Simultaneous Confidence Band for Partially Linear Panel Data Models with Fixed Effects

TL;DR

This paper develops a bootstrap-based method to construct simultaneous confidence bands for the nonparametric component in partially linear panel data models with fixed effects, addressing computational and bias estimation challenges.

Contribution

It introduces a bootstrap approach for SCB construction in partially linear panel models, removing fixed effects and overcoming bias and variance estimation issues.

Findings

Bootstrap method outperforms traditional methods in simulations

Provides reliable confidence bands with limited sample sizes

Addresses computational complexity in SCB construction

Abstract

In this paper, we construct the simultaneous confidence band (SCB) for the nonparametric component in partially linear panel data models with fixed effects. We remove the fixed effects, and further obtain the estimators of parametric and nonparametric components, which do not depend on the fixed effects. We establish the asymptotic distribution of their maximum absolute deviation between the estimated nonparametric component and the true nonparametric component under some suitable conditions, and hence the result can be used to construct the simultaneous confidence band of the nonparametric component. Based on the asymptotic distribution, it becomes difficult for the construction of the simultaneous confidence band. The reason is that the asymptotic distribution involves the estimators of the asymptotic bias and conditional variance, and the choice of the bandwidth for estimating the second derivative of nonparametric function. Clearly, these will cause computational burden and accumulative errors. To overcome these problems, we propose a Bootstrap method to construct simultaneous confidence band. Simulation studies indicate that the proposed Bootstrap method exhibits better performance under the limited samples.

Tables

Table 1. Table 1: The bias, standard deviation (SD) and mean squared error (MSE) of 𝜷 ^ ^ 𝜷 \hat{\bm{\beta}}

\ML		200	100	150	200
	Bias	0.0063	0.0059	0.0048	0.0045	0.0046	0.0023
${\hat{\beta }}_1$	SD	0.0859	0.0720	0.0682	0.0841	0.0647	0.0635
	MSE	0.0074	0.0052	0.0046	0.0071	0.0042	0.0040
	Bias	0.0057	0.0046	0.0031	0.0053	0.0027	0.0022
${\hat{\beta }}_2$	SD	0.0901	0.0696	0.0620	0.0906	0.0687	0.0601
	MSE	0.0081	0.0049	0.0038	0.0082	0.0048	0.0036
	Bias	0.0062	0.0049	0.0042	0.0041	0.0029	0.0026
${\hat{\beta }}_3$	SD	0.0912	0.0770	0.0650	0.0857	0.0679	0.0545
	MSE	0.0083	0.0059	0.0042	0.0074	0.0046	0.0031

Table 2. Table 2: Coverage probabilities of nonparametric component with the nominal level 95%

	$n$	$c=0$	$c=0.5$	$c=1$
	100	0.926	0.933	0.941
method one	150	0.933	0.940	0.949
	200	0.946	0.951	0.953
	100	0.928	0.934	0.942
Bootstrap	150	0.937	0.946	0.950
	200	0.948	0.952	0.954

Equations

$$Y_{it}={\mathbf{X}}_{it}^{\tau }\mathbf{\beta }+g(Z_{it})+{\alpha }_i+V_{it},i=1,\cdots ,n,t=1,\cdots ,T,$$

$$\mathbf{Y}=(Y_{11},\cdots ,Y_{1T},Y_{21},\cdots ,Y_{2T},\cdots ,Y_{n1},\cdots ,Y_{nT}{)}^{\tau },$$

$$\displaystyle\bm{g}=\Big{(}g(Z_{11}),\cdots,g(Z_{1T}),g(Z_{21}),\cdots,g(Z_{2T}),\cdots,g(Z_{n1}),\cdots,g(Z_{nT})\Big{)}^{\tau},$$

$$\mathbf{V}=(V_{11},\cdots ,V_{1T},V_{21},\cdots ,V_{2T},\cdots ,V_{n1},\cdots ,V_{nT}{)}^{\tau },$$

$${\mathbf{\alpha }}_0=({\alpha }_1,\cdots ,{\alpha }_n{)}^{\tau }$$

$$\mathbf{Y}=\mathbf{X}\mathbf{\beta }+\mathbf{g}+({\mathbf{I}}_n\otimes {\mathbf{e}}_T){\mathbf{\alpha }}_0+\mathbf{V},$$

$$\mathbf{Y}=\mathbf{X}\mathbf{\beta }+\mathbf{g}+\mathbf{D}\mathbf{\alpha }+\mathbf{V}.$$

$$\mathbf{Y}-\mathbf{X}\mathbf{\beta }-\mathbf{D}\mathbf{\alpha }=\mathbf{g}+\mathbf{V}.$$

$$g(Z_{it})\approx g(z)+g^{\prime }(z)(Z_{it}-z).$$

$$\mathbf{Z}_{z}=\left(\begin{array}[]{ccccccc}1&\cdots&1&\cdots&1&\cdots&1\\ Z_{11}-z&\cdots&Z_{1T}-z&\cdots&Z_{n1}-z&\cdots&Z_{nT}-z\end{array}\right)^{\tau},$$

