Quantile regression for general spatial panel data models with fixed effects

ABSTRACT

This paper considers the quantile regression model with both individual fixed effect and time period effect for general spatial panel data. Fixed effects quantile regression estimators based on instrumental variable method will be proposed. Asymptotic properties of the proposed estimators will be developed. Simulations are conducted to study the performance of the proposed method. We will illustrate our methodologies using a cigarettes demand data set.

1. Introduction

Spatial econometric models have been widely used in many areas (e.g. economics, political science, and public health) to deal with spatial interaction effects among geographical units (e.g. jurisdictions, regions, and states). Recently, the spatial econometrics literature has exhibited a growing interest in the specification and estimation of econometric relationships based on spatial panels, which typically refer to data containing time series observations of a number of spatial units. For instance, Kapoor et al. [11] developed a generalized moments (GM) estimator for a space–time model with error components that are both spatially and time-wise correlated. Lee and Yu [16] proposed the maximum likelihood (ML) estimator for the spatial autoregressive (SAR) panel model with both spatial lag and spatial disturbances. All these works were developed based on (conditional) mean regression methods. Reich et al. [17] reviewed nonparametric Bayesian approaches to inference for spatial data. Compared with mean regression methods, the quantile regression (QR) method is more robust and can be adopted to deal with data characterized by different error distributions.

Recently, there has been a growing literature on estimating and testing of QR panel data models. Koenker [14] introduced a novel approach for the estimation of a QR model for longitudinal data. Galvao [8] studied the quantile regression dynamic panel model with fixed effects. Galvao et al. [9] investigated the estimation of censored QR models with fixed effects. Galvao and Wang [10] developed a new minimum distance quantile regression (MD-QR) estimator for panel data models with fixed effects. However, quantile regression estimation for spatial econometric panel models has not been studied in the existing literature.

This paper focuses on the QR estimations in the general SAR panel data model with both individual fixed effects and time period specific effects [16]. We employ the fixed effects quantile regression (FEQR) estimator based on instrumental variable (IV) method to estimate the parameters. The asymptotic properties of the IV-FEQR estimator are also developed.

We apply our theoretical results for the demand for cigarettes. In our model, the spatial lag of real per capita sales of cigarettes is viewed as endogenous. The model presented in this paper can be viewed as a variant of the model proposed by Baltagi and Levin [3]. However, their model did not contain spatial lags, which neglects the spatial dependence in data, and therefore did not involve the ‘endogeneity’ problem. Besides, in the existing literature, most of the related researches upon this example are established in the mean regression framework. As an overview of our results, we found that the cigarette sales between neighbor states are significantly spatial correlated, and the estimation of the spatial correlation coefficients is affected by the endogeneity problem. In addition, the effects of the log average cigarettes retail price and the log disposable income to the cigarettes sales are differed at different quantile levels.

The rest of the paper is organized as follows. Section 2 introduces the SAC panel data model with both individual fixed effects and time period fixed effects, and proposes the IV-FEQR estimation procedure. The asymptotic properties of the IV-FEQR estimators are also discussed. Proofs of the theorems in Sections 2 are given in Appendix. Section 3 reports a simulation study for assessing the finite sample performance of the proposed estimators. An empirical illustration is considered in Section 4. Section 5 concludes the paper.

2. General spatial autoregressive panel data quantile regression model with both individual and time effects

Lee and Yu [16, Equation 19] considered the following general spatial autoregressive panel data model with both individual and time effects

$$\begin{array}{rl}{y}_{it}& =\rho \sum _{j\ne i}{W}_{ij}{y}_{jt}+{\mathit{X}}_{it}^{\mathrm{\top }}\mathit{\beta }+{\nu }_i+{\psi }_t+u_{it},i=1,\dots ,N,t=1,\dots ,T,\\ u_{it}& =\lambda \sum _{j\ne i}{M}_{ij}{u}_{jt}+{\epsilon }_{it},\end{array}$$ (1)

where $y_{it}$ is the dependent variable for subject i at time t, ${\mathit{X}}_{it}$ is a $p\times 1$ vector of non-stochastic time varying explanatory variables, $W_{ij}$ is the $(i,j)$th element of the spatial weight matrix $\mathit{W}$ reflecting spatial dependence on $y_{it}$ among cross sectional units, and ${\epsilon }_{it}$ is independent and identically distributed across i and t. Similarly, $M_{ij}$ is the $(i,j)$th element of the spatial weight matrix $\mathit{M}$ for the disturbances. The parameters ${\nu }_i,i=1,\dots ,N$ are fixed effects for the regions while the parameters ${\psi }_t,t=1,\dots ,T$ are fixed time effects. Interaction effects are reflected in the spatial–temporal lag variable ${\displaystyle \sum _{j\ne i}{W}_{ij}{y}_{jt}}$ (and associated scalar parameter ρ). Here, we consider the case $\mathit{M}\ne \mathit{W}$.

The model (1) can also be written in an alternative form as

$$y_{it}=\rho d_{it}^{\left(1\right)}+\lambda d_{jt}^{\left(2\right)}+\kappa d_{it}^{\left(3\right)}+{\mathit{X}}_{it}^{\ast \mathrm{\top }}{\mathit{\beta }}^{\ast }+{\mathit{Z}}_{1i}^{\ast \mathrm{\top }}{\mathit{\nu }}^{\ast }+{\mathit{Z}}_{2t}^{\ast \mathrm{\top }}{\mathit{\psi }}^{\ast }+{\epsilon }_{it},$$ (2)

where $d_{it}^{\left(1\right)}=\sum _{j\ne i}{W}_{ij}{y}_{jt}$, $d_{it}^{\left(2\right)}=\sum _{j\ne i}{M}_{ij}{y}_{jt}$, $d_{it}^{\left(3\right)}=-\sum _{j\ne i}\sum _{k\ne j}{M}_{ij}{W}_{jk}{y}_{kt}$, $\kappa =\rho \lambda$, ${\mathit{X}}_{it}^{\ast }=[{\mathit{X}}_{it}^{\mathrm{\top }},-\sum _{j\ne i}{M}_{ij}{\mathit{X}}_{jt}^{\mathrm{\top }}{]}^{\mathrm{\top }}$, ${\mathit{\beta }}^{\ast }=({\mathit{\beta }}^{\mathrm{\top }},\lambda {\mathit{\beta }}^{\mathrm{\top }}{)}^{\mathrm{\top }}$, $\mathit{\nu }=({\nu }_1,\dots ,{\nu }_N{)}^{\mathrm{\top }}$, $\mathit{\psi }=({\psi }_1,\dots ,{\psi }_T{)}^{\mathrm{\top }}$, ${\mathit{\nu }}^{\ast }=({\mathit{\nu }}^{\mathrm{\top }},\lambda {\mathit{\nu }}^{\mathrm{\top }}{)}^{\mathrm{\top }}$, ${\mathit{\psi }}^{\ast }=({\mathit{\psi }}^{\mathrm{\top }},\lambda {\mathit{\psi }}^{\mathrm{\top }}{)}^{\mathrm{\top }}$, ${\mathit{Z}}_1={\mathbf{1}}_T\otimes {\mathit{I}}_N$ is an $NT\times N$ matrix, ${\mathit{Z}}_2={\mathit{I}}_T\otimes {\mathbf{1}}_N$ is an $NT\times T$ matrix, ${\mathbf{1}}_J$ is the $J\times 1$ vector with all the elements being 1, ${\mathit{Z}}_{1i}^{\ast }=[{\mathit{Z}}_{1i}^{\mathrm{\top }},-\sum _{j\ne i}{M}_{ij}{\mathit{Z}}_{1j}^{\mathrm{\top }}{]}^{\mathrm{\top }}$, ${\mathit{Z}}_{2t}^{\ast }=[{\mathit{Z}}_{2t}^{\mathrm{\top }},-\sum _{j\ne i}{M}_{ij}{\mathit{Z}}_{2t}^{\mathrm{\top }}{]}^{\mathrm{\top }}$, ${\mathit{Z}}_{1i}={\mathit{Z}}_1^{\mathrm{\top }}{\mathit{h}}_{1i}$ is an indicator variable for the individual effect ${\nu }_i$, ${\mathit{h}}_{1i}$ is an $NT\times 1$ vector with the ith element equal to 1 and the rest equal to 0, $i=1,\dots ,N$, ${\mathit{Z}}_{2t}={\mathit{Z}}_2^{\mathrm{\top }}{\mathit{h}}_{2t}$ is an indicator variable for the time effect ${\psi }_t$, and ${\mathit{h}}_{2t}$ is an $NT\times 1$ vector with the $(t-1)N+1$th element equal to 1 and the rest equal to 0, $t=1,\dots ,T$.

Matrix form of model (2) is

$$\mathit{y}=\mathit{D}\mathit{\alpha }+{\mathit{X}}^{\ast }{\mathit{\beta }}^{\ast }+{\mathit{Z}}_1^{\ast }{\mathit{\nu }}^{\ast }+{\mathit{Z}}_2^{\ast }{\mathit{\psi }}^{\ast }+\mathit{\epsilon },$$ (3)

where $\mathit{y}=({\mathit{y}}_1,\dots ,{\mathit{y}}_T{)}^{\mathrm{\top }}$ is an $NT\times 1$ vector with ${\mathit{y}}_t=(y_{1t},\dots ,y_{Nt}{)}^{\mathrm{\top }}$, ${\mathit{D}}^{\left(i\right)}=({\mathit{D}}_1^{\left(i\right)},\dots ,{\mathit{D}}_T^{\left(i\right)}{)}^{\mathrm{\top }}$ is an $NT\times 1$ vector with ${\mathit{D}}_t^{\left(i\right)}=(d_{1t}^{\left(i\right)},\dots ,d_{Nt}^{\left(i\right)}{)}^{\mathrm{\top }}$, i=1,2,3, $\mathit{X}=({\mathit{X}}_1,\dots ,{\mathit{X}}_T{)}^{\mathrm{\top }}$ is an $NT\times p$ matrix, ${\mathit{X}}_t=({\mathit{X}}_{1t},\dots ,{\mathit{X}}_{Nt}{)}^{\mathrm{\top }}$, $\mathit{\epsilon }=({\mathit{\epsilon }}_1,\dots ,{\mathit{\epsilon }}_T{)}^{\mathrm{\top }}$ is an $NT\times 1$ vector with ${\mathit{\epsilon }}_t=({\epsilon }_{1t},\dots ,{\epsilon }_{Nt}{)}^{\mathrm{\top }}$, ${\mathit{X}}^{\ast }=[\mathit{X},-{\mathit{M}}^{\ast }\mathit{X}]$, ${\mathit{Z}}_1^{\ast }=[{\mathit{Z}}_1,-{\mathit{M}}^{\ast }{\mathit{Z}}_1]$, and ${\mathit{Z}}_2^{\ast }=[{\mathit{Z}}_2,-{\mathit{M}}^{\ast }{\mathit{Z}}_2]$, ${\mathit{W}}^{\ast }={\mathit{I}}_T\otimes \mathit{W}$, ${\mathit{M}}^{\ast }={\mathit{I}}_T\otimes \mathit{M}$, $\mathit{W}$ and $\mathit{M}$ are both $N\times N$ spatial weight matrices. Here we denote ${\mathit{\theta }}^{\ast }=(\rho ,\lambda ,\kappa ,{\mathit{\beta }}^{\ast \mathrm{\top }},{\mathit{\nu }}^{\ast \mathrm{\top }},{\mathit{\psi }}^{\ast \mathrm{\top }}{)}^{\mathrm{\top }}$.

We consider the following conditional τ-quantile of response variable:

$$\begin{array}{rl}& Q_{\tau }(y_{it}|d_{it}^{\left(1\right)},d_{it}^{\left(2\right)},d_{it}^{\left(3\right)},{\mathit{X}}_{it}^{\ast },{\mathit{Z}}_{1i}^{\ast },{\mathit{Z}}_{2t}^{\ast })\\ & =\rho \left(\tau \right)d_{it}^{\left(1\right)}+\lambda \left(\tau \right)d_{it}^{\left(2\right)}+\kappa \left(\tau \right)d_{it}^{\left(3\right)}+{\mathit{X}}_{it}^{\ast \mathrm{\top }}{\mathit{\beta }}^{\ast }\left(\tau \right)+{\mathit{Z}}_{1i}^{\ast \mathrm{\top }}{\mathit{\nu }}^{\ast }\left(\tau \right)+{\mathit{Z}}_{2t}^{\ast \mathrm{\top }}{\mathit{\psi }}^{\ast }\left(\tau \right),\end{array}$$ (4)

where τ is a quantile in the interval $(0,1)$, $Q_{\tau }({\epsilon }_{it}|d_{it}^{\left(1\right)},d_{it}^{\left(2\right)},d_{it}^{\left(3\right)},{\mathit{X}}_{it}^{\ast },{\mathit{Z}}_{1i}^{\ast },{\mathit{Z}}_{2t}^{\ast })=0$. We define the objection function by

$$\begin{array}{rl}R(\tau ,{\mathit{\theta }}^{\ast })& =\sum _{i=1}^N\sum _{t=1}^T{\rho }_{\tau }(y_{it}-\rho (\tau )d_{it}^{\left(1\right)}-\lambda (\tau )d_{it}^{\left(2\right)}-\kappa (\tau )d_{it}^{\left(3\right)}-{\mathit{X}}_{it}^{\ast \mathrm{\top }}{\mathit{\beta }}^{\ast }(\tau )\\ & -{\mathit{Z}}_{1i}^{\ast \mathrm{\top }}{\mathit{\nu }}^{\ast }\left(\tau \right)-{\mathit{Z}}_{2t}^{\ast \mathrm{\top }}{\mathit{\psi }}^{\ast }\left(\tau \right)),\end{array}$$ (5)

where ${\rho }_{\tau }\left(u\right)=u(\tau -I(u\le 0\left)\right)$ is the check function and $I(\cdot )$ is the indicator function [15]. The FEQR estimator $\widehat{{\mathit{\theta }}^{\ast }}\left(\tau \right)$ can then be obtained by

$$\widehat{{\mathit{\theta }}^{\ast }}\left(\tau \right)=\underset{{\mathit{\theta }}^{\ast }}{\arg min}R(\tau ,{\mathit{\theta }}^{\ast }).$$ (6)

Remark 1

However, in practice, the spatial weight matrices $\mathit{M}$ and $\mathit{W}$ may be equal, which will bring difficulty on the identification of ρ and λ in the quantile regression framework. Hence, we consider quantile regression for the following two special cases of model (1):

1. SAR panel data models:

   $$y_{it}=\rho \sum _{j\ne i}{W}_{ij}{y}_{jt}+{\mathit{X}}_{it}^{\mathrm{\top }}\mathit{\beta }+{\nu }_i+{\psi }_t+u_{it},i=1,\dots ,N,t=1,\dots ,T.$$ (7)
2. SEM panel data models  [6]:

   $$\begin{array}{rl}{y}_{it}& ={\mathit{X}}_{it}^{\mathrm{\top }}\mathit{\beta }+{\nu }_i+{\psi }_t+u_{it},\\ u_{it}& =\lambda \sum _{j\ne i}{M}_{ij}{u}_{jt}+{\epsilon }_{it},i=1,\dots ,N,t=1,\dots ,T,\end{array}$$ (8)

where the variables and parameters are the same as those in model (1).

