Inferences on the regression coefficients in panel data models: parametric bootstrap approach

Abstract

This article presents a parametric bootstrap approach to inference on the regression coefficients in panel data models. We aim to propose a method that is easily applicable for implement hypothesis testing and construct confidence interval of the regression coefficients vector of balanced and unbalanced panel data models. We show the results of our simulation study to compare of our parametric bootstrap approach with other approaches and approximated methods based on a Monte Carlo simulation study.

Introduction

Panel data are the combination of observations on a cross section of individuals, cities, factories, etc., over many time periods. Panel data have been applied in economics extensively. The Panel Study of Income Dynamics (PSID) from the Survey Research Center at the University of Michigan and the Gasoline demand panel of annual observations across 18 Organisation for Economic Co-operation and Development (OECD) countries, covering the period 1960–1978, are two famous examples of panel data. Analysis of panel data models has worked in statistics and econometrics by many researchers ([ 1 , 5 , 14 ], and references therein). Baltagi [ 2 ] presents an overview for panel data models excellently. Precious inferences in panel data are difficult when these models have nuisance parameters. Zhao [ 23 ] suggested generalized p value inferences in panel data models of generalized p values. When nuisance parameters are present in models, generalized p values are effective to solve the testing problems [ 8 , 11 , 16 , 17 , 18 – 19 , 24 ]. Parametric bootstrap approaches are used as another method for inferences in panel data models, when unknown parameters are present. Xu et al. [ 21 ] and [ 22 ] provided parametric bootstrap inferences for parameters of the linear combination of regression coefficients in balanced and unbalanced panel data models, respectively.

In this article, we aim to propose a method that is easily applicable for implement hypothesis testing and construct confidence interval of the regression coefficient vector of balanced and unbalanced panel data models. Our procedure is based on a new parametric bootstrap pivot variable . The performance of our PB method is compared with generalized p value approaches introduced by [ 23 ]. The numerical results in section “Simulation study” show that in terms of the type I error rate and power, the performance of our method is better than generalized p value (GPV) inferences and approximate (AP) method.

The rest of this paper is organized as follows. Our PB approaches for hypothesis testing and constructing confidence region about the regression coefficients vector are presented for the balanced and unbalanced panel data models in section “PB inferences for the regression coefficients”. In section “Simulation study”, the proposed PB methods are evaluated in terms of type I error rates and powers. The suggested PB approaches are illustrated with a real data example in section “Example”. The some conclusions are assumed in section “Conclusions”.

PB inferences for the regression coefficients

Balanced panel data models

Panel data regression models show the behaviour of several explanatory variables on the response variable between N individuals over T time periods. A panel data model is

Yit=α+xit′β+uit,
with
uit=μi+νit,i=1,2,…,N;t=1,2,…,T,
where Yit and xit are the response value and K explanatory variables on the i th individual for the t th time period, respectively. uit is the regression disturbance, μi denotes the unobservable individual specific effect and νit denotes the remainder disturbance. Usually, in the random effects model, we suppose that μi∼N(0,σμ2) and νit∼N(0,σν2) vary independently. α is the intercept and β is K×1 vector of unknown coefficients. Let yit denote the observed values of Yit for i=1,2,…,N;t=1,2,…,T .

Equation ( 2.1 ) can also be expressed as matrix notations,

Y=α1NT+Xβ+Zμμ+ν=Zδ+u,
where Y=(Y11,…,Y1T,…,YN1,…,YNT)′ , X is a NT×K matrix, Z=[1NT,X] , δ=(α,β′)′ is a unknown regression coefficients vector, Zμ=IN⊗1T , μ=(μ1,μ2,…,μN)′ , ν=(ν11,…,ν1T,…,νN1,…,νNT)′ , u=Zμμ+ν , IN is an identity matrix of order N , 1T denotes the T×1 vector whose elements are all ones and ⊗ denotes Kronecker product.