$$(\mathbf{Y}-\mathbf{X}\mathbf{\beta }-{\mathbf{Z}}_z\mathbf{G}(z)-\mathbf{D}\mathbf{\alpha }{)}^{\tau }{\mathbf{W}}_z(\mathbf{Y}-\mathbf{X}\mathbf{\beta }-{\mathbf{Z}}_z\mathbf{G}(z)-\mathbf{D}\mathbf{\alpha }).$$

$$\tilde{\mathbf{G}}(z,\mathbf{\eta })=({\mathbf{Z}}_z^{\tau }{\mathbf{W}}_z{\mathbf{Z}}_z{)}^{-1}{\mathbf{Z}}_z^{\tau }{\mathbf{W}}_z(\mathbf{Y}-\mathbf{X}\mathbf{\beta }-\mathbf{D}\mathbf{\alpha }).$$

$$\mathbf{M}(z)=({\mathbf{Z}}_z^{\tau }{\mathbf{W}}_z{\mathbf{Z}}_z{)}^{-1}{\mathbf{Z}}_z^{\tau }{\mathbf{W}}_z.$$

$$\tilde{g}(z,\mathbf{\eta })={\mathbf{m}}^{\tau }(z)(\mathbf{Y}-\mathbf{X}\mathbf{\beta }-\mathbf{D}\mathbf{\alpha }),$$

$$(\mathbf{Y}-\mathbf{X}\mathbf{\beta }-{\tilde{\mathbf{g}}}_{\mathbf{\eta }}(z)-\mathbf{D}\mathbf{\alpha }{)}^{\tau }(\mathbf{Y}-\mathbf{X}\mathbf{\beta }-{\tilde{\mathbf{g}}}_{\mathbf{\eta }}(z)-\mathbf{D}\mathbf{\alpha })$$

$$\mathbf{M}=\left(\begin{array}[]{c}(1,0)(\mathbf{Z}_{Z_{11}}^{\tau}\mathbf{W}_{Z_{11}}\mathbf{Z}_{Z_{11}})^{-1}\mathbf{Z}_{Z_{11}}^{\tau}\mathbf{W}_{Z_{11}}\\ \vdots\\ (1,0)(\mathbf{Z}_{Z_{1T}}^{\tau}\mathbf{W}_{Z_{1T}}\mathbf{Z}_{Z_{1T}})^{-1}\mathbf{Z}_{Z_{1T}}^{\tau}\mathbf{W}_{Z_{1T}}\\ \vdots\\ (1,0)(\mathbf{Z}_{Z_{nT}}^{\tau}\mathbf{W}_{Z_{nT}}\mathbf{Z}_{Z_{nT}})^{-1}\mathbf{Z}_{Z_{nT}}^{\tau}\mathbf{W}_{Z_{nT}}\\ \end{array}\right).$$

$$\tilde{\mathbf{\alpha }}(\mathbf{\beta })=({\tilde{\mathbf{D}}}^{\tau }\tilde{\mathbf{D}}{)}^{-1}{\tilde{\mathbf{D}}}^{\tau }(\tilde{\mathbf{Y}}-\tilde{\mathbf{X}}\mathbf{\beta }).$$

$$\hat{\mathbf{\beta }}=({\tilde{\mathbf{X}}}^{\tau }\tilde{\mathbf{Q}}\tilde{\mathbf{X}}{)}^{-1}{\tilde{\mathbf{X}}}^{\tau }\tilde{\mathbf{Q}}\tilde{\mathbf{Y}}.$$

$$\hat{\mathbf{\alpha }}=({\hat{\alpha }}_2,\cdots ,{\hat{\alpha }}_n)=({\tilde{\mathbf{D}}}^{\tau }\tilde{\mathbf{D}}{)}^{-1}{\tilde{\mathbf{D}}}^{\tau }(\tilde{\mathbf{Y}}-\tilde{\mathbf{X}}\hat{\mathbf{\beta }}).$$

$$\hat{\mathbf{G}}(z)$$

$$\hat{g}(z)={\mathbf{m}}^{\tau }(z)({\mathbf{I}}_{nT}-{\mathbf{D}({\mathbf{D}}^{\tau }\mathbf{PD})}^{-1}{\mathbf{D}}^{\tau }\mathbf{P})(\mathbf{Y}-\mathbf{X}\hat{\mathbf{\beta }}).$$

$$\Vert \hat{\mathbf{\beta }}-\mathbf{\beta }\Vert =O_p(n^{-1/2})$$

$$\sqrt{nh}\{\hat{g}(z)-g(z)-b(z)\}\stackrel{L}{⟶}N(0,{\Sigma }_g),$$

$$\sqrt{n{h}^3}\{{\hat{g}}^{\prime }(z)-g^{\prime }(z)\}\stackrel{L}{⟶}N(0,{\Sigma }_{g^{\prime }}),$$

$$\displaystyle P\left\{(-2\log h)^{1/2}\Big{(}\sup\limits_{z\in[0,1]}\left|(nh\Sigma_{g}^{-1})^{1/2}(\hat{g}(z)-g(z)-b(z))\right|-d_{n}\Big{)}<u\right\}$$

$$⟶\exp \left(-2\exp (-u)\right),asn\to \infty ,$$

$$d_n=(-2\log h{)}^{1/2}+\frac{1}{(-2\log h{)}^{1/2}}\left\{\log \frac{K^2(A)}{{\nu }_0{\pi }^{1/2}}+\frac{1}{2}\log \log h^{-1}\right\},$$

