ESSAYS ON TIME SERIES ECONOMETRICS
By
Cheol-Keun Cho

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
Economics – Doctor of Philosophy
2014

ABSTRACT
ESSAYS ON TIME SERIES ECONOMETRICS
By
Cheol-Keun Cho
Chapter 1 develops an asymptotic theory for testing the presence of structural change
in a weakly dependent time series regression model. The cases of full structural change
and partial structural change are considered. A HAC estimator is involved in the construction of the test statistics. Depending on how the long run variance for pre- and postbreak regimes is estimated, two types of heteroskedasticity and autocorrelation (HAC)
robust Wald statistics, denoted by Wald( F) and Wald(S) are analyzed. The fixed-b asymptotics established by Kiefer and Vogelsang (2005) is applied to derive the limits of the
statistics with the break date being treated as a priori. The fixed-b limits turn out to depend on the location of break fraction and the bandwidth ratio as well as on the kernel
being used. For both Wald statistics the limits capture the finite-sample randomness existing in the HAC estimators for the pre- and post-break regimes. The limit of Wald( F)
further captures the finite-sample covariance between the pre-break estimators of regression parameters and the post-break estimators of the regression parameters. The fixed-b
limit stays the same and is pivotal for Wald( F) irregardless of whether some of the regressors are not subject to structural change. Critical values for the tests are obtained by
simulation methods. Monte Carlo simulations compare the finite sample size properties
of the two Wald statistics and a local power analysis is conducted to provide guidance
on the power properties of the tests. This Chapter extends its analysis to cover the case
of the break date being unknown. Supremum, mean and exponential Wald statistics are
considered and finite sample size distortions are examined via simulations with newly
tabulated fixed-b critical values for these statistics.
Chapter 2 generalizes the structural change test developed in Chapter 1 while allowing

for a shift in the mean and(or) variance of the explanatory variable. Chapter 2 assumes
the break date for the mean/variance is different from the possible break date for the regression parameters. The test is robust to serial correlation and heteroskedasticity of the
error term and the explanatory variables. The fixed-b theory is applied to derive the limits
of the statistics. The asymptotic theory in this paper is based on a new set of high level
conditions which incorporates the possibility of the moments shift and serves to provide
pivotal limits of the test statistics.
Chapter 3 proposes a test of the null hypothesis of integer integration against the alternative of fractional integration. The null of integer integration is satisfied if the series is
either I (0) or I (1), while the alternative is that it is I (d) with 0 < d < 1. The test is based
on two statistics, the KPSS statistic and a new unit root test statistic . The null is rejected if
the KPSS test rejects I (0) and the unit root test rejects I (1). The newly proposed unit root
test is a lower-tailed KPSS test based on the first differences of the original data, so the
test of the null of integer integration is called the "Double KPSS" test. Chapter 3 shows
that the test has asymptotically correct size under the null that the series is either I (0)
or I (1) and the test is consistent against I (d) alternatives for all d between zero and one.
These statements are true under the assumption that the number of lags used in long-run
variance estimation goes to infinity with sample size, but more slowly than sample size.
Chapter 3 refers to this as "standard asymptotics." This requires some original asymptotic
theory for the new unit root test, and also for the KPSS short memory test for the case
that d = 1/2. Chapter 3 also considers "fixed-b asymptotics" as in Kiefer and Vogelsang
(2005). Finite-sample size and power of the Double KPSS test is investigated using both
the critical values based on standard asymptotics and the critical values based on fixed-b
asymptotics. The new test is more accurate when it uses the fixed-b critical values. The
conclusion is that one can distinguish integer integration from fractional integration using
the Double KPSS test, but it takes a rather large sample size to do so reliably.

To my beloved wife, Eun-Young,
and my daughter, Ellin

iv

ACKNOWLEDGMENTS

Over the past five years in East Lansing, I have owed to many people for their support
and help. Without them I couldn’t get through the times I had. I want to deeply thank
to Timothy J. Vogelsang, my advisor as well as coauthor, for his continuous support and
excellent guidance. I can’t simply imagine how I would be now without him. He is the
most influential person for me at MSU. I learned so many things from him in teaching
and research and a lot of meetings we had were among the best memorable moments
for me. I will always miss those times with him after I leave East Lansing. I am also so
grateful to Peter Schmidt, my another excellent advisor and Christine Amsler. They are
both my coauthors. It was greatly pleasing and exciting to work with them. I remember
how warm they welcomed their TA’s including me at the appreciation dinner meeting
every semester. Also, I am not forgetting how much they were helpful and considerate
when I was in need of help. I could survive my second year with their support. Special
thanks to Christine Amsler. I worked for her as research assistant but she was more than
a supervisor and I deeply thank her again for encouraging me during those days. I also
thank Leslie E. Papke, the graduate program director. She was so much responsive to
what I was going through. I am also indebted to two other committee members, Jeffrey
M. Wooldridge and Hira Koul at department of statistics and probability for what I have
learned from their lectures and for their invaluable comments on my research. What I
learned from them has become really important academic assets for me. I thank Margaret
Lynch and Lori Jean Nichols for their support and tips in every important administration work. I had many good friends while I was in East Lansing. I could take a break
from work and research and get refreshed while I was hanging out with them. They are
Jaemin Baik, Mikayla Bowen, Sarah Brown, Hon Fong Cheah, Yeonjei Jung, Myoung-Jin
Keay, Sang Hyun Kim, Soobin Kim, Sangjoon Lee, Seunghwa Rho and Sun Yu. I want
to specially thank Reverend Borin Cho, Geum Jang, and many more at Lansing Korean

v

UMC. Speicial thanks also go to Sangryun Lee, Myungsook Kim, Bernhard G. Bodmann
and Raul Susmel in Houston, Texas. I also deeply thank my parents, my parents in law
and sisters and brother in Korea for their unceasing support and trust toward me. Last
but not least I owe my deepest gratitude to my wife, Eun-Young Song and my daughter,
Ellin for their love and for being together with me.

vi

TABLE OF CONTENTS

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

ix

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii
CHAPTER 1

Fixed-b Inference for Testing Structural Break in a Time Series
Regresison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Review of the Fixed-b Asymptotics . . . . . . . . . . . . . . . . . . . . . . .
1.3 Model of Structural Change and Preliminary Results . . . . . . . . . . . . .
1.4 Asymptotic Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4.1 Asymptotic Limits under Traditional Approach . . . . . . . . . . .
1.4.2 Asymptotic Limits under Fixed-b Approach . . . . . . . . . . . . . .
1.5 Critical Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.6 Finite Sample Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.6.1 Wald(S) and Wald( F) : Traditional Inference vs. Fixed-b Inference . .
1.6.2 Fixed-b Inference: Wald(S) vs. Wald( F) . . . . . . . . . . . . . . . . .
1.7 Local Power Analysis of Fixed-b Inference . . . . . . . . . . . . . . . . . . .
1.7.1 Comparison of the Local Asymptotic Power of Wald(S) and Wald( F)
1.7.2 Impact of Breakpoint Location and Bandwidth on Power . . . . . .
1.8 Partial Structural Change Model . . . . . . . . . . . . . . . . . . . . . . . .
1.8.1 Setup and Assumptions . . . . . . . . . . . . . . . . . . . . . . . . .
1.8.2 Asymptotic Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.9 When the Break Date is Unknown . . . . . . . . . . . . . . . . . . . . . . .
1.10 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
APPENDIX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A Test of Parameter Instability Allowing for Change in the Moments of Explanatory Variables . . . . . . . . . . . . . . . . . . . .
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Model of Structural Change and Preliminary Results . . . . . . . . . . . . .
2.3 Asymptotic Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Stability of β . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.2 Stability of α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
APPENDIX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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CHAPTER 2

CHAPTER 3

A Test of the Null of Integer Integration against
the Alternative of Fractional Integration . . . . . . . . . . . . . . . 121

vii

3.1
3.2

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Setup and Assumptions . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Null Hypothesis . . . . . . . . . . . . . . . . . . . . . . .
3.2.2 Alternative Hypothesis . . . . . . . . . . . . . . . . . . .
3.2.3 Test Statistics and the Rejection Rule . . . . . . . . . . .
3.3 Asymptotic Results . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 Asymptotic Results for ηµ . . . . . . . . . . . . . . . . .
3.3.2 Asymptotic Results for ηµd . . . . . . . . . . . . . . . . .
3.3.3 Correct Size and Consistency of the Double-KPSS Test .
3.4 Fixed-b Asymptotic Results . . . . . . . . . . . . . . . . . . . .
3.5 Monte Carlo Simulations . . . . . . . . . . . . . . . . . . . . . .
3.5.1 Design of the Experiment . . . . . . . . . . . . . . . . .
3.5.2 Results with Standard Critical Values . . . . . . . . . .
3.5.3 Results with Fixed-b Critical Values . . . . . . . . . . .
3.5.4 Comparison of the ηµd Test and the ADF Test . . . . . .
3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
APPENDIX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

viii

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166

LIST OF TABLES

Table 1.1

95% Fixed-b critical values of Wald( F) with Bartlett kernel, l = 1 . . . 44

Table 1.2

95% Fixed-b critical values of Wald( F) with Bartlett kernel, l = 2 . . . 44

Table 1.3

95% Fixed-b critical values of Wald( F) with Parzen kernel, l = 1 . . . . 45

Table 1.4

95% Fixed-b critical values of Wald( F) with Parzen kernel, l = 2 . . . . 45

Table 1.5

95% Fixed-b critical values of Wald( F) with QS kernel, l = 1 . . . . . . 46

Table 1.6

95% Fixed-b critical values of Wald( F) with QS kernel, l = 2 . . . . . . 46

Table 1.7

95% Fixed-b critical values of Wald(S) with Bartlett kernel, l = 1,
b1 = b2 = b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

Table 1.8

95% Fixed-b critical values of Wald(S) with Bartlett kernel, l = 2,
b1 = b2 = b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

Table 1.9

95% Fixed-b critical values of Wald(S) with Parzen kernel, l = 1, b1 =
b2 = b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

Table 1.10 95% Fixed-b critical values of Wald(S) with Parzen kernel, l = 2, b1 =
b2 = b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
Table 1.11 95% Fixed-b critical values of Wald(S) with QS kernel, l = 1, b1 =
b2 = b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
Table 1.12 95% Fixed-b critical values of Wald(S) with QS kernel, l = 2, b1 =
b2 = b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
Table 1.13 95% Fixed-b critical values of Wald(S) , l = 2, b1 = b2 . . . . . . . . . . 49
Table 1.14 The Finite Sample Size associated with Wald(S) , l = 2, b1 = b2 = b. . . 50
Table 1.15 The Finite Sample Size associated with Wald(S) , l = 2, b1 = b2 = b.

. 51

Table 1.16 The Finite Sample Size associated with Wald(S) , l = 2, b1 = b2 = b.

. 52

Table 1.17 The Finite Sample Size associated with Wald(S) , l = 2, b1 = b2 = b.

. 53

ix

Table 1.18 The Finite Sample Size associated with Wald(S) , l = 2, b1 = b2 = b. . . 54
Table 1.19 The Finite Sample Size associated with Wald(S) , l = 2, b1 = b2 = b.

. 55

Table 1.20 The Finite Sample Size of the Tests Based on Wald(S) and Wald( F) , . . 56
Table 1.21 The Finite Sample Size of the Tests Based on Wald(S) and Wald( F) , . . 57
Table 1.22 The Finite Sample Size of the Tests Based on Wald(S) and Wald( F) , . . 58
Table 1.23 The Finite Sample Size of the Tests Based on Wald(S) and Wald( F) , . . 59
Table 1.24 The Finite Sample Size of the Tests Based on Wald(S) and Wald( F) , . . 60
Table 1.25 The Finite Sample Size of the Tests Based on Wald(S) and Wald( F) , . . 61
Table 1.26 Fixed-b 95% Critical Values of Wald( F) Unknown Break Date, Bartlett
kernel, l =2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
Table 1.27 Fixed-b 95% Critical Values of Wald( F) Unknown Break Date, QS kernel, l =2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
Table 1.28 The Finite Sample Size of the SupW ( F) Test with 5% Nominal Size . . 63
Table 1.29 The Finite Sample Size of the SupW ( F) Test with 5% Nominal Size . . 64
Table 1.30 The Finite Sample Size of the SupW ( F) Test with 5% Nominal Size . . 65
Table 1.31 The Finite Sample Size of the MeanW ( F) Test with 5% Nominal Size . 66
Table 1.32 The Finite Sample Size of the MeanW ( F) Test with 5% Nominal Size . 67
Table 1.33 The Finite Sample Size of the MeanW ( F) Test with 5% Nominal Size . 68
Table 1.34 The Finite Sample Size of the ExpW ( F) Test with 5% Nominal Size . . 69
Table 1.35 The Finite Sample Size of the ExpW ( F) Test with 5% Nominal Size . . 70
Table 1.36 The Finite Sample Size of the ExpW ( F) Test with 5% Nominal Size . . 71
Table 2.1

Size and Power in Finite Samples, T=100, 300, λ x = 0.4, = .1,
Bartlett kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

x

Table 3.1

Summary of the existing asymptotic results for ηµ . . . . . . . . . . . . 146

Table 3.2

Fixed-b Critical Values for ηµ and ηµd , Bartlett kernel, l = bT − 1, l =

b ( T − 1) − 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
Table 3.3

Size and Power Using Standard 5% Critical Values with Traditional Lag
Choices,T = 50 and 100 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

Table 3.4

Size and Power Using Standard 5% Critical Values with Traditional Lag
Choices, T = 200 and 500 . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

Table 3.5

Size and Power Using Standard 5% Critical Values with Traditional Lag
Choices, T = 1,000 and 2,000 . . . . . . . . . . . . . . . . . . . . . . . . . 150

Table 3.6

Size and Power Using 5% Fixed-b Critical Values with Traditional Lag Choices,
T = 50 and 100 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

Table 3.7

Size and Power Using 5% Fixed-b Critical Values with Traditional Lag Choices,
T = 200 and 500 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

Table 3.8

Size and Power Using 5% Fixed-b Critical Values with Traditional Lag Choices,
T = 1,000 and 2,000 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

Table 3.9

Size and Power of ADF and ηµ × ADF Tests Using Standard Critical Values with Traditional Lag Choices, T = 50, 100, 200, and 500 . . . . . . . . . 154

xi

LIST OF FIGURES

Figure 1.1 Empirical Null Rejection Probability of Structural Break Test using
Wald( F) β 1 = β 2 = 0, θ = 0.1, ρ = 0.5, φ = 0.5, b = 0.2, T = 50,
Replications=2, 500 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
Figure 1.2 Local Power, Bartlett and QS, λ = 0.2, b = 0.2, b1 = 1, b2 = 0.25 . . . . 28
Figure 1.3 Local Power, Bartlett and QS, λ = 0.2, b = 0.5, b1 = 2.5, b2 = 0.625 . . 28
Figure 1.4 Local Power, Bartlett and QS, λ = 0.2, b = 1, b1 = 5, b2 = 1.25 . . . . . 29
Figure 1.5 Local Power, Bartlett and QS, λ = 0.5, b = 1, b1 = 2, b2 = 1 . . . . . . 30
Figure 1.6 Local Power, Wald( F) , Bartlett, b = 0.1 . . . . . . . . . . . . . . . . . . . 31
Figure 1.7 Local Power, Wald( F) , QS, b = 0.1 . . . . . . . . . . . . . . . . . . . . . 31
Figure 1.8 Local Power, Wald(S) , Bartlett . . . . . . . . . . . . . . . . . . . . . . . 32
Figure 1.9 Local Power, Wald(S) , QS . . . . . . . . . . . . . . . . . . . . . . . . . . 32
Figure 1.10 Local Power, Wald( F) , Bartlett, λ = 0.5 . . . . . . . . . . . . . . . . . . 33
Figure 1.11 Local Power, Wald( F) , QS, λ = 0.5 . . . . . . . . . . . . . . . . . . . . . 33
Figure 1.12 Local Power, Wald(S) , Bartlett, λ = 0.5 . . . . . . . . . . . . . . . . . . 34
Figure 1.13 Local Power, Wald(S) , QS, λ = 0.5 . . . . . . . . . . . . . . . . . . . . . 34

xii

CHAPTER 1
Fixed-b Inference for Testing Structural
Break in a Time Series Regresison
1.1

Introduction

Chapter 1 focuses on fixed-b inference of heteroskedasticity and autocorrelation (HAC)
robust Wald statistics for testing for a structural break in a time series regression. Commonly used kernel-based nonparametric HAC estimators are considered to estimate the
asymptotic variance. HAC estimators allow for arbitrary structure of the serial correlation and heteroskedasticity of weakly dependent time series and are consistent estimators
of the long run variance under the assumption that the bandwidth (M) is growing at a
certain rate slower than the sample size (T). Under this assumption, the Wald statistics
converge to the usual chi-square distributions. However, because the critical values from
the chi-square distribution are based on a consistency approximation for the HAC estimator, the chi-square limit does not reflect the often substantial finite sample randomness
of the HAC estimator. Furthermore, the chi-square approximation does not capture the
impact of the choice of the kernel or the bandwidth on the Wald statistics. The sensitivity of the statistics to the finite sample bias and variability of the HAC estimator is
well known in the literature; Kiefer and Vogelsang (2005) among others have illustrated
by simulation that the traditional inference with a HAC estimator can have poor finite
sample properties.
Departing from the traditional approach, Kiefer and Vogelsang (2002a), Kiefer and
Vogelsang (2002b) , and Kiefer and Vogelsang (2005) obtain an alternative asymptotic ap-

1

proximation by assuming that the ratio of the bandwidth to the sample size, b = M/T,
is held constant as the sample size increases. Under this alternative nesting of the bandwidth, they obtain pivotal asymptotic distributions for the test statistics which depend
on the choice of kernel and bandwidth tuning parameter. Simulation results indicate that
the resulting fixed-b approximation has less size distortions in finite samples than the
traditional approach especially when the bandwidth is not small.
Theoretical explanations for the finite sample properties of the fixed-b approach include the studies by Hashimzade and Vogelsang (2008), Jansson (2004), Sun, Phillips, and
Jin (2008, hereafter SPJ), Gonçalves and Vogelsang (2011) and Sun (2013). Hashimzade
and Vogelsang (2008) provide an explanation for the better performance of the fixed-b
asymptotics by analyzing the bias and variance of the HAC estimator. Gonçalves and Vogelsang (2011) provide a theoretical treatment of the asymptotic equivalence between the
naive bootstrap distribution and the fixed-b limit. Higher order theory is used by Jansson
(2004), SPJ (2008) and Sun (2013) to show that the error in rejection probability using the
fixed-b approximation is more accurate than the traditional approximation. In a Gaussian location model Jansson (2004) proves that for the Bartlett kernel with bandwidth
equal to sample size (i.e. b = 1), the error in rejection probability of fixed-b inference is
O( T −1 log T ) which is smaller than the usual rate of O( T −1/2 ). The results in SPJ (2008)
complement Jansson’s result by extending the analysis for a larger class of kernels and
focusing on smaller values of bandwidth ratio b. In particular they find that the error
in rejection probability of the fixed-b approximation is O( T −1 ) around b = 0. They also
show that for positively autocorrelated series, which is typical for economic time series,
the fixed-b approximation has smaller error than the chi-square or standard normal approximation even when b is assumed to decrease to zero although the stochastic orders
are same.
In this chapter fixed-b asymptotics is applied to testing for structural change in a
weakly dependent time series regression. The structural change literature is now enor-

2

mous and no attempt will be made here to summarize the relevant literature. Some key
references include Andrews (1993) and Andrews and Ploberger (1994). Andrews (1993)
treats the issue of testing for a structural break in the GMM framework when the one-time
break date is unknown and Andrews and Ploberger (1994) derive asymptotically optimal
tests. Bai and Perron (1998) consider multiple structural change occurring at unknown
dates and cover the issues of estimation of break dates, testing for the presence of structural change and for the number of breaks. For a comprehensive survey of the recent
structural break literature see Perron (2006).
Because a structural change in a regression relationship effectively divides the sample
into two regimes, two different HAC estimators are considered when constructing Wald
statistics. Asymptotically, estimators of the regression parameters are uncorrelated across
the two regimes. One HAC estimator imposes this zero covariance restriction while the
other estimator does not and this leads to two possible Wald statistics that can be used in
practice denoted by Wald(S) and Wald( F) . When the break date is known and coefficients
of all regressors are subject to structural change (i.e. full structural change), both Wald
statistics have pivotal fixed-b limits but these limits are different. For the two Wald statistics the fixed-b critical values increase as the bandwidth gets bigger and as the hypothesized break date is closer to the boundary of the sample. When some of the regressors are
not subject to structural change (i.e. partial structural change), the Wald statistic based on
the unrestricted HAC estimator (Wald( F) ) still has the same pivotal fixed-b limit as in the
case of full structural change.
A local power analysis is carried out under the fixed-b approach and several patterns
are reported. The local power of both Wald statistics is lower with bigger b especially
for the QS kernel. Power also improves as the break date gets closer to the middle of the
sample regardless of bandwidth or kernel. With the within-regime effective bandwidths
matched across the two statistics, the power difference is more evident when a big value
of b is used with the QS kernel.

3

A simulation study on the finite sample performance of fixed-b inference reveals that
the Wald statistic based on the unrestricted HAC estimator has better overall size performance especially when the quadratic spectral (QS) kernel and large bandwidths are
used in the case of persistent data. In the comparison of the performance of fixed-b inference with traditional inference, it is found that the latter is subject to substantially more
size distortions. Similar to the known patterns in models without structural change, size
distortions decrease as one uses bigger bandwidth or as the hypothesized break date is
closer to the middle of the sample. When there is strong persistence in the data, rejections
using fixed-b critical values can be above nominal levels although these distortions are
much smaller compared to traditional inference.
The remainder of this chapter is organized as follows. Section 1.2 reviews fixed-b
asymptotic theory in a regression with no structural change. In Section 1.3 the basic set
up of the full structural-change model. Some preliminary results are laid out and the two
HAC estimators are introduced. Section 1.4 derives the fixed-b limits of the two Wald
statistics. Section 1.5 explores patterns in the fixed-b critical values when the break date
is treated as a priori. Section 1.6 compares finite sample size across different approach of
inferences, different choices of kernel and HAC estimators under various DGP specifications. Section 1.7 examines local power. Section 1.8 generalizes the results in Section 1.4
to a model with partial structural change. It is shown that the fixed-b limit of Wald( F) in
the partial structural change model is same as in the full structural change model. Section
1.9 covers the case where the break date is unknown providing the fixed-b critical values.
Section 1.10 summarizes and concludes. Proofs and supplemental results are collected in
the Appendix.

4

1.2

Review of the Fixed-b Asymptotics

Consider a weakly dependent time series regression with p regressors given by

yt = xt β + ut .

(1.1)

Model (1.1) is estimated by ordinary least squares (OLS) giving
−1

T

∑ xt xt

β=

T

∑ xt yt

t =1

,

t =1

and ut = yt − xt β are the OLS residuals. The centered OLS estimator is given by
−1

T

β−β =

∑ xt xt

t =1

T

∑ vt

,

t =1

where vt = xt ut . The asymptotic theory is based on the following two assumptions.
[rT ]

p

Assumption 1. T −1 ∑t=1 xt xt → rQ, uniformly in r ∈ [0, 1], and Q−1 exists.
[rT ]

[rT ]

Assumption 2. T −1/2 ∑t=1 xt ut = T −1/2 ∑t=1 vt ⇒ ΛWp (r ), r ∈ [0, 1], where ΛΛ = Σ,
and Wp (r ) is a p × 1 standard Wiener process.
For a more detailed discussion about the regularity conditions under which Assumptions
1 and 2 hold, refer to Kiefer and Vogelsang (2002b). See Davidson (1994), Phillips and
Durlauf (1986), Phillips and Solo (1992), and Wooldridge and White (1988) for more details. The matrix Q is the non-centered variance-covariance matrix of xt and is typically
estimated using the sample variance Q =

1
T

∑tT=1 xt xt . The matrix Σ ≡ ΛΛ is the asymp-

totic variance of T −1/2 ∑tT=1 vt , which is, for a stationary series, given by
∞

Σ = Γ0 + ∑ (Γ j + Γ j ) with Γ j = E(vt vt− j ).
j =1

5

Being a long run variance, Σ is commonly estimated by the kernel-based nonparametric
HAC estimator
Σ=T

−1

T

T

∑∑K

t =1 s =1

|t − s|
M

v t v s = Γ0 +

T −1

∑K

j =1

j
M

Γj + Γj ,

where Γ j = T −1 ∑tT= j+1 vt vt− j , vt = xt ut , M is a bandwidth, and K (·) is a kernel ighting
function.
Under some regularity conditions (see Andrews (1991), De Jong and Davidson (2000),
Hansen (1992), Jansson (2002) or Newey and West (1987)), Σ is a consistent estimator of
p

Σ, i.e. Σ → Σ. These regularity conditions include the assumption that M/T → 0 as
M, T → ∞. This asymptotics is called ‘traditional asymptotics’ throughout this chapter.
In contrast to the traditional approach, fixed-b asymptotics assumes b = M/T is held
constant as T increases. Assumptions 1 and 2 are the only regularity conditions required
to obtain a fixed-b limit for Σ. Under the fixed-b approach, for b ∈ (0, 1], Kiefer and
Vogelsang (2005) show that
Σ ⇒ ΛP(b, Wp )Λ ,

(1.2)

where Wp (r ) = Wp (r ) − rWp (1) is a p-vector of standard Brownian bridges and the form
of the random matrix P(b, Wp ) depends on the kernel. Following Kiefer and Vogelsang
(2005) three classes of kernels are considered. Let H p (r ) denote a generic vector of stochastic processes. H p (r ) denotes its transpose. Then P(b, H p ) is defined as follows:
case 1 If K ( x ) is twice continuously differentiable everywhere (Class 1) such as the
Quadratic Spectral kernel (QS), then

P b, H p ≡ −

1
0

1
0

r−s
b

1
K
b2

H p (r ) H p (s) drds,

where K (·) is the second derivative of the kernel K (·).

6

(1.3)

case 2 If K ( x ) is the Bartlett kernel (Class 2), then

P b, H p ≡

2
b

1
0

H p (r ) H p (r ) dr −

1
b

1− b

H p (r ) H p (r + b) + H p (r + b) H p (r ) dr.

0

(1.4)
case 3 If K ( x ) is continuous, K ( x ) = 0 for | x | ≥ 1, and K ( x ) is twice continuously
differentiable everywhere except for | x | = 1 (Class 3) like Parzen kernel, then

P b, H p ≡ −

+

K

|r − s |
b

1
K
b2

|r −s|<b
_ (1 ) 1− b

b

0

H p (r ) H p (s) drds

(1.5)

H p (r + b) H p (r ) + H p (r ) H p (r + b) dr,

where K _(1) = limh↓0 [(K (1) − K (1 − h)) /h] , i.e., K _(1) is the derivative of K ( x ) from
the left at x = 1.
Inference regarding β is based on the asymptotic normality of the OLS estimator given
by the result

√

T ( β − β) =

T

−1

T

∑ xt xt

−1

T −1/2

t =1

T

∑ xt ut

d

→ Q−1 ΛWp (1) ∼ N (0, Q−1 ΣQ−1 ),

t =1

which follows directly from Assumptions 1 and 2. Suppose the null hypothesis of interest
is, H0 : Rl × p β = r, where l is the number of the restrictions. Define the Wald statistic as
Wald = T R β − r

R Q −1 Σ Q −1 R

−1

Rβ − r .

Under traditional asymptotics the well known result is obtained:
d

Wald → χ2l ,

7

whereas under fixed-b asymptotics,

Wald ⇒ Wl (1)

P(b, Wl )

−1

Wl (1).

(1.6)

The limit of the Wald statistic depends on the sequence of bandwidths by which Σ is
indexed. It is important to note that we are not viewing the choice of sequence as a bandwidth rule for the choice of M in practice. Rather, the point is that different asymptotic
approximations are obtained for the two assumptions regarding M. Under the fixed-b
approach the random matrix P(b, Wl ) approximates the randomness in Σ and its dependence on M (through b) and the kernel K (·). In contrast the traditional approach approximates Σ by a constant that does not depend on M or the kernel.
Once structural change is allowed in the model, existing results in the fixed-b literature
no longer apply and new results are required.

1.3

Model of Structural Change and Preliminary Results

Consider a weakly dependent time series regression model with a structural break given
by
yt = wt β + u t ,

(1.7)

wt = x1t , x2t , β = β 1 , β 2 ,
x1t = xt · 1(t ≤ λT ), x2t = xt · 1(t > λT ),
where xt is p × 1 regressor vector, λ ∈ (0, 1) is a break point, and 1( · ) is the indicator
function. Let [λT ] denote the integer part of λT. Note that x2t = 0 for t = 1, 2, ..., [λT ] and
x1t = 0 for t = [λT ] + 1, ..., T. For the time being the potential break point λT is assumed
to be known. The case of λ being unknown is discussed in Section 1.9. The regression
model (1.7) implies that coefficients of all explanatory variables are subject to potential
structural change and this model is labeled the ’full’ structural change model.
8

Of interest it is that the presence of a structural change in the regression parameters.
Consider null hypothesis of the form

H0 : Rβ = 0,

(1.8)

where
R

(l ×2p)

= ( R1 , − R1 ) ,

and R1 is an l × p matrix with l ≤ p. Under the null hypothesis, it is being tested that one
or more linear relationships on the regression parameter(s) do not experience structural
change before and after the break point. Tests of the null hypothesis of no structural
change about a subset of the slope parameters are special cases. For example one can
test the null hypothesis that the slope parameter on the first regressor did not change by
setting R1 = (1, 0, . . . , 0). One can test the null hypothesis that none of the regression
parameters have structural change by setting R1 = I p .
In order to establish the asymptotic limits of the HAC estimators and the Wald statistics, Assumptions 1 and 2 given in previous Section are sufficient. Those Assumptions
imply that there is no heterogeneity in the regressors across the segments and the covariance structure of the errors are assumed to be same across segments as well.
notation For later use, define a l × l nonsingular matrix A such that
R1 Q−1 ΛΛ Q−1 R1 = AA ,

(1.9)

and
d

R1 Q−1 ΛWp (r ) = AWl (r ),
where Wl (r ) is l × 1 standard Wiener process.

9

(1.10)

Focus on the OLS estimator of β given by
−1

T

β=

∑ wt wt

T

∑ wt yt

t =1

,

t =1

which can be written for each segment as
−1

T

β1 =

∑ x1t x1t

∑ x1t yt

t =1

−1

T

β2 =

∑ x2t x2t



∑ x2t yt

t =1

t =1

∑ xt yt

∑

 −1 
xt xt 

,

(1.11)

t =1

t =1

T

=

[λT ]

∑ xt xt

=

t =1

T

−1

[λT ]

T

∑



t=[λT ]+1



T

xt yt  .

(1.12)

t=[λT ]+1

Fixed-b results depend on the limiting behavior of the following partial sum process
t

St =

∑ wj uj =

j =1

t

∑ wj

y j − x1j β 1 − x2j β 2

j =1

t

=

∑ wj

u j − x1j β 1 − β 1 − x2j β 2 − β 2

.

(1.13)

j =1

Under Assumptions 1 and 2 the limiting behavior of β and the partial sum process St are
easily obtained.
Proposition 1. Let λ ∈ (0, 1) be given. Suppose the data generation process is given by (1.7)
and let [rT ] denote the integer part of rT where r ∈ [0, 1]. Then under Assumptions 1 and 2 as
T → ∞,


√

and

√







−1

(λQ) ΛWp (λ)
 T β1 − β1  d 

T ( β − β) = √
→
,
T β2 − β2
((1 − λ) Q)−1 Λ Wp (1) − Wp (λ)







(1)
0   Fp (r, λ)





Λ
Λ 0 
T −1/2 S[rT ] ⇒ 

≡
 Fp (r, λ) ,
(2)
0 Λ
Fp (r, λ)
0 Λ
10

where
(1)

Fp (r, λ) = Wp (r ) −

r
Wp ( λ ) · 1 (r ≤ λ ),
λ

and
(2)

Fp (r, λ) =

Wp (r ) − Wp ( λ ) −

r−λ
Wp (1) − Wp ( λ )
1−λ

· 1 (r > λ ).

Proof: See the Appendix.
It is easily seen that the asymptotic distributions of β 1 and β 2 are Gaussian and are
independent of each other. Hence the asymptotic covariance of β 1 and β 2 is zero. The
√
1
−1
asymptotic variance of T ( β − β) is given by Q−
λ ΩQλ , where








0
0
λQ

λΣ

Qλ ≡ 
 and Ω ≡ 
.
0 (1 − λ ) Q
0 (1 − λ ) Σ
In order to test the null hypothesis (1.8) HAC robust Wald statistisc are considered. These
statistics are robust to heteroskedasticity and autocorrelation in the vector process, vt =
xt ut . The generic form of the robust Wald statistic is given by
1
−1
RQ−
λ ΩQλ R

Wald = T R β

where

−1

Rβ ,





−1 [λT ]
 T ∑ t =1 x t x t

Qλ = 

0
T −1 ∑tT=[λT ]+1

0

xt xt


,

and Ω is an HAC robust estimator of Ω.
Two estimators for Ω are analyzed. The first one, denoted by Ω( F) , is newly introduced in this chapter and it is constructed using the residuals directly from the dummy
regression (1.7):
Ω ( F ) = T −1

T

T

∑∑K

t =1 s =1

11

|t − s|
M

vt vs ,

(1.14)

where vt = wt ut =

x1t ut , x2t ut
(2)

xt ut 1(t ≤ λT ) and vt

(1)

2p×1

. Denote the components of vt as vt

= x1t ut =

= x2t ut = xt ut 1(t > λT ).

The other estimator is the HAC estimator appearing in existing literature (e.g. Bai and
Perron (2006)) which is given by


Ω(S) = 

where
Σ (1) =

Σ (2) =

1
[λT ]



λ Σ (1)

0

0

( 1 − λ ) Σ (2)

[λT ] [λT ]

∑ ∑K

t =1 s =1

|t − s|
M1

T
T
1
K
∑
∑
T − [λT ] t=[λT ]+1 s=[λT ]+1

(1)

Note that Σ(1) is constructed using vt


,

(1.15)

(1) (1)

(1.16)

vt vs ,

|t − s|
M2

(2) (2)

vt vs .

