ESTIMATION AND INFERENCE IN COINTEGRATED PANELS
By
Yi Li

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
Economics—Doctor of Philosophy
2017

ABSTRACT
ESTIMATION AND INFERENCE IN COINTEGRATED PANELS
By
Yi Li
This dissertation investigates parameter estimation and inference in cointegrated panel
data model. In Chapter 1, for homogeneous cointegrated panels, a simple, new estimation
method is proposed based on Vogelsang and Wagner [2014]. The estimator is labeled panel
integrated modified ordinary least squares (panel IM-OLS). Similar to panel fully modified
ordinary least squares (panel FM-OLS) and panel dynamic ordinary least squares (panel
DOLS), the panel IM-OLS estimator has a zero mean Gaussian mixture limiting distribution. However, panel IM-OLS does not require estimation of long run variance matrices and
avoids the need to choose tuning parameters such as kernel functions, bandwidths, leads and
lags. Inference based on panel IM-OLS estimates does require an estimator of a scalar long
run variance, and critical values for test statistics are obtained from traditional and fixed-b
methods. The properties of panel IM-OLS are analyzed using asymptotic theory and finite sample simulations. Panel IM-OLS performs well relative to other estimators. Chapter
2 compares asymptotic and bootstrap hypothesis tests in cointegrated panels with crosssectional uncorrelated units and endogenous regressors. All the tests are based on the panel
IM-OLS estimator from Chapter 1. The aim of using the bootstrap tests is to deal with the
size distortion problems in the finite samples of fixed-b tests. Finite sample simulations show
that the bootstrap method outperforms the asymptotic method in terms of having lower size
distortions. In general, the stationary bootstrap is better than the conditional-on-regressors
bootstrap, although in some cases, the conditional-on-regressors bootstrap has less size distortions. The improvement in size comes with only minor power losses, which can be ignored

when the sample size is large. Chapter 3 is concerned with parameter estimation and inference in a more general case than Chapter 1 with endogenous regressors and heterogeneous
long run variances in the cross section. In addition, the model allows a limited degree of
cross-sectional dependence due to a common time effect. The panel IM-OLS estimator is
provided for this less restricted model. Similar as in Chapter 1, this panel IM-OLS estimator has a zero mean Gaussian mixture limiting distribution. However, standard asymptotic
inference is infeasible due to the existence of nuisance parameters. Inference based on panel
IM-OLS relies on the stationary bootstrap. The properties of panel IM-OLS are analyzed
using the stationary bootstrap in finite sample simulations.

This dissertation is dedicated to my parents and my wife.
Thank you for nursing me with love and always believing in me.

iv

ACKNOWLEDGMENTS

The past 5 years have been a wonderful journey for me and I would never have been able
to get this far without the support from my professors, my friends and my family. It is my
pleasure to acknowledge people who have given me guidance, help and encouragement.
First and foremost I would like to gratefully and sincerely thank my PhD advisor, Professor Tim Vogelsang, for his guidance, understanding and patience. His Time Series course
is definitely my favorite class, which led my way to research field. I really appreciate all
the support and encouragement he has given me in my research. In particular, I will never
forget the effort he spent on revising this dissertation. For everything you have done for me,
Professor Vogelsang, I thank you.
I would also like to express my deepest gratitude to my committee members Professor
Peter Schmidt and Professor Jeffrey Wooldridge. Thank both of you for your guidance and
help during my graduate study, I cannot obtain my PhD without your support. It has been
such an honor for me being your student. Special thanks go to Professor Martin Wagner for
his important suggestions and remarks on my dissertation as well as the effort he spent as
my committee member. I must thank all of the professors from Economics Department who
I learned from during the last five years for showing me what is a researcher meant to be, I
will keep it in mind in my future career.
I would like to give special thanks to my cohort in Economics Department. I really
enjoyed the time we studied together. We discussed, debated, exchanged ideas and learned
from each other. It is definitely one of the best time periods in my life. My thanks also go
to my friends in Math Department. In particular, I would like to thank Xin Yang for his
expertise and patience when I had tough math problems.
v

Lastly, I would like to thank my family for all their love and the faith they put in me. For
my parents who raised me with endless love and supported me in all my pursuits. For my
grandfather who gave me all his trust and encouraged me all the time. Most importantly, I
would like to express my deepest appreciation to my wife, Yanbo Shen. Without her love,
understanding and encouragement, I would not have been able to finish this work.

vi

TABLE OF CONTENTS

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

KEY TO ABBREVIATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv
Chapter 1

Integrated modified OLS estimation and fixed-b inference for
homogeneous cointegrated panels . . . . . . . . . . . . . . . . . .
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Homogeneous cointegrated panels for benchmark estimators . . . . . . . . .
1.3 Panel integrated modified OLS . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.1 Panel IM-OLS estimator . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.2 Inference using Panel IM-OLS . . . . . . . . . . . . . . . . . . . . .
1.4 Finite sample bias and root mean squared error . . . . . . . . . . . . . . . .
1.4.1 Sample size N = 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4.2 Sample size N = 10 . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4.3 Sample size N = 25 . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4.4 Summary of finite sample bias and RMSE . . . . . . . . . . . . . . .
1.5 Finite sample performance of test statistics . . . . . . . . . . . . . . . . . . .
1.6 Summary and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
APPENDIX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 2

Hypothesis testing in cointegrated panels: Asymptotic
Bootstrap method . . . . . . . . . . . . . . . . . . . . . . . . .
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 The model, assumptions and asymptotic inference . . . . . . . . . . . .
2.2.1 The model and assumptions . . . . . . . . . . . . . . . . . . . .
2.2.2 Inference based on panel IM-OLS . . . . . . . . . . . . . . . . .
2.3 Bootstrap hypothesis tests . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Conditional-on-regressors bootstrap . . . . . . . . . . . . . . . .
2.3.2 Stationary bootstrap . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Finite sample simulations . . . . . . . . . . . . . . . . . . . . . . . . .
2.5 Summary and conclusion . . . . . . . . . . . . . . . . . . . . . . . . . .
APPENDIX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1
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and
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76
77
79
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82
88
88
90
92
97
99

Estimation and Inference for Heterogeneous Cointegrated Panels with Limited Cross Sectional Dependence . . . . . . . . . .
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Model set up and estimation . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 The model and assumptions . . . . . . . . . . . . . . . . . . . . . . .
3.2.2 Panel IM-OLS estimator . . . . . . . . . . . . . . . . . . . . . . . . .

120
121
123
123
126

Chapter 3
3.1
3.2

vii

3.3

Inference about θ . . . . . . . . . . . . . . . . .
3.3.1 Inference using panel IM-OLS . . . . . .
3.3.2 Inference using the stationary bootstrap
3.4 Finite sample simulation . . . . . . . . . . . . .
3.5 Summary and conclusions . . . . . . . . . . . .
APPENDIX . . . . . . . . . . . . . . . . . . . . . . .

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BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

viii

LIST OF TABLES

Table 1.1: Finite sample bias and RMSE of the various estimator of β1 , N = 5,
T = 50, Bartlett kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . .

34

Table 1.2: Finite sample bias and RMSE of the various estimator of β1 , N = 5,
T = 100, Bartlett kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

Table 1.3: Finite sample bias and RMSE of the various estimator of β1 , N = 10,
T = 50, Bartlett kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . .

36

Table 1.4: Finite sample bias and RMSE of the various estimator of β1 , N = 10,
T = 100, Bartlett kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

Table 1.5: Finite sample bias and RMSE of the various estimator of β1 , N = 25,
T = 50, Bartlett kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . .

38

Table 1.6: Finite sample bias and RMSE of the various estimator of β1 , N = 25,
T = 100, Bartlett kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39

Table 1.7: Empirical null rejection probabilities, 0.05 level, t-tests for h0 : β1 = 1,
N = 5, data dependent bandwidths and lag lengths. . . . . . . . . . . . .

40

Table 1.8: Empirical null rejection probabilities, 0.05 level, t-tests for h0 : β1 = 1,
N = 10, data dependent bandwidths and lag lengths. . . . . . . . . . . . .

40

Table 1.9: Empirical null rejection probabilities, 0.05 level, t-tests for h0 : β1 = 1,
N = 25, data dependent bandwidths and lag lengths. . . . . . . . . . . . .

41

Table 1.10: Empirical null rejection probabilities, 0.05 level, Wald-tests for h0 : β1 =
1, β2 = 1, N = 5, data dependent bandwidths and lag lengths. . . . . . . .

41

Table 1.11: Empirical null rejection probabilities, 0.05 level, Wald-tests for h0 : β1 =
1, β2 = 1, N = 10, data dependent bandwidths and lag lengths. . . . . . .

42

Table 1.12: Empirical null rejection probabilities, 0.05 level, Wald-tests for h0 : β1 =
1, β2 = 1, N = 25, data dependent bandwidths and lag lengths. . . . . . .

42

Table 1.13: Fixed-b asymptotic critical value for t-test of β in regression with intercept
and two regressors, N = 25, Bartlett kernel . . . . . . . . . . . . . . . . .

43

Table 1.14: Fixed-b asymptotic critical value for t-test of β in regression with intercept
and two regressors, N = 25, QS kernel . . . . . . . . . . . . . . . . . . . .

44

ix

Table 3.1: Empirical null rejection probabilities, 5% level, t-tests for H0 : β1 = 1,
N = 5, ρ = 0.6, Bartlett kernel . . . . . . . . . . . . . . . . . . . . . . . . 150
Table 3.2: Empirical null rejection probabilities, 5% level, t-tests for H0 : β1 = 1,
N = 5, ρ = 0.9, Bartlett kernel . . . . . . . . . . . . . . . . . . . . . . . . 150
Table 3.3: Empirical null rejection probabilities, 5% level, Wald-tests for H0 : β1 =
1, β2 = 1, N = 5, ρ = 0.6, Bartlett kernel . . . . . . . . . . . . . . . . . . 151
Table 3.4: Empirical null rejection probabilities, 5% level, Wald-tests for H0 : β1 =
1, β2 = 1, N = 5, ρ = 0.9, Bartlett kernel . . . . . . . . . . . . . . . . . . 151

x

LIST OF FIGURES

Figure 1.1: Empirical null rejections, t-test, N = 5, T = 100, ρ1 = ρ2 = 0.3, Bartlett
kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45

Figure 1.2: Empirical null rejections, t-test, N = 10, T = 100, ρ1 = ρ2 = 0.3, Bartlett
kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

46

Figure 1.3: Empirical null rejections, t-test, N = 25, T = 100, ρ1 = ρ2 = 0.3, Bartlett
kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47

Figure 1.4: Empirical null rejections, t-test, N = 5, T = 50, ρ1 = ρ2 = 0.9, Bartlett
kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

48

Figure 1.5: Empirical null rejections, t-test, N = 10, T = 50, ρ1 = ρ2 = 0.9, Bartlett
kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49

Figure 1.6: Empirical null rejections, t-test, N = 25, T = 50, ρ1 = ρ2 = 0.9, Bartlett
kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

50

Figure 1.7: Empirical null rejections, t-test, N = 5, T = 100, ρ1 = ρ2 = 0.9, Bartlett
kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51

Figure 1.8: Empirical null rejections, t-test, N = 10, T = 100, ρ1 = ρ2 = 0.9, Bartlett
kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

52

Figure 1.9: Empirical null rejections, t-test, N = 25, T = 100, ρ1 = ρ2 = 0.9, Bartlett
kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53

Figure 1.10: Empirical null rejections, t-test, N = 5, T = 100, ρ1 = ρ2 = 0.3, QS kernel 54
Figure 1.11: Empirical null rejections, t-test, N = 5, T = 50, ρ1 = ρ2 = 0.9, QS kernel

55

Figure 1.12: Empirical null rejections, t-test, N = 5, T = 100, ρ1 = ρ2 = 0.9, QS kernel 56
Figure 1.13: Size adjusted power, Wald test, N = 5, T = 50, ρ1 = ρ2 = 0.6, b = 0.3,
QS kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57

Figure 1.14: Size adjusted power of panel IM, Wald test, N = 5, T = 50, ρ1 = ρ2 = 0.6,
QS kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

58

Figure 1.15: Size adjusted power of panel IM, Wald test, N = 5, T = 50, ρ1 = ρ2 = 0.6,
Bartlett kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59

xi

Figure 1.16: Size adjusted power, Wald test, N = 5, T = 50, ρ1 = ρ2 = 0.6, QS kernel

60

Figure 1.17: Size adjusted power, Wald test, N = 10, T = 50, ρ1 = ρ2 = 0.6, b = 0.3,
QS kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61

Figure 1.18: Size adjusted power of panel IM, Wald test, N = 10, T = 50, ρ1 = ρ2 =
0.6, QS kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

62

Figure 1.19: Size adjusted power, Wald test, N = 10, T = 50, ρ1 = ρ2 = 0.6, QS kernel 63
Figure 2.1: Empirical null rejections, t-test, N = 5, T = 50, ρ1 = ρ2 = 0.6, Bartlett
kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
Figure 2.2: Empirical null rejections, Wald test, N = 5, T = 50, ρ1 = ρ2 = 0.6,
Bartlett kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
Figure 2.3: Empirical null rejections, t-test, N = 5, T = 100, ρ1 = ρ2 = 0.6, Bartlett
kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
Figure 2.4: Empirical null rejections, Wald test, N = 5, T = 100, ρ1 = ρ2 = 0.6,
Bartlett kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
Figure 2.5: Empirical null rejections, t-test, N = 5, T = 50, ρ1 = ρ2 = 0.9, Bartlett
kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
Figure 2.6: Empirical null rejections, Wald test, N = 5, T = 50, ρ1 = ρ2 = 0.9,
Bartlett kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
Figure 2.7: Empirical null rejections, t-test, N = 5, T = 100, ρ1 = ρ2 = 0.9, Bartlett
kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
Figure 2.8: Empirical null rejections, Wald test, N = 5, T = 100, ρ1 = ρ2 = 0.9,
Bartlett kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
Figure 2.9: Raw power, Wald test, N = 5, T = 50, ρ1 = ρ2 = 0.6, b = 0.1, Bartlett
kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
Figure 2.10: Raw power, Wald test, N = 5, T = 100, ρ1 = ρ2 = 0.6, b = 0.1, Bartlett
kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
Figure 2.11: Empirical null rejections, t-test, N = 5, T = 50, ρ1 = ρ2 = 0.6, Bartlett
kernel, residuals from non-augmented partial sum regression . . . . . . . 110
Figure 2.12: Empirical null rejections, Wald test, N = 5, T = 50, ρ1 = ρ2 = 0.6,
Bartlett kernel, residuals from non-augmented partial sum regression . . 111
xii

Figure 2.13: Empirical null rejections, t-test, N = 5, T = 100, ρ1 = ρ2 = 0.6, Bartlett
kernel, residuals from non-augmented partial sum regression . . . . . . . 112
Figure 2.14: Empirical null rejections, Wald test, N = 5, T = 100, ρ1 = ρ2 = 0.6,
Bartlett kernel, residuals from non-augmented partial sum regression . . 113
Figure 2.15: Empirical null rejections, t-test, N = 5, T = 50, ρ1 = ρ2 = 0.9, Bartlett
kernel, residuals from non-augmented partial sum regression . . . . . . . 114
Figure 2.16: Empirical null rejections, Wald test, N = 5, T = 50, ρ1 = ρ2 = 0.9,
Bartlett kernel, residuals from non-augmented partial sum regression . . 115
Figure 2.17: Empirical null rejections, t-test, N = 5, T = 100, ρ1 = ρ2 = 0.9, Bartlett
kernel, residuals from non-augmented partial sum regression . . . . . . . 116
Figure 2.18: Empirical null rejections, Wald test, N = 5, T = 100, ρ1 = ρ2 = 0.9,
Bartlett kernel, residuals from non-augmented partial sum regression . . 117
Figure 2.19: Raw power, Wald test, N = 5, T = 50, ρ1 = ρ2 = 0.6, b = 0.1, Bartlett
kernel, residuals from non-augmented partial sum regression . . . . . . . 118
Figure 2.20: Raw power, Wald test, N = 5, T = 100, ρ1 = ρ2 = 0.6, b = 0.1, Bartlett
kernel, residuals from non-augmented partial sum regression . . . . . . . 119
Figure 3.1: Power of bootstrap Stat-BS IM (D), Wald test, N = 5, ρ1 = ρ2 = 0.6,
b = 0.5, Bartlett kernel with different T and pT . . . . . . . . . . . . . . 152
Figure 3.2: Power of bootstrap Stat-BS IM (D), Wald test, N = 5, T = 500, b = 0.5,
Bartlett kernel with different ρ and pT . . . . . . . . . . . . . . . . . . . 153
Figure 3.3: Size adjusted power, Wald-tests, N = 5, T = 50, ρ1 = ρ2 = 0.6, b = 0.5,
Bartlett kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
Figure 3.4: Size adjusted power, Wald-tests, N = 15, T = 50, ρ1 = ρ2 = 0.6, b = 0.5,
Bartlett kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

xiii

KEY TO ABBREVIATIONS

OLS Ordinary Least Square
IM-OLS Integrated Modified Ordinary Least Square
FM-OLS Fully Modified Ordinary Least Square
DOLS Dynamic Ordinary Least Square
HAC Heteroskedastic and Autocorrelation Consistent RMSE Root Mean Squared Error
Fb Fixed-b
QS Quadratic Spectral

xiv

Chapter 1
Integrated modified OLS estimation
and fixed-b inference for homogeneous
cointegrated panels
This paper is concerned with parameter estimation and inference in homogeneous cointegrated panels. We propose a simple, new estimation method originated from Vogelsang and
Wagner [2014]. The estimator is labeled panel integrated modified ordinary least squares
(panel IM-OLS). Similar to panel fully modified ordinary least squares (panel FM-OLS) and
panel dynamic ordinary least squares (panel DOLS), the panel IM-OLS estimator has a zero
mean Gaussian mixture limiting distribution. However, panel IM-OLS does not require estimation of long run variance matrices and avoids the need to choose tuning parameters such
as kernel functions, bandwidths, leads and lags. Inference based on panel IM-OLS estimates
does require an estimator of a scalar long run variance, and we propose both traditional
and fixed-b methods for obtaining critical values for test statistics. The properties of panel
IM-OLS are analyzed using asymptotic theory and finite sample simulations. Panel IM-OLS
performs well relative to other estimators.

1

1.1

Introduction

This paper considers the extension of the pure time series integrated modified ordinary
least squares (IM-OLS) method of Vogelsang and Wagner [2014] for estimating and testing
hypotheses about a cointegrating vector to a balanced panel of N individuals observed over
T time periods. We call the estimator panel IM-OLS. We derive its limiting distribution and
provide a finite sample simulation of panel IM-OLS compared with pooled OLS, panel fully
modified OLS (panel FM-OLS) and panel dynamic OLS (panel DOLS).
It is well-known that in panel cointegration regression, when the regressors are endogenous, the limiting distribution of the pooled OLS estimator is contaminated by second order
bias terms. Inference is difficult in this situation because the nuisance parameters cannot be
removed by simple scaling methods. Consequently, panel FM-OLS and panel DOLS were
proposed, which both deal with the endogeneity problem and lead to zero mean Gaussian
mixture limiting distributions and in turn make standard asymptotic inference available.
The panel IM-OLS estimator is based on pooled OLS estimation of a partial sum transformation of the cointegrating panel regression. Similar to the panel FM-OLS and panel DOLS
estimators, the panel IM-OLS estimator also has a zero mean Gaussian mixture limiting
distribution, but it has advantage compared with its two counterparts. Panel IM-OLS estimator avoids kernel function and bandwidth choices for long run variance estimation, which
is required by panel FM-OLS, and leads and lags choices to expand the regression, which
is required by panel DOLS. However, for inference, panel IM-OLS does need to estimate a
scalar long run variance parameter.
The limit theory considered here is obtained for a fixed number of cross-sectional units N ,
letting T → ∞. This limit theory is widely used in empirical macroeconomics, empirical en2

ergy economics and empirical finance problems. In this case, even though the panel IM-OLS
estimator converges to a zero mean Gaussian mixture distribution, inference based on this
estimator still requires the estimation of a long run variance parameter. As in Vogelsang and
Wagner [2014], there are two solutions for this problem. First, standard asymptotic inference based on a consistent estimator of the long run variance and second, fixed-b inference.
The latter solution has its own benefit over standard asymptotic theory because fixed-b inference captures the impact of kernel and bandwidth choices on test statistics based upon
them, whereas standard asymptotic theory does not. As will be discussed in detail later, the
pooled OLS residuals of the panel IM-OLS regression need to be further adjusted to obtain
pivotal fixed-b test statistics.
All estimators and tests in this paper are derived for a cross-sectionally uncorrelated
homogeneous panel. For many applications this unrealistic assumption is still commonly
employed when developing panel cointegration methods, especially for estimation procedures. Only a few and partial results concerning both cointegration estimation and inference
are available for cross-sectionally dependent panels to date. One branch of the literature
considers panel data with spatial interaction among cross-sectional units (e.g., Kapoor et al.
[2007]; Yu et al. [2008], [2010], [2012]). An alternative to the spatial approach is the factor
structure approach, which can capture common stochastic shocks and trends (e.g., Bai and
Ng [2004]). Bai and Kao [2006] derive an extension of FM-OLS estimation to panels with
short-run cross-sectional correlation. Pesaran [2006] proposes the Common Correlated Effects (CCE) approach to estimation of panel data models with multi-factor error structure,
which is further developed by Kapetanios et al. [2011] allowing for nonstationary common
factors. The estimation and inference is challenging for cross-sectionally dependent heterogeneous panel with endogenous regressors, but our ongoing work shows that the methods
3

developed in this paper, with some modifications, will be able to estimate the parameter and
make valid inference in that scenario.
After the theoretical analysis, we provide a finite sample simulation study to assess the
performance of the estimators and tests. Benchmarks are given by pooled OLS, panel FMOLS and panel DOLS. In the simulations, panel IM-OLS performs relatively well with smaller
bias and only slightly larger RMSE than other estimators. The simulations of size and power
of the tests show that fixed-b test statistics based on the panel IM-OLS estimator lead to
the smallest size distortions at the price of only minor losses in size-corrected power.
The remainder of the paper is organized as follows. In the next section we present a
standard panel cointegrating regression and review several key results of the benchmark estimators. Section 1.3 describes the panel IM-OLS estimator and its asymptotic distribution.
Inference using the panel IM-OLS parameter estimator is discussed. Section 1.4 reports
the finite sample bias and root mean squared error of the various estimators. Section 1.5
assesses the finite sample performance of the test statistics described in Section 1.3. Section
1.6 concludes the paper. Appendix contains the proofs of this paper.

4

1.2

Homogeneous cointegrated panels for benchmark
estimators

Consider the following data generating process

yit = µ + xit β + uit

(1.1)

xit = xit−1 + vit

(1.2)

where yit and uit are scalars, xit and vit are k × 1 vectors with sub-index i = 1, 2, · · · , N for
the ith cross sectional unit, sub-index t = 1, 2, · · · T for the time period; β is k × 1 vector
of the slope parameters. For notational brevity here we only include the intercept µ as the
deterministic component (later when we discuss the panel IM-OLS estimator, we will extend
it into more general deterministic time trends such as µ0 + µ1 t + · · · + µp−1 tp−1 ). Define
the error vector as ηit = uit , v . It is assumed that ηit is a vector of I(0) processes, in
it
which case xit is a non-cointegrating vector of I(1) processes and there exists a cointegrating
relationship among yit , x
it

with cointegrating vector 1, −β

.

Assumption 1. Assume that {ηit }N
i=1 are cross-sectionally uncorrelated and theirs 2nd order
moment is constant.
Note that the Assumption 1 only requires the panels are homogeneous in the 2nd order
moment, it’s possible that the higher order moment structure are heterogeneous across i.
Assumption 2. Assume that ηit satisfies a functional central limit theorem (FCLT) of the
form
[rT ]

T −1/2

ηit ⇒ Bi (r) = Ω1/2 Wi (r),
t=1

5

r ∈ [0, 1].

In Assumption 2, [rT ] represents the integer part of rT , and Wi (r) is a (k + 1) × 1 vector
of independent standard Brownian motions. Ω1/2 is a (k + 1) × (k + 1) matrix that satisfies:
Ω = Ω1/2 Ω1/2 , and



Ωuu Ωuv 
Ω=
E(ηit ηit−j ) = 
 > 0,
j=−∞
Ωvu Ωvv
∞

where it is clear that Ωvu = Ωuv . The assumption Ωvv > 0 rules out cointegration in xit .
Partition Bi (r) as


Bu,i (r)
Bi (r) = 
,
Bv,i (r)
and likewise partition Wi (r) as




 wu,i (r) 
Wi (r) = 
,
Wv,i (r)
where wu,i (r) and Wv,i (r) are a scalar and a k-dimensional standard Brownian motion respectively. Using the Cholesky form of Ω1/2 ,




 σu·v λuv 
Ω1/2 = 
,
1/2
0k×1 Ωvv
−1/2

2 = Ω − Ω Ω−1 Ω
it can be shown that σu·v
uu
uv vv vu and λuv = Ωuv Ωvv

6

. By this Cholesky

decomposition, we can write

 

Bu,i (r) σu·v wu,i (r) + λuv Wv,i (r)
Bi (r) = 
=
.
1/2
Bv,i (r)
Ωvv Wv,i (r)
Next define the one-sided long run covariance matrix. For each i ∈ [1, 2, · · · , N ],


∞

Λ=



Λuu Λuv 
E(ηi,t−j ηit ) = 
.
j=1
Λvu Λvv

Also define the contemporary covariances, that is, for each i ∈ [1, 2, · · · , N ],




Σuu Σuv 
Σ = E(ηit ηit ) = 
.
Σvu Σvv
Note that ∆ = Σ + Λ is half long run variance, and it is likewise partitioned as


∞

∆=



∆uu ∆uv 
E(ηi,t−j ηit ) = 
.
j=0
∆vu ∆vv

The long run variance, Ω, is related to Λ and Σ as Ω = Σ + Λ + Λ .
Remark 1.

1. If we do have heterogeneity in the 2nd order moment structure, i.e. Ωi , Λi ,

Σi and ∆i are varied for different i, then even though we can estimate those moments
individual by individual, however, finding pivotal fixed-b statistics is challenging, and
we haven’t found it yet. In this case, one possible way to make valid inference is using
bootstrap to mimic those non-pivotal distributions. The stationary bootstrap is one
method that we could apply in this scenario.
7

2. If ηit are cross-sectionally dependent, then the inference is much more complicated.
Spatial approach and factor structure approach are possible solutions according to different dependence assumptions. We haven’t been able to find a way to deal with the
general cross-sectional dependence case. However, if the dependence only originates
from time fixed-effect dummy variables, then the methods developped in this paper will
go through with some natural modifications, and the bootstrap, both over time as well
as across units, is needed for valid inference.
As mentioned before, the benchmark estimators are the pooled OLS, the panel FM-OLS
and the panel DOLS estimators. To conserve space, we don’t provide detail results of all
those estimators. But we do want to review several key results for those estimators. For
the pooled OLS estimator, when the regressors are endogenous, it has an asymptotic bias
due to the nuisance parameters ∆vu , which cannot be removed by simple scaling methods.
The panel FM-OLS estimator as considered here is an extension of the FM-OLS estimator of
Phillips and Hansen [1990], which is designed to asymptotically remove ∆vu and to deal with
the correlation between Bu,i (r) and Bv,i (r). Conditional on Bv,i (r) for all i = 1, 2, . . . , N ,
the limit of the scaled panel FM-OLS estimator is a mean zero mixture of normals. Asymptotically pivotal t and Wald statistics with N (0, 1) and chi-square limiting distributions can
2 . The panel DOLS estimator considered here is almost
be constructed by estimating σu·v

identical to Mark and Sul [2003]. The only difference is that there is no fixed effect in the
data generating process (1.1). The homogeneous panel DOLS estimator of β has the same
limiting distribution as the homogeneous panel FM-OLS estimator. Hence, they are asymptotically equivalent. This result was shown by Kao and Chiang [2000], and it also can be
extended to heterogeneous panels.

8

1.3

Panel integrated modified OLS

1.3.1

Panel IM-OLS estimator

In this section, we present a new estimator for homogeneous cointegrated panels. This
estimator is an extension of Vogelsang and Wagner [2014], who propose the IM-OLS estimator for the time series case. The transformation used by IM-OLS provides an asymptotically
unbiased estimator with a zero mean Gaussian mixture limiting distribution. Compared
with panel FM-OLS, the transformation does not require estimators of Ω, so the choice of
bandwidth and kernel is avoided for parameter estimation. We consider a slightly more
general version of (1.1) given by

yit = Dt δi + xit β + uit ,

(1.3)

where δi and β are p × 1 and k × 1 parameter vectors respectively, xit continues to follow
(1.2) and for the deterministic component, Dt , we assume that there is a p × p matrix GD
and a vector of functions, D(s), such that
√
lim

T →∞

r

T G−1
D D[sT ] = D(s) with 0 <

D(s)D(s) ds < ∞, 0 < r

1.

(1.4)

0

The deterministic component Dt could include an intercept, time trend and polynomials
of the time trend.
Remark 2.

