A Parametric Bootstrap Approach for a One-Way Error Component Regression Model with Measurement Errors

Abstract

In this paper, a one-way error component regression model with measurement errors is considered. The unknown parameter vector is estimated by using the bias-corrected method, and its corresponding asymptotic properties are also developed. For the hypothesis testing problem of the vector of the coefficient parameter in the model, a parametric bootstrap (PB) method is proposed. Under various sample sizes and parameter configurations, the effectiveness of our proposed PB test method is discussed by using some numerical simulations and a real data analysis.

1. Introduction

In this paper, we will consider the one-way error component regression model of Baltagi [1] and assume that the true covariates cannot be exactly or directly observed, that is, the model has measurement errors, which take the following form

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{Y}_{ij}={\alpha }_0+{\mathit{x}}_{ij}^{\mathrm{T}}\mathit{\beta }+{\mu }_i+{\epsilon }_{ij},\hfill \\ {\mathit{\omega }}_{ij}={\mathit{x}}_{ij}+{\mathbf{\delta }}_{ij},i=1,\dots ,n,j=1,\dots ,m,\hfill \end{array}\right.\end{array}$$

(1)

where $Y_{ij}$ denotes the j-th time response variable of the i-th subject; $\mathit{\theta }={({\alpha }_0,{\mathit{\beta }}^{\mathrm{T}})}^{\mathrm{T}}={({\alpha }_0,{\beta }_1,\dots ,{\beta }_p)}^{\mathrm{T}}$ is the unknown vector of the coefficient parameter; ${\mathit{x}}_{ij}={(x_{ij1},\dots ,x_{ijp})}^{\mathrm{T}}$ is the true fixed covariate; ${\mathit{\omega }}_{ij}$ is the observed surrogate variable; and the individual effect ${\mu }_i$, random error ${\epsilon }_{ij}$ and measurement error ${\mathbf{\delta }}_{ij}={({\delta }_{ij1},\dots ,{\delta }_{ijp})}^{\mathrm{T}}$ are independent. Assume that ${\mu }_i$, ${\epsilon }_{ij}$ and ${\mathbf{\delta }}_{ij}$ are independent and identically distributed (i.i.d.) as $N(0,{\sigma }_{\mu }^2)$, $N(0,{\sigma }_{\epsilon }^2)$ and $N({\mathbf{0}}_p,{\Sigma }_{\delta })$, respectively, ${\mathbf{0}}_p$ is the $p\times 1$ vector of zeros and ${\sigma }_{\mu }^2\ge 0$, ${\sigma }_{\epsilon }^2>0$, ${\Sigma }_{\delta }$ is known. The information of ${\Sigma }_{\delta }$ may be obtained by replicating and observing ${\mathit{\omega }}_{ij}$; the details can be found in Lin and Carroll [2], Carroll et al. [3], Li et al. [4], Dong et al. [5] and Guo et al. [6]. If the measurement errors do not exist, model (1) has been discussed by refs. [7,8], etc. When the individual effect and the time index of $j=1,\dots ,m$ do not exist, model (1) reduces to the classical linear errors-in-variables (EV) model, which has been studied by refs. [9,10]. For the linear mixed effects EV model, Cui et al. [11] developed some moment estimation procedures, Xiao et al. [12] provided a GMM estimation approach and Yue et al. [13,14] proposed bias-corrected least square estimators for an unbalanced and balanced panel data model with measurement errors, respectively.

In practice, the hypothesis testing problem for the vector of the coefficient parameter

$$H_0:\mathit{\theta }={\mathit{\theta }}^{*}⟷H_1:\mathit{\theta }\ne {\mathit{\theta }}^{*}$$

(2)

is interested, where ${\mathit{\theta }}^{*}$ is an arbitrarily specified vector. For the nonhomogenous linear hypothesis testing problems ($H_0:\mathit{H}\mathit{\theta }=d$), Yue et al. [14] provided an adjusted testing method for model (1) by considering the discrepancies of the corrected residual sums of squares between the null and alternative hypotheses. When the measurement errors do not exist in model (1) (i.e., ${\mathit{x}}_{ij}$ is observed directly), Esmaeli-Ayan et al. [15] and Yue et al. [16] proposed a parametric bootstrap approach. For the hypothesis testing problem (2) in the linear EV model, Huwang et al. [17] developed a uniformly robust (RT) test method. However, when the sample size is small, the Type I error rates of the RT test in Huwang et al. [17] seem to be conservative. To solve this problem, many researchers suggested adopting the PB test to control the Type I error rates, such as Efron and Tibshirani [18], Krishnamoorthy et al. [19], Xu et al. [20], Xu [21], Ye et al. [22], Sun and Fisher [23], among others.

In this paper, we consider the hypothesis testing problem (2) of model (1) and develop a parametric bootstrap (PB) approach. The main contributions of this paper are three-fold. First, the biased-corrected estimator of $\mathit{\theta }$ is obtained for a one-way error component regression model with measurement errors in (1), and its asymptotic properties are also established. Second, a PB test is developed by using the obtained estimator of $\mathit{\theta }$. Then, we provide the corresponding algorithm for computing the p-value of the proposed PB test. Third, our proposed PB test can control the Type I error rates well, which can be found in finite sample numerical simulations.