2.1. IV-FEQR Estimator

However, there exist several lagged dependent variables (${\mathit{d}}^{\left(1\right)}$, ${\mathit{d}}^{\left(2\right)}$, ${\mathit{d}}^{\left(3\right)}$) in model (2), which are also endogenous variables. And the presence of endogenous variables will cause biased estimation. Thus the FEQR estimation of model (1) is biased as in the OLS case especially for ρ, λ and κ. The problem of bias for quantile regression for general spatial autoregressive panel model (1) ameliorated through the use of instrumental variables. Therefore, we employ the instrumental variable quantile regression (IVQR) method for estimation in this section. Let $\mathit{\alpha }=(\rho ,\lambda ,\kappa {)}^{\mathrm{\top }}$ and ${\mathit{D}}_{it}=[d_{it}^{\left(1\right)},d_{it}^{\left(2\right)},d_{it}^{\left(3\right)}{]}^{\mathrm{\top }}$ is a vector of endogenous variables, which are related to a vector of instruments ${\mathit{\omega }}_{it}$. The instruments ${\mathit{\omega }}_{it}$ are independent of ${\epsilon }_{it}$. Consider the objection function for the conditional instrumental quantile relationship:

$$\begin{array}{rl}& R_{IV}(\tau ,{\mathit{\theta }}^{\ast },\mathit{\gamma })=\sum _{i=1}^N\sum _{t=1}^T{\rho }_{\tau }(y_{it}-{\mathit{D}}_{it}^{\mathrm{\top }}\mathit{\alpha }(\tau )-{\mathit{X}}_{it}^{\ast \mathrm{\top }}{\mathit{\beta }}^{\ast }(\tau )-{\mathit{Z}}_{1i}^{\ast \mathrm{\top }}{\mathit{\nu }}^{\ast }(\tau )\\ & -{\mathit{Z}}_{2t}^{\ast \mathrm{\top }}{\mathit{\psi }}^{\ast }\left(\tau \right)-{\mathit{\omega }}_{it}^{\mathrm{\top }}\mathit{\gamma }\left(\tau \right)).\end{array}$$ (9)

Following Chernozhukov and Hansen [4,5] and Galvao [8], and assuming the availability of instrumental variables ${\mathit{\omega }}_{it}$, we can derive the IV-FEQR estimator via the following three steps:

Step 1: For a given quantile τ, define a suitable set of values $\{{\mathit{\alpha }}_i,i=1,\dots ,J;|\rho |,|\lambda |,|\kappa |<1\}$. One then minimizes the objective function for ${\mathit{\theta }}^{\ast },\mathit{\gamma }$ to obtain the ordinary QR estimators of ${\mathit{\beta }}^{\ast },{\mathit{\nu }}^{\ast },{\mathit{\psi }}^{\ast },\mathit{\gamma }$:

$$\left({\hat{\mathit{\beta }}}^{\ast }\right(\mathit{\alpha },\tau ),{\hat{\mathit{\nu }}}^{\ast }(\mathit{\alpha },\tau ),{\hat{\mathit{\psi }}}^{\ast }(\mathit{\alpha },\tau ),\hat{\mathit{\gamma }}(\mathit{\alpha },\tau \left)\right)=\underset{{\mathit{\beta }}^{\ast },{\mathit{\nu }}^{\ast },{\mathit{\psi }}^{\ast },\mathit{\gamma }}{\arg min}{R}_{IV}(\tau ,{\mathit{\theta }}^{\ast },\mathit{\gamma }).$$ (10)

Step 2: Choose $\hat{\mathit{\alpha }}\left(\tau \right)$ among $\{{\mathit{\alpha }}_i,i=1,\dots ,J\}$ which makes a weighted distance function defined on $\mathit{\gamma }$ closest to $\mathbf{0}$:

$$\hat{\mathit{\alpha }}\left(\tau \right)=\underset{\mathit{\alpha }\in \mathcal{A}}{\arg min}\left\{\hat{\mathit{\gamma }}(\mathit{\alpha },\tau {)}^{\mathrm{\top }}\hat{\mathit{A}}(\tau \left)\hat{\mathit{\gamma }}\right(\mathit{\alpha },\tau )\right\},$$ (11)

where $\mathit{A}$ is a positive definite matrix, $\mathcal{A}$ is the paramter space of $\mathit{\alpha }$.

Step 3: The estimation of ${\mathit{\beta }}^{\ast },{\mathit{\nu }}^{\ast },{\mathit{\psi }}^{\ast }$ can be obtained, which is respectively ${\hat{\mathit{\beta }}}^{\ast }\left(\hat{\mathit{\alpha }}\right(\tau ),\tau )$, ${\hat{\mathit{\nu }}}^{\ast }\left(\hat{\mathit{\alpha }}\right(\tau ),\tau )$, ${\hat{\mathit{\psi }}}^{\ast }\left(\hat{\mathit{\alpha }}\right(\tau ),\tau )$.

Similarly, the IV-FEQR estimator of SAR panel data model (7) can be obtained by minimizing the following objective function:

$$\begin{array}{rl}& R_{IV}(\tau ,\rho ,\mathit{\beta },\mathit{\nu },\mathit{\psi },\mathit{\gamma })=\sum _{i=1}^N\sum _{t=1}^T{\rho }_{\tau }(y_{it}-\rho (\tau )d_{it}^{\left(1\right)}-{\mathit{X}}_{it}^{\mathrm{\top }}\mathit{\beta }(\tau )-{\mathit{Z}}_{1i}^{\mathrm{\top }}\mathit{\nu }(\tau )\\ & -{\mathit{Z}}_{2t}^{\mathrm{\top }}\mathit{\psi }\left(\tau \right)-{\mathit{\omega }}_{it}^{\mathrm{\top }}\mathit{\gamma }\left(\tau \right)).\end{array}$$ (12)

And we can derive the IV-FEQR estimator via the following three steps:

Step 1: For a given quantile τ, define a suitable set of values $\{{\rho }_i,i=1,\dots ,J;|\rho |<1\}$. One then minimizes the objective function for $\mathit{\beta },\mathit{\nu },\mathit{\psi },\mathit{\gamma }$ to obtain the ordinary QR estimators of $\mathit{\beta },\mathit{\nu },\mathit{\psi },\mathit{\gamma }$:

$$\left(\hat{\mathit{\beta }}\right(\rho ,\tau ),\hat{\mathit{\nu }}(\rho ,\tau ),\hat{\mathit{\psi }}(\rho ,\tau ),\hat{\mathit{\gamma }}(\rho ,\tau \left)\right)=\underset{\mathit{\beta },\mathit{\nu },\mathit{\psi },\mathit{\gamma }}{\arg min}{R}_{IV}(\tau ,\rho ,\mathit{\beta },\mathit{\nu },\mathit{\psi },\mathit{\gamma }).$$ (13)

Step 2: Choose $\hat{\rho }\left(\tau \right)$ among $\{{\rho }_i,i=1,\dots ,J\}$ which makes a weighted distance function defined on $\mathit{\gamma }$ closest to 0:

$$\hat{\rho }\left(\tau \right)=\underset{\rho \in \mathcal{R}}{\arg min}\left\{\hat{\mathit{\gamma }}(\rho ,\tau {)}^{\mathrm{\top }}\hat{\mathit{A}}(\tau \left)\hat{\mathit{\gamma }}\right(\rho ,\tau )\right\},$$ (14)

where $\mathit{A}$ is a positive definite matrix, $\mathcal{R}$ is the paramter space of ρ.

Step 3: The estimation of $\mathit{\beta },\mathit{\nu },\mathit{\psi }$ can be obtained, which is respectively $\hat{\mathit{\beta }}\left(\hat{\rho }\right(\tau ),\tau )$, $\hat{\mathit{\nu }}\left(\hat{\rho }\right(\tau ),\tau )$, $\hat{\mathit{\psi }}\left(\hat{\rho }\right(\tau ),\tau )$.

The IV-FEQR estimation of SEM panel data model (8) can be found in Dai et al. [6].

Remark 2

In practice, we can let the instrumental variable either be ${\omega }_{it}$ or the predicted value of ${\mathit{\omega }}_{it}$ from a least squares projection of ${\mathit{D}}_{it}$ on ${\mathit{X}}_{it}^{\ast }$.

Remark 3

For an IV-FEQR estimation, we need instruments for the endogenous variables $\mathit{D}$. The instruments need to satisfy the following two conditions: (i) instruments ω can impact the endogenous variables D; (ii) instruments ω are independent of the random error ϵ. In practice, for general spatial autoregressive panel data model (1), we can choose ${\mathit{y}}_{-1}$ (i.e. the lag of dependent variable), ${\mathit{W}}^{\ast }{\mathit{X}}^{\ast }$, ${\mathit{M}}^{\ast }{\mathit{X}}^{\ast }$, ${\mathit{W}}^{\ast 2}{\mathit{X}}^{\ast }$, etc. as instrumental variable matrices; for SAR panel data model (7), we can choose ${\mathit{y}}_{-1}$, ${\mathit{W}}^{\ast }\mathit{X}$, ${\mathit{W}}^{\ast 2}\mathit{X}$, etc. as instrumental variable matrices; for SEM panel data model (8), we can choose ${\mathit{y}}_{-1}$, ${\mathit{M}}^{\ast }{\mathit{X}}^{\ast }$, ${\mathit{M}}^{\ast 2}{\mathit{X}}^{\ast }$, etc. as instrumental variable matrices.

2.2. Asymptotic theory

In this section, we investigate the asymptotic properties of the IV-FEQR estimator in Model (1). We impose the following regularity conditions:

A1 $\left\{\right(y_{it},X_{it}\left)\right\}$ is independent and identically distributed (i.i.d.) for each fixed i with conditional distribution function $F_{it}$ and differentiable conditional densities, $0<f_{it}<\mathrm{\infty }$, with bounded derivatives $f_{it}^{\prime }$ for $i=1,\dots ,N$ and $t=1,\dots ,T$.

A2 For all $\tau \in \mathcal{T}$, $\left(\mathit{\alpha }\right(\tau ),{\mathit{\beta }}^{\ast }(\tau \left)\right)$ is in the interior of the set $\mathcal{A}\times \mathcal{B}$, and $\mathcal{A}\times \mathcal{B}$ is compact and convex.

A3 Let $\mathit{\vartheta }=({\mathit{\theta }}^{\ast \mathrm{\top }},{\mathit{\gamma }}^{\mathrm{\top }}{)}^{\mathrm{\top }}$,

$$\mathrm{\Pi }(\mathit{\vartheta },\tau )=\mathbb{E}\left[\right(\tau -I(\mathit{y}<\mathit{D}\mathit{\alpha }+{\mathit{X}}^{\ast }{\mathit{\beta }}^{\ast }+{\mathit{Z}}_1^{\ast }{\mathit{\nu }}^{\ast }+{\mathit{Z}}_2^{\ast }{\mathit{\psi }}^{\ast }+\mathit{E}\mathit{\gamma })\left)\mathbf{\Delta }\right(\tau \left)\right],$$ (15)

and

$$\mathrm{\Pi }({\mathit{\theta }}^{\ast },\tau )=\mathbb{E}\left[\right(\tau -I(\mathit{y}<\mathit{D}\mathit{\alpha }+{\mathit{X}}^{\ast }{\mathit{\beta }}^{\ast }+{\mathit{Z}}_1^{\ast }{\mathit{\nu }}^{\ast }+{\mathit{Z}}_2^{\ast }{\mathit{\psi }}^{\ast })\left)\right)\mathbf{\Delta }\left(\tau \right)],$$ (16)

where $\mathbf{\Delta }\left(\tau \right)=[\mathit{E},{\mathit{X}}^{\ast },{\mathit{Z}}_1^{\ast },{\mathit{Z}}_2^{\ast }]$, $\mathit{E}=({\mathit{\omega }}_1,\dots ,{\mathit{\omega }}_T)$, ${\mathit{\omega }}_t=({\mathit{\omega }}_{1t},\dots ,{\mathit{\omega }}_{Nt}{)}^{\mathrm{\top }}$. The Jacobian matrices $\mathrm{\partial }\mathrm{\Pi }({\mathit{\theta }}^{\ast },\tau )/\mathrm{\partial }(\mathit{\alpha },{\mathit{\beta }}^{\ast },{\mathit{\nu }}^{\ast },{\mathit{\psi }}^{\ast })$ and $\mathrm{\partial }\mathrm{\Pi }(\mathit{\vartheta },\tau )/\mathrm{\partial }({\mathit{\beta }}^{\ast },{\mathit{\nu }}^{\ast },{\mathit{\psi }}^{\ast },\mathit{\gamma })$ are continuous and have full rank uniformly over $\mathcal{B}\times \mathcal{N}\times \mathcal{P}\times \mathcal{G}\times \mathcal{T}$. The parameter space $\mathcal{A}\times \mathcal{B}\times \mathcal{N}\times \mathcal{P}$ is a connected set and the image of $\mathcal{A}\times \mathcal{B}\times \mathcal{N}\times \mathcal{P}$ under the map ${\mathit{\theta }}^{\ast }\mapsto \mathrm{\Pi }({\mathit{\theta }}^{\ast },\tau )$ is simply connected.