$$d_n=(-2\log h{)}^{1/2}+\frac{1}{(-2\log h{)}^{1/2}}\log \left\{\frac{1}{4{\nu }_0\pi }\int (K^{\prime }(z){)}^{2}dz\right\}.$$

$$\displaystyle P\left\{(-2\log h/(d-c))^{1/2}\Big{(}\sup\limits_{z\in[c,d]}\left|(nh\Sigma_{g}^{-1})^{1/2}(\hat{g}(z)-g(z)-b(z))\right|-\widetilde{d}_{n}\Big{)}<u\right\}$$

$$⟶\exp \left(-2\exp (-u)\right),$$

$$\displaystyle P\left\{(-2\log h)^{1/2}\Big{(}\sup\limits_{z\in[0,1]}\left|(nh^{3}\Sigma_{g^{\prime}}^{-1})^{1/2}(\hat{g}^{\prime}(z)-g^{\prime}(z))\right|-d_{n_{1}}\Big{)}<u\right\}$$

$$⟶\exp \left(-2\exp (-u)\right),asn\to \infty ,$$

$$(h^2{\mu }_2/2)g^{\prime \prime }(z)(1+o_p(1)).$$

Full text

Key words: partially linear model, simultaneous confidence band, panel data, fixed effects, asymptotic property

2000 MR Subject Classification: 62G08; 62H12; 62G20

1 Introduction

In the literature, there were a large amount of studies about parametric linear and nonlinear panel data models, and Arellano (2003), Baltigi (2005), and Hsiao (2003) had provided excellent overview of parametric panel data model analysis. To relax the strong restrictions assumed in the parametric panel data models, nonparametric and semiparametric panel data models have received a lot of attention in recent years. Compared to traditional parametric panel data model, nonparametric or semiparametric panel data models are better and more flexible to fit the actual data. Thus, this kind of models have become the hot research topic for the econometricians and statisticians. For example, Henderson, Carroll and Li (2008), and Li, Peng and Tong (2013) considered the fixed effects nonparametric panel data model. Henderson and Ullah (2005), Lin and Ying (2001), and Wu and Zhang (2002) considered the random effects nonparametric panel data models. Li and Stengos (1996) considered a partially linear panel data model with some regressors being endogenous via IV approach. Su and Ullah (2006) investigated the fixed effects partially linear panel data model with exogenous regressors. Zhang et al. (2011) considered the empirical likelihood inference for the fixed effects partially linear panel data model. Sun, Carroll and Li (2009) considered the problem of estimating a varying coefficient panel data model with fixed effects using a local linear regression approach. Chen, Gao and Li (2013a, 2013b) and Lai, Li and Lian (2013) studied the semiparametric estimation for a single-index panel data model, and among others.

Recently, the fixed effects models are frequently used in econometrics and biometrics. In this paper, we consider the following partially linear panel data models with fixed effects:

[TABLE]

where $\{\bm{X}_{it}\}$ are $p\times 1$ vector of observable regressors, $\{Z_{it}\}$ are explanatory variables in [0,1], $\bm{\beta}$ is a $p\times 1$ vector of unknown coefficients, $g(\cdot)$ is an unknown smooth function in [0,1], $\{V_{it}\}$ are random errors with zero mean, and $\{\alpha_{i}\}$ are fixed effects. In addition, $T$ is the time series length, $n$ is the cross section size.

For model (1.1), we assume that $\{\alpha_{i}\}$ are unobserved time-invariant individual effects. Model (1.1) is called as a partially linear fixed effects model if $\{\alpha_{i}\}$ are correlated with $\{\bm{X}_{it},Z_{it}\}$ with an unknown correlation structure. For identification purpose, we impose $\sum\limits_{i=1}^{n}\alpha_{i}=0$. An application of fixed effects models is the study of individual wage rate, $\alpha_{i}$ represents different unobserved abilities of individual $i$, such as the unmeasured skills or unobservable characteristics of individual $i$, which maybe correlate with some observed covariates: age, educational level, job grade, gender, work experience and et al.. As a special case, when $\{\alpha_{i}\}$ are uncorrelated with $\{\bm{X}_{it},Z_{it}\}$, model (1.1) becomes a partially linear random effects model.

Baltagi and Li (2002) applied the first-order difference to eliminate the fixed effects and used the series method to estimate the parametric and nonparametric components, and they further established the asymptotic properties. Su and Ullah (2006) considered the estimation of partially linear panel data models with fixed effects. Zhang et al. (2011) applied the empirical likelihood method to model (1.1).

For the partially linear panel data models, the existing literatures considered the pointwise asymptotic normality of the estimator for the nonparametric component, and the result can be used to construct the pointwise confidence bands. In practice, we need to construct the simultaneous confidence band of the nonparametric function in the model. The simultaneous confidence band is a powerful tool to check the graphical representation of the nonparametric function during the practical applications. Therefore, there are extensive literatures on the construction of the simultaneous confidence band. For example, Fan and Zhang (2000), and Zhang and Peng (2010) considered the simultaneous confidence bands for the coefficient functions in varying-coefficient models; Li, Peng and Tong (2013) considered the simultaneous confidence band for nonparametric fixed effects panel data model; Li et al. (2014) and Yang et al. (2014) studied the simultaneous confidence band and hypothesis testing for the link function in single-index models, and more literatures see Yothers and Sampson (2011), Brabanter et al. (2012), Cao et al. (2012), Liu et al. (2013), and Li and Yang (2015).

