[e^{{\rm i}tX}]}{f^{{\rm ft}}(t)}$ implies that $\hat{f}_{\epsilon}^{{\rm ft}}$ defined in Section (ref) is a plug-in estimator of $f_{\epsilon}^{{\rm ft}}$ based on the sample analogs of $E[e^{{\rm i}sX}]$ and $E[e^{{\rm i}tW}]$, and $E[Xe^{{\rm i}sW}]$. Therefore, to estimate $\theta_{c}$ when $f_{\epsilon}$ is unknown, it is natural to replace $f_{\epsilon}^{{\rm ft}}$ in $\check{\theta}_{c}$ by $\hat{f}_{\epsilon}^{{\rm ft}}$, which gives \[ \hat{\theta}_{c}=-\frac{2}{n^{2}b_{n}^{3}}\sum_{j=1}^{n}\sum_{k=1}^{n}(Y_{j}-cW_{j})\int\mathbb{\hat{K}}\left(\frac{x-X_{j}}{b_{n}}\right)\hat{\mathbb{K}}^{\prime}\left(\frac{x-X_{k}}{b_{n}}\right)dx, \] where $\hat{\mathbb{K}}(u)=\frac{1}{2\pi}\int e^{-\mathrm{i}tu}\frac{K^{\mathrm{ft}}(t)}{\hat{f}_{\epsilon}^{\mathrm{ft}}(t/b_{n})}dt$, defined in Section (ref), is the deconvolution kernel function based on the estimated measurement error characteristic function $\hat{f}_{\epsilon}^{{\rm ft}}$.

Main result

In this section, we presents a formal theory that provides the asymptotic validity of the test procedure proposed in Section (ref). Specifically, we derive the asymptotic distribution for $\hat{\theta}_{c}$ and propose an estimator for its asymptotic variance. To this end, we make the following assumptions.

Assumption (1) requires random sampling and imposes conditions on the distribution of $(Y,X^{*})$ and measurement errors $(\epsilon,\nu)$. In particular, under the classical measurement error assumptions, $E[Y|X^{*},\epsilon]=E[Y|X^{*}]$ and $E[\nu|\epsilon]=E[\nu]$ are imposed for the identification of $E[Y-cW|X^{*}]$\footnote{\doublespacing The full independence between the measurement error in regressor and dependent variable has been commonly adopted in the literature of nonparametric regression with errors-in-variables, (e.g., Fan and Truong, 1993; Delaigle and Meister, 2007; Meister, 2009). In our setting, however, it is sufficient to require the mean independence. In particular, observe that the identification of $E[Y-cW|X^{*}]$ hinges on $$\{E[Y-cW|X]f_{X}\}(x)=\int\{E[Y-cW|X^{*}]f\}(x-e)f_{\epsilon}(e)de,$$ which holds if $E[Y-cW|X^{*},\epsilon]=E[Y-cW|X^{*}]$, or equivalently $E[Y|X^{*},\epsilon]=E[Y|X^{*}]$ and $E[\nu|\epsilon]=E[\nu]$, under the classical measurement error assumption. }, $E[\nu]=0$ is imposed for $\theta_{c}=-2E[(Y-cW)f^{\prime}(X^{*})]$, and the non-vanishing characteristic functions and $E[\epsilon|\nu]=0$ is imposed for the Kotlarski's identity\footnote{\doublespacing The full independence between $\epsilon$ and $\nu$ has been commonly adopted in the literature on identification and estimation based on Kotlarski's identity (e.g., Li and Vuong, 1998; Kato, Sasaki and Ura, 2021; Dong, Otsu and Taylor, 2022), but it can be relaxed to the mean independence; See Schennach(2004).}. $E[|X^{*}|^{2+\eta}]<\infty$ and $E[|\epsilon|^{2+\eta}]<\infty$ are regularity conditions required by Lemma (ref), which is used to characterize the uniform convergence rate of the empirical characteristic function of $(X,W)$ and their first-order derivatives over an expanding region. We remark in the context of ((ref))--((ref)) that our conditions, $E[\epsilon|\nu]=0$ and $E[\nu|\epsilon]=0$, do not rule out the typical assumptions about the permanent-transitory models of income and earnings dynamics in which the permanent shocks $\left\{ \eta_{jt}\right\} _{t}$ and the transitory components $\left\{ \tau_{jt}\right\} _{t}$ are white noise processes (e.g., Bonhomme and Robin, 2010).