A4 Denote $\mathbf{\Omega }=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left(f_{it}\right({\xi }_{it}\left(\tau \right)\left)\right)$, where ${\xi }_{it}\left(\tau \right)={\mathit{D}}_{it}^{\mathrm{\top }}\mathit{\alpha }\left(\tau \right)+{\mathit{X}}_{it}^{\ast \mathrm{\top }}{\mathit{\beta }}^{\ast }\left(\tau \right)+{\mathit{Z}}_{1i}^{\ast \mathrm{\top }}{\mathit{\nu }}^{\ast }\left(\tau \right)+{\mathit{Z}}_{2t}^{\ast \mathrm{\top }}{\mathit{\psi }}^{\ast }\left(\tau \right)+{\mathit{\omega }}_{it}^{\mathrm{\top }}\mathit{\gamma }\left(\tau \right)$. Let $\tilde{\mathit{X}}=[{\mathit{X}}^{\ast },\mathit{E}]$. Then, the following matrices are positive definite:

$$\begin{array}{rl}{\mathit{J}}_{\mathit{\zeta }}& =\underset{N,T\to \mathrm{\infty }}{lim}\frac{1}{NT}{\tilde{\mathit{X}}}^{\mathrm{\top }}{\mathit{M}}_{\tilde{\mathit{Z}}}^{\mathrm{\top }}\mathbf{\Omega }{\mathit{M}}_{\tilde{\mathit{Z}}}\tilde{\mathit{X}},\end{array}$$ (17)

$$\begin{array}{rl}{\mathit{J}}_{\mathit{\alpha }}& =\underset{N,T\to \mathrm{\infty }}{lim}\frac{1}{NT}{\tilde{\mathit{X}}}^{\mathrm{\top }}{\mathit{M}}_{\tilde{\mathit{Z}}}^{\mathrm{\top }}\mathbf{\Omega }{\mathit{M}}_{\tilde{\mathit{Z}}}\mathit{D},\end{array}$$ (18)

$$\begin{array}{rl}\mathit{S}& =\underset{N,T\to \mathrm{\infty }}{lim}\frac{\tau (1-\tau )}{NT}{\tilde{\mathit{X}}}^{\mathrm{\top }}{\mathit{M}}_{\tilde{\mathit{Z}}}^{\mathrm{\top }}{\mathit{M}}_{\tilde{\mathit{Z}}}\tilde{\mathit{X}},\end{array}$$ (19)

where ${\mathit{M}}_{\tilde{\mathit{Z}}}=\mathit{I}-{\mathit{P}}_{\tilde{\mathit{Z}}}$ and ${\mathit{P}}_{\tilde{\mathit{Z}}}=\tilde{\mathit{Z}}({\tilde{\mathit{Z}}}^{\mathrm{\top }}\mathbf{\Omega }\tilde{\mathit{Z}}{)}^{-1}{\tilde{\mathit{Z}}}^{\mathrm{\top }}\mathbf{\Omega }$, $\tilde{\mathit{Z}}=[{\mathit{Z}}_1^{\ast },{\mathit{Z}}_2^{\ast }]$. Let $[{\overline{\mathit{J}}}_{{\mathit{\beta }}^{\ast }}^{\mathrm{\top }},{\overline{\mathit{J}}}_{\mathit{\gamma }}^{\mathrm{\top }}]$ be a conformable partition of ${\mathit{J}}_{\mathit{\zeta }}^{-1}$ and $\mathit{H}={\overline{\mathit{J}}}_{\mathit{\gamma }}^{\mathrm{\top }}\mathit{A}{\overline{\mathit{J}}}_{\mathit{\gamma }}$. Hence, ${\mathit{J}}_{\mathit{\zeta }}$ is invertible and ${\mathit{J}}_{\mathit{\alpha }}^{\mathrm{\top }}\mathit{H}{\mathit{J}}_{\mathit{\alpha }}$ is also invertible.

A5 $max\parallel y_{it}\parallel =O\left(\sqrt{NT}\right)$, $max\parallel {\mathit{X}}_{it}\parallel =O\left(\sqrt{NT}\right)$ and $max\parallel {\mathit{\omega }}_{it}\parallel =O\left(\sqrt{NT}\right)$.

A6 $\mathit{W}$ and $\mathit{M}$ are non-stochastic spatial weights matrices with zero diagonals.

A7 $\mathit{W}$ and $\mathit{M}$ are uniformly bounded in both row and column sums in absolute value, i.e. $\underset{N\ge 1}{sup}\parallel W{\parallel }_{\mathrm{\infty }}<\mathrm{\infty }$ and $\underset{N\ge 1}{sup}\parallel W{\parallel }_1<\mathrm{\infty }$, $\underset{N\ge 1}{sup}\parallel M{\parallel }_{\mathrm{\infty }}<\mathrm{\infty }$ and $\underset{N\ge 1}{sup}\parallel M{\parallel }_1<\mathrm{\infty }$.

Lemma 1

Denote ${\epsilon }_{it}\left(\tau \right)=y_{it}-{\xi }_{it}\left(\tau \right)$, and let $\mathit{\vartheta }=({\mathit{\alpha }}^{\mathrm{\top }},{\mathit{\beta }}^{\ast \mathrm{\top }},{\mathit{\nu }}^{\ast \mathrm{\top }},{\mathit{\psi }}^{\ast \mathrm{\top }},{\mathit{\gamma }}^{\mathrm{\top }}{)}^{\mathrm{\top }}$ be a parameter vector in $\mathcal{V}=\mathcal{A}\times \mathcal{B}\times \mathcal{N}\times \mathcal{P}\times \mathcal{G}$. Let

$$\delta =\left(\begin{array}{c}{\mathit{\delta }}_{\mathit{\alpha }}\\ {\mathit{\delta }}_{{\mathit{\beta }}^{\ast }}\\ {\mathit{\delta }}_{{\mathit{\nu }}^{\ast }}\\ {\mathit{\delta }}_{{\mathit{\psi }}^{\ast }}\\ {\mathit{\delta }}_{\mathit{\gamma }}\end{array}\right)=\left(\begin{array}{c}\sqrt{NT}\left(\hat{\mathit{\alpha }}\right(\tau )-\mathit{\alpha }(\tau \left)\right)\\ \sqrt{NT}\left({\hat{\mathit{\beta }}}^{\ast }\right(\tau )-{\mathit{\beta }}^{\ast }(\tau \left)\right)\\ \sqrt{T}\left({\hat{\mathit{\nu }}}^{\ast }\right(\tau )-{\mathit{\nu }}^{\ast }(\tau \left)\right)\\ \sqrt{N}\left({\hat{\mathit{\psi }}}^{\ast }\right(\tau )-{\mathit{\psi }}^{\ast }(\tau \left)\right)\\ \sqrt{NT}\left(\hat{\mathit{\gamma }}\right(\tau )-\mathit{\gamma }(\tau \left)\right)\end{array}\right).$$ (20)

Under conditions A1–A7, we have

$$\begin{array}{rl}& \underset{\mathit{\vartheta }\in \mathcal{V}}{sup}\frac{1}{NT}\left|\sum _{i=1}^N\sum _{t=1}^T\left[{\rho }_{\tau }\left({\epsilon }_{it}\left(\tau \right)-\frac{{\mathit{D}}_{it}^{\mathrm{\top }}{\mathit{\delta }}_{\mathit{\alpha }}}{\sqrt{NT}}-\frac{{\mathit{X}}_{it}^{\ast \mathrm{\top }}{\mathit{\delta }}_{{\mathit{\beta }}^{\ast }}}{\sqrt{NT}}-\frac{{\mathit{Z}}_{1i}^{\ast \mathrm{\top }}{\mathit{\delta }}_{{\mathit{\nu }}^{\ast }}}{\sqrt{T}}-\frac{{\mathit{Z}}_{2t}^{\ast \mathrm{\top }}{\mathit{\delta }}_{{\mathit{\psi }}^{\ast }}}{\sqrt{N}}-\frac{{\mathit{\omega }}_{it}^{\mathrm{\top }}{\mathit{\delta }}_{\mathit{\gamma }}}{\sqrt{NT}}\right)\right.\right.\\ & -{\rho }_{\tau }\left({\epsilon }_{it}\right(\tau \left)\right)-E\left[{\rho }_{\tau }\left({\epsilon }_{it}\left(\tau \right)-\frac{{\mathit{D}}_{it}^{\mathrm{\top }}{\mathit{\delta }}_{\mathit{\alpha }}}{\sqrt{NT}}-\frac{{\mathit{X}}_{it}^{\ast \mathrm{\top }}{\mathit{\delta }}_{{\mathit{\beta }}^{\ast }}}{\sqrt{NT}}-\frac{{\mathit{Z}}_{1i}^{\ast \mathrm{\top }}{\mathit{\delta }}_{{\mathit{\nu }}^{\ast }}}{\sqrt{T}}-\frac{{\mathit{Z}}_{2t}^{\ast \mathrm{\top }}{\mathit{\delta }}_{{\mathit{\psi }}^{\ast }}}{\sqrt{N}}-\frac{{\mathit{\omega }}_{it}^{\mathrm{\top }}{\mathit{\delta }}_{\mathit{\gamma }}}{\sqrt{NT}}\right)\right.\\ & \left.\left.\left.-{\rho }_{\tau }\left({\epsilon }_{it}\right(\tau \left)\right)\right]\right]\right|=o_p\left(1\right).\end{array}$$

To further comment on the nature of the correlation between ${\mathit{y}}_{-1}$ and $\mathit{\omega }$ required by A3, note that, for a given quantile τ, by A1 we have that

$$\begin{array}{rl}& \frac{\mathrm{\partial }\mathbb{E}\left[(\tau -I(\mathit{y}<\mathit{D}\mathit{\alpha }+{\mathit{X}}^{\ast }{\mathit{\beta }}^{\ast }+{\mathit{Z}}_1^{\ast }{\mathit{\nu }}^{\ast }+{\mathit{Z}}_2^{\ast }{\mathit{\psi }}^{\ast }\left)\right)\left)\mathbf{\Delta }\right(\tau )\right]}{\mathrm{\partial }(\mathit{\alpha },{\mathit{\beta }}^{\ast },{\mathit{\nu }}^{\ast },{\mathit{\psi }}^{\ast })}\\ & =\mathbb{E}\left[\left({\mathit{E}}^{\mathrm{\top }},{\mathit{X}}^{\ast \mathrm{\top }},{\mathit{Z}}_1^{\ast \mathrm{\top }},{\mathit{Z}}_2^{\ast \mathrm{\top }}\right)\mathbf{\Omega }\left({\mathit{D}}^{\mathrm{\top }}{\mathit{X}}^{\ast \mathrm{\top }},{\mathit{Z}}_1^{\ast \mathrm{\top }},{\mathit{Z}}_2^{\ast \mathrm{\top }}\right)\right].\end{array}$$

Hence, the Jacobian in A3 takes a form of density-weighted covariance matrix for $\mathit{D}$, ${\mathit{Z}}_1^{\ast }$, ${\mathit{Z}}_2^{\ast }$, $\mathit{E}$ variables, and requires that this matrix has full rank. In addition, A3 imposes that global identifiability must hold; hence, the impact of $\mathit{E}$ should be rich enough to guarantee that the equations are solved uniquely.

Theorem 1 Consistency —

Under conditions A1–A7, $\left(\mathit{\alpha }\right(\tau ),{\mathit{\beta }}^{\ast }(\tau ),{\mathit{\nu }}^{\ast }(\tau ),{\mathit{\psi }}^{\ast }(\tau \left)\right)$ uniquely solves the equation $\mathbb{E}\left[\right(\tau -I(\mathit{y}<\mathit{D}\mathit{\alpha }+{\mathit{X}}^{\ast }{\mathit{\beta }}^{\ast }+{\mathit{Z}}_1^{\ast }{\mathit{\nu }}^{\ast }+{\mathit{Z}}_2^{\ast }{\mathit{\psi }}^{\ast })\left)\mathbf{\Delta }\right(\tau \left)\right]=0$ over $\mathcal{A}\times \mathcal{B}\times \mathcal{N}\times \mathcal{P}$ and $\left(\mathit{\alpha }\right(\tau ),{\mathit{\beta }}^{\ast }(\tau ),{\mathit{\nu }}^{\ast }(\tau ),{\mathit{\psi }}^{\ast }(\tau \left)\right)$ is consistently estimable. Therefore, the parameters $\mathit{\beta }\left(\tau \right),\mathit{\nu }\left(\tau \right),\mathit{\psi }\left(\tau \right)$ are also consistently estimable.