1. Note that, in regression (1.3), the intercept from Dt and δi together allow

fixed effect estimation of the system. In a simpler case, suppose that δi is the same

9

constant for all i so that
yit = Dt δ + xit β + uit .
In this case, the estimation and inference procedures introduced later in the paper go
through with minor changes.
2. In regression (1.3), β is a constant for all i, which means the same long-run relation
between yit and xit applies for all i. As in Philips and Moon [1999], we could also
allow this coefficient differs randomly across i, which leads to the heterogeneous panel
cointegration model. In that case, as long as the error vectors are uncorrelated across
i and their 2nd order moments are constant, then the results will be similar as what
we have in this paper. Otherwise, if the panel has heterogeneity in both cointegration
relation and 2nd moment structure, then inference is challenging and might need to
apply boostrap.
3. We could consider a more traditional panel data setting model like

yit = µi + λt + xit β + uit .

In this case, after the time effect λt being eliminated by cross-sectional demeaning, and
if it is also a homogeneous panel, then the estimation and inference procedures will be
similar as in this paper. But if there is heterogeneity in the sencond moment structure,
even though the estimation of the β will not be affected, however, the inference is much
more complicated as we disscussed in Remark 1, and bootstrap method could be used
for the inference.

10

Computing the partial sum of both sides of (1.3) gives

y

x β + Su ,
Sit = StD δi + Sit
it

y

t
j=1 yij ,

where Sit =

(1.5)

x and S u are defined analogously. As in Vogelsang and
and StD , Sit
it

Wagner [2014], we need to add xit as regressors in (1.5) to deal with correlation between uit
and vit , which leads to

y

x β + x γ + Su − x γ
Sit = StD δi + Sit
it
it
it
x β + x γ + Su
˜
= StD δi + Sit
it
it .

(1.6)

We now focus on the asymptotic behavior of the pooled OLS estimators of δi , β and γ
from (1.6), which we label the panel IM-OLS estimators of δi , β and γ. Define the stacked
vectors and matrices as follows:




β
 
 


 
y
y
u
˜
u
γ
 
 Si1 
 S1 
 Si1 − xi1 γ 
 S1 

 


 

 
 
 .. 
 .. 
 .. 


y
.
u
˜
u
˜
y
.
S =  .  , Si =  .  ; θ =  δ1  ; S =  .  , Si = 
;
.
 
 


 


 


 


 
 .. 
y
y
u
˜
u
 . 
SiT
SiT − xiT γ
SN
SN
 
 
δN









x
xi1
 Si1

x
˜
 .
 S1 
..
 ..
 
.

 
S x˜ =  ...  , Six˜ = 

 
S x x
 
 iT
iT
x
˜

SN







01×p

···

S1D

···

01×p

..
.

···

..
.

···

..
.

01×p

···

STD

···

01×p

1st block
11

ith block

N th block






.






With the above notation, the matrix form of (1.6) is given by

S y = S x˜ θ + S u˜ ,

(1.7)

and the OLS estimator of (1.7) is given by

−1

θ˜ = S x˜ S x˜

S x˜ S y ,

(1.8)

which leads to

θ˜ − θ = S x˜ S x˜

N

−1

T

S x˜ S u˜
−1 

qit qit 

=



T

u − x γ ,
qit Sit
it
i=1 t=1

xit 01×p · · · StD

submatrix of qit , 01×p · · · S D
t

N



i=1 t=1

where qit = S x
it

(1.9)

· · · 01×p

for i = 1, 2, . . . , N , t = 1, 2, . . . , T . The

D
th
· · · 01×p , consists of St as its i block and other

N − 1 zero vector blocks. Define the scaling matrix


0
T Ik





=
A−1


Ik
P IM




0
IN ⊗ GD
as a (2k + N p) × (2k + N p) diagonal matrix.
˜ γ˜ .
The following theorem gives the asymptotic distribution of δ˜i , β,
Theorem 1. Assume that the data are generated by (1.2) and (1.3), that the deterministic
components satisfy (1.4) for all i ∈ [1, 2, · · · , N ], and that Assumptions 1 and 2 hold. Define
12

θ by stacking the vectors δi , β and Ω−1
vv Ωvu . Then for fixed N , as T → ∞


 T β˜ − β


 γ˜ − Ω−1 Ωvu
vv



 GD δ˜1 − δ1


..


.


GD δ˜N − δN









˜
 = A−1
P IM θ − θ










 0k×1 




Ω−1 Ωvu 
 vv



−1


= AP IM S x˜ S x˜ AP IM
AP IM S x˜ S u˜ −  0p×1 




..




.




0p×1
⇒ σu·v Π

−1

N
i=1

= σu·v Π

−1

N
i=1

−1
1
0 g1,i (s)g1,i (s)

i=1
−1

1
0 g1,i (s)g1,i (s)

N

ds
N

ds

≡Ψ



1/2
0 
Ωvv




1/2
where Π = 
,
Ωvv




0
IN ⊗ Ip

i=1



1
0 g1,i (s)wu,i (s)ds
1
0 [G1,i (1) − G1,i (s)]dwu,i (s)




 w (r)
v,i




0p×1



..
g1,i (r) = 
.


 r
 0 D(s)ds


..

.



0p×1

13



r
0 wv,i (s)ds











r
, G1,i (r) = 0 g1,i (s)ds.











Conditional on g1,i (r) for all i ∈ {1, 2, . . . , N }, it holds that

Ψ ∼ N (0, VP IM ) ,

(1.10)

where VP IM is given by

−1 

2
VP IM = σu·v
Π






N

−1

1

i=1 0

×

g1,i (s)g1,i (s) ds



1

[G1,i (1) − G1,i (s)][G1,i (1) − G1,i (s)] ds ×

i=1 0
N

N

−1

1

i=1 0

(1.11)

g1,i (s)g1,i (s) ds

Π−1

It is clear that this conditional asymptotic variance differs from the conditional asymptotic variance of the panel FM-OLS and panel DOLS estimator of δ and β. Denoting with
mi (s) = D(s) , wv,i (s)

and with ΠP F M = diag Ip , Ω1/2
vv


2
VP F M = σu·v
ΠP F M

−1 

−1

1 N
0 i=1

the latter is given by

mi (s)mi (s) ds

(ΠP F M )−1 .

(1.12)

It is important to note that we can extend Theorem 1 further to obtain a sequential
result. Since the parameters in θ require different scaling for the sequential limits, and our
interests are mainly on β, therefore we only provide the result for β˜ − β . That is, if all
the assumptions in Theorem 1 hold, and first T → ∞, then N → ∞, we will have following
asymptotic distribution
√

N T β˜ − β ⇒ Φ

14

(1.13)

β

β

where Φ ∼ N 0, Vseq , and the asymptotic variance Vseq is given by

β
Vseq

=

2
σu·v

1/2
Ωvv

−1

−1

1
0

A1 (r)dr

1
0

−1

1

A2 (r)dr

2 Ω−1
= 5.6σu·v
vv

0

A1 (r)dr

1/2 −1

Ωvv

(1.14)

where A1 (r) = r3 /3 Ik and A2 (r) =

1 − r − r4 + r5 /12 Ik . The importance of this

result is that it leads to standard inference based on the large T and large N approximation.
The details of the derivation are in the Appendix.
Remark 3. Above is the sequential limit for the homogeneous panel. If our panel has heteroβ

geneity in the 2nd moment structure, then the variance, Vseq , will take the same expression,
2 = lim 1
but σu·v
N →∞ N

1.3.2

N
i=1

2
σu·v,i
and Ωvv = lim N1
N →∞

N

Ωvv,i .
i=1

Inference using Panel IM-OLS

This section provides a discussion of hypothesis testing using the panel IM-OLS estimator.
The zero mean Gaussian mixture limiting distribution of the panel IM-OLS estimator given
in Theorem 1 and the conditional asymptotic variance given in (1.11) offer the theoretical
basis for this discussion. In particular we consider Wald tests for testing multiple linear
hypotheses of the form
H0 : Rθ = r
where R ∈ Rq×(2k+N p) with full rank q and r ∈ Rq . Because the vector θ˜ has elements
that converge at different rates, we need restriction on R to get formal Wald statistics. We
15

assume that there exists a nonsingular q × q scaling matrix AR such that

lim A−1 RAP IM
T →∞ R

= R∗

where R∗ has rank q.
In order to carry out statistical inference, we need to scale out the asymptotic variance
2 is an estimator for σ 2 . Then an estimator for V
of panel IM-OLS. Suppose that σ
˘u·v
P IM
u·v

is given by


T

−1

N

2 T −2
V˘P IM = σ
˘u·v

×

AP IM qit qit AP IM 
t=1 i=1



T



N
q

T −4

q

q

q

SiT − Si,t−1 AP IM  ×

AP IM SiT − Si,t−1
t=1 i=1



−1

N

T

T −2

AP IM qit qit AP IM 

,

t=1 i=1
q

where Sit =

t
j=1 qij ,

q

and Si0 = 0 for all i = 1, 2, · · · , N .

2 . The first is based on the pooled OLS
There are several obvious candidates for σ
˘u·v

residuals. Let the pooled OLS residuals be uˆit = yit − xit βˆ − Dt δˆi , where δˆi and βˆ are the
pooled OLS estimators. Using these residuals, we can define estimators for the error vector
ˆ ij = T −1
ηˆit = uˆit , ∆x . Then Γ
it

T
t=j+1 ηˆit ηˆi,t−j ,

and



T −1
ˆ
ˆ
j
Ωuu,i Ωuv,i  ˆ
ˆ
ˆ ij + Γ
ˆ
Ωi = 
k( ) Γ
 = Γi0 +
ij .
M
ˆ
ˆ
j=1
Ωvu,i Ωvv,i

16

2 is given by
The estimator for σu·v

2
σ
ˆu·v

1
=
N

N

ˆ uu,i − Ω
ˆ uv,i Ω
ˆ vv,i
Ω

−1

ˆ vu,i .
Ω

i=1

u , the first differences of the pooled OLS
The second estimation approach is to use ∆S˜it
2 :
residuals of the panel IM-OLS regression (1.6), to directly estimate σu·v

2 =
σ
˜u·v

1
N

N



T

T

T −1

k
j=2 h=2

i=1

|j − h|
M


u ∆S
˜u  .
∆S˜ij
ih

2 is not a
Extending a result in Vogelsang and Wagner [2014], it can be shown that σ
˜u·v

consistent estimator under traditional assumptions on the bandwidth and kernel functions.
2 is larger than
Under traditional bandwidth assumptions, we can show that the limit of σ
˜u·v
2
2 which leads to asymptotically conservative results when we build test statistics by σ
˜u·v
σu·v

and use critical values from the standard normal or a chi-square distribution.
The third estimation approach is based on OLS residuals from a further augmented
2 based on
regression. As discussed in Vogelsang and Wagner [2014], an estimator of σu·v
2 , independent of
these residuals defined below has a fixed-b limit that is proportional to σu·v
2
˜ and does not depend upon additional nuisance parameters, whereas the estimators σ
θ,
ˆu·v
2 both fail those requirements. Define this estimator as:
and σ
˜u·v

2∗ =
σ
˜u·v

1
N

N



T

T

T −1
i=1

k
j=2 h=2

|j − h|
M


u∗ ∆S
˜u∗ 
∆S˜ij
ih

u∗ = S
˜u∗ − S˜u∗ is the difference of the residuals S˜u∗ , obtained by running the
where ∆S˜it
it
i,t−1
it
u∗ = S y − S D δ˜ − S x β˜ −
further augmented IM-OLS regression individual by individual, S˜it
t i
it i
it

17

˜ i . The augmented regressors, zit , are given by zit = t
xit γ˜i − zit λ
x =
x, x
qit
StD , Sit
it

T
j=1

x −
qij

t−1 j
j=1 s=1

x , where
qis

for all i = 1, 2, . . . , N , t = 1, 2, . . . , T .

2 , we can define the t and Wald statistic as
Using an estimator of σu·v

Rθ˜ − r

t˘ =

RAP IM V˘P IM AP IM R

˘ =
W

Rθ˜ − r

RAP IM V˘P IM AP IM R

−1

Rθ˜ − r

2 , which defines tˆ and W
2 , which
ˆ , or V˜P IM using σ
where V˘P IM could be VˆP IM using σ
ˆu·v
˜u·v
2∗ , which defines t˜∗ and W
˜ ∗ . The asymptotic null
˜ , or V˜ ∗
defines t˜ and W
˜u·v
P IM using σ

distribution of these test statistics are given in Theorem 2. Standard asymptotic results
ˆ , t˜ and W
˜ , whereas
based on traditional bandwidth and kernel assumptions are given for tˆ, W
˜ ∗.
a fixed-b result is given for t˜∗ and W
Theorem 2. Assume that the data are generated by (1.2) and (1.3), that the deterministic
components satisfy (1.4), and that Assumptions 1 and 2 hold. Suppose that the bandwidth,
2 is consistent. Then for fixed
M , and kernel function, k(·), satisfy conditions such that σ
ˆu·v

N , as T → ∞
ˆ ⇒ χ2 , where χ2 is a chi-square random variable with q degrees of freedom. When
• W
q
q
q = 1, tˆ ⇒ Z, where Z is a standard normal distribution.
2 ⇒ σ2
• Under the above assumptions, σ
˜u·v
u·v 1 + dγ dγ , with dγ denoting the second k×1
N

block of
i=1

−1

1
0 g1,i (s)g1,i (s)

˜ ⇒
it follows that W

χ2
q
,
1+dγ dγ

N

ds
i=1

1
0 [G1,i (1) − G1,i (s)]dwu,i (s)

. Therefore,

where χ2q is a chi-square random variable with q degrees of
18

freedom that is correlated with dγ . When q = 1, t˜ ⇒

Z
,
1+dγ dγ

where Z is distributed

standard normal and is correlated with dγ .
• If M = bT , where b ∈ (0, 1] is held fixed as T → ∞, then
χ2q

˜∗⇒
W
1
N

N
i=1

Q∗i (b)

where Q∗i (b) is exactly same form as Qb P˜ ∗ , P˜ ∗

in Vogelsang and Wagner [2014],1

and χ2q is a chi-square random variable with q degrees freedom independent of N1

N
i=1

Q∗i (b).

When q = 1,
Z

t˜∗ ⇒
1
N

N
i=1

Q∗i (b)

where Z is standard normal distribution independent of N1

N
i=1

Q∗i (b).

˜∗
• Due to the independence between the numerator and denominator of the limits of W
and t˜∗ , we can further obtain a sequential limit result, where T grows large, followed
sequentially by the limit as N grows large. Define

˜ ∗ = µQ · W
˜ ∗,
W
µQ
1 See Vogelsang and Wagner (2014) page 744 for Q (P , P ), and formula (30) for P˜ ∗ (r).
b 1 2

19

where µQ = E Q∗i (b) . Then as T → ∞,
χ2q

˜ µ∗ ⇒
W
Q

1
N

N

1
∗
µQ Qi (b)
i=1

P

−→ χ2q

as N → ∞.

When q = 1,

t˜∗µ

Q

=

µQ · t˜∗
Z

⇒
1
N
P

−→ Z

N

1
∗
µQ Qi (b)
i=1

as N → ∞.

2 , inference using W
2 is
ˆ is standard. In contrast σ
When appealing to consistency of σ
ˆu·v
˜u·v

inconsistent under traditional bandwidth assumptions. Since dγ dγ > 0, the critical values of
˜ are smaller than those of the χ2q distribution. Thus, using χ2q critical values for W
˜ leads
W
to a conservative test under the traditional bandwidth assumptions. The fixed-b limiting
˜ ∗ is complicated due to the presence of 1
distribution of W
N

N
i=1

Q∗i (b), which depends on

wv,i (r) for i = 1, 2, · · · , N . Therefore, critical values should be simulated taking into account
the cross-sectional sample size, the specifications of deterministic components, the number
of integrated regressors, the kernel function and bandwidth choice. For sake of brevity, in
Table 1.13 and Table 1.14, we only tabulate critical values for the t-statistic for the parameter
associated with xit in models with an intercept and 2 integrated regressors for the Bartlett
and QS kernels and a grid of bandwidths indexed by b for N = 25. However, when both T
20

˜ ∗ , with µ
and N large, inference using W
ˆQ as an estimator of µQ , requires merely χ2q critical
µ
Q

values rather than simulated critical values, which is quite convenient. The t statistics have
similar results.

1.4

Finite sample bias and root mean squared error

In this section, we compare the performance of the pooled OLS, panel FM-OLS, panel
DOLS, and panel IM-OLS estimators as measured by bias and root mean squared error
(RMSE) within a small simulation study. We provide results that the individual dummy
variable is not included. The data generating process is given by

yit = µ + x1it β1 + x2it β2 + uit
x1it = x1i,t−1 + v1it
x2it = x2i,t−1 + v2it

where, for ∀ i ∈ [1, 2, · · · , N ], x1i0 = 0, x2i0 = 0, and uit = ρ1 ui,t−1 + it + ρ2 (e1it + e2it ),
ui0 = 0, and v1it = e1it + 0.5e1i,t−1 , v2it = e2it + 0.5e2i,t−1 , where it , e1it and e2it are
i.i.d. standard normal random variables independent of each other. The parameter values
chosen are µ = 3, β1 = β2 = 1. Note that the estimators of β1 and β2 are exactly invariant
to the value of µ, so the value of µ has no effect on our results. In addition, we use ρ1 and ρ2
from the set {0, 0.3, 0.6, 0.9}. The parameter ρ1 controls serial correlation in the regression
error, and ρ2 determines whether the regressors are endogenous or not. The kernel function
chosen for panel FM-OLS are the Bartlett and the Quadratic Spectral kernels, and the
21

bandwidths are given by M = bT with b ∈ {0.06, 0.1, 0.3, 0.5, 0.7, 0.9}. We also use the data
dependent bandwidth from Andrews [1991]. The panel DOLS estimator is implemented
using the information criterion based lead and lag length choice as developed in Kejriwal
and Perron [2008], where we use the more flexible version discussed in Choi and Kurozumi
[2012] in which the numbers of leads and lags included are not exactly same. The sample
sizes are N ∈ {5, 10, 25}, T ∈ {50, 100} and the number of replications is 5000.
In Tables 1.1-1.6 we display the results for N = 5, 10, 25, T = 50, 100 using the Bartlett
kernel only. General patterns are similar for the QS kernel. In each of those tables, Panel A
reports bias and Panel B reports RMSE.

1.4.1

Sample size N = 5

Table 1.1 shows the results for N = 5 with T = 50 case. When ρ2 = 0 (no endogeneity),
none of the estimators show much bias for any value of ρ1 . When the bandwidth is relatively
small, panel FM-OLS has a little bit larger RMSE than pooled OLS. But, as the bandwidth
increases, the RMSE of panel FM-OLS tends to first increase and then decreases, indicating
a hump-shape in the RMSE, and the turning point is around b = 0.5, i.e. M = 0.5T . Pooled
OLS and panel FM-OLS have smaller RMSE than panel IM-OLS and this holds regardless
of bandwidth for panel FM-OLS. Panel IM-OLS has the largest RMSE in any cases. The
RMSE of panel DOLS is a little bit smaller than that of panel IM-OLS, but it is still larger
than that of pooled OLS and panel FM-OLS.
When ρ2 = 0 (there is endogeneity), the estimators show different patterns. For a given
value of ρ1 , as ρ2 increases, the bias of pooled OLS increases. Panel FM-OLS shows the

22

same pattern, but its bias is smaller than that of pooled OLS, which is expected from the
theory. In addition, the bias of panel FM-OLS also depends on the bandwidth and value
of ρ1 , as ρ1 is relatively small, the bias of panel FM-OLS increases as bandwidth increases,
however, as ρ1 is far away from zero, the bias of panel FM-OLS is seen to initially fall as
the bandwidth increases and then tends to increase as the bandwidth become large. But
no matter how large the bandwidth is, the bias of panel FM-OLS does not exceed that of
pooled OLS. On the contrary, the biases of panel IM-OLS and panel DOLS are much less
sensitive to ρ2 , especially when ρ1 is relatively small, and are always smaller than those of
pooled OLS and panel FM-OLS. The bias of panel DOLS is similar to the bias of panel
IM-OLS when ρ1 is small whereas for larger values of ρ1 , the bias of panel DOLS tends to
be larger than that of panel IM-OLS. The overall picture in this case is that panel IM-OLS
has smaller bias than panel DOLS which in turn has lower bias than both pooled OLS and
panel FM-OLS. The magnitude of the bias of both panel IM-OLS and panel DOLS are less
sensitive to the values of ρ2 than for pooled OLS and panel FM-OLS.
Considering the RMSE when there is endogeneity, we see that for given value of ρ1 , as ρ2
increases, the RMSE of pooled OLS increases. Panel FM-OLS shows the same pattern, but
its RMSE is smaller than that of pooled OLS, especially when ρ1 is relatively small. Focusing
on the bandwidth we see that the RMSE of panel FM-OLS has the same pattern as its bias,
if ρ1 is small, the RMSE of panel FM-OLS increases as bandwidth increases, whereas if ρ1 is
relatively large, the RMSE of panel FM-OLS is seen to initially fall as bandwidth increases
and then tends to increase as the bandwidth becomes large. For RMSE of panel IM-OLS,
it is larger than that of pooled OLS when both ρ1 and ρ2 are small, otherwise, it is smaller
than that of pooled OLS. The RMSE of panel DOLS also has similar pattern but it is smaller
than that of panel IM-OLS. The RMSE of panel IM-OLS does not vary with ρ2 when ρ1 is
23

small. The comparison of RMSE for panel IM-OLS and panel FM-OLS depend on value of
ρ1 , ρ2 and b. When both ρ1 and ρ2 are small, the RMSE of panel IM-OLS is larger than
that of panel FM-OLS, no matter what bandwidth used. However, when both ρ1 and ρ2 are
large, the RMSE of panel IM-OLS could be smaller than that of panel FM-OLS with very
large bandwidth.
Also, in Table 1.1, we can see that when there is endogeneity but no serial correlation, then
panel FM-OLS using the data dependent bandwidth performs better than all other estimators
with very small bias and smallest RMSE. And this is true for all different combinations of
N and T , which we can see from Table 1.2 to Table 1.6.
When we increase T to 100, all the estimators tend to have smaller bias than the T = 50
case, which is expected because the estimators are consistent. Almost all of the results are
similar to the T = 50 case. One slight difference is that when there is endogeneity and
both ρ1 and ρ2 are large, the bias of panel IM-OLS is a little bit larger than that panel
DOLS, even though they are still less biased than pooled OLS. The other difference when we
increase T from 50 to 100, is that when both ρ1 and ρ2 are very large, the RMSEs of panel
IM-OLS and panel DOLS are very similar and both are smaller than that of panel FM-OLS,
no matter what bandwidth used, and in turn smaller than RMSE of pooled OLS.

1.4.2

Sample size N = 10

The results of bias in N = 10 case are similar to the N = 5 case. From Panel B of Table
1.3, most of the results for RMSEs are similar as N = 5 case except that when both ρ1 and
ρ2 are large, the RMSEs of panel IM-OLS and panel DOLS are very similar and both are

24

smaller than that of panel FM-OLS for both T = 50 and T = 100. Also, when ρ1 and ρ2
are very large, the RMSEs of panel IM-OLS is slightly smaller than that of panel DOLS for
T = 50 and this relation is reversed when T = 100.

1.4.3

Sample size N = 25

When we increase the cross section sample size to N = 25, the bias results are similar,
but a different pattern emerges in the RMSEs. From Panel B of Tables 1.5 and 1.6, when
there is endogeneity, in both cases T = 50 and T = 100, the RMSE of pooled OLS is the
largest in any cases. This implies that when there is endogeneity, pooled OLS will have the
largest bias and largest RMSE. Also, we can see that when both ρ1 and ρ2 are large, the
RMSEs of panel IM-OLS and panel DOLS are very similar and both are smaller than that
of panel FM-OLS with any bandwidth for both T = 50 and T = 100. Similar as the N = 10
case, when both ρ1 and ρ2 are very large, the RMSEs of panel IM-OLS is slightly smaller
than that of panel DOLS when T = 50, and the RMSEs of panel IM-OLS is slightly larger
than that of panel DOLS when T = 100.

1.4.4

Summary of finite sample bias and RMSE

The simulation shows that, when there is no endogeneity (ρ2 = 0), pooled OLS dominates
other estimators with no bias and smallest variance. When there is no serial correlation
(ρ1 = 0), panel FM-OLS with the data dependent bandwidth performs better than other
estimators. When both serial correlation and endogeneity exist (ρ1 = 0, ρ2 = 0), the relative
25

performance of the estimators depends on the values of N , T , ρ1 and ρ2 . Panel IM-OLS
is more effective in reducing bias than the other estimators and both bias and RMSE of
panel IM-OLS are less sensitive to the parameters ρ1 and ρ2 than are the bias and RMSE
of panel FM-OLS. For N small (N = 5, 10) and T small (T = 50), panel IM-OLS has the
smallest bias but with larger RMSE as a cost, except that when both ρ1 and ρ2 are large
where panel IM-OLS has the smallest RMSE. For N small and T relatively large (T = 100),
panel IM-OLS and panel DOLS are similar, and dominate pooled OLS and panel FM-OLS.
When N is relatively large (N = 25), pooled OLS has the largest bias and largest RMSE in
all cases, and if T is small and ρ1 , ρ2 are relatively large, then panel IM-OLS is better than
the other estimators in reducing bias and has smallest RMSE. When N and T are large, and
ρ1 , ρ2 are large, then panel DOLS is a little bit better than panel IM-OLS, which in turn is
better than pooled OLS and panel FM-OLS.