The organization of this paper is listed as follows. In Section 2, a parametric bootstrap (PB) approach for the hypothesis testing problem (2) of model (1) is developed, and the corresponding algorithm is provided. In Section 3, the effectiveness of the proposed PB test is assessed by some simulation studies, and the numerical results indicate that the proposed PB test performs better than the uniformly robust (RT) test. In Section 4, we apply the PB test to analyse Grunfeld data. In Section 5, we make some summaries and concluding remarks for this paper. In Appendix A, we give the corresponding proofs of Theorems and needed lemmas.

  Notations: For a matrix $\mathbf{A}$, let ${\mathbf{A}}^{\mathrm{T}}$ denote the transpose for $\mathbf{A}$ and $\mathbf{A}\ge \mathbf{B}$ denote that $\mathbf{A}-\mathbf{B}$ is positive semi-definite. Let $\parallel ·\parallel$ be the Euclidean norm and “$\stackrel{\mathcal{L}}{⟶}$” and “$\stackrel{P}{⟶}$” be the convergence in distribution and probability, respectively.

2. Methodology

2.1. Parameter Estimator and Its Theoretical Properties

For each term in model (1), we obtain its means for all the time $j=1,\dots ,m$ of each subject i. Combining this with model (1), we have

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\overline{Y}}_{i·}={\alpha }_0+{\overline{\mathit{x}}}_{i·}^{\mathrm{T}}\mathit{\beta }+{\mu }_i+{\overline{\epsilon }}_{i·},\hfill \\ {\overline{\mathit{\omega }}}_{i·}={\overline{\mathit{x}}}_{i·}+{\overline{\mathbf{\delta }}}_{i·},i=1,\dots ,n,\hfill \end{array}\right.\end{array}$$

(3)

where ${\overline{Y}}_{i·}={\displaystyle \frac{1}{m}}{\displaystyle \sum _{j=1}^m}{Y}_{ij}$, ${\overline{\mathit{x}}}_{i·}={\displaystyle \frac{1}{m}}{\displaystyle \sum _{j=1}^m}{\mathit{x}}_{ij}$, ${\overline{\epsilon }}_{i·}={\displaystyle \frac{1}{m}}{\displaystyle \sum _{j=1}^m}{\epsilon }_{ij}$, ${\overline{\mathit{\omega }}}_{i·}={\displaystyle \frac{1}{m}}{\displaystyle \sum _{j=1}^m}{\mathit{\omega }}_{ij}$ and ${\overline{\mathbf{\delta }}}_{i·}={\displaystyle \frac{1}{m}}{\displaystyle \sum _{j=1}^m}{\mathbf{\delta }}_{ij}$. Here and after, the following notations are used, ${\overline{\mathbf{\tau }}}_{i·}={(0,{\overline{\mathbf{\delta }}}_{i·}^{\mathrm{T}})}^{\mathrm{T}}$, ${\overline{\mathit{u}}}_{i·}={(1,{\overline{\mathit{x}}}_{i·}^{\mathrm{T}})}^{\mathrm{T}}$ and ${\overline{\mathit{z}}}_{i·}={(1,{\overline{\mathit{\omega }}}_{i·}^{\mathrm{T}})}^{\mathrm{T}}={\overline{\mathit{u}}}_{i·}+{\overline{\mathbf{\tau }}}_{i·}$, and let

$$\begin{array}{c}\hfill \begin{array}{c}{\mathrm{M}}_{\overline{Y}\overline{Y}}={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n}{\overline{Y}}_{i·}^2,{\mathbf{M}}_{\overline{z}\overline{z}}={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n}{\overline{\mathit{z}}}_{i·}{\overline{\mathit{z}}}_{i·}^{\mathrm{T}},{\mathbf{M}}_{\overline{u}\overline{u}}={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n}{\overline{\mathit{u}}}_{i·}{\overline{\mathit{u}}}_{i·}^{\mathrm{T}},\\ {\mathbf{M}}_{\overline{z}\overline{Y}}={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n}{\overline{\mathit{z}}}_{i·}^{\mathrm{T}}{\overline{Y}}_{i·},{\mathit{M}}_{\overline{Y}\overline{z}}={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n}{\overline{Y}}_{i·}{\overline{\mathit{z}}}_{i·},\end{array}\end{array}$$

where ${\mathbf{M}}_{\overline{u}\overline{u}}$ is assumed to be nonsingular, $\underset{n\to \infty }{\lim }{\mathbf{M}}_{\overline{u}\overline{u}}={\mathbf{M}}^0$ and ${\mathbf{M}}^0$ is a nonsingular matrix. Then, we have

$$E\left(\begin{array}{cc}{\mathrm{M}}_{\overline{Y}\overline{Y}}& {\mathit{M}}_{\overline{z}\overline{Y}}\\ {\mathit{M}}_{\overline{Y}\overline{z}}& {\mathbf{M}}_{\overline{z}\overline{z}}\end{array}\right)=\left(\begin{array}{cccc}{\mathit{\theta }}^{\mathrm{T}}{\mathbf{M}}_{\overline{u}\overline{u}}\mathit{\theta }+{\sigma }_{\mu }^2+\frac{1}{m}{\sigma }_{\epsilon }^2& & & {\mathit{\theta }}^{\mathrm{T}}{\mathbf{M}}_{\overline{u}\overline{u}}\\ {\mathbf{M}}_{\overline{u}\overline{u}}\mathit{\theta }& & & {\mathbf{M}}_{\overline{u}\overline{u}}+\frac{1}{m}{\Sigma }_{\tau }\end{array}\right),$$