Theorem 2 Asymptotic distribution —

Under conditions A1–A7 and Lemma 1, for a given $\tau \in (0,1)$, $\hat{\mathit{\theta }}=({\hat{\mathit{\alpha }}}^{\mathrm{\top }},{\hat{\mathit{\beta }}}^{\ast \mathrm{\top }}{)}^{\mathrm{\top }}$ converges to a Gaussian distribution:

$$\sqrt{NT}\left(\hat{\mathit{\theta }}\right(\tau )-\mathit{\theta }(\tau \left)\right)\stackrel{d}{\to }N(\mathbf{0},\mathbf{\Lambda }(\tau \left)\right),$$ (21)

where $\mathbf{\Lambda }\left(\tau \right)={\mathit{J}}^{\mathrm{\top }}\mathit{S}\mathit{J}$, $\mathit{S}=\underset{N,T\to \mathrm{\infty }}{lim}\tau (1-\tau )/NT{\tilde{\mathit{X}}}^{\mathrm{\top }}{\mathit{M}}_{\tilde{\mathit{Z}}}^{\mathrm{\top }}{\mathit{M}}_{\tilde{\mathit{Z}}}\tilde{\mathit{X}}$, $\tilde{\mathit{X}}=[{\mathit{X}}^{\ast },\mathit{E}]$, $\mathit{J}=({\mathit{K}}^{\mathrm{\top }},{\mathit{L}}^{\mathrm{\top }})$, ${\mathit{M}}_{\tilde{\mathit{Z}}}=\mathit{I}-{\mathit{P}}_{\tilde{\mathit{Z}}}$, ${\mathit{P}}_{\tilde{\mathit{Z}}}=\tilde{\mathit{Z}}({\tilde{\mathit{Z}}}^{\mathrm{\top }}\mathbf{\Omega }\tilde{\mathit{Z}}{)}^{-1}{\tilde{\mathit{Z}}}^{\mathrm{\top }}\mathbf{\Omega }$, $\tilde{\mathit{Z}}=[{\mathit{Z}}_1^{\ast },{\mathit{Z}}_2^{\ast }]$, $\mathbf{\Omega }=diag\left(f_{it}\right({\xi }_{it}\left(\tau \right)\left)\right)$ , $\mathit{L}={\overline{\mathit{J}}}_{{\mathit{\beta }}^{\ast }}\mathit{M}$, $\mathit{M}=\mathit{I}-{\mathit{J}}_{\mathit{\alpha }}\mathit{K}$, $\mathit{K}=({\mathit{J}}_{\mathit{\alpha }}^{\mathrm{\top }}\mathit{H}{\mathit{J}}_{\mathit{\alpha }}{)}^{-1}{\mathit{J}}_{\mathit{\alpha }}^{\mathrm{\top }}\mathit{H}$, $\mathit{H}={\overline{\mathit{J}}}_{\mathit{\gamma }}^{\mathrm{\top }}\mathit{A}{\overline{\mathit{J}}}_{\mathit{\gamma }}$, ${\mathit{J}}_{\mathit{\alpha }}=\underset{N,T\to \mathrm{\infty }}{lim}1/NT{\tilde{\mathit{X}}}^{\mathrm{\top }}{\mathit{M}}_{\tilde{\mathit{Z}}}^{\mathrm{\top }}\mathbf{\Omega }{\mathit{M}}_{\tilde{\mathit{Z}}}\mathit{D}$, ${\mathit{J}}_{\mathit{\zeta }}=\underset{N,T\to \mathrm{\infty }}{lim}1/NT{\tilde{\mathit{X}}}^{\mathrm{\top }}{\mathit{M}}_{\tilde{\mathit{Z}}}^{\mathrm{\top }}\mathbf{\Omega }{\mathit{M}}_{\tilde{\mathit{Z}}}\tilde{\mathit{X}}$, and $[{\overline{\mathit{J}}}_{{\mathit{\beta }}^{\ast }}^{\mathrm{\top }},{\overline{\mathit{J}}}_{\mathit{\gamma }}^{\mathrm{\top }}]$ is a conformable partition of ${\mathit{J}}_{\mathit{\zeta }}^{-1}$.

3. Monte Carlo simulations

In this section, we report the results of a Monte Carlo study in which we assess the finite sample performance of the IV-FEQR estimators proposed in Section 2. For comparison purpose, we generate the samples being considered in the design of Lee and Yu [16]:

$$\begin{array}{rl}\mathit{y}& =\rho {\mathit{W}}^{\ast }\mathit{y}+\mathit{X}\mathit{\beta }+{\mathit{Z}}_1\mathit{\nu }+{\mathit{Z}}_2\mathit{\psi }+\mathit{u},\\ \mathit{u}& =\lambda {\mathit{M}}^{\ast }\mathit{u}+\mathit{\epsilon },\end{array}$$

where ${\rho }_0=0.2$, ${\lambda }_0=0.5$ and ${\beta }_0=1$. Here, $\mathit{X}$, $\mathit{\nu }$, $\mathit{\psi }$ are drawn independently from $N(0,1)$ and both the spatial weights matrices $\mathit{W}$ and $\mathit{M}$ are the same rook matrices. We use some combinations of T=20,50, and N=49,100. For the disturbance errors, we consider the standard normal (i.e. N(0, 1)) and Cauchy (i.e. $t_1$) distributions.

For each set of generated sample observations, we calculate the IV-FEQR estimators. This step is repeated for 1000 times. We consider the bias and root mean squared error (RMSE) for the MLE  [16, Section 2.1], QMLE [16, Section 2.2.], OLS and IV-FEQR. The quantile regression based estimators are calculated for quantiles $\tau =(0.25,0.5,0.75)$. For the IV-FEQR estimator, we employed ${\mathit{y}}_{-1}$ and ${\mathit{W}}^{\ast }{\mathit{X}}^{\ast }$ as instrument. The results are summarized in Tables 1 and 2.

Table 1. Bias and RMSE of various estimators (with both individual and time effects) when ${\epsilon }_{it}\sim N(0,1)$. The table shows the bias, RMSE (in parentheses) and t-statistic [in brackets].

			IV-FEQR
			${\mathit{y}}_{-1}$				${\mathit{W}}^{\ast }{\mathit{X}}^{\ast }$
N	T	Para.	$\tau =0.25$	$\tau =0.5$	$\tau =0.75$		$\tau =0.25$	$\tau =0.5$	$\tau =0.75$	MLE	QMLE	OLS
49	20	ρ	0.008	0.007	0.001		−0.001	0.001	−0.007	−0.035	−0.036	0.107
			(0.140)	(0.141)	(0.138)		(0.126)	(0.127)	(0.130)	(0.085)	(0.083)	(0.129)
			[1.806]	[1.569]	[0.229]		[0.251]	[0.249]	[1.702]	[13.015]	[13.709]	[26.217]
		λ	0.008	0.008	0.006		0.005	0.007	0.007	−0.023	−0.024	0.223
			(0.137)	(0.138)	(0.140)		(0.125)	(0.132)	(0.130)	(0.174)	(0.172)	(0.232)
			[1.846]	[1.832]	[1.355]		[1.264]	[1.676]	[1.702]	[4.178]	[4.410]	[30.381]
		β	−0.002	−0.001	0.002		−0.001	0.001	0.002	0.001	0.001	−0.029
			(0.044)	(0.043)	(0.049)		(0.042)	(0.042)	(0.046)	(0.044)	(0.047)	(0.045)
			[1.437]	[0.735]	[1.290]		[0.753]	[0.753]	[1.374]	[0.718]	[0.673]	[20.369]
	50	ρ	0.001	−0.001	0.003		0.002	0.003	0.005	−0.034	−0.033	0.107
			(0.136)	(0.133)	(0.135)		(0.145)	(0.140)	(0.144)	(0.080)	(0.080)	(0.115)
			[0.232]	[0.238]	[0.702]		[0.436]	[0.677]	[1.098]	[13.433]	[13.038]	[29.408]
		λ	0.002	−0.002	0.003		0.008	0.005	0.003	−0.022	−0.023	0.230
			(0.136)	(0.137)	(0.138)		(0.142)	(0.142)	(0.142)	(0.166)	(0.169)	(0.234)
			[0.465]	[0.461]	[0.671]		[1.781]	[1.113]	[0.668]	[4.189]	[4.302]	[31.067]
		β	0.001	−0.001	0.001		−0.001	0.001	0.002	−0.002	0.001	−0.029
			(0.030)	(0.027)	(0.029)		(0.029)	(0.027)	(0.029)	(0.028)	(0.029)	(0.036)
			[1.054]	[1.171]	[1.090]		[1.090]	[1.171]	[2.180]	[2.258]	[1.090]	[25.461]
100	20	ρ	0.003	−0.003	0.000		−0.004	0.002	−0.001	−0.016	−0.014	0.104
			(0.136)	(0.136)	(0.137)		(0.138)	(0.139)	(0.138)	(0.057)	(0.056)	(0.115)
			[0.697]	[0.697]	[0.000]		[0.916]	[0.455]	[0.229]	[8.872]	[7.902]	[28.584]
		λ	0.008	0.006	−0.002		0.005	0.008	0.005	−0.008	−0.008	0.263
			(0.136)	(0.137)	(0.137)		(0.137)	(0.133)	(0.140)	(0.126)	(0.128)	(0.266)
			[1.859]	[1.384]	[0.461]		[1.154]	[1.901]	[1.129]	[2.007]	[1.975]	[31.251]
		β	0.001	−0.001	−0.001		0.001	0.001	−0.001	−0.001	−0.001	−0.031
			(0.033)	(0.030)	(0.033)		(0.033)	(0.029)	(0.033)	(0.031)	(0.032)	(0.039)
			[0.958]	[1.054]	[0.958]		[0.958]	[1.090]	[0.958]	[1.020]	[0.988]	[25.124]
	50	ρ	−0.001	0.002	0.003		−0.003	−0.008	−0.001	−0.014	−0.014	0.103
			(0.082)	(0.133)	(0.081)		(0.148)	(0.140)	(0.145)	(0.053)	(0.055)	(0.108)
			[0.386]	[0.475]	[1.171]		[0.641]	[1.806]	[0.218]	[8.349]	[8.045]	[30.144]
		λ	0.003	0.001	0.001		0.007	0.006	0.003	−0.006	−0.006	0.269
			(0.087)	(0.132)	(0.083)		(0.143)	(0.137)	(0.141)	(0.121)	(0.123)	(0.270)
			[1.090]	[0.239]	[0.381]		[1.547]	[1.384]	[0.673]	[1.567]	[1.542]	[31.490]
		β	−0.001	0.001	−0.000		−0.001	−0.000	−0.000	0.001	0.001	−0.033
			(0.015)	(0.019)	(0.013)		(0.020)	(0.019)	(0.020)	(0.021)	(0.020)	(0.036)
			[2.107]	[1.664]	[0.000]		[1.580]	[0.000]	[0.000]	[1.505]	[1.580]	[28.973]

Table 2. Bias and RMSE of various estimators (with both individual and time effects) when ${\epsilon }_{it}\sim t\left(1\right)$. The table shows the bias, RMSE (in parentheses) and t-statistic [in brackets].