(1.17)

(data from the pre-break regime) and uses the
(2)

bandwidth M1 and pre-break sample size [λT ]. Likewise Σ(2) is constructed using vt

(data from the post-break regime) and uses the bandwidth M2 and post-break sample
size T − [λT ].
In Andrews (1993) and Bai and Perron (2006), the estimator (1.15) is used to allow
for a potential structural change in the long run variance Σ itself. In this chapter the
assumption is maintained that Σ does not have structural change because allowing for
heterogeneity in Σ results in a non-pivotal fixed-b limit of the Wald statistic. Finding an
estimator of Ω that has a pivotal fixed-b limit when Σ has structural change is a topic of
ongoing research.
At a superficial level, the estimators Ω( F) and Ω(S) look different but they are directly

12

(1)

related. Using vt = (vt


Ω( F) = 


=



=

( F)
Ω11
( F)
Ω21

(2)

, vt

( F)
Ω12
( F)
Ω22

) one can write Ω( F) as





∑tT=1 ∑sT=1 K

|t−s|
M

(1) (1)
vt vs

T −1 ∑tT=1 ∑sT=1 K

|t−s|
M

(2) (1)
vt vs

T −1

(1) (1)
[λT ] [λT ]
|t−s|
∑ t =1 ∑ s =1 K M v t v s


[λT ]
(2) (1)
|t−s|
T −1 ∑tT=[λT ]+1 ∑s=1 K M vt vs



=

(1) (2)
vt vs

T −1 ∑tT=1 ∑sT=1 K

|t−s|
M

(2) (2)
vt vs

[λT ]
∑t=1 ∑sT=[λT ]+1 K

[λT ]

|t−s|
M

|t−s|
M

T −1 ∑tT=[λT ]+1 ∑sT=[λT ]+1 K

T −1 ∑t=1 ∑sT=[λT ]+1 K

λ Σ (1)
[λT ]

|t−s|
M

T −1

T −1

T −1 ∑tT=[λT ]+1 ∑s=1 K

∑tT=1 ∑sT=1 K

T −1

(2) (1)

|t−s|
M

( 1 − λ ) Σ (2)

vt vs





(1) (2)
vt vs

|t−s|
M

(2) (2)
vt vs

(1) (2)

vt vs





(1.18)




It is seen that the diagonal blocks of Ω( F) are the same HAC estimators used by Ω(S)
except that the same bandwidth, M, is used for each diagonal block of Ω( F) whereas the
diagonal blocks of Ω(S) can have different bandwidths. Ω(S) is a restricted version of
Ω( F) with the off-diagonal blocks set to 0, i.e. Ω(S) imposes a zero covariance between
β 1 and β 2 matching the zero asymptotic covariance between β 1 and β 2 . In contrast Ω( F)
does not impose this zero covariance which is consistent with the possibility that the
finite sample covariance between β 1 and β 2 is not equal to zero (which is true in general).
Note however that Ω( F) uses the same bandwidth for both regimes whereas Ω(S) allows
different bandwidths in the two regimes. Thus, from the bandwidth perspective, Ω( F) is
restrictive relative to Ω(S) .
The next Section provides asymptotic results for the two HAC robust Wald statistics
under the traditional asymptotics and under the fixed-b asymptotics.

13

1.4
1.4.1

Asymptotic Results
Asymptotic Limits under Traditional Approach

The goal of the traditional approach is to find conditions under which the HAC estimator
is consistent. One requirement for consistency is that M grows with the sample but at
a slower rate. Then under additional regularity conditions, Σ(1) and Σ(2) are consistent
estimators of Σ and the limit of Ω(S) is straightforwardly given by


Ω(S) = 

λ Σ (1)
0







0
 p λΣ

→
.
0 (1 − λ ) Σ
( 1 − λ ) Σ (2)
0

(2p×2p)

However establishing consistency of Ω( F) requires some additional calculation beyond
existing results in the literature.
Proposition 2. Under regularity conditions for the consistency of the HAC estimators Σ(1) and
Σ(2) , as T → ∞,





0
p  λΣ

Ω( F) → 

0 (1 − λ ) Σ

.
2p×2p

Proof: See the Appendix.
Let Wald(S) denote the Wald statistic based on Ω(S) and let Wald( F) denote the Wald
statistic based on Ω( F) . Then the results given in this subsection, combined with Proposition 1, give us the limits of the test statistics under the traditional approach:
Wald(S) , Wald( F) ⇒

λ (1 − λ )

1
1
Wl (λ) −
(Wl (1) − Wl (λ))
λ
1−λ

×

1
1
Wl (λ) −
(Wl (1) − Wl (λ)) ,
λ
1−λ

where Wl is a l × 1 standard Wiener process. Note that for any given value of λ the limit
14

follows a chi-square distribution with degrees of freedom l. While convenient, the chisquare limit does not capture the impact of the randomness of Ω(S) and Ω( F) on the Wald
statistics in finite samples.

1.4.2

Asymptotic Limits under Fixed-b Approach

Now fixed-b limits for the HAC estimators and the test statistics are provided. The fixed-b
limits presented in next Lemma and Corollary approximate the diagonal blocks of Ω( F)
by random matrices. Also, it is shown that fixed-b approach gives nonzero limit for the
off-diagonal block, which further distinguishes fixed-b asymptotics from the traditional
asymptotics.
Lemma 1. Let λ ∈ (0, 1) and b ∈ (0, 1] be given. Suppose M = bT. Then under Assumptions 1
and 2, as T → ∞,






Λ 0 
Λ
Ω( F) ⇒ 
 × P b, Fp (r, λ) × 
0 Λ
0
where



Fp (r, λ) = 
(1)

Fp (r, λ) = Wp (r ) −
(2)

Fp (r, λ) =

Wp (r ) − Wp ( λ ) −

(1)
Fp (r, λ)
(2)
Fp (r, λ)


0
,
Λ

(1.19)



,

(1.20)

r
Wp ( λ ) 1 (0 ≤ r ≤ λ ) ,
λ

r−λ
Wp (1) − Wp ( λ )
1−λ

1 ( λ < r ≤ 1) ,

(1.21)
(1.22)

and P p b, Fp (r, λ) is defined by (1.3), (1.4), and (1.5) with H p (r ) = Fp (r, λ).
Proof: See the Appendix.
Extra algebra leads to an alternative representation for P b, Fp (r, λ) . The proof for
this Corollary is omitted.

15

Corollary 1.


P b, Fp (r, λ) = 

P
C

(1)
b, Fp (r, λ)

C

(1)
(2)
b, Fp (r, λ) , Fp (r, λ)

(1)
(2)
b, Fp (r, λ) , Fp (r, λ)

P

(2)
b, Fp (r, λ)



 , (1.23)

where
(1)

(2)

C b, Fp (r, λ) , Fp (r, λ)

=








−





 −

|r −s|<b

K

(1)
(2)
1 1 1
|r − s |
Fp (r, λ) Fp (s, λ) drds,
b
0 0 b2 K
(2)
1 1− b (1)
Fp (r, λ) Fp (r + b, λ) dr,
b 0
(1)
(2)
|r − s |
(2)
1− b (1)
Fp (r,λ) Fp (s,λ) drds
K _(1) 0 Fp (r,λ) Fp (r +b,λ) dr
b
+
,
b
b2

for Class 1,2 and 3 kernels respectively.
The expression for P b, Fp (r, λ) in this Corollary makes it easier to compare the fixedb limit of Ω( F) with the standrad fixed-b limit (see (1.2)) appearing in a non-structural
change settings. Since each diagonal block of Ω( F) is basically a HAC estimator (up to a
scale factor; see (1.18)) based on one of the pre- or post- break data, its limit should take
the same form as (1.2), which is verified in this Corollary. So each diagonal component
of P b, Fp (r, λ) serves to reflect the randomness and bandwidth/kernel-dependence of
the associated HAC estimator. Second, unlike the traditional approach, the fixed-b limit
of the off-diagonal component is nonzero. This implies the fixed-b inference is able to take
account of the covariance between β 1 and β 2 which is generally nonzero in finite samples.
Next Lemma contains the parallel result for Ω(S) .
Lemma 2. Let λ ∈ (0, 1) and b1 , b2 ∈ (0, 1] be given. Suppose M1 = b1 (λT ) and M2 =
b2 (1 − λ) T. Then under Assumptions 1 and 2, as T → ∞,
Σ (1) ⇒ Λ

1
(1)
P b1 λ, Fp (r, λ)
λ

Λ , Σ (2) ⇒ Λ

16

1
(2)
P b2 (1 − λ) , Fp (r, λ)
1−λ

Λ,

and






Λ 0  P
Ω(S) ⇒ 
×
0 Λ

(1)
b1 λ, Fp (r, λ)

0





 Λ
×
(2)
P b2 (1 − λ) , Fp (r, λ)
0
0


0
,
Λ

Proof: See the Appendix.
The limits of the Wald statistics can be derived by using Lemmas 1 and 2 respectively,
Theorem 1 delivers the result for Wald( F) .
Theorem 1. Let λ ∈ (0, 1) and b ∈ (0, 1] be given. Suppose M = bT. Then under Assumptions
1 and 2, as T → ∞,
Wald( F) ⇒

1
1
Wl (λ) −
(Wl (1) − Wl (λ))
λ
1−λ

1 (1)
1
(2)
× P b, Fl (r, λ) −
F (r, λ)
λ
1−λ l

−1

×

1
1
Wl (λ) −
(Wl (1) − Wl (λ))
λ
1−λ
(1.24)

Proof: See the Appendix.
The next Corollary provides an alternative representation for the limit given in (1.24).
Corollary 2. For a given value of λ ∈ (0, 1), the fixed-b limit of Wald( F) has the same distribution
as
1
W (1)
λ (1 − λ ) l
1
+
P
1−λ

1
P
λ

b
, W (r ) + CP (λ, b)
λ l

b
, W ∗ (r ) + CP (λ, b)
1−λ l

17

−1

Wl (1),

(1.25)

where

CP (λ, b) =












√ √
λ 1− λ

1 1
0 0

K

|λt−(1−λ)s−λ|
b
b2

1− b
b−λ
Wl ( λr )Wl∗ r+1−
0
√ √λ

(

√ √
λ 1− λ









 −

1− b
0

1 1
0 0

Wl (t)Wl∗ (s) dtds

) 1(λ−b<r≤λ)dr

b λ 1− λ

K

|λt−(1−λ)s−λ|
b

for Class-1 kernels,

for Class-2 kernels,

Wl (t)Wl∗ (s) 1(|λt−(1−λ)s−λ|<b)dtds

b2
b−λ
K_ (1)Wl ( λr )Wl∗ ( r+1−
1
λ
−b<r ≤λ)dr
(
)
√ √ λ
b λ 1− λ

for Class-3 kernels,

and Wl (r ) and Wl∗ (r ) are l × 1 Brownian Bridge processes which are independent of each other
and of Wl (1).
Proof: See the Appendix.
The limit in (1.25) shows how the components of Ω( F) affect the distribution of Wald( F) .
As mentioned earlier the random matrix P

b
λ , Wl (r )

( F)

reflects the random nature of Ω11

which is a part of asymptotic variance estimator of β 1 . But as the first argument of
b
λ , Wl (r ) manifests, this Corollary additionally reveals that the (effective) bandwidth
( F)
for Ω11 turns out to be λb not b. This picks up the fact that one implicitly uses the band( F)
width ratio λb for Ω11 when the researcher applies a bandwidth ratio b for constructing
( F)
Ω( F) . The second component P 1−b λ , Wl∗ (r ) is related with Ω22 (and β 2 ) in the exactly

P

same fashion. Finally, having the third component CP (λ, b) as a part of the limit indicates that the fixed-b inference captures the impact of finite-samples covariance on the
variance of β 1 − β 2 and on the inference for a structural change. The results for Wald(S)
are available in the following Theorem and Corollary.
Theorem 2. Let λ ∈ (0, 1) and b1 , b2 ∈ (0, 1] be given. Suppose M1 = b1 (λT ) and M2 =
b2 (1 − λ) T. Then under Assumptions 1 and 2, as T → ∞,
Wald(S) ⇒

×

1
1
Wl (λ) −
(Wl (1) − Wl (λ))
λ
1−λ

1
1
(1)
(2)
P b1 λ, Fl (r, λ) +
P b2 (1 − λ) , Fl (r, λ)
λ
1−λ
18

−1

(1.26)

1
1
Wl (λ) −
(Wl (1) − Wl (λ)) .
λ
1−λ

×
Proof: See the Appendix.

Corollary 3. For a given value of λ ∈ (0, 1), the fixed-b limit of Wald(S) has the same distribution
as
1
W (1)
λ (1 − λ ) l

1
1
P b1 , Wl (r ) +
P b2 , Wl∗ (r )
λ
1−λ

−1

Wl (1).

(1.27)

Proof: See the Appendix.
The major difference between (1.27) and (1.25) lies in the term CP (λ, b) which is
the limit associated with the cross product term of Ω( F) . The second difference is the
bandwidth ratios applied to the HAC estimator. In constructing the HAC estimator
Ω( F) , one can only choose a single value of bandwidth M and accordingly a single value
of bandwidth ratio b ≡

M
T.

But this bandwidth M implicitly determines the effective

bandwidth ratios within each regimes:
M
(1− λ ) T

=

bT
(1− λ ) T

=

b
(1− λ )

M
λT

=

bT
λT

=

b
λ

for the pre-break regime and

for the post-break regime. On the other hand, when one uses

Ω(S) , the researcher chooses two bandwidths M1 and M2 for each regime respectively
and accordingly the two bandwidth ratios are set as b1 =

M1
λT

and b2 =

M2
.
(1− λ ) T

In order

to accurately compare the test perfomance of Wald( F) and Wald(S) , it should be ensured
that the effective within-regime bandwidth ratios are the same across the two statistics.
Specifically, if the bandwidth ratio used for Wald( F) is b, then one should pick bandwidth
ratios as b1 =

b
λ

and b2 =

b
1− λ

for Wald(S) . By doing this one can isolate the impact of the

presence of CP (λ, b) on the inference.

1.5

Critical Values

The fixed-b limiting distributions are nonstandard. Asymptotic critical values are easily
obtained via simulations. The Wiener processes in the limiting distributions are approximated using scaled partial sums of 1,000 i.i.d. N (0, 1) random variables. Critical values

19

are tabulated based on 50,000 replications. The 95% critical values for l=1 and 2 are provided for selected values of b and λ in Tables 1.1 through Table 1.13. The break fraction,
λ, runs from 0.1 to 0.9 with 0.1 increments. The bandwidths considered are 0.02, 0.04,
0.06, 0.08, 0.1, 0.2, . . . , 1. Tables 1.1 through 1.6 contain the critical values for Wald( F) and
Table 1.7 through 1.13 provide the critical values for Wald(S) . The critical value tables for
Wald(S) only cover the case where b1 = b2 although Table 1.13 provides some critical
values with b1 = b2 .
Tables 1.1 through 1.6 display two main patterns of the critical values. First, for each
given λ the critical values increase as the bandwidth gets bigger. This is expected given
the well known downward bias induced into HAC estimators from estimation error. The
fixed-b approximation captures this downward bias and reflects it through larger critical
values. Second, for a given value of the bandwidth the critical values display a V-shaped
pattern as a function of λ. As the break point moves closer to zero or one, the critical
values increase and the minimum critical values occur at λ = 0.5. This V-shaped pattern
is present but is less pronounced in the critical values for Wald(S) .

1.6

Finite Sample Size

This Section reports the results of a finite sample simulation study that illustrates the
performance of the fixed-b critical values relative to the traditional critical values under
the null hypothesis of no structural change. The data generating process (DGP) used in
this simulation study is given by

yt = β 1 + β 2 xt + ut ,
xt = θxt−1 + t ,
ut = ρut−1 + ηt + ϕηt−1 ,

20

where

t

and ηt are independent of each other with

t , ηt

∼ i.i.d. N (0, 1). The following

parameter values are used:

( β 1 , β 2 ) = (0, 0),
λ ∈ {0.2, 0.4, 0.5},
θ ∈ {0.5, 0.8, 0.9},

(ρ, ϕ) ∈ {(0, 0), (0.5, 0), (0.5, 0.5), (0.9, 0.5), (0.9, 0.9)},

The DGP specifications for (θ, ρ, ϕ) include:

A: θ = 0.5, ρ = 0, ϕ = 0
B: θ = 0.5, ρ = 0.5, ϕ = 0
C: θ = 0.5, ρ = 0.5, ϕ = 0.5
D: θ = 0.8, ρ = 0.5, ϕ = 0.5
E: θ = 0.8, ρ = 0.9, ϕ = 0.5
F: θ = 0.9, ρ = 0.9, ϕ = 0.9

The value of θ measures the persistence of the time varying regressor xt . The parameters ρ
and ϕ jointly determine the serial correlation structure of the error term ut . Bigger values
of these three parameters leads to higher persistence of the time series vt ≡ xt ut except
for specification A. Results for sample sizes T = 50, 100, 500 are reported and the number
of replications is 2,500. The nominal level of all tests is 5%. Wald(S) and Wald( F) statistics
are computed for testing the joint null hypothesis of no structural change in both the β 1
and β 2 parameters.

21

1.6.1 Wald(S) and Wald( F) : Traditional Inference vs. Fixed-b Inference
First the empirical null rejection probabilities of the two statistics are examined. The performance of the traditional chi-square critical values are compared with that of fixed-b
critical values. Tables 1.14 through 1.19 report empirical null rejection probabilities for
Wald(S) test using the two critical values. In practice one can construct Ω(S) using different bandwidths ratios b1 and b2 in the two regimes. To conserve space, only results
for homogeneous bandwidths choices (i.e. b1 = b2 ) are provided. Tables 1.14 and 1.15
give results for DGP A for λ = 0.2, 0.4, Tables 1.16 and 1.17 give results for DGP D for
λ = 0.2, 0.4, and Tables 1.18 and 1.19 give results for DGP E for λ = 0.2, 0.4. Each table
reports results using the Bartlett and QS kernels for a range of values of b.
Consider the results for DGP A (no serial correlation in vt ) given in Tables 1.14 and
1.15. When T = 50, it is seen that both the Bartlett and QS kernel deliver tests that
over-reject when the chi-square critical value is used. As the bandwidth gets bigger, this
tendency to over-reject becomes more and more pronounced. Rejections using fixed-b
critical values are similar when small bandwidths are used. As the bandwidth increases,
rejections using fixed-b critical values systematically decrease towards the nominal level
of 0.05. Increasing T reduces over-rejections for both critical values when b is small. In
contrast, when b is large, rejections using the chi-square critical value tend to persist as
T increases but rejections tends towards 0.05 when fixed-b critical values are used. With
T = 500, the QS kernel has empirical rejection probabilities close to 0.06 for all values
of b when fixed-b critical values are used. It is clear that the fixed-b approximation is often a substantial improvement over the traditional approximation. This is not surprising
given that the fixed-b approximation captures much of the randomness in the HAC estimator whereas the traditional approach treats the HAC estimator as a constant equal to
its population value.
Tables 1.16-1.19 show the results for the data with stronger dependence. Patterns are

22

similar to Tables 1.14-1.15 except that over-rejection problems tend to be higher in all
cases. Using chi-square critical values leads to severe over-rejection problems regardless of bandwidth or kernel. Using fixed-b critical values tends to reduce over-rejection
problems especially when larger values of b are used. The QS kernel tends to be much
less over-sized than the Bartlett kernel. These finite sample patterns are very similar to
patterns found by Kiefer and Vogelsang (2005) in non-structural change settings.
Comparing Tables 1.14, 1.16, 1.18 (λ = 0.2) with Tables 1.15, 1.17, 1.19 (λ = 0.4) it
is seen that over-rejection problems tend to be greater with λ = 0.2 than with λ = 0.4.
This makes sense because the λ = 0.2 case has a relatively small sample size for regime 1
compared to regime 2 whereas for λ = 0.4 the regime sample sizes are similar.
Next null rejection probabilities of the Wald( F) statistic are examined. In Figure 1.1
rejection probabilities for Wald( F) is plotted for the DGP given by ( β 1 , β 2 , θ, ρ, ϕ) = (0,
0, 0.1, 0.5, 0.5) for the case of T = 50. Results are given for two kernels: Bartlett and QS
and bandwidth ratio b = 0.2. For each kernel, rejections are computed using the chisquare critical value and the fixed-b critical value. Rejections are plotted across a grid
of break points given by λ ∈ {0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9}. Figure 1.1 shows the
large size distortions associated with traditional inference. Over-rejections are substantial
especially when λ is close to the endpoints of the sample. In general a V-shaped pattern
appears in over-rejections as a function of λ with the least over-rejections occurring at the
middle of the sample (λ = 0.5). Rejections using fixed-b critical values are much closer
to 0.05 and rejections are less sensitive to the location of the break point. Figure 1.1 also
shows that the kernel matters. Rejections using the QS kernel when fixed-b critical values
are used are closer to 0.05.
Tables 1.20 through 1.25 provide additional results for Wald( F) for most of the DGPs
given in Tables 1.14-1.19. Again rejections are reported using traditional chi-square critical
values and fixed-b critical values. The table also reports some rejections for Wald(S) which
are discussed in the next subsection. The patterns in Tables 1.20-1.25 are generally very

23

0.9
Fixed−b 5% c.v., Bartlett
Chi square 5% c.v., Bartlett
Fixed−b 5% c.v., QS
Chi square 5% c.v., QS

0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0.1

0.2

0.3

0.4

0.5
0.6
Break Fraction (λ)

0.7

0.8

0.9

Figure 1.1: Empirical Null Rejection Probability of Structural Break Test using Wald( F)
β 1 = β 2 = 0, θ = 0.1, ρ = 0.5, φ = 0.5, b = 0.2, T = 50, Replications=2, 500
similar to the patterns seen in Tables 1.14-1.19. It is clear that the fixed-b approximation is
a substantial improvement over the traditional approximation.

1.6.2

Fixed-b Inference: Wald(S) vs. Wald( F)

In this subsection the finite sample rejections of the two Wald statistics are compared
under the fixed-b framework. As equation (1.25) and (1.27) manifest, the fixed-b limits
of the two statistics are different in two ways. First, the limit of Wald( F) has the extra
component CP (λ, b). Second, the limit of Wald(S) depends on b1 and b2 which are the
bandwidths for the two regimes respectively while the limit of Wald( F) depends on a
single bandwidth b implying effective bandwidths

b
λ

and

b
1− λ

in each regime. It is well

known in the fixed-b literature that larger bandwidths lead to less over-rejection problems
when fixed-b critical values are used. This serves to hedge against differences in effective
bandwidths resulting in differences in over-rejections between Wald(S) and Wald( F) so
that the impact of inclusion of the off-diagonal blocks used by Wald( F) can be isolated.
24

To keep bandwidth effects constant between Wald(S) and Wald( F) , the bandwidths for
Wald(S) are set as
b1 =

b
b
and b2 =
.
λ
1−λ

(1.28)

where b is the bandwidth for Wald( F) . By using (1.28) it is ensured that the within regime
bandwidth to (regime) sample size ratios for Wald(S) are the same as the bandwidth to
(full) sample size ratio used by Wald( F) .
The following eight bandwidths specifications satisfying (1.28) are used:

A’: λ = 0.2, b = 0.04, b1 = 0.2, b2 = 0.05,
B’: λ = 0.2, b = 0.1, b1 = 0.5, b2 = 0.125,
C’: λ = 0.2, b = 0.2, b1 = 1.0, b2 = 0.25,
D’: λ = 0.5, b = 0.5, b1 = 1.0, b2 = 1.0,
E’: λ = 0.5, b = 1.0, b1 = 2.0, b2 = 2.0,
F’: λ = 0.4, b = 0.5, b1 = 1.25, b2 = 0.83,
G’: λ = 0.2, b = 0.5, b1 = 2.5, b2 = 0.625,
H’: λ = 0.2, b = 1.0, b1 = 5.0, b2 = 1.25.

Tables 1.20 through 1.22 give results for relatively small bandwidths (A’, B’, and C’).
With small bandwidths Wald(S) and Wald( F) have very similar size distortions. Tables
1.23 through 1.25 report results for the relatively large bandwidths (D’, E’, F’, G’, and H’).
Note that compared to the small bandwidths case, there is noticeable difference in the
performance of the two statistics specially when the QS kernel is used. As the DGP becomes more persistent (such as DGP F or G) the differences between Wald(S) and Wald( F)
become more clear. Table 1.23 shows that under DGP G and bandwidth E’ the empirical
null rejection probability is 19.08% for Wald(S) and 13.6% for Wald( F) when T=50 whereas
in the Bartlett kernel case rejections are similar. The differences when T=100 are smaller

25

(15.96% vs. 11.92%); see Table 1.24. Once T reaches 500 (Table 1.25), the differences between Wald(S) and Wald( F) are very small for either kernel. In general Wald( F) is less size
distorted than Wald(S) when T is relatively small and persistence in the data is not weak.
For larger sample sizes the two tests have similar null rejection probabilities. From a size
perspective, Wald( F) is preferred over Wald(S) .
The better size performance of Wald( F) can be explained by examining CP (λ, b) in
( F)

( F)

( F)

(1.25) and the corresponding finite sample counterparts Ω21 (or Ω12 ) in (1.18). Ω21 is
estimating the sample covariance between β 1 and β 2 which is driven by finite sample
(1)

covariance between vt

(2)

and vt

( F)

across the two regimes. Larger bandwidths lead to Ω21

( F)

and Ω12 estimators that better reflect the finite sample covariance between β 1 and β 2 and
( F)

( F)

CP (λ, b) captures the impact of Ω21 and Ω12 on the sampling distribution of Wald( F) .
As the data becomes more persistent, the covariance between β 1 and β 2 becomes more
( F)

( F)

pronounced in small samples so there is benefit to including Ω21 and Ω12 in Ω. As T
increases, the covariance between β 1 and β 2 and it becomes more reasonable to impose
the restriction that the off-diagonal blocks of Ω are zero as done by Wald(S) .

1.7

Local Power Analysis of Fixed-b Inference

This Section investigates the local power of Wald(S) and Wald( F) under the fixed-b approach. The local alternative is given by
H1 : Rβ = T −1/2 δ∗

( l ×1)

26

(1.29)

with R = ( R1 , − R1 ) . Under the local alternative (1.29), the fixed-b limits of the Wald
statistics are easily obtained as
Wald( F) ⇒

1
1
Wl (λ) −
(Wl (1) − Wl (λ)) − A−1 δ∗
λ
1−λ

−1
1 (1)
1
(2)
× P b, Fl (r, λ) −
F (r, λ)
λ
1−λ l
1
1
×
Wl (λ) −
(Wl (1) − Wl (λ)) − A−1 δ∗ , .
λ
1−λ

Wald(S) ⇒

×
×

1
1
Wl (λ) −
(Wl (1) − Wl (λ)) − A−1 δ∗
λ
1−λ
1
1
(1)
(2)
P b1 λ, Fl (r, λ) +
P b2 (1 − λ) , Fl (r, λ)
λ
1−λ
1
1
Wl (λ) −
(Wl (1) − Wl (λ)) − A−1 δ∗ ,
λ
1−λ

−1

where the nonsingular matrix A is defined in equation (1.10).

1.7.1

Comparison of the Local Asymptotic Power of Wald(S) and Wald( F)

In this subsection the local asymptotic power of Wald(S) and Wald( F) are compared for the
case R = ( I2 , − I2 ). The bandwidth specifications are the same as those used in Section
1.6.2. Local asymptotic power was computed using the same simulation methods used
to compute asymptotic fixed-b critical values with A−1 δ∗ = (δ, δ) , where δ is a scalar
parameter. Figures 1.2-1.5 plot local asymptotic power for the Bartlett and QS kernels for
a selection breakpoint and bandwidth specifications. As the figures illustrate, there is no
substantial difference in the local asymptotic power between Wald(S) and Wald( F) for the
case of the Bartlett kernel. However, when QS kernel is used, differences in power emerge
with large bandwidths as depicted by Figures 1.3-1.5. For example, using b = 1 as shown
in Figures 1.4 and 1.5, one can see substantial power loss associated with Wald( F) with
the QS kernel. Comparing the Bartlett and QS kernels, the Bartlett kernel delivers higher
27

1
0.9
0.8
0.7
0.6

Wald(F), Bartlett
Wald(F), QS

0.5

Wald(S), Bartlett
Wald(S), QS

0.4
0.3
0.2
0.1
0

0

5

10
δ

15

20

Figure 1.2: Local Power, Bartlett and QS,
λ = 0.2, b = 0.2, b1 = 1, b2 = 0.25

1
0.9
0.8
0.7
0.6

Wald(F), Bartlett
Wald(F), QS

0.5

Wald(S), Bartlett
Wald(S), QS

0.4
0.3
0.2
0.1
0

0

5

10
δ

15

20

Figure 1.3: Local Power, Bartlett and QS,
λ = 0.2, b = 0.5, b1 = 2.5, b2 = 0.625

28

1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
Wald(F), Bartlett

0.2

Wald(F), QS
Wald(S), Bartlett

0.1
0

Wald(S), QS
0

5

10
δ

15

20

Figure 1.4: Local Power, Bartlett and QS,
λ = 0.2, b = 1, b1 = 5, b2 = 1.25
power than the QS kernel for both Wald statistics. The loss of power associated with
Wald( F) relative to Wald(S) occurs exactly for the kernel and bandwidth specifications that
resulted in Wald( F) having less size distortions than Wald(S) . Similarly, the higher power
of the Bartlett kernel relative to the QS kernel comes at the cost of greater size distortions.
As has been documented repeatedly in the fixed-b literature, there is an inherent tradeoff between reduction of over-rejection problems and loss of power. Bandwidth/kernel
combinations that reduce over-rejection problems also reduce power.

1.7.2

Impact of Breakpoint Location and Bandwidth on Power

The next focus is on the impact of the breakpoint location, λ, and bandwidths on local
asymptotic power. Figures 1.6-1.9 display local asymptotic power for a range of λ. Each
figure depicts power curves for the range of λ = 0.1, 0.2, 0.3, 0.4, 0.5 for the bandwidth
ratio b = 0.1. Figures 1.6 and 1.7 depict power for the Wald( F) statistic for the Bartlett
and QS kernels respectively. Figures 1.8 and 1.9 depict power for Wald(S) for the Bartlett
and QS kernels respectively. For a given λ, the bandwidths b1 and b2 for Wald(S) are
chosen so that they match the within-regime effective bandwidths of Wald( F) ,
For a given b, smaller values of λ implies larger values of max
29

b
b
λ , 1− λ

b
λ

and

b
1− λ .

and the local

1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
Wald(F), Bartlett

0.2

Wald(F), QS
Wald(S), Bartlett

0.1
0

Wald(S), QS
0

5

10
δ

15

20

Figure 1.5: Local Power, Bartlett and QS,
λ = 0.5, b = 1, b1 = 2, b2 = 1
power for Wald( F) decreases as max

b
b
λ , 1− λ

increases. First notice that regardless of

bandwidth or kernel, power is lowest for λ = 0.1 and steadily rises as λ increases to
0.5. It should be noted that power is symmetric in λ and power for values of λ larger
than 0.5 can be inferred from the figures. Second, comparing Figures 1.6 and 1.8 (Bartlett)
with Figures 1.7 and 1.9 (QS) the Bartlett kernel gives higher local power than the QS
kernel especially for large within-regime effective bandwidths. A similar finding was
documented by Kiefer and Vogelsang (2005) in models without structural change.
To isolate the impact of bandwidths on power, Figures 1.10 through 1.13 depict power
for a range of b values for a given kernel and λ = 0.5. These figures illustrate that increasing the bandwidth generally reduces power and this is especially true for the QS kernel.
For the Bartlett kernel, once b = 0.5, further increases in b have little effect on power.
In contrast, increasing b from 0.5 to 1.0 significantly reduces power when using the QS
kernel although the drop in power is even more dramatic when b increases from 0.1 to
0.5. That power of Wald( F) and Wald(S) is decreasing in the bandwidth is not surprising
given that this pattern holds in models without structural change.

30

1
0.9
0.8
0.7
0.6
0.5
0.4
0.3

λ=0.1
λ=0.2
λ=0.3
λ=0.4
λ=0.5

0.2
0.1
0

0

2

4

δ

6

8

10

Figure 1.6: Local Power, Wald( F) , Bartlett, b = 0.1

1
0.9
0.8
0.7
0.6
0.5
0.4
0.3

λ=0.1
λ=0.2
λ=0.3
λ=0.4
λ=0.5

0.2
0.1
0

0

2

4

δ

6

8

10

Figure 1.7: Local Power, Wald( F) , QS, b = 0.1

31

1
0.9
0.8
0.7
0.6
0.5
0.4
λ=.1, b1=1, b2=.11

0.3

λ=.2, b1=.5, b2=.125

0.2

λ=.3, b1=.33, b2=.14

0.1

λ=.4, b1=.25, b2=.17

0

λ=.5, b1=.2, b2=.2
0

2

4

δ

6

8

10

Figure 1.8: Local Power, Wald(S) , Bartlett

1
0.9
0.8
0.7
0.6
0.5
0.4
λ=.1, b1=1, b2=.11

0.3

λ=.2, b1=.5, b2=.125

0.2

λ=.3, b1=.33, b2=.14

0.1

λ=.4, b1=.25, b2=.17

0

λ=.5, b1=.2, b2=.2
0

2

4

δ

6

8

Figure 1.9: Local Power, Wald(S) , QS

32

10

1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
b=0.04
b=0.08
b=0.1
b=0.5
b=1

0.2
0.1
0

0

2

4

6

8

10

Figure 1.10: Local Power, Wald( F) , Bartlett, λ = 0.5

1
b=0.04
b=0.08
b=0.1
b=0.5
b=1

0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

0

2

4

δ

6

8

10

Figure 1.11: Local Power, Wald( F) , QS, λ = 0.5

33

1
0.9
0.8
0.7
0.6
0.5
0.4
b1=b2=.08

0.3

b1=b2=.16
0.2

b1=b2=.2
b1=b2=1

0.1
0

b1=b2=2
0

2

4

δ

6

8

10

Figure 1.12: Local Power, Wald(S) , Bartlett, λ = 0.5

1
0.9
0.8
0.7
0.6
0.5
0.4
b1=b2=.08

0.3

b1=b2=.16
0.2

b1=b2=.2
b1=b2=1

0.1
0

b1=b2=2
0

2

4

δ

6

8

10

Figure 1.13: Local Power, Wald(S) , QS, λ = 0.5

34

1.8

Partial Structural Change Model

This Section derives the fixed-b limit of Wald( F) in the partial structural change model.
The main result of this Section is that the limit is the same as the limit for the full structural
change model. In contrast, the Wald(S) statistic in the partial structural change case has a
non-pivotal fixed-b limit. A modificaiton of the Wald(S) statistic that has a pivotal fixed-b
limit is briefly discussed.