In this paper, combining the idea of least-squares dummy-variable approach in parametric panel data models with the local linear regression technique in nonparametric models, we use the profile least-squares dummy-variable method proposed in Su and Ullah (2006) to remove the fixed effects, and further obtain the estimators of parametric and nonparametric components, which do not depend on the fixed effects. Under some suitable conditions, we establish the asymptotic distribution of their maximum absolute deviation between the estimated nonparametric component and the true nonparametric component, and hence the result can be used to construct the simultaneous confidence band of the nonparametric component. In order to construct the simultaneous confidence band based on the asymptotic distribution, we first need to estimate the asymptotic bias and conditional variance, and choose the bandwidth for estimating the second derivative of nonparametric function. These will cause computational burden and accumulative errors, and it becomes difficult for the construction of the simultaneous confidence band. To overcome these problems, we further propose a Bootstrap method to construct the simultaneous confidence band of the nonparametric component in model (1.1).

The rest of the paper is organized as follows. In Section 2, we use the profile least-squares dummy-variable approach to obtain the estimators of the parametric and nonparametric components, and present the asymptotic properties. In Section 3, we propose the Bootstrap method to construct the simultaneous confidence band. In Section 4, simulation studies are used to illustrate the proposed method under the limited samples. The technical proofs of the main theorems are presented in the Appendix.

2 Estimation procedure and asymptotic properties

2.1 Estimation procedure

Let $\{(Y_{it};\bm{X}_{it}^{\tau},Z_{it}),i=1,\cdots,n,t=1,\cdots,T\}$ be an independent identically distributed (i.i.d.) random sample which comes from model (1.1). In this paper, we consider the asymptotic theories by letting $n$ approach infinity and holding $T$ fixed. In this section, we consider the estimation procedure to first remove the fixed effects, and further obtain the efficient estimators of parametric and nonparametric components.

For ease of notation, let

[TABLE]

and $\mathbf{X}=(\bm{X}_{11},\cdots,\bm{X}_{1T},\bm{X}_{21},\cdots,\bm{X}_{2T},\cdots,\bm{X}_{n1},\cdots,\bm{X}_{nT})^{\tau}$ is an $nT\times p$ matrix, where $\bm{X}_{it}=(X_{it1},\cdots,X_{itp})^{\tau}$. Then model (1.1) can be written as the following matrix form,

[TABLE]

where $\mathbf{I}_{n}$ is an $n\times n$ identity matrix, $\bm{e}_{T}$ is a $T$-dimensional column vector with all elements being 1, and $\otimes$ denotes the Kronecker product. Furthermore, by the identification assumption $\sum\limits_{i=1}^{n}\alpha_{i}=0$, we have $\alpha_{1}=-\sum\limits_{i=2}^{n}\alpha_{i}$. Define the $(nT)\times(n-1)$ matrix $\mathbf{D}=[-\bm{e}_{n-1},\mathbf{I}_{n-1}]^{\tau}\otimes\bm{e}_{T}$, and $\bm{\alpha}=(\alpha_{2},\cdots,\alpha_{n})^{\tau}$, model (2.1) can be rewritten as

[TABLE]

Given $\bm{\alpha}$ and $\bm{\beta}$, model (2.2) is a version of the usual nonparametric fixed effects panel data model

[TABLE]

We first apply the local polynomial method (see the details in Fan and Gijbels, 1996) to estimate the nonparametric function $g(\cdot)$. For $Z_{it}$ in a small neighborhood of $z\in[0,1]$, approximate $g(Z_{it})$ by

[TABLE]

Let $K(\cdot)$ is a kernel function in $\mathbb{R}$, $K_{h}(z)=K(z/h)/h$, where $h$ is a bandwidth, and let

[TABLE]

$\mathbf{W}_{z}=\mathrm{diag}(K_{h}(Z_{11}-z),\cdots,K_{h}(Z_{1T}-z),K_{h}(Z_{21}-z),\cdots,K_{h}(Z_{2T}-z),\cdots,K_{h}(Z_{n1}-z),\cdots,K_{h}(Z_{nT}-z))$ is an $(nT)\times(nT)$ diagonal matrix. Let $\bm{G}(z)=(g(z),(g^{\prime}(z)))^{\tau}$, $\bm{\eta}=(\bm{\alpha}^{\tau},\bm{\beta}^{\tau})^{\tau}$.

In what follows, we outline the estimation procedure for $\bm{\beta}$ and $g(\cdot)$.

Given $\bm{\eta}=(\bm{\alpha}^{\tau},\bm{\beta}^{\tau})^{\tau}$, we define the following weighted least-squares objective function

[TABLE]

Minimizing the above objective function (2.5) with respect to $\bm{G}(z)$, we can obtain the solution of $\bm{G}(z)$ as follows

[TABLE]

Define the smoothing operator by

[TABLE]

Then, we can define the estimator of $g(z)$ by

[TABLE]

where $\bm{m}^{\tau}(z)=\bm{e}^{\tau}\mathbf{M}(z)$, $\bm{e}=(1,0)^{\tau}$ is a $2\times 1$ vector.