Assumption (2) constitutes mild assumptions on the smoothness of the density function $f$ and the regression function $g$, which is equivalent to Assumptions 3 and 5 in Powell, Stock and Stoker (1989). Assumption (3) concerns the kernel function $K$. Specifically, we require $K$ to be a symmetric $\alpha$-th order kernel, which together with Assumption (2) can be used to control the magnitude of the estimation bias. In addition, we also require $K^{{\rm ft}}$ to be compactly supported, which is to regularize the deconvolution problem that is well-known to be ill-posed.

Assumption (4) gives two conditions on the bandwidth $b_{n}$. In particular, the first condition is needed to control the estimation variance, and the second condition requires $g$ and $f$ to be smooth enough so that the estimation bias is asymptotically negligible. Here, we maintain a general expression without specifying the decay rates of the tails of $f^{{\rm ft}}$, $f_{\epsilon}^{{\rm ft}}$ and $f_{\nu}^{{\rm ft}}$ as is typical in the deconvolution literature. By doing so, we can apply our result to a larger set of measurement error distributions, including both ordinary smooth distributions and supersmooth distributions. Assumption (5) is a high-level assumption on the boundedness of the asymptotic variance of $\hat{\theta}_{c}$; an analogous assumption is made in Fan (1995) and Dong, Otsu and Taylor (2021).

This is the main result of the paper. To understand it, we can decompose $\xi_{c}$ into two parts as $\xi_{c}(Y,X,W)=\xi_{c,1}(Y,X,W)+\xi_{c,2}(Y,X,W)$, where

and consider, for ease of illustration, a special case in which $f_{\epsilon}^{{\rm ft}}$ is of the form \[ f_{\epsilon}^{{\rm ft}}(t)=\frac{1}{c_{0}+c_{1}t+\cdots+c_{\beta}t^{\beta}}, \] where $\beta$ is a positive integer, and $c_{0}=1$, $c_{1},\ldots,c_{\beta}$ are complex numbers, which includes the Laplace distribution as a special case when $c_{1}=0$ and $\beta=2$.

In such cases, using $\{a^{(k)}\}^{{\rm ft}}(t)=(-{\rm i}t)^{k}a^{{\rm ft}}(t)$ for a positive integer $k$, we obtain that

which when $c=0$ coincides with $2r(X,Y)$ defined as in equation (25) of Fan (1995). Thus, $\xi_{c,1}(Y,X,W)$ characterizes the randomness of $\check{\theta}_{c}$, which is the estimator of $\theta_{c}$ when $f_{\epsilon}$ is known. Furthermore, compared to $2r(X,Y)$ in Fan (1995), $\xi_{c,1}(Y,X,W)$ allows non-zero value of $c$ and can cover a larger set of measurement error distributions.

Since $\xi_{c,1}(Y,X,W)$ characterizes the randomness in the estimation of $\theta_{c}$ when $f_{\epsilon}$ is known, the additional randomness introduced by using $\hat{f}_{\epsilon}^{{\rm ft}}$ in the place of $f_{\epsilon}^{{\rm ft}}$ is reflected by $\xi_{c,2}(Y,X,W)$. It is worthy to note that the structure of $\xi_{c,2}(Y,X,W)$ is similar to that of $\xi_{c,1}(Y,X,W)$, but is more complicated. In general, it is difficult to simplify $\xi_{c,2}(Y,X,W)$ as we did for $\xi_{c,1}(Y,X,W)$ even when $f^{{\rm ft}}$, $f_{\epsilon}^{{\rm ft}}$ and $f_{\nu}^{{\rm ft}}$ are all specified. However, there are special cases in which $\xi_{c,2}(Y,X,W)$ can be completely ignored. To see a case in point, note that $\{h_{c}^{\prime}\}^{{\rm ft}}(-t)f^{{\rm ft}}(t)-h_{c}^{{\rm ft}}(t)\{f^{\prime}\}^{{\rm ft}}(-t)={\rm i}t\{h_{c}^{{\rm ft}}(-t)f^{{\rm ft}}(t)-h_{c}^{{\rm ft}}(t)f^{{\rm ft}}(-t)\}$, which implies that if $h_{c}^{{\rm ft}}(-t)=h_{c}^{{\rm ft}}(t)$ and $f^{{\rm ft}}(-t)=f^{{\rm ft}}(t)$, for example when both $h_{c}$ and $f$ are symmetric around zero, $\xi_{c,2}(Y,X,W)=0$, i.e. the estimation error brought by using $\hat{f}_{\epsilon}^{{\rm ft}}$ in the place of $f_{\epsilon}^{{\rm ft}}$ is exactly zero and $\hat{\theta}_{c}$ has exactly the same asymptotic distribution as that of $\check{\theta}_{c}$.