(4)

where ${\Sigma }_{\tau }/m$ is a known covariance matrix of ${\overline{\mathbf{\tau }}}_{i·}$ (since ${\Sigma }_{\delta }$ is known). From ref. [9], we obtain a biased-corrected estimator of $\mathit{\theta }$ as follows,

$$\begin{array}{c}\hfill \widehat{\mathit{\theta }}={\left({\mathbf{M}}_{\overline{z}\overline{z}}-\frac{1}{m}{\Sigma }_{\tau }\right)}^{-1}{\mathit{M}}_{\overline{Y}\overline{z}}.\end{array}$$

Denote $\mathbf{W}={\mathbf{M}}_{\overline{z}\overline{z}}-{\Sigma }_{\tau }/m$, then

$$\begin{array}{c}\hfill \widehat{\mathit{\theta }}={\mathbf{W}}^{-1}\left(\mathbf{W}\mathit{\theta }+{\mathit{M}}_{\overline{Y}\overline{z}}-\mathbf{W}\mathit{\theta }\right)=\mathit{\theta }+{\mathbf{W}}^{-1}\left({\mathit{M}}_{\overline{Y}\overline{z}}-\mathbf{W}\mathit{\theta }\right).\end{array}$$

(5)

That is, $\mathbf{W}(\widehat{\mathit{\theta }}-\mathit{\theta })={\mathit{M}}_{\overline{Y}\overline{z}}-\mathbf{W}\mathit{\theta }$.

Proposition 1.

Let ${\overline{\upsilon }}_{i·}={\mu }_i+{\overline{\epsilon }}_{i·}-{\overline{\mathbf{\tau }}}_{i·}^{\mathrm{T}}\mathit{\theta }={\overline{Y}}_{i·}-{\overline{\mathit{z}}}_{i·}^{\mathrm{T}}\mathit{\theta }$ and ${\mathit{M}}_{\overline{\upsilon }\overline{z}}=n^{-1}{\sum }_{i=1}^n{\overline{\upsilon }}_{i·}{\overline{\mathit{z}}}_{i·}={\mathit{M}}_{\overline{Y}\overline{z}}-{\mathbf{M}}_{\overline{z}\overline{z}}\mathit{\theta }$. Together with (4), we have

$$\begin{array}{c}\hfill \mathbf{W}(\widehat{\mathit{\theta }}-\mathit{\theta })={\mathit{M}}_{\overline{\upsilon }\overline{z}}-E{\mathit{M}}_{\overline{\upsilon }\overline{z}}=\frac{1}{n}{\displaystyle \sum _{i=1}^n}\left\{{\overline{\upsilon }}_{i·}{\overline{\mathit{z}}}_{i·}-E\left({\overline{\upsilon }}_{i·}{\overline{\mathit{z}}}_{i·}\right)\right\},\end{array}$$

(6)

and the covariance matrix of  $\mathbf{W}(\widehat{\mathit{\theta }}-\mathit{\theta })$ is

$$\begin{array}{c}\hfill \mathrm{Cov}\left(\mathbf{W}(\widehat{\mathit{\theta }}-\mathit{\theta })\right)=\frac{1}{n}\left\{{\sigma }_{\overline{\upsilon }}^2\left({\mathbf{M}}_{\overline{u}\overline{u}}+\frac{1}{m}{\Sigma }_{\tau }\right)+\frac{1}{m^2}{\Sigma }_{\tau }\mathit{\theta }{\mathit{\theta }}^{\mathrm{T}}{\Sigma }_{\tau }\right\}\triangleq {\displaystyle \frac{\mathbf{S}}{n}},\end{array}$$

(7)

where $E{\mathit{M}}_{\overline{\upsilon }\overline{z}}=-{\Sigma }_{\tau }\mathit{\theta }/m$ and $\mathbf{S}={\sigma }_{\overline{\upsilon }}^2\left({\mathbf{M}}_{\overline{u}\overline{u}}+m^{-1}{\Sigma }_{\tau }\right)+m^{-2}{\Sigma }_{\tau }\mathit{\theta }{\mathit{\theta }}^{\mathrm{T}}{\Sigma }_{\tau }$ is positive definite.

Theorem 1.