Let JT=1T1T′ , J¯T=1TJT and ET=IT-J¯T . Then, the covariance matrix of Y is

Cov(Y)=Σ=σμ2(IN⊗JT)+σν2(IN⊗IT)=σ12P+σν2Q,
where σ12=Tσμ2+σν2 , P=IN⊗J¯T and Q=IN⊗ET . ( 2.3 ) is the spectral decomposition representation of Σ , which is the main key to the following inferences. Both P and Q are symmetric and idempotent matrices, such that PQ=QP=0 . [ 15 ] using the properties of P and Q show that
Σr=σ12rP+σν2rQ,
where r is an arbitrary scalar. Hence,
Σ-1=σ12-1P+σν2-1Q.
The generalized least squares estimator (GLSE) of δ is obtained by [ 12 ] as
δ^σ12,σν2,Y=(Z′Σ-1Z)-1Z′Σ-1Y.
It is easy to verify
δ^σ12,σν2,Y∼N(δ,(Z′Σ-1Z)-1).
To attain the estimators of σν2 and σ12 , transformed model ( 2.2 ) is as follows:
QYPY=QZPZδ+QuPu=QXβPZδ+QνPu.
It is easy to show that QY∼N(QXβ,σν2Q) and PY∼N(PZδ,σ12P) , such that PY and QY are mutually independent, since
CovQYPY=σν2Q00σ12P.
Therefore, we can define
S12=Y′PY-Y′PZ(Z′PZ)-1Z′PY,Sν2=Y′QY-Y′QX(X′QX)-1X′QY,
such that S12 and Sν2 are independently distributed as
S12σ12∼χ(N-K-1)2,Sν2σν2∼χ(N(T-1)-K)2,
where χ(m)2 denotes a central Chi-square random variable with m degree of freedom. Then, the unbiased estimators of σ12 , σν2 and σμ2 can be given
σ~12=S12N-K-1,σ~ν2=Sν2N(T-1)-K,andσ~μ2=1Tσ~12-σ~ν2.
According to ( 2.4 ) and ( 2.5 ), the natural estimators of Σ and Σ-1 are, respectively,
Σ~=σ~12P+σ~ν2QandΣ~-1=σ~12-1P+σ~ν2-1Q.
When Σ is known, a natural pivotal quantity for inferences on δ is given by
H∗=(δ^-δ)′(Z′Σ-1Z)(δ^-δ)∼χ(K+1)2.
Then,
Rδ=δ|(δ^0-δ)′(Z′Σ-1Z)(δ^0-δ)<χ(γ,K+1)2
is an exact 100(1-γ)% confidence region for δ , where δ^0 is the observed value of δ^ by replacing Y in ( 2.6 ) by y and χ(γ,m)2 stands for the lower (1-γ) th quantile of the central Chi-square distribution with m degree of freedom.

The values of σν2 , σμ2 and then Σ are usually unknown in practice. Therefore, we propose to replace σν2 and σμ2 with their unbiased estimators, which leads to

H=(δ~-δ)′(Z′Σ~-1Z)(δ~-δ),
where δ~=(Z′Σ~-1Z)-1Z′Σ~-1Y is a feasible GLSE.

We can construct a approximated (AP) confidence region as

RδAP={δ|(δ~0-δ)′(Z′Σ~-1Z)(δ~0-δ)<χ(γ,K+1)2},
where δ~0 is a observed value of δ~ . This approximated method is applicable while the sample size is large. Since the distribution of H is unknown and approximated method has poor performance (based on simulation results), we use a parametric bootstrap approach to approximate distribution of H .

Let sν2 and s12 be the observed values of Sν2 and S12 in ( 2.9 ), respectively. For a given (δ~0,s12,sν2) , let YB∼N(Zδ~0,Σ~0) , where Σ~0 is the observed value of Σ~ . Then, the PB pivot variable based on the random quantity ( 2.15 ) is

HB=(δ~B-δ~0)′(Z′Σ~B-1Z)(δ~B-δ~0),
where
δ~B=(Z′Σ~B-1Z)-1Z′Σ~B-1YB,Σ~B=σ~1B2P+σ~νB2Q,
σ~1B2=S1B2N-K-1,σ~νB2=SνB2N(T-1)-K,
S1B2=YB′PYB-YB′PZ(Z′PZ)-1Z′PYB
and
SνB2=YB′QYB-YB′QX(X′QX)-1X′QYB.
Distribution of HB for a given (δ~0,s12,sν2) in ( 2.16 ) does not depend on any unknown parameters. Therefore, we can construct a PB confidence region for the parameter δ based on the distribution of HB, where HγB denotes the lower (1-γ) th quantile of HB . Then, we propose a 100(1-γ)% confidence region for δ by
RδB=δ|(δ~0-δ)′(Z′Σ~0-1Z)(δ~0-δ)<HγB.
Next, we consider the problem of hypothesis testing about δ as
H0:δ=δ∗vs.H1:δ≠δ∗,
where δ∗=(α∗,β1∗,…,βK∗)′ is a pre-specified values vector. Our proposed test statistic is
D=(δ~-δ∗)′(Z′Σ~-1Z)(δ~-δ∗).
The null hypothesis ( 2.18 ) is rejected at level γ when D0>HγBH_\gamma ^B$$\end{document}]]> , where D0 is the observed value of D . Also, it can be defined a PB p value as
p=P(HB>D0).D_0). \end{aligned}$$\end{document}]]>
Therefore, H0 is rejected at level γ when p<γ .