1.8.1

Setup and Assumptions

The regression model with partial structural change is given by

yt = zt α + x1t β 1 + x2t β 2 + ut

(1.30)

= z t α + Xt β + u t ,
where xt is p × 1 and zt is q × 1 vector and

x1t = xt 1(t ≤ λT ), x2t = xt 1(t > λT ),
Xt = ( x1t x2t ) , and β = ( β 1 β 2 ) .
The coefficients on the xt regressors are unrestricted in terms of a structural change whereas
the coefficients on the zt regressors are assumed to not have structural change. Denote

y = (y1, y2 , . . . , y T ) , X = ( X1 , X2 , . . . XT ) ,
Z = ( z1 , z2 , . . . , z T ) , u = ( u1 , u2 , . . . , u T ) .

35

The parameters (α, β) are estimated by OLS and the OLS residual vector can be written
as
u = y − X β = u − X β − β − PZ u,
where
y = ( I − PZ ) y, X = ( I − PZ ) X, and PZ = Z ( Z Z )−1 Z .
The residual for a individual observation is given by

u t = u t − Xt β − β − z t Z Z
Also, note that

−1


X t = X t − X Z ( Z Z ) −1 z t =

Z u.

(1.31)



(1)
 Xt 
 p ×1 

.
 X (2) 
t
p ×1

The following assumptions
replace
 Assumptions 1 and
2 inSection 1.2:

 Λ1 
[rT ]  xt ut 
Assumption 1’. T −1/2 ∑t=1 
 ⇒ ΛWp+q (r ) ≡   Wp+q (r ), where Λ1 is a
zt ut
Λ2
p × ( p + q) matrix, Λ2 is a q × ( p + q) matrix, and Wp+q (r ) is a ( p + q) × 1 vector of
independent Wiener process.
[rT ]

[rT ]

[rT ]

Assumption 2’. p lim T1 ∑t=1 zt zt = rQ ZZ , p lim T1 ∑t=1 xt xt = rQ xx , and p lim T1 ∑t=1 xt zt =
1
−1
rQ xZ uniformly in r ∈ [0, 1], and there exist Q−
ZZ and Q xx .

1.8.2

Asymptotic Limits

Continue to focus on tests of the null hypothesis of no structural change in the xt slope
parameters of the form
H0 : Rβ = r

36

with
R1 , − R1

R =
l ×2p

l× p

Recall that the OLS estimator, β = β 1 , β 2

β=

(1.32)

can be rewritten as
−1

T

and r = 0.

l× p

∑ Xt Xt

T

∑ Xt y t

t =1

.

(1.33)

t =1

Proposition 3. Under Assumptions 1’ and 2’, as T → ∞

d

T 1/2 β − β → Q−1 
XX



1
Λ1 Wp+q (λ) − λQ xZ Q−
ZZ Λ2 Wp+q (1)

Λ1 Wp + q (1) − Wp + q ( λ )

1
− (1 − λ) Q xZ Q−
ZZ Λ2 Wp+q (1)


,

and

√

H0

1
T R β − r ⇒ R1 Q −
xx Λ1

1
1
Wp + q ( λ ) −
Wp + q (1) − Wp + q ( λ )
λ
1−λ

,

(1.34)

where Q X X = p lim T −1 ∑tT=1 Xt Xt .
Proof: See the Appendix.
As seen from the above proposition, β 1 and β 2 are not asymptotically independent in
the partial structural change regression model. This is true because we are projecting out
the variation of explanatory variables zt so that β 1 and β 2 depend on the entire series of
xt and zt . The dichotomy that β 1 is dependent only on the pre-break data and β 2 depends
only on the post-break data no longer holds in the partial structural change model. The
1
dependence manifests in the common term, Q xZ Q−
ZZ Λ2 Wp+q (1) in the above Proposition.

However, this term cancels out in (1.34) when the restriction matrix takes the form of
(1.32). As a result, and also as suggested by equation (1.34), one only needs to estimate
Λ1 Λ1 for the inference on structural change. But, Λ1 Λ1 can be easily estimated using
x1t ut or x2t ut and the corresponding version of Wald(S) can be defined by constructing

37

Σ(1) with x1t ut and constructing Σ(2) with x2t ut . By doing this, we are ignoring the need
to project zt out of the xt variables but this ignorance does not make the inference invalid.
The next question is whether in conducting inference one can still take into account the
dependence between β 1 and β 2 which is not due to the presence of zt . This can be verified
by finding a version of Wald( F) which has a pivotal limit in the partial structural change
model. The answer is positive as below.
The Wald statistic is given by
R Q −1 Ω Q −1 R

Wald = T R β

XX

XX

−1

Rβ ,

(1.35)

where Q X X = T −1 ∑tT=1 Xt Xt . For constructing Wald( F) , consider a HAC estimator Ω( F)
which is computed using Xt ut

T
t =1

:

Ω ( F ) = T −1

T

T

∑∑K

t =1 s =1

|t − s|
M

ξtξs,

(1.36)

where ξ t = Xt ut . This is a straightforward extension of Wald( F) to the case of partial
structural change.
Next Lemma provides the limit of the the scaled partial sum process of ξ t premultiplied by an appropriate term.
Lemma 3. Let St = ∑tj=1 ξ j . Under Assumptions 1’ and 2’, as T → ∞,
ξ

1 (1)
1
(2)
Fp+q (r, λ) −
Fp+q (r, λ) ,
λ
1−λ

1
R Q−1 T −1/2 S[rT ] ⇒ R1 Q−
xx Λ1
ξ

XX

where
(1)

Fp+q (r, λ) = Wp+q (r ) −
(2)

Fp+q (r, λ) =

Wp + q (r ) − Wp + q ( λ ) −

r
Wp + q ( λ ) 1 (0 ≤ r ≤ λ ) ,
λ

r−λ
Wp + q (1) − Wp + q ( λ )
1−λ

Proof: See the Appendix.
38

1 ( λ < r ≤ 1) .

As Lemma 3 shows, the partial sums of the inputs to Ω( F) are asymptotically propor√
tional to the same nuisance parameters as T R β − r . This is the key condition for an
asymptotic pivotal fixed-b limit. The next Theorem provides the fixed-b limit of Wald( F) .
Theorem 3. Let λ ∈ (0, 1) and b ∈ (0, 1] be given. Suppose M = bT. Then under Assumptions
1’ and 2’, Wald( F) weakly converges to the same limit in (1.24), i.e. as T → ∞,
Wald( F) ⇒

1
1
Wl (λ) −
(Wl (1) − Wl (λ))
λ
1−λ

1
1 (1)
(2)
Fl (r, λ)
× P b, Fl (r, λ) −
λ
1−λ

−1

×

1
1
Wl (λ) −
(Wl (1) − Wl (λ)) .
λ
1−λ

Proof: See the Appendix.
According to Theorem 3, the limit of Wald( F) in the partial structural change model is
the same as in the full structural change model.
Getting back to Wald(S) , as mentioned earlier, β 1 and β 2 are no longer asymptotically
uncorrelated in the partial structural change model. Therefore, forcing the covariance
between β 1 and β 2 to be zero cannot be justified any longer. Even though forcing the
covariance to be zero is not theoretically justified, even asymptotically, Wald(S) can be
(1)

modified so that it has a pivotal fixed-b limit. This is obtained by using vt
(2)

xt ut 1(t ≤ λT ) for constructing Σ(1) and vt

= x1t ut =

= x2t ut = xt ut 1(t > λT ) for Σ(2) with ut

being defined in (1.31). One can easily show that the limit of this Wald(S) is the same
as in (1.26). It is tempted to use Xt ut 1(t ≤ λT ) for constructing Σ(1) (and similarly for
Σ(2) using Xt ut 1(t > λT )). But this version of Wald(S) turns out to have a non-pivotal
fixed-b limit. This can be easily verified by deriving the limits of the associated partial
sum processes.

39

1.9

When the Break Date is Unknown

Tests for a potential structural break with a unknown break date are well studied in Andrews (1993) and Andrews and Ploberger (1994). Andrews (1993) considers several tests
based on the supremums across break points of Wald and LM statistics and shows they
are asymptotically equivalent. Andrews and Ploberger (1994) derive tests that maximize
average power across potential breakpoints. Wald( F) statistic is the only focus of this Section. Given a value of b, the test statistic is computed for a range of λ. This implicitly
changes the effective bandwidths ( λb , 1−b λ ) as λ varies. This results in a built-in mechanism where bigger bandwidth ratio is used as the sample size of a regime shrinks. One
might want to make a comparison with the test based on Wald(S) . But note that one needs
to adjust the bandwidth ratios (b1 , b2 ) every time λ changes so that (b1 , b2 ) stay equal to
( λb , 1−b λ ). The implementation of tests with Wald(S) is omitted in this chapter. Denote
the Wald( F) statistic computed using break date Tb ≡ [λT ] by Wald( F) ( Tb ). Also de( F)

( F)

note the limit of Wald( F) ( Tb ) as Wald∞ (λ) where the form of Wald∞ (λ) depends on
whether traditional or fixed-b asymptotic theory is being used. In the case of fixed-b the( F)

ory, Wald∞ (λ) depends on P b, Fp (r, λ) . As argued by Andrews (1993) and Andrews
and Ploberger (1994), break dates close to the end points of the sample cannot be used
and so some trimming is needed. To that end define Ξ∗ = [ T, T − T ] with 0 <
to be the set of admissible break dates. The tuning parameter

<1

denotes the amount of

trimming of potential break dates. Consider the three statistics following Andrews (1993)
and Andrews and Ploberger (1994) defined as

SupW ( F) ≡ sup Wald( F) ( Tb ),
Tb ∈Ξ∗

MeanW ( F) ≡

1
T

∑ ∗ Wald(F) (Tb ),

Tb ∈Ξ

40

(1.37)

(1.38)

1
T

ExpW ( F) ≡ log

∑ ∗ exp

Tb ∈Ξ

1
Wald( F) ( Tb )
2

.

(1.39)

The asymptotic limits of these statistics follow from the continuous mapping theorem and
are given by
SupW ( F) ⇒

( F)

Wald∞ (λ),

sup
λ∈( ,1− )

MeanW ( F) ⇒

1−

ExpW ( F) ⇒ log

( F)

Wald∞ (λ)dλ,
1−

exp

1
( F)
Wald∞ (λ) dλ .
2

In Tables 1.26 and 1.27 fixed-b critical values for SupW ( F) , MeanW ( F) , and ExpW ( F)
are provided for l = 2,

= 0.05, 0.1, 0.2 and for b ∈ {0.02, 0.04, 0.06, 0.08, 0.1, 0.2, 0.3,

..., 0.9, 1}.
Tables 1.28 through 1.36 presents the simulation result for some of the DGP specifications introduced in Section 1.6 and for T=100, 500 and 1000. Fixed-b critical values
are used for the Bartlett and QS kernels. For the Bartlett kernel, results are also reported using the traditional critical values obtained by Andrews (1993) and Andrews
and Ploberger (1994). Several patterns stand out for the null rejection probabilities associated with SupW ( F) in Table 1.28 to 1.30. First, rejections using the traditional critical
values are often substantially above the 5% nominal level unless persistence is very weak
and a small bandwidth is used. Rejections can be close to 100%. The situation is much
improved by the use of fixed-b critical values but severe over-rejections are still possible. Size distortions are higher with more persistence in the data, with a smaller value
of , and a smaller value of b. As was true in the case of a known break date, the QS
kernel gives less size distortion than the Bartlett kernel although the use of large bandwidths causes under-rejections. But over-rejections and under-rejections dissipate as T
grows. The Bartlett kernel can suffer from over-rejections that are not easily removed just
by using a big bandwidth. A larger value of T helps along with more trimming. Similar
41

patterns hold for the MeanW ( F) and ExpW ( F) statistics; see Tables 1.31-1.33 and 1.34-1.36
respectively.

1.10

Summary and Conclusions

In this chapter fixed-b asymptotics was applied to the problem of testing for the presence
of a structural break in a weakly dependent time series regression. Two different HAC
estimators and accordingly two different Wald statistics were investigated. The Wald( F)
statistic is the Wald statistic that one obtains when structural change is expressed in terms
of dummy variables interacted with regressors. The Wald(S) statistic is a restricted version
of Wald( F) where the off-diagonal blocks of the HAC estimator are set to zero mimicking
the asymptotic zero covariance between OLS estimators in the two regimes. The fixed-b
limits of the two statistics were derived, and the fixed-b inference was compared with the
traditional inference. In a model with full structural change, both Wald statistics have
pivotal fixed-b limits. However, in models with partial structural change, the straightforwardly adapted version of Wald( F) has the same pivotal fixed-b limit as in the full
structural change case whereas the straightforwardly adapted version of Wald(S) does
not have pivotal fixed-b limit.
In simulation study the finite sample size distortions associated with the fixed-b approach and the traditional approach were examined. The simulation results indicate that
the traditional inference is more subject to severe size distortions. When small bandwidths are used, the gap is not huge but as b gets bigger, the difference becomes substantial. Overall, rejections obtained when using fixed-b critical values are closer to the
nominal level compared to using the traditional chi-square critical values. When fixed-b
critical values are used, finite sample size distortions becomes more pronounce as b gets
smaller or as λ gets closer to 0 or 1. Local asymptotic power is decreasing in b and power
is highest for structural change located near the center of the sample.

42

In a comparison of the Wald( F) and Wald(S) statistics for full structural change models
it was found that Wald( F) tends to be less size distorted than Wald(S) when using fixed-b
critical values especially when serial correlation is strong and a large bandwidth is used.
The better size performance of Wald( F) comes at the cost of lower power reflecting the
usual trade-off between size robustness and power typically found in fixed-b analyses.
The choice between Wald( F) and Wald(S) becomes a choice between tolerance for overrejections relative to desire for high power. At a practical level, Wald( F) is appealing because it retains the same asymptotic pivotal fixed-b limit in models with partial structural
change whereas Wald(S) becomes nonpivotal.
Finally, some fixed-b critical values tabulated for SupW, MeanW, and ExpW statistics
which are commonly used for testing the presence of a structural break when the break
date is not known a priori. A simulation study revealed that over-rejections are a bigger
concern when the break date is treated as unknown. Critical values based on traditional
asymptotics can lead to very severe over-rejection problems. Rejections using fixed-b
critical values are less distorted especially when the QS kernel is used with a bandwidth
that is not too small. When the Bartlett kernel is used, fixed-b rejections show substantial
over-rejections.

43

Table 1.1: 95% Fixed-b critical values of Wald( F) with Bartlett kernel, l = 1
b=0.02
0.04
0.06
0.08
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1

λ=0.1
5.94
8.36
10.69
12.68
14.77
23.82
32.48
41.16
50.17
58.98
68.93
77.9
87.48
97.25

0.2
4.68
5.59
6.45
7.37
8.26
13
17.66
22.74
27.59
32.89
37.77
42.82
47.7
53.15

0.3
4.34
4.85
5.33
5.94
6.55
9.75
13.6
17.5
21.61
25.87
29.8
33.54
37.35
41.77

0.4
4.22
4.58
4.97
5.39
5.86
8.49
11.79
15.22
18.81
22.28
25.79
29.52
32.84
36.3

0.5
4.2
4.52
4.89
5.21
5.61
8.06
11.19
14.59
18.85
21.89
25.3
28.36
31.88
35.58

0.6
4.21
4.56
4.95
5.33
5.78
8.37
11.78
15.18
18.75
22.39
25.97
29.37
32.59
36.3

0.7
4.31
4.78
5.32
5.96
6.52
9.74
13.64
17.62
21.77
25.86
29.98
34.03
37.78
41.77

0.8
4.59
5.44
6.26
7.21
8.21
12.89
17.73
22.72
27.98
32.9
38.06
43.43
48.04
53.15

0.9
5.92
8.28
10.55
12.67
14.81
24.15
32.85
41.63
50.61
59.99
69.38
78.76
88.14
97.25

Table 1.2: 95% Fixed-b critical values of Wald( F) with Bartlett kernel, l = 2
b=0.02
0.04
0.06
0.08
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1

λ=0.1
0.2
10.45
7.49
15.43
9.36
20.01
11.3
23.63
13.23
27.45
15.17
45.04
24.5
62.32
34.33
79.15
44.31
97.5
54.75
115.81 64.44
134.73 75.73
153.2
85.47
171.39 94.64
197.92 111.33

0.3
6.74
7.74
8.86
10.08
11.16
18.28
25.71
34.01
41.9
50.3
58.44
65.2
71.96
84.82

0.4
6.49
7.2
7.98
8.92
9.77
15.5
22.44
29.69
37.15
44.32
50.96
56.85
63.23
74.22

44

0.5
6.53
7.18
7.93
8.76
9.62
15.22
21.98
28.71
36
42.62
49.05
55.35
61.05
71.25

0.6
6.6
7.34
8.14
9
9.97
15.8
22.62
29.56
36.28
44.31
51.09
56.79
63.14
74.22

0.7
0.8
7.04
7.78
8.1
9.74
9.21 11.64
10.41 13.68
11.86 15.61
18.77 25.29
26.46 35.39
34.46 46.08
42.98 56.62
51.09 67.64
59.28 78.46
66.47
88.5
73.97 98.19
84.82 111.33

0.9
10.73
15.9
20.46
24.46
28.41
45.98
63.27
81.68
100.45
119.13
137.84
155.94
174.38
197.92

Table 1.3: 95% Fixed-b critical values of Wald( F) with Parzen kernel, l = 1
b=0.02
0.04
0.06
0.08
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1

λ=0.1
0.2
0.3
0.4
5.53
4.48 4.22 4.12
7.74
5.21
4.63 4.43
10.05
5.99
5.08 4.71
12.8
6.78
5.48 5.04
15.39
7.61
5.93 5.39
28.72
12.88
8.9 7.53
43.41
18.92 12.78 10.38
59.19
26.32 17.63 14.4
77.93
35.19 24.52 19.48
101.1
45.41 32.72 25.83
126.46 58.14 41.87 33.32
156.18
72.5 53.07 42.13
193.5
88.33 65.16 52.59
236.76 106.98
78.8 64.15

0.5
4.13
4.36
4.66
4.93
5.19
7.13
9.81
13.6
18.73
25.14
32.72
42.21
52.45
65.05

0.6
4.14
4.41
4.7
5.01
5.34
7.47
10.52
14.79
19.75
26.31
34.24
43.55
53.96
65.94

0.7
0.8
4.2
4.42
4.61
5.11
5
5.83
5.46
6.59
5.97
7.44
8.82 12.53
12.78 18.77
17.71 26.38
24.37 35.61
32.14 46.83
41.36 58.84
52.62 74.23
64.97 91.91
79.76 112.21

0.9
5.45
7.62
10.05
12.6
15.31
29.14
44.02
60.09
78.96
101.31
126.34
154.86
189.83
229.3

Table 1.4: 95% Fixed-b critical values of Wald( F) with Parzen kernel, l = 2
b=0.02
0.04
0.06
0.08
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1

λ=0.1
0.2
9.68
7.16
14.7
8.68
20.78
10.44
26.9 12.28
32.86
14.32
61.88
26.88
95.45
42.82
139.66
65.29
196.8
95.6
278.36 134.62
382.08 186.56
514.04 248.31
680.53 326.98
892.26 423.95

0.3
0.4
0.5
6.54
6.32
6.43
7.33
6.9
6.92
8.22
7.5
7.46
9.19
8.17
8.13
10.3
8.96
8.73
17.11
13.96
13.6
27.94
22.48 21.23
42.75
35.2 32.89
63.74
53.21 49.54
92.64
77.72 72.15
131.83 110.23 101.57
179.83
150.7 139.6
234.32
200.3 184.72
303.13 260.04 240.38

45

0.6
6.44
7.01
7.62
8.37
9.13
14.16
22.38
34.8
51.62
75.69
105.94
145.76
194.26
251.93

0.7
6.81
7.68
8.6
9.59
10.68
17.92
28.87
43.65
64.56
92.71
128.55
173.65
230.37
303.92

0.8
7.48
9.04
10.85
12.71
14.79
27.25
44.02
66.36
97.41
137.41
190.44
259.83
341.18
435.32

0.9
9.87
15.26
21.32
27.69
33.72
62.55
97.73
144.3
201.36
282.96
385.63
520.59
702.39
914.14

Table 1.5: 95% Fixed-b critical values of Wald( F) with QS kernel, l = 1
λ=0.1
b=0.02
7.43
0.04
12.21
0.06
17.56
0.08
22.64
0.1
27.85
0.2
57.72
0.3
99.95
0.4
158.39
0.5
249.48
0.6
367.72
0.7
532.6
0.8
726.47
0.9
970.42
1 1261.34

0.2
5.07
6.48
8.08
9.95
12.16
25.61
45.22
74.89
114.08
168.98
241.2
326.56
431.92
559.48

0.3
4.52
5.28
6.17
7.23
8.43
16.86
32.2
54.62
83.55
122.86
176.66
245.78
323.43
416.24

0.4
4.36
4.91
5.54
6.23
7.08
13.44
25.09
43.04
68.85
103.11
143.09
197.2
262.52
339.54

0.5
4.29
4.79
5.3
5.99
6.76
12.75
24.18
43.29
70.81
103.31
146.09
199.56
262.71
340.36

0.6
4.34
4.86
5.46
6.23
6.99
13.88
25.58
44.15
70.42
103.7
148.06
201.11
262.84
334.59

0.7
4.53
5.27
6.22
7.24
8.43
16.89
31.73
53.58
85.95
128.44
182.71
247.71
325.08
423.93

0.8
0.9
4.98
7.31
6.34
12.04
7.97
17.35
9.82
22.78
11.95
28.47
25.34
59.62
46.97 101.36
76.5 158.68
118.39 239.27
175.59 353.19
247.64 501.66
343.69 697.25
452.05 926.69
585.82 1216.77

Table 1.6: 95% Fixed-b critical values of Wald( F) with QS kernel, l = 2
b=0.02
0.04
0.06
0.08
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1

λ=0.1
14.06
26.97
38.59
50.18
62.26
146.13
312.09
658.6
1286.25
2448.94
4375.15
7288.27
11611.96
17823.91

0.2
0.3
0.4
8.45
7.16
6.81
11.83
8.83
7.95
15.95
11.06
9.39
20.7
13.43
11.1
25.97
16.55
13.27
67.5
43.02
35.24
158.42
108.51
87.98
331.17
241.54
196.85
655.18
499.07
420.69
1237.59
916.23
798.85
2209.76 1622.99 1431.74
3760.93 2728.88 2359.81
5871.16
4433.2 3809.97
8876.78 6763.38 5763.21

0.5
6.82
7.88
9.18
10.88
12.82
33.63
80.79
184.84
378.64
722.79
1296.51
2208.22
3646.67
5547.04

46

0.6
0.7
0.8
0.9
6.89
7.56
8.73
14.73
8.08
9.28
12.26
27.58
9.58
11.44
16.41
39.04
11.25
14.14
21.28
50.94
13.42
17.24
26.42
62.99
35.2
44.36
69.12
149.5
87.85
106.46
163.47
314.22
195.45
233.84
339.63
665.6
404.62
478.49
685.69
1357.87
773.38
897.04 1264.11
2526.8
1368.93
1595.1 2240.27
4503.02
2208.18 2662.06 3808.74
7606.67
3495.78 4198.07 6017.23 12269.85
5223.44 6474.23 9157.09 18845.07

Table 1.7: 95% Fixed-b critical values of Wald(S) with Bartlett kernel, l = 1, b1 = b2 = b
b = 0.02
0.04
0.06
0.08
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1

λ = 0.1
4.101
4.313
4.544
4.71
4.933
6.14
7.65
9.16
10.88
12.62
14.39
16.23
18.28
20.31

0.2
4.025
4.207
4.38
4.569
4.773
5.925
7.112
8.422
9.928
11.477
13.215
14.886
16.586
18.425

0.3
0.4
0.5
4.027 4.022 4.033
4.174 4.183 4.202
4.365 4.323 4.334
4.543 4.503 4.517
4.757 4.668 4.691
5.723 5.682 5.571
6.991 6.727 6.728
8.281 8.086 7.931
9.666 9.453 9.387
11.063 10.91 10.886
12.677 12.493 12.541
14.418 14.16 14.231
16.083 15.794 16.021
17.886 17.582 17.792

0.6
4.046
4.195
4.363
4.513
4.702
5.6
6.566
7.835
9.317
10.879
12.409
14.103
15.764
17.521

0.7
0.8
0.9
3.991 4.019 4.115
4.147 4.189 4.267
4.327 4.326
4.46
4.507 4.519 4.621
4.714 4.697 4.862
5.67 5.768 6.107
6.93 6.965
7.51
8.198 8.339 9.141
9.697 9.823 10.774
11.258 11.472 12.538
12.916 13.08 14.292
14.562 14.771 16.041
16.414 16.613 17.992
18.177 18.41 20.008

Table 1.8: 95% Fixed-b critical values of Wald(S) with Bartlett kernel, l = 2, b1 = b2 = b
b=0.02
0.04
0.06
0.08
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1

λ=0.1
0.2
6.416
6.199
6.788
6.557
7.241
6.916
7.695
7.34
8.224
7.774
10.957 10.138
14.159 12.953
17.909 16.119
21.74
19.6
25.701 22.974
29.558 26.374
33.527 29.887
37.692 33.535
41.749 37.265

0.3
6.196
6.459
6.786
7.114
7.461
9.728
12.297
15.067
18.337
21.6
24.714
28.008
31.375
34.985

0.4
6.122
6.404
6.739
7.061
7.395
9.466
11.799
14.606
17.7
20.891
23.974
27.221
30.147
33.507

47

0.5
6.246
6.527
6.851
7.183
7.518
9.526
11.982
14.84
17.746
20.907
24.045
27.304
30.581
33.784

0.6
6.197
6.553
6.867
7.184
7.561
9.587
12.059
14.835
17.997
20.952
23.829
26.967
30.212
33.558

0.7
6.428
6.702
7.06
7.471
7.845
10.041
12.699
15.726
18.904
22.073
25.325
28.709
32.187
35.802

0.8
6.481
6.835
7.22
7.588
8.024
10.526
13.346
16.787
20.126
23.75
27.618
30.806
34.566
38.393

0.9
6.557
6.987
7.422
7.865
8.388
11.256
14.675
18.563
22.678
26.415
30.453
34.488
38.466
42.774

Table 1.9: 95% Fixed-b critical values of Wald(S) with Parzen kernel, l = 1, b1 = b2 = b
b=0.02
0.04
0.06
0.08
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1

λ= 0.1
0.2
4.049
3.982
4.218
4.121
4.378
4.258
4.549
4.407
4.709
4.559
5.673
5.439
6.89
6.457
8.337
7.569
9.952
8.899
11.766 10.479
14.238 12.371
16.803 14.548
19.604
17.11
22.963 19.776

0.3
3.999
4.124
4.258
4.38
4.532
5.326
6.209
7.332
8.604
10.063
11.714
13.621
15.697
18.188

0.4
3.986
4.106
4.236
4.356
4.46
5.242
6.132
7.127
8.408
9.749
11.269
13.113
15.141
17.469

0.5
0.6
0.7
4.002 4.003 3.963
4.136 4.111 4.096
4.251 4.245 4.194
4.353 4.371 4.349
4.509 4.492 4.496
5.188 5.187
5.27
6.057 6.013
6.2
7.094 6.996 7.323
8.21 8.098 8.571
9.586 9.516 10.086
11.267 11.291 11.839
13.106 13.188 13.911
15.394 15.483 16.075
17.794 17.65 18.511

0.8
3.978
4.1
4.245
4.346
4.503
5.307
6.282
7.478
8.742
10.371
12.212
14.335
16.557
19.084

0.9
4.047
4.185
4.347
4.484
4.646
5.582
6.811
8.278
9.905
11.863
14.041
16.533
19.562
22.964

Table 1.10: 95% Fixed-b critical values of Wald(S) with Parzen kernel, l = 2, b1 = b2 = b
b=0.02
0.04
0.06
0.08
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1

λ=0.1
0.2
6.318
6.118
6.593
6.427
6.926
6.723
7.314
7.027
7.734
7.344
9.978
9.293
13.036 11.711
16.699 14.885
21.601 18.733
27.984 23.528
35.876 29.316
45.449 36.522
56.114 44.869
68.506 54.481

0.3
6.118
6,368
6.558
6.83
7.109
8.823
11.071
13.781
17.057
21.279
26.426
32.699
40.393
48.77

0.4
6.064
6.266
6.519
6.796
7.08
8.604
10.663
13.07
16.273
20.453
25.412
31.458
38.288
45.915

48

0.5
6.157
6.425
6.658
6.918
7.194
8.775
10.725
13.217
16.441
20.275
25.186
30.822
37.563
45.317

0.6
6.145
6.377
6.658
6.911
7.169
8.794
10.809
13.365
16.498
20.393
25.205
30.985
37.654
45.607

0.7
6.333
6.587
6.841
7.14
7.438
9.174
11.466
14.299
17.645
22.011
27.459
34.199
41.742
49.919

0.8
6.423
6.688
6.992
7.269
7.597
9.51
12.068
15.271
19.301
24.447
30.67
38.416
47.109
57.401

0.9
6.435
6.782
7.14
7.52
7.923
10.252
13.438
17.304
22.368
28.489
36.487
45.707
57.801
70.667

Table 1.11: 95% Fixed-b critical values of Wald(S) with QS kernel, l = 1, b1 = b2 = b
b=0.02
0.04
0.06
0.08
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1

0.1
0.2
4.176
4.097
4.466
4.324
4.789
4.629
5.109
4.91
5.461
5.239
7.908
7.158
11.14
9.864
15.772 13.486
21.421
18.5
28.285 24.521
37.351 32.408
48.093 41.538
60.429 52.591
75.711 64.723

0.3
0.4
0.5
4.082
4.082 4.098
4.33
4.303 4.299
4.572
4.515 4.542
4.858
4.756 4.792
5.15
5.055 5.045
6.91
6.746 6.723
9.397
9.007 8.873
12.522
12.13 12.109
16.806 16.16 16.533
22.728 21.041 21.838
29.597
27.6 28.351
38.324 35.403 36.689
48.937 44.754 46.413
60.735
55.78 57.464

0.6
4.078
4.298
4.534
4.776
5.031
6.603
8.798
12.24
16.272
21.285
27.798
35.926
46.106
56.974

0.7
4.065
4.281
4.537
4.816
5.095
6.924
9.357
12.867
17.009
22.432
29.577
38.133
48.34
60.451

0.8
4.073
4.301
4.551
4.817
5.121
7.033
9.723
13.339
17.961
24.054
32.03
41.491
52.617
66.712

0.9
4.162
4.418
4.709
5.027
5.376
7.796
11.1
15.716
21.766
29.2
38.786
49.723
62.802
78.41

Table 1.12: 95% Fixed-b critical values of Wald(S) with QS kernel, l = 2, b1 = b2 = b
b=0.02
0.04
0.06
0.08
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1

0.1
6.545
7.194
7.939
8.715
9.665
16.041
27.467
45.53
72.151
108.625
158.481
221.964
304.042
404.653

0.2
6.371
6.916
7.533
8.208
8.968
13.952
22.471
35.924
54.831
82.168
117.545
163.498
220.842
296.357

0.3
0.4
0.5
0.6
0.7
6.319
6.233
6.359
6.326
6.53
6.741
6.662
6.792
6.83
7.014
7.247
7.182
7.315
7.316
7.55
7.811
7.69
7.836
7.875
8.242
8.486
8.306
8.462
8.466
8.922
13.07
12.468
12.481
12.526
13.544
20.156
19.41
19.266
19.249
21.214
31.708
30.315
29.839
30.121
33.261
49.151
45.982
45.157
45.123
50.044
72.852
66.928
66.757
66.24
72.815
103.562
96.266
94.481
93.173 104.123
143.644
133.73 129.628 129.225 144.891
193.648 182.573 174.469 174.765 193.792
256.175 243.928 231.015 229.871 256.899

0.8
6.635
7.19
7.747
8.427
9.2
14.544
23.57
37.764
57.892
86.59
125.598
175.599
242.272
316.291

Table 1.13: 95% Fixed-b critical values of Wald(S) , l = 2, b1 = b2
λ
0.2
0.2
0.2
0.4
0.5
0.5

b1
b2
0.2
0.05
0.5 0.125
1 0.25
1.25
0.83
1
1
2
2

Bartlett kernel
9.2936
14.9354
23.8607
34.864
33.7844
67.5688

49

QS kernel
11.8473
24.8345
59.9221
256.3
231.0145
1677.88

0.9
6.69
7.38
8.162
8.986
9.933
16.558
27.843
46.608
74.435
112.953
162.34
227.359
308.793
412.891

Table 1.14: The Finite Sample Size associated with Wald(S) , l = 2, b1 = b2 = b.