Since the fixed effects is an $n$-dimensional unobserved variable, it is difficult to obtain the consistent estimator for the fixed effects. Therefore, we first need to remove the fixed effects from the model, and further obtain the estimators of parametric and nonparametric components. By (2.7), we define the following objective function

[TABLE]

where $\widetilde{\bm{g}}_{\bm{\eta}}(z)=(\widetilde{g}(Z_{11},\bm{\eta}),\cdots,\widetilde{g}(Z_{1T},\bm{\eta}),\cdots,\widetilde{g}(Z_{n1},\bm{\eta}),\cdots,\widetilde{g}(Z_{nT},\bm{\eta})$, $\widetilde{\bm{Y}}=(\mathbf{I}_{nT}-\mathbf{M})\bm{Y}$, $\widetilde{\mathbf{X}}=(\mathbf{I}_{nT}-\mathbf{M})\mathbf{X}$, $\widetilde{\mathbf{D}}=(\mathbf{I}_{nT}-\mathbf{M})\mathbf{D}$, $\widetilde{\mathbf{Q}}=\mathbf{I}_{nT}-\widetilde{\mathbf{D}}(\widetilde{\mathbf{D}}^{\tau}\widetilde{\mathbf{D}})^{-1}\widetilde{\mathbf{D}}^{\tau}$, and $\mathbf{M}$ is an $(nT)\times(nT)$ smoothing matrix, that is

[TABLE]

In addition, let $\mathbf{P}=(\mathbf{I}_{nT}-\mathbf{M})^{\tau}(\mathbf{I}_{nT}-\mathbf{M})$ be an $(nT)\times(nT)$ matrix.

Taking derivative of (2.8) with respect to $\bm{\alpha}$ and setting it equal to zero, we have

[TABLE]

Obviously, the estimator of the fixed effects depends on $\bm{\beta}$. Based on the idea of least-squares dummy-variable approach in panel data parametric models and the nonparametric local linear regression technique, we then apply the profile least-squares dummy variable method to estimate parameter vector $\bm{\beta}$.

Plugging (2.9) into (2.8), we then minimize the profile least-squares objective function with respect to $\bm{\beta}$. Thus, we obtain the profile least-squares estimator of $\bm{\beta}$ as

[TABLE]

By (2.10) and (2.9), we have

[TABLE]

By $\sum\limits_{i=1}^{n}\alpha_{i}=0$ and (2.11), the estimator of $\alpha_{1}$ is $\hat{\alpha}_{1}=-\sum\limits_{i=2}^{n}\hat{\alpha}_{i}$.

By (2.6), (2.10) and (2.11), and some simple calculations, we can obtain the estimator of $\bm{{G}}(z)$ as follows

[TABLE]

By (2.7) and (2.12), we get the estimator of $g(z)$ as

[TABLE]

Remark 1**.**

From (2.10) and (2.13), it is easy to see that the estimators of $\bm{\beta}$ and $g(\cdot)$ do not depend on the fixed effects.

2.2 Asymptotic properties

Let $\mu_{l}=\int z^{l}K(z)dz$ and $\nu_{l}=\int z^{l}K^{2}(z)dz$ for $l=0,1,2$. Define the observed covariate set by $\mathcal{D}=\{\bm{X}_{it},Z_{it},1\leq i\leq n,1\leq t\leq T\}$. In order to obtain the main results, we first present the following technical conditions.

(C1)  $(\alpha_{i},\bm{V}_{i},\mathbf{X}_{i},\bm{Z}_{i}),i=1,\cdots,n$, are i.i.d., where $\bm{V}_{i}=({V}_{i1},{V}_{i2},\cdots,{V}_{iT})^{\tau}$, and $\mathbf{X}_{i}$ and $\bm{Z}_{i}$ can be defined similarly. $E\|\bm{X}_{it}\|^{2+\delta}<\infty$ and $E\|V_{it}\|^{2+\delta}<\infty$ for some $\delta>0$. Let $\sigma^{2}(\bm{x},z)=\mathrm{Var}(Y_{it}|\bm{X}_{it}=\bm{x},Z_{it}=z)$, $\sigma^{2}(z)=\mathrm{Var}(Y_{it}|Z_{it}=z)$, and $0<\sigma^{2}(\bm{x},z),\sigma^{2}(z)<\infty$.

(C2)  $E(Y_{it}|\mathbf{X}_{i},\bm{Z}_{i},\alpha_{i})=E(Y_{it}|\bm{X}_{it},Z_{it},\alpha_{i})=\bm{X}_{it}^{\tau}\bm{\beta}+g(Z_{it})+\alpha_{i},i=1,\cdots,n,t=1,\cdots,T$.

(C3)  Let $f(z)=\sum\limits_{t=1}^{T}f_{t}(z)$, where $f_{t}(z)$ is the continuous density function of $Z_{it}$, and $f_{t}(z)$ is bounded away from zero and infinity on $[0,1]$ for each $t=1,\cdots,T$. Let $\widetilde{V}_{it}=V_{it}-\frac{1}{T}\sum_{s=1}^{T}V_{is}$, $\sigma_{t}^{2}(z)=E[\widetilde{V}^{2}_{it}|Z_{it}=z]$ and $\bar{\sigma}^{2}(z)=\sum_{t=1}^{T}\sigma_{t}^{2}(z)f(z)$.