To test hypotheses on $\theta_{c}$ like $H_{0}$, besides the asymptotic distribution of $\hat{\theta}_{c}$, we also need to estimate $s_{c}^{2}=Var[\xi_{c}(Y,X,W)]$. Observe that we can rewrite

where the first step uses $\{a^{(k)}\}^{{\rm ft}}(t)=(-{\rm i}t)^{k}a^{{\rm ft}}(t)$ and the second step follows from $f_{X}^{{\rm ft}}=f^{{\rm ft}}f_{\epsilon}^{{\rm ft}}$, $f_{W}^{{\rm ft}}=f^{{\rm ft}}f_{\nu}^{{\rm ft}}$ and $f^{{\rm ft}}(t)=\exp\left(\int_{0}^{t}\frac{{\rm i}E[Xe^{{\rm i}sW}]}{E[e^{{\rm i}sW}]}ds\right)$, which implies $E[\xi_{c}(Y,X,W)]=0$ and $s_{c}^{2}=E[\xi_{c}^{2}(Y,X,W)]$. In practice,$f^{{\rm ft}}$, $f_{\epsilon}^{{\rm ft}}$, $f_{X}^{{\rm ft}}$, $f_{W}^{{\rm ft}}$, $E[Xe^{{\rm i}tW}]$ and $h_{c}^{{\rm ft}}$ are all unknown, and we have to estimate them. $f^{{\rm ft}}$, $f_{\epsilon}^{{\rm ft}}$ and $f_{X}^{{\rm ft}}$ can be estimated by $\hat{f}^{{\rm ft}}$, $\hat{f}_{\epsilon}^{{\rm ft}}$, and $f_{X}^{{\rm ft}}$ defined in Section (ref), $f_{W}^{{\rm ft}}$ can be estimated by $\hat{f}_{W}^{{\rm ft}}(t)=\frac{1}{n}\sum_{j=1}^{n}e^{{\rm i}tW_{j}}$, $E[Xe^{{\rm i}tW}]$ can be estimated by $\hat{E}[Xe^{{\rm i}tW}]=\frac{1}{n}\sum_{j=1}^{n}X_{j}e^{{\rm i}sW_{j}}$, and $h_{c}^{{\rm ft}}$ can be estimated by $\hat{h}_{c}^{{\rm ft}}(t)=\frac{\frac{1}{n}\sum_{j=1}^{n}(Y_{j}-cW_{j})e^{{\rm i}tX_{j}}}{\hat{f}_{\epsilon}^{\mathrm{ft}}(t)}$. Therefore, we can estimate $s_{c}^{2}$ by $\hat{s}_{c}^{2}=\frac{1}{n}\sum_{j=1}^{n}\hat{\xi}_{c}^{2}(Y_{j},X_{j},W_{j})$ with \[ \hat{\xi}_{c}(y,x,w)=\frac{{\rm 1}}{\pi}\int{\rm i}t\left\{

\right\} K^{{\rm ft}}(tb_{n})dt, \] where $K^{{\rm ft}}(tb_{n})$ is introduced to regularize the integration.

Simulation

This section presents simulation studies to analyze the finite sample performance of the proposed method of inference about $\theta_{c}$. We generate $N$ independent copies of the observed variables $(Y,X,U)$ via the structural equations

where the latent variables $(X^{\ast},U,\epsilon,\nu)$ are in turn generated independently from the standard normal distribution. Note in this setting that the null hypothesis $H_{0}:\theta_{1}\ge0$ is true if and only if $\delta\le0$. Furthermore, positive values of the design parameter $\delta$ in this data generating process measures deviations from the null hypothesis. We run sets of simulations across combinations of the values of $\delta\in[0.0,0.5]$ and $N\in\left\{ 250,500\right\} $, where each set of simulations consists of 2,500 Monte Carlo iterations.

Figure (ref) plots the Monte Carlo frequencies of rejecting the null hypothesis $H_{0}:\theta_{1}\ge0$ against the alternative $H_{1}:\theta_{1}<0$ based on the one-sided test with our estimator $\hat{\theta}_{1}$ and its standard error estimator $\hat{s}_{1}$. The nominal size of the test is set to 0.05. The horizontal axis of the figure measures the deviation $\delta\in[0.0,0.5]$ away from the null hypothesis $H_{0}$. The dashed (respectively, solid) curve indicates the results with $N=250$ (respectively, $500$). Observe that the rejection frequency at $\delta=0$ is close to the nominal size, 0.05. As $\delta$ becomes larger, on the other hand, the rejection frequencies increase. For a given value of $\delta>0$, the rejection frequency is larger for the larger sample size, demonstrating the power of the test as well as the size control.