			IV-FEQR
			${\mathit{y}}_{-1}$				${\mathit{W}}^{\ast }{\mathit{X}}^{\ast }$
N	T	Para.	$\tau =0.25$	$\tau =0.5$	$\tau =0.75$		$\tau =0.25$	$\tau =0.5$	$\tau =0.75$	MLE	QMLE	OLS
49	20	ρ	−0.004	0.002	−0.001		0.003	0.001	−0.001	−0.039	−0.041	0.227
			(0.120)	(0.119)	(0.115)		(0.118)	(0.117)	(0.115)	(0.051)	(0.056)	(0.241)
			[1.054]	[0.531]	[0.275]		[0.804]	[0.270]	[0.275]	[24.170]	[23.141]	[29.771]
		λ	−0.003	0.002	−0.002		−0.001	0.002	0.004	−0.036	−0.033	0.241
			(0.120)	(0.119)	(0.119)		(0.114)	(0.114)	(0.116)	(0.078)	(0.079)	(0.260)
			[0.790]	[0.531]	[0.531]		[0.277]	[0.555]	[1.090]	[14.588]	[13.203]	[29.297]
		β	0.006	−0.001	0.006		0.006	−0.002	0.005	−0.593	1.915	−0.653
			(0.110)	(0.074)	(0.111)		(0.113)	(0.076)	(0.108)	(17.774)	(30.814)	(25.040)
			[1.724]	[0.427]	[1.709]		[1.678]	[0.832]	[1.463]	[1.055]	[1.964]	[0.824]
	50	ρ	−0.002	−0.001	0.003		−0.002	0.004	−0.003	−0.040	−0.042	0.225
			(0.115)	(0.117)	(0.121)		(0.115)	(0.117)	(0.119)	(0.046)	(0.048)	(0.230)
			[0.550]	[0.270]	[0.784]		[0.550]	[1.081]	[0.797]	[27.484]	[27.656]	[30.920]
		λ	−0.001	0.001	−0.001		0.002	0.001	−0.002	−0.033	−0.035	0.247
			(0.121)	(0.119)	(0.121)		(0.121)	(0.119)	(0.123)	(0.070)	(0.073)	(0.260)
			[0.261]	[0.266]	[0.261]		[0.522]	[0.266]	[0.514]	[14.900]	[15.154]	[30.027]
		β	−0.002	0.001	−0.001		−0.002	−0.001	−0.001	0.791	−0.978	0.767
			(0.065)	(0.044)	(0.066)		(0.068)	(0.046)	(0.067)	(14.184)	(30.731)	(16.594)
			[0.973]	[0.718]	[0.479]		[0.930]	[0.687]	[0.472]	[1.763]	[1.006]	[1.461]
100	20	ρ	0.005	0.008	0.006		0.004	0.005	0.005	−0.020	−0.021	0.204
			(0.121)	(0.121)	(0.119)		(0.121)	(0.122)	(0.120)	(0.030)	(0.040)	(0.211)
			[1.306]	[2.089]	[1.594]		[1.045]	[1.295]	[1.317]	[21.071]	[16.594]	[30.558]
		λ	−0.003	0.007	−0.004		−0.002	0.006	−0.005	−0.012	−0.013	0.286
			(0.119)	(0.116)	(0.119)		(0.119)	(0.116)	(0.121)	(0.042)	(0.045)	(0.292)
			[0.797]	[1.907]	[1.062]		[0.531]	[1.635]	[1.306]	[9.031]	[9.131]	[30.958]
		β	0.001	−0.001	0.001		−0.001	0.001	0.002	0.361	0.154	0.609
			(0.077)	(0.054)	(0.077)		(0.076)	(0.056)	(0.078)	(20.749)	(9.490)	(60.260)
			[0.411]	[0.585]	[0.411]		[0.416]	[0.564]	[0.810]	[0.550]	[0.513]	[0.319]
	50	ρ	−0.001	0.001	−0.001		−0.001	0.001	0.001	−0.019	−0.020	0.207
			(0.091)	(0.119)	(0.093)		(0.092)	(0.118)	(0.095)	(0.027)	(0.027)	(0.211)
			[0.347]	[0.266]	[0.340]		[0.343]	[0.268]	[0.333]	[16.231]	[23.413]	[31.008]
		λ	0.001	0.005	0.004		0.001	0.004	0.005	−0.011	−0.012	0.285
			(0.094)	(0.112)	(0.094)		(0.096)	(0.111)	(0.098)	(0.036)	(0.037)	(0.291)
			[0.336]	[1.411]	[1.345]		[0.329]	[1.139]	[1.613]	[9.658]	[10.251]	[30.955]
		β	−0.001	−0.001	−0.002		0.001	−0.001	0.001	2.562	−0.399	−0.082
			(0.039)	(0.032)	(0.040)		(0.040)	(0.033)	(0.040)	(54.012)	(10.464)	(10.549)
			[0.810]	[0.988]	[1.580]		[0.790]	[0.958]	[0.790]	[1.499]	[1.205]	[0.246]

Tables 1 and 2 show that the IV-FEQR estimator performs better than the other estimators in $t_1$ settings. In general, we find that the IV-FEQR estimator based on two different instruments has similar bias and RMSEs. Under normal disturbance errors, the IV-FEQR estimators for λ perform better than the other estimators, the IV-FEQR estimators for ρ have smaller biases and larger RMSEs than the MLE, QMLE and OLS estimators, while the IV-FEQR estimators for $\mathit{\beta }$ have similar biases and RMSEs as the MLE, QMLE and OLS estimators. For Cauchy disturbance errors, our proposed IV-FEQR estimators outperform the other estimators as we do not impose any finite moment assumption on the disturbance errors. Therefore, we conclude that the proposed IV-FEQR is more robust in practice. To confirm the asymptotic properties, we employ the t-statistic to test whether the bias of parameters ρ, λ and β is 0. The critical value is $t_{1-\alpha /2}\left(999\right)=1.962$ with the significant level $\alpha =0.05$. From Tables 1 and 2, we can see that the approximate t-statistic values for IV-FEQR estimators are generally all smaller than the critical value.

4. Illustration

In this section, we use the cigarette demand data set (https://spatial-panels.com/software/) to illustrate our methodologies. The data set is based on a panel of 46 states over 30 time periods (1963–1992) including the spatial weight matrix $\mathit{W}$, which has been analyzed by many authors (see Baltagi and Levin [3], Baltagi [1], Baltagi, Griffin, and Xiong [2], Yang [19], Elhorst [7], Kelejian and Piras [12]). We employ the general spatial autoregressive panel data model (1), the SAR panel data model (7) and the SEM panel data model (8) for fitting the date set. Table 3 gives the SIC values (see Kim [13]) of the three fitted models. The top half of table presents the SIC values with both the individual and time-period effects while the bottom half of table shows the SIC values with individual effects only. From which we can see that the general spatial autoregressive panel data model with individual effects only has the smallest SIC value at most of the quantile levels and is employed for analysis.

Table 3. The SIC values of the three spatial panel data models at quantile $\tau =0.1,\dots ,0.9$. The first two smallest values are marked in bold.

	τ
	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9
With both individual and time period effects
Model (1)	9.469	13.438	10.413	9.796	9.703	10.067	8.862	9.027	8.129
Model (7)	9.463	8.854	6.593	9.385	8.420	8.681	8.362	8.282	4.471
Model (8)	9.342	6.665	8.338	10.025	9.652	7.968	8.172	8.077	7.976
With individual effects only
Model (1)	8.585	6.254	8.811	8.324	7.969	7.354	6.985	9.184	11.329
Model (7)	5.643	8.968	10.345	8.716	7.782	9.199	7.761	8.042	8.759
Model (8)	10.058	7.999	12.639	8.386	10.284	8.867	9.262	8.374	6.997

The fitted QR model takes the form:

$$Q_{\tau }(\log C_{it}|{\mathit{D}}_{it},{\mathit{X}}_{it}^{\ast })={\mathit{D}}_{it}^{\mathrm{\top }}\mathit{\alpha }\left(\tau \right)+{\mathit{X}}_{it}^{\ast \mathrm{\top }}{\mathit{\beta }}^{\ast }\left(\tau \right)+{\mathit{Z}}_{1i}^{\ast \mathrm{\top }}{\mathit{\nu }}^{\ast }\left(\tau \right),i=1,\dots ,46,t=2,\dots ,30,$$ (22)

where $C_{it}$ is real per capita sales of cigarettes by persons of smoking age (14 years and older), ${\mathit{D}}_{it}=[d_{it}^{\left(1\right)},d_{it}^{\left(2\right)},d_{it}^{\left(3\right)}]$, ${\mathit{X}}_{it}^{\ast }=[{\mathit{X}}_{it},-\sum _{j\ne i}{M}_{ij}{\mathit{X}}_{jt}]$, $d_{it}^{\left(1\right)}=\sum _{j\ne i}{W}_{ij}\log C_{jt}$, $d_{it}^{\left(2\right)}=\sum _{j\ne i}{M}_{ij}\log C_{jt}$, $d_{it}^{\left(3\right)}=-\sum _{j\ne i}\sum _{k\ne j}{M}_{ij}{W}_{jk}\log C_{kt}$, ${\mathit{X}}_{it}=[\log P_{it},\log Y_{it}]$, $P_{it}$ is the average retail price of a pack of cigarettes measured in real terms, and $Y_{it}$ is real per capita disposable income. Here, the spatial weight matrix $\mathit{M}$ is a second-order spatial weight matrix, which can be generated based on $\mathit{W}$ via $\mathrm{s}\mathrm{l}\mathrm{a}\mathrm{g}$ function from the Econometrics Toolbox provided by LeSage (http://www.spatial-econometrics.com). Considering the computational cost, we choose $\log C_{it-1}$ as instruments.

We estimate the parameters using the IV-FEQR, MLE, and OLS methods. The results are presented in Table 4. The first three columns are the IV-FEQR estimates for $\tau =0.25,0.5,0.75$, and the last two columns correspond to the MLE and OLS estimates respectively. We can see that the IV-FEQR estimates vary at different quantiles (i.e. $\tau =0.25,0.5,0.75$). The signs of the estimates for ρ, $\mathit{\beta }$ are the same among IV-FEQR, MLE, and OLS methods, which show that at quantiles 0.25,0.5 and 0.75, the cigarette sales between neighbor states have a positive effect to each other, the log average cigarettes retail price has a negative effect to the cigarette sales, and the log disposable income has a positive effect to the cigarette sales.

Table 4. Estimation results of cigarette demand using general spatial panel data models.

	IV-FEQR
Parameter	$\tau =0.25$	$\tau =0.50$	$\tau =0.75$	MLE	OLS
ρ	0.022	0.292	0.492	0.391	0.308
λ	−0.890	0.432	0.850	0.729	2.176
Log average cigarettes retail price	−1.009	−0.810	−0.686	−3.236	−0.815
Log disposable income	0.878	0.557	0.591	0.527	0.540

Figure 1 presents a complete analysis, which considers other quantiles of the conditional cigarettes demand distribution. The x-axis presents the quantiles and y-axis presents the estimation of parameters (red lines) and their corresponding confidence intervals (blue lines). We find that the cigarette retail price has negative effect to the capita sales of cigarettes and disposable income has positive effect to the capita sales of cigarettes at all quantile levels. Besides, the estimates of capita sales of cigarettes are larger at extreme and middle quantiles than those at other quantiles. On the contrary, the estimates of disposable income are smaller at the extreme and middle quantiles.

Figure 1.

Quantile effects of the log average retail price of a pack of cigarettes and the log disposable income with individual effects only. The areas represent 95% point-wise confidence intervals.

5. Conclusion

In this paper, we investigate the IVQR estimation of general spatial autoregressive panel data model with fixed effects. The model with both individual and time-period effects is considered. The asymptotic properties are studied. Monte Carlo results are provided to show that the proposed methodology is robust to error distributions with undefined moments.

Appendix: Proofs.

Proof of Lemma 1 is similar to that of Lemma 2 in Galvao [8] and is hence omitted here.

A.1. Proof of Theorem 1

Proof.

First, following Chernozhukov and Hansen [4], $\left(\mathit{\alpha }\right(\tau ),{\mathit{\beta }}^{\ast }(\tau ),{\mathit{\nu }}^{\ast },{\mathit{\psi }}^{\ast })$ uniquely solves the problem for each τ.

To prove the consistency of the parameter, we need to show that under conditions A1–A7, ${\hat{\mathit{\theta }}}^{\ast }\left(\tau \right)={\mathit{\theta }}^{\ast }\left(\tau \right)+o_p\left(1\right)$. Let

$$\mathcal{P}:\mathit{\vartheta }\mapsto {\rho }_{\tau }(\mathit{y}-\mathit{D}\mathit{\alpha }-{\mathit{X}}^{\ast }{\mathit{\beta }}^{\ast }-{\mathit{Z}}_1^{\ast }{\mathit{\nu }}^{\ast }-{\mathit{Z}}_2^{\ast }{\mathit{\psi }}^{\ast }),$$

and $\mathcal{P}$ is continuous. Under condition Lemma 1, we have that $\parallel {\hat{\mathit{\vartheta }}}^{\ast }(\mathit{\alpha },\tau )-{\mathit{\vartheta }}^{\ast }(\mathit{\alpha },\tau )\parallel \stackrel{P}{\to }0$ for ${\mathit{\vartheta }}^{\ast }=(\mathit{\alpha },{\mathit{\beta }}^{\ast },{\mathit{\nu }}^{\ast },{\mathit{\psi }}^{\ast },\mathit{\gamma })$, which implies that $\parallel \parallel \hat{\mathit{\gamma }}(\mathit{\alpha },\tau )\parallel -\parallel \mathit{\gamma }(\mathit{\alpha },\tau )\parallel \parallel \stackrel{P}{\to }0$. By Corollary 3.2.3 in van der Vaart and Wellner [18], we have $\parallel \hat{\rho }\left(\tau \right)-\rho \left(\tau \right)\parallel \stackrel{P}{\to }0$ and $\parallel \hat{\lambda }\left(\tau \right)-\lambda \left(\tau \right)\parallel \stackrel{P}{\to }0$. Therefore, $\parallel {\hat{\mathit{\beta }}}^{\ast }\left(\hat{\mathit{\alpha }}\right(\tau ),\tau )-{\mathit{\beta }}^{\ast }\left(\tau \right)\parallel \stackrel{P}{\to }0$, $\parallel {\hat{\mathit{\nu }}}^{\ast }\left(\hat{\mathit{\alpha }}\right(\tau ),\tau )-{\mathit{\nu }}^{\ast }\parallel \stackrel{P}{\to }0$, $\parallel {\hat{\mathit{\psi }}}^{\ast }\left(\hat{\mathit{\alpha }}\right(\tau ),\tau )-{\mathit{\psi }}^{\ast }\parallel \stackrel{P}{\to }0$, and $\parallel \hat{\mathit{\gamma }}\left(\hat{\mathit{\alpha }}\right(\tau ),\tau )-0\parallel \stackrel{P}{\to }0$. Hence, $\parallel {\hat{\mathit{\theta }}}^{\ast }\left(\tau \right)-{\mathit{\theta }}^{\ast }\left(\tau \right)\parallel \stackrel{P}{\to }0$ and the theorem follows. □