For model (3), when ${\mathbf{M}}_{\overline{u}\overline{u}}$ is nonsingular, the moment generating function

$$\begin{array}{c}\hfill M\left(\mathit{t}\right)=E\left[\exp \left(\sqrt{n}{(\widehat{\mathit{\theta }}-\mathit{\theta })}^{\mathrm{T}}{\mathbf{W}}^{\mathrm{T}}{\mathbf{S}}^{-\frac{1}{2}}\mathit{t}\right)\right]⟶e^{{\mathit{t}}^{\mathrm{T}}\mathit{t}/2}\end{array}$$

uniformly over the parameter space Ω for any $\mathit{t}\in {\mathbb{R}}^{p+1}$, and ${({\mathit{\theta }}^{\mathrm{T}},{\mathit{\varphi }}^{\mathrm{T}})}^{\mathrm{T}}\in \Omega$, $\mathit{\varphi }$ is the nuisance parameter, which consists of ${\overline{\mathit{x}}}_{1·},\dots ,{\overline{\mathit{x}}}_{n·}$ in model (3), ${\sigma }_{\mu }^2$ and ${\sigma }_{\epsilon }^2$. As $n\to \infty$, we have

$$\begin{array}{c}\hfill \sqrt{n}{\mathbf{S}}^{-\frac{1}{2}}\mathbf{W}(\widehat{\mathit{\theta }}-\mathit{\theta })\stackrel{\mathcal{L}}{⟶}N\left({\mathbf{0}}_{p+1},{\mathbf{I}}_{p+1}\right)\end{array}$$

uniformly over the parameter space Ω.

Remark 1.

Theorem 1 shows the asymptotic normality of $\widehat{\mathit{\theta }}-\mathit{\theta }$. Based on this result and assuming the null hypothesis of (2) holds, we have

$$\begin{array}{c}\hfill T_0=n{(\widehat{\mathit{\theta }}-{\mathit{\theta }}^{*})}^{\mathrm{T}}{\mathbf{W}}^{\mathrm{T}}{\mathbf{S}}^{-1}\mathbf{W}(\widehat{\mathit{\theta }}-{\mathit{\theta }}^{*})\stackrel{\mathcal{L}}{⟶}{\chi }_{p+1}^2\end{array}$$

uniformly over the parameter space ${\Omega }^{*}$; ${\Omega }^{*}$ is a subspace of Ω, with $\mathit{\theta }$ being fixed at ${\mathit{\theta }}^{*}$.

Theorem 2.

Under model (3), when ${\mathbf{M}}_{\overline{u}\overline{u}}$ is nonsingular and $n\to \infty$, we have

$$T_R=n{(\widehat{\mathit{\theta }}-\mathit{\theta })}^{\mathrm{T}}{\mathbf{W}}^{\mathrm{T}}{\tilde{\mathbf{S}}}^{-1}\mathbf{W}(\widehat{\mathit{\theta }}-\mathit{\theta })\stackrel{\mathcal{L}}{⟶}{\chi }_{p+1}^2$$

(8)

uniformly over the parameter space Ω, where

$$\begin{array}{c}\hfill \tilde{\mathbf{S}}={\tilde{\sigma }}_{\overline{\upsilon }}^2{\mathbf{M}}_{\overline{z}\overline{z}}+\frac{1}{m^2}{\Sigma }_{\tau }\mathit{\theta }{\mathit{\theta }}^{\mathrm{T}}{\Sigma }_{\tau },{\tilde{\sigma }}_{\overline{\upsilon }}^2=\frac{1}{n}{\displaystyle \sum _{i=1}^n}{\left({\overline{Y}}_{i·}-{\overline{\mathit{z}}}_{i·}^{\mathrm{T}}\mathit{\theta }\right)}^2.\end{array}$$

(9)

Thus, when $T_R{|}_{\mathit{\theta }={\mathit{\theta }}^{*}}>{\chi }_{p+1,\alpha }^2$ holds, the null hypothesis $H_0$ in (2) should be rejected, where $P({\chi }_{p+1}^2>{\chi }_{p+1,\alpha }^2)=\alpha$.

Remark 2.

Theorem 2 gives a tool to assess whether the null hypothesis in (2) holds; the test method is viewed as the existing uniformly robust (RT) test method of Huwang et al. [17]. In this method, $\tilde{\mathbf{S}}$ is a “pseudo” estimator of $\mathbf{S}$ because the parameter $\mathit{\theta }$ therein is not estimated.

2.2. The Parametric Bootstrap Approach

There are unknown parameters $\mathit{\theta }$ in $T_0$ and $T_R$. In such cases, the test statistic can be defined as

$$\begin{array}{c}\hfill T=n{(\widehat{\mathit{\theta }}-{\mathit{\theta }}^{*})}^{\mathrm{T}}{\mathbf{W}}^{\mathrm{T}}{\widehat{\mathbf{S}}}^{-1}\mathbf{W}(\widehat{\mathit{\theta }}-{\mathit{\theta }}^{*}),\end{array}$$

(10)

where

$$\begin{array}{c}\hfill \widehat{\mathbf{S}}={\widehat{\sigma }}_{\overline{\upsilon }}^2{\mathbf{M}}_{\overline{z}\overline{z}}+\frac{1}{m^2}{\Sigma }_{\tau }\widehat{\mathit{\theta }}{\widehat{\mathit{\theta }}}^{\mathrm{T}}{\Sigma }_{\tau },{\widehat{\sigma }}_{\overline{\upsilon }}^2=\frac{1}{n}{\displaystyle \sum _{i=1}^n}{\left({\overline{Y}}_{i·}-{\overline{\mathit{z}}}_{i·}^{\mathrm{T}}\widehat{\mathit{\theta }}\right)}^2.\end{array}$$

Unfortunately, the distribution of T in (10) is not known. To solve this problem, a PB test method can be developed similar to that in Yue et al. [16], Xu et al. [20] and Xu [21], where we can generate samples from estimated models by replacing the unknown parameter with its estimator.