Unbalanced panel data Models

The unbalanced panel data model is given by:

Yit=α+xit′β+uit,
with
uit=μi+νit,i=1,2,…,N;t=1,2,…,Ti,
where Yit , xit and so on are similar to the balanced case which is defined, with the difference that in unbalanced case, the time period for each i th cross section is different and equal to the time Ti . In matrix notations, equation ( 2.21 ) can also be expressed as
Y=α1n+Xβ+Zμμ+ν=Zδ+u,
where n=Σi=1NTi,Y=(Y11,…,Y1T1,…,YN1,…,YNTN)′ , X is a n×K matrix, Z=[1n,X] , δ=(α,β′)′ , Zμ=diag(1T1,…,1TN) , μ=(μ1,μ2,…,μN)′ , ν=(ν11,…,ν1T1,…,νN1,…,νNTN)′ and u=Zμμ+ν .

JTi=1Ti1Ti′ , J¯Ti=1TiJTi and ETi=ITi-J¯Ti , for i=1,…,N . Then, the covariance matrix of Y is

Cov(Y)=Σ=σμ2diag(JT1,…,JTN)+σν2Indiag[(T1σμ2+σν2)J¯T1,…,(TNσμ2+σν2)J¯TN]+σν2Q,
where Q=diag(ET1,…,ETN) . It is established that
Σ-1=diag(T1σμ2+σν2)-1J¯T1,…,(TNσμ2+σν2)-1J¯TN+(σν2)-1Q.
Then, the generalized least square estimator (GLSE) of δ is
δ^(σ12,σν2,Y)=(Z′Σ-1Z)-1Z′Σ-1Y.
Also, the GLSE of δ is distributed as
δ^(σ12,σν2,Y)∼N(δ,(Z′Σ-1Z)-1).
Similar to the balanced case, we consider the following two quadratic forms defining the Between and Within residuals sums of squares to obtain the estimators of σμ2 and σν2 .
S12=Y′PY-Y′PZ(Z′PZ)-1Z′PY,S22=Y′QY-Y′QX(X′QX)-1X′QY,
where P=diag(J¯T1,...,J¯TN) and S22/σν2∼χ(n-N-K)2 . According to [ 12 ], the unbiased estimators of σν2 and σμ2 can be given as
σ~ν2=S22n-N-K,σ~μ2=S12-(N-K-1)σ~ν2n-tr((Z′PZ)-1Z′ZμZμ′Z).
Therefore, the natural estimators of Σ and Σ-1 are
Σ~=diag[(T1σ~μ2+σ~ν2)J¯T1,…,(TNσ~μ2+σ~ν2)J¯TN]+σ~ν2Q,Σ~-1=diag[(T1σ~μ2+σ~ν2)-1J¯T1,…,(TNσ~μ2+σ~ν2)-1J¯TN]+(σ~ν2)-1Q.
To construct a confidence region for δ in this case, we propose to use a similar random quantity H in ( 2.15 ) and PB approach to approximated its distribution.

Simulation study

In this section, we present the results of our simulation study to compare the size and powers of our PB approach with generalized p values by [ 23 ] and approximated methods based on a Monte Carlo simulation study. we use the abbreviation PB, GPV and AP to refer these three methods. At first, we briefly review the GPV method.

[ 23 ] only proposed a generalized p value method for testing H0:δ=δ∗ v.s H1:δ≠δ∗ in balanced panel data state. He proposed the generalized F test for testing the null hypothesis as

T~T(Y;y,σ12,σν2,δ)=δ^(σ12,σν2,Y)-δ∗)′Sδ-2(σ12,σν2)(δ^(σ12,σν2,Y)-δ∗δ^(σ12ss1SS1,σν2ssνSSν,y)-δ∗)′Sδ-2(σ12ss1SS1,σν2ssνSSν)(δ^(σ12ss1SS1,σν2ssνSSν,y)-δ∗.
Subsequently, the generalized p value can be computed as
p=PTT≥1∣H0)=P(χ2(δ^(ss1U,ssνV,y)-δ∗)′Sδ-2(ss1U,ssνV)(δ^(ss1U,ssνV,y)-δ∗)≥1,
where Sδ2(σ12,σν2)=(Z′Σ-1Z)-1,U∼χ(N-K-1)2,V∼χ(N(T-1)-K)2,χ2∼χ(K+1)2 and χ2,U,V are mutually independent.