A.2. Proof of Theorem 2

For any $\hat{\mathit{\alpha }}\left(\tau \right)\stackrel{P}{\to }\mathit{\alpha }\left(\tau \right)\left({\mathit{\delta }}_{\mathit{\alpha }}\stackrel{P}{\to }\mathbf{0}\right)$, we can write the objective function defined in Equation (9) as

$$V_{IV}\left(\mathit{\delta }\right)=\sum _{i=1}^N\sum _{t=1}^T\left[{\rho }_{\tau }\left({\epsilon }_{it}\left(\tau \right)-\frac{D_{it}^{\mathrm{\top }}{\mathit{\delta }}_{\mathit{\alpha }}}{\sqrt{NT}}-\frac{{\mathit{X}}_{it}^{\ast \mathrm{\top }}{\mathit{\delta }}_{{\mathit{\beta }}^{\ast }}}{\sqrt{NT}}-\frac{{\mathit{Z}}_{1i}^{\ast \mathrm{\top }}{\mathit{\delta }}_{{\mathit{\nu }}^{\ast }}}{\sqrt{T}}-\frac{{\mathit{Z}}_{2t}^{\ast \mathrm{\top }}{\mathit{\delta }}_{{\mathit{\psi }}^{\ast }}}{\sqrt{N}}-\frac{{\mathit{\omega }}_{it}^{\mathrm{\top }}{\mathit{\delta }}_{\mathit{\gamma }}}{\sqrt{NT}}\right)-{\rho }_{\tau }\left({\epsilon }_{it}\right(\tau \left)\right)\right]$$

where ${\epsilon }_{it}\left(\tau \right)=y_{it}-{\xi }_{it}\left(\tau \right)$, ${\xi }_{it}\left(\tau \right)={\mathit{D}}_{it}^{\mathrm{\top }}\mathit{\alpha }\left(\tau \right)+{\mathit{X}}_{it}^{\ast \mathrm{\top }}{\mathit{\beta }}^{\ast }\left(\tau \right)+{\mathit{Z}}_{1i}^{\ast \mathrm{\top }}{\mathit{\nu }}^{\ast }\left(\tau \right)+{\mathit{Z}}_{2t}^{\ast \mathrm{\top }}{\mathit{\psi }}^{\ast }\left(\tau \right)+{\mathit{\omega }}_{it}^{\mathrm{\top }}\mathit{\gamma }\left(\tau \right)$, and

$$\delta =\left(\begin{array}{c}{\mathit{\delta }}_{\mathit{\alpha }}\\ {\mathit{\delta }}_{{\mathit{\beta }}^{\ast }}\\ {\mathit{\delta }}_{{\mathit{\nu }}^{\ast }}\\ {\mathit{\delta }}_{{\mathit{\psi }}^{\ast }}\\ {\mathit{\delta }}_{\mathit{\gamma }}\end{array}\right)=\left(\begin{array}{c}\sqrt{NT}\left(\hat{\mathit{\alpha }}\right(\tau )-\mathit{\alpha }(\tau \left)\right)\\ \sqrt{NT}\left({\hat{\mathit{\beta }}}^{\ast }\right(\tau )-{\mathit{\beta }}^{\ast }(\tau \left)\right)\\ \sqrt{T}\left({\hat{\mathit{\nu }}}^{\ast }\right(\tau )-{\mathit{\nu }}^{\ast }(\tau \left)\right)\\ \sqrt{N}\left({\hat{\mathit{\psi }}}^{\ast }\right(\tau )-{\mathit{\psi }}^{\ast }(\tau \left)\right)\\ \sqrt{NT}\left(\hat{\mathit{\gamma }}\right(\tau )-\mathit{\gamma }(\tau \left)\right)\end{array}\right).$$

For fixed $({\mathit{\delta }}_{\mathit{\alpha }},{\mathit{\delta }}_{{\mathit{\beta }}^{\ast }},{\mathit{\delta }}_{{\mathit{\psi }}^{\ast }},{\mathit{\delta }}_{\mathit{\gamma }})$, we can consider the behavior of ${\mathit{\delta }}_{{\mathit{\nu }}^{\ast }}$. Let ${\varphi }_{\tau }\left(u\right)=\tau -I(u<0)$ and

$$\begin{array}{rl}& g_{it}({\mathit{\delta }}_{\mathit{\alpha }},{\mathit{\delta }}_{{\mathit{\beta }}^{\ast }},{\mathit{\delta }}_{{\mathit{\nu }}^{\ast }},{\mathit{\delta }}_{{\mathit{\psi }}^{\ast }},{\mathit{\delta }}_{\mathit{\gamma }})\\ & =\frac{-1}{\sqrt{T}}\sum _{t=1}^T{\varphi }_{\tau }\left({\epsilon }_{it}\left(\tau \right)-\frac{{\mathit{D}}_{it}^{\mathrm{\top }}{\mathit{\delta }}_{\mathit{\alpha }}}{\sqrt{NT}}-\frac{{\mathit{X}}_{it}^{\ast \mathrm{\top }}{\mathit{\delta }}_{{\mathit{\beta }}^{\ast }}}{\sqrt{NT}}-\frac{{\mathit{Z}}_{1i}^{\ast \mathrm{\top }}{\mathit{\delta }}_{{\mathit{\nu }}^{\ast }}}{\sqrt{T}}-\frac{{\mathit{Z}}_{2t}^{\ast \mathrm{\top }}{\mathit{\delta }}_{{\mathit{\psi }}^{\ast }}}{\sqrt{N}}-\frac{{\mathit{\omega }}_{it}^{\mathrm{\top }}{\mathit{\delta }}_{\mathit{\gamma }}}{\sqrt{NT}}\right).\end{array}$$

Let

$$\begin{array}{rl}& sup\parallel g_{it}({\mathit{\delta }}_{\mathit{\alpha }},{\mathit{\delta }}_{{\mathit{\beta }}^{\ast }},{\mathit{\delta }}_{{\mathit{\nu }}^{\ast }},{\mathit{\delta }}_{{\mathit{\psi }}^{\ast }},{\mathit{\delta }}_{\mathit{\gamma }})-g_{it}(0,0,0,0,0)\\ & -\mathbb{E}\left[g_{it}\right({\mathit{\delta }}_{\mathit{\alpha }},{\mathit{\delta }}_{{\mathit{\beta }}^{\ast }},{\mathit{\delta }}_{{\mathit{\nu }}^{\ast }},{\mathit{\delta }}_{{\mathit{\psi }}^{\ast }},{\mathit{\delta }}_{\mathit{\gamma }})-g_{it}(0,0,0,0,0\left)\right]\parallel =o_p\left(1\right).\end{array}$$

Expanding $g_{it}$, we obtain

$$\begin{array}{rl}& \mathbb{E}\left[g_{it}\right({\mathit{\delta }}_{\mathit{\alpha }},{\mathit{\delta }}_{{\mathit{\beta }}^{\ast }},{\mathit{\delta }}_{{\mathit{\nu }}^{\ast }},{\mathit{\delta }}_{{\mathit{\psi }}^{\ast }},{\mathit{\delta }}_{\mathit{\gamma }})-g_{it}(0,0,0,0,0\left)\right]\\ & =\frac{-1}{\sqrt{T}}\sum _{t=1}^T\mathbb{E}({\varphi }_{\tau }({\epsilon }_{it}\left(\tau \right)-\frac{{\mathit{D}}_{it}^{\mathrm{\top }}{\mathit{\delta }}_{\mathit{\alpha }}}{\sqrt{NT}}-\frac{{\mathit{X}}_{it}^{\ast \mathrm{\top }}{\mathit{\delta }}_{{\mathit{\beta }}^{\ast }}}{\sqrt{NT}}-\frac{{\mathit{Z}}_{1i}^{\ast \mathrm{\top }}{\mathit{\delta }}_{{\mathit{\nu }}^{\ast }}}{\sqrt{T}}-\frac{{\mathit{Z}}_{2t}^{\ast \mathrm{\top }}{\mathit{\delta }}_{{\mathit{\psi }}^{\ast }}}{\sqrt{N}}-\frac{{\mathit{\omega }}_{it}^{\mathrm{\top }}{\mathit{\delta }}_{\mathit{\gamma }}}{\sqrt{NT}})-{\varphi }_{\tau }\left({\epsilon }_{it}\right(\tau ))\\ & =\frac{-1}{\sqrt{T}}\sum _{t=1}^T[\tau -F({\xi }_{it}\left(\tau \right)+\frac{{\mathit{D}}_{it}^{\mathrm{\top }}{\mathit{\delta }}_{\mathit{\alpha }}}{\sqrt{NT}}+\frac{{\mathit{X}}_{it}^{\ast \mathrm{\top }}{\mathit{\delta }}_{{\mathit{\beta }}^{\ast }}}{\sqrt{NT}}+\frac{{\mathit{Z}}_{1i}^{\ast \mathrm{\top }}{\mathit{\delta }}_{{\mathit{\nu }}^{\ast }}}{\sqrt{T}}+\frac{{\mathit{Z}}_{2t}^{\ast \mathrm{\top }}{\mathit{\delta }}_{{\mathit{\psi }}^{\ast }}}{\sqrt{N}}+\frac{{\mathit{\omega }}_{it}^{\mathrm{\top }}{\mathit{\delta }}_{\mathit{\gamma }}}{\sqrt{NT}})]\\ & =\frac{1}{\sqrt{T}}\sum _{t=1}^T{f}_{it}\left({\xi }_{it}\right(\tau \left)\right)[\frac{{\mathit{D}}_{it}^{\mathrm{\top }}{\mathit{\delta }}_{\mathit{\alpha }}}{\sqrt{NT}}+\frac{{\mathit{X}}_{it}^{\ast \mathrm{\top }}{\mathit{\delta }}_{{\mathit{\beta }}^{\ast }}}{\sqrt{NT}}+\frac{{\mathit{Z}}_{1i}^{\ast \mathrm{\top }}{\mathit{\delta }}_{{\mathit{\nu }}^{\ast }}}{\sqrt{T}}+\frac{{\mathit{Z}}_{2t}^{\ast \mathrm{\top }}{\mathit{\delta }}_{{\mathit{\psi }}^{\ast }}}{\sqrt{N}}+\frac{{\mathit{\omega }}_{it}^{\mathrm{\top }}{\mathit{\delta }}_{\mathit{\gamma }}}{\sqrt{NT}}]+o_p\left(1\right),\end{array}$$

where $F(\cdot )$ is the conditional distribution of $y_{it}$. At the minimizer, $g_{it}({\hat{\mathit{\delta }}}_{\mathit{\alpha }},{\hat{\mathit{\delta }}}_{{\mathit{\beta }}^{\ast }},{\hat{\mathit{\delta }}}_{{\mathit{\nu }}^{\ast }},{\hat{\mathit{\delta }}}_{{\mathit{\psi }}^{\ast }},{\hat{\mathit{\delta }}}_{\mathit{\gamma }})\to 0$, and thus $\mathbb{E}\left[g_{it}\right({\mathit{\delta }}_{\mathit{\alpha }},{\mathit{\delta }}_{{\mathit{\beta }}^{\ast }},{\mathit{\delta }}_{{\mathit{\nu }}^{\ast }},{\mathit{\delta }}_{{\mathit{\psi }}^{\ast }},{\mathit{\delta }}_{\mathit{\gamma }})-g_{it}(0,0,0,0,0\left)\right]=-g_{it}(0,0,0,0,0)$, i.e. the last equation has the following equivalent expression:

$$\frac{1}{\sqrt{T}}\sum _{t=1}^T{f}_{it}\left({\xi }_{it}\right(\tau \left)\right)[\frac{{\mathit{D}}_{it}^{\mathrm{\top }}{\mathit{\delta }}_{\mathit{\alpha }}}{\sqrt{NT}}+\frac{{\mathit{X}}_{it}^{\ast \mathrm{\top }}{\mathit{\delta }}_{{\mathit{\beta }}^{\ast }}}{\sqrt{NT}}+\frac{{\mathit{Z}}_{1i}^{\ast \mathrm{\top }}{\mathit{\delta }}_{{\mathit{\nu }}^{\ast }}}{\sqrt{T}}+\frac{{\mathit{Z}}_{2t}^{\ast \mathrm{\top }}{\mathit{\delta }}_{{\mathit{\psi }}^{\ast }}}{\sqrt{N}}+\frac{{\mathit{\omega }}_{it}^{\mathrm{\top }}{\mathit{\delta }}_{\mathit{\gamma }}}{\sqrt{NT}}]=\frac{1}{\sqrt{T}}\sum _{t=1}^T{\varphi }_{\tau }\left({\epsilon }_{it}\right(\tau \left)\right).$$