Note that, when the null hypothesis $H_0:\mathit{\theta }={\mathit{\theta }}^{*}$ holds, we have ${\overline{Y}}_{i·}\sim N({\overline{\mathit{z}}}_{i·}^{\mathrm{T}}{\mathit{\theta }}^{*},{\sigma }_{\overline{\upsilon }}^2)$. The PB pivot variable can be developed as follows, ${\overline{Y}}_{B_{i·}}\sim N({\overline{\mathit{z}}}_{i·}^{\mathrm{T}}{\mathit{\theta }}^{*},{\widehat{\sigma }}_{{\overline{\upsilon }}_0}^2)$, and ${\widehat{\sigma }}_{{\overline{\upsilon }}_0}^2$ is the observed value of ${\widehat{\sigma }}_{\overline{\upsilon }}^2$. Then,

$${\widehat{\mathit{\theta }}}_B={\left({\mathbf{M}}_{\overline{z}\overline{z}}-\frac{1}{m}{\Sigma }_{\tau }\right)}^{-1}{\mathit{M}}_{{\overline{Y}}_B\overline{z}},{\mathit{M}}_{{\overline{Y}}_B\overline{z}}=\frac{1}{n}{\displaystyle \sum _{i=1}^n}{\overline{Y}}_{B_{i·}}{\overline{\mathit{z}}}_{i·},$$

(11)

$${\widehat{\mathbf{S}}}_B={\widehat{\sigma }}_{{\overline{\upsilon }}_{{}_B}}^2{\mathbf{M}}_{\overline{z}\overline{z}}+\frac{1}{m^2}{\Sigma }_{\tau }{\widehat{\mathit{\theta }}}_B{\widehat{\mathit{\theta }}}_B^{\mathrm{T}}{\Sigma }_{\tau },{\widehat{\sigma }}_{{\overline{\upsilon }}_{{}_B}}^2={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n}{\left({\overline{Y}}_{B_{i·}}-{\overline{\mathit{z}}}_{i·}^{\mathrm{T}}{\widehat{\mathit{\theta }}}_B\right)}^2.$$

(12)

According to (10), the PB pivot variable can be constructed as

$$\begin{array}{c}\hfill T_B=n{({\widehat{\mathit{\theta }}}_B-{\mathit{\theta }}^{*})}^{\mathrm{T}}{\mathbf{W}}^{\mathrm{T}}{\widehat{\mathbf{S}}}_B^{-1}\mathbf{W}({\widehat{\mathit{\theta }}}_B-{\mathit{\theta }}^{*}),\end{array}$$

(13)

where the PB test rejects $H_0$ in (2) whenever

$$\begin{array}{c}\hfill P(T_B>t_{obs})<\alpha ,\end{array}$$

(14)

where $t_{obs}$ is an observed T-value of (10), the significance level $\alpha$ is always given. The probability in (14) can be obtained using the following Algorithm.

3. Simulation Studies

In this section, the performance of our proposed PB test is studied using the Type I error rates and powers. The Type I error rates of the PB test are evaluated with following steps. Firstly, we evaluate the p-values by using Algorithm 1 based on generated samples $({\overline{Y}}_{1·},\dots ,{\overline{Y}}_{n·})$ and $({\overline{\mathit{z}}}_{1·},\dots ,{\overline{\mathit{z}}}_{n·})$, the above step is repeated M times and the M p-values are kept; then, the Type I error rate can be obtained by computing the ratio of M p-values less than $\alpha$. Here, M and r are taken to be 2500 and 5000, respectively. The RT test method is evaluated by (8) in Section 2, which is similar to that in ref. [17]. Each test method is carried out at a given significance level $\alpha =0.05$.

Algorithm 1 Given samples $({\overline{Y}}_{1·},\dots ,{\overline{Y}}_{n·})$ and $({\overline{\mathit{z}}}_{1·},\dots ,{\overline{\mathit{z}}}_{n·})$,

Step 1. Obtain estimations $\widehat{\mathit{\theta }}$ and ${\widehat{\sigma }}_{\overline{\upsilon }}^2$, then obtain the T-value in (10) and denote it $t_{obs}$;      Step 2. For $i=1,\dots ,n$, generate bootstrap samples ${\overline{Y}}_{B_{i·}}\sim N({\overline{\mathit{z}}}_{i·}^{\mathrm{T}}{\mathit{\theta }}^{*},{\widehat{\sigma }}_{\overline{\upsilon }}^2)$, and provide ${\widehat{\mathit{\theta }}}_B$ and ${\widehat{\mathbf{S}}}_B$ based on (11) and (12), respectively;      Step 3. Using (13) to obtain the value of PB pivot variable $T_B$. Let $R=1$ if $T_B>t_{obs}$, $R=0$ if $T_B<t_{obs}$;      Step 4. Repeat r times for Steps 2–3, keep the r values of R and marked as $R_1,\dots ,R_r$;      Step 5. Using (14), the p-value can be evaluated as $r^{-1}{\sum }_{t=1}^r{R}_t$.