1.5

Finite sample performance of test statistics

In this section we provide some finite sample results about the tests’ performance using
the simulation design from Section 1.4. Here, we only report results for cases where ρ1 = ρ2 .
The results include t-statistics for testing the null hypothesis H0 : β1 = 1 and Wald statistics
for testing the joint null hypothesis H0 : β1 = 1, β2 = 1. The pooled OLS statistics serve as a
benchmark. The panel FM-OLS statistics were implemented using σ
ˆ 2+ . The panel IM-OLS
u

2 and is labeled panel IM(O),
statistics were implemented in three ways. The first uses σ
ˆu·v
2 and is labeled panel IM(D) and the third uses σ
2∗ and is labeled panel
the second uses σ
˜u·v
˜u·v

IM(Fb). We report results for both the Bartlett and Quadratic Spectral kernels. As for the
26

choice of bandwidth for panel FM-OLS and panel IM-OLS statistics, we follow Vogelsang
and Wagner [2014]. One bandwidth choice is the data dependent bandwidth rule of Andrews
[1991]. The other choice is the fixed-b bandwidth, that is b = M/T , where M = 1, 2, · · · , T .
Rejections for the pooled OLS, panel FM-OLS, panel DOLS, panel IM(O) and panel IM(D)
are carried out using N (0, 1) critical values for all values of M . From Theorem 2, the panel
IM(D) test statistic is asymptotically conservative under traditional asymptotic theory. In
contrast, rejections for panel IM(Fb) are carried out using fixed-b asymptotic critical values.
The empirical rejection probabilities were computed using 5000 replications, and the nominal
level is 0.05 in all cases.
Tables 1.7 to 1.9 and Tables 1.10 to 1.12 report empirical null rejection probabilities using
data dependent bandwidth choices for Bartlett and QS kernel. Tables 1.7 to 1.9 show results
for the t-tests and Tables 1.10 to 1.12 contain results for the Wald tests. In each table Panel
A corresponds to T = 50 and Panel B to T = 100. We only briefly summarize some main
findings in the tables. When ρ1 = ρ2 = 0 (no serial correlation and no endogeneity), we
can see that pooled OLS tests have rejection probabilities close to 0.05, but there are huge
over-rejections as the value of ρ1 and ρ2 increase. For ρ1 = ρ2 = 0, when using the QS
kernel, panel IM(Fb) tests tend to have rejection probabilities less than 0.05, whereas other
tests show some over-rejections. For ρ1 = ρ2 = 0, when using the Bartlett kernel, all the
tests show some over-rejections, but the over-rejection problem is less severe when T = 100
than T = 50. Note that both panel IM(O) and panel IM(D) show some over-rejections, but
those are less severe than panel FM-OLS, especially when there is strong serial correlation
and strong endogeneity. Generally, panel IM(D) tests have rejection probabilities that are
smaller than those of panel IM(O), which is what we expected because the panel IM(D) test
is conservative under standard asymptotic theory. In addition, increasing the values of ρ1
27

and ρ2 leads to over-rejection problems for all the tests. The problem with panel IM-OLS is
that the data-dependent bandwidth is too small to give less size distortions. In contrast to
the pure time series case, there is no test that dominates the others in that scenario.
In order to see the impact of bandwidth and kernel choices on over-rejection problem,
we plot in Figures 1.1-1.3 null rejection probabilities of the t-tests as a function of b ∈ (0, 1].
The first three figures give the results for N ∈ {5, 10, 25}, T = 100 using the Bartlett
kernel and ρ1 = ρ2 = 0.3. In Figure 1.1, with cross-section sample size N = 5, we can
see that with small bandwidths, all tests have some over-rejection problems. Panel IM(D)
is less severe than the other tests because it is conservative. As the bandwidth increases,
all rejection probabilities increase substantially except for panel IM(Fb). The panel IM(Fb)
rejection probabilities are close to 10% for all values of b, which indicates that the fixed-b
approximation performs relatively well for panel IM(Fb). In Figures 1.2 and 1.3, the crosssection sample size increases to 10 and 25 and the pattern of rejection probabilities are similar
as Figure 1.1. However, when the bandwidth is small, like b = 0.08, panel FM(Fb) has the
least rejection probabilities, around 8% and 7.5%, respectively. In addition, as N increases,
panel IM(Fb) rejection probabilities are close to 10% and 12% when large bandwidth used.
As the values of ρ1 , ρ2 increase to 0.9, there exists strong serial correlation and endogeneity. We can see from Figures 1.4-1.6 that all the tests have serious over-rejection problems
regardless of bandwidth. Interestingly, for small N (N = 5, 10), panel FM(Fb) has less of
an over-rejection problem than panel IM(Fb) although both tests are severely size distorted.
As N increases to 25, the rejection probabilities of panel IM(Fb) tend to smaller than that of
panel FM(Fb). In Vogelsang and Wagner [2014], it was pointed out that the over-rejection
problems of IM(Fb) becomes less problematic as T increases. We find similar patterns in
our simulations. In Figures 1.7 to 1.9, we show the results with T = 100, and it is clear that
28

the panel IM(Fb) over-rejections are reduced although they are still large. We believe that,
similar to the pure time series case, if we further increase T to 500 or 1000, the rejections
for panel IM(Fb) with non-small bandwidths will be substantially reduced to reasonable size
whereas the other statistics will continue to have over-rejection problems.
Figures 1.10-1.12 give some results for the QS kernel. For brevity we only report results
for N = 5, T = 50, 100, ρ1 = ρ2 ∈ {0.3, 0.9}. Compared with results using Bartlett kernel,
panel IM(Fb) has less over-rejection problems using QS kernel. In Figures 1.10, when serial
correlation and endogeneity are not that strong, panel IM(Fb) tends to have rejections close
to 10%, whereas all other tests have over-rejection problems. In Figures 1.11 and 1.12,
when there is strong serial correlation and endogeneity, all the tests have some over-rejection
problems, however, the QS kernel leads to less size distorted results than the Bartlett kernel.
The overall picture is that the panel IM(Fb) test is the most robust statistic in terms
of controlling over-rejection problems although for given sample sizes, N , T , increasing the
values of ρ1 , ρ2 causes over-rejections to emerge. Large sample sizes of both N and T in
conjunction with large bandwidths and the QS kernel are desirable when serial correlation
and endogeneity are strong.
Now, we turn to the analysis of the power properties of the tests. For the sake of brevity
we only display results for the case ρ1 = ρ2 = 0.6 for the Wald test for N = 5, T = 50
and using the QS kernel. Patterns are similar for other values of ρ1 , ρ2 for t tests for other
combinations of N, T with the Bartlett kernel. Starting from the null values of β1 and β2
equal to 1, we consider under the alternative β1 = β2 = β ∈ (1, 1.4], using (including the
null value) a total of 21 values on a grid with mesh 0.02. We focus on size-corrected power
because of the potential over-rejection problems under the null hypothesis. This allows us
to see power differences across tests while holding null rejection probabilities constant at
29

0.05. This is useful for the theoretical power comparisons, but such size-corrections are not
feasible in practice.
In Figure 1.13 we display size adjusted power of the panel FM(Fb) and panel IM-OLS
Wald tests using the QS kernel with b = 0.3. For all other values of b, the patterns are very
similar. From Figure 1.13, we can see that panel IM(Fb) has the least power across the four
2∗ to obtain asymptotically fixed-b inference and less finite sample size
tests. The use of σ
˜u·v

distortions comes at the price of a small reduction in power.
Figure 1.14 shows the effect of the bandwidth on size adjusted power of the panel IM(Fb)
test by plotting power curves for eight values of b = 0.02, 0.06, 0.1, 0.3, 0.5, 0.7, 0.9, 1.0. We
can see that panel IM(Fb) power depends on the bandwidth and tends to decrease as bandwidth increases, but power is not that sensitive to the bandwidth. In addition, when b ≥ 0.5,
the power of panel IM(Fb) is almost constant. In Figure 1.15, we display power using the
Bartlett kernel, and it is clear that all tests almost have similar power, and the power is not
sensitive to b.
Figure 1.16 gives power comparisons across the various tests: pooled OLS, panel FM-OLS,
panel DOLS, panel IM-OLS. In Figure 1.16, panel IM(Fb) test is shown for b = 0.06, 0.1, 1.0,
and using the Andrews data dependent bandwidth. The panel FM-OLS test is implemented
with the Andrews data dependent bandwidth. We note that the pooled OLS and panel FMOLS tests have the largest size-adjusted power, with the power of panel DOLS test being
slightly smaller and panel IM-OLS tests have the smallest power. But the power difference
between panel IM-OLS and all other tests are relatively small.
Finally, Figures 1.17-1.19 provide size adjusted power comparisons similar to Figures
1.13, 1.14 and 1.16 but with N = 10. The main feature is that size adjusted power increases
with N . In addition, as N increases, the power of panel IM(Fb) becomes less sensitive to the
30

bandwidth. With a larger N , the power rankings are the same as before, but the difference
of the power between panel IM-OLS and panel FM-OLS is smaller.

1.6

Summary and conclusions

This paper considers the extension of the integrated modified ordinary least squares
(IM-OLS) method of Vogelsang and Wagner [2014] for estimation and inference about a
cointegrating vector in homogeneous cointegrated panels. We label the estimator panel
IM-OLS. It is a tuning parameter free estimator that is based on a partial sum transformed
regression augmented by the original integrated regressors themselves. The advantage is that
it leads to a zero mean mixed Gaussian limiting distribution without requiring the choice
of tuning parameters (like bandwidth, kernel, numbers of leads and lags). For inference
based on panel IM-OLS estimates, a long run variance still needs to be scaled out. Using a
consistent estimator of the corresponding long run variance leads to tests having standard
asymptotic distributions. Fixed-b inference is another way to obtain pivotal test statistics.
Critical values of fixed-b t and Wald tests need to be simulated taking into account the
specification of deterministic components, the number of integrated regressors, the kernel
function and the bandwidth choice.
We provide a finite sample simulation study in which the performance of the panel IMOLS estimator and test statistics are compared with pooled OLS, panel FM-OLS and panel
DOLS. Typically, panel IM-OLS shows good performance in terms of bias and RMSE especially in the following two scenarios: (i) the panel has large sample size; (ii) small sample size
panel with strong serial correlation and endogeneity. The size and power analysis of the tests
31

show that the fixed-b test statistics are more robust, in terms of having lower size distortions
than all other test statistics, especially for larger sample sizes. This robustness comes at the
cost of minor power losses provided serial correlation and endogeneity is not that strong.
When there is strong serial correlation and endogeneity, all tests have severe over-rejection
problems, and we prefer panel IM-OLS test with QS kernel and large bandwidth in this case.
Further research will study panel IM-OLS estimator for panels that have identical dependent unit in cross section, panels that have non-identical dependent unit in cross section,
for higher order cointegrating regressions and for nonlinear cointegration relationships.

32

APPENDIX

33

Tables and Figures
Table 1.1: Finite sample bias and RMSE of the various estimator of β1 , N = 5, T = 50,
Bartlett kernel
ρ1

ρ2

P-OLS

P-IM

P-DOLS

Panel FM-OLS
b=0.06

0.1

0.3

0.5

0.7

0.9

AND

Panel A: Bias
0

0.3

0.6

0.9

0

-0.0002

-0.0001

-0.0004

-0.0003

-0.0003

-0.0003

-0.0003

-0.0003

-0.0003

-0.0003

0.3

0.0057

-0.0004

-0.0006

-0.0004

0.0003

0.0021

0.0030

0.0036

0.0040

0.0001

0.6

0.0116

-0.0008

-0.0003

-0.0006

0.0009

0.0044

0.0063

0.0075

0.0082

0.0005

0.9

0.0175

-0.0011

-0.0002

-0.0007

0.0015

0.0068

0.0096

0.0113

0.0124

0.0010

0

-0.0003

-0.0002

-0.0004

-0.0004

-0.0004

-0.0004

-0.0004

-0.0004

-0.0004

-0.0004

0.3

0.0100

-0.0001

-0.0001

0.0013

0.0017

0.0039

0.0055

0.0064

0.0071

0.0016

0.6

0.0202

0.0001

0.0001

0.0029

0.0038

0.0083

0.0113

0.0132

0.0145

0.0035

0.9

0.0304

0.0002

0.0002

0.0045

0.0059

0.0126

0.0171

0.0200

0.0219

0.0055

0

-0.0004

-0.0003

-0.0008

-0.0005

-0.0005

-0.0006

-0.0005

-0.0005

-0.0005

-0.0005

0.3

0.0200

0.0024

0.0034

0.0083

0.0075

0.0091

0.0116

0.0134

0.0146

0.0076

0.6

0.0404

0.0051

0.0068

0.0172

0.0155

0.0188

0.0238

0.0274

0.0297

0.0158

0.9

0.0608

0.0078

0.0090

0.0260

0.0235

0.0285

0.0360

0.0413

0.0448

0.0240

0

-0.0009

-0.0007

-0.0007

-0.0011

-0.0011

-0.0012

-0.0011

-0.0010

-0.0010

-0.0011

0.3

0.0715

0.0404

0.0521

0.0598

0.0568

0.0511

0.0529

0.0561

0.0587

0.0575

0.6

0.1438

0.0815

0.1031

0.1206

0.1147

0.1035

0.1069

0.1131

0.1183

0.1162

0.9

0.2162

0.1226

0.1534

0.1815

0.1726

0.1559

0.1610

0.1702

0.1780

0.1749

0

0.0115

0.0202

0.0147

0.0119

0.0120

0.0124

0.0124

0.0124

0.0124

0.0120

0.3

0.0134

0.0202

0.0151

0.0119

0.0121

0.0128

0.0131

0.0133

0.0134

0.0121

0.6

0.0180

0.0202

0.0156

0.0121

0.0124

0.0141

0.0151

0.0157

0.0160

0.0123

0.9

0.0239

0.0203

0.0158

0.0123

0.0130

0.0160

0.0179

0.0190

0.0197

0.0127

Panel B: RMSE
0

0.3

0.6

0.9

0

0.0161

0.0286

0.0211

0.0168

0.0170

0.0175

0.0175

0.0175

0.0175

0.0169

0.3

0.0198

0.0286

0.0213

0.0169

0.0172

0.0183

0.0189

0.0192

0.0194

0.0171

0.6

0.0283

0.0286

0.0215

0.0173

0.0179

0.0208

0.0227

0.0238

0.0246

0.0177

0.9

0.0386

0.0287

0.0217

0.0180

0.0192

0.0244

0.0279

0.0300

0.0313

0.0188

0

0.0270

0.0490

0.0358

0.0283

0.0287

0.0297

0.0297

0.0296

0.0295

0.0286

0.3

0.0352

0.0491

0.0365

0.0298

0.0301

0.0318

0.0330

0.0337

0.0341

0.0300

0.6

0.0529

0.0494

0.0378

0.0345

0.0341

0.0379

0.0416

0.0439

0.0455

0.0341

0.9

0.0735

0.0499

0.0391

0.0411

0.0401

0.0464

0.0529

0.0571

0.0598

0.0402

0

0.0843

0.1638

0.1090

0.0887

0.0910

0.0967

0.0973

0.0967

0.0958

0.0904

0.3

0.1131

0.1689

0.1231

0.1089

0.1091

0.1113

0.1131

0.1142

0.1149

0.1090

0.6

0.1736

0.1839

0.1563

0.1552

0.1518

0.1475

0.1512

0.1557

0.1591

0.1526

0.9

0.2432

0.2068

0.1986

0.2111

0.2041

0.1935

0.1992

0.2071

0.2134

0.2059

34

Table 1.2: Finite sample bias and RMSE of the various estimator of β1 , N = 5, T = 100,
Bartlett kernel
ρ1

ρ2

P-OLS

P-IM

P-DOLS

Panel FM-OLS
b=0.06

0.1

0.3

0.5

0.7

0.9

AND

Panel A: Bias
0

0.3

0.6

0.9

0

0.0001

0.0001

0.0000

0.0001

0.0001

0.0000

0.0000

0.0000

0.0000

0.0001

0.3

0.0030

0.0000

0.0001

0.0002

0.0005

0.0012

0.0017

0.0020

0.0021

0.0001

0.6

0.0060

0.0000

0.0002

0.0003

0.0009

0.0024

0.0033

0.0039

0.0043

0.0002

0.9

0.0089

-0.0001

0.0001

0.0004

0.0013

0.0036

0.0050

0.0058

0.0064

0.0003

0

0.0001

0.0002

0.0002

0.0001

0.0001

0.0001

0.0000

0.0000

0.0000

0.0001

0.3

0.0052

0.0002

0.0003

0.0008

0.0010

0.0022

0.0029

0.0034

0.0038

0.0007

0.6

0.0104

0.0002

0.0004

0.0015

0.0020

0.0043

0.0059

0.0069

0.0075

0.0014

0.9

0.0156

0.0003

0.0004

0.0022

0.0030

0.0064

0.0088

0.0103

0.0113

0.0021

0

0.0002

0.0003

0.0002

0.0002

0.0002

0.0001

0.0001

0.0000

0.0000

0.0002

0.3

0.0107

0.0011

0.0011

0.0034

0.0032

0.0047

0.0061

0.0071

0.0078

0.0035

0.6

0.0212

0.0018

0.0015

0.0067

0.0062

0.0093

0.0122

0.0142

0.0155

0.0069

0.9

0.0317

0.0026

0.0016

0.0099

0.0093

0.0139

0.0183

0.0212

0.0232

0.0102

0

0.0008

0.0012

0.0002

0.0008

0.0008

0.0006

0.0005

0.0004

0.0004

0.0008

0.3

0.0431

0.0174

0.0174

0.0325

0.0294

0.0261

0.0286

0.0314

0.0335

0.0331

0.6

0.0854

0.0336

0.0335

0.0642

0.0580

0.0515

0.0567

0.0623

0.0666

0.0654

0.9

0.1277

0.0499

0.0486

0.0959

0.0866

0.0769

0.0849

0.0933

0.0997

0.0976

0

0.0056

0.0102

0.0082

0.0058

0.0059

0.0061

0.0062

0.0062

0.0061

0.0058

0.3

0.0067

0.0102

0.0085

0.0058

0.0059

0.0064

0.0066

0.0067

0.0067

0.0058

0.6

0.0091

0.0102

0.0087

0.0059

0.0061

0.0070

0.0076

0.0079

0.0081

0.0059

0.9

0.0121

0.0102

0.0089

0.0061

0.0065

0.0080

0.0090

0.0096

0.0099

0.0060

Panel B: RMSE
0

0.3

0.6

0.9

0

0.0080

0.0145

0.0131

0.0083

0.0083

0.0087

0.0088

0.0088

0.0087

0.0082

0.3

0.0101

0.0145

0.0133

0.0083

0.0085

0.0092

0.0096

0.0098

0.0099

0.0083

0.6

0.0145

0.0145

0.0135

0.0086

0.0089

0.0105

0.0115

0.0121

0.0125

0.0085

0.9

0.0198

0.0145

0.0137

0.0090

0.0096

0.0123

0.0142

0.0153

0.0160

0.0089

0

0.0136

0.0251

0.0254

0.0142

0.0144

0.0150

0.0151

0.0151

0.0150

0.0142

0.3

0.0184

0.0252

0.0258

0.0149

0.0150

0.0162

0.0170

0.0174

0.0177

0.0148

0.6

0.0278

0.0252

0.0262

0.0164

0.0166

0.0193

0.0214

0.0228

0.0237

0.0165

0.9

0.0387

0.0253

0.0264

0.0187

0.0188

0.0235

0.0272

0.0296

0.0311

0.0188

0

0.0474

0.0932

0.0855

0.0501

0.0513

0.0541

0.0545

0.0541

0.0537

0.0499

0.3

0.0669

0.0950

0.0881

0.0616

0.0609

0.0619

0.0637

0.0648

0.0656

0.0618

0.6

0.1046

0.0997

0.0957

0.0865

0.0822

0.0800

0.0848

0.0888

0.0917

0.0874

0.9

0.1471

0.1071

0.1045

0.1164

0.1086

0.1031

0.1111

0.1182

0.1234

0.1180

35

Table 1.3: Finite sample bias and RMSE of the various estimator of β1 , N = 10, T = 50,
Bartlett kernel
ρ1

ρ2

P-OLS

P-IM

P-DOLS

Panel FM-OLS
b=0.06

0.1

0.3

0.5

0.7

0.9

AND

Panel A: Bias
0

0.3

0.6

0.9

0

-0.0001

0.0000

-0.0001

-0.0001

-0.0001

-0.0001

0.0000

0.0000

0.0000

-0.0001

0.3

0.0055

-0.0002

-0.0003

-0.0003

0.0003

0.0017

0.0026

0.0033

0.0037

0.0002

0.6

0.0111

-0.0005

0.0000

-0.0005

0.0006

0.0035

0.0053

0.0065

0.0074

0.0004

0.9

0.0167

-0.0008

0.0001

-0.0007

0.0010

0.0053

0.0080

0.0098

0.0111

0.0006

0

-0.0001

0.0001

-0.0001

-0.0001

-0.0001

-0.0001

0.0000

-0.0001

0.0000

-0.0001

0.3

0.0092

0.0001

0.0002

0.0012

0.0014

0.0032

0.0046

0.0056

0.0063

0.0014

0.6

0.0186

0.0002

0.0004

0.0024

0.0030

0.0064

0.0092

0.0112

0.0126

0.0028

0.9

0.0279

0.0003

0.0005

0.0037

0.0045

0.0097

0.0138

0.0168

0.0189

0.0043

0

-0.0001

0.0001

-0.0003

-0.0002

-0.0002

-0.0001

-0.0001

-0.0001

0.0000

-0.0002

0.3

0.0182

0.0024

0.0036

0.0074

0.0064

0.0074

0.0096

0.0114

0.0127

0.0066

0.6

0.0365

0.0047

0.0066

0.0149

0.0130

0.0150

0.0193

0.0229

0.0254

0.0134

0.9

0.0548

0.0070

0.0085

0.0224

0.0196

0.0225

0.0290

0.0343

0.0382

0.0202

0

0.0001

0.0005

-0.0001

0.0001

0.0000

0.0001

0.0003

0.0004

0.0005

0.0001

0.3

0.0672

0.0386

0.0488

0.0560

0.0530

0.0469

0.0482

0.0512

0.0541

0.0538

0.6

0.1343

0.0768

0.0959

0.1119

0.1060

0.0937

0.0960

0.1020

0.1077

0.1075

0.9

0.2014

0.1149

0.1419

0.1679

0.1590

0.1405

0.1439

0.1528

0.1613

0.1612

0

0.0069

0.0118

0.0088

0.0071

0.0072

0.0074

0.0075

0.0075

0.0074

0.0072

0.3

0.0091

0.0118

0.0090

0.0071

0.0072

0.0077

0.0081

0.0083

0.0084

0.0072

0.6

0.0138

0.0118

0.0092

0.0072

0.0073

0.0086

0.0097

0.0104

0.0110

0.0073

0.9

0.0193

0.0119

0.0094

0.0073

0.0076

0.0099

0.0119

0.0133

0.0142

0.0075

Panel B: RMSE
0

0.3

0.6

0.9

0

0.0098

0.0168

0.0125

0.0101

0.0102

0.0105

0.0106

0.0106

0.0105

0.0101

0.3

0.0139

0.0168

0.0127

0.0102

0.0103

0.0111

0.0118

0.0122

0.0125

0.0103

0.6

0.0220

0.0168

0.0128

0.0105

0.0108

0.0129

0.0148

0.0161

0.0171

0.0107

0.9

0.0312

0.0168

0.0130

0.0110

0.0116

0.0154

0.0187

0.0212

0.0229

0.0114

0

0.0166

0.0290

0.0215

0.0172

0.0174

0.0180

0.0181

0.0181

0.0179

0.0173

0.3

0.0253

0.0291

0.0221

0.0189

0.0187

0.0198

0.0210

0.0218

0.0225

0.0187

0.6

0.0418

0.0294

0.0233

0.0233

0.0223

0.0244

0.0278

0.0305

0.0325

0.0225

0.9

0.0600

0.0299

0.0244

0.0293

0.0273

0.0307

0.0365

0.0411

0.0444

0.0277

0

0.0548

0.0994

0.0688

0.0568

0.0579

0.0610

0.0616

0.0612

0.0606

0.0576

0.3

0.0878

0.1068

0.0854

0.0806

0.0792

0.0776

0.0790

0.0807

0.0820

0.0795

0.6

0.1477

0.1261

0.1210

0.1276

0.1227

0.1138

0.1163

0.1213

0.1259

0.1240

0.9

0.2129

0.1529

0.1620

0.1805

0.1724

0.1564

0.1602

0.1684

0.1760

0.1744

36

Table 1.4: Finite sample bias and RMSE of the various estimator of β1 , N = 10, T = 100,
Bartlett kernel
ρ1

ρ2

P-OLS

P-IM

P-DOLS

Panel FM-OLS
b=0.06

0.1

0.3

0.5

0.7

0.9

AND

Panel A: Bias
0

0.3

0.6

0.9

0

0.0000

-0.0001

0.0000

0.0000

0.0000

-0.0001

-0.0001

-0.0001

-0.0001

0.0000

0.3

0.0027

-0.0001

0.0000

0.0000

0.0002

0.0009

0.0013

0.0016

0.0018

0.0000

0.6

0.0055

-0.0002

0.0001

0.0001

0.0005

0.0018

0.0026

0.0033

0.0037

0.0000

0.9

0.0083

-0.0003

0.0000

0.0002

0.0008

0.0027

0.0040

0.0049

0.0056

0.0000

0

-0.0001

-0.0001

0.0000

-0.0001

-0.0001

-0.0001

-0.0001

-0.0001

-0.0001

-0.0001

0.3

0.0046

-0.0001

0.0001

0.0005

0.0006

0.0015

0.0022

0.0027

0.0031

0.0004

0.6

0.0093

-0.0001

0.0002

0.0010

0.0013

0.0031

0.0045

0.0055

0.0063

0.0009

0.9

0.0140

0.0000

0.0002

0.0015

0.0020

0.0048

0.0068

0.0084

0.0094

0.0014

0

-0.0001

-0.0002

0.0000

-0.0001

-0.0001

-0.0001

-0.0002

-0.0002

-0.0002

-0.0001

0.3

0.0092

0.0005

0.0009

0.0026

0.0023

0.0034

0.0046

0.0056

0.0063

0.0027

0.6

0.0186

0.0011

0.0012

0.0052

0.0047

0.0069

0.0094

0.0113

0.0127

0.0055

0.9

0.0280

0.0017

0.0013

0.0079

0.0071

0.0104

0.0142

0.0170

0.0191

0.0082

0

-0.0001

-0.0005

-0.0005

-0.0001

-0.0002

-0.0003

-0.0003

-0.0004

-0.0004

-0.0001

0.3

0.0381

0.0141

0.0146

0.0281

0.0251

0.0213

0.0233

0.0260

0.0282

0.0286

0.6

0.0763

0.0287

0.0287

0.0563

0.0504

0.0429

0.0470

0.0523

0.0568

0.0574

0.9

0.1145

0.0433

0.0417

0.0846

0.0757

0.0645

0.0707

0.0787

0.0853

0.0862

0

0.0034

0.0059

0.0047

0.0035

0.0035

0.0036

0.0037

0.0036

0.0036

0.0035

0.3

0.0045

0.0059

0.0049

0.0035

0.0035

0.0038

0.0039

0.0040

0.0041

0.0035

0.6

0.0068

0.0059

0.0050

0.0035

0.0036

0.0042

0.0048

0.0051

0.0054

0.0035

0.9

0.0096

0.0059

0.0051

0.0036

0.0038

0.0050

0.0059

0.0066

0.0071

0.0036

Panel B: RMSE
0

0.3

0.6

0.9

0

0.0048

0.0084

0.0075

0.0050

0.0050

0.0052

0.0052

0.0052

0.0051

0.0049

0.3

0.0069

0.0084

0.0075

0.0050

0.0051

0.0055

0.0058

0.0060

0.0061

0.0050

0.6

0.0110

0.0084

0.0077

0.0051

0.0053

0.0064

0.0073

0.0080

0.0085

0.0051

0.9

0.0157

0.0084

0.0078

0.0053

0.0057

0.0076

0.0093

0.0105

0.0114

0.0053

0

0.0083

0.0146

0.0145

0.0086

0.0087

0.0089

0.0090

0.0089

0.0089

0.0085

0.3

0.0128

0.0146

0.0148

0.0090

0.0090

0.0097

0.0103

0.0108

0.0111

0.0090

0.6

0.0213

0.0147

0.0151

0.0102

0.0101

0.0119

0.0137

0.0152

0.0162

0.0103

0.9

0.0307

0.0148

0.0152

0.0121

0.0117

0.0148

0.0181

0.0206

0.0223

0.0123

0

0.0299

0.0548

0.0519

0.0311

0.0317

0.0331

0.0333

0.0330

0.0326

0.0310

0.3

0.0494

0.0566

0.0540

0.0425

0.0410

0.0399

0.0414

0.0428

0.0439

0.0429

0.6

0.0845

0.0620

0.0608

0.0661

0.0611

0.0559

0.0598

0.0641

0.0676

0.0671

0.9

0.1223

0.0703

0.0692

0.0930

0.0848

0.0755

0.0818

0.0890

0.0949

0.0946

37

Table 1.5: Finite sample bias and RMSE of the various estimator of β1 , N = 25, T = 50,
Bartlett kernel
ρ1

ρ2

P-OLS

P-IM

P-DOLS

Panel FM-OLS
b=0.06

0.1

0.3

0.5

0.7

0.9

AND

Panel A: Bias
0

0.3

0.6

0.9

0

-0.0001

0.0001

-0.0001

-0.0001

-0.0001

0.0000

0.0000

0.0000

0.0000

-0.0001

0.3

0.0054

-0.0002

-0.0002

-0.0003

0.0002

0.0015

0.0023

0.0030

0.0034

0.0001

0.6

0.0108

-0.0004

0.0000

-0.0005

0.0005

0.0030

0.0047

0.0060

0.0069

0.0003

0.9

0.0162

-0.0007

0.0001

-0.0007

0.0008

0.0045

0.0071

0.0090

0.0104

0.0004

0

-0.0001

0.0001

-0.0001

-0.0001

-0.0001

-0.0001

0.0000

0.0000

-0.0001

-0.0001

0.3

0.0088

0.0002

0.0002

0.0010

0.0012

0.0027

0.0040

0.0050

0.0057

0.0012

0.6

0.0177

0.0003

0.0004

0.0021

0.0026

0.0055

0.0080

0.0100

0.0115

0.0024

0.9

0.0266

0.0003

0.0005

0.0033

0.0039

0.0083

0.0121

0.0151

0.0173

0.0037

0

-0.0001

0.0002

-0.0002

-0.0002

-0.0002

-0.0001

-0.0001

-0.0001

-0.0001

-0.0002

0.3

0.0172

0.0024

0.0034

0.0067

0.0058

0.0065

0.0085

0.0102

0.0116

0.0060

0.6

0.0345

0.0045

0.0062

0.0137

0.0118

0.0131

0.0170

0.0205

0.0232

0.0122

0.9

0.0518

0.0066

0.0080

0.0206

0.0178

0.0197

0.0256

0.0308

0.0348

0.0184

0

0.0000

0.0008

-0.0002

-0.0001

-0.0002

-0.0001

0.0000

0.0001

0.0001

-0.0002

0.3

0.0644

0.0378

0.0467

0.0534

0.0505

0.0441

0.0451

0.0480

0.0510

0.0512

0.6

0.1289

0.0747

0.0919

0.1070

0.1011

0.0884

0.0901

0.0959

0.1018

0.1026

0.9

0.1933

0.1116

0.1354

0.1605

0.1518

0.1327

0.1351

0.1439

0.1527

0.1540

0

0.0040

0.0068

0.0049

0.0041

0.0041

0.0042

0.0042

0.0042

0.0042

0.0041

0.3

0.0068

0.0068

0.0051

0.0041

0.0041

0.0045

0.0049

0.0052

0.0055

0.0041

0.6

0.0118

0.0068

0.0052

0.0042

0.0042

0.0054

0.0065

0.0075

0.0083

0.0042

0.9

0.0171

0.0068

0.0053

0.0042

0.0043

0.0066

0.0086

0.0103

0.0115

0.0043

Panel B: RMSE
0

0.3

0.6

0.9

0

0.0056

0.0097

0.0071

0.0058

0.0058

0.0059

0.0060

0.0059

0.0059

0.0058

0.3

0.0106

0.0097

0.0072

0.0059

0.0060

0.0066

0.0073

0.0079

0.0083

0.0059

0.6

0.0190

0.0097

0.0072

0.0062

0.0065

0.0084

0.0103

0.0120

0.0132

0.0064

0.9

0.0278

0.0097

0.0074

0.0067

0.0072

0.0107

0.0140

0.0167

0.0188

0.0071

0

0.0096

0.0167

0.0122

0.0099

0.0100

0.0102

0.0102

0.0101

0.0100

0.0099

0.3

0.0199

0.0169

0.0128

0.0120

0.0116

0.0122

0.0135

0.0146

0.0155

0.0117

0.6

0.0364

0.0173

0.0141

0.0171

0.0157

0.0170

0.0204

0.0234

0.0258

0.0160

0.9

0.0536

0.0180

0.0152

0.0232

0.0208

0.0229

0.0284

0.0333

0.0370

0.0213

0

0.0322

0.0578

0.0397

0.0332

0.0337

0.0351

0.0352

0.0349

0.0345

0.0336

0.3

0.0725

0.0691

0.0618

0.0633

0.0610

0.0567

0.0576

0.0597

0.0619

0.0616

0.6

0.1338

0.0946

0.1012

0.1128

0.1073

0.0959

0.0976

0.1030

0.1084

0.1087

0.9

0.1975

0.1260

0.1427

0.1651

0.1566

0.1384

0.1410

0.1494

0.1578

0.1587

38

Table 1.6: Finite sample bias and RMSE of the various estimator of β1 , N = 25, T = 100,
Bartlett kernel
ρ1