T
50

100

500

DGP A: (θ, ρ, ϕ) = (0.5, 0, 0).
H0 : No Structural Change in both β 1 and β 2 at t = λT, λ = 0.2
b
kernel Inference 0.020 0.040 0.060 0.080 0.100 0.200
Bartlett
Bartlett
QS
QS
Bartlett
Bartlett
QS
QS
Bartlett
Bartlett
QS
QS

Fixed-b
chi square
Fixed-b
chi square
Fixed-b
chi square
Fixed-b
chi square
Fixed-b
chi square
Fixed-b
chi square

0.1784
0.1892
0.1704
0.1888
0.1096
0.1176
0.106
0.1216
0.066
0.0724
0.0628
0.0756

0.1636
0.1916
0.1548
0.194
0.1024
0.1216
0.0936
0.1204
0.064
0.0832
0.064
0.0916

0.1544
0.1948
0.1372
0.1936
0.098
0.1296
0.0916
0.1356
0.0636
0.0924
0.0628
0.1052

0.1444
0.1984
0.1236
0.2012
0.1
0.1428
0.0884
0.1596
0.0632
0.1036
0.062
0.1224

0.1348
0.2008
0.1124
0.2132
0.098
0.1544
0.0896
0.1764
0.064
0.1104
0.062
0.142

0.1256
0.2544
0.1064
0.2988
0.0928
0.2096
0.0892
0.274
0.0632
0.1664
0.0668
0.2284

0.122
0.3088
0.0964
0.3936
0.0888
0.2692
0.0804
0.3688
0.0696
0.23
0.062
0.3304

0.800

0.900

1.000

0.1168
0.5416
0.0864
0.742
0.0864
0.4964
0.0672
0.726
0.0652
0.4664
0.062
0.7204

0.116
0.5724
0.0856
0.7776
0.0868
0.5296
0.0672
0.7676
0.0652
0.5068
0.0628
0.7692

0.116
0.5932
0.0848
0.8084
0.086
0.5596
0.0656
0.7984
0.0652
0.5396
0.0604
0.8024

T

kernel

Inference

0.400

0.500

0.600

b
0.700

50

Bartlett
Bartlett
QS
QS
Bartlett
Bartlett
QS
QS
Bartlett
Bartlett
QS
QS

Fixed-b
chi square
Fixed-b
chi square
Fixed-b
chi square
Fixed-b
chi square
Fixed-b
chi square
Fixed-b
chi square

0.1212
0.364
0.0916
0.4932
0.0876
0.322
0.072
0.4636
0.0696
0.2824
0.0608
0.4216

0.114
0.4152
0.0892
0.5764
0.0836
0.3644
0.072
0.5408
0.0668
0.3316
0.0616
0.5144

0.1144
0.464
0.0856
0.6404
0.0832
0.4116
0.0636
0.6176
0.0704
0.3816
0.062
0.5964

0.1176
0.5052
0.0868
0.7016
0.0876
0.4508
0.0644
0.6808
0.0672
0.4232
0.0628
0.6564

100

500

50

0.300

Table 1.15: The Finite Sample Size associated with Wald(S) , l = 2, b1 = b2 = b.

T
50

100

500

DGP A: (θ, ρ, ϕ) = (0.5, 0, 0).
H0 : No Structural Change in both β 1 and β 2 at t = λT, λ = 0.4
b
kernel Inference 0.020 0.040 0.060 0.080 0.100 0.200
Bartlett
Bartlett
QS
QS
Bartlett
Bartlett
QS
QS
Bartlett
Bartlett
QS
QS

Fixed-b
chi square
Fixed-b
chi square
Fixed-b
chi square
Fixed-b
chi square
Fixed-b
chi square
Fixed-b
chi square

0.1036
0.1088
0.0988
0.1088
0.0796
0.0816
0.0776
0.0812
0.0672
0.0692
0.0656
0.0712

0.0976
0.1096
0.0852
0.1108
0.08
0.092
0.078
0.0972
0.0684
0.0764
0.0676
0.0868

0.088
0.116
0.0848
0.1252
0.078
0.0996
0.0772
0.1084
0.066
0.088
0.0656
0.1

0.092
0.1288
0.0864
0.1376
0.0772
0.1068
0.0776
0.1244
0.0668
0.0968
0.0644
0.1124

0.09
0.136
0.088
0.1588
0.0748
0.12
0.0772
0.1432
0.068
0.1044
0.0668
0.126

0.0852
0.194
0.084
0.2504
0.076
0.1688
0.074
0.2268
0.0644
0.1508
0.0628
0.21

0.0844
0.2492
0.0792
0.3452
0.0716
0.2268
0.0696
0.3124
0.0676
0.2052
0.0572
0.3032

0.800

0.900

1.000

0.0848
0.4708
0.0764
0.6884
0.066
0.4468
0.0616
0.656
0.0664
0.432
0.0596
0.658

0.0848
0.506
0.076
0.7368
0.0664
0.4764
0.0596
0.7004
0.0656
0.4648
0.0588
0.7072

0.0852
0.5372
0.0728
0.776
0.066
0.5028
0.0556
0.7464
0.0648
0.4976
0.0576
0.7476

T

kernel

Inference

0.400

0.500

0.600

b
0.700

50

Bartlett
Bartlett
QS
QS
Bartlett
Bartlett
QS
QS
Bartlett
Bartlett
QS
QS

Fixed-b
chi square
Fixed-b
chi square
Fixed-b
chi square
Fixed-b
chi square
Fixed-b
chi square
Fixed-b
chi square

0.0832
0.306
0.0824
0.42
0.0716
0.272
0.0624
0.3948
0.0588
0.2628
0.056
0.3828

0.0844
0.348
0.0808
0.5048
0.066
0.3268
0.0604
0.4736
0.062
0.3084
0.0644
0.4608

0.0852
0.39
0.0832
0.5756
0.064
0.374
0.0604
0.5372
0.0636
0.3512
0.064
0.5348

0.0828
0.4348
0.0804
0.6368
0.0656
0.4156
0.0608
0.598
0.0656
0.39
0.0612
0.6028

100

500

51

0.300

Table 1.16: The Finite Sample Size associated with Wald(S) , l = 2, b1 = b2 = b.

T
50

100

500

DGP C: (θ, ρ, ϕ) = (0.5, 0.5, 0.5).
H0 : No Structural Change in both β 1 and β 2 at t = λT, λ = 0.2
b
kernel Inference 0.020 0.040 0.060 0.080 0.100 0.200
Bartlett
Bartlett
QS
QS
Bartlett
Bartlett
QS
QS
Bartlett
Bartlett
QS
QS

Fixed-b
chi square
Fixed-b
chi square
Fixed-b
chi square
Fixed-b
chi square
Fixed-b
chi square
Fixed-b
chi square

0.4812
0.4908
0.4712
0.4928
0.3904
0.4032
0.3596
0.3832
0.1892
0.1976
0.1492
0.164

0.424 0.3864
0.4528 0.4348
0.3904 0.346
0.4388 0.4184
0.334 0.286
0.3648 0.334
0.3028 0.2332
0.3516 0.3088
0.124 0.106
0.1476
0.14
0.0956 0.082
0.1244 0.126

0.3548
0.4224
0.3108
0.4092
0.2468
0.3144
0.192
0.2872
0.0948
0.1428
0.076
0.1408

0.336
0.4172
0.2756
0.3972
0.2136
0.302
0.1756
0.2808
0.092
0.1488
0.0716
0.1544

0.2444
0.4068
0.178
0.4156
0.1704
0.3148
0.1172
0.3348
0.08
0.194
0.0708
0.2328

0.2168
0.4428
0.1408
0.4812
0.1504
0.368
0.1044
0.422
0.0804
0.2472
0.0672
0.3396

0.800

0.900

1.000

0.2008
0.6372
0.1024
0.7904
0.144
0.5944
0.082
0.764
0.078
0.4956
0.0644
0.7236

0.2008
0.664
0.0996
0.8248
0.1456
0.6232
0.082
0.8048
0.0804
0.5268
0.0636
0.7648

0.2024
0.6888
0.096
0.8524
0.148
0.6508
0.0804
0.8332
0.0796
0.5588
0.0608
0.7908

T

kernel

Inference

0.400

0.500

0.600

b
0.700

50

Bartlett
Bartlett
QS
QS
Bartlett
Bartlett
QS
QS
Bartlett
Bartlett
QS
QS

Fixed-b
chi square
Fixed-b
chi square
Fixed-b
chi square
Fixed-b
chi square
Fixed-b
chi square
Fixed-b
chi square

0.2084
0.4832
0.1236
0.562
0.1464
0.4184
0.0944
0.5104
0.0832
0.3048
0.0696
0.4332

0.206
0.5308
0.1136
0.6236
0.1452
0.47
0.0868
0.5856
0.08
0.36
0.0676
0.5192

0.2036
0.5668
0.1056
0.6888
0.1476
0.5176
0.0844
0.6468
0.0804
0.4036
0.0668
0.594

0.2048
0.6088
0.1064
0.7476
0.1464
0.5576
0.084
0.71
0.0824
0.4532
0.0664
0.6664

100

500

52

0.300

Table 1.17: The Finite Sample Size associated with Wald(S) , l = 2, b1 = b2 = b.

T
50

100

500

DGP C: (θ, ρ, ϕ) = (0.5, 0.5, 0.5).
H0 : No Structural Change in both β 1 and β 2 at t = λT, λ = 0.4
b
kernel Inference 0.020 0.040 0.060 0.080 0.100 0.200
Bartlett
Bartlett
QS
QS
Bartlett
Bartlett
QS
QS
Bartlett
Bartlett
QS
QS

Fixed-b
chi square
Fixed-b
chi square
Fixed-b
chi square
Fixed-b
chi square
Fixed-b
chi square
Fixed-b
chi square

0.4076
0.416
0.4076
0.422
0.3544
0.362
0.336
0.3492
0.1116
0.1168
0.0856
0.094

0.372
0.3968
0.35
0.3836
0.2464
0.2692
0.208
0.24
0.0856
0.0968
0.0688
0.0888

0.3132
0.3532
0.2572
0.3284
0.2036
0.2328
0.1528
0.2076
0.0776
0.0956
0.068
0.0936

0.2564
0.328
0.2184
0.2956
0.1712
0.2192
0.1296
0.1992
0.0764
0.1016
0.0688
0.1044

0.2368
0.3092
0.1896
0.284
0.1528
0.2168
0.1184
0.2012
0.0772
0.1072
0.0688
0.1204

0.1776
0.3096
0.1264
0.3272
0.122
0.2364
0.0904
0.2596
0.0736
0.162
0.0636
0.202

0.158
0.3584
0.108
0.4092
0.1164
0.2884
0.0836
0.342
0.07
0.2124
0.0576
0.2896

0.800

0.900

1.000

0.1512
0.5744
0.0992
0.7484
0.1124
0.506
0.0728
0.6884
0.0656
0.4308
0.0608
0.6616

0.1556
0.6032
0.0952
0.788
0.112
0.5388
0.0736
0.7252
0.0692
0.4676
0.0592
0.7016

0.1548
0.6356
0.096
0.8152
0.1116
0.5744
0.0728
0.7696
0.07
0.5052
0.0588
0.7372

T

kernel

Inference

0.400

0.500

0.600

b
0.700

50

Bartlett
Bartlett
QS
QS
Bartlett
Bartlett
QS
QS
Bartlett
Bartlett
QS
QS

Fixed-b
chi square
Fixed-b
chi square
Fixed-b
chi square
Fixed-b
chi square
Fixed-b
chi square
Fixed-b
chi square

0.1536
0.4104
0.1004
0.4924
0.1156
0.3372
0.0792
0.422
0.0664
0.2548
0.0564
0.374

0.1512
0.4628
0.1012
0.5732
0.1072
0.3864
0.0784
0.5052
0.066
0.2988
0.0572
0.4588

0.1476
0.5112
0.0996
0.6324
0.1056
0.428
0.0748
0.5776
0.068
0.348
0.0576
0.5344

0.1492
0.5428
0.0992
0.6976
0.1108
0.472
0.0748
0.6392
0.0668
0.388
0.0588
0.602

100

500

53

0.300

Table 1.18: The Finite Sample Size associated with Wald(S) , l = 2, b1 = b2 = b.

T
50

100

500

DGP D: (θ, ρ, ϕ) = (0.8, 0.5, 0.5).
H0 : No Structural Change in both β 1 and β 2 at t = λT, λ = 0.2
b
kernel Inference 0.020 0.040 0.060 0.080 0.100 0.200
Bartlett
Bartlett
QS
QS
Bartlett
Bartlett
QS
QS
Bartlett
Bartlett
QS
QS

Fixed-b
chi square
Fixed-b
chi square
Fixed-b
chi square
Fixed-b
chi square
Fixed-b
chi square
Fixed-b
chi square

0.5204
0.5296
0.5164
0.5312
0.4504
0.4624
0.4264
0.444
0.2132
0.2248
0.1668
0.1828

0.462
0.4936
0.426
0.4756
0.3984
0.4256
0.3676
0.4124
0.144
0.1676
0.108
0.1428

0.4212
0.4712
0.3788
0.4548
0.3492
0.3948
0.292
0.3688
0.1212
0.1576
0.0948
0.1444

0.3956
0.46
0.3428
0.4436
0.3044
0.3728
0.2464
0.3464
0.1084
0.1632
0.0872
0.1604

0.3728
0.4524
0.3052
0.4356
0.2728
0.3608
0.2184
0.3404
0.1028
0.1692
0.0808
0.1772

0.2832
0.4472
0.204
0.4608
0.2152
0.38
0.1528
0.39
0.0948
0.218
0.0804
0.2624

0.2532
0.4836
0.1676
0.5188
0.1944
0.4168
0.1292
0.4588
0.0916
0.28
0.0804
0.3588

0.800

0.900

1.000

0.2284
0.6768
0.12
0.8076
0.1828
0.6328
0.102
0.7836
0.0908
0.5124
0.0692
0.7344

0.2308
0.7028
0.1188
0.8356
0.1828
0.6616
0.1008
0.8232
0.092
0.5476
0.0692
0.7744

0.2308
0.726
0.118
0.8644
0.1836
0.6908
0.0964
0.8488
0.0928
0.5816
0.0676
0.8136

T

kernel

Inference

0.400

0.500

0.600

b
0.700

50

Bartlett
Bartlett
QS
QS
Bartlett
Bartlett
QS
QS
Bartlett
Bartlett
QS
QS

Fixed-b
chi square
Fixed-b
chi square
Fixed-b
chi square
Fixed-b
chi square
Fixed-b
chi square
Fixed-b
chi square

0.2392
0.5252
0.1436
0.5948
0.1884
0.4624
0.1184
0.5432
0.0936
0.3304
0.0772
0.4512

0.2328
0.5744
0.1332
0.656
0.1844
0.5084
0.11
0.6216
0.0912
0.3836
0.0768
0.5352

0.2336
0.6128
0.1268
0.716
0.1844
0.5572
0.1064
0.6868
0.0916
0.4296
0.0752
0.6156

0.2292
0.6436
0.1244
0.7652
0.182
0.596
0.1016
0.744
0.0904
0.474
0.0712
0.6836

100

500

54

0.300

Table 1.19: The Finite Sample Size associated with Wald(S) , l = 2, b1 = b2 = b.

T
50

100

500

DGP D: (θ, ρ, ϕ) = (0.8, 0.5, 0.5).
H0 : No Structural Change in both β 1 and β 2 at t = λT, λ = 0.4
b
kernel Inference 0.020 0.040 0.060 0.080 0.100 0.200
Bartlett
Bartlett
QS
QS
Bartlett
Bartlett
QS
QS
Bartlett
Bartlett
QS
QS

Fixed-b
chi square
Fixed-b
chi square
Fixed-b
chi square
Fixed-b
chi square
Fixed-b
chi square
Fixed-b
chi square

0.4664
0.4768
0.4668
0.4856
0.4148
0.422
0.3964
0.4084
0.1368
0.142
0.0992
0.1092

0.4304
0.4528
0.406
0.4396
0.292
0.3128
0.2464
0.2796
0.1
0.1144
0.0812
0.0992

0.3632
0.406
0.3092
0.3708
0.2368
0.2732
0.1772
0.2416
0.09
0.1152
0.0768
0.1088

0.3124
0.3692
0.2536
0.3392
0.2008
0.2552
0.1532
0.2256
0.0896
0.1228
0.0748
0.1236

0.2828
0.3512
0.2188
0.326
0.1868
0.2492
0.1408
0.2288
0.0848
0.1292
0.0736
0.1396

0.2144
0.3652
0.1576
0.3696
0.1528
0.2784
0.1168
0.296
0.0744
0.1764
0.0648
0.218

0.2064
0.4104
0.142
0.4496
0.1452
0.328
0.104
0.3792
0.074
0.2296
0.0608
0.3068

0.800

0.900

1.000

0.1908
0.6244
0.1088
0.7688
0.1344
0.5528
0.0876
0.7196
0.074
0.45
0.0648
0.6716

0.1916
0.6524
0.106
0.8028
0.1364
0.58
0.0868
0.7588
0.0756
0.482
0.0656
0.7188

0.1904
0.6808
0.1052
0.8328
0.1372
0.608
0.0864
0.7928
0.0756
0.5128
0.0648
0.7568

T

kernel

Inference

0.400

0.500

0.600

b
0.700

50

Bartlett
Bartlett
QS
QS
Bartlett
Bartlett
QS
QS
Bartlett
Bartlett
QS
QS

Fixed-b
chi square
Fixed-b
chi square
Fixed-b
chi square
Fixed-b
chi square
Fixed-b
chi square
Fixed-b
chi square

0.1956
0.4584
0.132
0.5276
0.1404
0.376
0.0968
0.4664
0.0708
0.2796
0.0612
0.4004

0.1916
0.5012
0.1236
0.6188
0.1336
0.4192
0.0924
0.5576
0.0724
0.3228
0.0572
0.4768

0.1872
0.5492
0.1184
0.674
0.1336
0.4732
0.0924
0.624
0.0736
0.3732
0.058
0.5536

0.1888
0.59
0.1112
0.7256
0.1344
0.5148
0.0912
0.6752
0.0744
0.4096
0.0604
0.6176

100

500

55

0.300

Table 1.20: The Finite Sample Size of the Tests Based on Wald(S) and Wald( F) ,
l = 2, b1 = λb , b2 =

b
1− λ ,

T = 50, H0 : No Structural Change in both β 1 and β 2 at t = λT
Bartlett kernel
Wald( F)
Wald( F)
Fixed-b Fixed-b chi square
0.1184
0.1168
0.2284
0.11
0.112
0.346
0.1132
0.1116
0.4924

Wald(S)
DGP λ
A
.2

b
.04
.1
.2

b1
b2
.2
.05
.5 .125
1.0 .25

.04
.1
.2

QS kernel
Wald( F)
Wald( F)
Fixed-b Fixed-b chi square
0.0968
0.098
0.2664
0.0992
0.0928
0.464
0.0952
0.0932
0.6596

Wald(S)

B

.2

.2
.05
.5 .125
1.0 .25

0.2444
0.1704
0.1548

0.244
0.17
0.1544

0.3804
0.436
0.558

0.1868
0.1168
0.096

0.1884
0.1108
0.0932

0.3904
0.4936
0.6704

C

.2 .04 .2
.05
.1
.5 .125
.2 1.0 .25

0.2948
0.1932
0.1668

0.2928
0.1952
0.1676

0.4244
0.4592
0.5728

0.2092
0.1216
0.0924

0.2116
0.1184
0.0928

0.4152
0.506
0.6668

D

.2 .04
.1
.2

.2
.05
.5 .125
1.0 .25

0.3296
0.2312
0.2024

0.3292
0.2304
0.2036

0.47
0.514
0.6164

0.2428
0.1464
0.1156

0.2444
0.1416
0.1168

0.4596
0.5552
0.6984

E

.2

.2
.05
.5 .125
1.0 .25

0.5672
0.3696
0.2768

0.5664
0.3688
0.284

0.6812
0.6444
0.682

0.4576
0.2148
0.1428

0.46
0.2128
0.1384

0.6612
0.6412
0.712

F

.2

.04
.1
.2

.04 .2
.05
0.582
0.5804
0.6948
0.4816
0.4872
0.6732
.1
.5 .125
0.3924
0.3928
0.6728
0.2492
0.2452
0.6684
.2 1.0 .25
0.3064
0.3136
0.7064
0.1696
0.1632
0.7364
Note: The DGP labels are given by A: (θ, ρ, ϕ) = (0.5, 0.0, 0.0), B: (θ, ρ, ϕ) = (0.5, 0.5, 0.0),
C: (θ, ρ, ϕ) = (0.5, 0.5, 0.5), D: (θ, ρ, ϕ) = (0.8, 0.5, 0.5), E: (θ, ρ, ϕ) = (0.8, 0.9, 0.5), and
F: (θ, ρ, ϕ) = (0.9, 0.9, 0.9).

56

Table 1.21: The Finite Sample Size of the Tests Based on Wald(S) and Wald( F) ,
l = 2, b1 = λb , b2 =

b
1− λ ,

T = 100, H0 : No Structural Change in both β 1 and β 2 at t = λT
Bartlett kernel
Wald( F)
Wald( F)
Fixed-b Fixed-b chi square
0.0928
0.092
0.1924
0.0876
0.0884
0.3044
0.0868
0.086
0.4472

Wald(S)
DGP λ
A
.2

b
.04
.1
.2

b1
b2
.2
.05
.5 .125
1.0 .25

.04
.1
.2

QS kernel
Wald( F)
Wald( F)
Fixed-b Fixed-b chi square
0.0872
0.0892
0.2408
0.08
0.0796
0.43
0.082
0.0784
0.6308

Wald(S)

B

.2

.2
.05
.5 .125
1.0 .25

0.1672
0.1264
0.1168

0.1668
0.1312
0.1152

0.2916
0.3872
0.5144

0.1204
0.088
0.0768

0.122
0.086
0.0716

0.3004
0.4556
0.6344

C

.2 .04 .2
.05
.1
.5 .125
.2 1.0 .25

0.1816
0.1396
0.1244

0.1828
0.1376
0.1264

0.3132
0.4028
0.5288

0.1316
0.0904
0.0792

0.1344
0.092
0.0812

0.3124
0.4624
0.6424

D

.2 .04
.1
.2

.2
.05
.5 .125
1.0 .25

0.2312
0.1772
0.1624

0.2296
0.1784
0.1636

0.3792
0.4528
0.58

0.1636
0.114
0.1116

0.168
0.1108
0.106

0.3696
0.5044
0.6892

E

.2

.2
.05
.5 .125
1.0 .25

0.4656
0.276
0.2192

0.4672
0.28
0.23

0.6104
0.5848
0.6472

0.3404
0.1476
0.1048

0.3428
0.1448
0.1032

0.5716
0.586
0.6908

F

.2

.04
.1
.2

.04 .2
.05
0.4864
0.4872
0.6308
0.3588
0.364
0.5996
.1
.5 .125
0.3008
0.3036
0.618
0.1764
0.1748
0.6156
.2 1.0 .25
0.25
0.2524
0.6784
0.1292
0.1232
0.7152
Note: The DGP labels are given by A: (θ, ρ, ϕ) = (0.5, 0.0, 0.0), B: (θ, ρ, ϕ) = (0.5, 0.5, 0.0),
C: (θ, ρ, ϕ) = (0.5, 0.5, 0.5), D: (θ, ρ, ϕ) = (0.8, 0.5, 0.5), E: (θ, ρ, ϕ) = (0.8, 0.9, 0.5), and
F: (θ, ρ, ϕ) = (0.9, 0.9, 0.9).

57

Table 1.22: The Finite Sample Size of the Tests Based on Wald(S) and Wald( F) ,
l = 2, b1 = λb , b2 =

b
1− λ ,

T = 500, H0 : No Structural Change in both β 1 and β 2 at t = λT
Bartlett kernel
Wald( F)
Wald( F)
Fixed-b Fixed-b chi square
0.0676
0.0672
0.1468
0.066
0.0628
0.274
0.0632
0.06
0.4128

Wald(S)
DGP λ
A
.2

b
.04
.1
.2

b1
b2
.2
.05
.5 .125
1.0 .25

.04
.1
.2

QS kernel
Wald( F)
Wald( F)
Fixed-b Fixed-b chi square
0.0628
0.0636
0.1948
0.0592
0.0616
0.3872
0.0588
0.0604
0.6104

Wald(S)

B

.2

.2
.05
.5 .125
1.0 .25

0.0788
0.0768
0.078

0.078
0.076
0.076

0.174
0.2932
0.432

0.0676
0.0664
0.0668

0.0688
0.0616
0.0624

0.1996
0.3904
0.6008

C

.2 .04 .2
.05
.1
.5 .125
.2 1.0 .25

0.0816
0.0724
0.0748

0.0816
0.0732
0.0756

0.1804
0.2928
0.4324

0.0724
0.0648
0.0604

0.0732
0.0632
0.0644

0.202
0.3908
0.6028

D

.2 .04
.1
.2

.2
.05
.5 .125
1.0 .25

0.0976
0.0888
0.0892

0.096
0.0864
0.0884

0.2032
0.3212
0.4556

0.0788
0.08
0.0732

0.0796
0.078
0.0732

0.228
0.4216
0.62

E

.2

.2
.05
.5 .125
1.0 .25

0.1964
0.1316
0.122

0.1972
0.1308
0.1208

0.3256
0.3876
0.5104

0.1236
0.0804
0.0788

0.1276
0.0772
0.0768

0.3092
0.436
0.6228

F

.2

.04
.1
.2

.04 .2
.05
0.2196
0.2204
0.3652
0.142
0.1448
0.3468
.1
.5 .125
0.1568
0.1564
0.4344
0.098
0.0944
0.4756
.2 1.0 .25
0.1432
0.1436
0.548
0.0884
0.0836
0.6524
Note: The DGP labels are given by A: (θ, ρ, ϕ) = (0.5, 0.0, 0.0), B: (θ, ρ, ϕ) = (0.5, 0.5, 0.0),
C: (θ, ρ, ϕ) = (0.5, 0.5, 0.5), D: (θ, ρ, ϕ) = (0.8, 0.5, 0.5), E: (θ, ρ, ϕ) = (0.8, 0.9, 0.5), and
F: (θ, ρ, ϕ) = (0.9, 0.9, 0.9).

58

Table 1.23: The Finite Sample Size of the Tests Based on Wald(S) and Wald( F) ,
l = 2, b1 = λb , b2 =

DGP
A

b
.5
1.0
.4 .5
.2 .5
1.0
λ
.5

b
1− λ ,

T = 50, H0 : No Structural Change in both β 1 and β 2 at t = λT

Bartlett kernel
Wald(S) Wald( F)
Wald( F)
b1
b2
Fixed-b Fixed-b chi square
1.0
1.0
0.0888
0.0856
0.552
2.0
2.0
0.0888
0.0804
0.7256
1.25 .83
0.0828
0.0824
0.5572
2.5 .625
0.11
0.1116
0.6936
5.0 1.25
0.1116
0.1032
0.8172

QS kernel
Wald(S) Wald( F)
Wald( F)
Fixed-b Fixed-b chi square
0.0672
0.068
0.808
0.0644
0.0636
0.9396
0.0724
0.0716
0.8072
0.0828
0.08
0.8828
0.0744
0.0628
0.9652

B

.5

.5
1.0
.4 .5
.2 .5
1.0

1.0
1.0
2.0
2.0
1.25 .83
2.5 .625
5.0 1.25

0.144
0.144
0.1384
0.15
0.1492

0.1476
0.1328
0.1316
0.1456
0.1376

0.6348
0.7748
0.636
0.736
0.8504

0.0964
0.0832
0.0876
0.0816
0.0832

0.0844
0.062
0.0784
0.076
0.0744

0.824
0.9464
0.8236
0.882
0.9648

C

.5

.5
1.0
.4 .5
.2 .5
1.0

1.0
1.0
2.0
2.0
1.25 .83
2.5 .625
5.0 1.25

0.1516
0.1516
0.1492
0.1564
0.1624

0.1548
0.1412
0.148
0.1596
0.146

0.6508
0.7912
0.6516
0.7412
0.8592

0.09
0.0848
0.092
0.0892
0.086

0.0864
0.0712
0.0844
0.08
0.082

0.8332
0.9528
0.8368
0.8752
0.9692

D

.5

.5
1.0
.4 .5
.2 .5
1.0

1.0
1.0
2.0
2.0
1.25 .83
2.5 .625
5.0 1.25

0.192
0.192
0.19
0.1908
0.1908

0.1944
0.182
0.1852
0.1928
0.1756

0.6788
0.8192
0.694
0.7792
0.8756

0.1124
0.1012
0.1056
0.102
0.0976

0.1032
0.0816
0.0976
0.0912
0.0824

0.8592
0.9588
0.8504
0.9008
0.9788

E

.5

1.0
1.0
2.0
2.0
1.25 .83
2.5 .625
5.0 1.25

0.3756
0.3756
0.354
0.25
0.2564

0.3856
0.3608
0.354
0.2512
0.2412

0.7872
0.872
0.7876
0.7972
0.8752

0.2044
0.1724
0.1728
0.1108
0.1148

0.174
0.132
0.156
0.0964
0.0892

0.8788
0.964
0.8896
0.8852
0.9692

F

.5

.5
1.0
.4 .5
.2 .5
1.0

.5
1.0
1.0
0.4008
0.4044
0.7988
0.2216
0.1904
0.8956
1.0 2.0
2.0
0.4008
0.388
0.8816
0.1908
0.136
0.97
.4 .5 1.25 .83
0.3832
0.3796
0.8168
0.186
0.1608
0.9032
.2 .5
2.5 .625
0.2876
0.2836
0.8148
0.1284
0.1144
0.8984
1.0 5.0 1.25
0.2892
0.2756
0.8948
0.1272
0.0972
0.974
Note: The DGP labels are given by A: (θ, ρ, ϕ) = (0.5, 0.0, 0.0), B: (θ, ρ, ϕ) = (0.5, 0.5, 0.0),
C: (θ, ρ, ϕ) = (0.5, 0.5, 0.5), D: (θ, ρ, ϕ) = (0.8, 0.5, 0.5), E: (θ, ρ, ϕ) = (0.8, 0.9, 0.5), and
F: (θ, ρ, ϕ) = (0.9, 0.9, 0.9).

59

Table 1.24: The Finite Sample Size of the Tests Based on Wald(S) and Wald( F) ,
l = 2, b1 = λb , b2 =

DGP
A

b
.5
1.0
.4 .5
.2 .5
1.0
λ
.5

b
1− λ ,

T = 100, H0 : No Structural Change in both β 1 and β 2 at t = λT

Wald(S)
b1
b2
Fixed-b
1.0
1.0
0.0668
2.0
2.0
0.0668
1.25 .83
0.0664
2.5 .625 0.0832
5.0 1.25 0.0872

Bartlett kernel
Wald( F)
Wald( F)
Fixed-b chi square
0.058
0.5344
0.0568
0.7052
0.0664
0.5272
0.0832
0.666
0.076
0.7996

QS kernel
Wald(S) Wald( F)
Wald( F)
Fixed-b Fixed-b chi square
0.0544
0.056
0.7856
0.052
0.054
0.9316
0.0556
0.052
0.7896
0.0688
0.0684
0.8724
0.0676
0.0628
0.9668

B

.5

.5
1.0
.4 .5
.2 .5
1.0

1.0
1.0
2.0
2.0
1.25 .83
2.5 .625
5.0 1.25

0.0856
0.0856
0.1004
0.1108
0.1144

0.088
0.0784
0.1008
0.1092
0.1048

0.5776
0.7428
0.5844
0.6988
0.8288

0.0604
0.0576
0.0692
0.0712
0.0724

0.0636
0.0508
0.0668
0.0716
0.0656

0.804
0.9412
0.8012
0.8716
0.9672

C

.5

.5
1.0
.4 .5
.2 .5
1.0

1.0
1.0
2.0
2.0
1.25 .83
2.5 .625
5.0 1.25

0.0988
0.0988
0.1112
0.1228
0.1216

0.0956
0.0852
0.108
0.1228
0.1076

0.5816
0.7492
0.5892
0.7184
0.8356

0.0648
0.0616
0.07
0.0816
0.0764

0.0588
0.0552
0.0696
0.0772
0.0664

0.8076
0.9428
0.8028
0.872
0.97

D

.5

.5
1.0
.4 .5
.2 .5
1.0

1.0
1.0
2.0
2.0
1.25 .83
2.5 .625
5.0 1.25

0.1168
0.1168
0.134
0.1644
0.166

0.1228
0.1104
0.1312
0.162
0.152

0.61
0.7588
0.6272
0.7584
0.856

0.082
0.078
0.0804
0.0872
0.0832

0.074
0.064
0.0804
0.084
0.0696

0.8096
0.9384
0.8204
0.8912
0.9696

E

.5

1.0
1.0
2.0
2.0
1.25 .83
2.5 .625
5.0 1.25

0.2756
0.2756
0.2852
0.2128
0.216

0.2852
0.2596
0.2792
0.2112
0.2

0.742
0.8532
0.7604
0.7768
0.8808

0.1412
0.126
0.1364
0.0948
0.0992

0.1304
0.0932
0.1276
0.088
0.0812

0.87
0.9584
0.8852
0.8852
0.974

F

.5

.5
1.0
.4 .5
.2 .5
1.0

.5
1.0
1.0
0.3044
0.3028
0.7664
0.1708
0.154
0.8764
1.0 2.0
2.0
0.3044
0.2872
0.8612
0.1596
0.1192
0.96
.4 .5 1.25 .83
0.2936
0.2948
0.776
0.1528
0.136
0.884
.2 .5
2.5 .625 0.2384
0.2408
0.7892
0.1152
0.0976
0.8864
1.0 5.0 1.25 0.2408
0.2252
0.8804
0.1168
0.0872
0.9728
Note: The DGP labels are given by A: (θ, ρ, ϕ) = (0.5, 0.0, 0.0), B: (θ, ρ, ϕ) = (0.5, 0.5, 0.0),
C: (θ, ρ, ϕ) = (0.5, 0.5, 0.5), D: (θ, ρ, ϕ) = (0.8, 0.5, 0.5), E: (θ, ρ, ϕ) = (0.8, 0.9, 0.5), and
F: (θ, ρ, ϕ) = (0.9, 0.9, 0.9).