In the simulation, the data are generated from the model

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{Y}_{ij}={\alpha }_0+{\mathit{x}}_{ij}^{\mathrm{T}}\mathit{\beta }+{\mu }_i+{\epsilon }_{ij},\hfill \\ {\mathit{\omega }}_{ij}={\mathit{x}}_{ij}+{\mathbf{\delta }}_{ij},i=1,\dots ,n,j=1,\dots ,m,\hfill \end{array}\right.\end{array}$$

where the sample size $n=10,20,30,50,$ and $100$; $m=10$; $p=2$; ${\alpha }_0=1.0$; $\mathit{\beta }={(2.0,1.0)}^{\mathrm{T}}$; ${\mathit{x}}_{ij}\sim N({\mathbf{0}}_p,{\mathbf{I}}_p)$; ${\mu }_i\sim N(0,{\sigma }_{\mu }^2)$; ${\epsilon }_{ij}\sim N(0,{\sigma }_{\epsilon }^2)$; ${\mathbf{\delta }}_{ij}\sim N({\mathbf{0}}_p,{\sigma }_{\delta }^2{\mathbf{I}}_p)$; and ${\sigma }_{\epsilon }^2=1.0$. Six different variances of individual effect, ${\sigma }_{\mu }^2=0.25,0.5,1.0,2.0,4.0,$ and $8.0$, are chosen to compare the estimated Type I error rates of our proposed PB and the existing RT tests. To study the effect of measurement errors on the performance of the PB test method, four different standard deviations of measurement error ${\sigma }_{\delta }$ are adopted as 0.1, $0.2$, $0.3$ and $0.5$ in our simulation studies.

In Table 1, we calculate the Type I error rates of the proposed PB and existing RT tests for different sample sizes n, variances in individual effect ${\sigma }_{\mu }^2$ and standard deviations of measurement error ${\sigma }_{\delta }$.

Table 1. Evaluated Type I error rates of PB and RT test methods.

n	${\mathit{\sigma }}_{\mathit{\mu }}^2$	${\mathit{\sigma }}_{\mathit{\delta }}=0.1$		${\mathit{\sigma }}_{\mathit{\delta }}=0.2$		${\mathit{\sigma }}_{\mathit{\delta }}=0.3$		${\mathit{\sigma }}_{\mathit{\delta }}=0.5$
		PB	RT	PB	RT	PB	RT	PB	RT
10	0.25	0.0464	0.0104	0.0488	0.0140	0.0504	0.0188	0.0520	0.0392
	0.5	0.0460	0.0086	0.0476	0.0108	0.0464	0.0162	0.0524	0.0338
	1.0	0.0472	0.0084	0.0488	0.0102	0.0460	0.0132	0.0524	0.0244
	2.0	0.0468	0.0084	0.0476	0.0098	0.0524	0.0120	0.0504	0.0198
	4.0	0.0468	0.0084	0.0472	0.0098	0.0516	0.0120	0.0480	0.0160
	8.0	0.0472	0.0080	0.0468	0.0092	0.0496	0.0108	0.0496	0.0150
20	0.25	0.0464	0.0378	0.0476	0.0384	0.0484	0.0370	0.0488	0.0432
	0.5	0.0460	0.0400	0.0492	0.0388	0.0488	0.0366	0.0480	0.0414
	1.0	0.0468	0.0374	0.0508	0.0390	0.0460	0.0386	0.0496	0.0364
	2.0	0.0472	0.0386	0.0496	0.0394	0.0484	0.0394	0.0492	0.0362
	4.0	0.0476	0.0386	0.0496	0.0386	0.0488	0.0386	0.0460	0.0356
	8.0	0.0476	0.0392	0.0488	0.0378	0.0496	0.0376	0.0472	0.0354
30	0.25	0.0504	0.0384	0.0496	0.0388	0.0468	0.0374	0.0488	0.0442
	0.5	0.0508	0.0388	0.0508	0.0388	0.0476	0.0386	0.0488	0.0446
	1.0	0.0504	0.0388	0.0512	0.0386	0.0488	0.0394	0.0480	0.0432
	2.0	0.0500	0.0388	0.0500	0.0390	0.0500	0.0392	0.0460	0.0434
	4.0	0.0500	0.0386	0.0496	0.0388	0.0504	0.0388	0.0464	0.0428
	8.0	0.0500	0.0386	0.0504	0.0392	0.0492	0.0396	0.0468	0.0414
50	0.25	0.0496	0.0464	0.0476	0.0430	0.0492	0.0418	0.0500	0.0474
	0.5	0.0488	0.0448	0.0500	0.0426	0.0468	0.0440	0.0492	0.0462
	1.0	0.0492	0.0446	0.0476	0.0438	0.0476	0.0444	0.0484	0.0448
	2.0	0.0492	0.0452	0.0488	0.0468	0.0492	0.0460	0.0500	0.0466
	4.0	0.0500	0.0466	0.0500	0.0472	0.0512	0.0470	0.0488	0.0474
	8.0	0.0504	0.0466	0.0512	0.0468	0.0508	0.0472	0.0488	0.0484

Table 1 shows that:

(1)

The Type I error rate of our proposed PB test is very close to $\alpha =0.05$, but the existing RT test is unstable. For the smaller sample size $n=10$, the RT test seems to be very conservative, but the PB test can control Type I error rates within the significance level very well.