ρ2

P-OLS

P-IM

P-DOLS

Panel FM-OLS
b=0.06

0.1

0.3

0.5

0.7

0.9

AND

Panel A: Bias
0

0.3

0.6

0.9

0

0.0000

-0.0001

0.0000

0.0000

0.0000

0.0000

0.0000

-0.0001

-0.0001

0.0000

0.3

0.0027

-0.0002

0.0000

0.0000

0.0002

0.0007

0.0011

0.0015

0.0017

0.0000

0.6

0.0054

-0.0002

0.0001

0.0000

0.0004

0.0015

0.0023

0.0030

0.0034

0.0000

0.9

0.0081

-0.0003

0.0000

0.0001

0.0006

0.0023

0.0035

0.0045

0.0052

0.0000

0

-0.0001

-0.0001

0.0000

0.0000

0.0000

-0.0001

-0.0001

-0.0001

-0.0001

-0.0001

0.3

0.0044

-0.0001

0.0001

0.0004

0.0005

0.0013

0.0019

0.0024

0.0028

0.0004

0.6

0.0089

-0.0001

0.0002

0.0008

0.0011

0.0027

0.0040

0.0050

0.0057

0.0008

0.9

0.0134

-0.0001

0.0002

0.0013

0.0017

0.0040

0.0060

0.0075

0.0086

0.0012

0

-0.0001

-0.0002

-0.0001

-0.0001

-0.0001

-0.0001

-0.0001

-0.0001

-0.0001

-0.0001

0.3

0.0087

0.0004

0.0008

0.0023

0.0020

0.0029

0.0040

0.0050

0.0057

0.0024

0.6

0.0175

0.0009

0.0011

0.0047

0.0041

0.0059

0.0082

0.0101

0.0115

0.0049

0.9

0.0264

0.0015

0.0012

0.0071

0.0062

0.0089

0.0123

0.0151

0.0173

0.0074

0

-0.0003

-0.0009

-0.0001

-0.0003

-0.0002

-0.0003

-0.0003

-0.0003

-0.0003

-0.0003

0.3

0.0360

0.0130

0.0142

0.0263

0.0235

0.0194

0.0212

0.0238

0.0261

0.0269

0.6

0.0722

0.0270

0.0271

0.0530

0.0472

0.0391

0.0427

0.0479

0.0525

0.0540

0.9

0.1084

0.0409

0.0390

0.0796

0.0709

0.0588

0.0642

0.0720

0.0789

0.0811

0

0.0020

0.0035

0.0026

0.0020

0.0021

0.0021

0.0021

0.0021

0.0021

0.0020

0.3

0.0034

0.0035

0.0027

0.0020

0.0021

0.0023

0.0025

0.0026

0.0027

0.0020

0.6

0.0059

0.0035

0.0028

0.0021

0.0021

0.0027

0.0033

0.0038

0.0041

0.0021

0.9

0.0085

0.0035

0.0029

0.0021

0.0023

0.0033

0.0043

0.0051

0.0058

0.0021

Panel B: RMSE
0

0.3

0.6

0.9

0

0.0028

0.0050

0.0041

0.0029

0.0029

0.0030

0.0030

0.0030

0.0030

0.0029

0.3

0.0053

0.0050

0.0042

0.0029

0.0030

0.0033

0.0037

0.0040

0.0042

0.0029

0.6

0.0095

0.0050

0.0043

0.0031

0.0032

0.0042

0.0052

0.0060

0.0066

0.0030

0.9

0.0140

0.0050

0.0044

0.0032

0.0035

0.0053

0.0070

0.0083

0.0094

0.0032

0

0.0049

0.0087

0.0081

0.0050

0.0051

0.0052

0.0053

0.0052

0.0051

0.0050

0.3

0.0102

0.0087

0.0083

0.0056

0.0055

0.0061

0.0068

0.0073

0.0078

0.0056

0.6

0.0186

0.0087

0.0085

0.0070

0.0067

0.0081

0.0100

0.0116

0.0129

0.0072

0.9

0.0273

0.0088

0.0086

0.0089

0.0083

0.0107

0.0139

0.0165

0.0185

0.0092

0

0.0181

0.0328

0.0297

0.0186

0.0189

0.0196

0.0197

0.0195

0.0192

0.0186

0.3

0.0407

0.0354

0.0333

0.0326

0.0304

0.0279

0.0293

0.0311

0.0327

0.0330

0.6

0.0753

0.0426

0.0412

0.0568

0.0515

0.0445

0.0479

0.0526

0.0567

0.0578

0.9

0.1113

0.0527

0.0504

0.0827

0.0743

0.0631

0.0684

0.0759

0.0824

0.0843

39

Table 1.7: Empirical null rejection probabilities, 0.05 level, t-tests for h0 : β1 = 1, N = 5,
data dependent bandwidths and lag lengths.
ρ1 , ρ2 P-OLS

Bartlett kernel
P-

P-FM

DOLS

QS kernel

P-

P-

P-

P-

IM(O)

IM(D)

IM(Fb)

DOLS

P-FM

P-

P-

P-

IM(O)

IM(D)

IM(Fb)

Panel A: T = 50
0

0.0514

0.1294

0.0762

0.0724

0.0544

0.0744

0.1264

0.0836

0.0808

0.0624

0.0422

0.3

0.2178

0.1792

0.1032

0.0974

0.0796

0.132

0.1632

0.0984

0.0936

0.0746

0.101

0.6

0.6064

0.2452

0.2214

0.1462

0.1212

0.2694

0.2176

0.1908

0.1246

0.109

0.1924

0.9

0.9366

0.6068

0.777

0.4804

0.422

0.7628

0.5736

0.7336

0.4462

0.4042

0.7052

0.0524

0.0458

Panel B: T = 100
0

0.0472

0.1416

0.0588

0.0586

0.0516

0.0524

0.1402

0.062

0.0626

0.3
0.6

0.2082

0.227

0.0794

0.0774

0.0678

0.1038

0.2158

0.0756

0.0726

0.0602

0.0866

0.5914

0.3256

0.1584

0.111

0.0952

0.1776

0.2976

0.1326

0.094

0.0784

0.1308

0.9

0.934

0.4804

0.7266

0.3376

0.286

0.6198

0.4494

0.6974

0.3112

0.2626

0.561

Table 1.8: Empirical null rejection probabilities, 0.05 level, t-tests for h0 : β1 = 1, N = 10,
data dependent bandwidths and lag lengths.
ρ1 , ρ2 P-OLS

Bartlett kernel
P-

P-FM

DOLS

QS kernel

P-

P-

P-

P-

IM(O)

IM(D)

IM(Fb)

DOLS

P-FM

P-

P-

P-

IM(O)

IM(D)

IM(Fb)

Panel A: T = 50
0

0.051

0.1142

0.0654

0.0708

0.0642

0.0762

0.1116

0.0712

0.08

0.072

0.0516

0.3

0.2846

0.1652

0.0902

0.0982

0.0908

0.1418

0.1454

0.0848

0.0912

0.0852

0.1054

0.6

0.7924

0.2188

0.2422

0.1482

0.1352

0.3306

0.1898

0.1994

0.1256

0.1222

0.2892

0.9

0.9924

0.7518

0.9332

0.5832

0.5276

0.8576

0.7094

0.908

0.5464

0.5094

0.8362

0

0.0476

0.139

0.0542

0.0582

0.0534

0.0536

0.137

0.0558

0.0628

0.057

0.046

0.3

0.2882

0.2252

0.0758

0.0794

0.075

0.1096

0.21

0.0662

0.0728

0.0688

0.0908

Panel B: T = 100

0.6

0.787

0.308

0.1802

0.1098

0.0984

0.1776

0.281

0.1466

0.0938

0.0848

0.1366

0.9

0.9896

0.4982

0.9002

0.3888

0.3396

0.6724

0.4598

0.8742

0.3588

0.316

0.6254

40

Table 1.9: Empirical null rejection probabilities, 0.05 level, t-tests for h0 : β1 = 1, N = 25,
data dependent bandwidths and lag lengths.
ρ1 , ρ2 P-OLS

Bartlett kernel
P-

P-FM

DOLS

QS kernel

P-

P-

P-

P-

IM(O)

IM(D)

IM(Fb)

DOLS

P-FM

P-

P-

P-

IM(O)

IM(D)

IM(Fb)

Panel A: T = 50
0

0.0482

0.1022

0.0566

0.0622

0.0592

0.0664

0.099

0.0622

0.0706

0.0682

0.0404

0.3

0.5116

0.1534

0.0822

0.0848

0.0826

0.1292

0.1358

0.0724

0.0796

0.079

0.0972

0.6

0.9786

0.2362

0.3852

0.1428

0.1328

0.335

0.2042

0.2952

0.1204

0.1202

0.2966

0.9

1

0.962

0.9976

0.8058

0.7674

0.9542

0.9448

0.9966

0.7804

0.7474

0.9476

0.1332

0.0562

0.0638

0.0612

0.046

Panel B: T = 100
0

0.0476

0.1354

0.0554

0.0612

0.0588

0.0558

0.3

0.5124

0.2032

0.0766

0.08

0.079

0.1106

0.188

0.067

0.0726

0.0716

0.0954

0.6

0.9744

0.2752

0.2708

0.1128

0.1056

0.1844

0.2478

0.2064

0.0988

0.095

0.1424

0.9

1

0.6172

0.9958

0.5474

0.5132

0.7944

0.5736

0.993

0.5224

0.4876

0.7528

Table 1.10: Empirical null rejection probabilities, 0.05 level, Wald-tests for h0 : β1 = 1, β2 =
1, N = 5, data dependent bandwidths and lag lengths.
ρ1 , ρ2 P-OLS

Bartlett kernel
P-

P-FM

DOLS

QS kernel

P-

P-

P-

P-

IM(O)

IM(D)

IM(Fb)

DOLS

P-FM

P-

P-

P-

IM(O)

IM(D)

IM(Fb)

0.0696

0.04

Panel A: T = 50
0

0.05

0.1552

0.3

0.3026

0.2344

0.6

0.822

0.3362

0.9

0.9972

0.8098

0

0.049

0.3

0.0794

0.0804

0.0574

0.0794

0.1514

0.0938

0.1006

0.1254

0.1188

0.3098

0.1996

0.0952

0.1746

0.2086

0.1168

0.1156

0.088

0.122

0.161

0.3802

0.297

0.2578

0.167

0.1408

0.2664

0.9486

0.7018

0.6244

0.9414

0.7728

0.9202

0.6646

0.6006

0.9068

0.185

0.0626

0.0636

0.0478

0.056

0.1832

0.0686

0.071

0.052

0.0418

0.2898

0.3258

0.0908

0.0912

0.0772

0.1302

0.3048

0.085

0.0812

0.067

0.105

0.6
0.9

0.8276

0.4574

0.2068

0.1372

0.1142

0.2356

0.4216

0.1696

0.1168

0.0952

0.169

0.9982

0.6786

0.9156

0.5056

0.4282

0.8274

0.6352

0.8962

0.4684

0.3962

0.7724

Panel B: T = 100

41

Table 1.11: Empirical null rejection probabilities, 0.05 level, Wald-tests for h0 : β1 = 1, β2 =
1, N = 10, data dependent bandwidths and lag lengths.
ρ1 , ρ2 P-OLS

Bartlett kernel
P-

P-FM

DOLS

QS kernel

P-

P-

P-

P-

IM(O)

IM(D)

IM(Fb)

DOLS

P-FM

P-

P-

P-

IM(O)

IM(D)

IM(Fb)

Panel A: T = 50
0

0.0486

0.1396

0.0708

0.0798

0.0704

0.0874

0.134

0.0766

0.0928

0.078

0.049

0.3

0.4166

0.2162

0.109

0.1148

0.1038

0.1782

0.1886

0.099

0.1094

0.0976

0.127

0.6

0.9562

0.3076

0.3512

0.1914

0.1706

0.4722

0.2586

0.2786

0.1558

0.1494

0.4046

0.9

1

0.916

0.9952

0.8002

0.7398

0.9756

0.8894

0.9888

0.765

0.7182

0.9672

0.0608

0.0444

Panel B: T = 100
0

0.0484

0.1718

0.0602

0.0636

0.0564

0.0546

0.169

0.065

0.068

0.3

0.4212

0.311

0.085

0.6

0.9602

0.4354

0.2532

0.091

0.082

0.1342

0.2854

0.0762

0.0806

0.073

0.1084

0.1358

0.1186

0.2432

0.3916

0.1936

0.1106

0.0964

0.1782

0.9

1

0.688

0.9916

0.566

0.5024

0.8694

0.6454

0.9854

0.528

0.468

0.8304

Table 1.12: Empirical null rejection probabilities, 0.05 level, Wald-tests for h0 : β1 = 1, β2 =
1, N = 25, data dependent bandwidths and lag lengths.
ρ1 , ρ2 P-OLS

Bartlett kernel
P-

P-FM

DOLS

QS kernel

P-

P-

P-

P-

IM(O)

IM(D)

IM(Fb)

DOLS

P-FM

P-

P-

P-

IM(O)

IM(D)

IM(Fb)

Panel A: T = 50
0

0.0528

0.132

0.0656

0.073

0.0702

0.0836

0.1292

0.0702

0.085

0.0806

0.0472

0.3
0.6

0.7142

0.208

0.1128

0.9998

0.3328

0.5454

0.1078

0.105

0.1686

0.1796

0.0984

0.1016

0.0982

0.1202

0.1914

0.1786

0.484

0.2736

0.4302

0.1598

0.1608

0.4234

0.9

1

0.9986

1

0.9474

0.9264

0.9978

0.9972

1

0.9344

0.9138

0.9962

0

0.0538

0.1706

0.0606

0.0628

0.0596

0.058

0.1674

0.0628

0.069

0.065

0.046

0.3

0.7098

0.2822

0.0868

0.0876

0.0866

0.135

0.257

0.076

0.08

0.0786

0.1074

0.6

0.9998

0.4058

0.3832

0.1362

0.125

0.2498

0.3598

0.2922

0.1124

0.1054

0.1796

0.9

1

0.8108

1

0.7578

0.7186

0.9488

0.774

1

0.7304

0.6874

0.93

Panel B: T = 100

42

Table 1.13: Fixed-b asymptotic critical value for t-test of β in regression with intercept and
two regressors, N = 25, Bartlett kernel
b

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

95%

1.7329

1.9363

2.1731

2.4357

2.7227

3.0220

3.3221

3.6112

3.8807

4.1140

97.5%

2.0630

2.3079

2.5845

2.8934

3.2298

3.5864

3.9396

4.2836

4.5986

4.8755

99%

2.4683

2.7561

3.0980

3.4713

3.8776

4.3055

4.7293

5.1508

5.5243

5.8506

99.5%

2.7275

3.0599

3.4300

3.8481

4.3085

4.7773

5.2548

5.7118

6.1411

6.5021

b

0.22

0.24

0.26

0.28

0.3

0.32

0.34

0.36

0.38

0.40

95%

4.3072

4.4702

4.6004

4.7217

4.8296

4.9306

5.0294

5.1291

5.2242

5.3160

97.5%

5.1085

5.2956

5.4627

5.5963

5.7331

5.8559

5.9673

6.0789

6.1903

6.3132

99%

6.1355

6.3751

6.5593

6.7345

6.8818

7.0326

7.1687

7.3157

7.4607

7.5930

99.5%

6.8015

7.0752

7.2862

7.4722

7.6352

7.7980

7.9644

8.1320

8.2883

8.4333

b

0.42

0.44

0.46

0.48

0.5

0.52

0.54

0.56

0.58

0.60

95%

5.4079

5.5070

5.5967

5.6805

5.7679

5.8447

5.9164

5.9942

6.0680

6.1366

97.5%

6.4189

6.5329

6.6388

6.7461

6.8517

6.9494

7.0410

7.1285

7.2094

7.2829

99%

7.7228

7.8553

7.9922

8.1188

8.2377

8.3674

8.4834

8.5767

8.6779

8.7842

99.5%

8.5820

8.7258

8.8732

9.0041

9.1516

9.2895

9.4089

9.5389

9.6542

9.7574

b

0.62

0.64

0.66

0.68

0.7

0.72

0.74

0.76

0.78

0.80

95%

6.2004

6.2597

6.3208

6.3811

6.4329

6.4885

6.5452

6.5982

6.6506

6.7020

97.5%

7.3600

7.4337

7.5122

7.5804

7.6560

7.7180

7.7900

7.8533

7.9185

7.9721

99%

8.8838

8.9709

9.0515

9.1402

9.2231

9.3065

9.3759

9.4616

9.5297

9.5959

99.5%

9.8619

9.9699

10.0559

10.1605

10.2384

10.3414

10.4381

10.5210

10.6046

10.6947

b

0.82

0.84

0.86

0.88

0.9

0.92

0.94

0.96

0.98

1.00

95%

6.7480

6.7906

6.8332

6.8817

6.9216

6.9658

7.0077

7.0485

7.0834

7.1205

97.5%

8.0248

8.0815

8.1347

8.1868

8.2422

8.2949

8.3433

8.3876

8.4348

8.4781

99%

9.6626

9.7342

9.8024

9.8573

9.9243

9.9824

10.0446

10.1029

10.1630

10.2177

99.5%

10.7659

10.8418

10.9134

10.9985

11.0667

11.1332

11.1945

11.2611

11.3375

11.4049

43

Table 1.14: Fixed-b asymptotic critical value for t-test of β in regression with intercept and
two regressors, N = 25, QS kernel
b

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

95%

1.7870

2.0829

2.4752

2.9884

3.6602

4.5278

5.5699

6.6972

7.7744

8.6329

97.5%

2.1270

2.4815

2.9371

3.5436

4.3513

5.3846

6.6572

8.0320

9.3128

10.3492

99%

2.5440

2.9713

3.5281

4.2744

5.2392

6.5058

8.0225

9.6946

11.2728

12.5370

99.5%

2.8123

3.2967

3.9182

4.7454

5.8424

7.2470

8.9767

10.8163

12.6241

14.0653

b

0.22

0.24

0.26

0.28

0.3

0.32

0.34

0.36

0.38

0.40

95%

9.2475

9.6378

9.8866

10.0412

10.1446

10.2071

10.2604

10.2899

10.3154

10.3393

97.5%

11.1166

11.5874

11.8763

12.0800

12.2085

12.2906

12.3468

12.3913

12.4277

12.4530

99%

13.4557

14.0671

14.4533

14.7158

14.8841

14.9774

15.0857

15.1470

15.1848

15.2096

99.5%

15.1063

15.7767

16.2224

16.5170

16.6887

16.8283

16.9298

16.9828

17.0212

17.0680

b

0.42

0.44

0.46

0.48

0.5

0.52

0.54

0.56

0.58

0.60

95%

10.3527

10.3643

10.3739

10.3828

10.3897

10.3990

10.4029

10.4086

10.4133

10.4200

97.5%

12.4740

12.4893

12.5025

12.5136

12.5235

12.5329

12.5409

12.5420

12.5483

12.5543

99%

15.2323

15.2485

15.2564

15.2734

15.2875

15.2977

15.3124

15.3196

15.3307

15.3435

99.5%

17.0886

17.1114

17.1343

17.1512

17.1645

17.1798

17.1817

17.1869

17.2133

17.2270

b

0.62

0.64

0.66

0.68

0.7

0.72

0.74

0.76

0.78

0.80

95%

10.4231

10.4271

10.4279

10.4262

10.4278

10.4319

10.4349

10.4357

10.4382

10.4410

97.5%

12.5603

12.5697

12.5720

12.5781

12.5785

12.5849

12.5881

12.5913

12.5929

12.5952

99%

15.3526

15.3495

15.3554

15.3513

15.3575

15.3655

15.3730

15.3769

15.3827

15.3920

99.5%

17.2313

17.2400

17.2423

17.2405

17.2307

17.2383

17.2374

17.2485

17.2494

17.2503

b

0.82

0.84

0.86

0.88

0.9

0.92

0.94

0.96

0.98

1.00

95%

10.4437

10.4459

10.4456

10.4464

10.4469

10.4463

10.4480

10.4482

10.4474

10.4481

97.5%

12.5982

12.5994

12.6013

12.6013

12.6019

12.6023

12.6019

12.6021

12.6028

12.6043

99%

15.3955

15.3959

15.3964

15.3969

15.3977

15.4018

15.4060

15.4099

15.4127

15.4150

99.5%

17.2619

17.2727

17.2749

17.2764

17.2739

17.2792

17.2822

17.2859

17.2892

17.2920

44

Figure 1.1: Empirical null rejections, t-test, N = 5, T = 100, ρ1 = ρ2 = 0.3, Bartlett kernel

45

Figure 1.2: Empirical null rejections, t-test, N = 10, T = 100, ρ1 = ρ2 = 0.3, Bartlett kernel

46

Figure 1.3: Empirical null rejections, t-test, N = 25, T = 100, ρ1 = ρ2 = 0.3, Bartlett kernel

47

Figure 1.4: Empirical null rejections, t-test, N = 5, T = 50, ρ1 = ρ2 = 0.9, Bartlett kernel

48

Figure 1.5: Empirical null rejections, t-test, N = 10, T = 50, ρ1 = ρ2 = 0.9, Bartlett kernel

49

Figure 1.6: Empirical null rejections, t-test, N = 25, T = 50, ρ1 = ρ2 = 0.9, Bartlett kernel

50

Figure 1.7: Empirical null rejections, t-test, N = 5, T = 100, ρ1 = ρ2 = 0.9, Bartlett kernel

51

Figure 1.8: Empirical null rejections, t-test, N = 10, T = 100, ρ1 = ρ2 = 0.9, Bartlett kernel

52

Figure 1.9: Empirical null rejections, t-test, N = 25, T = 100, ρ1 = ρ2 = 0.9, Bartlett kernel

53

Figure 1.10: Empirical null rejections, t-test, N = 5, T = 100, ρ1 = ρ2 = 0.3, QS kernel

54

Figure 1.11: Empirical null rejections, t-test, N = 5, T = 50, ρ1 = ρ2 = 0.9, QS kernel

55

Figure 1.12: Empirical null rejections, t-test, N = 5, T = 100, ρ1 = ρ2 = 0.9, QS kernel

56

Figure 1.13: Size adjusted power, Wald test, N = 5, T = 50, ρ1 = ρ2 = 0.6, b = 0.3, QS
kernel

57

Figure 1.14: Size adjusted power of panel IM, Wald test, N = 5, T = 50, ρ1 = ρ2 = 0.6, QS
kernel

58

Figure 1.15: Size adjusted power of panel IM, Wald test, N = 5, T = 50, ρ1 = ρ2 = 0.6,
Bartlett kernel

59

Figure 1.16: Size adjusted power, Wald test, N = 5, T = 50, ρ1 = ρ2 = 0.6, QS kernel

60

Figure 1.17: Size adjusted power, Wald test, N = 10, T = 50, ρ1 = ρ2 = 0.6, b = 0.3, QS
kernel

61

Figure 1.18: Size adjusted power of panel IM, Wald test, N = 10, T = 50, ρ1 = ρ2 = 0.6,
QS kernel

62

Figure 1.19: Size adjusted power, Wald test, N = 10, T = 50, ρ1 = ρ2 = 0.6, QS kernel

63

Proof of Theorem 1
By Assumption 2 above, we can define stacked innovation vector
ηt = u1t , · · · uN t , v1t , · · · vN t ,
which dimension is (N + N k) × 1, and assume that
[rT ]

Bu (r)
Λ11 W1 (r) + Λ12 W2 (r)
=
,
Bv (r)
Λ22 W2 (r)

1/2

T −1/2

ηt ⇒ Ωη W (r) =
t=1

where
1/2

Ωη

=

W (r) =

Λ11

Λ12
0N k×N Λ22
W1 (r)
.
W2 (r)

The dimension of the above matrix are as follows: Λ11 is N × N , Λ12 is N × N k, 0N k×N is
N k × N zero matrix, Λ22 is N k × N k; W1 (r) = wu,1 (r), · · · , wu,N (r) is N × 1 vector,
is N k × 1 vector.

W2 (r) = Wv,1 (r) , · · · , Wv,N (r)
Long run variance of ηt is:
1/2

Ωη = Ωη

1/2

Ωη

∞

=

E ηt ηt−j .
j=−∞

Also we have:
η

η

Ω11 Ω12
Λ11
Λ12
=
η
η
Ω21 Ω22
0N k×N Λ22

Ωη =

Λ11 0N ×N k
Λ11 Λ11 + Λ12 Λ12 Λ12 Λ22
=
,
Λ12
Λ22
Λ22 Λ12
Λ22 Λ22
η

η

where Ω11 = Λ11 Λ11 + Λ12 Λ12 is long run variance of ut = u1t · · · uN t , Ω22 = Λ22 Λ22
η

η

is long run variance of vt = v1t · · · vN t , Ω12 = Ω21 = Λ12 Λ22 is long run covariance
is N × N diagonal
of ut and vt . From assumption 1, we know that Λ11 = IN ⊗ σu·v
N ×N
matrix, Λ12 = IN ⊗ λuv
is N × N k diagonal matrix, Λ22 = IN ⊗ Ω1/2
vv N k×N k is
N ×N k
1/2
N k × N k diagonal matrix, where σu·v is scalar, λuv is 1 × k vector, and Ωvv is k × k matrix.

64

Using diagonal scaling matrix

 3/2
T Ik
0


A1T = 
T 1/2 Ik
,
0
T 1/2 IN ⊗ GD
then we have:


T β˜ − β


 γ˜ − Ω−1 Ωvu
vv


−1/2 A
˜
˜
˜

A−1
1T θ − θ =  GD δ1 − δ1
P IM θ − θ = T

..

.

GD δ˜N − δN



T

−1 

N

−1 
A−1
1T qit qit A1T

= T −1

T













N

−1/2 S u − x γ  .
A−1
it
it
1T qit T

T −1

t=1 i=1



t=1 i=1

By our assumptions,


3
x
T − 2 Si[rT
]
1
T − 2 xi[rT ]



  1/2 r

 r
Ω
W
(s)ds
B
(s)ds
vv
v,i
v,i


0

 

 0



 Bv,i (r)   Ω1/2

W
(r)
vv
v,i

 




 



0
0
p×1

 



0p×1
p×1






..
.
−1
.



 = Πg1,i (r),


.
..
A1T qit = 
=
⇒
.
.

 


 1
 

 r

r
 − 2 −1 D 

 0 D(s)ds  
0 D(s)ds
T GD St 




..
.






.
.
.



 

.
.
.


0p×1
0p×1
0p×1
and it follows that
T

N
−1
A−1
1T qit qit A1T

T −1

1 N

⇒
0 i=1

t=1 i=1

65

Πg1,i (r)g1,i (r) Π dr.

Also, by previous assumptions,
u − T −1/2 x γ
= T −1/2 Sit
it

u −x γ
T −1/2 Sit
it

⇒ Bu,i (r) − Bv,i (r)γ
1/2

= σu·v wu,i (r) + λuv Wv,i (r) − Ωvv Wv,i (r) γ
1/2

1/2

γ−

Ωvv

1/2

γ−

Ωvv

1/2

γ − Ω−1
vv Ωvu

= σu·v wu,i (r) − Wv,i (r) Ωvv
= σu·v wu,i (r) − Wv,i (r) Ωvv

= σu·v wu,i (r) − Wv,i (r) Ωvv

1/2

−1

λuv
−1

−1/2

Ωvv

Ωvu

so if γ = Ω−1
vv Ωvu , then
u − x γ ⇒ σ w (r).
T −1/2 Sit
u·v u,i
it

Combining above results, we have:




T β˜ − β


 γ˜ − Ω−1 Ωvu
vv


 GD δ˜1 − δ1


..

.

GD δ˜N − δN

⇒

−1 

1 N
0 i=1

Πg1,i (s)g1,i (s) Π dr


= σu·v Π


= σu·v Π

−1 





˜
 = A−1

P IM θ − θ




−1 

N
i=1 0

1



0 i=1

0 i=1

−1 

1 N

g1,i (s)g1,i (s) ds

−1 
g1,i (s)g1,i (s) ds





1 N



N
i=1 0

1

Πg1,i (s)σu·v wu,i (s)ds
1 N
0 i=1


g1,i (s)wu,i (s)ds


[G1,i (1) − G1,i (s)]dwu,i (s) = Ψ.

For the sequential limit of β˜ − β , we first let T → ∞, then let N → ∞, so we have

66

√

√
˜
N T β˜ − β = N Ik 0k×k 0k×p · · · 0k×p A−1
P IM θ − θ

−1
T N
1
−1 
= Ik 0k×k 0k×p · · · 0k×p 
A−1
×
1T qit qit A1T
NT
t=1 i=1


T N
1
u −x γ 
√ 1
A−1
Sit
it
1T qit √
N T t=1 i=1
T

−1
N
T
1
1
−1 
= Ik 0k×k 0k×p · · · 0k×p 
A−1
×
1T qit qit A1T
N
T
t=1
i=1


N
T
1
1
u −x γ 
 √1
A−1
qit √ Sit
it
1T
N i=1 T t=1
T
T →∞

−1

=⇒ σu·v Ik 0k×k 0k×p · · · 0k×p Π
×
−1 


N
N
1
1
1
1
g1,i (r)g1,i (r) dr  √
[G1,i (1) − G1,i (r)]dwu,i (r)
N
N
i=1 0
i=1 0

N →∞

=⇒ Φ

In order to get the distribution of Φ, we need to know the limit of the upper and left
k × k block of N1

N
i=1

1
0 g1,i (r)g1,i (r)

N
1
√1
[G1,i (1) − G1,i (r)]dwu,i (r)
N i=1 0

dr and the distribution of the upper k × 1 block of

as N → ∞.