Optimality of ${\hat{\mathit{\delta }}}_{{\mathit{\nu }}^{\ast }}$ implies that $g_{it}({\mathit{\delta }}_{\mathit{\alpha }},{\mathit{\delta }}_{{\mathit{\beta }}^{\ast }},{\mathit{\delta }}_{{\mathit{\nu }}^{\ast }},{\mathit{\delta }}_{{\mathit{\psi }}^{\ast }},{\mathit{\delta }}_{\mathit{\gamma }})=o\left(T^{-1}\right)$, and thus

$$\begin{array}{rl}{\hat{\delta }}_{{\mathit{\nu }}_i^{\ast }}& ={\overline{f}}_i^{-1}(\frac{1}{\sqrt{T}}\sum _{t=1}^T{\varphi }_{\tau }\left({\epsilon }_{it}\right(\tau \left)\right)-\frac{1}{\sqrt{T}}\sum _{t=1}^T{f}_{it}\left({\xi }_{it}\right(\tau \left)\right)(\frac{{\mathit{D}}_{it}^{\mathrm{\top }}{\mathit{\delta }}_{\mathit{\alpha }}}{\sqrt{NT}}+\frac{{\mathit{X}}_{it}^{\ast \mathrm{\top }}{\mathit{\delta }}_{{\mathit{\beta }}^{\ast }}}{\sqrt{NT}}\\ & +\frac{{\mathit{Z}}_{2t}^{\ast \mathrm{\top }}{\mathit{\delta }}_{{\mathit{\psi }}^{\ast }}}{\sqrt{N}}+\frac{{\mathit{\omega }}_{it}^{\mathrm{\top }}{\mathit{\delta }}_{\mathit{\gamma }}}{\sqrt{NT}}))+o_p\left(1\right),\end{array}$$

where ${\overline{f}}_i=T^{-1}\sum _{t=1}^T{f}_{it}\left({\xi }_{it}\right(\tau \left)\right)$. Substituting ${\mathit{Z}}_{1i}^{\ast }{\hat{\mathit{\delta }}}_{{\mathit{\nu }}^{\ast }}$'s, we denote

$$g_t({\mathit{\delta }}_{\mathit{\alpha }},{\mathit{\delta }}_{{\mathit{\beta }}^{\ast }},{\mathit{\delta }}_{{\mathit{\psi }}^{\ast }},{\mathit{\delta }}_{\mathit{\gamma }})=\frac{-1}{\sqrt{N}}\sum _{i=1}^N{\varphi }_{\tau }({\epsilon }_{it}\left(\tau \right)-\frac{{\mathit{D}}_{it}^{\mathrm{\top }}{\mathit{\delta }}_{\mathit{\alpha }}}{\sqrt{NT}}-\frac{{\mathit{Z}}_{2t}^{\ast \mathrm{\top }}{\mathit{\delta }}_{{\mathit{\psi }}^{\ast }}}{\sqrt{N}}-\frac{{\mathit{\omega }}_{it}^{\mathrm{\top }}{\mathit{\delta }}_{\mathit{\gamma }}}{\sqrt{NT}}).$$

Let

$$sup\parallel g_t({\mathit{\delta }}_{\mathit{\alpha }},{\mathit{\delta }}_{{\mathit{\beta }}^{\ast }},{\mathit{\delta }}_{{\mathit{\psi }}^{\ast }},{\mathit{\delta }}_{\mathit{\gamma }})-g_t(0,0,0,0)-\mathbb{E}\left[g_t\right({\delta }_{\rho },{\delta }_{\lambda },{\delta }_{\kappa },{\mathit{\delta }}_{{\mathit{\beta }}^{\ast }},{\mathit{\delta }}_{{\mathit{\psi }}^{\ast }},{\mathit{\delta }}_{\mathit{\gamma }})-g_t(0,0,0,0\left)\right]\parallel =o_p\left(1\right).$$

Expanding $g_t$, we obtain

$$\begin{array}{rl}& \mathbb{E}\left[g_t\right({\mathit{\delta }}_{\mathit{\alpha }},{\mathit{\delta }}_{{\mathit{\beta }}^{\ast }},{\mathit{\delta }}_{{\mathit{\psi }}^{\ast }},{\mathit{\delta }}_{\mathit{\gamma }})-g_t(0,0,0,0\left)\right]\\ & =\frac{-1}{\sqrt{N}}\sum _{i=1}^N\mathbb{E}({\varphi }_{\tau }({\epsilon }_{it}\left(\tau \right)-\frac{{\mathit{D}}_{it}^{\mathrm{\top }}{\mathit{\delta }}_{\mathit{\alpha }}}{\sqrt{NT}}-\frac{{\mathit{X}}_{it}^{\ast \mathrm{\top }}{\mathit{\delta }}_{{\mathit{\beta }}^{\ast }}}{\sqrt{NT}}\frac{{\mathit{Z}}_{1i}^{\ast }{\hat{\mathit{\delta }}}_{{\mathit{\nu }}^{\ast }}}{\sqrt{T}}-\frac{{\mathit{Z}}_{2t}^{\ast \mathrm{\top }}{\mathit{\delta }}_{{\mathit{\psi }}^{\ast }}}{\sqrt{N}}-\frac{{\mathit{\omega }}_{it}^{\mathrm{\top }}{\mathit{\delta }}_{\mathit{\gamma }}}{\sqrt{NT}})-{\varphi }_{\tau }\left({\epsilon }_{it}\right(\tau ))\\ & =\frac{1}{\sqrt{N}}\sum _{i=1}^N{f}_{it}\left({\xi }_{it}\right(\tau \left)\right)[(1-T^{-1}{\overline{f}}_i^{-1}\sum _{t=1}^T{f}_{it}\left({\xi }_{it}\right(\tau \left)\right))(\frac{{\mathit{D}}_{it}^{\mathrm{\top }}{\mathit{\delta }}_{\mathit{\alpha }}}{\sqrt{NT}}+\frac{{\mathit{X}}_{it}^{\ast \mathrm{\top }}{\mathit{\delta }}_{{\mathit{\beta }}^{\ast }}}{\sqrt{NT}}+\frac{{\mathit{Z}}_{2t}^{\ast \mathrm{\top }}{\mathit{\delta }}_{{\mathit{\psi }}^{\ast }}}{\sqrt{N}}+\frac{{\mathit{\omega }}_{it}^{\mathrm{\top }}{\mathit{\delta }}_{\mathit{\gamma }}}{\sqrt{NT}})\\ & +T^{-1}{\overline{f}}_i^{-1}\sum _{t=1}^T{\varphi }_{\tau }\left({\epsilon }_{it}\right(\tau \left)\right)]+o_p\left(1\right).\end{array}$$

At the minimizer, $g_t({\hat{\mathit{\delta }}}_{\mathit{\alpha }},{\hat{\mathit{\delta }}}_{{\mathit{\beta }}^{\ast }},{\hat{\mathit{\delta }}}_{{\mathit{\psi }}^{\ast }},{\hat{\mathit{\delta }}}_{\mathit{\gamma }})\to 0$, thus $\mathbb{E}\left[g_t\right({\mathit{\delta }}_{\mathit{\alpha }},{\mathit{\delta }}_{{\mathit{\beta }}^{\ast }},{\mathit{\delta }}_{{\mathit{\psi }}^{\ast }},{\mathit{\delta }}_{\mathit{\gamma }})-g_t(0,0,0,0\left)\right]=-g_t(0,0,0,0)$, i.e. the last equation has the following equivalent expression:

$$\begin{array}{rl}& \frac{1}{\sqrt{N}}\sum _{i=1}^N{f}_{it}\left({\xi }_{it}\right(\tau \left)\right)[(1-T^{-1}{\overline{f}}_i^{-1}\sum _{t=1}^T{f}_{it}\left({\xi }_{it}\right(\tau \left)\right))(\frac{{\mathit{D}}_{it}^{\mathrm{\top }}{\mathit{\delta }}_{\mathit{\alpha }}}{\sqrt{NT}}+\frac{{\mathit{X}}_{it}^{\ast \mathrm{\top }}{\mathit{\delta }}_{{\mathit{\beta }}^{\ast }}}{\sqrt{NT}}\\ & +\frac{{\mathit{Z}}_{2t}^{\ast \mathrm{\top }}{\mathit{\delta }}_{{\mathit{\psi }}^{\ast }}}{\sqrt{N}}+\frac{{\mathit{\omega }}_{it}^{\mathrm{\top }}{\mathit{\delta }}_{\mathit{\gamma }}}{\sqrt{NT}})+T^{-1}{\overline{f}}_i^{-1}\sum _{t=1}^T{\varphi }_{\tau }\left({\epsilon }_{it}\right(\tau \left)\right)]=\frac{1}{\sqrt{N}}\sum _{i=1}^N{\varphi }_{\tau }\left({\epsilon }_{it}\right(\tau \left)\right).\end{array}$$

Optimality of ${\hat{\mathit{\delta }}}_{{\mathit{\psi }}^{\ast }}$ implies that $g_t({\mathit{\delta }}_{\mathit{\alpha }},{\mathit{\delta }}_{{\mathit{\beta }}^{\ast }},{\mathit{\delta }}_{{\mathit{\psi }}^{\ast }},{\mathit{\delta }}_{\mathit{\gamma }})=o\left(N^{-1}\right)$, and thus

$$\begin{array}{rl}{\hat{\delta }}_{{\mathit{\psi }}_t^{\ast }}& ={\overline{f}}_t^{-1}(\frac{1}{\sqrt{N}}(1-T^{-1}{\overline{f}}_i^{-1}\sum _{t=1}^T{f}_{it}\left({\xi }_{it}\right(\tau \left)\right){)}^{-1}\sum _{i=1}^N{\varphi }_{\tau }\left({\epsilon }_{it}\right(\tau \left)\right)-\sqrt{N}{T}^{-1}{\overline{f}}_t{\overline{f}}_i^{-1}\sum _{t=1}^T{\varphi }_{\tau }\left({\epsilon }_{it}\right(\tau \left)\right)\\ & -\frac{1}{\sqrt{N}}\sum _{i=1}^N{f}_{it}\left({\xi }_{it}\right(\tau \left)\right)(\frac{{\mathit{D}}_{it}^{\mathrm{\top }}{\mathit{\delta }}_{\mathit{\alpha }}}{\sqrt{NT}}+\frac{{\mathit{X}}_{it}^{\ast \mathrm{\top }}{\mathit{\delta }}_{{\mathit{\beta }}^{\ast }}}{\sqrt{NT}}+\frac{{\mathit{\omega }}_{it}^{\mathrm{\top }}{\mathit{\delta }}_{\mathit{\gamma }}}{\sqrt{NT}}))+o_p\left(1\right),\end{array}$$

where ${\overline{f}}_t=N^{-1}\sum _{i=1}^N{f}_{it}\left({\xi }_{it}\right(\tau \left)\right)$. Substituting ${\mathit{Z}}_{2t}^{\ast }{\hat{\mathit{\delta }}}_{{\mathit{\psi }}^{\ast }}$, we denote

$$\begin{array}{rl}G({\mathit{\delta }}_{\mathit{\alpha }},{\mathit{\delta }}_{{\mathit{\beta }}^{\ast }},{\mathit{\delta }}_{\mathit{\gamma }})& =\frac{-1}{\sqrt{NT}}\sum _{i=1}^N\sum _{t=1}^T{\tilde{\mathit{X}}}_{it}^{\mathrm{\top }}{\varphi }_{\tau }({\epsilon }_{it}\left(\tau \right)-\frac{{\mathit{D}}_{it}^{\mathrm{\top }}{\mathit{\delta }}_{\mathit{\alpha }}}{\sqrt{NT}}-\frac{{\mathit{X}}_{it}^{\ast \mathrm{\top }}{\mathit{\delta }}_{{\mathit{\beta }}^{\ast }}}{\sqrt{NT}}-\frac{{\mathit{Z}}_{1i}^{\ast }{\hat{\mathit{\delta }}}_{{\mathit{\nu }}^{\ast }}}{\sqrt{T}}\\ & -\frac{{\mathit{Z}}_{2t}^{\ast }{\hat{\mathit{\delta }}}_{{\mathit{\psi }}^{\ast }}}{\sqrt{N}}-\frac{{\mathit{\omega }}_{it}^{\mathrm{\top }}{\mathit{\delta }}_{\mathit{\gamma }}}{\sqrt{NT}}),\end{array}$$

where ${\tilde{\mathit{X}}}_{it}=[{\mathit{X}}_{it}^{\ast },{\mathit{\omega }}_{it}]$. Let

$$sup\parallel G({\mathit{\delta }}_{\mathit{\alpha }},{\mathit{\delta }}_{{\mathit{\beta }}^{\ast }},{\mathit{\delta }}_{\mathit{\gamma }})-G(0,0,0)-\mathbb{E}\left[G\right({\mathit{\delta }}_{\mathit{\alpha }},{\mathit{\delta }}_{{\mathit{\beta }}^{\ast }},{\mathit{\delta }}_{\mathit{\gamma }})-G(0,0,0\left)\right]\parallel =o_p\left(1\right).$$

Expanding G, we obtain

$$\begin{array}{rl}& \mathbb{E}\left[G\right({\mathit{\delta }}_{\mathit{\alpha }},{\mathit{\delta }}_{{\mathit{\beta }}^{\ast }},{\mathit{\delta }}_{\mathit{\gamma }})-G(0,0,0\left)\right]\\ & =\frac{1}{\sqrt{NT}}\sum _{i=1}^N\sum _{t=1}^T{\tilde{\mathit{X}}}_{it}^{\mathrm{\top }}{f}_{it}\left({\xi }_{it}\right(\tau \left)\right)[\frac{{\mathit{D}}_{it}^{\mathrm{\top }}{\mathit{\delta }}_{\mathit{\alpha }}}{\sqrt{NT}}+\frac{{\mathit{X}}_{it}^{\ast \mathrm{\top }}{\mathit{\delta }}_{{\mathit{\beta }}^{\ast }}}{\sqrt{NT}}+\frac{{\mathit{Z}}_{1i}^{\ast }{\hat{\mathit{\delta }}}_{{\mathit{\nu }}^{\ast }}}{\sqrt{T}}+\frac{{\mathit{Z}}_{2t}^{\ast }{\hat{\mathit{\delta }}}_{{\mathit{\psi }}^{\ast }}}{\sqrt{N}}+\frac{{\mathit{\omega }}_{it}^{\mathrm{\top }}{\mathit{\delta }}_{\mathit{\gamma }}}{\sqrt{NT}}]+o_p\left(1\right),\\ & =\frac{1}{\sqrt{NT}}\sum _{i=1}^N\sum _{t=1}^T{\tilde{\mathit{X}}}_{it}^{\mathrm{\top }}{f}_{it}\left({\xi }_{it}\right(\tau \left)\right)[(1-N^{-1}{\overline{f}}_t^{-1}\sum _{i=1}^N{f}_{it}\left({\xi }_{it}\right(\tau \left)\right))(1-T^{-1}{\overline{f}}_i^{-1}\sum _{t=1}^T{f}_{it}\left({\xi }_{it}\right(\tau \left)\right))\\ & \times (\frac{{\mathit{D}}_{it}^{\mathrm{\top }}{\mathit{\delta }}_{\mathit{\alpha }}}{\sqrt{NT}}+\frac{{\mathit{X}}_{it}^{\ast \mathrm{\top }}{\mathit{\delta }}_{{\mathit{\beta }}^{\ast }}}{\sqrt{NT}}+\frac{{\mathit{\omega }}_{it}^{\mathrm{\top }}{\mathit{\delta }}_{\mathit{\gamma }}}{\sqrt{NT}})+N^{-1}{\overline{f}}_t^{-1}\sum _{i=1}^N{\varphi }_{\tau }\left({\epsilon }_{it}\right(\tau \left)\right)+T^{-1}{\overline{f}}_i^{-1}\sum _{t=1}^T{\varphi }_{\tau }\left({\epsilon }_{it}\right(\tau \left)\right)]\\ & +o_p\left(1\right).\end{array}$$