We ran additional simulations with alternative data generating designs, only to find very similar simulation results to the baseline results presented above. Overall, the simulation outcomes demonstrate excellent small-sample performance of the estimation and inference methods. Our observations that the asymptotic approximations are already accurate even at such small sample sizes $N$ as $250$ demonstrate the practical merit of our root-n consistent test under the highly sophisticated problem of errors-in-variables nonparametric regressions.

Application

This section revisits the analysis of the MPCP introduced in Sections (ref)--(ref). Recall from ((ref)) and ((ref)) that we require four time periods of panel data $\left\{ \left\{ \iota_{j\tau}\right\} _{\tau=t-2}^{t+1}\right\} _{j=1}^{n}$ on observed income to construct the variables $\left\{ (X_{j},W_{j})\right\} _{j=1}^{n}$ that we use as inputs for our proposed method of inference. Using the U.S. Panel Study of Income Dynamics (PSID) for the four most recent survey years 2013, 2015, 2017, and 2019, we aim to test the null hypothesis $H_{0}:\theta_{1}\ge0$ of (super-) unit MPCP against the alternative hypothesis $H_{1}:\theta_{1}<0$ of sub-unit MPCP. Rejecting the null hypothesis $H_{0}$ supports the buffer-stock model that arises with inpatient consumers having a standard precautionary saving motive.

The income variable $\iota_{jt}$ is defined by the log total family income of unit $j$ reported in year $t$. The consumption variable $C_{jt}$ is similarly defined by the log family expenditures of unit $j$ reported in year $t$, where the categories of consumption consist of food, housing, telephone/internet, transportation, vehicle, education, child care, health care, household repairs, household furnishing, clothing, and recreation. The first three columns of Table (ref) display summary statistics of this data set. Displayed values are the sample means. Parentheses enclose sample standard deviations.

Adapting ((ref)) and ((ref)) to this panel data set, we construct $X_{j}=\iota_{j2017}-\iota_{j2013}$ and $W_{j}=\iota_{j2019}-\iota_{j2015}$. Likewise, we construct the outcome variable by $Y_{j}=C_{j2017}-C_{j2015}$. We drop units that experience a missing value or an infinite value for $X_{j}$, $W_{j}$ or $Y_{j}$. Consequently, we obtain a balanced panel of 5976 household units. The last three columns of Table (ref) display summary statistics of the constructed variables $\left\{ (X_{j},W_{j},Y_{j})\right\} _{j=1}^{n}$. Again, displayed values are the sample means, and parentheses enclose sample standard deviations.

Applying our proposed method of estimation and inference, we obtain the point estimate of $\hat{\theta}_{1}=-0.0607$ with the estimated standard error of $\hat{s}_{1}=0.0052$. Using our asymptotic normality results along with these estimates, we formally reject the one-sided test of the null hypothesis $H_{0}:\theta_{1}\ge0$ of (super-) unit MPCP in favor of the alternative hypothesis $H_{1}:\theta_{1}<0$ of sub-unit MPCP. Our test result supports the buffer-stock model that arises with inpatient consumers having a standard precautionary saving motive. Even though a number of prior studies have calibrated or estimated the MPCP under various models of income dynamics, to our best knowledge, our result is the first formal statistical inference result about the MPCP using flexible nonparametric distribution in the permanent-transitory model of income processes.

Conclusion

In this paper, we propose a density-weighted average derivative estimator when two noisy measures of a latent regressor is available. Both measures have classical errors, and the error distributions are possibly asymmetric. We show that this estimator achieves the root-$n$ rate of convergence and is asymptotically normal. Simulation studies demonstrate excellent small-sample performance, and support the merit of the root-$n$ asymptotic normality. Based on the proposed estimator, we construct a test on the sub-unity of MPCP under a nonparametric consumption model. In particular, under a permanent-transitory model of income dynamics, we construct two noisy measures of a permanent income shock using four periods data. With an application using recent waves of the U.S. PSID, we reject the null hypothesis of unit MPCP in favor or a sub-unit MPCP, supporting the buffer-stock model of saving.