First, consider the limit of the upper and left k × k block of N1
N

N

N
i=1

1
0 g1,i (r)g1,i (r)

dr.

1
1 1
Note that, N1
g1,i (r)g1,i (r) dr. The related compo0 g1,i (r)g1,i (r) dr = 0 N
i=1
i=1
nents for the sequential limits is the integral of the upper and left k × k block of the limit of
1
N

N

g1,i (r)g1,i (r) , which is given by
i=1

67

N

1
N
=

=

=

r

i=1 0
N
r

1
N
1
N

r

i=1 0 0
N
r r
i=1 0
r s

→
0
3
r

0

Wv,i (u)du Wv,i (s) ds

0

i=1 0 0
N
r s

1
N

Wv,i (s) ds

0

r

i=1 0
N
r

1
N

=

r

Wv,i (s)ds

s

Wv,i (u)Wv,i (s) duds

Wv,i (u)Wv,i (s) duds+

Wv,i (u)Wv,i (s) duds
r

ududs · Ik +

0

r
s

sduds · Ik

I = A1 (r)
3 k

Therefore, 01 A1 (r)dr = (1/12)Ik .
Second, consider the distribution of the upper k × 1 block of
N

1
1
√
[G1,i (1) − G1,i (r)]dwu,i (r).
N i=1 0

It has an asymptotic normal distribution, with zero mean, when conditional on G1,i (r)
for all i = 1, 2, . . . , N . So we only need to find its asymptotic variance. Also, recall that the
units are cross-sectional independent, so we have


N



1

1
[G1,i (1) − G1,i (r)]dwu,i (r)
var  √
N i=1 0
1

N

1
=
[G1,i (1) − G1,i (r)][G1,i (1) − G1,i (r)] dr
0 N i=1
as N fixed.

68

Note that the variance of the upper k × 1 block of
N

1
1
√
[G1,i (1) − G1,i (r)]dwu,i (r)
N i=1 0

is just the upper and left k × k block of
N

1

1
[G1,i (1) − G1,i (r)][G1,i (1) − G1,i (r)] dr,
0 N i=1
which is given by
1
N
=

=

1
N
1
N
1
N

N

1

s

i=1 r 0
N
1 s

1

Wv,i (u)duds
1

i=1 r 0 r 0
N
1 s 1 s
0

r

1

u

→
r

0
1

r

=

r
s

0

r

Wv,i (u) duds

u

Wv,i (v)Wv,i (u) dvdtduds

Wv,i (v)Wv,i (u) dvdtduds+

Wv,i (v)Wv,i (u) dvdtduds

vdvdtduds Ik +

0
1

0

s

i=1 r 0 r 0
N
1 s 1 u

i=1 r
1 s

r

s

s
u

udvdtduds Ik

1
(1 − r) 1 − r4 Ik = A2 (r)
12

So, we have 01 A2 (r)dr = (7/180)Ik .
Using above notations, the sequential asymptotic distribution Φ is given by
−1

Φ∼N

2
0, σu·v

−1

1

Ω1/2
vv

1

A1 (r)dr
0

−1

1

A2 (r)dr
0

A1 (r)dr

Ω1/2
vv

−1

.

0

We denote its variance as
β
Vseq

1/2
Ωvv

=

2
σu·v

=

2 Ω−1 .
5.6 · σu·v
vv

−1

−1

1
0

A1 (r)dr

1
0

69

−1

1

A2 (r)dr

0

A1 (r)dr

1/2 −1

Ωvv

Proof of Theorem 2
˘ , with W
˘ ∈ {W
ˆ ,W
˜ ,W
˜ ∗ }. Those
In Theorem 2, the Wald statistics considered was W
statistics only differ with respect to the used estimator of the long run variance parameter,
2∗ . As in the proof of Theorem 2, θ˜ represents the vector of panel
2 ,σ
2 ,σ
2 ∈ σ
˜u·v
˜u·v
ˆu·v
σ
˘u·v
IM-OLS estimators δ˜ , β˜ , γ˜
estimator for VP IM is given by


, and θ denotes the vector

T

−1

N

2 T −2
V˘P IM = σ
˘u·v

δ , β , Ωvu Ω−1
vv . The

×

AP IM qit qit AP IM 
t=1 i=1



T



N
q

T −4

q

q

AP IM SiT − Si,t−1

q

SiT − Si,t−1 AP IM  ×

t=1 i=1



T

−1

N

T −2

AP IM qit qit AP IM 
t=1 i=1

=

2
σ
˘u·v

· V˘ ,

where V˘ is the estimator for

V

=

Π





−1 

−1

1 N
0 i=1

g1,i (s)g1,i (s) ds



1 N
0 i=1

[G1,i (1) − G1,i (s)][G1,i (1) − G1,i (s)] ds ×
−1

1 N
0 i=1

×

g1,i (s)g1,i (s) ds

Π−1 .

Under the null hypothesis the Wald statistics and t statistics can be written as

70

˘ = Rθ˜ − r
W
= R θ˜ − θ
=

−1

RAP IM V˘P IM AP IM R

RAP IM V˘P IM AP IM R

−1
A−1
R RAP IM AP IM

Rθ˜ − r
−1

R θ˜ − θ

˘
A−1
R RAP IM VP IM AP IM R

θ˜ − θ

−1
˜
A−1
R RAP IM AP IM θ − θ

A−1
R

−1

×

,

and

t˘ = Rθ˜ − r /

RAP IM V˘P IM AP IM R

−1
˜
= A−1
R RAP IM AP IM θ − θ

˘
A−1
R RAP IM VP IM AP IM R

/

A−1
R

.

Now, by assumption the restriction matrix fulfills
lim A−1 RAP IM
T →∞ R

= R∗ ,

and
˜
A−1
P IM θ − θ ⇒ Ψ (VP IM )
under the null hypothesis. Therefore, in case of consistent estimation of the conditional long
2 using V
ˆP IM it follows that
run variance σu·v
ˆ
W

=

−1
A−1
R RAP IM AP IM

θ˜ − θ

ˆ
A−1
R RAP IM VP IM AP IM R

A−1
R

2 ˘
A−1
R RAP IM σu·v V AP IM R

A−1
R

−1

−1
˜
× A−1
R RAP IM AP IM θ − θ

=

−1
˜
A−1
R RAP IM AP IM θ − θ

σ2
× u·v
2
σ
ˆu·v

−1
˜
× A−1
R RAP IM AP IM θ − θ

⇒ (R∗ Ψ (VP IM )) R∗ VP IM R∗

−1

(R∗ Ψ (VP IM ))

∼ χ2q
and for q = 1, we have
71

−1

R∗ Ψ (VP IM )
∼ Z.
tˆ ⇒
R∗ VP IM R∗
2 . From the
˜ using σ
Next, we consider the asymptotic behavior of the test statistic W
˜u·v
2 , we know that it is an estimator based on ∆S
˜u , which is the difference
construction of σ
˜u·v
it
u , where S
˜u = S y − S D δ˜i − S x β˜ − x γ˜ . Then, we have
of S˜it
t
it
it
it
it
u = ∆S y − ∆S D δ˜ − ∆S x β˜ − ∆x γ
∆S˜it
t i
it ˜
it
it
= yit − Dt δ˜i − xit β˜ − vit γ˜
= Dt δi + xit β + uit − Dt δ˜i − xit β˜ − vit γ˜
γ − γ) − Dt δ˜i − δi − xit β˜ − β
= uit − vit γ − vit (˜

γ − γ) − Dt δ˜i − δi − xit β˜ − β .
= u+
it − vit (˜
It can be shown that the last two parts of the formula can be neglected for long run
u . Thus, the long run variance estimator based on ∆S
˜u , that is
variance estimation of ∆S˜it
it
+
2
σ
˜u·v , asymptotically coincides with long run variance estimator of uit − vit (˜
γ − γ).
2
+=
+ = σu·v
Let’s define ηit
,
and
then
its
long
run
variance
is
Ω
, so an
u+
,
v
i
it
it
Ω22
p

+
+ → Ω+ .
infeasible long run variance estimator Ω+
i , using unobserved ηit is consistent: Ωi −
i
1
+
˜ + , for u+ −
Note that: u+
γ − γ) = ηit
, then HAC estimator, Ω
it − vit (˜
i
it
− (˜
γ − γ)
vit (˜
γ − γ) can be written as:

1 − (˜
γ − γ) Ω+
i

1
− (˜
γ − γ)

with
(˜
γ − γ) ⇒

0k×k Ik 0k×p · · · 0k×p σu·v Π

−1 
N
N
1



i=1 0

g1,i (s)g1,i (s) ds

−1/2

= σu·v Ωvv

dγ ,

72



i=1 0

−1
1

×


[G1,i (1) − G1,i (s)]dwu,i (s)

where dγ is the second k × 1 block of



N

−1 

1

g1,i (s)g1,i (s) ds

i=1 0



N



1

i=1 0

[G1,i (1) − G1,i (s)]dwu,i (s) .

+
This implies that Ω˜i will converge to:

−1/2

1 − σu·v Ωvv

=

2
σu·v

2
+ σu·v

dγ

2 ⇒ 1
So we have: σ
˜u·v
N

˜
W

=

dγ

1/2 −1 1/2
Ωvv
Ωvv
N
i=1



2
σu·v

0 
−1/2
Ωvv −σu·v Ωvv

0

1/2
Ωvv

1/2
Ωvv

−1

dγ



2
dγ = σu·v
1 + dγ dγ .

2
1 + dγ dγ . This implies that
= σu·v

2
1 + dγ dγ
σu·v

−1
˜
A−1
R RAP IM AP IM θ − θ



1

˜
A−1
R RAP IM VP IM AP IM R

A−1
R

2 ˘
A−1
R RAP IM σu·v V AP IM R

A−1
R

−1

−1
˜
× A−1
R RAP IM AP IM θ − θ

=

−1
A−1
R RAP IM AP IM

θ˜ − θ

σ2
× u·v
2
σ
˜u·v

−1
˜
× A−1
R RAP IM AP IM θ − θ

⇒
∼

(R∗ Ψ (VP IM )) R∗ VP IM R∗
1 + dγ dγ

−1

(R∗ Ψ (VP IM ))

χ2q
1 + dγ dγ

and when q = 1, we have
t˜ ⇒

Z
1 + dγ dγ

73

.

−1

2∗ , we have
For the result of the fixed-b test statistic, using σ
˜u·v

˜∗ =
W

−1
˜
A−1
R RAP IM AP IM θ − θ

˜∗
A−1
R RAP IM VP IM AP IM R

A−1
R

2 ˘
A−1
R RAP IM σu·v V AP IM R

A−1
R

−1

−1
˜
× A−1
R RAP IM AP IM θ − θ

=

−1
˜
A−1
R RAP IM AP IM θ − θ

σ2
× u·v
2∗
σ
˜u·v

−1
˜
× A−1
R RAP IM AP IM θ − θ

⇒

−1

(R∗ Ψ (VP IM )) R∗ VP IM R∗
1
N

∼
1
N

N
i=1

−1

(R∗ Ψ (VP IM ))

Q∗i (b)

χ2q
N
i=1

Q∗i (b)

and when q = 1, we have
Z

t˜∗ ⇒
1
N

N
i=1

.
Q∗i (b)

Note that numerator and the denominator of the limiting distribution are independent,
because Vogelsang and Wagner [2014] have proved that the numerator is independent with
Q∗i (b) for all i = 1, 2, · · · , N , then it follows that the numerator is independent with the sum
1
N

N
i=1

Q∗i (b).

Due to the independence of numerator and denominator in above limiting distribution,
if we know µQ , which given by µQ = E Q∗i (b) , then as T → ∞ followed by N → ∞, we
have the following sequentially limit results:
˜∗
W
µ

Q

=
=

˜∗
µQ · W
χ2q

⇒
1
N
P

2∗
σ
˜u.v
R∗ Vˆ R∗
µQ

T Rβ˜ − r

−→ χ2q

N

1
∗
µQ Qi (b)
i=1

as N → ∞.

74

−1

T Rβ˜ − r

Also, when q = 1, similarly, we have:
t˜∗µ

Q

=

µQ · t˜∗
Z

⇒
1
N
P

−→ Z

N

1
∗
µQ Qi (b)
i=1

as N → ∞.

75

Chapter 2
Hypothesis testing in cointegrated
panels: Asymptotic and Bootstrap
method
This paper compares asymptotic and bootstrap hypothesis tests in cointegrated panels
with cross-sectional uncorrelated units and endogenous regressors. All the tests are based
on the panel integrated modified ordinary least square (panel IM-OLS) estimator from Vogelsang et al. [2016]. The aim of using the bootstrap tests is to deal with the size distortion
problems in the finite samples of fixed-b tests. Finite sample simulations show that the bootstrap method is better than the asymptotic method in terms of having lower size distortions.
In general, the stationary bootstrap is better than the conditional-on-regressors bootstrap,
although in some cases, the conditional-on-regressors bootstrap has less size distortions. The
improvement in size comes with only minor power losses, which can be ignored when the
sample size is large.

76

2.1

Introduction

The bootstrap has become common in econometric analysis, especially in performing
hypothesis tests. The basic idea of hypothesis tests is to compare the observed value of
a test statistic with the distribution that it would follow if the null hypothesis were true.
If the distribution is known, then we can perform exact tests. However, in many cases of
interest, the distribution of the test statistic is only known asymptotically or is dependent
upon nuisance parameters. In many cases, bootstrap hypothesis testing works well since the
bootstrap statistics converge to the same asymptotic distributions as the sample statistics
do. Therefore, the nuisance parameter dependent limit distributions can be approximated
by the bootstrap simulations, which makes inference available.
The purpose of the present paper is to compare the fixed-b asymptotic hypothesis test
with two bootstrap hypothesis tests, conditional-on-regressors bootstrap test and stationary
bootstrap test, for panel cointegrated regressions with endogenous regressors. When the
regressors are endogenous, it is well known that a variety of different methods, such as
panel fully modified Ordinary Least Square (panel FM-OLS), panel dynamic Ordinary Least
Square (panel DOLS) and panel integrated modified Ordinary Least Square (panel IMOLS), will deliver estimators that have zero mean Gaussian mixture limiting distributions,
which in turn allow asymptotic inference to be carried out (see Kao and Chiang [2000],
Pedroni [2000], Bai et al. [2009], Mark and Sul [2003], Vogelsang et al. [2016]). Among those
methodologies, panel IM-OLS relies on the fixed-b asymptotic theory. Compared with the
traditional asymptotic theory, the fixed-b asymptotic theory can capture the impact of kernel
and bandwidth choices on the sampling distributions of HAC-type test statistics. However,
both of those asymptotic theories often provide poor approximations to the distributions
77

of associated test statistics in finite samples, which leads to size distortion problems. To
improve the quality of finite sample inference, in terms of decreasing size distortions, the
bootstrap method is considered in this paper.
Although bootstrap methods are widely employed for analyzing nonstationary time series
data, a surprisingly small proportion are devoted to bootstrap inference in cointegrated
regressions. Li and Maddala [1997] investigated the usefulness of bootstrap methods for small
sample inference in cointegrated regression models. Their simulation results showed that the
substantial size distortions of the asymptotic tests can be corrected by properly implemented
bootstrap methods. Psaradakis [2001] applied the sieve bootstrap procedure to cointegrated
regressions, and his simulation study demonstrated the small-sample superiority of the sieve
bootstrap over both the traditional asymptotic approximation and the blockwise bootstrap.
Chang et al. [2006] considered the sieve bootstrap based on a VAR model for the cointegrated
regressions. They established the bootstrap consistency for both OLS and DOLS, which
leads to valid bootstrap inference. Shin and Hwang [2013] applied the stationary bootstrap to
cointegrated regressions. They established the limiting distribution of the bootstrap ordinary
least square estimator (OLSE) as well as the limiting null distribution of the bootstrap
Wald-type test regarding the cointegration parameter. Also, finite sample size and power
properties of the bootstrap test were studied by a Monte Carlo simulation. Note that in the
above literature, the bootstrap methods are applied in pure time series setting.
The contribution of this paper is twofold. First, the results complement the existing
literature by applying bootstrap inference to panel cointegrated regressions, and second,
comparisons are made between bootstrap and fixed-b methods for inference using panel IMOLS. Bootstrap methods are applied to a cointegrated panel with uncorrelated cross sectional
units and homogeneous 2nd order moments. Finite sample size and power properties of the
78

bootstrap test are studied by a Monte Carlo simulation. The bootstrap methods applied in
this paper are the conditional-on-regressors bootstrap and stationary bootstrap. We do not
consider the sieve bootstrap for two reasons. First, even though the sieve bootstrap can be
applied in fairly general models and performs well in pure time series setting, Smeekes and
Urbain [2014] questioned the validity of the use of VAR sieve bootstrap in panels with a
moderate cross-sectional dimension. In addition, when estimating the models and carrying
out inference, we do not assume the error terms follow AR or VAR models.
The rest of the paper is organized as follows. Section 2.2 introduces the model, assumptions and asymptotic inference based on the panel IM-OLS estimators. In Section 2.3, the
conditional-on-regressors bootstrap and stationary bootstrap procedures are presented. Section 2.4 provides a simulation study to compare the size and the power of the bootstrap
tests with the fixed-b asymptotic test. Section 2.5 summarizes the results and concludes the
paper.

2.2

2.2.1

The model, assumptions and asymptotic inference

The model and assumptions

Consider the panel data model given by

yit = Dt δ + xit β + uit

(2.1)

xit = xit−1 + vit

(2.2)

79

where i = 1, 2, · · · , N and t = 1, 2, · · · , T index the cross-sectional and time series units
respectively; yit and uit are scalars; Dt is the deterministic component, and δ is a p × 1
vector; xit , vit and β are k ×1 vectors. Suppose that ηit = uit v
it

is a (k +1) dimensional

stationary vector process across i, then the model introduced in (2.1) describes a system of
panel cointegrated regressions, i.e. yit is cointegrated with xit .
In the above system, we are interested in inference about β based on the panel IMOLS estimator. Before we define the panel IM-OLS estimator of β, we make following
assumptions.
Assumption 3. Assume that {ηit }N
i=1 are cross-sectionally uncorrelated and 2nd order moments are constant across i.
Note that the Assumption 3 only requires that the panels are homogeneous in the 2nd
order moment; it’s possible the higher order moment structures are heterogeneous across i.
Assumption 4. Assume that for all i, ηit is a stationary process and it satisfies a functional
central limit theorem (FCLT) of the form
T

T −1/2

ηit ⇒ Bi (r) = Ω1/2 Wi (r),

r ∈ (0, 1].

t=1

In Assumption 4, [rT ] represents the integer part of rT , and Wi (r) is a (k + 1) × 1 vector
of independent standard Brownian motions. Ω1/2 is a (k + 1) × (k + 1) matrix that satisfies
Ω = Ω1/2 Ω1/2 , and



∞

Ω=



Ωuu Ωuv 
E ηit ηit−j = 
 > 0,
j=−∞
Ωvu Ωvv

80

where it is clear that Ωuv = Ωvu . The assumption Ωvv > 0 rules out cointegration in xit .
Partition Bi (r) as Bi (r) =

Bu,i (r) Bv,i (r) , and likewise partition Wi (r) as Wi (r) =

wu,i (r) Wv,i (r) , where wu,i (r) and Wv,i (r) are a scalar and a k-dimensional standard
Brownian motion respectively. Using the Cholesky form of Ω1/2 ,




 σu·v λuv 
Ω1/2 = 
,
1/2
0k×1 Ωvv
−1/2 −1

2 = Ω −Ω Ω−1 Ω , and λ = Ω
it can be shown that σu·v
uv
uv Ωvv
uu
uv vv vu

. By this Cholesky

decomposition, we can write

 

Bu,i (r) σu·v wu,i (r) + λuv Wv,i (r)
Bi (r) = 
=
.
1/2
Bv,i (r)
Ωvv Wv,i (r)
Assumption 5. For the deterministic component, Dt , assume that there is a p × p matrix
GD and a vector of functions, D(s), such that
√
lim

T →∞

T G−1
D D[sT ] = D(s)

r

with

D(s)D(s) ds < ∞,

0<

0<r

1

0

The deterministic component Dt could include an intercept, time trend and polynomial
of time and other functions of time.

81

2.2.2

Inference based on panel IM-OLS

This section provides some key results about the panel IM-OLS estimator and inference
based on it. To conserve space, we don’t provide all the details. After applying a partial
sum transformation and adding the original regressors, xit , to regression (2.1) , the system
becomes
y

˜
x β + x γ + Su
Sit = StD δ + Sit
it .
it

(2.3)

Vogelsang et al. [2016] pointed out that the parameter β in panel cointegrated regression
ˆ which is the OLS
(2.1) can be consistently estimated by the panel IM-OLS estimator θ,
estimator of regression (2.3). Its limiting distribution is given by


ˆ
 GD δ − δ

ˆ− θ = 
A−1
θ
 T βˆ − β
P IM


γˆ − Ω−1
vv Ωvu

⇒ σu·v Π



N
i=1 0

−1 








1 N
0 i=1

−1
g1,i (s)g1,i (s)ds


1

G1,i (1) − G1,i (s) dwu,i (s)

= Ψ1

82

×

with N fixed and T → ∞. The scaling matrix A−1
P IM is a diagonal matrix given by


0
GD




−1
AP IM = 
,
T
·
I
k




0
Ik
and Π is a diagonal matrix given by




0 
Ip




1/2
Π=
.
Ω
vv




1/2
0
Ωvv
In addition,
r

G1,i (r) =

0

g1,i (s)ds

where g1,i (r) is defined as


r
0 D(s)ds









g1,i (r) =  r Wv,i (s)ds .

 0


Wv,i (r)

83

Conditional on g1,i (r) for all i, it can be shown that Ψ1 ∼ N (0, VP IM ) where VP IM is given
by

−1 

2
Π
VP IM = σu·v



−1

1 N
0 i=1

g1,i (s)g1,i (s)ds

×


1 N



G1,i (1) − G1,i (s) G1,i (1) − G1,i (s) ds ×

0 i=1



−1

1 N



0 i=1

Π −1 .

g1,i (s)g1,i (s)ds

Consider the null hypothesis H0 : Rθ = r, where R ∈ Rq×(p+2k) with full rank q and
r ∈ Rq . Define qit = S D
t

x
Sit

q

xit , Sit =

q

t
j=1 qij ,

and Si0 is a zero vector for all i.

2 is an estimator for σ 2 , then an estimator for V
Suppose that σ
˘u·v
P IM is given by
u·v



T

−1

N

2 T −2
V˘P IM = σ
˘u·v

AP IM qit qit AP IM 

×

t=1 i=1



T



N
q

T −4

q

AP IM SiT − Si,t−1

q

q

SiT − Si,t−1 AP IM  ×

t=1 i=1



T

−1

N

T −2

AP IM qit qit AP IM 

.

t=1 i=1
2 are considered. The first candidate, σ
2 , is based
Here, two potential candidates for σ
˘u·v
˜u·v

on the first differences of the residuals of the augmented partial sum regression (2.3), i.e.

u = S y − S D δˆ − S x βˆ − x γ
S˜it
t
it
it ˆ ,
it

ˆ βˆ and γˆ are the panel IM-OLS estimators. Define a HAC estimator using the first
where δ,

84

u
difference of S˜it

2 =
σ
˜u·v

1
N



N

T

T

T −1
i=1

k
j=2 h=2



|j − h|
M

u ∆S
˜u  .
∆S˜ij
ih

Another HAC-type estimator is based on the first difference of the residuals from the further
augmented partial sum regression

2∗ =
σ
˜u·v

1
N

N



T

T

T −1

k
j=2 h=2

i=1

|j − h|
M


u∗ ∆S
˜u∗  ,
∆S˜ij
ih

where
u∗ = S y − S D δˆ − S x βˆ − x γ
ˆ
S˜it
t i
it i
it ˆi − zit λi ,
it

zit is given by
t−1 j

T

qij −

zit = t

qis ,
j=1 s=1

j=1

ˆ i (i = 1, 2, · · · , N ) are the OLS estimators from the further augmented
and δˆi , βˆi , γˆi and λ
regressions given by
y

xβ +x γ +z λ .
Sit = StD δi + Sit
i
it i
it i

(2.4)

u∗ is obtained by estimating regression (2.4) individual by individual.
Note that the residual S˜it
2∗ has a fixed-b limit that is proportional
As discussed in Vogelsang and Wagner [2014], σ
˜u·v
2 , independent of θ,
ˆ and does not depend upon additional nuisance parameters.
to σu·v
2 to construct V
˜ denote statistics defined using σ
˜P IM , and likewise t˜∗ and
Let t˜ and W
˜u·v
2∗ to construct V
˜ ∗ denote statistics defined using σ
˜ ∗ . Letting t˘ and W
˘ denote either
W
˜u·v
P IM

85

˜ or t˜∗ and W
˜ ∗ , define the t and W ald statistics as:
t˜ and W
Rθˆ − r

t˘ =

RAP IM V˘P IM AP IM R
˘ = Rθˆ − r
W

RAP IM V˘P IM AP IM R

−1

Rθˆ − r .

The limiting distributions of above test statistics are discussed in Vogelsang et al. [2016].

1. Under traditional bandwidth and kernel assumptions, with N fixed as T → ∞,

˜ ⇒
W

χ2q
1 + dγ dγ

and when q = 1,
Z

t˜ ⇒

1 + dγ dγ
where χ2q is a chi-square random variable with q degrees of freedom that is correlated
with dγ , Z is distributed standard normal and is correlated with dγ , and dγ denotes
the second k × 1 block of



1 N
0 i=1

−1 
g1,i (s)g1,i (s)ds



N
i=1 0

1


G1,i (1) − G1,i (s) dwu,i (s) .

2. Under fixed-b asymptotics where M = bT , b ∈ (0, 1] is held fixed as T → ∞, the

86

˜ and t˜ are given by
fixed-b limits of W
χ2q

˜ ⇒
W
1
N

N

Qb P˜i (r)

i=1

Z

t˜ ⇒
1
N

N
i=1

Qb P˜i (r)

where

Qb P˜i (r)

=

2 1−b ˜
2 1 ˜2
Pi (r)dr −
Pi (r)P˜i (r + b)dr
b 0
b 0
2 1 ˜
P (r)P˜i (1)dr + P˜i2 (r)
−
b 1−b i

and

P˜i (r) = wu,i (r) − g1,i (r) 



N

−1

1 N
0 i=1

g1,i (s)g1,i (s)ds


1

i=1 0

×

G1,i (1) − G1,i (s) dwu,i (s)

3. Under fixed-b asymptotics where M = bT , b ∈ (0, 1] is held fixed as T → ∞, the
˜ ∗ and t˜∗ are given by
fixed-b limits of W
χ2q

˜∗⇒
W
1
N

N
i=1

Qb P˜i∗ (r)
Z

t˜∗ ⇒
1
N

87

N
i=1

Qb P˜i∗ (r)

where Qb (·) is the same as above, and P˜i∗ (r) is similar as P˜i (r) but its component is
from the further augmented regression (2.4), which is a complicated stochastic process
depend on the kernel function, bandwidth, and Wi (r)1 . In addition, Qb P˜i∗ (r)

is

independent of χ2q and Z.

2.3

Bootstrap hypothesis tests

In this section, we introduce two different bootstrap procedures based on the panel IMOLS estimator. For each of these bootstrap procedures, bootstrap test statistics are computed using the same formulas as the original test statistics but with resampled data.

2.3.1

Conditional-on-regressors bootstrap

A formal description of the conditional-on-regressors bootstrap is given below.

1. Calculate the residuals as

u = S y − S D δˆ − S x βˆ − x γ
Sˆit
t
it
it ˆ
it

ˆ βˆ and γˆ are the panel IM-OLS estimators.
where δ,
u , ∆S
ˆu , · · · , ∆Sˆu
2. Obtain the bootstrap resamples u∗i1 , u∗i2 , · · · , u∗iT from ∆Sˆi1
i2
iT

i.i.d. sampling with replacement.
1 For more details about the Q P˜ ∗ (r) function, please see Vogelsang and Wagner [2014].
b i

88

by

t
∗
j=1 uij .

u∗ =
3. Define Sit

y∗

And generate the bootstrap samples Sit from

y∗
u∗
x βˆ + x γ
Sit = StD δˆ + Sit
it ˆ + Sit .

(2.5)

4. Define the bootstrap statistics as

t˘∗CBS =

Rθˆ∗ − Rθˆ
RAP IM V˘P∗IM AP IM R

ˆ∗
ˆ
˘∗
W
CBS = Rθ − Rθ

RAP IM V˘P∗IM AP IM R

−1

Rθˆ∗ − Rθˆ

where θˆ∗ is the bootstrap panel IM-OLS estimator for regression (2.5), V˘P∗IM is constructed exactly as V˘P IM but using the bootstrap data.
5. Repeat above steps 2-4 independently B times to obtain samples
˘∗
W
CBS,j

B
j=1

t˘∗CBS,j

B
j=1

and

.

6. Compute the equal tail bootstrap p-value as

1
p∗ t˘ = 2min 
B
p∗

˘
W

1
=
B

B
j=1

1
I t˘∗CBS,j ≤ t˘ ,
B

B


I t˘∗CBS,j > t˘ 

b=1

B

˘∗
˘
I W
CBS,j > W
j=1

where I(·) is the indicator function. Reject the null hypothesis if the equal tail bootstrap p-value is less than 5%.