(2)

The magnitude of variance in individual effect ${\sigma }_{\mu }^2$ has little effect on the performance of each test method (PB and RT).

(3)

When the sample sizes n and ${\sigma }_{\mu }^2$ are fixed, the Type I error rates of both tests are robust to the variation in the standard deviations of measurement error.

In Table 2, we provide the estimated powers for hypothesis testing problem $H_0:\mathit{\theta }={\mathit{\theta }}^{*}$ versus $H_1:\mathit{\theta }\ne {\mathit{\theta }}^{*}$ of the PB and RT tests under moderate and large sample sizes, where $\mathit{\theta }=({\alpha }_0,{\mathit{\beta }}^{\mathrm{T}})={({\alpha }_0,{\beta }_1,{\beta }_2)}^{\mathrm{T}}$ and ${\mathit{\theta }}^{*}={(1.0,2.0,1.0)}^{\mathrm{T}}$. To evaluate the powers of each test method, nine different values of $d={\sum }_{j=1}^3|{\mathit{\theta }}_j-{\mathit{\theta }}_j^{*}|$ are chosen, which is similar to that in Huwang et al. [17]. The null hypothesis holds when $d=0$. In the simulations of Table 2, the variance in individual effect ${\sigma }_{\mu }^2=4.0$, ${\sigma }_{\epsilon }^2=1.0$ and the standard deviation of measurement error ${\sigma }_{\delta }$ is chosen as 0.1, 0.2, 0.3 and 0.5.

Table 2. Simulated powers of the test methods when ${\sigma }_{\mu }^2=4.0$.

n	${\mathit{\sigma }}_{\mathit{\delta }}$	Methods	$\mathit{d}=0.0$	$\mathit{d}=0.2$	$\mathit{d}=0.6$	$\mathit{d}=0.8$	$\mathit{d}=1.0$	$\mathit{d}=1.2$	$\mathit{d}=1.4$	$\mathit{d}=1.6$	$\mathit{d}=1.8$
30	$0.1$	PB	0.0500	0.0548	0.0616	0.0708	0.0780	0.0892	0.1000	0.1092	0.1256
		RT	0.0386	0.0418	0.0548	0.0618	0.0742	0.0858	0.1018	0.1216	0.1454
	$0.2$	PB	0.0496	0.0532	0.0628	0.0712	0.0800	0.0908	0.0988	0.1136	0.1248
		RT	0.0388	0.0418	0.0534	0.0630	0.0750	0.0862	0.1020	0.1202	0.1394
	$0.3$	PB	0.0504	0.0520	0.0624	0.0732	0.0824	0.0924	0.1020	0.1128	0.1268
		RT	0.0388	0.0410	0.0560	0.0628	0.0738	0.0864	0.1018	0.1176	0.1378
	$0.5$	PB	0.0464	0.0532	0.0652	0.0736	0.0848	0.0968	0.1112	0.1244	0.1424
		RT	0.0428	0.0456	0.0556	0.0644	0.0720	0.0824	0.0948	0.1126	0.1280
50	$0.1$	PB	0.0536	0.0540	0.0656	0.0760	0.0920	0.1124	0.1376	0.1656	0.1928
		RT	0.0466	0.0466	0.0600	0.0718	0.0886	0.1102	0.1330	0.1664	0.2036
	$0.2$	PB	0.0524	0.0548	0.0696	0.0828	0.1008	0.1212	0.1460	0.1700	0.2028
		RT	0.0472	0.0482	0.0598	0.0720	0.0884	0.1072	0.1308	0.1634	0.1970
	$0.3$	PB	0.0512	0.0568	0.0756	0.0932	0.1056	0.1284	0.1552	0.1820	0.2096
		RT	0.0470	0.0492	0.0594	0.0736	0.0868	0.1040	0.1268	0.1548	0.1864
	$0.5$	PB	0.0532	0.0576	0.0804	0.0976	0.1156	0.1372	0.1652	0.1936	0.2284
		RT	0.0474	0.0498	0.0608	0.0706	0.0800	0.0954	0.1188	0.1424	0.1622
100	$0.1$	PB	0.0504	0.0512	0.0752	0.1020	0.1300	0.1740	0.2192	0.2768	0.3528
		RT	0.0466	0.0472	0.0724	0.1032	0.1338	0.1754	0.2304	0.2916	0.3630
	$0.2$	PB	0.0504	0.0512	0.0848	0.1100	0.1452	0.1812	0.2360	0.2952	0.3732
		RT	0.0460	0.0484	0.0728	0.1014	0.1296	0.1704	0.2232	0.2848	0.3498
	$0.3$	PB	0.0464	0.0572	0.0948	0.1248	0.1544	0.2036	0.2564	0.3208	0.3912
		RT	0.0444	0.0478	0.0738	0.0960	0.1254	0.1604	0.2112	0.2700	0.3346
	$0.5$	PB	0.0488	0.0668	0.1108	0.1376	0.1800	0.2288	0.2836	0.3496	0.4120
		RT	0.0450	0.0474	0.0706	0.0862	0.1104	0.1432	0.1842	0.2352	0.2960

From Table 2, we can see that

(1)

When the sample is large ($n=50,100$), both the PB and RT tests can control the Type I error rates.