60

Table 1.25: The Finite Sample Size of the Tests Based on Wald(S) and Wald( F) ,
l = 2, b1 = λb , b2 =

DGP
A

b
.5
1.0
.4 .5
.2 .5
1.0
λ
.5

b
1− λ ,

T = 500, H0 : No Structural Change in both β 1 and β 2 at t = λT

Wald(S)
b1
b2
Fixed-b
1.0
1.0
0.0488
2.0
2.0
0.0488
1.25 .83
0.064
2.5 .625 0.0664
5.0 1.25 0.0636

Bartlett kernel
Wald( F)
Wald( F)
Fixed-b chi square
0.0532
0.4988
0.0428
0.6856
0.0612
0.5144
0.0676
0.6488
0.056
0.7952

Wald(S)
Fixed-b
0.0532
0.0528
0.0568
0.056
0.0556

QS kernel
Wald( F)
Wald( F)
Fixed-b chi square
0.0496
0.7708
0.0492
0.9264
0.058
0.7848
0.058
0.8664
0.0568
0.958

B

.5

.5
1.0
.4 .5
.2 .5
1.0

1.0
1.0
2.0
2.0
1.25 .83
2.5 .625
5.0 1.25

0.06
0.06
0.0692
0.0752
0.0724

0.0652
0.0524
0.0688
0.0764
0.0652

0.5024
0.6948
0.5148
0.6552
0.7932

0.0548
0.05
0.0576
0.054
0.0564

0.0472
0.0496
0.0564
0.0592
0.0576

0.776
0.9312
0.784
0.8624
0.9612

C

.5

.5
1.0
.4 .5
.2 .5
1.0

1.0
1.0
2.0
2.0
1.25 .83
2.5 .625
5.0 1.25

0.0596
0.0596
0.0668
0.076
0.0736

0.0632
0.0556
0.0688
0.076
0.066

0.514
0.6884
0.528
0.656
0.798

0.0552
0.0492
0.0572
0.0556
0.0584

0.0488
0.0428
0.06
0.0584
0.0516

0.7744
0.9308
0.7868
0.858
0.9572

D

.5

.5
1.0
.4 .5
.2 .5
1.0

1.0
1.0
2.0
2.0
1.25 .83
2.5 .625
5.0 1.25

0.0652
0.0652
0.0744
0.0868
0.0884

0.0644
0.0568
0.07
0.086
0.0804

0.5264
0.7
0.534
0.6704
0.8148

0.0528
0.0504
0.0604
0.066
0.0612

0.0528
0.0464
0.0548
0.0696
0.058

0.7824
0.9364
0.792
0.8728
0.9596

E

.5

1.0
1.0
2.0
2.0
1.25 .83
2.5 .625
5.0 1.25

0.1032
0.1032
0.11
0.1188
0.1196

0.1024
0.0964
0.1076
0.1188
0.1056

0.5728
0.7288
0.582
0.7032
0.8352

0.0724
0.0692
0.0728
0.0796
0.0708

0.0676
0.0648
0.0652
0.0652
0.0624

0.7968
0.9304
0.794
0.8704
0.9624

F

.5

.5
1.0
.4 .5
.2 .5
1.0

.5
1.0
1.0
0.12
0.1216
0.5904
0.0784
0.07
0.8052
1.0 2.0
2.0
0.12
0.1116
0.7452
0.0688
0.06
0.932
.4 .5 1.25 .83
0.1224
0.1212
0.6112
0.0712
0.0692
0.812
.2 .5
2.5 .625 0.1416
0.1424
0.7324
0.0868
0.0796
0.872
1.0 5.0 1.25
0.138
0.1288
0.8432
0.0804
0.0732
0.9692
Note: The DGP labels are given by A: (θ, ρ, ϕ) = (0.5, 0.0, 0.0), B: (θ, ρ, ϕ) = (0.5, 0.5, 0.0),
C: (θ, ρ, ϕ) = (0.5, 0.5, 0.5), D: (θ, ρ, ϕ) = (0.8, 0.5, 0.5), E: (θ, ρ, ϕ) = (0.8, 0.9, 0.5), and
F: (θ, ρ, ϕ) = (0.9, 0.9, 0.9).

61

Table 1.26: Fixed-b 95% Critical Values of Wald( F) Unknown Break Date, Bartlett kernel,
l =2
b
0.02
0.04
0.06
0.08
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1

SupW
30.293
48.447
61.976
73.862
84.848
138.92
193.94
254.14
313.06
374.36
433.71
491.83
549.63
608.99

= 0.05
MeanW
4.861
5.9489
7.0183
8.001
8.973
14.018
19.113
24.443
29.999
35.304
40.902
46.205
51.450
57.142

ExpW
9.588
18.1938
24.816
30.656
36.109
63.068
90.408
120.71
149.85
180.46
210.22
239.08
268.05
297.78

= 0.1
MeanW
4.235
4.974
5.729
6.496
7.278
11.323
15.596
20.009
24.565
29.202
33.625
38.016
42.238
46.623

SupW
18.230
26.034
33.172
39.957
46.263
76.971
109.11
142.31
176.51
212.05
245.66
279.65
311.37
344.26

ExpW
5.051
8.173
11.483
14.695
17.653
32.706
48.657
65.120
82.037
99.596
116.32
133.32
149.22
165.51

SupW
13.542
16.313
19.496
22.812
26.323
46.122
67.262
89.241
111.18
134.00
153.93
173.96
192.52
212.76

= 0.2
MeanW
3.263
3.688
4.162
4.617
5.146
8.052
11.216
14.464
17.912
21.386
24.666
27.702
30.670
33.936

ExpW
3.539
4.654
5.967
7.364
8.998
18.156
28.446
39.161
49.818
61.205
70.991
81.134
90.145
100.36

Table 1.27: Fixed-b 95% Critical Values of Wald( F) Unknown Break Date, QS kernel, l =2
b SupW
0.02
64.848
0.04
122.00
0.06
161.74
0.08
207.65
0.1
257.31
0.2
832.93
0.3
3339.8
0.4
13932
0.5
47253
0.6 136211
0.7 328737
0.8 719812
0.9 1444833
1 2647520

= 0.05
MeanW
5.678
8.102
10.617
13.202
16.139
40.501
99.975
239.82
537.89
1115.4
2170.5
3982.4
7015.5
11566

ExpW
SupW
26.200
24.831
54.483
46.350
74.329
68.158
97.163
91.258
122.02
118.67
409.56
452.33
1663.0
2055.3
6959.4
8975.9
23620
31752
68099
91828
164361
224463
359899
488008
722409
970172
1323754 1829406

62

= 0.1
MeanW
4.641
6.059
7.630
9.461
11.671
30.155
77.012
185.18
411.53
850.69
1674.7
3100.4
5395.5
9072.3

ExpW
SupW
7.548
15.051
17.433
20.670
28.148
28.305
39.595
38.905
53.066
52.759
219.29
240.65
1020.8
1144.7
4481.1
4771.4
15869
16684
45907
49492
112225 128234
243997 283267
485079 565285
914696 1062685

= 0.2
MeanW
3.458
4.205
5.060
6.143
7.491
19.924
51.677
124.22
276.98
580.43
1140.0
2099.3
3626.6
5951.4

ExpW
4.111
6.401
9.666
14.409
20.987
113.55
565.45
2378.8
8334.9
24740
64110
141627
282635
531336

Table 1.28: The Finite Sample Size of the SupW ( F) Test with 5% Nominal Size
H0 : No Structural Change in both β 1 and β 2 , DGP A: (θ, ρ, ϕ) = (0.5, 0.0, 0.0)
T=100
kernel
b = 0.02
0.04
0.06
0.08
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
T=500
kernel
b = 0.02
0.04
0.06
0.08
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
T=1000
kernel
b = 0.02
0.04
0.06
0.08
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1

= 0.05
Fixed-b
Andrews
Bartlett
QS
Bartlett
0.331 0.184
0.721
0.290
0.191
0.847
0.284
0.212
0.898
0.284
0.211
0.936
0.277
0.212
0.954
0.262
0.153
0.992
0.264
0.084
0.998
0.259
0.048
1.000
0.250
0.036
1.000
0.250
0.026
1.000
0.250
0.024
1.000
0.243
0.020
1.000
0.248
0.015
1.000
0.253 0.012
1.000

= 0.1
Fixed-b
Andrews
Bartlett
QS
Bartlett
0.160 0.111
0.347
0.136 0.096
0.494
0.131 0.104
0.601
0.127 0.108
0.696
0.124 0.106
0.756
0.125 0.081
0.908
0.124 0.050
0.962
0.123 0.031
0.983
0.120 0.023
0.994
0.118 0.017
0.997
0.120 0.016
0.999
0.119 0.014
1.000
0.123 0.010
1.000
0.121 0.007
1.000

= 0.2
Fixed-b
Andrews
Bartlett
QS
Bartlett
0.104 0.099
0.163
0.108 0.098
0.241
0.103 0.082
0.308
0.098 0.078
0.384
0.094 0.071
0.447
0.084 0.056
0.678
0.081 0.043
0.812
0.081 0.028
0.885
0.081 0.019
0.923
0.080 0.017
0.950
0.080 0.015
0.968
0.079 0.012
0.984
0.079 0.010
0.989
0.080 0.009
0.994

= 0.05
Fixed-b
Andrews
Bartlett
QS
Bartlett
0.093 0.084
0.472
0.086
0.080
0.704
0.086
0.082
0.810
0.087
0.079
0.865
0.084
0.077
0.904
0.078
0.063
0.983
0.081
0.055
0.995
0.078
0.047
0.999
0.083
0.046
1.000
0.081
0.034
1.000
0.080
0.026
1.000
0.078
0.025
1.000
0.079
0.026
1.000
0.081 0.026
1.000

= 0.1
Fixed-b
Andrews
Bartlett
QS
Bartlett
0.069 0.070
0.217
0.069 0.063
0.376
0.069 0.063
0.507
0.065 0.058
0.607
0.064 0.057
0.679
0.058 0.052
0.878
0.057 0.052
0.946
0.056 0.041
0.974
0.057 0.038
0.986
0.059 0.034
0.994
0.062 0.028
0.998
0.058 0.027
1.000
0.058 0.025
1.000
0.061 0.024
1.000

= 0.2
Fixed-b
Andrews
Bartlett
QS
Bartlett
0.062 0.062
0.111
0.060 0.060
0.179
0.057 0.058
0.247
0.060 0.056
0.315
0.062 0.056
0.381
0.054 0.044
0.641
0.061 0.052
0.786
0.054 0.050
0.865
0.052 0.042
0.908
0.050 0.041
0.935
0.055 0.035
0.959
0.054 0.031
0.977
0.050 0.030
0.987
0.055 0.028
0.992

= 0.05
Fixed-b
Andrews
Bartlett
QS
Bartlett
0.0772 0.0784
0.4132
0.078
0.072
0.6584
0.0784 0.0712
0.7832
0.0756 0.0592
0.8528
0.0696 0.0664
0.8956
0.0704 0.0584
0.9804
0.07 0.0472
0.994
0.0696 0.0448
0.9984
0.0716 0.0468
1
0.072 0.0456
1
0.0724 0.0472
1
0.0664 0.0436
1
0.0696
0.04
1
0.0708
0.04
1

= 0.1
Fixed-b
Andrews
Bartlett
QS
Bartlett
0.056 0.0512
0.1932
0.0544 0.0536
0.3664
0.05 0.052
0.4984
0.0464 0.0484
0.5956
0.0496 0.0532
0.6788
0.05 0.0556
0.866
0.0488 0.0476
0.934
0.0432 0.0464
0.9688
0.052 0.0452
0.9852
0.0472 0.0484
0.9912
0.046 0.0492
0.996
0.054 0.0484
0.9992
0.0508 0.0456
0.9996
0.0524 0.044
1

= 0.2
Fixed-b
Andrews
Bartlett
QS
Bartlett
0.0592 0.0572
0.116
0.0592 0.0592
0.1808
0.0572 0.0584
0.2452
0.056
0.054
0.3048
0.0564 0.0516
0.3712
0.0492 0.0472
0.6172
0.0488 0.0436
0.7744
0.05
0.05
0.8536
0.054 0.0548
0.8964
0.0464
0.05
0.928
0.0496 0.0468
0.956
0.0472 0.0468
0.974
0.0484 0.0456
0.9852
0.0484 0.0424
0.9928

63

Table 1.29: The Finite Sample Size of the SupW ( F) Test with 5% Nominal Size
H0 : No Structural Change in both β 1 and β 2 , DGP D: (θ, ρ, ϕ) = (0.8, 0.5, 0.5)
T=100
kernel
b = 0.02
0.04
0.06
0.08
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
T=500
kernel
b = 0.02
0.04
0.06
0.08
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
T=1000
kernel
b = 0.02
0.04
0.06
0.08
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1

= 0.05
Fixed-b
Andrews
Bartlett
QS
Bartlett
0.665
0.308
0.947
0.466
0.173
0.955
0.398
0.166
0.966
0.365
0.159
0.978
0.351 0.161
0.985
0.316 0.114
0.996
0.308 0.071
1.000
0.304 0.044
1.000
0.302 0.034
1.000
0.310 0.024
1.000
0.310 0.020
1.000
0.307 0.021
1.000
0.304 0.021
1.000
0.309
0.020
1.000

= 0.1
Fixed-b
Andrews
Bartlett
QS
Bartlett
0.598
0.395
0.801
0.417
0.200
0.800
0.355
0.166
0.842
0.319
0.156
0.881
0.308 0.153
0.906
0.283 0.101
0.968
0.271 0.061
0.990
0.266 0.036
0.997
0.272 0.026
0.999
0.276 0.023
1.000
0.266 0.020
1.000
0.265 0.021
1.000
0.265 0.020
1.000
0.273
0.021
1.000

= 0.2
Fixed-b
Andrews
Bartlett
QS
Bartlett
0.493
0.376
0.604
0.341
0.227
0.542
0.292
0.172
0.587
0.268
0.145
0.638
0.255
0.128
0.674
0.224
0.077
0.832
0.216
0.054
0.907
0.213
0.037
0.948
0.213
0.034
0.966
0.208
0.028
0.980
0.210
0.024
0.989
0.211
0.020
0.993
0.209
0.020
0.995
0.209
0.018
0.999

= 0.05
Fixed-b
Andrews
Bartlett
QS
Bartlett
0.247
0.109
0.697
0.207
0.097
0.832
0.186
0.096
0.888
0.181
0.086
0.930
0.170 0.089
0.956
0.144 0.074
0.991
0.146 0.054
0.998
0.140 0.036
0.999
0.144 0.037
1.000
0.145 0.032
1.000
0.146 0.031
1.000
0.144 0.032
1.000
0.146 0.031
1.000
0.149
0.032
1.000

= 0.1
Fixed-b
Andrews
Bartlett
QS
Bartlett
0.193
0.112
0.391
0.149
0.085
0.514
0.139
0.083
0.616
0.126
0.078
0.685
0.129 0.077
0.756
0.114 0.058
0.908
0.110 0.050
0.964
0.106 0.036
0.985
0.107 0.034
0.993
0.105 0.038
0.998
0.103 0.033
0.999
0.106 0.034
1.000
0.106 0.030
1.000
0.108
0.031
1.000

= 0.2
Fixed-b
Andrews
Bartlett
QS
Bartlett
0.131
0.090
0.216
0.108
0.084
0.251
0.102
0.077
0.318
0.098
0.070
0.382
0.096
0.070
0.447
0.088
0.051
0.684
0.089
0.044
0.824
0.088
0.041
0.888
0.083
0.041
0.926
0.083
0.038
0.949
0.087
0.036
0.970
0.090
0.033
0.984
0.087
0.030
0.990
0.087
0.030
0.994

= 0.05
Fixed-b
Andrews
Bartlett
QS
Bartlett
0.1628 0.0928
0.5716
0.144 0.0888
0.7504
0.1344 0.0844
0.844
0.1308 0.0796
0.8956
0.122 0.0672
0.9248
0.1196 0.0648
0.9828
0.1044 0.0512
0.9964
0.1072 0.0436
0.9996
0.1056 0.0424
0.9996
0.1056 0.042
1
0.1084 0.0412
1
0.1048 0.0416
1
0.1068 0.0404
1
0.1088 0.0404
1

= 0.1
Fixed-b
Andrews
Bartlett
QS
Bartlett
0.1176 0.0836
0.2892
0.0996 0.0636
0.4304
0.0924 0.0576
0.5604
0.0856 0.058
0.654
0.0864 0.056
0.724
0.0828 0.052
0.8888
0.0752 0.058
0.9516
0.0768 0.046
0.9748
0.076
0.04
0.9884
0.0772 0.0392
0.9952
0.0804 0.0392
0.9992
0.0804 0.0404
0.9992
0.082 0.0404
1
0.0788 0.0404
1

= 0.2
Fixed-b
Andrews
Bartlett
QS
Bartlett
0.0952 0.0744
0.1596
0.0884 0.0732
0.2132
0.0836 0.0732
0.2776
0.0828
0.066
0.3492
0.0816 0.0592
0.4104
0.0624 0.0504
0.6548
0.0648 0.0544
0.7856
0.0664 0.0516
0.862
0.0648 0.0512
0.9076
0.0692 0.0436
0.936
0.0688 0.0412
0.9568
0.066 0.0408
0.9788
0.0672 0.0364
0.9884
0.0668 0.0396
0.9912

64

Table 1.30: The Finite Sample Size of the SupW ( F) Test with 5% Nominal Size
H0 : No Structural Change in both β 1 and β 2 , DGP F: (θ, ρ, ϕ) = (0.9, 0.9, 0.9)
T=100
kernel
b = 0.02
0.04
0.06
0.08
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
T=500
kernel
b = 0.02
0.04
0.06
0.08
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
T=1000
kernel
b = 0.02
0.04
0.06
0.08
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1

= 0.05
Fixed-b
Andrews
Bartlett
QS
Bartlett
0.934
0.586
0.998
0.672
0.198
0.994
0.508
0.136
0.991
0.405
0.100
0.991
0.346 0.092
0.990
0.272 0.073
0.994
0.286 0.060
0.998
0.299 0.045
0.999
0.307 0.036
1.000
0.300 0.034
1.000
0.296 0.033
1.000
0.291 0.030
1.000
0.290 0.026
1.000
0.296
0.026
1.000

= 0.1
Fixed-b
Andrews
Bartlett
QS
Bartlett
0.963
0.879
0.992
0.806
0.472
0.976
0.680
0.293
0.971
0.578
0.227
0.973
0.528 0.184
0.974
0.445 0.122
0.986
0.440 0.081
0.995
0.444 0.057
0.998
0.447 0.046
1.000
0.448 0.038
1.000
0.442 0.036
1.000
0.442 0.034
1.000
0.443 0.033
1.000
0.446
0.030
1.000

= 0.2
Fixed-b
Andrews
Bartlett
QS
Bartlett
0.942
0.886
0.967
0.810
0.652
0.914
0.700
0.487
0.895
0.632
0.369
0.895
0.596
0.304
0.900
0.507
0.160
0.948
0.503
0.104
0.972
0.492
0.082
0.986
0.488
0.064
0.994
0.498
0.052
0.997
0.499
0.042
0.999
0.492
0.038
0.999
0.492
0.035
1.000
0.494
0.032
1.000

= 0.05
Fixed-b
Andrews
Bartlett
QS
Bartlett
0.591
0.195
0.945
0.347
0.093
0.941
0.276
0.075
0.950
0.239
0.070
0.962
0.212 0.059
0.971
0.184 0.054
0.990
0.178 0.050
0.998
0.180 0.040
1.000
0.176 0.041
1.000
0.180 0.039
1.000
0.176 0.042
1.000
0.170 0.038
1.000
0.173 0.038
1.000
0.174
0.038
1.000

= 0.1
Fixed-b
Andrews
Bartlett
QS
Bartlett
0.608
0.351
0.812
0.385
0.142
0.779
0.306
0.107
0.812
0.260
0.090
0.846
0.239 0.083
0.874
0.208 0.062
0.952
0.202 0.056
0.983
0.200 0.050
0.994
0.201 0.043
0.998
0.192 0.042
1.000
0.198 0.044
1.000
0.197 0.042
1.000
0.198 0.040
1.000
0.200
0.039
1.000

= 0.2
Fixed-b
Andrews
Bartlett
QS
Bartlett
0.512
0.354
0.632
0.313
0.190
0.527
0.252
0.139
0.545
0.227
0.114
0.587
0.204
0.098
0.630
0.177
0.066
0.802
0.183
0.062
0.878
0.184
0.059
0.930
0.180
0.050
0.956
0.178
0.048
0.973
0.176
0.044
0.987
0.179
0.043
0.993
0.174
0.043
0.995
0.174
0.044
0.997

= 0.05
Fixed-b
Andrews
Bartlett
QS
Bartlett
0.4004 0.1216
0.8364
0.2688 0.0724
0.888
0.2176 0.0688
0.9216
0.1956 0.0564
0.9436
0.1728 0.0516
0.9568
0.1504 0.046
0.9908
0.1464 0.0444
0.9964
0.1464 0.0424
0.9996
0.1532
0.04
0.9996
0.1556 0.0396
0.9996
0.1532 0.0408
1
0.1416 0.042
1
0.1452 0.042
1
0.1448 0.0444
1

= 0.1
Fixed-b
Andrews
Bartlett
QS
Bartlett
0.3504 0.1736
0.5964
0.2204 0.0892
0.6408
0.1764 0.068
0.7052
0.1628 0.0576
0.772
0.1504 0.054
0.8152
0.132 0.0528
0.9236
0.1332 0.0448
0.9712
0.132 0.0352
0.9864
0.1288 0.0352
0.9924
0.1364 0.0388
0.9968
0.1332 0.0392
0.9992
0.1372 0.0412
0.9996
0.1292 0.0424
0.9996
0.1296 0.0408
1

= 0.2
Fixed-b
Andrews
Bartlett
QS
Bartlett
0.2788 0.1832
0.3852
0.192 0.1188
0.3712
0.156 0.0928
0.4092
0.1492 0.0812
0.478
0.1392 0.0696
0.5344
0.1232 0.0476
0.7312
0.118 0.0452
0.8332
0.122
0.036
0.8908
0.1232
0.036
0.924
0.126
0.038
0.9536
0.1164 0.0376
0.9736
0.1192 0.0388
0.9856
0.1204 0.0388
0.9936
0.1244
0.04
0.9968

65

Table 1.31: The Finite Sample Size of the MeanW ( F) Test with 5% Nominal Size
H0 : No Structural Change in both β 1 and β 2 , DGP A: (θ, ρ, ϕ) = (0.5, 0.0, 0.0)
T=100
kernel
b = 0.02
0.04
0.06
0.08
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
T=500
kernel
b = 0.02
0.04
0.06
0.08
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
T=1000
kernel
b = 0.02
0.04
0.06
0.08
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1

= 0.05
Fixed-b
AP
Bartlett
QS
Bartlett
0.162 0.148
0.290
0.175 0.190
0.446
0.182 0.202
0.586
0.192 0.215
0.686
0.190 0.226
0.759
0.204 0.194
0.941
0.209 0.165
0.984
0.208 0.132
0.996
0.216 0.120
0.999
0.212 0.112
1.000
0.216 0.110
1.000
0.216 0.102
1.000
0.218 0.097
1.000
0.217 0.097
1.000

= 0.1
Fixed-b
AP
Bartlett
QS
Bartlett
0.101
0.094
0.157
0.105
0.105
0.243
0.108
0.114
0.328
0.107
0.114
0.406
0.103 0.116
0.482
0.109 0.109
0.750
0.114 0.096
0.895
0.117 0.087
0.950
0.119 0.080
0.977
0.121 0.080
0.991
0.120 0.075
0.996
0.123 0.071
1.000
0.121 0.068
1.000
0.122
0.066
1.000

= 0.2
Fixed-b
AP
Bartlett
QS
Bartlett
0.084 0.082
0.120
0.089 0.090
0.156
0.090 0.087
0.201
0.090 0.088
0.243
0.086 0.081
0.291
0.086 0.079
0.518
0.081 0.068
0.674
0.086 0.069
0.771
0.089 0.065
0.850
0.087 0.058
0.902
0.084 0.059
0.931
0.086 0.058
0.955
0.088 0.056
0.971
0.087 0.055
0.982

= 0.05
Fixed-b
AP
Bartlett
QS
Bartlett
0.064 0.066
0.134
0.063 0.066
0.248
0.060 0.069
0.363
0.063 0.068
0.471
0.064 0.070
0.570
0.069 0.066
0.853
0.071 0.065
0.956
0.068 0.057
0.986
0.065 0.057
0.995
0.065 0.056
1.000
0.068 0.058
1.000
0.069 0.058
1.000
0.068 0.055
1.000
0.067 0.055
1.000

= 0.1
Fixed-b
AP
Bartlett
QS
Bartlett
0.063
0.060
0.103
0.058
0.060
0.163
0.058
0.064
0.235
0.059
0.061
0.302
0.059 0.054
0.372
0.056 0.054
0.666
0.056 0.057
0.827
0.055 0.053
0.911
0.054 0.052
0.954
0.059 0.055
0.975
0.057 0.052
0.990
0.058 0.048
0.996
0.058 0.047
0.998
0.054
0.050
1.000

= 0.2
Fixed-b
AP
Bartlett
QS
Bartlett
0.056 0.060
0.084
0.054 0.056
0.113
0.056 0.058
0.151
0.059 0.056
0.189
0.061 0.055
0.231
0.056 0.056
0.465
0.055 0.060
0.628
0.062 0.055
0.741
0.060 0.051
0.818
0.054 0.052
0.869
0.057 0.051
0.912
0.056 0.050
0.940
0.056 0.049
0.961
0.060 0.050
0.971

= 0.05
Fixed-b
AP
Bartlett
QS
Bartlett
0.0576 0.0576
0.1296
0.0604 0.0616
0.224
0.0596 0.058
0.3244
0.0588 0.0584
0.4352
0.0576
0.06
0.5264
0.0556 0.0588
0.8344
0.0556 0.0512
0.942
0.0532 0.0536
0.974
0.0592 0.056
0.9912
0.0584 0.0516
0.9976
0.056 0.0492
0.9988
0.058 0.0516
1
0.0556 0.0516
1
0.056 0.0528
1

= 0.1
Fixed-b
AP
Bartlett
QS
Bartlett
0.0508 0.0524
0.0988
0.0524 0.0532
0.1516
0.0516 0.0484
0.2188
0.0504
0.05
0.2896
0.0504 0.0484
0.352
0.052 0.0488
0.6536
0.0492 0.046
0.8236
0.0472 0.0472
0.9012
0.0544 0.0508
0.9424
0.052 0.0508
0.9696
0.0512 0.0504
0.9868
0.0496 0.048
0.9956
0.0516
0.05
0.9984
0.0512 0.0468
1

= 0.2
Fixed-b
AP
Bartlett
QS
Bartlett
0.0468 0.0476
0.0812
0.0476
0.05
0.112
0.0496 0.0484
0.1508
0.0504
0.05
0.1904
0.0512 0.0464
0.2332
0.048 0.0428
0.4424
0.0508 0.0492
0.6132
0.048 0.0496
0.736
0.0504 0.052
0.8072
0.0476 0.0452
0.8616
0.0488 0.0432
0.9028
0.0472 0.0476
0.9328
0.0492 0.0472
0.958
0.0504 0.0476
0.9712

66

Table 1.32: The Finite Sample Size of the MeanW ( F) Test with 5% Nominal Size
H0 : No Structural Change in both β 1 and β 2 , DGP D: (θ, ρ, ϕ) = (0.8, 0.5, 0.5)
T=100
kernel
b = 0.02
0.04
0.06
0.08
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
T=500
kernel
b = 0.02
0.04
0.06
0.08
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
T=1000
kernel
b = 0.02
0.04
0.06
0.08
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1

= 0.05
Fixed-b
AP
Bartlett
QS
Bartlett
0.664 0.530
0.806
0.523 0.365
0.806
0.476 0.312
0.852
0.456 0.301
0.893
0.445 0.291
0.926
0.432 0.234
0.985
0.440 0.201
0.996
0.448 0.178
1.000
0.462 0.155
1.000
0.457 0.146
1.000
0.460 0.143
1.000
0.460 0.134
1.000
0.456 0.129
1.000
0.455 0.121
1.000

= 0.1
Fixed-b
AP
Bartlett
QS
Bartlett
0.527
0.420
0.632
0.386
0.273
0.593
0.343
0.245
0.646
0.328
0.231
0.700
0.317 0.219
0.752
0.310 0.176
0.914
0.325 0.156
0.973
0.333 0.138
0.990
0.347 0.127
0.997
0.342 0.130
0.998
0.339 0.122
1.000
0.342 0.115
1.000
0.339 0.111
1.000
0.341
0.105
1.000

= 0.2
Fixed-b
AP
Bartlett
QS
Bartlett
0.420 0.324
0.488
0.283 0.198
0.414
0.235 0.167
0.431
0.229 0.158
0.475
0.216 0.154
0.521
0.208 0.136
0.712
0.214 0.121
0.822
0.223 0.118
0.897
0.224 0.114
0.938
0.224 0.106
0.958
0.222 0.099
0.977
0.226 0.094
0.984
0.226 0.090
0.992
0.229 0.089
0.996

= 0.05
Fixed-b
AP
Bartlett
QS
Bartlett
0.160 0.110
0.291
0.142 0.103
0.412
0.133 0.106
0.519
0.134 0.102
0.614
0.128 0.103
0.694
0.136 0.086
0.915
0.136 0.088
0.976
0.139 0.075
0.994
0.148 0.078
0.999
0.147 0.078
1.000
0.144 0.077
1.000
0.144 0.075
1.000
0.144 0.074
1.000
0.141 0.073
1.000

= 0.1
Fixed-b
AP
Bartlett
QS
Bartlett
0.127
0.091
0.201
0.110
0.084
0.255
0.103
0.087
0.330
0.101
0.088
0.396
0.099 0.084
0.468
0.103 0.074
0.737
0.105 0.077
0.879
0.101 0.071
0.941
0.104 0.072
0.974
0.106 0.072
0.989
0.105 0.070
0.996
0.108 0.068
0.999
0.104 0.070
1.000
0.105
0.067
1.000

= 0.2
Fixed-b
AP
Bartlett
QS
Bartlett
0.109 0.085
0.152
0.089 0.070
0.172
0.086 0.070
0.214
0.083 0.068
0.254
0.081 0.070
0.295
0.080 0.061
0.502
0.082 0.063
0.675
0.084 0.061
0.776
0.082 0.065
0.848
0.083 0.062
0.900
0.085 0.062
0.929
0.084 0.061
0.955
0.084 0.062
0.970
0.086 0.065
0.982

= 0.05
Fixed-b
AP
Bartlett
QS
Bartlett
0.1044 0.082
0.202
0.0932 0.0784
0.3076
0.0952 0.0732
0.434
0.0928 0.0732
0.5352
0.0912 0.0696
0.6228
0.094 0.068
0.8792
0.0952 0.0688
0.9568
0.0956 0.0648
0.9844
0.0948 0.0616
0.996
0.0984 0.058
0.9984
0.0968 0.0608
1
0.0984 0.0604
1
0.0952 0.0616
1
0.096 0.062
1

= 0.1
Fixed-b
AP
Bartlett
QS
Bartlett
0.086 0.0672
0.1416
0.0768 0.0632
0.1972
0.0768 0.0624
0.2692
0.0716 0.0588
0.3376
0.0676 0.0624
0.4068
0.0716 0.0636
0.6992
0.072 0.0624
0.8524
0.0744 0.0624
0.9208
0.0736 0.054
0.9552
0.0736 0.0532
0.9772
0.072 0.0536
0.9912
0.0728 0.0532
0.9972
0.072 0.054
0.9992
0.0716 0.0512
1

= 0.2
Fixed-b
AP
Bartlett
QS
Bartlett
0.0732 0.0612
0.1096
0.0696 0.0628
0.1332
0.066 0.0552
0.1768
0.0628 0.0544
0.2144
0.0608 0.0528
0.2552
0.0588 0.0536
0.464
0.0608
0.062
0.6356
0.0664 0.0596
0.7512
0.0612 0.0616
0.822
0.0624 0.056
0.8664
0.0604 0.0552
0.9092
0.0592
0.054
0.938
0.0616 0.0548
0.9604
0.0628 0.0568
0.974

67

Table 1.33: The Finite Sample Size of the MeanW ( F) Test with 5% Nominal Size
H0 : No Structural Change in both β 1 and β 2 , DGP F: (θ, ρ, ϕ) = (0.9, 0.9, 0.9)
T=100
kernel
b = 0.02
0.04
0.06
0.08
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
T=500
kernel
b = 0.02
0.04
0.06
0.08
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
T=1000
kernel
b = 0.02
0.04
0.06
0.08
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1

= 0.05
Fixed-b
AP
Bartlett
QS
Bartlett
0.985 0.948
0.996
0.910 0.758
0.985
0.825 0.591
0.979
0.771 0.478
0.978
0.723 0.407
0.979
0.650 0.269
0.992
0.646 0.243
0.997
0.655 0.218
0.999
0.668 0.207
1.000
0.670 0.201
1.000
0.659 0.194
1.000
0.659 0.184
1.000
0.662 0.182
1.000
0.664 0.174
1.000

= 0.1
Fixed-b
AP
Bartlett
QS
Bartlett
0.964
0.924
0.982
0.859
0.735
0.945
0.783
0.610
0.932
0.724
0.513
0.937
0.687 0.448
0.942
0.628 0.302
0.976
0.619 0.260
0.991
0.634 0.233
0.997
0.641 0.228
0.998
0.642 0.215
1.000
0.638 0.204
1.000
0.631 0.195
1.000
0.640 0.189
1.000
0.641
0.181
1.000

= 0.2
Fixed-b
AP
Bartlett
QS
Bartlett
0.916 0.854
0.943
0.772 0.658
0.858
0.680 0.538
0.824
0.627 0.462
0.822
0.586 0.409
0.838
0.533 0.302
0.912
0.543 0.252
0.955
0.545 0.235
0.975
0.552 0.223
0.988
0.546 0.207
0.996
0.537 0.194
0.998
0.546 0.180
0.998
0.546 0.175
0.998
0.547 0.171
0.999

= 0.05
Fixed-b
AP
Bartlett
QS
Bartlett
0.653 0.451
0.788
0.458 0.241
0.761
0.382 0.181
0.794
0.342 0.160
0.833
0.324 0.145
0.868
0.296 0.118
0.963
0.312 0.108
0.992
0.306 0.107
0.998
0.316 0.099
1.000
0.320 0.097
1.000
0.310 0.095
1.000
0.310 0.094
1.000
0.306 0.093
1.000
0.304 0.091
1.000