(C4)  Let $\bm{p}(z)=E(\bm{X}_{it}|Z_{it}=z)$. The functions $g(\cdot)$ and $\bm{p}(\cdot)$ have the bounded and continuous second derivatives on $[0,1]$.

(C5)  The kernel function $K(\cdot)$ is a symmetric density function, and is absolutely continuous on its support set $[-A,A]$.

(C5a)    $K(A)\neq 0$ or

(C5b)    $K(A)=0$, $K(t)$ is absolutely continuous and $K^{2}(t)$, $[K^{\prime}(t)]^{2}$ are integrable on the $(-\infty,+\infty)$.

(C6)  The bandwidth $h$ satisfies that $nh^{3}/\log n\rightarrow\infty$, $nh^{5}\log n\rightarrow 0$, as $n\rightarrow\infty$.

Theorem 1**.**

Assume that conditions (C1)–(C6) hold. Let $b(z)=h^{2}\mu_{2}g^{\prime\prime}(z)/2$, $\Sigma_{g}=\nu_{0}\bar{\sigma}^{2}(z)f^{-2}(z)$, $\Sigma_{g^{\prime}}=\nu_{2}\bar{\sigma}^{2}(z)/(f^{2}(z)\mu^{2}_{2})$, Then uniformly for $z\in[0,1]$, we have

[TABLE]

and

[TABLE]

[TABLE]

where “$\stackrel{{\scriptstyle L}}{{\longrightarrow}}$” denotes the convergence in distribution.

Theorem 2**.**

Assume that conditions (C1)–(C6) hold and $h=O(n^{-\rho})$ for $1/5\leq\rho<1/3$. Then for all $z\in[0,1]$, we have

[TABLE]

where if $K(A)\neq 0$,

[TABLE]

and if $K(A)=0$,

[TABLE]

Theorem 2 gives the asymptotic distribution of the maximum absolute deviation between the estimated nonparametric component $\hat{g}(\cdot)$ and the true nonparametric component $g(\cdot)$ when the estimator of $\bm{\beta}$ is $\sqrt{n}-$consistent. It provides us the theoretical foundation for constructing the simultaneous confidence band of the nonparametric function in model (1.1).

Remark 2**.**

If the supremum in Theorem 2 is taken on an interval of $[c,d]$ instead of $[0,1]$, Theorem 2 still holds under certain conditions by transformation. The asymptotic distribution is represented as

[TABLE]

where $\widetilde{d}_{n}$ is the same as $d_{n}$ in the Theorem 2 except that $h$ is replaced by $h/(d-c)$.

Theorem 3**.**

Assume that conditions (C1)–(C6) hold and $\Sigma_{g^{\prime}}=\nu_{2}\bar{\sigma}^{2}(z)/(f^{2}(z)\mu^{2}_{2})$. Then for all $z\in[0,1]$, we have

[TABLE]

where $d_{n_{1}}=(-2\log h)^{1/2}+\frac{1}{(-2\log h)^{1/2}}\log\left\{\frac{1}{2\pi\sqrt{\nu_{2}}}(\int z^{2}(K^{\prime}(z))^{2}dz)^{1/2}\right\}.$ If $K(c_{0})=0$, $K(z)$ is absolutely continuous and $K^{2}(z)$, $(K^{\prime}(z))^{2}$ are integrable on $(-\infty,+\infty)$

.

Theorem 3 presents the asymptotic distribution of the maximum absolute deviation for $\hat{g}^{\prime}(\cdot)$

2.3 Simultaneous confidence band for the nonparametric

function

Since the asymptotic bias and variance of $\hat{g}(\cdot)$ in Theorem 2 involve some unknown quantities, we cannot apply Theorem 2 to construct simultaneous confidence band of $g(\cdot)$ directly. In order to construct the simultaneous confidence band of $g(\cdot)$, we first need to get the consistent estimators of the asymptotic bias and variance of $\hat{g}(\cdot)$. By Theorem 1, the asymptotic bias of $\hat{g}(z)$ is

[TABLE]

Thus, the consistent estimator of the asymptotic bias is $\mathrm{\widehat{bias}}(\hat{g}(z))=h^{2}\mu_{2}\hat{g}^{\prime\prime}(z)/2$, where the estimator $\hat{g}^{\prime\prime}(z)$ of $g^{\prime\prime}(z)$ is obtained by using local cubic fit with an appropriate pilot bandwidth $h_{*}=O(n^{-1/7})$, which is optimal for estimating $g^{\prime\prime}(z)$ and can be chosen by the residual squares criterion proposed in Fan and Gijbels (1996).