89

2.3.2

Stationary bootstrap

The stationary bootstrap, proposed by Politis and Romano [1994], is a special type of
block bootstrap where the block size follows a geometric distribution instead of a fixed
number. For a geometric distribution with parameter pT , the expected block size of the
stationary bootstrap is 1/pT . The stationary bootstrap has been used in the literature of
unit root tests, cointegration tests and cointegrated regression inference (see Swensen [2003],
Paparoditis and Politis [2005], Parker et al. [2006], Shin [2015] and Shin and Hwang [2013]).
It can capture the serial correlation structure in the original sample by block resampling, and
it produces stationary bootstrap samples. A formal description of the stationary bootstrap
inference procedure is given below.

1. Calculate the residuals as

u = S y − S D δˆ − S x βˆ − x γ
Sˆit
t
it
it ˆ
it

ˆ βˆ and γˆ are the panel IM-OLS estimators.
where δ,
u , ∆x
ˆu
2. Define ηˆit = ∆Sˆit
it for t = 1, 2, · · · , T , where Si0 = 0, and xi0 is zero vector for

all i.
T

∗
3. Resample the series {ˆ
ηit }Tt=1 via the stationary bootstrap, obtaining ηˆit
t=1 .
∗ =
4. Partition ηˆit

∗
u∗it vit

analogously as ηit =

T

samples x∗it t=1 by

t

x∗it

∗,
vij

=
j=1

90

uit vit . Obtain the bootstrap

T

∗
and generate the bootstrap samples yit
t=1 from

∗ = D δˆ + x∗ βˆ + u∗ .
yit
t
it
it

(2.6)

5. Define the bootstrap statistics as

t˘∗SBS

Rθ˜∗ − Rθˆ
=

RAP IM V˘P∗IM AP IM R

˜∗
ˆ
˘∗
W
SBS = Rθ − Rθ

RAP IM V˘P∗IM AP IM R

−1

Rθ˜∗ − Rθˆ

where θ˜∗ is the bootstrap panel IM-OLS estimator from regression (2.6), V˘P∗IM is
constructed exactly as V˘P IM but using the bootstrapping data.
6. Repeat above steps 3-5 independently B times to obtain samples
˘∗
W
SBS,j

B
j=1

t˘∗SBS,j

B
j=1

and

.

7. Compute the equal tail bootstrap p-value as

1
p∗ t˘ = 2min 
B
p∗

˘
W

1
=
B

B
j=1

1
I t˘∗SBS,j ≤ t˘ ,
B

B


I t˘∗SBS,j > t˘ 

j=1

B

˘∗
˘
I W
SBS,j > W
j=1

where I(·) is the indicator function. Reject the null hypothesis if the equal tail bootstrap p-value is less than 5%.

Note that the step 1 of Section 2.3.1 and Section 2.3.2 are both based on the regression

91

(2.3), which includes the augmented regressor xit in the regression. It might worth exploring
how the bootstrap works if the residuals are calculated from non-augmented partial sum
regression. That is, for both stationary and conditional-on-regressors bootstrap, the residuals
are obtained from
u = S y − S D δˆ − S x βˆ
Sˆit
t
it
it

(2.7)

where δˆ and βˆ are the panel IM-OLS estimators, and all other steps are the same as its
corresponding procedures. Using these residuals, the stationary bootstrap resampling and
the conditional-on-regressors bootstrap resampling will be more comparable, and it could
capture some of endogeneity in the bootstrap resamples. In next section, we will provide the
bootstrap results based on Section 2.3.1 and Section 2.3.2 as well as the bootstrap results
based on the residuals from regression (2.7).

2.4

Finite sample simulations

In this section, we compare finite sample size and power performance of the bootstrap
tests with the asymptotic tests based on the panel IM-OLS estimators. The data generating
process is the same as in Vogelsang et al. [2016], which is given by

yit = µ + x1it β1 + x2it β2 + uit
x1it = x1i,t−1 + v1it
x2it = x2i,t−1 + v2it

92

where for all i = 1, 2, · · · , N , ui0 = 0, x1i0 and x2i0 are zero vectors, and

uit = ρ1 ui,t−1 + it + ρ2 (e1it + e2it )
v1it = e1it + 0.5e1i,t−1
v2it = e2it + 0.5e2i,t−1

where it , e1it and e2it are i.i.d. standard normal random variables independent of each
other. The parameter values are µ = 3, β1 = β2 = 1. In addition, we use ρ1 , ρ2 ∈ {0.6, 0.9}.
The parameter ρ1 controls serial correlation in the regression error, and ρ2 determines the
endogeneity of the regressors. In this paper, we only provide results where both ρ1 and ρ2 are
relatively large because according to the findings in Vogelsang et al. [2016], if ρ1 and ρ2 are
relatively small (ρ1 = ρ2 = 0.3), there are only minor size distortions for fixed-b asymptotic
tests. Therefore, the bootstrap method is not necessary when ρ1 and ρ2 are small. The kernel
function used in this simulation study is the Bartlett kernel, and the bandwidths are given by
M = bT with b ∈ {0.06, 0.1, 0.3, 0.5, 0.7, 0.9, 1}. We use pT = 0.02(T /50)−1/3 as the block
length parameter in the stationary bootstrap2 . The sample sizes are N = 5, T ∈ {50, 100}.
The number of bootstrap replications is B = 399, and the number of simulation replications
is 1000.
Using the simulation designed above, we only report results for cases where ρ1 = ρ2 . The
results include t-statistics for testing the null hypothesis H0 : β1 = 1 and Wald statistics
2 Politis and White (2004, 2009) considered estimators constructed via stationary bootstrap to obtain an
approximation to the sampling distribution of the mean of a finite sample from the (strictly) stationary realvalued sequence. They showed that the optimal block length parameter minimizing MSE of the stationary
bootstrap sample mean is cT −1/3 for some constant c. In addition, Shin and Hwang (2013) considered
the bootstrap ordinary least square estimator for cointegrating regressions. They established large sample
validity of a bootstrap test regarding cointegration parameters and showed that the block length parameter
0.02(T /50)−1/3 would provide stable size performance for the stationary bootstrap test.

93

for testing the joint null hypothesis H0 : β1 = 1, β2 = 1. The asymptotic panel IM2 and is labeled panel
OLS statistics were implemented in two ways. The first uses σ
˜u·v
2∗ and is labeled panel IMOLS(fb). The bootstrap panel
IMOLS(D), and the second uses σ
˜u·v

IM-OLS statistics were implemented in four ways. The first two statistics are based on the
2 and are labeled Cond-BS IMOLS (D) and Stat-BS IMOLS(D) respectively
bootstrapped σ
˜u·v

for the conditional-on-regressors bootstrap and the stationary bootstrap. The second two
2∗ and are labeled Cond-BS IMOLS(fb) and Statstatistics are based on the bootstrapped σ
˜u·v

BS IMOLS(fb) respectively. Rejections for panel IMOLS(D) are carried out using N (0, 1)
critical values for the t test and χ22 critical values for the Wald test. Rejections for panel
IMOLS(fb) are carried out using fixed-b asymptotic critical values. In contrast, rejections
for the bootstrap statistics are carried out by comparing the bootstrap p-value with the
nominal level, which is 5% in this simulation.
In order to see if the bootstrap methods can help solve the over-rejection problem of the
asymptotic tests in finite sample, we plot in Figures 2.1-2.8 null rejection probabilities of
the t and Wald tests as a function of b ∈ (0, 1]. The first two figures give the results for
N = 5, T = 50 using the Bartlett kernel and ρ1 = ρ2 = 0.6. In Figure 2.1, all t-tests have
some over-rejection problems, and there is no test that dominates the others in this scenario.
When the bandwidth is small (b = 0.1), panel IMOLS(D) is better than the other tests
because it is conservative. But when bandwidth is relative large (b > 0.2), it turns out that
Cond-BS IMOLS(D) is the best. Even though it is better than all other tests, the Cond-BS
IMOLS(D) rejection probabilities are close to 15%, which is much larger than nominal level
5%. In Figure 2.2, for Wald tests, the stationary bootstrap tests dominate the other tests
for all values of b. Its rejection probabilities are close to 10% for all values of b, which is
much better than the asymptotic tests. Also, Cond-BS IMOLS(D) has rejection probabilities
94

around 12% as long as the bandwidth is not very small (b > 0.1).
In Figures 2.3 and 2.4, all the settings are the same as in Figure 2.1 and 2.2 except that
the time series sample size increases from T = 50 to T = 100. Comparing Figures 2.1 and
2.3, Figures 2.2 and 2.4, we see that both t and Wald tests have less size distortion when
sample size increases. In Figure 2.3, the patterns of the rejection probabilities of all t tests
are similar as those in Figure 2.1. And still, there is no test that dominates the others for t
tests. For Wald tests, the pattern is very clear. As we can see from Figure 2.4, for all values
of b, the asymptotic tests have the highest size distortions. But the rejection probabilities
of the stationary bootstrap tests are stable and close to 5%, which implies that stationary
bootstrap successfully solves the over-rejection problem in this scenario. The null rejection
probabilities of the conditional-on-regressors bootstrap tests are higher than 5% but less
than those of the asymptotic tests.
As the values of ρ1 , ρ2 increase to 0.9, there exists strong serial correlation and endogeneity. We can see from Figures 2.5-2.8 that all the tests have serious over-rejection problems
regardless of bandwidth. For N = 5, a time series sample size T = 100 is not large enough for
the stationary bootstrap to obtain reasonable size that is close to 5%. But among all three
tests, the stationary bootstrap tests are better than conditional on regressor bootstrap and
asymptotic fixed-b tests. And this is true for both t and Wald tests, which is not the case
when ρ1 = ρ2 = 0.6. In addition, unlike the results before, Stat-BS IMOLS(D) and Stat-BS
IMOLS(fb) rejection probabilities are not that close any more. Generally speaking, when
both ρ1 and ρ2 are very large, Stat-BS IMOLS(D) tends to have the smaller size distortion
than Stat-BS IMOLS(fb). Therefore, when ρ1 , ρ2 are large, in order to obtain reasonable
size, we need a very large time series sample size and to use the Stat-BS IMOLS(D) statistics.
From the above, we see that the bootstrap tests generally have less size distortions than
95

the asymptotic tests. However, if the power of the bootstrap testing is low, then the bootstrap
methods are less useful. When the alternative is true, some bootstrap methods fail to
simulate critical values that are valid under the null in which case the tests have no power.
Therefore, the analysis of the power properties of the bootstrap tests is necessary. For the
sake of brevity we only display results of the stationary bootstrap for the case ρ1 = ρ2 = 0.6
for the Wald test for N = 5, T ∈ {50, 100} and using the Bartlett kernel. Starting from the
null values of β1 and β2 equal to 1, we consider under the alternative β1 = β2 = β ∈ (1, 1.25],
using (including the null value) a total of 13 values on a grid with mesh 0.02. We focus on
raw power using bootstrapped critical values.
Using N = 5, T = 50, with Bartlett kernel and b = 0.1, Figure 2.9 provides power
comparisons between Stat-BS IMOLS(D) and Stat-BS IMOLS(fb). The power plots indicate
that when the alternative is true, the stationary bootstrap is still simulating critical values
that are valid under the null. Figure 2.10 displays the same power comparisons as in Figure
2.9 but with T = 100. The main finding is that power increases as T increases. From Figures
2.9 and 2.10, we can see that the bootstrap tests have good power.
As mentioned in the end of Section 2.3, we also consider the stationary bootstrap and the
conditional-on-regressors bootstrap based on the residuals from the non-augmented partial
sum regression. Null rejection probabilities of the t and Wald tests as a function of b ∈ (0, 1]
are shown in Figures 2.11-2.18. The general patterns in Figures 2.11-2.18 are close to those
in Figures 2.1-2.8. Overall, the bootstrap methods based on the residuals from the nonaugmented partial sum regression have less size distortion problems than the asymptotic
methods especially for the Wald test with large sample size. But when serial correlation
and endogeneity are both large, it seems that the bootstrap results are depend little on the
choice of the residuals. The power results in this case are displayed in Figure 2.19 and 2.20,
96

which are very similar to the power results in Figures 2.9 and 2.10.

2.5

Summary and conclusion

This paper compares bootstrap tests with fixed-b asymptotic tests based on the panel
IM-OLS estimator of Vogelsang et al. [2016] for a homogeneous panel cointegrated regression
with endogenous regressors. The bootstrap methods used are the conditional-on-regressors
bootstrap and the stationary bootstrap. The purpose of using the bootstrap tests is to
improve the quality of finite sample inference. The Monte Carlo simulations show that the
bootstrap methods can effectively reduce size distortions in finite samples. In general, the stationary bootstrap has less size distortions than the conditional-on-regressors bootstrap and
asymptotic fixed-b tests, especially when there is strong serial correlation and endogeneity
(ρ1 = ρ2 = 0.9). It is necessary to have a large time series sample size to obtain reasonable
size of the tests. When the serial correlation and endogeneity is medium (ρ1 = ρ2 = 0.6), the
bootstrap methods still have less size distortion, but t and Wald tests have different results.
For Wald tests, the stationary bootstrap is always better than the other two methods. In
contrast, for t-tests, Cond-BS IMOLS(D), the statistic constructed using a HAC estimator
based on the first differences of the residuals from the augmented partial sum regression for
2 , has less size distortions when the bandwidth is relatively large (b > 0.25). In addition,
σu·v

the stationary bootstrap statistics are more robust than all other test statistics for all values
of bandwidth. Finally, the power plots from the simulation show that the bootstrap tests
have good power.
Further research will study the panel IM-OLS method for estimation and inference in a
heterogeneous cointegrating panel with endogenous regressors. In that more general scenario,
97

finding a fixed-b asymptotic pivotal statistic based on panel IM-OLS will be challenging.
However, the results in this paper indicate that the bootstrap method could be an alternative
solution for hypothesis tests. In addition, if the panel consists of cross-sectional dependent
units, then the bootstrap procedure will need to be modified to resample all individuals
together rather than resample individual by individual. Another topic of future research is
to establish the consistency of the bootstrap for panel IM-OLS tests.

98

APPENDIX

99

Figures

Figure 2.1: Empirical null rejections, t-test, N = 5, T = 50, ρ1 = ρ2 = 0.6, Bartlett kernel

100

Figure 2.2: Empirical null rejections, Wald test, N = 5, T = 50, ρ1 = ρ2 = 0.6, Bartlett
kernel

101

Figure 2.3: Empirical null rejections, t-test, N = 5, T = 100, ρ1 = ρ2 = 0.6, Bartlett kernel

102

Figure 2.4: Empirical null rejections, Wald test, N = 5, T = 100, ρ1 = ρ2 = 0.6, Bartlett
kernel

103

Figure 2.5: Empirical null rejections, t-test, N = 5, T = 50, ρ1 = ρ2 = 0.9, Bartlett kernel

104

Figure 2.6: Empirical null rejections, Wald test, N = 5, T = 50, ρ1 = ρ2 = 0.9, Bartlett
kernel

105

Figure 2.7: Empirical null rejections, t-test, N = 5, T = 100, ρ1 = ρ2 = 0.9, Bartlett kernel

106

Figure 2.8: Empirical null rejections, Wald test, N = 5, T = 100, ρ1 = ρ2 = 0.9, Bartlett
kernel

107

Figure 2.9: Raw power, Wald test, N = 5, T = 50, ρ1 = ρ2 = 0.6, b = 0.1, Bartlett kernel

108

Figure 2.10: Raw power, Wald test, N = 5, T = 100, ρ1 = ρ2 = 0.6, b = 0.1, Bartlett kernel

109

Figure 2.11: Empirical null rejections, t-test, N = 5, T = 50, ρ1 = ρ2 = 0.6, Bartlett kernel,
residuals from non-augmented partial sum regression

110

Figure 2.12: Empirical null rejections, Wald test, N = 5, T = 50, ρ1 = ρ2 = 0.6, Bartlett
kernel, residuals from non-augmented partial sum regression

111

Figure 2.13: Empirical null rejections, t-test, N = 5, T = 100, ρ1 = ρ2 = 0.6, Bartlett
kernel, residuals from non-augmented partial sum regression

112

Figure 2.14: Empirical null rejections, Wald test, N = 5, T = 100, ρ1 = ρ2 = 0.6, Bartlett
kernel, residuals from non-augmented partial sum regression

113

Figure 2.15: Empirical null rejections, t-test, N = 5, T = 50, ρ1 = ρ2 = 0.9, Bartlett kernel,
residuals from non-augmented partial sum regression

114

Figure 2.16: Empirical null rejections, Wald test, N = 5, T = 50, ρ1 = ρ2 = 0.9, Bartlett
kernel, residuals from non-augmented partial sum regression

115

Figure 2.17: Empirical null rejections, t-test, N = 5, T = 100, ρ1 = ρ2 = 0.9, Bartlett
kernel, residuals from non-augmented partial sum regression

116

Figure 2.18: Empirical null rejections, Wald test, N = 5, T = 100, ρ1 = ρ2 = 0.9, Bartlett
kernel, residuals from non-augmented partial sum regression

117

Figure 2.19: Raw power, Wald test, N = 5, T = 50, ρ1 = ρ2 = 0.6, b = 0.1, Bartlett kernel,
residuals from non-augmented partial sum regression

118

Figure 2.20: Raw power, Wald test, N = 5, T = 100, ρ1 = ρ2 = 0.6, b = 0.1, Bartlett kernel,
residuals from non-augmented partial sum regression

119

Chapter 3
Estimation and Inference for
Heterogeneous Cointegrated Panels
with Limited Cross Sectional
Dependence
This paper is concerned with parameter estimation and inference in a panel cointegrating
regression with endogenous regressors and heterogeneous long run variances in the cross
section. In addition, the model allows a limited degree of cross-sectional dependence due
to a common time effect. The estimator is labeled as panel integrated modified ordinary
least squares (panel IM-OLS). Similar to panel fully modified OLS (panel FM-OLS) and
panel dynamic OLS (panel DOLS), the panel IM-OLS estimator has a zero mean Gaussian
mixture limiting distribution. However, standard asymptotic inference is infeasible due the
existence of nuisance parameters. Inference based on panel IM-OLS relies on the stationary
bootstrap. The properties of panel IM-OLS are analyzed using the stationary bootstrap in
finite sample simulations.

120

3.1

Introduction

In the past decade, panel cointegration methods have drawn much attention in empirical
research. The attractive feature of panel cointegration methods is that they permit investigation of the long-run relationship among nonstationary variables more efficiently than
using time series data alone. However, panel cointegration is more complicated than single
time-series cointegration when cross-sectional dependence and heterogeneity exist. If the
cross-sectional dependence and heterogeneity were ignored, it might lead to poor inference
and inconsistent estimators. It is well known that the application of the first generation
panel unit root tests, which generally assume cross-sectional independence, to the series
with cross-sectional correlation leads to size distortion and low power. This might also be
the case for the panel cointegration estimation and testing. For example, Westerlund and
Edgerton [2008] claim that the tests of McCoskey and Kao [1998], Pedroni [1999], [2004] and
Westerlund [2005] all require independence among the cross-sectional units, and their size
properties become suspect when this assumption does not hold. The homogeneity assumption is often not well supported by the data. Therefore a framework that allows potential
heterogeneity is necessary.
In the panel cointegrated regression literature, the panel fully modified OLS (panel FMOLS) and the panel dynamic OLS (panel DOLS) methods are the most popular methods
(see Kao and Chiang [2000], Pedroni [2000], Bai et al. [2009] and Mark and Sul [2003]). They
are the extensions of the single time series fully modified OLS (FM-OLS) and dynamic OLS
(DOLS). Integrated modified OLS (IM-OLS), proposed by Vogelsang and Wagner [2014],
provides a fully parametric and computationally convenient alternative to the FM-OLS and
the DOLS estimators. Vogelsang et al. [2016] extend IM-OLS to panel data models with
121

individual dummies and homogeneous second moment structure. The present paper considers an extension of Vogelsang et al. [2016] by allowing time dummies and heterogeneous
variance structure in the model. The benefit of adding time dummies is twofold. First, time
dummies can handle deterministic components and common factor shocks, and second, time
dummies make the model robust to limited degrees of cross-sectional dependence. Allowing
heterogeneous, rather than homogeneous, variance structure makes the framework discussed
in this paper more applicable in empirical research. Bai et al. [2009] and Mark and Sul [2003]
consider similar problems using the panel FM-OLS and the panel DOLS estimators.
The limit theory considered here is obtained for a fixed number of cross-sectional units
N , letting the number of the time periods, T , go to infinity. The setting of N fixed and T →
∞ is widely used in empirical macroeconomics, empirical energy economics and empirical
finance problems (see Christopoulos and Tsionas [2004], Lee [2005], Apergis and Payne
[2009], Narayan and Smyth [2008] and Canzoneri et al. [1999]). Under this scenario, even
though the panel IM-OLS estimator converges to a zero mean Gaussian mixture distribution,
asymptotic inference is complicated by the presence of nuisance parameters. One way to
implement valid hypothesis tests is using bootstrap methods. Although bootstrap methods
are widely employed for analyzing nonstationary time series data, e.g. bootstrap unit root
tests and bootstrap cointegration tests (see Chang [2004], Paparoditis and Politis [2003],
Parker et al. [2006], Westerlund and Edgerton [2007]), surprisingly few papers are devoted
to bootstrap inference in cointegrated regressions. Psaradakis [2001] and Chang et al. [2006]
employ the sieve bootstrap procedure to cointegrated regressions. Li and Maddala [1997]
and Shin and Hwang [2013] apply the stationary bootstrap to cointegrated regression.
In the literature, the sieve bootstrap can be applied in fairly general models and performs
well in pure time series setting, but it requires fitting a finite order AR or VAR model to
122

the errors. Smeekes and Urbain [2014] questioned the validity of the use of VAR sieve
bootstrap in panels with a moderate cross-sectional dimension and showed that the AR
sieve bootstrap might be misleading when cross-sectional dependence is present. On the
contrary, the stationary bootstrap requires no parametric structure for drawing bootstrap
samples. In addition, a working paper by Li [2016] shows that the stationary bootstrap
performs well in panel cointegrated regressions with fixed effects and homogeneous variance
structure when cross sectional units are uncorrelated. The bootstrap method used in the
present paper is the stationary bootstrap.
The rest of the paper is organized as follows. Section 3.2 introduces the model, assumptions and the panel IM-OLS estimator. In Section 3.3 asymptotic inference and stationary
bootstrap inference are presented. Section 3.4 provides a Monte Carlo simulation to investigate the finite sample properties of the proposed bootstrap test. Section 3.5 summarizes
the results and concludes the paper. All proofs are collected in Appendices C - F.

3.2

3.2.1

Model set up and estimation

The model and assumptions

Consider the following panel data model

yit = αi + xit β + eit

(3.1)

xit = xit−1 + vit

(3.2)

123

where i = 1, 2, · · · , N and t = 1, 2, · · · , T index the cross-sectional and time series units,
respectively; yit , αi and eit are scalars; xit , β and vit are k × 1 vectors. The regressor, xit ,
is potentially endogenous for each individual i.
Assumption 6. Assume that the error term, eit , follows a special case of a factor model

eit = Ft λ + uit .

(3.3)

In Assumption 6, Ft is the common factor and uit is idiosyncratic component. Assumption 6 is a special case of factor model, as the factor loading λ is constant across i. Under
the above assumption, eit and ejt are correlated due to the common factor Ft , therefore
the panel data model is cross-sectional dependent. Because the regressor considered here is
endogenous, vit is assumed to be correlated with uit . In addition, there is no restriction on
the correlation between vit and Ft .
Define the error vector as ηit = uit v
it

and suppose that it is a (k + 1) dimensional

stationary vector for each i. In addition, assume that Ft is a I(0) process. This implies that
the model introduced in (3.1) describes a system of panel cointegrated regressions, i.e. yit
is cointegrated with xit . It might also be interesting to consider the case that Ft is a I(1)
process. Bai et al. [2009] consider the CupBC (continuously-updated and bias-corrected) and
the CupFM (continuously-updated and fully-modified) estimators for panel cointegration
models with cross-sectional dependence generated by unobserved global stochastic trends,
where Ft is non-stationary. In this paper, the interest is in estimation and inference about
β based on the panel IM-OLS estimator when Ft is stationary. In order to derive the panel
IM-OLS estimator’s limiting distribution, a second assumption is sufficient.

124

Assumption 7. Assume that ηit is independent across i, and satisfies the Functional CLT


1
1/2
Bu,i (r)
T −2
ηit ⇒ Bi (r) = 
 = Ωi Wi (r),
t=1
Bv,i (r)
[rT ]

where r ∈ (0, 1], and [rT ] denotes the largest integer value of rT .
1/2

In Assumption 7, Ωi

1/2

is a (k + 1) × (k + 1) matrix that satisfies Ωi = Ωi

1/2

Ωi

where


Ωuu,i Ωuv,i 
Ωi =
E ηit ηit−j = 
 > 0,
j=−∞
Ωvu,i Ωvv,i
∞

where it is obvious that Ωuv,i = Ωvu,i . Assume that Ωvv,i is non-singular, which implies that
{xit } are not cointegrated among themselves. Partition Bi (r) as Bi (r) = Bu,i (r) B (r) ,
v,i
and likewise partition Wi (r) as Wi (r) = wu,i (r) W (r) , where wu,i (r) and Wv,i (r) are
v,i
a scalar and a k-dimensional standard Brownian motion, respectively. Using the Cholesky
1/2

form of Ωi

,



1/2

Ωi

σu·v,i λuv,i 
=
,
1/2
0k×1 Ωvv,i
−1/2

2
it can be shown that σu·v,i
= Ωuu,i − Ωuv,i Ω−1
vv,i Ωvu,i and λuv,i = Ωuv,i Ωvv,i

. In

addition, it follows that


 
Bu,i (r) σu·v,i wu,i (r) + λuv,i Wv,i (r)
Bi (r) = 
.
=
1/2
Bv,i (r)
Ωvv,i Wv,i (r)
Note that λuv,i = 0 for all i because the regressors are allowed to be endogenous. Notice
that the 2nd order moment structure is heterogeneous across i.
125

3.2.2

Panel IM-OLS estimator

From regression (3.1) and Assumption 1, the system can be rewritten as

yit = αi + xit β + Ft λ + uit ,

and its cross-sectional mean is given by

y¯t = α
¯ + x¯t β + Ft λ + u¯t ,

where

1
y¯t =
N
α
¯ =

x¯t =

u¯t =

1
N
1
N
1
N

N

yit
i=1
N

αi
i=1
N

xit
i=1
N

uit .
i=1

Cross-sectional demeaning can be used to remove Ft λ and provides an estimation equation
that is exactly invariant to Ft λ. Note that the cross-sectional demeaning is exactly the
same as including time period dummies and projecting them out of the regression. Since
Ft could be unobserved time shock, therefore projecting it out before partial summing is

126

crucial. Cross-sectional demeaning gives

yit − y¯t = αi − α
¯ + xit − x¯t β + uit − u¯t ,

(3.4)

y¨it = µi + x¨it β + u¨it ,

(3.5)

which is denoted as

where

µi = α i − α
¯
y¨it = yit − y¯t
x¨it = xit − x¯t
u¨it = uit − u¯t .

Following Vogelsang and Wagner [2014], compute the partial sum of regression (3.5) to give

y¨

x
¨ β + Su
¨
Sit = tµi + Sit
it ,

where
t
y¨
Sit

=

y¨ij
j=1
t

x
¨ =
Sit

x¨ij
j=1
t

u
¨ =
Sit

u¨ij .
j=1

127

(3.6)

In order to deal with the endogeneity problem generated by the correlation between uit
and vit , it is sufficient to add additional regressors into regression (3.6). A natural candidate
is the demeaned regressor x¨it , however, this does not work due to the heterogeneity in
the model. This is formally shown in the Appendix. The endogeneity problem, which is
complicated by the heterogeneity in the variance structure, can be solved by adding the
decomposed x¨it into (3.6). The decomposed x¨it can be expressed as
−1
−1
N −1
−1
−1
x1t , · · ·
xi−1,t ,
xit ,
xi+1,t , · · ·
x .
N
N
N
N
N Nt

Adding these regressors separately will overcome the heterogeneous variance problem when
dealing with endogeneity. Details are given in the Appendix.
Remark 4. In regression (3.5), if β is the only parameter of interest, then it is possible to
demean across time to remove µi before partial summing. That is

+=x
y¨it
¨+
¨+
it β + u
it ,

where
+
y¨it

1
= y¨it −
T

T

y¨ik ,
k=1

x¨+
it

1
= x¨it −
T

T

x¨ik ,

u¨+
it

k=1

1
= u¨it −
T

Then regression (3.6) becomes to

y¨+

+

+

x
¨ β + Su
¨
Sit = Sit
it ,

where
y¨+

t

+

t

+, S x
¨
y¨ij
it =

Sit =
j=1

+

t

u
¨
x¨+
ij , Sit =
j=1

128

u¨+
ij .
j=1

T

u¨ik .
k=1

However, in this case, including the components of x¨it is not sufficient to deal with the
endogeneity problem. Finding the additional regressors for this partial sum regression is
much more challenging if not impossible, therefore this method is not considered in this
paper.
Remark 5. If the system does have homogeneous 2nd order moment structures, i.e. Ωi = Ωj
for any i, j for {1, 2, · · · , N }, then adding x¨it to the partial sum regression will be sufficient
for solving the endogeneity problem.
Including the additional regressors in (3.6) gives


y¨

x
¨β+
Sit = tµi + Sit

−1
N −1
xit γi + 
N
N



N

u
˜
xjt γj  + Sit

(3.7)

j=1,j=i

where
u
˜
Sit

=

u
¨
Sit

N −1
1
−
xit γi +
N
N

N

xjt γj .
j=1,j=i

Stacking all time periods and all individuals’ data together, the matrix form of the system
is given by
S y¨ = S x¨ θ + S u¨ ,

129

(3.8)

where



y¨
S11











u
˜



β 
 S11



 

 . 
 