= 0.1
Fixed-b
AP
Bartlett
QS
Bartlett
0.532
0.390
0.645
0.360
0.225
0.568
0.302
0.179
0.595
0.275
0.156
0.642
0.261 0.140
0.691
0.248 0.118
0.872
0.257 0.112
0.952
0.263 0.111
0.980
0.268 0.102
0.992
0.265 0.099
0.995
0.261 0.094
0.998
0.267 0.093
1.000
0.271 0.092
1.000
0.270
0.091
1.000

= 0.2
Fixed-b
AP
Bartlett
QS
Bartlett
0.429 0.318
0.502
0.284 0.193
0.404
0.235 0.157
0.408
0.217 0.137
0.432
0.206 0.124
0.469
0.188 0.104
0.656
0.198 0.110
0.792
0.204 0.108
0.867
0.204 0.105
0.918
0.203 0.094
0.947
0.204 0.089
0.968
0.200 0.088
0.983
0.204 0.089
0.989
0.204 0.084
0.992

= 0.05
Fixed-b
AP
Bartlett
QS
Bartlett
0.358
0.22
0.5164
0.2628 0.1344
0.5592
0.2288 0.1116
0.6384
0.2136 0.1004
0.7148
0.2052 0.0948
0.7752
0.1968 0.0804
0.9344
0.2072 0.0792
0.9848
0.2072 0.0716
0.996
0.2116 0.068
0.998
0.2176 0.068
1
0.2136
0.07
1
0.2108 0.072
1
0.2116 0.0684
1
0.2064 0.0728
1

= 0.1
Fixed-b
AP
Bartlett
QS
Bartlett
0.282 0.1912
0.3736
0.202 0.1224
0.3712
0.1768 0.1068
0.4324
0.1612 0.092
0.4948
0.1544 0.0892
0.5624
0.1508 0.0828
0.7948
0.1584 0.0772
0.906
0.1576 0.0664
0.9564
0.1604 0.0632
0.9788
0.1632 0.0664
0.9908
0.1596 0.068
0.998
0.1632 0.0664
0.9996
0.1608 0.0688
0.9996
0.162 0.0672
1

= 0.2
Fixed-b
AP
Bartlett
QS
Bartlett
0.2272 0.154
0.2804
0.1576 0.1032
0.2536
0.1328 0.0912
0.2776
0.1228 0.0852
0.312
0.1208
0.074
0.362
0.1104 0.0672
0.5728
0.1204 0.0724
0.7132
0.1232 0.0676
0.8116
0.1264 0.0652
0.8716
0.1228
0.06
0.914
0.118
0.06
0.9404
0.1224 0.0584
0.9596
0.1248 0.0564
0.9724
0.1212 0.0588
0.9856

68

Table 1.34: The Finite Sample Size of the ExpW ( F) Test with 5% Nominal Size
H0 : No Structural Change in both β 1 and β 2 , DGP A: (θ, ρ, ϕ) = (0.5, 0.0, 0.0)
T=100
kernel
b = 0.02
0.04
0.06
0.08
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
T=500
kernel
b = 0.02
0.04
0.06
0.08
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
T=1000
kernel
b = 0.02
0.04
0.06
0.08
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1

= 0.05
Fixed-b
AP
Bartlett
QS
Bartlett
0.368 0.198
0.712
0.316 0.201
0.839
0.301 0.218
0.900
0.303 0.221
0.938
0.291 0.217
0.956
0.271 0.154
0.994
0.271 0.084
0.999
0.266 0.048
1.000
0.256 0.036
1.000
0.253 0.026
1.000
0.254 0.024
1.000
0.246 0.020
1.000
0.250 0.015
1.000
0.254 0.012
1.000

= 0.1
Fixed-b
AP
Bartlett
QS
Bartlett
0.178
0.135
0.348
0.160
0.113
0.494
0.150
0.113
0.600
0.144
0.114
0.694
0.136 0.113
0.760
0.137 0.083
0.915
0.134 0.051
0.967
0.129 0.031
0.986
0.126 0.023
0.994
0.124 0.017
0.998
0.124 0.016
1.000
0.125 0.014
1.000
0.126 0.010
1.000
0.124
0.007
1.000

= 0.2
Fixed-b
AP
Bartlett
QS
Bartlett
0.108 0.104
0.179
0.114 0.108
0.248
0.111 0.094
0.314
0.111 0.090
0.386
0.104 0.080
0.454
0.090 0.058
0.690
0.089 0.043
0.817
0.086 0.028
0.889
0.086 0.019
0.929
0.084 0.017
0.954
0.082 0.015
0.976
0.082 0.012
0.987
0.084 0.010
0.993
0.082 0.009
0.996

= 0.05
Fixed-b
AP
Bartlett
QS
Bartlett
0.094 0.086
0.404
0.092 0.082
0.657
0.090 0.083
0.774
0.090 0.079
0.843
0.086 0.078
0.888
0.082 0.063
0.978
0.082 0.056
0.994
0.081 0.047
0.999
0.083 0.046
1.000
0.081 0.034
1.000
0.081 0.026
1.000
0.079 0.025
1.000
0.080 0.026
1.000
0.082 0.026
1.000

= 0.1
Fixed-b
AP
Bartlett
QS
Bartlett
0.070
0.072
0.196
0.070
0.064
0.345
0.072
0.064
0.470
0.068
0.060
0.572
0.064 0.058
0.655
0.060 0.053
0.866
0.060 0.052
0.946
0.058 0.041
0.974
0.058 0.038
0.987
0.059 0.034
0.995
0.062 0.028
1.000
0.059 0.027
1.000
0.059 0.025
1.000
0.062
0.024
1.000

= 0.2
Fixed-b
AP
Bartlett
QS
Bartlett
0.062 0.064
0.115
0.060 0.058
0.177
0.058 0.057
0.247
0.061 0.058
0.308
0.060 0.055
0.367
0.055 0.044
0.632
0.062 0.052
0.779
0.057 0.050
0.860
0.055 0.042
0.906
0.050 0.041
0.941
0.056 0.035
0.960
0.055 0.031
0.975
0.051 0.030
0.987
0.055 0.028
0.992

= 0.05
Fixed-b
AP
Bartlett
QS
Bartlett
0.0776 0.0776
0.3416
0.076 0.072
0.5936
0.0784 0.0716
0.736
0.076 0.0592
0.8136
0.0692 0.0664
0.8692
0.0696 0.0584
0.9728
0.0692 0.0472
0.9928
0.07 0.0448
0.9984
0.0712 0.0468
0.9996
0.072 0.0456
1
0.0724 0.0472
1
0.0668 0.0436
1
0.07
0.04
1
0.0708
0.04
1

= 0.1
Fixed-b
AP
Bartlett
QS
Bartlett
0.0536 0.0488
0.1668
0.0544 0.0552
0.308
0.0528 0.0508
0.4544
0.0464 0.0484
0.5532
0.0472 0.0536
0.6348
0.0496 0.0556
0.8584
0.0492 0.0476
0.93
0.0448 0.0464
0.9684
0.0512 0.0452
0.9832
0.0468 0.0484
0.992
0.0464 0.0492
0.9964
0.0544 0.0484
0.9996
0.0508 0.0456
0.9996
0.052
0.044
1

= 0.2
Fixed-b
AP
Bartlett
QS
Bartlett
0.0592 0.0572
0.1128
0.06 0.0604
0.174
0.0592
0.056
0.2368
0.0592 0.0564
0.298
0.056
0.052
0.3624
0.0488 0.0472
0.62
0.0488 0.0436
0.7656
0.0504
0.05
0.8516
0.0544 0.0548
0.8992
0.0464
0.05
0.9288
0.0504 0.0468
0.956
0.0476 0.0468
0.9768
0.0484 0.0456
0.9872
0.048 0.0424
0.9928

69

Table 1.35: The Finite Sample Size of the ExpW ( F) Test with 5% Nominal Size
H0 : No Structural Change in both β 1 and β 2 , DGP D: (θ, ρ, ϕ) = (0.8, 0.5, 0.5)
T=100
kernel
b = 0.02
0.04
0.06
0.08
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
T=500
kernel
b = 0.02
0.04
0.06
0.08
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
T=1000
kernel
b = 0.02
0.04
0.06
0.08
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1

= 0.05
Fixed-b
AP
Bartlett
QS
Bartlett
0.708 0.333
0.954
0.517 0.183
0.960
0.435 0.172
0.975
0.396 0.164
0.982
0.372 0.165
0.986
0.324 0.116
0.997
0.315 0.072
1.000
0.307 0.044
1.000
0.309 0.034
1.000
0.312 0.024
1.000
0.315 0.020
1.000
0.310 0.021
1.000
0.308 0.021
1.000
0.313 0.020
1.000

= 0.1
Fixed-b
AP
Bartlett
QS
Bartlett
0.646
0.452
0.818
0.467
0.226
0.817
0.398
0.182
0.861
0.353
0.164
0.896
0.332 0.159
0.917
0.300 0.102
0.973
0.283 0.062
0.994
0.280 0.036
0.998
0.281 0.026
1.000
0.281 0.023
1.000
0.273 0.020
1.000
0.272 0.021
1.000
0.270 0.020
1.000
0.278
0.021
1.000

= 0.2
Fixed-b
AP
Bartlett
QS
Bartlett
0.518 0.398
0.627
0.360 0.243
0.568
0.310 0.195
0.600
0.292 0.163
0.651
0.275 0.139
0.689
0.235 0.079
0.844
0.225 0.054
0.916
0.226 0.037
0.954
0.219 0.034
0.970
0.216 0.028
0.982
0.220 0.024
0.990
0.215 0.020
0.996
0.218 0.020
0.999
0.214 0.018
1.000

= 0.05
Fixed-b
AP
Bartlett
QS
Bartlett
0.260 0.113
0.647
0.213 0.100
0.794
0.193 0.097
0.868
0.185 0.088
0.912
0.172 0.090
0.946
0.148 0.074
0.990
0.149 0.054
0.999
0.141 0.036
1.000
0.144 0.037
1.000
0.146 0.032
1.000
0.147 0.031
1.000
0.144 0.032
1.000
0.148 0.031
1.000
0.150 0.032
1.000

= 0.1
Fixed-b
AP
Bartlett
QS
Bartlett
0.188
0.122
0.368
0.158
0.087
0.485
0.143
0.085
0.590
0.135
0.081
0.668
0.133 0.078
0.737
0.115 0.058
0.911
0.113 0.050
0.973
0.107 0.036
0.988
0.109 0.034
0.995
0.107 0.038
0.999
0.104 0.033
1.000
0.106 0.034
1.000
0.106 0.030
1.000
0.108
0.031
1.000

= 0.2
Fixed-b
AP
Bartlett
QS
Bartlett
0.132 0.094
0.211
0.110 0.084
0.253
0.104 0.080
0.318
0.102 0.076
0.382
0.096 0.072
0.446
0.092 0.051
0.683
0.088 0.044
0.822
0.089 0.041
0.893
0.085 0.041
0.925
0.084 0.038
0.954
0.088 0.036
0.972
0.091 0.033
0.985
0.089 0.030
0.991
0.087 0.030
0.997

= 0.05
Fixed-b
AP
Bartlett
QS
Bartlett
0.1652 0.0948
0.5116
0.1444 0.0892
0.7064
0.1344 0.084
0.8044
0.1312 0.0796
0.8684
0.1224 0.0676
0.9048
0.1184 0.0648
0.9792
0.1048 0.0512
0.996
0.1072 0.0436
0.9988
0.106 0.0424
0.9996
0.1056 0.042
1
0.1088 0.0412
1
0.1048 0.0416
1
0.1072 0.0404
1
0.1088 0.0404
1

= 0.1
Fixed-b
AP
Bartlett
QS
Bartlett
0.1212 0.0824
0.2648
0.108 0.0648
0.3912
0.0948 0.0564
0.52
0.0912 0.0572
0.6192
0.0868 0.0568
0.6972
0.0844 0.052
0.8804
0.0744 0.058
0.9476
0.0764 0.046
0.9788
0.0768
0.04
0.9896
0.0776 0.0392
0.9952
0.0808 0.0392
0.9988
0.0808 0.0404
0.9996
0.082 0.0404
1
0.0788 0.0404
1

= 0.2
Fixed-b
AP
Bartlett
QS
Bartlett
0.0924 0.0708
0.1576
0.0848 0.0712
0.2124
0.0792 0.0712
0.278
0.0824 0.0688
0.3432
0.0784
0.06
0.3988
0.064 0.0504
0.64
0.0656 0.0544
0.78
0.0656 0.0516
0.8604
0.0656 0.0512
0.902
0.0692 0.0436
0.9336
0.0692 0.0412
0.9636
0.0664 0.0408
0.9808
0.0672 0.0364
0.9888
0.0668 0.0396
0.9932

70

Table 1.36: The Finite Sample Size of the ExpW ( F) Test with 5% Nominal Size
H0 : No Structural Change in both β 1 and β 2 , DGP F: (θ, ρ, ϕ) = (0.9, 0.9, 0.9)
T=100
kernel
b = 0.02
0.04
0.06
0.08
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
T=500
kernel
b = 0.02
0.04
0.06
0.08
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
T=1000
kernel
b = 0.02
0.04
0.06
0.08
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1

= 0.05
Fixed-b
AP
Bartlett
QS
Bartlett
0.958 0.624
0.999
0.719 0.212
0.995
0.544 0.143
0.993
0.445 0.106
0.992
0.370 0.095
0.993
0.282 0.074
0.996
0.296 0.060
0.998
0.307 0.045
1.000
0.313 0.036
1.000
0.304 0.034
1.000
0.300 0.033
1.000
0.297 0.030
1.000
0.296 0.026
1.000
0.301 0.026
1.000

= 0.1
Fixed-b
AP
Bartlett
QS
Bartlett
0.975
0.909
0.994
0.850
0.514
0.981
0.721
0.323
0.977
0.624
0.238
0.977
0.558 0.191
0.981
0.465 0.123
0.992
0.458 0.081
0.997
0.456 0.057
0.998
0.456 0.046
1.000
0.456 0.038
1.000
0.451 0.036
1.000
0.449 0.034
1.000
0.451 0.033
1.000
0.452
0.030
1.000

= 0.2
Fixed-b
AP
Bartlett
QS
Bartlett
0.951 0.902
0.971
0.826 0.689
0.925
0.729 0.522
0.910
0.665 0.402
0.908
0.617 0.327
0.918
0.529 0.164
0.956
0.519 0.105
0.980
0.505 0.082
0.991
0.502 0.064
0.996
0.507 0.052
0.998
0.507 0.042
0.999
0.500 0.038
0.999
0.502 0.035
1.000
0.498 0.032
1.000

= 0.05
Fixed-b
AP
Bartlett
QS
Bartlett
0.634 0.204
0.942
0.375 0.096
0.938
0.292 0.076
0.952
0.251 0.070
0.965
0.220 0.060
0.970
0.188 0.054
0.992
0.181 0.050
0.999
0.181 0.040
1.000
0.177 0.041
1.000
0.182 0.039
1.000
0.178 0.042
1.000
0.172 0.038
1.000
0.174 0.038
1.000
0.174 0.038
1.000

= 0.1
Fixed-b
AP
Bartlett
QS
Bartlett
0.635
0.392
0.823
0.411
0.154
0.785
0.330
0.112
0.817
0.280
0.093
0.850
0.252 0.085
0.878
0.218 0.062
0.956
0.206 0.056
0.986
0.202 0.050
0.995
0.205 0.043
0.999
0.196 0.042
1.000
0.200 0.044
1.000
0.199 0.042
1.000
0.201 0.040
1.000
0.202
0.039
1.000

= 0.2
Fixed-b
AP
Bartlett
QS
Bartlett
0.517 0.372
0.641
0.328 0.206
0.542
0.269 0.150
0.554
0.242 0.121
0.596
0.218 0.102
0.637
0.188 0.068
0.807
0.190 0.062
0.885
0.188 0.059
0.933
0.183 0.050
0.961
0.182 0.048
0.976
0.179 0.044
0.988
0.180 0.043
0.993
0.175 0.043
0.996
0.177 0.044
0.998

= 0.05
Fixed-b
AP
Bartlett
QS
Bartlett
0.426 0.1272
0.8092
0.2788 0.0728
0.8676
0.2228 0.0684
0.9092
0.198 0.0568
0.9348
0.1764 0.052
0.9592
0.1528 0.046
0.9916
0.1488 0.0444
0.9972
0.1472 0.0424
0.9996
0.1532
0.04
1
0.1564 0.0396
1
0.1536 0.0408
1
0.1416 0.042
1
0.1456 0.042
1
0.1448 0.0444
1

= 0.1
Fixed-b
AP
Bartlett
QS
Bartlett
0.3728 0.1908
0.5816
0.236 0.0948
0.626
0.1928 0.0704
0.6932
0.1712 0.0588
0.7588
0.1572 0.0544
0.8084
0.1364 0.0528
0.9288
0.1368 0.0448
0.9696
0.1312 0.0352
0.9884
0.1292 0.0352
0.9956
0.1356 0.0388
0.998
0.134 0.0392
0.9996
0.1384 0.0412
0.9996
0.1308 0.0424
1
0.1292 0.0408
1

= 0.2
Fixed-b
AP
Bartlett
QS
Bartlett
0.2912 0.1968
0.3884
0.1944 0.1196
0.3728
0.1612 0.0968
0.4176
0.1512
0.088
0.486
0.1436 0.074
0.536
0.1248 0.0476
0.7312
0.1212 0.0452
0.8356
0.1236
0.036
0.8956
0.1248 0.036
0.9328
0.1264
0.038
0.9576
0.1196 0.0376
0.9768
0.1184 0.0388
0.99
0.1208 0.0388
0.9944
0.1248
0.04
0.9972

71

APPENDIX

72

Appendix for Chapter 1
The following expression is a general representation of the HAC estimators:
Ω=T

−1

T

T

∑∑K

t =1 s =1

|t − s|
M

vt vs .

This representation can be rewritten in terms of the partial sum processes St = ∑tj=1 v j
following Kiefer and Vogelsang (2005) and Hashimzade and Vogelsang (2008) as follows.
Let M = bT. Then for the kernels in Class 1,
Ω = T −2

T −1 T −1

∑ ∑ T −1/2 St

T 2 ∆2t,s T −1/2 Ss ,

(1.40)

t =1 s =1

where
∆2t,s ≡ (Kt,s − Kt,s+1 ) − (Kt+1,s − Kt+1,s+1 ) ,

|t − s|
.
bT

Kt,s = K
For the Class 2 kernel (Bartlett),
Ω=

2
bT

T −1

∑

T −1 S t S t −

t =1

1
bT

T − M −1

∑

T −1 St+bT St + T −1 St St+bT .

(1.41)

t =1

For the kernels in Class 3,
Ω = T −2

∑∑

T −1 St T 2 ∆2t,s Ss

(1.42)

|t−s|<bT

1
+
bT
1
−
bT

T −bT

∑

T

−1/2

Ss T

−1/2

Ss+bT

K (1) − K (1 −
1
bT

s =1

T −bT

∑

T

−1/2

Ss T

−1/2

Ss+bT

s =1

73

1
bT )

K (−1 +

1
bT ) − K (−1)
1
bT

.

Proof of Proposition 1: From equation (1.11) and (1.12) the limit of the fi follows immediately by Assumption 1 and 2. Plugging the limits of β 1 and β 2 in equation (1.13) yields,
for r ≤ λ,





 Λ 0   Wp (r ) −
T −1/2 S[rT ] ⇒ 

0
0 Λ

r
λ Wp ( λ )



,

and for r > λ,




0
Λ 0  
T −1/2 S[rT ] ⇒ 

λ
Wp (r ) − Wp (λ) − 1r−
0 Λ
− λ Wp (1) − Wp ( λ )



.

Thus one can rewrite this result by using indicator functions as






(1)
0   Fp (r, λ)





Λ
Λ 0 
T −1/2 S[rT ] ⇒ 
 Fp (r, λ) ,

≡
(2)
0 Λ
Fp (r, λ)
0 Λ
where
(1)

Fp (r, λ) = Wp (r ) −

r
Wp ( λ ) · 1 (r ≤ λ ),
λ

and
(2)

Fp (r, λ) =

Wp (r ) − Wp ( λ ) −

r−λ
Wp (1) − Wp ( λ )
1−λ

· 1 (r > λ ).

Proof of Proposition 2: From (1.18),


Ω( F) = 


=

λ Σ (1)
T −1 ∑tT=λT +1 ∑λT
s =1 K

T −1
|t−s|
M

(2) (1)
vt vs

λ Σ (1)
T −1 ∑tT=λT +1 ∑λT
s =1 K

xt ut us xs

|t−s|
M

( 1 − λ ) Σ (2)
T −1

|t−s|
M

T
∑λT
t=1 ∑s=λT +1 K

T
∑λT
t=1 ∑s=λT +1 K

|t−s|
M

( 1 − λ ) Σ (2)

(1) (2)
vt vs






xt ut us xs 
.

The diagonal blocks converge respectively to λΣ and (1 − λ)Σ in probability under traditional asymptotics. One needs to show that the off-diagonal blocks converge to zero.

74

|t−s|
M

First, for the Bartlett K

= 0 for |t − s| ≥ M. Therefore, the upper right off-diagonal

block can be rewritten as
( F)

Ω1,2 ≡

1
T

|t−s|
M

where Kt,s = K

T

λT

∑ ∑

(1) (2)

Kt,s vt vs

=

t=1 s=λT +1

(1)

. Set at = vt

1
T

λT

∑ vt

T

∑

(1)

t =1

(2)

Kt,s vs ,

s=λT +1

(2)

and bt = ∑sT=λT +1 Kt,s vs

and apply the partial

summation formula
λT

∑ a t bt =

t =1

λT −1

t

∑

∑

t =1

λT

( bt − bt + 1 ) +

as

s =1

∑ as

bλT

s =1

to get
1
T

λT

∑ vt

T

(1)

t =1

∑

(2)

Kt,s vs

=

s=λT +1

1
T

λT −1

∑

t =1

(1)

+ SλT
=

(1)

where St

(1)

= ∑tj=1 v j

(1)

1
T

T

∑

(1)

St

(2)

Kt,s vs

T

s=λT +1
T

∑

∑

−

(2)

Kt+1,s vs

(1.43)

s=λT +1

(2)

KλT,s vs

s=λT +1

λT −1

∑

T

∑

(1)

St

t =1

(2)

Kt,s vs

s=λT +1

(2)

for t ≤ λT and St

T

−

∑

(2)

Kt+1,s vs

,

s=λT +1

(1)

= ∑tj=λT +1 v j

for t ≥ λT + 1. The above lines

(2)

used SλT = 0 and ST = 0.
One can rewrite each component of the summands of this equation:
T

∑

(2)

Kt,s vs

T −1

=

s=λT +1

T

∑

(Kt,s − Kt,s+1 ) Ss

∑

(Kt,s − Kt,s+1 ) Ss ,

∑

(Kt+1,s − Kt+1,s+1 ) Ss

∑

(Kt+1,s − Kt+1,s+1 ) Ss

(2)

s=λT +1
T −1

=
and

∑

(2)

Kt+1,s vs

s=λT +1

(2)

s=λT +1
T −1

=

(2)

(2)

s=λT +1
T −1

=

(2)

+ Kt,T ST − Kt,λT +1 SλT

(2)

s=λT +1

75

(2)

(2)

+ Kt+1,T ST − Kt+1,λT +1 SλT

Plugging these representations into (1.43) gives
( F)
Ω1,2

1
=
T

λT −1

∑

T −1

∑

(1)

(2)

St (Kt,s − Kt,s+1 − Kt+1,s + Kt+1,s+1 ) Ss

.

t=1 s=λT +1

From Hashimzade and Vogelsang (2008) for the Bartlett kernel, Kt,s − Kt,s+1 − Kt+1,s +
1
for {(t, s) : M + t = s} and zero otherwise. This yields
Kt+1,s+1 = − M

( F)

Ω1,2 =

1
T

λT −1

∑

t=λT − M+1

−1
(1) (2)
St St + M .
M

Taking the matrix-maximum norm yields
( F)
Ω1,2

λT −1
1
(1)
(2)
T −1/2 St
· T −1/2 St+ M
≤
∑
TM t=λT − M+1

M−1
(1)
(2)
max
T −1/2 St
·
max
T −1/2 Ss
TM λT − M+1≤t≤λT −1
λT +1≤s≤λT −1+ M
1
= O p (1) = o p (1).
T

≤

Here we used
(1)

= O p (1),

(2)

= O p (1),

max

T −1/2 St

max

T −1/2 Ss

λT − M+1≤t≤λT −1

and
λT +1≤s≤λT −1+ M

which is true given the result in Proposition 1 and the traditional assumption that M/T →
0. The same argument applies to the left lower off-diagonal block of Ω( F) yielding the
desired result.
Proof of Lemma 1: Plugging the limit of the partial sum process in Proposition 1 into the
HAC estimators in (1.40), (1.41), or (1.42) the desired result follows from direct application
of the continuous mapping theorem to obtain the desired result in (1.19).

76

Proof of Lemma 2: The proof is for the Bartlett kernel. Recall from (1.16) that
1
λT

Σ (1) =

[λT ] [λT ]

|t − s|
M1

∑ ∑K

t =1 s =1

(1) (1)

vt vs .

With the Bartlett kernel, one can rewrite this as:

Σ

1 1
−
b1 λT

(1)

[(1−b1 )λT ]−1

with b1 =

∑

[λT ]−1

∑

(1)

t =1

(1)

(1)

and St

(1)

(λT )−1/2 St (λT )−1/2 St

(1)

(λT )−1/2 St+b1 λT (λT )−1/2 St

t =1
M1
λT

2 1
=
b1 λT

(1)

(1)

+ (λT )−1/2 St (λT )−1/2 St+b1 λT

(1)

= ∑tj=1 v j . Apply the continuous mapping theorem using the

result on the limit of the partial sum process in Proposition 1 to obtain
Σ (1) ⇒

−
−

1
b1 λ
1
b1 λ

1

2
b1 λ

0

(1−b1 )λ
0

(1−b1 )λ
0

(1)

(1)

λ−1/2 ΛFp (r, λ) Fp (r, λ) λ−1/2 Λ dr

(1)

(1)

λ−1/2 ΛFp (r + b1 λ, λ) Fp (r, λ) λ−1/2 Λ dr
(1)

(1)

λ−1/2 ΛFp (r, λ) Fp (r + b1 λ, λ) λ−1/2 Λ dr


= Λ

(1)

P b1 λ, Fp (r, λ)
λ


Λ

by recalling (1.4). One can easily get the limit of Σ(2) in the same way and this completes
the proof.
Proof of Theorem 1: Recall that

Wald( F) = T R β

1 ( F ) −1
Qλ R
RQ−
λ Ω

77

−1

Rβ .

,

Using R = ( R1 , − R1 ) it follows that
H0

d

RT 1/2 β − β = R1 T 1/2 β 1 − β 1 − T 1/2 β 2 − β 2
R 1 Q −1 Λ

1
1
Wp ( λ ) −
Wp (1) − Wp ( λ )
λ
1−λ

→

.

Using Assumption 1 and Lemma 1,
1 ( F ) −1
RQ−
Qλ R ⇒
λ Ω

×

−1
1
R1 Q−1 Λ,
R1 Q−1 Λ × P b, Fp (r, λ)
λ
1−λ
−1
1
R1 Q−1 Λ,
R 1 Q −1 Λ
λ
1−λ
(1)

By writing out P b, Fp (r, λ) using Fp (r, λ) =

.
(2)

Fp (r, λ) , Fp (r, λ)

, one can obtain

the following expression for above limit after some algebra:
R1 Q−1 ΛP b,

1
1 (1)
(2)
Fp (r, λ) −
Fp (r, λ) Λ Q−1 R1 .
λ
1−λ

Now apply the transformation
d

R1 Q−1 ΛWp (r ) = AWl (r ),

with
R1 Q−1 ΛΛ Q−1 R1 = AA ,
and conclude
1 ( F ) −1
RQ−
Qλ R ⇒ AP b,
λ Ω

1
1 (1)
(2)
Fl (r, λ) −
F (r, λ)
λ
1−λ l

78

A,

giving the needed result:
1
1
Wl (λ) −
(Wl (1) − Wl (λ))
λ
1−λ

Wald( F) ⇒

−1

1
1 (1)
(2)
Fl (r, λ)
× P b, Fl (r, λ) −
λ
1−λ

1
1
Wl (λ) −
(Wl (1) − Wl (λ)) .
λ
1−λ

×

Proof of Corollary 2: Consider the Bartlerr kernel case. Other cases can be proved in
(1)

the same way. Look at P b, λ1 Fl
(1)

definitions of Fl

(2)

(r, λ) and Fl

P b,

−
−

1 (1)
1
(2)
Fl (r, λ) −
F (r, λ)
λ
1−λ l
1 (1)
1
(2)
Fl (r, λ) −
Fl (r, λ)
λ
1−λ

1 (1)
1
(2)
Fl (r, λ) −
Fl (r, λ)
λ
1−λ

0
1− b

1
1 (1)
(2)
Fl (r + b, λ) −
Fl (r + b, λ)
λ
1−λ

0

(1)

2
b

−

+

1− b

1
b

0

1

2
b

0

1

1

(2)

(1 − λ )
(2)

+ Fl

F
2 l

F
2 l

(2)

(r, λ) Fl
(2)

(r + b, λ) Fl
(1)

(r, λ) Fl

0

(2)

(r, λ) × Fl

dr.

(r, λ) = 0 as

1 (1)
(1)
Fl (r, λ) Fl (r, λ) dr
2
λ

(r, λ) dr −

(1)

1− b

1
b

(r, λ) dr dr +

(r + b, λ) + Fl

dr

1 (1)
1
(2)
Fl (r, λ) −
Fl (r, λ)
λ
1−λ

1 (1)
1 (1)
(1)
(1)
Fl (r, λ) Fl (r + b, λ) + 2 Fl (r + b, λ) Fl (r, λ)
2
λ
λ
(2)

(1 − λ )

1

dr

1 (1)
1
(2)
Fl (r + b, λ) −
Fl (r + b, λ)
λ
1−λ

Rewrite the above expression by noting that Fl

+

(r, λ) in (1.24). Rewrite it using the

(r, λ) in Lemma 1.

1
1 (1)
(2)
Fl (r, λ) −
Fl (r, λ)
λ
1−λ

0

1− b

1
b

1
b

1

2
b

=

(2)

(r, λ) − 1−1 λ Fl

(1 − λ )

0
1− b

1
b

0

(2)

(r + b, λ) Fl

79

1

(2)
F
2 l

(2)

(r, λ) Fl

dr

(r + b, λ)

1
(1)
(2)
Fl (r, λ) Fl (r + b, λ)
λ (1 − λ )
(2)

(r, λ) + Fl

(1)

(r + b, λ) Fl

(r, λ)

dr.

(1)

By reminding that λ is given fixed and noticing Fl (r, λ)

(2)

0≤r ≤ λ

and Fl (r, λ)

λ ≤r ≤1

are independent, one can still preserve the distribution of the above expression under the
following transformations

√
√

λWl∗

r
λ

d

(1)

= Wl (r ) for W (·) in Fl (r, λ),

1 − λWl∗∗

r−λ
1−λ

Then apply change of variables:

d

(2)

= Wl (r ) − Wl (λ) for W (·) in Fl (r, λ).

r
λ

= u for the first two integrals and

r −λ
1− λ

= v for the next

two integrals. Finally, notice the term outside the inverse in equation (1.24) is independent
of the term inside the inverse. So one can apply the following separate transformation to
the outside term:
1
1
d
Wl (λ) −
(Wl (1) − Wl (λ)) =
λ
1−λ

1
λ (1 − λ )

Wl (1),

which yields the desired result.
Proof of Theorem 2: Using Lemma 2 and the transformation shown in the proof of Theorem 1, the result in Theorem 2 immediately follows.
Proof of Corollary 3: Proof is similar to the proof of Corollary 2.
Proof of Proposition 3: Frisch-Waugh Theorem gives
T

β=

−1

∑ Xt Xt

t =1
T

=

−1

∑ Xt Xt
T

∑ Xt Xt

t =1

(1.44)

t =1
T

∑ Xt Xt β +

t =1

=

T

∑ Xt y t

−1

T

∑ Xt u t −

t =1

t =1

T

T

∑ Xt Xt β +

t =1

∑ Xt u t

t =1

80

,

T

∑ X t z t ( Z Z ) −1 Z u

t =1

and it immediately follows that

T

1/2

β−β =

T

−1

−1

T

∑ Xt Xt

T

−1/2

=

T

.

(1.45)

−1

T

∑

∑ Xt u t

t =1

t =1

−1

T

Xt − X Z ( Z Z )

−1

Xt − z t ( Z Z )

zt

−1

ZX

(1.46)

t =1

×

T −1/2

T

∑

X t − X Z ( Z Z ) −1 z t u t

.

(1.47)

t =1

Under Assumption 1’ and 2’ it follows in a straightforward manner that


√


T ( β − β ) ⇒ Q −1 
XX



1
Λ1 Wp+q (λ) − λQ xZ Q−
ZZ Λ2 Wp+q (1)
1
Λ1 Wp+q (1) − Wp+q (λ) − (1 − λ) Q xZ Q−
ZZ Λ2 Wp+q (1)

In order to derive the limit of

√

T ( R β − r ) following immediate results are useful:

T −1

Q XZ ≡ p lim

T

∑ Xt z t

t =1


=


λQ xZ

(1 − λ) Q xZ

T −1

T

∑ Xt Xt

t =1

Q X X = p lim

T

−1

,



2p×q




Q XX ≡ p lim


 (1.48)

0

λQ xx
=

0
(1 − λ) Q xx

T

∑ Xt Xt

,
2p×2p

1
= Q XX − Q XZ Q−
ZZ Q XZ .

t =1

Also by recalling from matrix algebra (see e.g. Schott (1997)) that
1
−1
−1
Q −1 = Q −
XX + Q XX Q XZ Q ZZ − Q XZ Q XX Q XZ
XX

81

−1

1
Q XZ Q−
XX

(1.49)

and using (1.49) one can further show that




1 −1
 λ Q xx

Q −1 = 
XX

+P

P
1
−1
1−λ Q xx

P

+P


,

(1.50)

where
1
−1
P = Q−
xx Q xZ Q ZZ − Q xZ Q xx Q xZ

−1

1
Q xZ Q−
xx .

Now plug (1.50) in (1.48) and conclude

√

H0

T ( R β − r) =

1
⇒ R1 Q −
xx Λ1

√

TR β − β

(1.51)

1
1
Wp + q ( λ ) +
Wp + q ( λ ) − Wp + q (1)
λ
1−λ

.