Next we will estimate the asymptotic variance of $\hat{g}(z)$. For simplicity, suppose that the random errors $V_{it}$ are i.i.d. for all $i$ and $t$. By the proofs of theorem, we have

[TABLE]

where $\mathbf{Q}_{1}=(\mathbf{I}_{nT}-\mathbf{D(D^{\tau}PD)}^{-1}\mathbf{D^{\tau}P})$ and $\Phi_{1}=\mathrm{diag}(\sigma^{2}(Z_{11}),\cdots,\sigma^{2}(Z_{1T}),\sigma^{2}(Z_{21}),\cdots,\\ \sigma^{2}(Z_{2T}),\cdots,\sigma^{2}(Z_{n1}),\cdots,\sigma^{2}(Z_{nT}))$. Using the similar approximate local homoscedasticity in Li, Peng and Tong (2013), the asymptotic variance of $\hat{g}(z)$ is defined by

[TABLE]

Let $\hat{\bm{V}}=\bm{Y}-\bm{\hat{Y}}$ be the residual, where $\bm{\hat{Y}}=\bm{\hat{g}}+\bf{X}\hat{\bm{\beta}}+\bf{D}\hat{\bm{\alpha}}$. By (2.10), (2.11) and (2.13), we have

[TABLE]

where $\mathbf{Q}_{2}=\mathbf{I}_{nT}-\mathbf{X}(\mathbf{X}^{\tau}\mathbf{PQ}_{1}\mathbf{X})^{-1}\mathbf{X}^{\tau}\mathbf{PQ}_{1}$. Obviously, the residual $\hat{\bm{V}}$ does not depend on the fixed effects, and is a linear function of $\bm{Y}$. By the normalized weighted residual sum of squares, $\sigma^{2}(z)$ can be estimated by

[TABLE]

Theorem 4**.**

Under the conditions in Theorem 2, and assume that $\hat{g}^{(3)}(\cdot)$ is continuous on $[0,1]$ and the pilot bandwidth $h_{\ast}$ satisfies that $h_{\ast}=O(n^{-1/7})$. Then for all $z\in[0,1]$, we have

[TABLE]

where $d_{n}$ is defined in Theorem 2.

By Theorem 4, we construct the $(1-\alpha)\times 100\%$ simultaneous confidence band of the nonparametric function $g(z)$ as

[TABLE]

where $\Delta_{1,\alpha}(z)=\left(d_{n}+[\log 2-\log\{-\log(1-\alpha)\}](-2\log h)^{-1/2}\right)\left[\widehat{\mathrm{Var}}\{\hat{g}(z)|\mathcal{D}\}\right]^{1/2}.$

3 The Bootstrap method

Despite the fact that Theorem 4 provides the asymptotic distribution to construct the simultaneous confidence band (2.15) for the nonparametric component, we need to estimate the asymptotic bias and the asymptotic conditional variance. First, the estimator of the asymptotic bias involves the estimator the second derivative $g^{\prime\prime}(\cdot)$ and the choice of the pilot bandwidth $h_{*}$ for estimating the second derivative $g^{\prime\prime}(\cdot)$. The estimator of the second derivative $g^{\prime\prime}(\cdot)$ has a slow convergence rate, and is very sensitive with the pilot bandwidth $h_{*}$. This will influence the estimator of the asymptotic bias. Second, the asymptotic variance estimation is very complicated, especially for panel data semiparametric fixed effects model. Finally, the asymptotic critical value $c_{\alpha}$ depends on the double exponential distribution, the estimators of asymptotic bias and the asymptotic conditional variance. These will not only cause computational burden and accumulative errors, but also lead to the difficulty to construct simultaneous confidence band. To overcome these problems, we extend the Bootstrap method used in Li, Peng and Tong (2013) to partially linear panel data fixed effects model (1.1).

Now we discuss how to use the Bootstrap procedure to construct simultaneous confidence band for $g(\cdot)$. Let

[TABLE]

Suppose that the upper $\alpha$ quantile of $T$ is $c_{\alpha}$. If $c_{\alpha}$ and $\mathrm{Var}(\hat{g}(z|\mathcal{D}))$ are known, the simultaneous confidence band of $g(\cdot)$ with $(1-\alpha)\times 100\%$ on the interval $[0,1]$ should be

[TABLE]

However, $c_{\alpha}$ and $\mathrm{Var}(\hat{g}(z|\mathcal{D}))$ are unknown. We will get their estimators using the bootstrap method. Suppose that we have the estimators $\hat{c}_{\alpha}$ and $\mathrm{Var}^{\ast}(\hat{g}(z|\mathcal{D}))$ of $c_{\alpha}$ and $\mathrm{Var}(\hat{g}(z|\mathcal{D}))$, respectively. Then we can obtain the $(1-\alpha)\times 100\%$ simultaneous confidence band of $g(\cdot)$ as follows

[TABLE]

The Bootstrap procedure is given as follows:

(1)  By (2.14), obtain the residuals $\bm{\hat{V}}=(\mathbf{I}_{nT}-\mathbf{M})\mathbf{Q}_{1}\mathbf{Q}_{2}\bm{Y}$, where $\bm{\hat{V}}=(\hat{V}_{11},\cdots,\hat{V}_{1T},\\ \hat{V}_{21},\cdots,\hat{V}_{2T},\cdots,\hat{V}_{n1},\cdots,\hat{V}_{nT})^{\tau}$.