Algorithm : We use the following steps to estimate powers of the PB and GPV methods.

1. For a given ( N ,  T ) and (Z,δ,σμ2,σν2) , generate y and compute s12,sν2,Σ~0 , δ~0 and observed value of H from ( 2.15 ), i.e. h0 , respectively.

2. Generate YB∼N(Zδ~0,Σ~0) , U∼χ(N-K-1)2,V∼χ(N(T-1)-K)2,χ2∼χ(K+1)2 .

3. Repeat step 2 many times ( n=5000 ) to obtain values of H1B ,..., HnB and TT1 ,... TTn and compute the estimations of the p values of PB and GPV methods.

4. Repeat steps ( 1 ) to ( 3 ) for m=5000 times to obtain estimations of the two test powers.

For power estimation of the AP method, we compute the fraction of times that the value of D0 is exceed χ(γ,K+1)2.

The results of simulation for the different values of N,T,σν2,σμ2 are shown in Table 1 . Also, we take δ∗ to be equal to (2, 3, 1, 5) and δ be various values of vectors. Notice that, in this simulation, we have used the three columns of the panel data as reported in Table 2 instead of the matrix X . That is, (lnY/N,lnPMG/PGDP,lnCar/N) , where we clarified this example in section 5. The first column of Table 1 shows estimated type I error rate (actually size) of the tests and other three columns show estimated powers. We consider the following reasonable criterion for comparing the methods: firstly, a method is preferred to the other methods when its estimated size is not significantly different than 0.05. We refer to such a method as a reliable method. Secondly, the candidate for the best method must have the largest power among reliable methods, see [ 7 , 9 , 10 , 20 ] and [ 6 ]. In addition, using the central limit theorem, 98% confidence intervals around estimates between 0.0428 and 0.0572 cover the nominal level 0.05. In other words, if the estimated size of a test is less than or greater than that of these bounds, we can conclude that that test is conservative or liberal, respectively. In Table 1 , the estimated sizes in boldface show that they are significantly less or greater than 0.05.

Table 1

Simulated powers of the GPV, PB and AP tests at 5% nominal level

(N, T)

(σμ2,σν2)

δ

Tests

(2,3,1,5)

(2.1,3.1,1.1,5.1)

(4,3,1,5)

(2,3.1,1,5.1)

(2,3,1.5,5.1)

(10, 6)

(0.01, 1)

GPV

0.0628

1.0000

1.0000

1.0000

1.0000

PB

0.0458

1.0000

1.0000

1.0000

1.0000

AP

0.1448

1.0000

1.0000

1.0000

1.0000

(1, 1)

GPV

0.0718

0.8582

0.9570

0.8784

0.8194

PB

0.0446

0.8700

0.9754

0.8956

0.7896

AP

0.1308

0.9530

0.9942

0.9654

0.9114

(10, 1)

GPV

0.0638

0.2612

0.2622

0.2578

0.3680

PB

0.0508

0.2388

0.2870

0.2416

0.3006

AP

0.0972

0.3654

0.4068

0.3550

0.4456

(100, 1)

GPV

0.0576

0.1322

0.0814

0.1336

0.2556

PB

0.0466

0.1074

0.0756

0.1078

0.2483

AP

0.0912

0.1788

0.1280

0.1782

0.3312

(12, 5)

(0.01, 1)

GPV

0.0518

1.0000

1.0000

1.0000

1.0000

PB

0.0486

1.0000

1.0000

1.0000

1.0000

AP

0.1408

1.0000

1.0000

1.0000

1.0000

(1, 1)

GPV

0.0494

0.8594

0.9796

0.8850

0.6903

PB

0.0496

0.8484

0.9760

0.8730

0.6812

AP

0.1300

0.9620

0.9964

0.9732

0.8734

(10, 1)

GPV

0.0628

0.1826

0.2724

0.1888

0.1386

PB

0.0508

0.1632

0.2362

0.1596

0.1200

AP

0.1352

0.3320

0.4498

0.3354

0.2662

(100, 1)

GPV

0.0772

0.0880

0.0654

0.0630

0.0710

PB

0.0547

0.0644

0.0484

0.0446

0.0528

AP

0.1318

0.1466

0.1106

0.1138

0.1286

(20, 3)