 . 
 .. 
γ 
 .. 



 1




 

 y¨ 

 . 

u
˜
.
S 
 . 
S 
 1T 
 
 1T 



 

 .. 
 
 .. 
y
¨
u
¨
S =  .  , θ =  γN  , S =  .  ,



 




 

 y¨ 
 
 u˜ 
 SN 1 
 µ1 
 SN 1 



 

 . 
 . 
 . 
 . 
 . 
 . 
 . 
 . 
 . 


 





 

y¨
u
˜
SN T
SN
µN
T



x
¨
S11



 x¨
S
 12

 ..
 .


 x¨
 S1T

 .
x
¨
S =
 ..


 S x¨
 N1

 x¨
S
 N2

 ..
 .


x
¨
SN
T

N −1 x
11
N

−1 x
N 21

···

N −1 x
12
N

−1 x
N 22

···

..
.

..
.

···

N −1 x
1T
N

−1 x
N 2T

···

..
.

..
.

..
.

−1 x
N 11

−1 x
N 21

···

−1 x
N 12

−1 x
N 22

···

..
.

..
.

..
.

−1 x
N 1T

−1 x
N 2T

···

−1 x
N N1



1 · · · 0


−1 x

2
·
·
·
0
N
2
N


..
..
.. 
.
. ··· . 



−1 x
T
·
·
·
0

N NT

..
.. .. .. 
.
. . .
.


N −1 x

0
·
·
·
1

N1
N


N −1 x

0
·
·
·
2

N2
N

..
.. .. .. 
.
. . .


N −1 x
0
·
·
·
T
NT
N

The panel IM-OLS estimator is the OLS estimator of regression (3.8), which is given by

θˆ = S x¨ S x¨

−1

130

S x¨ S y¨ .

It follows that

θˆ − θ =

S x¨ S x¨

N

−1

T

= 

S x¨ S u¨
−1 

qit qit 

N



T

u
˜
qit Sit


i=1 t=1

i=1 t=1

where

q1t =

x
¨
S1t

N −1 x
1t
N

q2t =

x
¨
S2t

−1 x
N 1t

..
.

..
.

qN t =

−1 x
N 2t

· · · −1
N xN t t 0 · · · 0

N −1 x
2t
N

· · · −1
N xN t 0 t · · · 0

..
.
−1
−1
N −1 x
x
¨
.
SN
t N x1t N x2t · · ·
Nt 0 0 · · · t
N

Define the scaling matrix


0
T · Ik





−1
AP IM = 

I
⊗
I
N
k



1
0
IN ⊗ T 2
as a (k + N k + N ) × (k + N k + N ) diagonal matrix.
The following theorem gives the asymptotic distribution of the panel IM-OLS estimator.
Theorem 3. Assume that the data are generated by (3.1) and (3.2), and that Assumptions
6 and 7 hold. Define θ by stacking β, γi and µi . Then for fixed N , as T → ∞

131



ˆ
A−1
P IM θ − θ



 T βˆ − β





 (ˆ

γ
−
γ
)
1
1






..


.






=  (ˆ

 γN − γN ) 
√



 T (ˆ
µ1 − µ1 ) 




..


.





√
T (ˆ
µN − µN )


N

−1 

T

−1 
A−1
1T qit qit A1T

= T −1

N

T −1

i=1 t=1


⇒ 




1 N


1

−2 u
Sit˜ 
A−1
1T qit T

i=1 t=1

−1

1 N
0 i=1

T

hi (r)hi (r)dr




0 i=1 

×


hi (r) σu·v,i wu,i (r) −

= Ψ

132

1
N

N
j=1

 


dr
σu·v,j wu,j (r)


where



1
1
N
r
2 W (s) − 1
2
Ω
Ω
0
vv,i v,i
vv,j Wv,j (s)
N
j=1
1
−1 Ω 2 W (r)
N vv,1 v,1




















hi (r) = 



















..
.
1
N −1 Ω 2 W (r)
vv,i v,i
N

..
.
1
−1 Ω 2
N vv,N Wv,N (r)

0
..
.
r
..
.
0


ds


















.



















Conditional on hi (r) for i = 1, 2, · · · , N , it can be shown that Ψ ∼ N (0, VP IM ), where
VP IM is given by

VP IM = 


−1

1 N
0 i=1

hi (r)hi (r)dr

N
2
σu·v,i


i=1




1 N
0 i=1

1
0

×


¨ i (1) − H
¨ i (r)
H

¨ i (1) − H
¨ i (r) dr ×
H

−1
hi (r)hi (r)dr

and
¨ i (r) = Hi (r) − 1
H
N

N

Hj (r).
j=1

The derivation of this conditional variance is given in the Appendix.
133

3.3

3.3.1

Inference about θ

Inference using panel IM-OLS

This section provides a discussion of hypothesis testing using the panel IM-OLS estimator.
In particular, the hypothesis being considered is given by

H0 : Rθ = r

where R ∈ Rq×(k+N k+N ) with full rank q and r ∈ Rq . Because the vector θˆ has elements
that converge at different rates, restrictions on R are necessary. Assume that there exists a
non-singular q × q matrix AR such that

lim A−1
RAP IM = R∗
R
T →∞
with R∗ has rank q.
In order to carry out statistical inference, the asymptotic variance, VP IM , needs to be
estimated. The outside parts of the sandwich form can be estimated by


N

−1

T

T −2

AP IM qit qit AP IM 

.

i=1 t=1

The tricky part is estimating the middle part of the sandwich form of the variance. Suppose
2
2
that σ
˘u·v,i
is an estimator for σu·v,i
, then an estimator for the middle part of the variance is

134

given by
T

N
q
q
2
AP IM S¨iT − S¨i,t−1
σ
˘u·v,i

T −4

q
q
AP IM S¨iT − S¨i,t−1

,

t=1 i=1
q
q
with S¨it = Sit − N1

q
N
j=1 Sjt

q

t
k=1 qik .

and Sit =

Therefore, the estimator of VP IM takes

the form


N

−1

T

V˘P IM = T −2

AP IM qit qit AP IM 

×

i=1 t=1



T



N
q
q
AP IM S¨iT − S¨i,t−1

q
q
2
σ
˘u·v,i
AP IM S¨iT − S¨i,t−1

T −4

×

t=1 i=1



N

−1

T

T −2

AP IM qit qit AP IM 

.

i=1 t=1
2
Here, two potential candidates for σ
˘u·v,i
are considered.
2
1. The first candidate, σ
ˆu·v,i
, is based on the residuals of regression (3.7), i.e.

1
u
˜ = S y¨ − tˆ
x
¨ βˆ − N − 1 x γ
Sˆit
µi − Sit
ˆi +
it
it
N
N

N

xjt γˆj
j=1,j=i

where µ
ˆi , βˆ and γˆi (i = 1, 2, · · · , N ) are the panel IM-OLS estimators. Define a HAC
u
˜:
estimator using the first difference of Sˆit
T
2
σ
ˆu·v,i

=

T

T −1

k
j=2 h=2

|j − h|
M

u
˜ S
ˆu˜ .
Sˆij
ih

2
2. The second candidate, σ
˜u·v,i
, is based on the residuals of a further augmented regres-

135

sion of the partial sum regression (3.6), i.e.

1
x
¨ β˜ − N − 1 x γ
u
˜ = S y¨ − t˜
˜i,i +
µi − Sit
S˜it
i
it
it
N
N

N

˜i
xjt γ˜i,j − zit λ
j=1,j=i

where
t−1 j

T

Dij −

zit = t

j=1 s=1

j=1

Dit = S x¨
it

Dis ,

N −1 x
−1
t −1
it · · · N xN t
N x1t · · ·
N

,

and µ
˜i , β˜i , γ˜i,i and γ˜i,j are OLS from the further augmented regression given by

y¨
Sit

=

x
¨β
tµi + Sit
i

N −1
1
+
xit γi,i −
N
N

N

xjt γi,j + zit λi .

(3.9)

j=1,j=i

Note that for given i, the estimators γi,i is the parameter associate with ith individual’s
xit regressor, and γi,j are the parameters associate with ith individual’s all xjt regressor for
j = 1, 2, · · · , N and j = i. They are allowed to be different across different individual because
the further augmented regressions are being done individual by individual. Therefore, the
u
˜ is defined as
HAC estimator using S˜it
T
2
σ
˜u·v,i

=

T

T −1

k
j=2 h=2

|j − h|
M

u
˜ S
˜u˜ .
S˜ij
ih

2
Remark 6. The reason for considering the second variance estimator, σ
˜u·v,i
, is that it

delivers an asymptotic pivotal limit in the following two cases: (i) N = 1 (See Vogelsang
and Wagner [2014]); (ii) N > 1 with homogeneous variance structure (See Vogelsang et
2
al. [2016]). However, when N > 1 and heterogeneous variance structure exists, σ
˜u·v,i
no

136

longer leads to an asymptotic pivotal limit. In practice, this estimator should be considered
for N = 1 case or N > 1 with homogeneous variance structure case.
2
ˆ denote statistics defined using σ
Let tˆ and W
ˆu·v,i
to construct V˘P IM , and likewise t˜ and
2
˜ denote statistics defined using σ
˘ denote either
W
˜u·v,i
to construct V˘P IM . Letting t˘ and W

ˆ or t˜ and W
˜ , define the t and W ald statistics as:
tˆ and W
Rθˆ − r

t˘ =

RAP IM V˘P IM AP IM R
˘ =
W

Rθˆ − r

RAP IM V˘P IM AP IM R

−1

Rθˆ − r .

Theorem 4. Assume that the data are generated by (3.1) and (3.2), and that Assumptions
6 and 7 hold. Under traditional bandwidth and kernel assumptions, with N fixed as T → ∞
•
ˆ ⇒
W

χ2q
1 + dγ dγ

and when q = 1,
Z

tˆ ⇒

(1 + dγ dγ )
where χ2q is a chi-square random variable with q degrees of freedom, Z is a standard

137

normal random variable,


dγ dγ =VP−1
IM


−1

1 N
0 i=1

hi (r)hi (r)dr

N
2
dγ dγ i
σu·v,i
i


i=1




1 N
0 i=1

1
0

×


¨ i (1) − H
¨ i (r)
H

¨ i (1) − H
¨ i (r) dr ×
H

−1
hi (r)hi (r)dr

,

and
−2 d Ω
dγ dγi = σu·v,i
Ψi vv,i dΨi ,
i

where dΨi is the (i + 1)th k × 1 block of the distribution Ψ.
• Under fixed-b asymptotics where M = bT , b ∈ (0, 1] is held fixed as T → ∞, then the
ˆ and tˆ are given by
fixed-b limits of W

ˆ ⇒
W

χ2q
ˆ
Q(b)

and when q = 1,
tˆ ⇒

Z
ˆ
Q(b)

138

where

ˆ

Q(b)
=VP−1
IM


−1

1 N
0 i=1

N
2
Qb Pˆi (r)
σu·v,i


i=1



0 i=1

1
0


¨ i (1) − H
¨ i (r)
H

¨ i (1) − H
¨ i (r) dr ×
H

−1

1 N



×

hi (r)hi (r)dr

hi (r)hi (r)dr

is a stochastic process that depends on the kernel function, bandwidth and Wi (r).
• Under fixed-b asymptotics where M = bT , b ∈ (0, 1] is held fixed as T → ∞, then

˜ ⇒
W

χ2q
˜
Q(b)

and when q = 1,
Z

t˜ ⇒

˜
Q(b)
where

˜

Q(b)
=VP−1
IM


−1

1 N
0 i=1

hi (r)hi (r)dr

N
2
σu·v,i
Qb P˜i (r)


i=1




1 N
0 i=1

1
0

×


¨ i (1) − H
¨ i (r)
H

¨ i (1) − H
¨ i (r) dr ×
H

−1
hi (r)hi (r)dr

is a stochastic process that depends on the kernel function, bandwidth and Wi (r).

139

3.3.2

Inference using the stationary bootstrap

˘ are not asymptotic pivotal, which makes asymptotic
Unfortunately, the statistics t˘ and W
inference infeasible. One possible solution is applying the stationary bootstrap to mimic the
non-pivotal asymptotic distribution of those statistics. The stationary bootstrap, proposed
by Politis and Romano [1994], is a special type of block bootstrap where the block size follows a geometric distribution instead of a fixed number. For a geometric distribution with
parameter pT , the expected block size of the stationary bootstrap is 1/pT . The stationary
bootstrap has been used in the literature of unit root tests, cointegration tests and cointegrated regression inference; see Swensen [2003], Paparoditis and Politis [2005] , Parker
et al. [2006], Shin [2015], and Shin and Hwang [2013]. It can capture the serial correlation
structure in the original sample by block resampling, and it produces stationary bootstrap
samples. A formal description of the stationary bootstrap inference procedure is given below.

1. Calculate the residuals based on regression (3.7) as

1
u
˜ = S y¨ − tˆ
x
¨ βˆ − N − 1 x γ
ˆi +
Sˆit
µi − Sit
it
it
N
N

N

xjt γˆj
j=1,j=i

ˆ γˆi and γˆj for j = 1, · · · , N and j = i are the panel IM-OLS estimators.
where µ
ˆi , β,
2. Define

u
˜ as a proxy for u
Sˆit
¨it , and

x¨it as a proxy for v¨it , which is the cross-sectional

demeaned vit . Based on those proxies, define η¨ˆit =
proxy for η¨it = u¨it v¨
it

u
˜
Sˆit

x¨it

= uˆ¨it vˆ¨
it

as a

u
˜ = 0 and x
for t = 1, 2, · · · , T , and set Sˆi0
¨i0 be zero vector

for all i.

140

3. Re-sample the series

ηˆ¨it

via the stationary bootstrap, obtaining

ηˆ¨

, which can

be partitioned the same as ηˆ¨it into ηˆ¨ = uˆ¨ vˆ¨ .
it
it
4. Obtain the bootstrap samples x¨it by
t

vˆ¨ij ,

x¨it =
j=1

and generate the bootstrap samples y¨it from 1

ˆi + x¨it βˆ + uˆ¨it .
y¨it = µ

5. After obtaining the bootstrap demeaned variables x¨it and y¨it , follow the same procedure as discussed before to estimate θ, denoted by θˆ , and compute the bootstrap
estimator of the limiting variance VP IM , say V˘P IM . Define the bootstrap statistics as
follows

t˘

Rθˆ − Rθˆ
=
RAP IM V˘P IM AP IM R

˘
W

=

Rθˆ − Rθˆ

RAP IM V˘P IM AP IM R

−1

6. Repeat steps 3-5 independently B times to obtain samples t˘j

Rθˆ − Rθˆ .

B
j=1

˘
and W
j

B
j=1

.

1 Note that there is another method to obtain x
¨it . One can resample directly from the original regressor
¨it . However, the size of the
xit to obtain xit and then apply cross-sectional demeaning to xit to obtain x
tests based on this method is higher than that of the method introduced in procedure 1 to 4. Therefore, this
method is not included in this paper.

141

7. Compute the equal tail bootstrap p-value as

p

p

t˘

˘
W

= 2min 

=

1
B

1
B

B
j=1

1
I t˘j ≤ t˘ ,
B



B

I t˘j > t˘ 
j=1

B

˘ ,
˘ >W
I W
j
j=1

where I(·) is the indicator function. Reject the null hypothesis if the equal tail bootstrap p-value is less than 5%.

3.4

Finite sample simulation

This section investigates finite sample size and power of the bootstrap tests based on the
panel IM-OLS estimators. The data generating process is given by

yit = x1it β1 + x2it β2 + uit
x1it = x1i,t−1 + v1it
x2it = x2i,t−1 + v2it

where for all i = 1, 2, · · · , N , ui0 = 0, x1i0 and x2i0 are zero vectors, and

uit = ρ1 ui,t−1 + ρ2 (e1it + e2it ) + εit
v1it = e1it + 0.5e1i,t−1
v2it = e2it + 0.5e2i,t−1

142

where for ith individual, εit , e1it and e2it are i.i.d. N 0, i2 random variables. There is
no individual effect term and common time effect term included in the data generating
process because the focus here is on β1 and β2 , and the estimates of β1 and β2 are exactly
invariant to those terms. The parameter values are β1 = β2 = 1. In addition, ρ1 and ρ2
are chosen from {0.6, 0.9}. The parameter ρ1 controls serial correlation in the regression
error, and ρ2 determines the endogeneity of the regressors. The kernel function used in
this simulation study is the Bartlett kernel, and the bandwidths are given by M = bT with
b ∈ {0.1, 0.5, 1}. For the block length parameter pT in the stationary bootstrap, two different
settings are presented. One is pT = 0.01(4 − j)(T /50)−1/3 with j ∈ {1, 2, 3}, the other is
pT = 0.04(4 − j)(T /50)−1/3 with j ∈ {1, 2, 3}. The sample sizes are N = 5, T ∈ {50, 500}.
The number of bootstrap replications is B = 399, and the number of simulation replications
is 1000.
Results only for cases where ρ1 = ρ2 are reported. The results include t-statistics for
testing the null hypothesis H0 :

β1 = 1 and Wald statistics for testing the joint null

hypothesis H0 : β1 = β2 = 1. The bootstrap panel IM-OLS statistics were implemented
in two ways. The first one uses the stationary bootstrap procedures with the bootstrap
2
version of σ
ˆu·v,i
and is labeled Stat-BS IM-OLS(D). The second one uses the stationary
2
bootstrap procedures with the bootstrap version of σ
˜u·v,i
and is labeled Stat-BS IM-OLS(fb).

Rejections for the bootstrap statistics are carried out by comparing the bootstrap p-value
with the nominal level, which is 5% in this simulation.
Tables 3.1 to 3.4 report empirical null rejection probabilities of the t and Wald tests. In
each table Panel A corresponds to T = 50 and Panel B to T = 500. Some common findings
about the t and Wald tests can be summarized as follows. For both t and Wald tests, Stat-BS
IM-OLS(D) statistics tend to have smaller null rejection probabilities than those of Stat-BS
143

IM-OLS(fb) statistics. When the bandwidth parameter b varies, the rejection probabilities
are relatively stable for both t and Wald tests, which shows that the bootstrap method can
successfully capture the impact of the bandwidth on the test statistics. In addition, when
the sample size T increases from 50 to 500, rejection probabilities approach 0.05 as expected.
As the values of ρ1 , ρ2 increase from 0.6 to 0.9, there exists strong serial correlation and
endogeneity. It can be seen from Tables 3.1-3.4 that the rejection probabilities in all cases
generally increase, but those increases depend on the sample size and the test statistics. If
the time sample is small (T = 50), the rejection probabilities increase quite a lot for all tests.
In contrast, if the time sample size is large (T = 500), the Stat-BS IM-OLS(D) statistics
have similar rejection probabilities as ρ1 = ρ2 = 0.6, whereas the rejection probabilities
increase quite a bit for the Stat-BS IMOLS(fb) statistics. This implies that when the time
sample size is large enough, the Stat-BS IM-OLS(D) statistics can effectively handle strong
serial correlation and endogeneity.
Another important pattern in Tables 3.1-3.4 is that the size of the tests depends heavily
on the tuning parameter pT . It is not a surprise because the stationary bootstrap is a moving
block bootstrap with changing block lengths. Theoretically, there is no rule of thumb for
choosing the value of pT to ensure the hypothesis test has correct size. In a given sample,
Politis and White [2004] and Patton et al. [2009] propose a method to obtain an optimal
block length parameter for the stationary bootstrap. However, that optimal block length
parameter is based on minimizing the MSE of the stationary bootstrap sample mean, which
doesn’t necessarily guarantee the correct size of the tests. Therefore, several different values
for pT were used in this simulation study. In Tables 3.1-3.4, for both t and Wald tests,
when pT is small, corresponding to large average block length, the tests tend to have over
rejection problems. As pT increases, the over rejection problem becomes less severe and
144

under-rejection problems appear in some of the cases. To obtain the correct size, the t tests
require pT to be larger than that of the Wald tests.
Next consider the power properties of the tests. When the alternative is true, some
bootstrap methods fail to simulate critical values that are valid under the null in which case
the tests have no power. Therefore, the analysis of the power properties of bootstrap tests
is important. Here, only results for the case ρ1 = ρ2 ∈ {0.6, 0.9} for the Wald test for
N ∈ {5, 15}, T ∈ {50, 500} with the Bartlett kernel are provided. If the power of the test
is not an issue for small sample size, like T = 50, then it will not be a concern when the
sample size is large, like T = 500. Starting from the null values of β1 and β2 equal to 1,
the alternative values being considered are β1 = β2 = β ∈ (1, 1.4], which are total of 21
values on a grid with mesh 0.02 including the null value. Power and size-adjusted power are
reported. Note that size-adjusted power is not feasible in practice, but it allows us to see the
theoretical power differences across tests while holding null rejection probabilities constant
at 0.05.
Figures 3.1-3.4 show that using the bootstrap method, the Stat-BS IM-OLS(D) and StatBS IM-OLS(fb) Wald tests do have power. Figure 3.1 shows the power comparison of the
Stat-BS IM-OLS(D) Wald test for small (T = 50) and large (T = 500) sample sizes and
using respective block size parameter values, pT ∈ {0.08, 0.00464}, give null rejections close
to 5%. It can be seen that the power of the tests with the larger sample size (T = 500)
and smaller pT (pT = 0.00464) grows dramatically fast. This implies that if the sample
size is large enough and the resampling block size parameter pT can be wisely chosen, the
Stat-BS IM-OLS(D) Wald test tends to have very high power. And even if the sample size
is relatively small (T = 50), the power of the test is still acceptable if pT is carefully chosen.
Next consider the impact of the serial correlation and endogeneity on the power of the
145

Stat-BS IM-OLS(D) Wald test. Figure 3.2 displays the power comparison of the Stat-BS IMOLS(D) Wald test for small (ρ1 = ρ2 = 0.6) and large (ρ1 = ρ2 = 0.9) serial correlation and
endogeneity with respective block size parameter values, pT ∈ {0.00464, 0.00696}, give null
rejections close to 5%. The power of the test with smaller serial correlation and endogeneity
(ρ1 = ρ2 = 0.6) is higher than that of the test with larger serial correlation and endogeneity
(ρ1 = ρ2 = 0.9). If the sample size is small, the power of the tests is lower as expected.
Figures 3.3 and 3.4 provide size-corrected power comparisons between the Stat-BS IMOLS(D) and Stat-BS IM-OLS(fb) Wald tests for the same values of T , ρ1 , ρ2 , b but using
different sample size N . In Figure 3.3, the sample size N is 5, while in Figure 3.4, the sample
size N is 15. These two figures allow us to see power differences across tests while holding
null rejection probabilities constant at 0.05. It can be seen that when the cross sectional
sample size is small, the Stat-BS IM-OLS(fb) test has slightly higher power than that of
the Stat-BS IM-OLS(D) test. However, when the cross sectional sample size increases, the
power of Stat-BS IM-OLS(D) test is much higher. This implies that we should not consider
using the Stat-BS IM-OLS(fb) test when N is large, because it has large size distortions and
lower power in this scenario.

3.5

Summary and conclusions

This paper considers the estimation and inference of a homogeneous cointegrated vector
in a panel data model with individual heterogeneity and heterogeneous variance structure.
In addition, the model allows a limited degree of cross-sectional dependence due to a common
time effect. The estimator is labeled as panel IM-OLS. It is a fully parametric estimator that
is based on a partial sum transformed regression augmented by the decomposed demeaned
146

original regressor. The advantage is that it leads to a zero mean mixed Gaussian limiting
distribution without requiring the choice of tuning parameters (like bandwidth, kernel function, numbers of leads and lags). Asymptotic inference is infeasible due to the presence of
nuisance parameters, and the stationary bootstrap is used for hypothesis testing. Monte
Carlo simulations show that the bootstrap method can deliver good size and power for t and
Wald tests, depending on the sample size, serial correlation, endogeneity and the stationary
bootstrap block length resampling parameter. When there is strong serial correlation and
endogeneity, for moderate time sample sizes, the size of the tests are close to nominal level
for certain values of pT .
Unlike in Vogelsang et al. [2016], the further augmented regression residuals do not lead
to an asymptotic pivotal test, and the bootstrap hypothesis test based on it has more size
distortion. When the cross sectional sample size N is small, the power of the test based on
the further augmented regression residuals is a little bit higher than that of the test based on
augmented regression residuals. However, when the cross sectional sample size N increases,
the power of the test based on the further augmented regression residuals is much lower than
that of the test based on augmented regression residuals. This power loss as N increases
is because the further augmented regression requires adding many additional regressors to
compute the residuals. Therefore, in practice, when N is large and the panel has cross
sectional dependence and heterogeneous variance structure, inference based on the further
augmented regression residuals is not recommended.
One limitation of the present paper is that the cross-sectional dependence is only coming
from a common time effect with a constant factor loading. This might be restrictive in
some applications. Therefore, a model with more general cross-sectional dependence may be
worth considering in the future. In that more general scenario, the theory of the inference
147

based on the panel IM-OLS type estimators will rely on more general bootstrap procedures.
If the stationary bootstrap can mimic the non-pivotal limit of the original statistics, then
formally proving the asymptotic equivalence between the stationary bootstrap statistics and
the original test statistics may be a viable research topic in the future.

148

APPENDIX

149

Tables and Figures
Table 3.1: Empirical null rejection probabilities, 5% level, t-tests for H0 : β1 = 1, N = 5,
ρ = 0.6, Bartlett kernel
pT
0.01
0.02
0.03
0.04
0.08
0.12
0.00464
0.00928
0.01393
0.01857
0.03713
0.05570

Stat-BS(D)
Stat-BS(fb)
b=0.1 b=0.5 b=1 b=0.1 b=0.5 b=1
Panel A: T = 50
0.246 0.214 0.226 0.346 0.335 0.336
0.219 0.191 0.198 0.31
0.309 0.31
0.199
0.17 0.173
0.3
0.308 0.303
0.184 0.162 0.157 0.285 0.286 0.287
0.127 0.114 0.115 0.231 0.239 0.239
0.102 0.094 0.095 0.213 0.209 0.207
Panel B: T = 500
0.152 0.145 0.138 0.174 0.176 0.162
0.099 0.101 0.102 0.123 0.122 0.106
0.074 0.075 0.081 0.086 0.074 0.085
0.053 0.062 0.065 0.066 0.065 0.068
0.032 0.044 0.033 0.037 0.039 0.033
0.023 0.028 0.024 0.03
0.02 0.025

Table 3.2: Empirical null rejection probabilities, 5% level, t-tests for H0 : β1 = 1, N = 5,
ρ = 0.9, Bartlett kernel
pT
0.01
0.02
0.03
0.04
0.08
0.12
0.00464
0.00928
0.01393
0.01857
0.03713
0.05570

Stat-BS(D)
Stat-BS(fb)
b=0.1 b=0.5 b=1 b=0.1 b=0.5 b=1
Panel A: T = 50
0.427 0.375 0.371 0.74
0.745 0.75
0.405 0.349 0.347 0.747 0.743 0.75
0.395 0.328 0.328 0.738 0.733 0.734
0.356 0.307 0.291 0.738
0.74
0.73
0.333 0.279 0.267 0.724 0.724 0.72
0.307 0.247 0.235 0.715 0.716 0.715
Panel B: T = 500
0.144 0.138 0.137 0.243
0.23
0.23
0.095 0.096 0.098 0.175 0.172 0.165
0.07
0.07 0.074 0.144 0.125 0.13
0.053 0.061 0.06 0.115 0.113 0.114
0.031 0.035 0.033 0.093
0.08 0.077
0.022 0.027 0.024 0.073 0.077 0.072

150

Table 3.3: Empirical null rejection probabilities, 5% level, Wald-tests for H0 : β1 = 1, β2 = 1,
N = 5, ρ = 0.6, Bartlett kernel
pT
0.01
0.02
0.03
0.04
0.08
0.12
0.00464
0.00928
0.01393
0.01857
0.03713
0.05570

Stat-BS(D)
Stat-BS(fb)
b=0.1 b=0.5 b=1 b=0.1 b=0.5 b=1
Panel A: T = 50
0.159 0.142 0.153 0.296 0.299 0.302
0.125 0.122 0.122 0.266 0.268 0.269
0.107 0.109 0.109 0.239 0.236 0.236
0.092 0.101 0.086 0.216
0.22 0.214
0.049 0.047 0.047 0.155 0.151 0.157
0.029 0.031 0.027
0.1
0.113 0.119
Panel B: T = 500
0.053 0.053 0.051 0.081 0.073 0.074
0.02
0.023 0.022 0.037 0.038 0.039
0.01
0.011 0.019 0.019 0.019 0.021
0.002
0.01 0.007 0.008
0.01 0.016
0.002 0.003 0.004 0.002 0.002 0.005
0
0.003 0.003
0
0.002 0.003

Table 3.4: Empirical null rejection probabilities, 5% level, Wald-tests for H0 : β1 = 1, β2 = 1,
N = 5, ρ = 0.9, Bartlett kernel
pT
0.01
0.02
0.03
0.04
0.08
0.12
0.00464
0.00928
0.01393
0.01857
0.03713
0.05570

Stat-BS(D)
Stat-BS(fb)
b=0.1 b=0.5 b=1 b=0.1 b=0.5 b=1
Panel A: T = 50
0.474 0.402 0.393 0.876 0.876 0.867
0.448 0.374 0.368 0.874 0.873 0.864
0.43
0.346 0.348 0.874
0.87 0.865
0.402 0.337 0.316 0.869 0.867 0.865
0.325 0.261 0.25 0.859 0.861 0.846
0.268 0.212 0.21 0.848 0.854 0.841
Panel B: T = 500
0.07
0.062 0.069 0.16
0.159 0.166
0.031
0.04 0.041 0.107 0.096 0.097
0.014 0.016 0.021 0.069 0.063 0.069
0.011 0.018 0.013 0.057 0.043 0.056
0.004 0.006 0.008 0.028 0.031 0.029
0.003 0.005 0.004 0.019 0.019 0.021

151

Figure 3.1: Power of bootstrap Stat-BS IM (D), Wald test, N = 5, ρ1 = ρ2 = 0.6, b = 0.5,
Bartlett kernel with different T and pT

152

Figure 3.2: Power of bootstrap Stat-BS IM (D), Wald test, N = 5, T = 500, b = 0.5, Bartlett
kernel with different ρ and pT

153

Figure 3.3: Size adjusted power, Wald-tests, N = 5, T = 50, ρ1 = ρ2 = 0.6, b = 0.5, Bartlett
kernel

154

Figure 3.4: Size adjusted power, Wald-tests, N = 15, T = 50, ρ1 = ρ2 = 0.6, b = 0.5,
Bartlett kernel

155

Proof of failure of using x¨it to solve endogeneity problem
This is the proof showing that directly adding x¨it to regression (3.6) cannot fully deal
with the endogeneity problem in the model considered in this paper. Suppose, we add x¨it
into the partial sum model, which gives
Sity¨ = tµi + Sitx¨ β + x
¨it γi + Situ¨ − x
¨it γi
1

.Consider the behavior of T − 2 Situ¨ − x
¨it γi as T → ∞,

T

− 21

Situ¨ − x
¨it γi = T

− 12

Situ − 1
N

N
u
Sjt
− xit γi +
j=1

1

= T − 2 Situ − xit γi −

1
N

1
N



N

xjt γi 
j=1

N
1

u
− xjt γi
T − 2 Sjt
j=1

1
⇒ Bu,i (r) − Bv,i (r)γi −
N

N

Bu,j (r) − Bv,j (r)γi
j=1
1

2
Wv,i (r) γi
= σu·v,i wu,i (r) + λuv,i Wv,i (r) − Ωvv,i

1
−
N


N

1
2
σu·v,j wu,j (r) + λuv,j Wv,j (r) − Ωvv,j
Wv,j (r)

γi

j=1

1
= σu·v,i wu,i (r) −
N



N

σu,v,j wu,j (r)
j=1

1

−1

2
2
γi − Ωvv,i
λuv,i
− Wv,i (r)Ωvv,i

1
+
N

N

1

−1

2
2
Wv,j (r)Ωvv,j
γi − Ωvv,j
λuv,j

j=1

1
= σu·v,i wu,i (r) −
N

N

σu,v,j wu,j (r)
j=1

Note that the last inequality holds because when there is heterogeneity in the 2nd moment
structure, it is almost impossible that
−1

−1

2
2
γi = Ωvv,i
λuv,i = Ωvv,j
λuv,j

for all j = 1, 2, · · · , N . Therefore, just adding x¨it to regression (3.6) cannot fully deal with
the endogeneity problem. Note that, if the 2nd moment structure is homogeneous, then only

156

adding x¨it to regression (3.6) will work, because
−1

−1

−1

2
2
γi = Ωvv,i
λuv,i = Ωvv,j
λuv,j = γj = Ωvv2 λuv

for all i, j.