At the minimizer, $G({\hat{\mathit{\delta }}}_{\mathit{\alpha }},{\hat{\mathit{\delta }}}_{{\mathit{\beta }}^{\ast }},{\hat{\mathit{\delta }}}_{\mathit{\gamma }})\to 0$, $\mathbb{E}\left[G\right({\mathit{\delta }}_{\mathit{\alpha }},{\mathit{\delta }}_{{\mathit{\beta }}^{\ast }},{\mathit{\delta }}_{\mathit{\gamma }})-G(0,0,0\left)\right]=-G(0,0,0)$, i.e. the last equation has the following equivalent expression:

$$\begin{array}{rl}& \frac{1}{\sqrt{NT}}\sum _{i=1}^N\sum _{t=1}^T{\tilde{\mathit{X}}}_{it}^{\mathrm{\top }}{\varphi }_{\tau }\left({\epsilon }_{it}\right(\tau \left)\right)=\frac{1}{\sqrt{NT}}\sum _{i=1}^N\sum _{t=1}^T{\tilde{\mathit{X}}}_{it}^{\mathrm{\top }}{f}_{it}\left({\xi }_{it}\right(\tau \left)\right)[(1-N^{-1}{\overline{f}}_t^{-1}\sum _{i=1}^N{f}_{it}\left({\xi }_{it}\right(\tau \left)\right))\\ & \times (1-T^{-1}{\overline{f}}_i^{-1}\sum _{t=1}^T{f}_{it}\left({\xi }_{it}\right(\tau \left)\right))(\frac{{\mathit{D}}_{it}^{\mathrm{\top }}{\mathit{\delta }}_{\mathit{\alpha }}}{\sqrt{NT}}+\frac{{\mathit{X}}_{it}^{\ast \mathrm{\top }}{\mathit{\delta }}_{{\mathit{\beta }}^{\ast }}}{\sqrt{NT}}+\frac{{\mathit{\omega }}_{it}^{\mathrm{\top }}{\mathit{\delta }}_{\mathit{\gamma }}}{\sqrt{NT}})\\ & +N^{-1}{\overline{f}}_t^{-1}\sum _{i=1}^N{\varphi }_{\tau }\left({\epsilon }_{it}\right(\tau \left)\right)+T^{-1}{\overline{f}}_i^{-1}\sum _{t=1}^T{\varphi }_{\tau }\left({\epsilon }_{it}\right(\tau \left)\right)].\end{array}$$

Letting ${\delta }_{\mathit{\zeta }}=({\mathit{\delta }}_{{\mathit{\beta }}^{\ast }}^{\mathrm{\top }},{\mathit{\delta }}_{\mathit{\gamma }}^{\mathrm{\top }}{)}^{\mathrm{\top }}$, we write the equation above as

$$\begin{array}{rl}& \frac{1}{\sqrt{NT}}\sum _{i=1}^N\sum _{t=1}^T{\tilde{\mathit{X}}}_{it}^{\mathrm{\top }}{f}_{it}\left({\xi }_{it}\right(\tau \left)\right)[(1-N^{-1}{\overline{f}}_t^{-1}\sum _{i=1}^N{f}_{it}\left({\xi }_{it}\right(\tau \left)\right))(1-T^{-1}{\overline{f}}_i^{-1}\sum _{t=1}^T{f}_{it}\left({\xi }_{it}\right(\tau \left)\right))\\ & \times (\frac{{\tilde{\mathit{X}}}_{it}{\mathit{\delta }}_{\mathit{\zeta }}}{\sqrt{NT}}+\frac{{\mathit{D}}_{it}^{\mathrm{\top }}{\mathit{\delta }}_{\mathit{\alpha }}}{\sqrt{NT}})+N^{-1}{\overline{f}}_t^{-1}\sum _{i=1}^N{\varphi }_{\tau }\left({\epsilon }_{it}\right(\tau \left)\right)+T^{-1}{\overline{f}}_i^{-1}\sum _{t=1}^T{\varphi }_{\tau }\left({\epsilon }_{it}\right(\tau \left)\right)]\\ & =\frac{1}{\sqrt{NT}}\sum _{i=1}^N\sum _{t=1}^T{\tilde{\mathit{X}}}_{it}^{\mathrm{\top }}{\varphi }_{\tau }\left({\epsilon }_{it}\right(\tau \left)\right).\end{array}$$

Alternatively, using more convenient notation, we write the last expression as

$${\mathit{J}}_{\mathit{\zeta }}{\mathit{\delta }}_{\mathit{\zeta }}+{\mathit{J}}_{\mathit{\alpha }}{\mathit{\delta }}_{\mathit{\alpha }}={\mathbb{J}}_{\phi },$$

where ${\mathit{J}}_{\mathit{\zeta }}=\underset{N,T\to \mathrm{\infty }}{lim}{\tilde{\mathit{X}}}^{\mathrm{\top }}{\mathit{M}}_{\tilde{\mathit{Z}}}^{\mathrm{\top }}\mathbf{\Omega }{\mathit{M}}_{\tilde{\mathit{Z}}}\tilde{\mathit{X}}$, ${\mathit{J}}_{\mathit{\alpha }}=\underset{N,T\to \mathrm{\infty }}{lim}{\tilde{\mathit{X}}}^{\mathrm{\top }}{\mathit{M}}_{\tilde{\mathit{Z}}}^{\mathrm{\top }}\mathbf{\Omega }{\mathit{M}}_{\tilde{\mathit{Z}}}\mathit{D}$, ${\mathbb{J}}_{\phi }$ is a mean zero r.v. with covariance $\tau (1-\tau ){\tilde{\mathit{X}}}^{\mathrm{\top }}{\mathit{M}}_{\tilde{\mathit{Z}}}^{\mathrm{\top }}{\mathit{M}}_{\tilde{\mathit{Z}}}\tilde{\mathit{X}}$, $\mathbf{\Omega }=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left(f_{it}\right({\xi }_{it}\left(\tau \right)\left)\right)$ and ${\mathbf{\Phi }}_{\tau }$ is an NT vector $\left({\phi }_{\tau }\right({\epsilon }_{it}\left(\tau \right)\left)\right)$, $\tilde{\mathit{Z}}=[{\mathit{Z}}_1^{\ast },{\mathit{Z}}_2^{\ast }]$, ${\mathit{M}}_{\tilde{\mathit{Z}}}=\mathit{I}-{\mathit{P}}_{\tilde{\mathit{Z}}}$, ${\mathit{P}}_{\tilde{\mathit{Z}}}=\tilde{\mathit{Z}}({\tilde{\mathit{Z}}}^{\mathrm{\top }}\mathbf{\Omega }\tilde{\mathit{Z}}{)}^{-1}{\tilde{\mathit{Z}}}^{\mathrm{\top }}\mathbf{\Omega }$.

Letting $[{\overline{\mathit{J}}}_{{\mathit{\beta }}^{\ast }}^{\mathrm{\top }},{\overline{\mathit{J}}}_{\mathit{\gamma }}^{\mathrm{\top }}]$ be a conformable partition of ${\mathit{J}}_{\mathit{\zeta }}^{-1}$ as in Galvao [8] and Chernozhukov and Hansen [4] yields ${\hat{\mathit{\delta }}}_{{\mathit{\beta }}^{\ast }}={\overline{\mathit{J}}}_{{\mathit{\beta }}^{\ast }}^{\mathrm{\top }}({\mathbb{J}}_{\phi }-{\mathit{J}}_{\mathit{\alpha }}{\mathit{\delta }}_{\mathit{\alpha }})$, and ${\hat{\mathit{\delta }}}_{\mathit{\gamma }}={\overline{\mathit{J}}}_{\mathit{\gamma }}^{\mathrm{\top }}({\mathbb{J}}_{\phi }-{\mathit{J}}_{\mathit{\alpha }}{\mathit{\delta }}_{\mathit{\alpha }})$. Letting $\mathit{H}={\overline{\mathit{J}}}_{\mathit{\gamma }}^{\mathrm{\top }}\mathit{A}{\overline{\mathit{J}}}_{\mathit{\gamma }}$ as in Chernozhukov and Hansen [4] gives ${\hat{\mathit{\delta }}}_{\mathit{\alpha }}=\mathit{K}{\mathbb{J}}_{\phi }$, where $\mathit{K}=({\mathit{J}}_{\mathit{\alpha }}^{\mathrm{\top }}\mathit{H}{\mathit{J}}_{\mathit{\alpha }}{)}^{-1}{\mathit{J}}_{\mathit{\alpha }}^{\mathrm{\top }}\mathit{H}$. Replacing it in the previous expression, ${\hat{\mathit{\delta }}}_{\mathit{\gamma }}={\overline{\mathit{J}}}_{\mathit{\gamma }}^{\mathrm{\top }}({\mathbb{J}}_{\phi }-{\mathit{J}}_{\mathit{\alpha }}{\mathit{\delta }}_{\mathit{\alpha }})={\overline{\mathit{J}}}_{\mathit{\gamma }}^{\mathrm{\top }}(\mathit{I}-{\mathit{J}}_{\mathit{\alpha }}({\mathit{J}}_{\mathit{\alpha }}^{\mathrm{\top }}\mathit{H}{\mathit{J}}_{\mathit{\alpha }}{)}^{-1}{\mathit{J}}_{\mathit{\alpha }}^{\mathrm{\top }}\mathit{H}){\mathbb{J}}_{\phi }={\overline{\mathit{J}}}_{\mathit{\gamma }}^{\mathrm{\top }}\mathit{M}{\mathbb{J}}_{\phi }$, where $\mathit{M}=\mathit{I}-{\mathit{J}}_{\mathit{\alpha }}({\mathit{J}}_{\mathit{\alpha }}^{\mathrm{\top }}\mathit{H}{\mathit{J}}_{\mathit{\alpha }}{)}^{-1}{\mathit{J}}_{\mathit{\alpha }}^{\mathrm{\top }}\mathit{H}$. Due to the invertibility of ${\mathit{J}}_{\mathit{\alpha }}{\overline{\mathit{J}}}_{\mathit{\gamma }}$, ${\hat{\mathit{\delta }}}_{\mathit{\gamma }}=\mathbf{0}\times O_p\left(1\right)+o_p\left(1\right)$. Similarly, substituting back ${\delta }_{\rho }$ and ${\delta }_{\lambda }$, we obtain that ${\hat{\mathit{\delta }}}_{{\mathit{\beta }}^{\ast }}={\overline{\mathit{J}}}_{{\mathit{\beta }}^{\ast }}^{\mathrm{\top }}\mathit{M}{\mathbb{J}}_{\phi }$. By the regularity conditions, we have that

$$\left(\begin{array}{c}{\hat{\mathit{\delta }}}_{\mathit{\alpha }}({\rho }_n,{\lambda }_n,{\kappa }_n,\tau )\\ {\hat{\mathit{\delta }}}_{{\mathit{\beta }}^{\ast }}({\rho }_n,{\lambda }_n,{\kappa }_n,\tau )\end{array}\right)=\left(\begin{array}{c}\sqrt{NT}\left(\hat{\mathit{\alpha }}\right({\rho }_n,{\lambda }_n,{\kappa }_n,\tau )-\mathit{\alpha }(\tau \left)\right)\\ \sqrt{NT}\left({\hat{\mathit{\beta }}}^{\ast }\right({\rho }_n,{\lambda }_n,{\kappa }_n,\tau )-{\mathit{\beta }}^{\ast }(\tau \left)\right)\end{array}\right)⇝\mathcal{N}(\mathbf{0},{\mathit{J}}^{\mathrm{\top }}\mathit{S}\mathit{J}).$$

Funding Statement

The work was partially supported by the National Natural Science Foundation of China (Nos. 11271368, 11861042), the major research projects of philosophy and social science of the Chinese Ministry of Education (No. 15JZD015), the Key Program of National Philosophy and Social Science Foundation Grant (No. 13AZD064), the Fundamental Research Funds for the Central Universities, and the Research Funds of Renmin University of China (No. 18XNL012), Shanghai Natural Science Foundation (No. 18ZR1427200), and National Science Foundation of China (No. 11801370).

Disclosure statement

No potential conflict of interest was reported by the authors.