The following lemma is used in the proof of Lemma 3.
1
−1
−1
−1
Lemma 4. Let K = Q xZ Q−
ZZ Q xZ . Then it holds that Q xx KP = P − Q xx KQ xx .

Proof of Lemma 4: From equation (1.46),








−1
1
2
0
λ(1 − λ) Q xZ Q−
λQ xx
  λ Q xZ Q ZZ Q xZ
ZZ Q xZ 
QX X = 
−
 
.
−1
−1
2
0
(1 − λ) Q xx
λ(1 − λ) Q xZ Q ZZ Q xZ (1 − λ) Q xZ Q ZZ Q xZ

The desired result comes from the identity Q X X Q−1 = I by substituting equation (1.50)
XX

for Q−1 .
XX

Proof of Lemma 3: First note that implicit in the proof of Proposition 3 is the result that
p lim Q−1 = Q−1 . For R = ( R1 , − R1 ) it follows that
XX

XX

p lim R Q−1 = R1
XX

1
1 −1
1
Q xx , −
Q−
xx
λ
1−λ

82

(1.52)

using (1.50). The scaled partial sum process is given by

T −1/2 S[rT ] = T −1/2
ξ

=T

−1/2

[rT ]

[rT ]

∑ Xt u t − T ∑ Xt Xt
−1

t =1

√

[rT ]

∑ Xt u t

t =1

T ( β − β) − T

−1

t =1

[rT ]

∑ Xt z t

t =1

ZZ
T

−1

T −1/2 Z u .
(1.53)

For 0 ≤ r < λ, the first term in (1.53) satisfies

T −1/2





−1
Λ1 Wp+q (r ) − λQ xZ Q ZZ Λ2 Wp+q (r )

[rT ]

∑ Xt u t ⇒ 

1
−(1 − λ) Q xZ Q−
ZZ Λ2 Wp+q (r )

t =1

.

(1.54)

Hence with R = ( R1 , − R1 ) , from (1.52) and (1.54) it follows that

R Q−1 T −1/2
XX



[rT ]

∑ Xt u t ⇒ R1

t =1



−1
Λ1 Wp+q (r ) − λQ xZ Q ZZ Λ2 Wp+q (r )

1
1 −1
1
Q xx , −
Q−

xx
λ
1−λ

=



1
−(1 − λ) Q xZ Q−
ZZ Λ2 Wp+q (r )

1
1
R1 Q −
xx Λ1 Wp+q (r ).
λ

For the first part of the second term in (1.53) it follows that

T −1



−

[rT ]







−1
rλQ xZ Q ZZ Q xZ

rQ xx 0 p× p 
X
X
⇒
−
t
∑ t 
t =1
0 p× p 0 p× p


1
rλQ xZ Q−
ZZ Q xZ
1
r (1 − λ) Q xZ Q−
ZZ Q xZ





1
r (1 − λ) Q xZ Q−
ZZ Q xZ 

0 p× p

λ2 Q

−1
xZ Q ZZ Q xZ

0 p× p 

+r
1
0 p× p
λ(1 − λ) Q xZ Q−
ZZ Q xZ



0 p× p



1
λ(1 − λ) Q xZ Q−
ZZ Q xZ 
1
(1 − λ)2 Q xZ Q−
ZZ Q xZ



2
2
rQ xx + (rλ − 2rλ)K −r (1 − λ) K 
=
,
2
2
−r (1 − λ ) K
r (1 − λ ) K

83





1
where K = Q xZ Q−
ZZ Q xZ . Hence with R = ( R1 , − R1 ) ,

R Q −1 T −1
XX



[rT ]

∑ Xt Xt ⇒ R1

t =1

+ (rλ2

− 2rλ)K
1 −1
1
rQ xx
1
Q xx , −
Q−
×
xx
λ
1−λ
−r (1 − λ )2 K

= rR1

−r (1 − λ )2 K
r (1 − λ )2 K

λ − 1 −1
1
1
I − Q−
Q xx K ,
xx K,
λ
λ

which combined with (1.48) and Lemma 4 immediately yields

R Q −1
XX

T

−1

[rT ]

√

∑ Xt Xt

T ( β − β) ⇒

t =1




rR1

1 −1
 λ Q xx

1
λ − 1 −1
1
I − Q−
Q xx K × 
xx K,
λ
λ



×

=

P
1
−1
1−λ Q xx

P

+P

1
Λ1 Wp+q (1) − Wp+q (λ) − (1 − λ) Q xZ Q−
ZZ Λ2 Wp+q (1)

1 −1
Q , 0 p× p
λ2 xx


×






1
Λ1 Wp+q (λ) − λQ xZ Q−
ZZ Λ2 Wp+q (1)



= rR1

+P




1
Λ1 Wp+q (λ) − λQ xZ Q−
ZZ Λ2 Wp+q (1)
1
Λ1 Wp+q (1) − Wp+q (λ) − (1 − λ) Q xZ Q−
ZZ Λ2 Wp+q (1)

r
r
1
−1
−1
R
Q
Λ
W
(
λ
)
−
R1 Q −
p
+
q
1
1
xx
xx Q xZ Q ZZ Λ2 Wp+q (1).
λ
λ2

Finally, premultiplying the third term in (1.53) by R Q−1 gives
XX

R Q −1 T −1
XX

=

[rT ]

∑ Xt z t

t =1

R Q −1 T −1
XX

ZZ
T

−1

T −1/2 Z u

[rT ]

∑

Xt − X Z ( Z Z )

t =1

84

−1

zt zt

ZZ
T

−1

T −1/2 Z u











⇒ R1
=

1
1 −1
1
Q xx , −
Q−
xx
λ
1−λ



 r (1 − λ) Q xZ  −1
×
 Q ZZ Λ2 Wp+q (1)
−r (1 − λ) Q xZ

r
1
−1
R1 Q −
xx Q xZ Q ZZ Λ2 Wp+q (1).
λ

Combining the results for the three terms gives
1
r
ξ
1
1
R Q−1 T −1/2 S[rT ] ⇒ R1 Q−
Λ1 Wp + q (r ) − R1 Q −
Λ1 2 Wp + q ( λ )
xx
xx
XX
λ
λ
r
r
1
−1
−1
−1
+ R1 Q −
xx Q xZ Q ZZ Λ2 Wp+q (1) − R1 Q xx Q xZ Q ZZ Λ2 Wp+q (1)
λ
λ
1
= R1 Q −
xx Λ1

r
1
Wp + q (r ) − 2 Wp + q ( λ )
λ
λ

1 (1)
1
= R1 Q −
xx Λ1 Fp+q (r, λ ) .
λ

(1.55)

Now consider λ ≤ r ≤ 1, for the first term in 1.53) it follows that

T −1/2

[rT ]



∑ Xt u t ⇒ 


t =1



1
Λ1 Wp+q (λ) − λQ xZ Q−
ZZ Λ2 Wp+q (r )
1
Λ1 Wp+q (r ) − Wp+q (λ) − (1 − λ) Q xZ Q−
ZZ Λ2 Wp+q (r )


.

Hence with R = ( R1 , − R1 ) ,
R Q−1 T −1/2
XX


R1

1
1 −1

Q xx , −
Q −1 × 
λ
1 − λ xx
Λ
1
= R1 Q −
xx Λ1

[rT ]

∑ Xt u t ⇒

t =1

1
Λ1 Wp+q (λ) − λQ xZ Q−
ZZ Λ2 Wp+q (r )
1

Wp + q (r ) − Wp + q ( λ )

1
− (1 − λ) Q xZ Q−
ZZ Λ2 Wp+q (r )

1
1
Wp + q ( λ ) −
Wp + q (r ) − Wp + q ( λ )
λ
1−λ

For the first part of the second term in (1.53) it follows that

T −1

[rT ]


λQ xx

∑ Xt Xt ⇒ 

t =1


0 p× p

85

0 p× p

(r − λ) Q xx




.



.



−

λ2 Q



1
λ(1 − λ) Q xZ Q−
ZZ Q xZ

−1
1
λ(r − λ) Q xZ Q−
ZZ Q xZ (r − λ )(1 − λ ) Q xZ Q ZZ Q xZ



−

−1
xZ Q ZZ Q xZ

λ2 Q

−1
xZ Q ZZ Q xZ



1
λ(r − λ) Q xZ Q−
ZZ Q xZ

−1
1
λ(1 − λ) Q xZ Q−
ZZ Q xZ (r − λ )(1 − λ ) Q xZ Q ZZ Q xZ



+r 


λ2 Q






−1
xZ Q ZZ Q xZ

1
λ(1 − λ) Q xZ Q−
ZZ Q xZ 

1
λ(1 − λ) Q xZ Q−
ZZ Q xZ

− 2) λ2 K

λQ xx + (r
=
(2 − r ) λ2 − λ K




1
(1 − λ)2 Q xZ Q−
ZZ Q xZ

(2 − r ) λ2




−λ K

(r − λ) Q xx + ((r − 2) λ + r ) (λ − 1) K


.

Hence with R = ( R1 , − R1 ) ,
R Q −1 T −1
XX


R1

[rT ]

∑ Xt Xt ⇒

t =1

− 2) λ2 K

1
1 −1
λQ xx + (r
1
Q xx , −
Q−
×
xx
λ
1−λ
(2 − r ) λ2 − λ K

= R1 I +

(2 − r ) λ2

It directly follows that

XX

T −1

[rT ]

∑

Xt Xt

t =1

86

√

−λ K

(r − λ) Q xx + ((r − 2) λ + r ) (λ − 1) K

r−λ
λ ( r − 1 ) −1
1
Q xx K, −
I + (r − 1) Q −
xx K .
1−λ
1−λ

R Q −1



T ( β − β) ⇒





I+

R1

1 −1
 λ Q xx

r−λ
λ ( r − 1 ) −1
1
Q xx K, −
I + (r − 1) Q −
xx K × 
1−λ
1−λ



×

P


1
Λ1 Wp+q (λ) − λQ xZ Q−
ZZ Λ2 Wp+q (1)
1
Λ1 Wp+q (1) − Wp+q (λ) − (1 − λ) Q xZ Q−
ZZ Λ2 Wp+q (1)

= R1


×



+P

P
1
−1
1−λ Q xx




r−λ
1 −1
Q −1
Q xx , −
λ
(1 − λ)2 xx


1
Λ1 Wp+q (λ) − λQ xZ Q−
ZZ Λ2 Wp+q (1)
1
Λ1 Wp+q (1) − Wp+q (λ) − (1 − λ) Q xZ Q−
ZZ Λ2 Wp+q (1)

1
= R1 Q −
xx Λ1




r−λ
r−λ
1
Wp + q ( λ ) +
Wp + q ( λ ) −
Wp + q (1)
2
λ
(1 − λ )
(1 − λ )2
1
−1
− R1 Q −
xx Q xZ Q ZZ Λ2

1−r
Wp + q (1).
1−λ

Finally, premultiplying the third term in (1.53) by R Q−1 gives
XX

R Q −1 T −1
XX

=

[rT ]

∑ Xt z t

t =1

R Q −1 T −1
XX

ZZ
T

−1

T −1/2 Z u

[rT ]

∑

Xt − X Z ( Z Z )

−1

t =1

zt zt

ZZ
T



⇒ R1
=

1 −1
1
Q xx , −
Q −1
λ
1 − λ xx

−1

T −1/2 Z u



 λ(1 − r ) Q xZ  −1
×
 Q ZZ Λ2 Wp+q (1)
−λ(1 − r ) Q xZ

1−r
−1
1
R1 Q −
xx Q xZ Q ZZ Λ2 Wp+q (1).
1−λ

Combining the results for the three terms gives
1
R Q−1 T −1/2 S[rT ] ⇒ R1 Q−
xx Λ1
ξ

XX

87

+P




× −

1
r−λ
r−λ
W (λ) +
W (1)
(W (r ) − W (λ)) −
2
1−λ
(1 − λ )
(1 − λ )2

1
−1
+ R1 Q −
xx Q xZ Q ZZ Λ2
1
= R1 Q −
xx Λ1 −

1−r
1−r
1
−1
W (1) − R1 Q −
W (1)
xx Q xZ Q ZZ Λ2 ·
1−λ
1−λ

1
r−λ
r−λ
W (λ) +
W (1)
(W (r ) − W (λ)) −
2
1−λ
(1 − λ )
(1 − λ )2
1
= − R1 Q −
xx Λ1 ·

1
(2)
F (r, λ).
1 − λ p+q

(1.56)

By combining (1.55) and (1.56), for r ∈ [0.1],
1
R Q−1 T −1/2 S[Tr] ⇒ R1 Q−
xx Λ1
ξ

XX

1
1 (1)
(2)
Fp+q (r, λ) −
Fp+q (r, λ) .
λ
1−λ

Proof of Theorem 3: To conserve on space the proof for this Theorem is provided only
for the case of the Bartlett kernel with M = T (i.e. b = 1). However, the proof given here
goes through for other kernels and different values of b. Note that with b = 1 the HAC
estimator in equation (1.41) can be rewritten as
( F)
Ω b =1

2
=
T

T −1

∑ T −1/2 St T −1/2 St .
ξ

ξ

t =1

With this HAC estimator the term within the inverse in (1.35) is given by


2
T

T −1 

∑ R

t =1

T −1

T

−1

∑ Xs Xs

T −1/2 St T −1/2 St
ξ

T −1

ξ

s =1

1
⇒ P 1, R1 Q−
xx Λ1

T

∑ Xs Xs

s =1

−1

R





1 (1)
1
(2)
Fp+q (r, λ) −
Fp+q (r, λ)
λ
1−λ

where the limit is obtained directly from Lemma 3 and the continuous mapping theorem.
The result for (1.35) can be obtained by using similar arguments as those used in Theorem
d

1
1 where the transformation is used: R1 Q−
xx Λ1 Wp+q (r ) = Ξ · Wl (r ), 0 ≤ r ≤ 1 for a p.d.
1
−1
matrix Ξ satisfying ΞΞ = R1 Q−
xx Λ1 Λ1 Q xx R1 .
l ×l

88

REFERENCES

89

REFERENCES

Andrews, D. W. K.: (1991), ‘Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimation’. Econometrica 59, 817–854.
Andrews, D. W. K.: (1993), ‘Tests for Parameter Instability and Structural Change with
Unknown Change Point’. Econometrica 61, 821–856.
Andrews, D. W. K. and W. Ploberger: (1994), ‘Optimal Tests When a Nuisance Parameter
is Present Only Under the Alternative’. Econometrica 62, 1383–1414.
Bai, J. and P. Perron: (2006), ‘Multiple structural change models: a simulation analysis’.
Econometric theory and practice: Frontiers of analysis and applied research pp. 212–237.
Bai, J. S. and P. Perron: (1998), ‘Estimating and Testing Linear Models with Multiple
Structural Breaks’. Econometrica 66, 47–78.
Davidson, J.: (1994), Stochastic Limit Theory. New York: Oxford University Press.
De Jong, R. M. and J. Davidson: (2000), ‘Consistency of Kernel Estimators of Heteroskedastic and Autocorrelated Covariance Matrices’. Econometrica 68, 407–424.
Gonçalves, S. and T. J. Vogelsang: (2011), ‘Block Bootstrap HAC Robust Tests: The Sophistication of the Naive Bootstrap’. Econometric Theory 27(4), 745–791.
Hansen, B. E.: (1992), ‘Consistent Covariance Matrix Estimation for Dependent Heterogenous Processes’. Econometrica 60, 967–972.
Hashimzade, N. and T. J. Vogelsang: (2008), ‘Fixed-b Asymptotic Approximation of the
Sampling Behavior of Nonparametric Spectral Density Estimators’. Journal of Time Series
Analysis 29, 142–162.
Jansson, M.: (2002), ‘Consistent Covariance Estimation for Linear Processes’. Econometric
Theory 18, 1449–1459.
Jansson, M.: (2004), ‘The Error Rejection Probability of Simple Autocorrelation Robust
Tests’. Econometrica 72, 937–946.
Kiefer, N. M. and T. J. Vogelsang: (2002a), ‘Heteroskedasticity-autocorrelation robust
standard errors using the Bartlett kernel without truncation’. Econometrica 70, 2093–2095.
Kiefer, N. M. and T. J. Vogelsang: (2002b), ‘Heteroskedasticity-Autocorrelation Robust
Testing Using Bandwidth Equal to Sample Size’. Econometric Theory 18, 1350–1366.
Kiefer, N. M. and T. J. Vogelsang: (2005), ‘A New Asymptotic Theory for
Heteroskedasticity-Autocorrelation Robust Tests’. Econometric Theory 21, 1130–1164.

90

Newey, W. K. and K. D. West: (1987), ‘A Simple, Positive Semi-Definite, Heteroskedasticity and Autocorrelation Consistent Covariance Matrix’. Econometrica 55, 703–708.
Perron, P.: (2006), ‘Dealing with structural breaks’. Palgrave handbook of econometrics 1,
278–352.
Phillips, P. C. B. and S. N. Durlauf: (1986), ‘Multiple Regression with Integrated Processes’. Review of Economic Studies 53, 473–496.
Phillips, P. C. B. and V. Solo: (1992), ‘Asymptotics for Linear Processes’. The Annals of
Statistics 20, 971–1001.
Schott, J. R.: (1997), ‘Matrix analysis for statistics’. A Wiley InterScience Publication, New
York.
Sun, Y.: (2013), ‘Fixed-smoothing Asymptotics in a Two-step GMM Framework’. Working Paper, Department of Economics, UCSD.
Sun, Y., P. C. B. Phillips, and S. Jin: (2008), ‘Optimal Bandwidth Selection in
Heteroskedasticity-Autocorrelation Robust Testing’. Econometrica 76, 175–194.
Wooldridge, J. M. and H. White: (1988), ‘Some invariance principles and central limit
theorems for dependent heterogeneous processes’. Econometric Theory pp. 210–230.

91

CHAPTER 2
A Test of Parameter Instability Allowing
for Change in the Moments of Explanatory
Variables
2.1

Introduction

Most of the structural change literature addresses the instability of the parameters in the
conditional model. The goal of this chapter is to develop a valid HAC-robust test when
there is a shift in the mean and second moment of the explanatory variables. The break
date, λ x T, for the mean and second moment is assumed to be different from the break
date λT for the regression parameters. The bootstrap approach in Hansen (2000) considers a setup where general form of the structural change in the marginal distribution of
the explanatory variables are allowed. But Hansen (2000) assumes that the error term
is a martingale difference sequence with respect to a certain information set. Thus serial
correlation of the product of x variables and the error, ut , is not allowed. This can be a limitation particularly in the static time series regression model where the series { xt ut }t=1,...,T
often displays serial correlation. In this Chapter a valid HAC-robust approach for testing structural change in the regression slope/intercept parameters is developed. To ease
the distribution theory a modified version of the standard set of high level conditions is
introduced. To make the test robust to serial correlation and heteroskedasticity, a HAC
estimator is used for constructing the test statistic and the fixed-b theory developed in
Kiefer and Vogelsang (2005) is applied to derive the asymptotic distribution. The limiting
distributions of the statistics are pivotal. The rest of the chapter is organized as follows.

92

Section 2.2 lays out the basic set up of the problem. Section 2.3 derives the limiting distributions of the appropriate test statistics under the fixed-b approach. Section 2.4 presents
fixed-b critical values for certain break point value and bandwidths. The finite sample
properties of the test is examined in Monte Carlo simulation experiments. Section 2.5
summarizes and concludes. Proofs are collected in the Appendix.

2.2

Model of Structural Change and Preliminary Results

Suppose the univariate series { xt } has a mean shift at t = λ x T. Denote

E ( x t ) = µi (t) ,

with
µi (t)



 µ1 for t ≤ λ x T
.
=

 µ2 for t ≥ λ x T + 1

(2.1)

The subscript i (t) indicates a regime for time t and µ1 is the mean in the first regime and
µ2 is the mean in the second regime. Suppose λ x , µ1 , and µ2 are known.
Now consider a simple time series regression model with a structural break given by

y t = α 1 Dt + α 2 ( 1 − Dt ) + β 1 x t Dt + β 2 x t ( 1 − Dt ) + u t ,

(2.2)

Dt = 1(t ≤ λT ),

where xt is a regressor, λ ∈ (0, 1) is a hypothetical break point for the regression parameters, and 1( · ) is the indicator function. For expositional simplicity let us suppose λT and
λ x T are integer-valued. The regression model (2.2) implies that both regression parameters are subject to potential structural change (full structural change model). Consider the

93

null hypothesis of no structural change in the slope parameter:

H0 : β 1 = β 2

(2.3)

The OLS estimators of α1 , α2 , β 1 , and β 2 are given by
−1

λT

β1 =

∑ ( xt − x1 )

t =1

T

β2 =

∑

λT

∑ ( xt − x1 ) yt

2

t =1
−1

t=λT +1

(2.4)

T

∑

( x t − x 2 )2

,

( xt − x2 ) yt ,

t=λT +1

α1 = y1 − β 1 x 1 ,
α2 = y2 − β 2 x 2 ,
where
1
y1 =
λT
x1 =

1
λT

λT

1
∑ y t , y2 = (1 − λ ) T
t =1
λT

∑ xt , x2 =

t =1

1
(1 − λ ) T

T

∑

yt , and

t=λT +1
T

∑

xt .

t=λT +1

Consider the case where λ > λ x . The asymptotic distribution of the slope estimator can
be obtained under a certain set of high level conditions. A typical set of conditions is:
[rT ]

p

Assumption 1. T −1 ∑t=1 xt2→ rq2x , 
uniformly in r ∈ [0, 1], and q2x is strictly positive.
[rT ]  ut 
Assumption 2. T −1/2 ∑t=1 
 ⇒ ΛW (r ), r ∈ [0, 1], where ΛΛ = Σ, and W (r ) is a
xt ut

2 × 1 standard Wiener process.
Assumptions 1 and 2 are the standard high level conditions used widely in the econometrics literature. Under these assumptions an immediate result is presented in the next
Proposition without proof.
Proposition 4. Suppose λ is known and λ > λ x . Under Assumptions 1 and 2 the OLS estimators
94

in (2.4) have the following asymptotic distributions:

T

1/2

β1 − β1 ⇒

λq2x

+ λx

λx
1−
λ

( µ1 − µ2 )

T 1/2 β 2 − β 2 ⇒ (1 − λ) q2x

−1

2

−1

× − µ2 −

λx
(µ1 − µ2 ) , 1 · ΛW (λ) ,
λ

× (−µ2 , 1) · Λ (W (1) − W (λ)) .

For simplicity assume q2x is known. To test the null hypothesis (2.3), one can consider
using a robust Wald statistic
−1

avar T 1/2 β 1 − β 2

Wald ( Tb ) = T β 1 − β 2

β1 − β2 ,

where Tb = λT denotes a hypothetical break date and
avar T 1/2 β 1 − β 2

λ λq2x + λ x 1 −

λx
λ

( µ1 − µ2 )2

× Σ × − µ2 −
+ (1 − λ) (1 − λ) q2x

−2

=

−2

× − µ2 −

λx
( µ1 − µ2 ) , 1
λ

λx
( µ1 − µ2 ) , 1
λ

× (−µ2 , 1) × Σ × (−µ2 , 1) ,

where Σ is a nonparametric kernel estimator of Σ = ΛΛ using a kernel K (·) and the
bandwidth M:
Σ = T −1

T

T

∑∑K

t =1 s =1



|t − s|
M

vt vs ,



 ut 
with vt = 
 . Unfortunately, Wald ( Tb ) does not have a pivotal asymptotic null
xt ut
distribution under fixed-b asymptotics as long as µ1 is not equal to µ2 and neither is
the supremum statistic. The reason is the vector, λ λq2x + λ x 1 −

− µ2 −

λx
λ

λx
λ

( µ1 − µ2 )2

(µ1 − µ2 ) , 1 , is generally different from (1 − λ) (1 − λ) q2x

95

−1

−1

×

× (−µ2 , 1)

unless µ1 is equal to µ2 .
In order to obtain a test statistic which has a pivotal fixed-b limiting distribution, a
modified regression equation and more general version of the high level conditions are
introduced. The following two high-level conditions replace Assumptions 1 and 2.


2 p 
rs21 uniformly in r ∈ [0, λ x ]
[
rT
]
−
1
→
Assumption 1’. T ∑t=1 xt − µi(t)

 λ x s2 + (r − λ x ) s2 uniformly in r ∈ [λ x , 1]
2
1
with s2i > 0 for i = 1, 2.
Obviously under Assumption 1’

T

−1

[rT ]

x t − µi (t)

∑

si (t)

t =1


[rT ] 
Assumption 2’. T −1/2 ∑t=1 

2
p

→ r uniformly in r ∈ [0, 1].



1

si (t) u t
x t − µi (t)
si (t)

ut


 ⇒ ΛW (r ), r ∈ [0, 1], where ΛΛ = Ω, W (r )

is a 2 × 1 standard Wiener process and si(t) = s1 1(t ≤ λ x T ) + s2 1(t ≥ λ x T + 1).
When µ1 = µ2 = µ and s21 = s22 = s2 , Assumption 1’ is equivalent to Assumption 1
and Assumption 2’ is equivalent to Assumption 2. Under µ1 = µ2 = µ and s21 = s22 = s2 ,
[rT ]

[rT ]

p

Assumption 1’ implies T −1 ∑t=1 ( xt − µt )2 = T −1 ∑t=1 ( xt − µ)2 → rs2 and it is easy to
show this is equivalent to Assumption 1 by defining q2x = s2 + µ2 . Also, one can show the
equivalence of Assumption 2’ and 2 through the relationship




 s 0

 Λ = Λ.
µs s
Note however that Assumption 1’ is more general than Assumption 1 since it allows the
mean or probability limit of (centered and noncentered) sample second moment of xt
to have a break. As a special case, if xt is a stationary process within each regime, then
s2i is the variance of xt process and Assumption 1’ is a robust version of Assumption 1
96

allowing for a shift of the variance. Assumption 2’ is also a more general version than
Assumption 2. To see this why, suppose one has evidence of shift in q2x (or s2i(t) ). Then it
would be hard to justify Assumption 2 as being still valid. For example, take the case
where xt is stationary and conditional homoskedasticity holds, E u2t | xt

= σu2 . For a

stationary variable, E xt2 = q2x . Then the variance of xt ut , denoted by Γ0 , is E xt2 u2t =
E E xt2 u2t | xt

= σu2 E xt2 = σu2 q2x . Hence Γ0 should have a structural break in general as

long as there is a shift in q2x . This in turn implies that the long-run variance matrix Ω ≡
ΛΛ also has structural break because Γ0 is a component of the long run variance matrix.
Therefore, it would not be appropriate to maintain Assumption 2 while allowing a shift
in q2x or s2i(t) as in Assumption 1’. One needs to match Assumption 1’ with Assumption
2’ due to this reason. However, note that Assumption 2’ does not allow the long run
variance to change in an arbitrary fashion. Assumption 2’ can only reflect the impact of
change in q2x or s2i(t) on Ω in a particular way.
In next Section tests of parameter instability of α or(and) β will be presented under a
particular specification of breaks in µi(t) , si(t) and breaks in (α, β), which is as follows:
µi(t) , si(t) has a structural break at λ x T and (α, β) is allowed to have a break at λT.
There may be an empirical application where a different specification is more relevant.
For example, only µi(t) has structural break at λ x T and (α, β) are allowed to have break at
λT. But analysis for these other cases is not pursued in this chapter.

2.3

Asymptotic Results

Suppose µi(t) , si(t) has a break at λ x T and (α, β) is allowed to change at an unknown
break date, t = λT. Assume λ x and µi(t) , si(t)

are given (known). Also, assume λ ∈

[ , (λ x − )] ∪ [(λ x + ) , 1 − ] following Andrews (1993). The admissible values for λ
is obtained by trimming the values at both ends of the sample period and the values in
the neighborhood of λ x T. It is necessarily implied that the break date for the regression

97

parameters (α, β) should not be same as the break date for the moments. Recall the regression equation (2.2). Define wt =

xt
si (t) .

Without parameter instability, the equation can

be written as

yt = α + βxt + ut = α + βsi(t)

xt
si (t)

+ ut = α + β (s1 1 (t ≤ λ x T ) + s2 1 (t > λ x T )) wt + ut

= α + βs1 1 (t ≤ λ x T ) wt + βs2 1 (t > λ x T ) wt + ut .
Once we allow (α, β) to have a break at λT, the above regression equation can be rewritten
as

yt = α1 1 (t ≤ λT ) + α2 1 (t > λT ) + ( β 1 1 (t ≤ λT ) + β 2 1 (t > λT )) s1 1 (t ≤ λ x T ) wt

+ ( β 1 1 (t ≤ λT ) + β 2 1 (t > λT )) s2 1 (t > λ x T ) wt + ut .
Note that there are four interaction terms made by the two time indicator functions and
one of these four interactions is identical to zero. For example, if λ < λ x , then 1 (t ≤ λT ) ×
1 (t > λ x T ) = 0. Depending on the relative magnitude of λ and λ x , one can rewrite the
regression equation by reparametrization. For λ < λ x

yt = α1 1 (t ≤ λT ) + α2 1 (λT < t ≤ λ x T ) + α3 1 (t > λ x T )

(2.5)

+ γ1 wt 1 (t ≤ λT ) + γ2 wt 1 (λT < t ≤ λ x T ) + γ3 wt 1 (t > λ x T ) + ε t ,
where γ1 = β 1 s1 , γ2 = β 2 s1 , γ3 = β 2 s2 , and α2 = α3 . For λ > λ x
yt = α1 1 (t ≤ λ x T ) + α2 1 (λ x T < t ≤ λT ) + α3 1 (t > λT )

+ γ1 wt 1 (t ≤ λ x T ) + γ2 wt 1 (λ x T < t ≤ λT ) + γ3 wt 1 (t > λT ) + ε t ,
where γ1 = β 1 s1 , γ2 = β 1 s2 , γ3 = β 2 s2 , and α1 = α2 .

98

(2.6)

Denote as W the T × 6 matrix which collects the six explanatory variables: for (2.5),
1 (t ≤ λT ) , wt 1 (t ≤ λT ) , 1 (λT < t ≤ λ x T ) , wt 1 (λT < t ≤ λ x T ) , 1 (t > λ x T ) , wt 1 (t > λ x T )
in this order and for (2.6), 1 (t ≤ λ x T ) , wt 1 (t ≤ λ x T ) , 1 (λ x T < t ≤ λT ) , wt 1 (λ x T < t ≤ λT ) ,
1 (t > λT ) , and wt 1 (t > λT ) in this order.
Notice that the extra parameter for α is introduced for each equation so that the regression model takes the form of full structural change model with three regimes.
The stability of α can be rewritten as α1 = α2 in (2.5) and α2 = α3 in (2.6). Likewise the
stability of slope parameter β can be rewritten as β 1 = β 2 in (2.5) and β 2 = β 3 in (2.6).
The parameters in (2.5) and (2.6) are estimated by OLS and robust Wald statistics will be
constructed based on these estimators. The OLS estimators in (2.5) are given by
−1

λT

γ1 =

∑ ( w t − w1 )

t =1
−1

t =1

λx T

∑

γ2 =

λT

∑ ( w t − w1 ) y t

2

−1

T

∑

( w t − w2 ) y t ,

t=λT +1

t=λT +1

γ3 =

λx T

∑

( w t − w2 )2

,

( w t − w3 )

T

∑

2

( w t − w3 ) y t ,

t = λ x T +1

t = λ x T +1

and

α1 = y1 − γ1 w1 ,
α2 = y2 − γ2 w2 ,
α3 = y3 − γ3 w3 ,
where
w1 =

1
λT

∑λT
t =1

xt
s1

y1 =

∑t=x λT +1

λ T

xt
s1

and y2 =

∑tT=λx T +1

xt
s2

y3 =

w2 =

1
λ x T −λT

w3 =

1
T −λ x T

1
λT

∑λT
t =1 y t

1
λ x T −λT
1
T −λ x T

λ T
∑t=x λT +1 yt .

∑tT=λx T +1 yt

Defining x1 , x2 , and x3 similarly as above immediately gives w1 =

99

x1
s1 ,

w2 =

x2
s1 ,

and w3 =

x3
s2 .

One can easily write down the expression for the OLS estimators (γ1 , γ2 , γ3 , α1 , α2 , α3 )

and all the other quantities when λ ∈ [(1 + ) λ x , 1 − ] . The next Proposition presents
the asymptotic limits of the OLS estimators.




 Λ1 
Proposition 5. Suppose λ ∈ [ , (1 − ) λ x ] . Denote Λ = 
 . Under Assumptions 1’
Λ2
and 2’, as T → ∞ the OLS estimators in (2.5) have the following limits:
W (λ)
,
λ
µ
W (λ)
T 1/2 (α1 − α1 ) ⇒ s1 Λ1 − 1 Λ2
,
s1
λ
W (λ x ) − W (λ)
T 1/2 (γ2 − γ2 ) ⇒ Λ2
, and
λx − λ
µ
W (λ x ) − W (λ)
.
T 1/2 (α2 − α2 ) ⇒ s1 Λ1 − 1 Λ2
s1
λx − λ
T 1/2 (γ1 − γ1 ) ⇒ Λ2

Suppose λ ∈ [(1 + ) λ x , 1 − ] . Then the OLS estimators in (2.6) have following limits:
1
Λ2 (W (λ) − W (λ x )) ,
λ − λx
W (λ) − W (λ x )
µ2
,
T 1/2 (α2 − α2 ) ⇒ s2 Λ1 − Λ2
s2
λ − λx
1
Λ2 (W (1) − W (λ)) , and
T 1/2 (γ3 − γ3 ) ⇒
1−λ
µ2
W (1) − W ( λ )
T 1/2 (α3 − α3 ) ⇒ s2 Λ1 − Λ2
.
s2
1−λ
T 1/2 (γ2 − γ2 ) ⇒

Proof: See the Appendix.

2.3.1

Stability of β

Consider the null hypothesis
H0 : β is stable.