(2)  For each $i=1,\cdots,n,~{}t=1,\cdots,T$, obtain the bootstrap error $V^{\ast}_{it}=\hat{V}_{it}\varepsilon_{it}$, where $\varepsilon_{it}$ are i.i.d. $\thicksim N(0,1)$ across $i$ and $t$. Generate the bootstrap sample member $Y^{\ast}_{it}$ by $Y^{\ast}_{it}=\hat{Y}_{it}+V^{\ast}_{it},~{}i=1,\cdots,n,~{}t=1,\cdots,T$.

(3)  Given the bootstrap resample $\{(Y^{\ast}_{it},\bm{X}_{it},Z_{it}),i=1,\cdots,n,~{}t=1,\cdots,T\}$, obtain the estimators of $\bm{\beta}$ and $g(\cdot)$, and denote the resulting estimate by $\bm{\hat{\beta}}^{\ast}$ and $\hat{g}^{\ast}(\cdot)$, as the bootstrap estimators of $\bm{\beta}$ and $g(\cdot)$, respectively.

(4)  Repeat (2)–(3) $N$ times to get a size $N$ bootstrap sample of $g(\cdot)$, $\hat{g}^{\ast}_{k}(\cdot),k=1,\cdots,N$. The estimator $\mathrm{Var}^{\ast}(\hat{g}(z))$ of $\mathrm{Var}(\hat{g}(\cdot))$ is taken as the sample variance of $\hat{g}^{\ast}_{k}(\cdot)$.

(5)  Compute the bootstrap sample of $T$ by

[TABLE]

Use the upper $\alpha$ percentile $\hat{c}_{\alpha}$ of $T^{\ast}_{k},k=1,\cdots,N,$ to estimate the upper $\alpha$ quantile $c_{\alpha}$ of $T$.

We can construct the $(1-\alpha)\times 100\%$ simultaneous confidence band of $g(\cdot)$ by (3.1) when we obtain the estimators of $c_{\alpha}$ and $\mathrm{Var}(\hat{g}(z|\mathcal{D}))$.

4 Simulation studies

We conduct simulation studies to assess the performance of our proposed method. Our simulated data are generated from the following model:

[TABLE]

where $\bm{\beta}=(-1,3,5)^{\tau}$, $\bm{X}_{it}$ are three dimensional i.i.d. random variables from uniform [-1,1], $Z_{it}$ are i.i.d. from uniform [-1,1], and the random errors $V_{it}$ are i.i.d. from $N(0,1)$. In this simulation, we only consider $\alpha_{i}$ are correlated with the covariate $Z_{i.}$, and generate $\alpha_{i}=\varepsilon_{i}+cZ_{i\cdot},i=2,\cdots,n$, where $\varepsilon_{i}\thicksim N(0,1),Z_{i\cdot}=\frac{1}{T}\sum\limits_{t=1}^{T}Z_{it}$ and $\alpha_{1}=-\sum\limits^{n}_{i=2}\alpha_{i},i=1,\cdots,n$. We consider three cases for $c=0,0.5,1$. When $c\neq 0$, $Z_{it}$ and $\alpha_{i}$ are correlated, model (4.1) is the partially linear fixed effects model. When $c=0$, model (4.1) leads to the usual partially linear random effects model.

In our simulation studies, we apply the Epanechnikov kernel $K(z)=0.75(1-z^{2})_{+}$ for estimating the nonparametric function. Finding an appropriate bandwidth can be of both theoretical and practical interest. To implement the estimation procedure described in Section 2, we need to choose the bandwidth $h$. One can select $h$ by minimizing the generalized cross validation criterion. Here we use the following cross validation method to automatically select the optimal bandwidth $h_{\rm CV}$.

[TABLE]

where $Y_{it}^{-it}$ denote the fitted values that are computed from data with measurements of the $\{Y_{it},{\bm{X}}_{it}\}$ observation deleted. $k=(i-1)T+t$, $\hat{V}_{it}=Y_{it}-\hat{Y}_{it}$ and $l_{kk}$ is the ${(k,k)}$ element of matrix $[\mathbf{I}_{nT}-(\mathbf{I}_{nT}-\mathbf{M})\mathbf{Q}_{1}\mathbf{Q}_{2}]$. The cross validation bandwidth $h_{\rm CV}$ is then defined to be the minimizer of ${\rm CV}(h)$.

We fix $T=5$ and examine the finite sample performance of the proposed method when the sample size is taken as $n=100,150$ and $200$. For each case, 1000 replicates of simulated realizations are generated, and the nominal level is taken as $1-\alpha=0.95$. The results are given in Tables 4–2 and Figure 4. Table 4 gives the bias, the standard deviation and the mean squared error of the estimator $\hat{\bm{\beta}}$ for $c=0$ and $c=1$. From Table 4, we can find that the bias, the standard deviation and the mean squared error are decreased as the sample size $n$ increases for two cases. For the same sample size $n$, the results of $c=1$ are better than those of $c=0$. Model (4.1) is reduced to partially linear random effects model when $c=0$. From (2.10) and (2.13), it is easy to see that, in order to remove the fixed effects from the model, we loss some sample information to obtain the estimators of parametric and nonparametric components. So the profile least-squares dummy-variable method is not suitable for the partially linear random effects model, and the resulting estimators of parametric and nonparametric components are not efficient. Thus, we need develop the effective estimation procedure to estimate the random effects models, such as the generalized profile least squares method or the generalized estimating equation (GEE).