(2)

When the sample sizes n and ${\sigma }_{\delta }$ are fixed, the power increases as the d-value increases, as expected. In most scenarios, the powers of our proposed PB test are slightly higher than those of the RT test.

(3)

For the same standard deviation of measurement error ${\sigma }_{\delta }$, the estimated power increases as the sample size increases.

Generally speaking, the proposed PB test is feasible.

4. A Real Data Analysis

In this section, our proposed PB method is applied to analyze Grunfeld data. The dataset was available in the R package “plm” and studied in Baltagi [1] and Grunfeld [24], and was a balanced panel dataset of 10 large US manufacturing firm observations from 1935 to 1954. For this dataset, we considered the following one-way error component regression model,

$$Y_{ij}=x_{ij1}{\beta }_1+x_{ij2}{\beta }_2+{\mu }_i+{\epsilon }_{ij},i=1,\dots ,n,j=1,\dots ,m,$$

(15)

where $n=10$, $m=20$, $Y_{ij}$ is the j-year gross investment of firm i, $x_{ij1}$ denotes firm value, $x_{ij2}$ is the capital stock of plant and equipment, ${\mu }_i$ represents the individual firm effect, ${\epsilon }_{ij}$ is the random error and we assume the observations of firm and capital stock cannot be observed accurately. That is, $x_{ijl}$ has additive errors ${\delta }_{ijl}$ (${\omega }_{ijl}=x_{ijl}+{\delta }_{ijl}$) for $l=1,2$, ${\delta }_{ijl}\sim N(0,{\sigma }_{{\delta }_l}^2)$. In the absence of validation or replication data, we conducted a sensitivity analysis by taking ${\sigma }_{{\delta }_l}^2=0.25\times \mathrm{Var}\left({\mathit{X}}_l\right)$ and ${\mathit{X}}_l={(x_{11l},\dots ,x_{1ml},\dots ,x_{nml})}^{\mathrm{T}}$, which is similar to that in Lin and Carroll [2], Yue et al. [13] and Yang et al. [25]. The interested hypothesis testing problem is $H_0:\mathit{\theta }={\mathit{\theta }}^{*}$ vs. $H_1:\mathit{\theta }\ne {\mathit{\theta }}^{*}$, where $\mathit{\theta }={({\beta }_1,{\beta }_2)}^{\mathrm{T}}$. To compare the performance of the proposed PB and existing RT tests, we chose three different values of ${\mathit{\theta }}^{*}$, and evaluate the p-values of the PB and RT tests in Table 3. In addition, the approximate 95% confidence intervals for ${\beta }_1$ and ${\beta }_2$ were also calculated as $(0.0891,0.1880)$ and $(-0.2501,0.2410)$, respectively.

Table 3. Estimated p-values for the PB and RT tests.

Methods	${\mathit{\theta }}^{*}$
	${(\mathbf{0.06}, \mathbf{0.25})}^{\mathbf{T}}$	${(\mathbf{0.09}, \mathbf{0.24})}^{\mathbf{T}}$	${(\mathbf{0.11}, \mathbf{0.11})}^{\mathbf{T}}$
PB	0.0494	0.2530	0.6050
RT	0.0725	0.2311	0.5593

From Table 3, we can see that the PB test rejects the null hypothesis when ${\mathit{\theta }}^{*}={(0.06,0.25)}^{\mathrm{T}}$ and $\alpha =0.05$, whereas the RT is in favor of the null hypothesis $H_0$. According to the 95% confidence intervals for ${\beta }_1$ and ${\beta }_2$, we should reject the null hypothesis $H_0$. That is, the PB test can make a right choice, but the RT test cannot. When ${\mathit{\theta }}^{*}={(0.09,0.24)}^{\mathrm{T}}$ and ${\mathit{\theta }}^{*}={(0.11,0.11)}^{\mathrm{T}}$, both the PB and RT tests are in favor of the null hypothesis, which is consistent with the results from the 95% confidence intervals for ${\beta }_1$ and ${\beta }_2$. Therefore, the proposed PB test is more informative.

5. Conclusions

In this paper, we consider a one-way error component regression model with measurement errors. When we ignore the measurement errors and simply adopt the observed surrogate variables to obtain the estimation of an unknown parameter vector, an inefficient testing method may be developed. To solve this problem, we combine the biased-corrected estimator of the unknown parameter and propose a parametric bootstrap (PB) method for the hypothesis testing problem of the unknown vector of the coefficient parameter. Through some simulation studies and a real data analysis, we find the proposed PB test can control the Type I error rates within the significance level very well, but the existing uniformly robust (RT) test is conservative when the sample size is small. When the sample size is larger, both test methods have similar performance. In a word, the proposed PB test is feasible.