157

Proof of Theorem 3
In order to derive the asymptotic distribution of the panel IM-OLS estimator, we start
u
˜ with N fixed and T → ∞.
with regression (3.7). First consider the limit of T −1/2 Sit

¨
u
˜ = T −1/2 S u
T −1/2 Sit
it −

N −1
1
xit γi +
N
N


u−
= T −1/2 Sit

1
N

j=1,j=i

u −x γ +
Sjt
it i
j=1

1

u −x γ −
= T − 2 Sit
it i

⇒

xjt γj 

N

1
N

Bu,i (r) − Bv,i (r)γi −

N



N

1
N



N

xjt γj 
j=1

1

u −x γ
T − 2 Sjt
jt j

j=1

1
N

N

Bu,j (r) − Bv,j (r)γj
j=1
1

2 γ
= σu·v,i wu,i (r) + λuv,i Wv,i (r) − Wv,i (r)Ωvv,i
i
N

1
1
2 γ
σu·v,j wu,j (r) + λuv,j Wv,i (r) − Wv,j (r)Ωvv,j
−
j
N
j=1


N
1
−1
1
2
2 λ
= σu·v,i wu,i (r) −
σu·v,j wu,j (r) − Wv,i (r)Ωvv,i
γi − Ωvv,i
uv,i
N
j=1

1
+
N

N

−1

1

2
2 λ
Wv,j (r)Ωvv,j
γj − Ωvv,j
uv,j
j=1
−1

−1
2 λ
Therefore, when γi = Ωvv,i
uv,i = Ωvv,i Ωvu,i , it follows that

u
˜
T −1/2 Sit

1
=⇒ σu·v,i wu,i (r) −
N

1

N

σu·v,j wu,j (r).
j=1

−1 = T − 2 A
−1
Define A1T
P IM . The next step of the proof is to obtain the limit of A1T qit for

158

N fixed and T → ∞.


N
3
3
−
−
1
x
x
T 2 Sjt 
T Sitx¨
T 2 Sit − N
j=1
 
 − 1 −1

 T 2 N x1t  

−1 − 21
 


x
T
1t
N
 


..




.
..
 
 1 .

 
 − 2 N −1

1
x
T



it
N −1 − 2
N
T
x




it
N
.
 


..
..
 


.
 


1
−
−1




=  T 2 N xN t  = 
−1 − 12

x
T
N
t
 


N




0
 


0
..
 


.




.
..
 


 


t




t
T




T
..




..
.

 

.


0
0

 

N
1
1
N
r
1
2
2
r
r
1
Ωvv,j Wv,j (s) ds
 0 Ωvv,i Wv,i (s) − N
 0 Bv,i (s)ds − N
0 Bv,j (s)ds

j=1
j=1
 


1
 



−1
−1
2
 

B
(r)
W
(r)
Ω
v,1

v,1
vv,1
N
N
 


 

..
..

 

.
.

 


1


N −1


N
−1
2
B
(r)


v,i
W
(r)
Ω
N


vv,i v,i
N




..
 

.

.
.
 

.


=
⇒ 


1
−1
 

B
(r)

−1
2
v,N
 

N
W
(r)
Ω

N vv,N v,N
 


 

0

0
 


..
 


.
 

.
.

.
 


 


r
 

r

 


.
 

..
.

..
 


0
0


A−1
1T qit

=

− 32



hi (r).

Therefore, as T → ∞,
A−1
1T qit ⇒ hi (r).

159

For fixed N , as T → ∞,


ˆ
A−1
P IM θ − θ



T βˆ − β





γ1 − γ1 ) 
 (ˆ


..


.




− 21
ˆ
= T A1T θ − θ =  (ˆ

γ
−
γ
)
N
N

√
 T (ˆ
µ1 − µ1 ) 




.


..

√
T (ˆ
µN − µN )
−1 

N

−1 
A−1
1T qit qit A1T

= T −1

⇒ 



1 N


− 21 u
A−1
q
T
Sit˜ 
it
1T

i=1 t=1

−1

1 N
0 i=1

T

T −1

i=1 t=1



N

T

hi (r)hi (r)dr



0 i=1 

×


hi (r) σu·v,i wu,i (r) −

= Ψ

160

1
N

N
j=1

 

σu·v,j wu,j (r) dr


Proof of the derivation of the form of the asymptotic variance Ψ
We start from rewriting 01

1 N




0 i=1 
1 N

N
i=1

hi (r) σu·v,i wu,i (r) − N1


hi (r) σu·v,i wu,i (r) −



1
N

N
j=1

N
j=1 σu·v,j wu,j (r)

dr as




σu·v,j wu,j (r)
dr




N

hi (r) − 1
hj (r) σu·v,i wu,i (r)dr
=
N
0 i=1
j=1


N
N
1
hi (r) − 1
=
σu·v,i
hj (r) wu,i (r)dr
N
0
i=1
j=1


N
N
1
1
wu,i (r)d Hi (r) −
σu·v,i
=
Hj (r)
N
0
i=1
j=1






N
N
N
1
1
Hi (r) − 1
σu·v,i wu,i (r) Hi (r) −
Hj (r) |10 −
Hj (r) dwu,i (r)
=
N
N
0
i=1
j=1
j=1






N
N
N
1
1
Hi (r) − 1
σu·v,i wu,i (1) Hi (1) −
=
Hj (1) −
Hj (r) dwu,i (r)
N
N
0
i=1
j=1
j=1






N
N
N
1
1
Hi (1) − 1
Hi (r) − 1
σu·v,i 
Hj (1) dwu,i (r) −
Hj (r) dwu,i (r)
=
N
N
0
0
i=1

j=1

N

1

σu·v,i

=
i=1

0

j=1

¨ i (1) − H
¨ i (r)]dwu,i (r).
[H

Therefore, the variance of
1 N




0 i=1 


hi (r) σu·v,i wu,i (r) −

1
N

N
j=1



σu·v,j wu,j (r) dr


will be same as the variance of
N

1

σu·v,i
i=1

0

¨ i (1) − H
¨ i (r)]dwu,i (r)
[H

161

which is

N
2
σu·v,i
i=1

1
0

¨ i (1) − H
¨ i (r)][H
¨ i (1) − H
¨ i (r)] dr.
[H

Then, the variance of Ψ is

VP IM = 


−1

1 N
0 i=1

hi (r)hi (r)dr

N
2
σu·v,i


i=1




1 N
0 i=1

1
0

×


¨ i (1) − H
¨ i (r)][H
¨ i (1) − H
¨ i (r)] dr ×
[H
−1

hi (r)hi (r)dr

162

.

Proof of Theorem 4
This is the proof of the null limiting distribution of the test statistics in Theorem 4.
2
u
˜ , where
First, consider the behavior of σ
ˆu·v,i
. It is based on Sˆit
y¨
Sit

u
˜ =
Sˆit

−

tˆ
µi −

= y¨it − µ
ˆi − x¨it βˆ −

x
¨ βˆ −
Sit

N −1
1
xit γˆi +
N
N

N −1
1
vit γˆi +
N
N

N

xjt γˆj
j=1,j=i

N

vjt γˆj
j=1,j=i

1
N −1
vit γˆi +
= µi + x¨it β + u¨it − µ
ˆi − x¨it βˆ −
N
N
1
= µi + x¨it β + uit −
N
1
= uit − vit γi −
N
1
+
N
=

N
j=1

N

vjt γˆj
j=1,j=i

1
ujt − µ
ˆi − x¨it βˆ − vit γˆi +
N

N

vjt γˆj
j=1

N

γi − γi )
ujt − vjt γj − vit (ˆ
j=1

N

vjt γˆj − γj − (ˆ
µi − µi ) − x¨it βˆ − β
j=1

u+
it

1
− vit (ˆ
γi − γi ) −
N

N

u+
ˆj − γj
jt − vjt γ

− (ˆ
µi − µi ) − x¨it βˆ − β

j=1

where u+
it = uit − vit γi . It can be shown that the last three parts of the formula can be
u
˜ . Thus, the long run variance estimator
neglected for long run variance estimation of Sˆit
u
˜ , asymptotically coincides with long run variance estimator based on u+ −
based on Sˆit
it
vit (ˆ
γi − γi ).
2
0
+ =
+ = σu·v,i
,
and
then
its
long
run
variance
is
Ω
. Using
Define ηit
u+
,
v
i
it
it
0
Ωvv,i
p
+ , an infeasible long run variance estimator, Ω+ , is consistent. That is Ω+ →
unobserved ηit
i
i
Ω+
i .
1
+
+
γi − γi ) = ηit
, then HAC estimator, Ω+
Note that: u+
it − vit (ˆ
i , for uit −
− (ˆ
γi − γi )

163

vit (ˆ
γi − γi ) can be written as
1 − (ˆ
γi − γi ) Ω+
i

1
− (ˆ
γi − γi )

with
(ˆ
γi − γi ) ⇒ 0k×k 0k×k · · · Ik · · · 0k×k 0 · · · 0 ×


−1 
N

1
1 N
¨ i (1) − H
¨ i (r) dwu,i (r)

σu·v,i
hi (r)hi (r)dr
H


0
0
i=1

i=1

= d Ψi
where dΨi represents the (i + 1)th k × 1 block of the distribution Ψ.
Combining the above results shows that Ω+
i converges to
1 −dΨ

2
σu·v,i

0

0

Ωvv,i

i

1
2
= σu·v,i
+ dΨ Ωvv,i dΨi
i
−dΨi
−2 d Ω
2
= σu·v,i
1 + σu·v,i
Ψ vv,i dΨi
i

=

2
σu·v,i

1 + dγ dγi ,
i

2
2
2
ˆu·v,i
, converges
1 + dγ dγi . This implies that VˆP IM , using σ
which leads to σ
ˆu·v,i
⇒ σu·v,i
i
to
−1

1 N

VˆP IM ⇒

×

hi (r)hi (r)dr
0

i=1

N

1
2
σu·v,i
1 + dγi dγi

i=1

¨ i (1) − H
¨ i (r)
H

0
−1

1 N

hi (r)hi (r)dr
0

i=1

=VP IM 1 + dγ dr

164

¨ i (1) − H
¨ i (r) dr ×
H

where


−1

1 N


dγ dγ = VP−1
hi (r)hi (r)dr ×
IM
0 i=1

N
1
2
¨ i (1) − H
¨ i (r)
dγ dγ i
σu·v,i
H
i

0
i=1
−1


¨ i (1) − H
¨ i (r) dr
H




×



1 N



0 i=1

hi (r)hi (r)dr

.

The null limiting distribution of Wald and t statistics can be computed as follows.
ˆ
W

=

Rθˆ − r

=

R θˆ − θ

=

−1
A−1
R RAP IM AP IM

−1

RAP IM V˘P IM AP IM R

RAP IM V˘P IM AP IM R
θˆ − θ

Rθˆ − r
−1

R θˆ − θ

˘
A−1
R RAP IM VP IM AP IM R

−1
ˆ
A−1
R RAP IM AP IM θ − θ
−1

⇒ [R∗ Ψ] R∗ VP IM 1 + dγ dγ (R∗ )
=

[R∗ Ψ]

χ2q
1 + dγ dγ

and for q = 1,
tˆ =

=

=

Rθˆ − r
RAP IM V˘P IM AP IM R
R θˆ − θ
RAP IM V˘P IM AP IM R
A−1 RAP IM A−1
θˆ − θ
R

P IM

−1
˘
A−1
R RAP IM VP IM AP IM R AR

R∗ Ψ

⇒

R∗ VP IM 1 + dγ dγ (R∗ )
=

Z

.

1 + dγ dγ

165

A−1
R

−1

×

ˆ . Recall that
Second, consider the fixed-b limit of the tˆ and W
1
u
˜ = S y¨ − tˆ
x
¨ βˆ − N − 1 x γ
ˆi +
Sˆit
µi − Sit
it
it
N
N
x
¨β+
= tµi + Sit

x
¨ βˆ −
− tˆ
µi − Sit

1
N −1
xit γi −
N
N

=

u
Sit

1
−
N

u
¨
xjt γj + Sit
j=1,j=i

xjt γˆj

N

xjt Ω−1
vv,j Ωvu,j
j=1,j=i
N

ˆ
xjt Ω−1
vv,j Ωvu,j − qit θ − θ
j=1,j=i

N
u
Sjt

j=1,j=i

N

j=1,j=i

N −1
1
xit Ω−1
Ωvu,i +
vv,i
N
N

u
¨−
= Sit

xjt γˆj

N

N −1
1
xit γˆi +
N
N

1
N −1
Ωvu,i +
xit Ω−1
−
vv,i
N
N

N

− xit Ω−1
vv,i Ωvu,i

j=1

1
+
N

N

ˆ
xjt Ω−1
vv,j Ωvu,j − qit θ − θ
j=1

where qit is defined above.
u
˜ can be written as
Then, the first difference of Sˆit
u
˜ =
Sˆit

u
Sit

1
−
N

1
= uit −
N

N
u
Sjt

−

xit Ω−1
vv,i Ωvu,i

j=1

N

ujt − vit Ω−1
vv,i Ωvu,i
j=1

1
+
N

1
+
N

N

xjt Ω−1
vv,j Ωvu,j −

qit θˆ − θ

j=1

N
−1 Ω
vjt Ωvv,j
vu,j −
j=1

166

qit θˆ − θ .

Consequently,
1

[rT ]

1

[rT ]

u
˜ = T −2
Sˆit

T −2
t=1

uit −
t=1

1
+
N
=

+

N

T − 2

1

ujt  − T − 2 xi[rT ] Ω−1
vv,i Ωvu,i
t=1

j=1

1

1

j=1

N

1
−
N

N

1

1

−2
u
T − 2 Sj[rT
xi[rT ] Ω−1
vv,i Ωvu,i
]−T

j=1

1

1

−2
ˆ
T − 2 xj[rT ] Ω−1
qi[rT ] AP IM A−1
vv,j Ωvu,j − T
P IM θ − θ

j=1

1
⇒ Bu,i (r) −
N
1
+
N

1



[rT ]

−2
T − 2 xj[rT ] Ω−1
qi[rT ] θˆ − θ
vv,j Ωvu,j − T

1 u
T − 2 Si[rT
]

1
N

1
N



N

N

Bu,j (r) − Bv,i (r)Ω−1
vv,i Ωvu,i
j=1

N

Bv,j (r)Ω−1
vv,j Ωvu,j − hi (r)Ψ
j=1

1
= σu·v,i wu,i (r) + λuv,i Wv,i (r) −
N
1

2 Ω−1 Ω
− Wv,i (r)Ωvv,i
vv,i vu,i +

1
N

N

N

N

σu·v,j wu,j (r) + λuv,j Wv,j (r)
j=1
1

2 Ω−1 Ω
Wv,j (r)Ωvv,j
vv,j vu,j − hi (r)Ψ
j=1

1
σu·v,j wu,j (r) − hi (r)Ψ
= σu·v,i wu,i (r) −
N
j=1


N
σu·v,j wu,j (r)
1
−1 h (r)Ψ
= σu·v,i wu,i (r) −
− σu·v,i
i
N
σu·v,i
j=1

= σu·v,i · Pˆi (r)

167

where
N

1
Pˆi (r) = wu,i (r) −
N
= wu,i (r) −

j=1

N
σu·v,j wu,j (r)
1
N
σu·v,i
j=1


−1 h (r) 
− σu·v,i
i




−1

1 N

0 i=1

×

hi (r)hi (r)dr



1 N
0 i=1

σu·v,j wu,j (r)
−1 h (r)Ψ
− σu·v,i
i
σu·v,i

hi (r) σu·v,i wu,i (r) −

1
N



N



σu·v,j wu,j (r) dr .
j=1

In sum,
T

−1
2

[rT ]

1

u
˜ = T −2 S
ˆ
ˆu˜
Sˆit
i[rT ] =⇒ σu·v,i Pi (r).

(3.10)

t=1
1
2
u
˜
Next, write σ
ˆu·v,i
in terms of T − 2 Sˆi[rT
. The kernel function used here is the Bartlett
]
kernel. Define

Kts = k
2K

ts

|t − s|
M

= (Kts − Kt,s+1 ) − (Kt+1,s − Kt+1,s+1 ).

Simple algebra gives
T
2
σ
ˆu·v,i

=

T

T −1

k
j=2 h=2
T

= T −1



|j − h|
M



T

where

T

aj =

u
˜.
Kjh Sˆih

bj =
h=2

168

aj b j
j=2

h=2

u
˜,
Sˆij

T

u
˜  = T −1
Kjh Sˆih

 Sˆu˜
ij
j=2

u
˜ S
ˆu˜
Sˆij
ih

Using summation by parts we can write


T



T

t

as  bT − a1 b2 +

at b t = 
t=2

T −1

s=1

as (bt − bt+1 )
t=2

(3.11)

s=1

which gives
T
2
σ
ˆu·v,i

=

u
˜
T −1 SˆiT

T
u
˜ − T −1 S
ˆu˜
KT h Sˆih
i1

h=2

T −1
Sˆu˜
+ T −1
ij
j=2



u
˜
K2h Sˆih
h=2
T

T
u
˜ −
Kjh Sˆih


h=2



u
˜ 
Kj+1,h Sˆih
h=2

We need to apply (3.11) to the sums over h:
T

T −1

T
u
˜ =
KT h Sˆih

(1)
h=2

u
˜K
ˆu˜
ˆu˜
Sˆih
T h = SiT KT T − Si1 KT 2 +
h=2

T

h=2

T −1

T
u
˜ =
K2h Sˆih

(2)
h=2

u
˜K =S
ˆu˜ K2T − Sˆu˜ K22 +
Sˆih
2h
i1
iT

T −1

T
u
˜ =
Kjh Sˆih

h=2

u
˜K =S
ˆu˜ KjT − Sˆu˜ Kj2 +
Sˆih
jh
i1
iT

T
u
˜K
Sˆih
j+1,h

u
˜ =
Kj+1,h Sˆih
h=2

u
˜ K −K
Sˆih
jh
j,h+1
h=2

h=2

T

(4)

u
˜ K −K
Sˆih
2h
2,h+1
h=2

h=2

T

(3)

u
˜ K
Sˆih
T h − KT,h+1

h=2
T −1
u
˜ K
ˆu˜
= SˆiT
j+1,T − Si1 Kj+1,2 +

u
˜ K
Sˆih
j+1,h − Kj+1,h+1 .
h=2

169

2
Plugging in these expressions to σ
ˆu·v,i
gives




T −1

u
˜ S
2
ˆu˜ KT T − Sˆu˜ KT 2 +
= T −1 SˆiT
σ
ˆu·v,i
i1
iT

u
˜ K

Sˆih
T h − KT,h+1
h=2





T −1

u
˜ S
ˆu˜ K2T − Sˆu˜ K22 +
−T −1 Sˆi1
i1
iT

u
˜ K −K

Sˆih
2h
2,h+1
h=2

T −1

+T −1



u
˜ K −K

Sˆih
jh
j,h+1

u
˜ S
ˆu˜ KjT − Sˆu˜ Kj2 +
Sˆij
i1
iT
j=2
T −1

−T −1



T −1
h=2





T −1

u
˜ S
ˆu˜ Kj+1,T − Sˆu˜ Kj+1,2 +
Sˆij
i1
iT
j=2

u
˜ K

Sˆih
j+1,h − Kj+1,h+1
h=2

T −1
u
˜ K S
−1
ˆu˜
= T −1 SˆiT
T T iT + T

u
˜ (K
ˆu˜
SˆiT
T h − KT,h+1 )Sih
h=2

T −1

+T −1

u
˜ (K
ˆu˜
ˆu˜
Sˆij
jT − Kj+1,T )SiT + terms related toSi1
j=2
T −1 T −1

+T −1

u
˜ (K − K
ˆu˜
Sˆij
jh
j,h+1 ) − (Kj+1,h − Kj+1,h+1 ) Sih

j=2 h=2
T −1 T −1

=

T −1

T −1

u
˜ 2K S
−1
ˆu˜
Sˆij
jh ih + T

j=2 h=2
T −1
−1
u
˜ (K
+T
SˆiT
Th
h=2

u
˜ (K
ˆu˜
Sˆij
jT − Kj+1,T )SiT
j=2

u
˜ + T −1 S
ˆu˜ KT T Sˆu˜ + terms related toSˆu˜
− KT,h+1 )Sˆih
i1
iT
iT
1

u
˜ vanish as T → ∞, because T − 2 S
ˆu˜ converges to σu·v,i Pˆi (0),
Note that the terms related to Sˆi1
i1
which equals zero. For the Bartlett kernel we have

Kts = k

|t − s|
M

|t−s|

=

1 − M , |t − s| M
0
|t − s| > M

Then it follows that

0,



1,
Kts − Kt,s+1 = M 1

− ,


 M
0,

170

t s−M
s+1−M t s
s+1 t s+M
t s+M +1


0,


1


t s−M −1
,
s−M t s−1
Kt+1,s − Kt+1,s+1 = M 1

− , s t s−1+M


 M
0,
t s+M
and


2

t=s
M ,
2K =
1
−M , t = s ± M
ts


0,
otherwise

Using these result we have


T −1
T −M −1
2
2
u
˜S
u
˜
u
˜ +S
ˆu˜ Sˆu˜
ˆu˜ − 1

= T −1 
σ
ˆu·v,i
Sˆij
Sˆi,j+M
Sˆij
ij i,j+M
ij
M
M
j=2
j=2


T −1
T −1
1
u
˜S
u
˜ S
ˆu˜ − 1
ˆu˜ 
Sˆij
SˆiT
+ T −1 −
iT
ih
M
M
j=T −M

u
˜ S
ˆu˜
+ T −1 SˆiT
iT
T −1

2
=
MT

h=T −M

u
˜
+ terms related to Sˆi1

u
˜S
ˆu˜ − 2
Sˆij
ij
MT

j=2
−1
u
˜ S
ˆu˜
+ T SˆiT
iT

T −M −1
j=2

+ terms related to

2
u
˜S
ˆu˜
Sˆij
i,j+M − M T

T −1
u
˜S
ˆu˜
Sˆij
iT
j=T −M

u
˜,
Sˆi1

where the last term follows from the fact that KT T = 1.
Under fixed-b asymptotics we set M = bT where b ∈ (0, 1] is held fixed as T → ∞.
2
Plugging in bT for M into σ
ˆu·v,i
gives
2
σ
ˆu·v,i
=

2
bT

T −1

1

1

u
˜ T −2 S
ˆu˜ −
T − 2 Sˆij
ij

j=2

2
−
bT

T −1

1

2
bT

1

T −bT −1

1

1

u
˜ T −2 S
ˆu˜
T − 2 Sˆij
i,j+M

j=2
1

1

u
˜ T −2 S
ˆu˜ + T − 2 Sˆu˜ T − 2 Sˆu˜
T − 2 Sˆij
iT
iT
iT

j=T −bT
1

u
˜.
+terms related to T − 2 Sˆi1

171

Using (3.10) and the continuous mapping theorem gives
2
⇒
σ
ˆu·v,i

=

2
2 1
2 1−b
σu·v,i Pˆi (r) dr −
σu·v,i Pˆi (r)σu·v,i Pˆi (r + b)dr
b 0
b 0
2
2 1
−
σu·v,i Pˆi (r)σu·v,i Pˆi (1)dr + σu·v,i Pˆi (1)
b 1−b
2
σu·v,i

2 1−b ˆ
2 1 ˆ2
2 1 ˆ
ˆ
Pi (r)dr −
Pi (r)Pi (r + b)dr −
Pi (r)Pˆi (1)dr + Pˆi2 (1)
b 0
b 0
b 1−b

2
Qb Pˆi (r)
= σu·v,i

where
2 1 ˆ2
2 1−b ˆ
2 1 ˆ
Qb Pˆi (r) =
Pi (r)dr −
Pi (r)Pˆi (r + b)dr −
Pi (r)Pˆi (1)dr + Pˆi2 (1).
b 0
b 0
b 1−b
2
Therefore, based on σ
ˆu·v,i
, the fixed-b limit of the covariance matrix is given by


VˆP IM ⇒ 

−1

1 N
0 i=1

×

hi (r)hi (r)dr


N

1

2
¨ i (1) − H
¨ i (r)
H
Qb Pˆi (r)
σu·v,i

0
i=1
−1


¨ i (1) − H
¨ i (r) dr
H




×



1 N



0 i=1

hi (r)hi (r)dr

ˆ
=VP IM · Q(b)
where


−1

1 N

ˆ

hi (r)hi (r)dr ×
Q(b)
= VP−1
IM
0 i=1

N
1
2
¨ i (1) − H
¨ i (r)
σu·v,i
Qb Pˆi (r)
H

0
i=1

−1
1 N



0 i=1

hi (r)hi (r)dr

.

172

¨ i (1) − H
¨ i (r) dr
H





×

This implies that
χ2q
,
ˆ
Q(b)

ˆ ⇒
W
and for q = 1,

Z

tˆ ⇒

.

ˆ
Q(b)
Lastly, consider the result for the fixed-b test statistics. Similar as above and Vogelsang
2
2
and Wagner [2014], σ
˜u·v,i
⇒ σu·v,i
Qb P˜i (r) where Qb (·) is the same as above, and P˜i (r) is
similar as Pˆi (r) but its component is from the further augment regression (3.9). Therefore,
the fixed-b limit of the covariance matrix is such that

V˜P IM ⇒ 

−1

1 N
0 i=1

×

hi (r)hi (r)dr


N

1

2
¨ i (1) − H
¨ i (r)
H
σu·v,i
Qb P˜i (r)

0
i=1
−1


¨ i (1) − H
¨ i (r) dr
H




×



1 N



0 i=1

hi (r)hi (r)dr

˜
=VP IM · Q(b)
where


−1

1 N

˜

hi (r)hi (r)dr ×
Q(b)
= VP−1
IM
0 i=1

N
1
2
¨ i (1) − H
¨ i (r)
σu·v,i
Qb P˜i (r)
H

0
i=1

−1
1 N



0 i=1

hi (r)hi (r)dr

.

This implies that
˜ ⇒
W

χ2q
,
˜
Q(b)

173

¨ i (1) − H
¨ i (r) dr
H





×

and for q = 1,
t˜ ⇒

Z
˜
Q(b)

174

.

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