(0.01, 1)

GPV

0.0588

1.0000

1.0000

1.0000

1.0000

PB

0.0492

1.0000

1.0000

1.0000

1.0000

AP

0.0964

1.0000

1.0000

1.0000

1.0000

(1, 1)

GPV

0.0616

0.9948

1.0000

0.9968

0.9726

PB

0.0528

0.9960

1.0000

0.9968

0.9698

AP

0.0952

0.9988

1.0000

0.9988

0.9840

(10, 1)

GPV

0.0606

0.3576

0.5180

0.3628

0.3818

PB

0.0516

0.3456

0.5166

0.3558

0.3548

AP

0.0828

0.4302

0.6010

0.4446

0.4446

(100, 1)

GPV

0.0510

0.0968

0.0950

0.0906

0.2002

PB

0.0456

0.0912

0.0900

0.0836

0.1906

AP

0.0704

0.1284

0.1266

0.1190

0.2452

Table 2

Data of motor gasoline consumption

Country

Year

lnGas/Car

lnY/N

lnPMG/PGDP

lnCar/N

Austria

1960

4.1732

-6.4743

-0.3345

-9.7668

1961

4.1010

-6.4260

-0.3513

-9.6086

1962

4.0732

-6.4073

-0.3795

-9.4573

1963

4.0595

-6.3707

-0.4143

-9.3432

1964

4.0377

-6.3222

-0.4453

-9.2377

Belgium

1960

4.1640

-6.2151

-0.1657

-9.4055

1961

4.1244

-6.1768

-0.1717

-9.3031

1962

4.0760

-6.1296

-0.2223

-9.2181

1963

4.0013

-6.0940

-0.2505

-9.1149

1964

3.9944

-6.0365

-0.2759

-9.0055

Canada

1960

4.8552

-5.8897

-0.9721

-8.3789

1961

4.8266

-5.8843

-0.9723

-8.3467

1962

4.8505

-5.8446

-0.9786

-8.3205

1963

4.8381

-5.7924

-1.0190

-8.2694

1964

4.8398

-5.7601

-1.0029

-8.2524

Denmark

1960

4.5020

-6.0617

-0.1957

-9.3262

1961

4.4828

-6.0009

-0.2536

-9.1931

1962

4.3854

-5.9875

-0.2188

-9.0473

1963

4.3540

-5.9731

-0.2480

-8.9528

1964

4.3264

-5.8947

-0.3065

-8.8526

France

1960

3.9077

-6.2644

-0.0196

-9.1457

1961

3.8856

-6.2209

-0.0239

-9.0443

1962

3.8237

-6.1736

-0.0689

-8.9301

1963

3.7890

-6.1371

-0.1379

-8.8186

1964

3.7671

-6.0872

-0.1978

-8.7110

Germany

1960

3.9170

-6.1598

-0.1859

-9.3425

1961

3.8853

-6.1209

-0.2310

-9.1838

1962

3.8715

-6.0943

-0.3438

-9.0373

1963

3.8488

-6.0684

-0.3746

-8.9136

1964

3.8690

-6.0134

-0.3997

-8.8110

Spain

1960

4.7494

-6.1661

1.1253

-11.5884

1961

4.5892

-6.0578

1.1096

-11.3840

1962

4.4291

-5.9805

1.0570

-11.1578

1963

4.3465

-5.9051

0.9768

-10.9845

1964

4.3006

-5.8585

0.9153

-10.7879

Sweden

1960

4.0630

-8.0725

-2.5204

-8.7427

1961

4.0619

-8.0196

-2.5715

-8.6599

1962

4.0064

-7.9972

-2.5345

-8.5774

1963

4.0028

-7.9667

-2.6051

-8.4943

1964

4.0249

-7.8976

-2.6580

-8.4335

Switzer

1960

4.3976

-6.1561

-0.8232

-9.2624

1961

4.4413

-6.1116

-0.8656

-9.1582

1962

4.2871

-6.0930

-0.8222

-9.0461

1963

4.3125

-6.0680

-0.8601

-8.9508

1964

4.3134

-6.0215

-0.8677

-8.8394

Turkey

1960

6.1296

-7.8011

-0.2534

-13.4752

1961

6.1062

-7.7867

-0.3425

-13.3847

1962

6.0846

-7.8363

-0.4082

-13.2459

1963

6.0751

-7.6312

-0.2250

-13.2550

1964

6.0646

-7.6269

-0.2522

-13.2103

U.K.

1960

4.1002

-6.1868

-0.3911

-9.1176

1961

4.0886

-6.1689

-0.4519

-9.0489

1962

4.0481

-6.1667

-0.4229

-8.9669

1963

3.9853

-6.1307

-0.4634

-8.8559

1964

3.9768

-6.0864

-0.4958

-8.7498

U.S.A.

1960

4.8240

-5.6984

-1.1211

-8.0195

1961

4.7963

-5.6952

-1.1462

-7.9993

1962

4.7989

-5.6488

-1.1619

-7.9864

1963

4.7879

-5.6269

-1.1799

-7.9595

1964

4.8083

-5.5871

-1.2003

-7.9299

Taken from [ 3 ]

Note that the estimated powers vary slightly from one simulation to another [ 9 ]. Therefore, we used the well-known z test to compare powers of two methods. One can conclude that the powers of two test procedures are statistically significant at 100 α % level when |p^1-p^2|>Zα/2p^(1-p^)/5000Z_{\alpha /2}\sqrt{{\hat{p}}(1-{\hat{p}})/5000}$$\end{document}]]> , where p^=(p^1+p^2)/2 and p^1 and p^2 denote the estimated powers of the two test procedures based on 5000 samples. In the following remarks, we discuss the results of simulation.

Remark 1

In all cases that we considered here, the estimated sizes of our PB test vary between 0.0446 and 0.0547 which shows that our proposed test behaves like the exact test.

Remark 2

The simulated size probabilities in the GPV and AP often exceed the upper limit of this range, and then, these methods are assumed to be liberal. Therefore, in this paper, the powers of these test methods cannot be comparable with our parametric bootstrap approach.

Remark 3

To compare the estimated power, in the cases that the estimated size of GPV is close to 0.05, the PB test and GPV have not significantly different powers.

Remark 4

Overall, it seems that the proposed PB method has better performance than two other methods in terms of both controlling the type I error rates and powers.

Example

To illustrate our suggested approach to inference on the regression coefficients of a panel data, we consider the following gasoline demand equation like [ 3 ] as

lnGasCar=α+β1lnYN+β2lnPMGPGDP+β3lnCarN+u,
where Gas/Car is motor gasoline consumption per auto, Y  /  N is real per capita income, PMG/PGDP is real motor gasoline price and Car/N denotes the stock of cars per capita. This panel consists of annual observations across 18 OECD countries, covering the period 1960–1978. We take a part of the panel as well as reported in Table 3 by [ 23 ]. At first, let δ=(α,β1,β2,β3)′ , then a computed GLSE of δ is given as δ~=(0.765,0.323,-0.469,-0.578)′ and unbiased estimators of σμ2 and σν2 are 0.0552 and 0.0012, respectively. The p values are computed using simulations consisting of 20,000 runs in the three methods.

For testing H0:δ=(1.7,0.55,-0.42,-0.61)′ , the p values of the PB, GPV and AP for the regression coefficients are computed to be 0.014, 0.0006 and 0.0002, respectively. Thus, for the problem of testing regression coefficients vector, the three methods made the same decision reject the corresponding null hypothesis at the nominal level of 5%, but PB method is not reject null hypothesis at level 1%.

In this example, for obtaining PB confidence region for the regression coefficient vector δ=(α,β1,β2,β3)′ , at the confidence level of 0.95 by ( 2.17 ), we have

RδB={δ|(δ~0-δ)′(Z′Σ~0-1Z)(δ~0-δ)<H0.05B},
where,
Z′Σ~0-1Z=216.53-1375.44-117.73-2041.19-1375.449036.17703.3513488.31-117.73703.35342.23667.97-2041.1913488.31667.9720852.25,
and the lower 0.95th quantile of HB, i.e. H0.05B , using 20,000 simulations, is computed to be around 13.74.

Conclusions

In this article, we propose a parametric bootstrap method for testing hypothesis as well as constructing confidence region on the regression coefficients vector ( δ ) in panel data models in balanced and unbalanced panels. We study performance our PB method with GPV and AP methods based on simulation study in balanced state. The simulation study is compared type I error rate and power of three methods. The simulation results show close estimated size of our PB test to the nominal level (0.05), in which two other methods are often liberal (significantly greater than 0.05). However, in the cases that the estimated size of GPV is close to 0.05, the PB test and GPV have not significantly different powers. Therefore, for testing or constructing confidence region about δ we propose PB method.

Acknowledgements

We are grateful to referees for their valuable comments and suggestions.

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