100

(2.7)

This null hypothesis implies γ1 = γ2 in (2.5) and γ2 = γ3 in (2.6). From Proposition 10,
for λ ∈ [ , λ x − ] ,
W (λ x ) − W (λ) W (λ)
−
λx − λ
λ
λx
∼ N 0,
Λ2 Λ2 ,
λ (λ x − λ)

T 1/2 (γ2 − γ1 ) ⇒ Λ2

(2.8)

and for λ ∈ [λ x + , 1 − ] ,
W (1) − W ( λ ) W ( λ ) − W ( λ x )
−
1−λ
λ − λx
1 − λx
Λ2 Λ2 .
∼ N 0,
(1 − λ ) ( λ − λ x )

T 1/2 (γ3 − γ2 ) ⇒ Λ2

(2.9)

Test Statistic T1

Denote Tb = λT and define robust Wald statistics:
β
Wald1

β

( Tb ) =

Wald2 ( Tb ) =

T (γ2 − γ1 )2
Λ2 Λ2
T (γ3 − γ2 )2
Λ2 Λ2

for Tb ∈ [ T, (λ x − ) T ] ≡ Ξ1 and

(2.10)

for Tb ∈ [(λ x + ) T, (1 − ) T ] ≡ Ξ2 ,

(2.11)

where Λ2 Λ2 is a nonparametric kernel HAC estimator given by
Λ 2 Λ 2 = T −1
with vt =

T

T

∑∑K

t =1 s =1

x t − µi (t)
si (t)

|t − s|
M

vt vs ,

(2.12)

ut .

The HAC estimator Λ2 Λ2 in Wald1 ( Tb ) is computed using the residuals ut from the reβ

gression of equation (2.5) and Λ2 Λ2 in Wald2 ( Tb ) is computed with the residuals ut from
β

the regression of equation (2.6). Under the assumption of a fixed bandwidth ratio this
101

HAC estimator can be rewritten as a function of partial sum processes (see Kiefer and
Vogelsang (2005)),
[rT ]
β

S[rT ] =

∑ vt =

t =1

[rT ]

x t − µi (t)

t =1

si (t)

∑

ut .

The next Proposition presents the limit of this (scaled) partial sum process.
Proposition 6. Under Assumptions 1’ and 2’, as T → ∞, the limit of the partial sum process is
given by
When 0 < λ < λ x ,

T 1/2 S[rT ] ⇒
β








Λ2 W (r ) − λr W (λ) for 0 ≤ r ≤ λ,

Λ2 W (r ) − W (λ) − λrx−−λλ (W (λ x ) − W (λ)) for λ ≤ r ≤ λ x ,




 Λ2 W (r ) − W (λ x ) − r−λx (W (1) − W (λ x )) for λ x ≤ r ≤ 1,
1− λ x

and
When λ x < λ < 1,

T 1/2 S[rT ] ⇒
β













Λ2 W (r ) −
Λ2 W (r ) − W ( λ x ) −

r
λx W (λ x )

r −λ x
λ−λ x

Λ2 W (r ) − W ( λ ) −

r −λ
1− λ

for 0 ≤ r ≤ λ x ,

(W (λ) − W (λ x ))
(W (1) − W (λ))

for λ x ≤ r ≤ λ,
for λ ≤ r ≤ 1.

Proof: See the Appendix.
Define
H1 ≡ H1 (r, λ, λ x )
r
= W1 (r ) − W1 (λ) · 1(r ≤ λ)
λ

+ W1 (r ) − W1 (λ) −

r−λ
(W1 (λ x ) − W1 (λ)) · 1(λ < r ≤ λ x )
λx − λ

+ W1 (r ) − W1 (λ x ) −

r − λx
(W1 (1) − W1 (λ x )) · 1(λ x < r ≤ 1),
1 − λx

and
H2 ≡ H2 (r, λ, λ x )
102

W1 (r ) −

=

+ W1 (r ) − W1 (λ x ) −

r
W1 (λ x ) · 1(r ≤ λ x )
λx

r − λx
(W1 (λ) − W1 (λ x )) · 1(λ x < r ≤ λ)
λ − λx

+ W1 (r ) − W1 (λ) −

r−λ
(W1 (1) − W1 (λ)) · 1(λ < r ≤ 1),
1−λ

where W1 (·) is a 1-dimensional Wiener process.
Theorem 4. Let λ ∈ (0, 1) and b ∈ (0, 1] be given. Suppose M = bT. Then under Assumptions
1’ and 2’, as T → ∞, the limits under the null hypothesis in (2.7) is given by

H0

β

Wald1 ( Tb ) ⇒

W1 (λ x )−W1 (λ)
λ x −λ

− W1λ(λ)

2

P (b, H1 )

and,
β

H0

Wald2 ( Tb ) ⇒

W1 (1)−W1 (λ)
1− λ

W1 (λ x )
− W1 (λλ)−
−λ x

P (b, H2 )

≡ Wald1∞ (λ, λ x ) ,

2

≡ Wald2∞ (λ, λ x ) ,

The definition of P (b, H1 ) and P (b, H2 ) can be found in Cho and Vogelsang (2014).
Proof: See the Appendix.
Finally, define a test statistic T1 :

T1 = max

β

β

max Wald1 ( Tb ) , max Wald2 ( Tb ) .
Tb ∈Ξ2

Tb ∈Ξ1

This statistic can be used when the break date is unknown. Its limit is given by

max

sup
λ∈[ ,(1− )λ x ]

Wald1∞ (λ, λ x ) ,

sup
λ∈[(1+ )λ x ,1− ]

103

Wald2∞ (λ, λ x ) .

(2.13)

( F)

Test Statistic T1

Alternatively, the inference in this setup can be based on different HAC estimators defined in Cho and Vogelsang (2014). These alternative HAC estimators are given by
Υ 1 = T −1

T

T

∑∑K

t =1 s =1


(1)
vt



=



x t − µ1
s1
x t − µ1
s1
x t − µ2
s2

|t − s|
M

(1) (1)

vt vs ,

1 (t ≤ λT )

(2.14)





1 (λT < t ≤ λ x T ) 
 ut ,

1 (λ x T ≤ t ≤ T )

where λ < λ x , and
Υ2 = T

−1

T

T

∑∑K

t =1 s =1


(2)
vt



=



x t − µ1
s1
x t − µ2
s2
x t − µ2
s2

|t − s|
M

(2) (2)

vt vs ,

1 (t ≤ λ x T )

(2.15)





1 (λ x T < t ≤ λT ) 
 ut ,

1 (λT ≤ t ≤ T )

where λ > λ x . Define robust Wald statistics based on the above HAC estimators:
T (γ2 − γ1 )2
for Tb ∈ [ T, (λ x − ) T ] ≡ Ξ1 ,
=
D1 Q1−1 Υ1 Q1−1 D1

(2.16)

T (γ3 − γ2 )2
=
for Tb ∈ [(λ x + ) T, (1 − ) T ] ≡ Ξ2 ,
D2 Q2−1 Υ2 Q2−1 D2

(2.17)

( F ),β
Wald1
( Tb )

( F ),β
Wald2
( Tb )

104

where




1 0 0
D1 ≡ (1, − 1) × 
,
0 1 0

2
λT
1
0
 T ∑ t =1 ( w t − w 1 )

2
λx T
1
Q1 = 
0

T ∑t=λT +1 ( wt − w2 )

0
0


0 1 0
D2 ≡ (1, − 1) × 
 , and
0 0 1

2
λx T
1
0
 T ∑ t =1 ( w t − w 1 )

2
λT
1
Q2 = 
0

T ∑ t = λ x T +1 ( w t − w 2 )

0
0


0
0
1
T

∑tT=λx T +1 (wt − w3 )

2



,




0
0
1
T

∑tT=λT +1 (wt − w3 )

2



.



Theorem 5. Let λ ∈ (0, 1) and b ∈ (0, 1] be given. Suppose M = bT. Then under Assumptions
1’ and 2’, as T → ∞, the limits under the null hypothesis in (2.7) is given by

( F ),β

Wald1

H

( Tb ) ⇒0

W1 (λ x )−W1 (λ)
λ x −λ

− W1λ(λ)

2

P (b, H3 )

and,
( F ),β

Wald2

W1 (1)−W1 (λ)
1− λ

H0

( Tb ) ⇒

W1 (λ x )
− W1 (λλ)−
−λ x

P (b, H4 )

( F ),∞

≡ Wald1

(λ, λ x ) ,

2

( F ),∞

≡ Wald2

(λ, λ x ) ,

where the processes H3 and H4 are defined as
H3 ≡ H3 (r, λ, λ x )

=
−

1
λx − λ

1
r
W1 (r ) − W1 (λ) · 1(r ≤ λ)
λ
λ

W1 (r ) − W1 (λ) −

r−λ
(W1 (λ x ) − W1 (λ)) · 1(λ < r ≤ λ x ),
λx − λ

105

and
H4 ≡ H4 (r, λ, λ x )

=

1
λ − λx

−

1
1 − λx

W1 (r ) − W1 (λ x ) −

r − λx
(W1 (λ) − W1 (λ x )) · 1(λ x < r ≤ λ)
λ − λx

W1 (r ) − W1 (λ) −

r−λ
(W1 (1) − W1 (λ)) · 1(λ < r ≤ 1).
1−λ

The test statistic for testing the null hypothesis in (2.7) is simply given by
( F)

T1

= max
⇒ max

( F ),β

max Wald1

Tb ∈Ξ1

( F ),β

( Tb ) , max Wald2
Tb ∈Ξ2

( F ),∞

sup
λ∈[ ,(1− )λ x ]

Wald1

(λ, λ x ) ,

( Tb )
sup

λ∈[(1+ )λ x ,1− ]

( F ),∞

Wald2

(λ, λ x )

≡ T ( F),∞ ( , λ x ) .

2.3.2

Stability of α

Consider the null hypothesis
H0 : α is stable.

(2.18)

With a hypothetical break point λT, the break in α may occur at λT. This implies α1 = α2
in (2.5) and α2 = α3 in (2.6) under the null hypothesis (2.18). From Proposition 10, for
λ ∈ [ , λx − ] ,
W (λ x ) − W (λ) W (λ)
−
λx − λ
λ


λx
µ

 s1  
∼ N 0,
s1 , − 1 ΛΛ 
 ,
λ (λ x − λ)
s1
− µs11

T 1/2 (α2 − α1 ) ⇒

µ
s1 Λ1 − 1 Λ2
s1


106

(2.19)

and for λ ∈ [λ x + , 1 − ] ,
W (1) − W ( λ ) W ( λ ) − W ( λ x )
−
1−λ
λ − λx


1 − λx
µ2
 s2  

s2 , −
ΛΛ 
∼ N 0,
 .
s2
(1 − λ ) ( λ − λ x )
− µs22
µ2
s2 Λ1 − Λ2
s2


T 1/2 (α3 − α2 ) ⇒

(2.20)

Test Statistic T2
Denote Tb = λT and define robust Wald statistics:
T ( α2 − α1 )2

Wald1α ( Tb ) =

s1 , −

µ1
s1

× ΛΛ × s1 , −

µ1
s1

for Tb ∈ [ T, (λ x − ) T ] ≡ Ξ1 and
(2.21)

Wald2α ( Tb ) =

T ( α3 − α2 )
s2 , −

µ2
s2

2

× ΛΛ × s2 , −

µ2
s2

for Tb ∈ [(λ x + ) T, (1 − ) T ] ≡ Ξ2 ,
(2.22)

where ΛΛ is a nonparametric kernel HAC estimator given by
ΛΛ = T −1

T

T

∑∑K

t =1 s =1



with ξ t = 

|t − s|
M

ξtξs,

(2.23)



1

si (t)
x t − µi (t)
si (t)


 ut .

As before, the HAC estimator ΛΛ in Wald1α ( Tb ) is computed using the residuals ut from
the regression of equation (2.5) and ΛΛ in Wald2α ( Tb ) is computed with the residuals
ut from the regression of equation (2.6). Under the assumption of fixed bandwidth ratio,
this HAC estimator can be rewritten as a function of partial sum processes (see Kiefer and
Vogelsang (2005)),
[rT ]

S[αrT ] =

∑ ξt =

t =1

[rT ]



∑


t =1

107

1

si (t)
x t − µi (t)
si (t)



 ut .

The next Proposition presents the limit of the (scaled) partial sum processes.
Proposition 7. Under Assumptions 1’ and 2’, as T → ∞, the limit of the partial sum process is
given by
When 0 < λ < λ x ,

T 1/2 S[αrT ] ⇒








Λ W (r ) − λr W (λ) for 0 ≤ r ≤ λ,

Λ W (r ) − W (λ) − λrx−−λλ (W (λ x ) − W (λ)) for λ ≤ r ≤ λ x ,




 Λ W (r ) − W (λ x ) − r−λx (W (1) − W (λ x )) for λ x ≤ r ≤ 1,
1− λ x

and
When λ x < λ < 1,

T 1/2 S[αrT ] ⇒













Λ W (r ) −
Λ W (r ) − W ( λ x ) −

r
λx W (λ x )

r −λ x
λ−λ x

Λ W (r ) − W ( λ ) −

r −λ
1− λ

for 0 ≤ r ≤ λ x ,

(W (λ) − W (λ x ))
(W (1) − W (λ))

for λ x ≤ r ≤ λ,
for λ ≤ r ≤ 1.

Proof: See the Appendix.
The next Theorem presents the limit of the statistics. The limit is the same as in Theorem 8.
Theorem 6. Let λ ∈ (0, 1) and b ∈ (0, 1] be given. Suppose M = bT. Then under Assumptions
1’ and 2’, as T → ∞, the limits under the null hypothesis in (2.7) is given by
H0

Wald1α ( Tb ) ⇒ Wald1∞ (λ, λ x ) ,
and,
H0

Wald2α ( Tb ) ⇒ Wald2∞ (λ, λ x ) .
The definition of P (b, H1 ) and P (b, H2 ) can be found in Cho and Vogelsang (2014).
Proof: See the Appendix.

108

Using Theorem 9, the test statistic T2 defined below has the following limit:

T2 = max

max Wald1α ( Tb ) , max Wald2α ( Tb )

Tb ∈Ξ1

⇒ max

Tb ∈Ξ2

sup
λ∈[ ,(1− )λ x ]

Wald1∞ (λ, λ x ) ,

sup
λ∈[(1+ )λ x ,1− ]

Wald2∞ (λ, λ x ) .

( F)

Test Statistic T2

Alternative statistic can be derived by a similar way as before. Construct HAC estimators
(1)

Υ3 , Υ4 using ξ t

(2)

and ξ t

respectively where


(1)

ξt








=








and

1
s1 1 ( t

(2)

ξt

≤ λT )




1 (t ≤ λT )



1
1
λT
<
t
≤
λ
T
(
)

x
s1
 × ut ,

x t − µ1
1 (λT < t ≤ λ x T ) 
s1



1
1
λ
T
≤
t
≤
T
(
)
x

s2

x t − µ2
1
λ
T
≤
t
≤
T
(
)
x
s2
x t − µ1
s1









=










1
s1 1 ( t



≤ λx T )




1 (t ≤ λ x T )



1
1
λ
T
<
t
≤
λT
(
)

x
s2
 × ut .

x t − µ2
1 (λ x T < t ≤ λT ) 
s2



1
1
λT
≤
t
≤
T
(
)

s2

x t − µ2
1
λT
≤
t
≤
T
(
)
s2
x t − µ1
s1

Define Wald statistics
( F ),α

Wald1

T ( α2 − α1 )2

( Tb ) =
s1 , −

µ1
s1

× D3

WW
T

−1

109

Υ3

WW
T

−1

(2.24)
D3 × s1 , −

µ1
s1

for Tb ∈ [ T, (λ x − ) T ] ≡ Ξ1 , and
( F ),α
Wald2
( Tb )

T ( α3 − α2 )2

=
s2 , −

µ2
s2

× D4

WW
T

−1

Υ4

WW
T

−1

(2.25)
D4 × s2 , −

µ2
s2

for Tb ∈ [(λ x + ) T, (1 − ) T ] ≡ Ξ2 , where
D3 ≡ ( I2 , − I2 ) × ( I4 , 04×1 ) ,
D4 ≡ ( I2 , − I2 ) × (04×1 , I4 ) .

Finally,
( F)

T2

≡ max
⇒ max

( F ),α

max Wald1

Tb ∈Ξ1

Tb ∈Ξ2

( F ),∞

sup
λ∈[ ,(1− )λ x ]

Wald1

(λ, λ x ) ,

( F ),α

( Tb ) , Wald2

The asymptotic limits of Wald1
( F ),∞

Wald1

2.4

( F ),∞

(λ, λ x ) , Wald1

( F ),α

( Tb ) , max Wald2

( Tb )
sup

λ∈[(1+ )λ x ,1− ]

( F ),α

( F ),∞

Wald2

( F)

( Tb ) and T2

(λ, λ x ) .

are the same as

(λ, λ x ) and T ( F),∞ ( , λ x ) respectively in previous Theorem.

Simulations

In this Section the finite sample properties of the test are examined via Monte Carlo simulation. The data generating process (DGP) is given by

yt = α1 1 (t ≤ λT ) + α2 1 (t > λT ) + β 1 1 (t ≤ λT ) xt + β 2 1 (t > λT ) xt + ε t ,
xt = si(t) ut + µi(t) , ut = ρut−1 + ηt ,
ε t = 0.5ε t−1 + νt ,

110

where ηt ∼ N (0, ση2 ) and νt ∼ N (0, 1) are independent. The values of α1 and α2 are set to
zero. The mean and variance of xt are allowed to have a single break at λ x = 0.4. Three
specifications on the mean and variance of xt are considered.

specification 0:

µ1 , µ2 , s21 , s22 = (0, 0, 1, 1)

specification 1:

µ1 , µ2 , s21 , s22 = (0, 0.5, 1, 1.5)

specification 2:

µ1 , µ2 , s21 , s22 = (0, 1, 1, 2)

specification 3:

µ1 , µ2 , s21 , s22 = (0, 2, 1, 4)

Specification 0 implies there is no break in the mean and variance. When the true values of
λ x , µ1 , µ2 , s21 , s22 are known, one can recover ut from xt and use ut to construct a HAC
estimator. So the break in mean and variance has no effect on inference and the empirical
rejection probabilities should be same across the specifications. The values of ρ, ση2 are
chosen so that the variance of xt becomes 1 for the first regime (before λ x T). The selected
sets of those values are (0.5, 0.75) and (0.8, 0.36). The DGP with ρ, ση2

= (0.8, 0.36)

has more persistent autocovariance function than the other and has the larger long run
variance of ut .
To examine the size property, the case where β 1 = β 2 = 1 is considered. To learn about
power, the following values for λ0 , β 1 , β 2 are considered: (0.2, 1, 1.5) , (0.25, 1, 1.5),
where λ0 denotes the true break point for the regression parameters. The value of the
trimming parameter to be used in this experiment is 0.1. Based on this particular value
of trimming parameter and the value of λ x , the admissible set of λ is [0.1, 0.4 − 0.1]

[0.4 + 0.1, 1 − 0.1] . To check against the case where the true value does not belong to this
admissible set, extra values of λ is considered: λ = 0.4. With

= 0.1 and λ x = 0.4 and the

Bartlett kernel being used, the 95% fixed-b critical values for the statistic T1 are 188.64 for
( F)

b = 0.1 and 717.15 for b = 0.5. For T1

with

= 0.1 and λ x = 0.4 the 95% fixed-b critical

values are 35.11 for b = 0.1 and 162.07 for b = 0.5. Table 2.1 reports the results for the

111

simulation experiments for T = 100 and 300 and for three different tests: SupW ( F) test in
( F)

Cho and Vogelsang (2014), and T1 and T1

tests proposed in this chapter.
( F)

Table 2.1 shows that the test results for T1 and T1

are invariant to the break in the

moments of the explanatory variable. This is, as mentined earlier, because the true value
of λ x , µ1 , µ2 , s21 , s22 is assumed to be observable and these two statistics are numerically
the same across different values of λ x , µ1 , µ2 , s21 , s22 . There are several points conveyed
by this table. First, in the absence of break in the moments, SupW ( F) test displays better
power property and comparable size property over the other two tests. But when there
exists a break in the moments, the size and power of SupW ( F) test would be sensitive to
the magnitude of the change in the moments and the size property does not get better as T
( F)

increases. When there is change in the moments and T1

test is used for the inference, the

size distortion is relatively small and the rejection frequency is not so much sensitive to
the persistence of the underlying process. However, this test has poor power compared to
( F)

SupW ( F) test. This is because for given a value of b, T1

uses bigger effective bandwidth

for estimating the variance matrix in each regime. The test based on T1 is seen to be
( F)

dominated by T1

2.5

test in terms of the size and power when T is 300.

Summary and Conclusions

This chapter proposes an inference procedure for testing stability of regression parameters allowing for a single break in the mean and second moments of the x variable. The
mean and second moments are assumed to have a break at λ x T. The break point for the
regression parameter (λT ) should be different from λ x T. The analysis focuses on a simple
linear regression model but the proposed test can be generalized to a multiple linear regression model. A new set of high level conditions are introduced which incorporates the
possibility of change in the mean and second moments of the x variable. Under fixed-b
asymptotics, the limiting distribution of the robust Wald statistic is pivotal so the criti-

112

cal values can be simulated and used for conducting inference. The simulation results in
this chapter show there is substantial size distortion in finite samples. This is not surprising because the three regimes induced by two different types of break points make even
smaller sample size for each regime. Also, whether the main results will be still valid
when λ x and the moments are unknown and need to be estimated should be a straightforward direction of future research.

113

Table 2.1: Size and Power in Finite Samples, T=100, 300, λ x = 0.4,
µ1 = 0, µ2 = 0.5
s21 = 1, s22 = 1.5

No break in (µ, s2 )
ρ, ση2
b

(.5, .75)
0.1
0.5

(.8, .36)
0.1 0.5

(.5, .75)
0.1 0.5

(.8, .36)
0.1 0.5

= .1, Bartlett kernel

µ1 = 0, µ2 = 1
s21 = 1, s22 = 2
(.5, .75)
0.1 0.5

µ1 = 0, µ2 = 2
s21 = 1, s22 = 4

(.8, .36)
0.1
0.5

(.5, .75)
0.1
0.5

(.8, .36)
0.1
0.5

63.8

58.8

SupW ( F) Test
T=100
Size
Power
λ0 = .2
.25
.4
T=300
Size
Power
λ0 = .2
.25
.4

13.3

11.9 23.0

20.7

16.5 14.1

25.3 21.1

24.6 19.4 30.8

ρ,

b
T=100
Size
Power
λ0 = .2
.25
.4
T=300
Size
Power
λ0 = .2
.25
.4

50.8

49.6

27.9 22.1 33.1 28.2
28.1 22.6 33.4 28.7
26.3 20.5 32.0 26.7

46.8 36.7 45.3 37.2
48.6 38.0 46.5 38.5
47.2 36.9 46.2 38.2

67.5 52.8 60.5 50.7
69.0 55.0 61.8 51.4
70.4 56.5 62.4 52.8

98.1 86.7 93.6 83.6
98.3 88.6 94.1 84.5
98.4 93.2 94.3 86.8

8.4

14.8 13.3

29.9 25.1 30.9

25.6

93.0

83.6

93.4 75.1 86.8 71.9
95.2 81.0 88.6 74.7
97.1 88.8 90.6 79.8

100
100
100

96.0 99.9 95.6
97.3 99.8 96.6
99.8 99.9 99

7.9

14.4

11.9

41.5 34.8 38.8 32.9
41.8 34.1 38.7 32.5
37.8 30.6 36.3 29.3

18.6 16.0

76.0 59.8 67.6 55.5
78.3 63.6 69.4 57.4
80.0 68.0 70.6 58.3

( F)

T1 Test
ση2

26.7

T1

Test

(.5, .75)
0.1
0.5

(.8, .36)
0.1 0.5

(.5, .75)
0.1
0.5

(.8, .36)
0.1
0.5

20.1

18.7

45.7 41.2

12.0

13.7

25.6 21.6
25.2 21.9
20.1 18.7

49.6 43.2
49.3 43.6
45.7 41.2

20.3 18.9
19.5 19.0
12.0 10.7

17.3 15.8
17.2 16.4
13.7 12.3

9.5

9.4

20.9 18.9

8.6

11.8

25.1 20.5
25.3 19.8
9.5 9.4

31.9 25.8
31.4 25.0
20.9 18.9

25.0 25.3
24.6 24.9
8.6
7.8

114

10.7

7.8

12.3

11.1

22.2 22.1
22.0 21.6
11.8 11.1

84.8

75.0

APPENDIX

115

Appendix for Chapter 2
Proof of Proposition 10: The proof is only provided for γ2 . One can easily prove the rest
of the results.
−1

λx T

∑

γ2 =

( w t − w2 )

λx T

∑

2

( w t − w2 ) y t
t=λT +1
2 −1
λx T
xt −

t=λT +1
λx T

∑

= γ2 +

t=λT +1

Since p lim x2 = p lim (λ

1

x −λ) T

xt − x2
s1

∑

s1

t=λT +1

x2

ut

.

λ T
∑t=x λT +1 xt = µ1 , it follows under Assumptions 1’ and 2’,

W (λ x ) − W (λ)
λx − λ

T 1/2 (γ2 − γ2 ) ⇒ Λ2

.

Proof of Proposition 11: Proof is only provided for r ∈ [λ, λ x ] when 0 < λ < λ x .
[rT ]
β

S[rT ] =

[rT ]

= ( α2 − α2 )

∑

t=λT +1

∑

t =1

x t − µ1
s1

x t − µ1
s1

[rT ]

ut =

∑

t=λT +1

x t − µ1
s1

[rT ]

∑

+ (γ2 − γ2 )

t=λT +1

x t − µ1
s1

yt − α2 − γ2

[rT ]

xt
s1

xt
+
s1 t =∑
λT +1

x t − µ1
s1

ut .

So,
W (λ x ) − W (λ)
× (r − λ) + Λ2 (W (r ) − W (λ))
λx − λ
r−λ
W (r ) − W ( λ ) −
(W (λ x ) − W (λ)) .
λx − λ

T −1/2 S[rT ] ⇒ −Λ2
β

= Λ2

Proof of Theorem 8: The results immediately follows from equation (2.8), (2.9), Proposition 3 and the transformation
d

Λ2 W (·) = AW1 (·),

116

where A is the positive constant satisfying Λ2 Λ2 = A2 .
Proof of Proposition 12: Proof is only provided for r ∈ [λ, λ x ] when 0 < λ < λ x .
[rT ]

S[αrT ] =

[rT ]

=

∑

t=λT +1





1
s1
x t − µ1
s1



 ut =


[rT ]

= ( α2 − α2 )

∑

[rT ]

t=λT +1




∑

t=λT +1

t =1



x t − µ1
s1


T −1/2 S[αrT ] ⇒ − (r − λ) 

µ1
s21

1



[rT ]

1
s1


 ut

xt

 yt − α2 − γ2
s1

x t − µ1
s1




− (r − λ ) 

t =1


 + (γ2 − γ2 )

Hence

x t − µ1
s1



1
s1






1
s1


∑



1
s1



[rT ]

∑ ξt =



∑

t=λT +1




1
s1
x t − µ1
s1



[rT ]



 xt

 + ∑ 
s1 t=λT +1

1
s1
x t − µ1
s1



 ut .



µ1

 s1 , −
s1
0

Λ

W (λ x ) − W (λ)
λx − λ



 Λ2

W (λ x ) − W (λ)
λx − λ

= Λ W (r ) − W ( λ ) −

+ Λ (W (r ) − W (λ))

r−λ
(W (λ x ) − W (λ)) .
λx − λ

Proof of Theorem 9: The results immediately follows from equation (2.19), (2.20), Proposition 4 and the transformations
µ1
s1
µ2
s2 , −
s2

s1 , −

d

ΛW (·) = B1 W1 (·),
d

ΛW (·) = B2 W1 (·),

117

where the constants B1 and B2 satisfies
s1 , −

µ1
s1

ΛΛ

s1 , −

µ1
s1

= B12 ,

s2 , −

µ2
s2

ΛΛ

s2 , −

µ2
s2

= B22 .

118

REFERENCES

119

REFERENCES

Andrews, D. W. K.: (1993), ‘Tests for Parameter Instability and Structural Change with
Unknown Change Point’. Econometrica 61, 821–856.
Cho, C. K. and T. J. Vogelsang: (2014), ‘Fixed-b Inference for Testing Structural Change
in a Time Series Regression’. Working paper, Department of Economics, Michigan State
University.
Hansen, B. E.: (2000), ‘Testing for structural change in conditional models’. Journal of
Econometrics 97(1), 93–115.
Kiefer, N. M. and T. J. Vogelsang: (2005), ‘A New Asymptotic Theory for
Heteroskedasticity-Autocorrelation Robust Tests’. Econometric Theory 21, 1130–1164.

120

CHAPTER 3
A Test of the Null of Integer Integration
against
the Alternative of Fractional Integration
3.1

Introduction

Chapter 2 proposes a test of the null hypothesis of “integer integration” against the alternative of fractional integration. More precisely, the null is that the series is either I (0) or
I (1), while the alternative is that it is I (d) with 0 < d < 1. The null of integer integration
is rejected in favor of the alternative of fractional integration if the KPSS test rejects the
null of I (0) and a unit root test rejects the null of I (1). A new unit root test is used as
the second part of the testing procedure, which is a lower-tailed KPSS test based on first
differences of the data, but other unit root tests like the ADF test could also have been
used. This two-part testing procedure will be called the “Double-KPSS” test because it
consists of two steps, but it should be pointed out that the test is treated as one test and is
evaluated for its properties (consistency and finite sample size and power) as such.
The KPSS test of Kwiatkowski et al. (1992) was originally suggested as a test of the null
of (short memory) stationarity against the alternative of a unit root. Conversely, standard
unit root tests like the Dickey-Fuller tests, the augmented Dickey-Fuller (ADF) test of Said
and Dickey (1984) or the Phillips-Perron test of Phillips and Perron (1988) were viewed
as tests of the null of a unit root against the alternative of short-memory stationarity. So
if the KPSS test rejected but the unit root test did not, the conclusion was that the series
had a unit root. If the unit root test rejected but the KPSS test did not, the conclusion was

121

that the series was short-memory stationary. If neither test rejected, the conclusion was
that the data were not informative enough to decide whether the series was I (0) or I (1).
However, if both tests rejected, there was in some sense a contradiction.
This apparent contradiction can be resolved by considering a wider class of processes,
specifically long-memory processes. The leading example considered in this chapter, is
the I (d) process (with 0 < d < 1) of Adenstedt (1974), Granger and Joyeux (1980), and
Hosking (1981). Since both the KPSS test and unit root tests have power against longmemory alternatives, the “double rejection” outcome can be taken as evidence that the
series has long memory, as opposed to being either I (0) or I (1). This is not a novel observation. However, the approach in this chapter is novel in its consideration of the doubletesting procedure as a single test, and its investigation of the size and power properties of
this test. In this regard, the basic observation is that if the nominal size of each of the two
tests is set to 5%, the double test also has size of 5% asymptotically. For example, if the
DGP is I (0), then asymptotically the KPSS test will reject with probability 5% while the
unit root test will reject with probability one, while if the DGP is I (1) the converse will
occur. So whether the data are I (0) or I (1), the probability of rejection of the double test
is asymptotically 5%.
The practical issue to be faced is to what extent one can be reasonably sure that the
double rejection outcome is due to fractional integration, as opposed to size distortions of
the test under the I (0) or I (1) null. For example, Caner and Kilian (2001) and Müller
(2005) have shown that the KPSS test has large size distortions if the DGP is AR(1)
with autoregressive coefficient near unity. Conversely, Dejong et al. (1992), Phillips and
Perron (1988) and Vogelsang and Wagner (2013), among others, have found that the
Dickey-Fuller test and its variants can have large size distortions, especially if the DGP
is ARI MA(0, 1, 1) with moving average root near (negative) unity. This does not imply
that the Double KPSS test will suffer from large size distortions in either of these cases,
since the cases for which the KPSS test has large size distortions correspond to cases in

122

which the unit root test may have low power, and conversely. However, it does argue for
a careful investigation of the size and power properties of the new test in finite samples.
As noted above, the specific unit root test used in this chapter is a lower-tail KPSS test
based on the data in differences. The KPSS unit root test suggested by Shin and Schmidt
(1992) and Breitung (2002) was considered. However, as shown by Lee and Amsler (1997),
the KPSS unit root test is not consistent against I (d) alternatives with 1/2 < d < 1. One
might also consider using the ADF test, but this test is known to have low power against
I (d) alternatives (e.g. Diebold and Rudebusch (1991), Hassler and Wolters (1994)), and
there is also the practical consideration that it is easier to prove the consistency of our
test against I (d) alternatives for all d between zero and one than it is for the ADF test. In
simulation, it makes little difference whether the new test or the ADF test is used.
The consistency of the Double KPSS test depends on the consistency of the KPSS test
and of the unit root test proposed in this chapter, and these in turn depend on the number
of lags used in the estimation of the long-run variance going to infinity, but more slowly
than sample size. Under this assumption a single critical value for each test (for each
significance level) obtains, and these will be referred to as the “standard asymptotics”
critical values. They do not depend on the kernel used to estimate the long-run variance
or on the bandwidth (so long as the number of lags behaves as assumed above). However,
following Kiefer and Vogelsang (2002a, 2002b, 2005), this chapter also considers “fixed-b
asymptotics,” where b, defined as the ratio of the number of lags to the sample size, is
constant as the sample size grows. The fixed-b critical values depend on the kernel and
on the value of b, and there is evidence in many settings that they yield tests with smaller
size distortions than the critical values based on the standard asymptotics.
The main theoretical contribution is that the consistency of the Double-KPSS test is
proved against I (d) for all d between zero and one. For the KPSS test, this can be shown
using existing results except for the case of d = 1/2, so the divergence of the statistic for
d = 1/2 is proven in this chapter. For the new unit root test, its asymptotic distribution

123

is established for d = 0, 0 < d < 1/2 and 1/2 < d < 1, and it is proven that the statistic
converges to zero in probability when d = 1/2. Besides these theoretical results, this
chapter contains substantial simulation results to show the extent to which this testing
procedure is likely to be useful in finite samples.
The plan of the chapter is as follows. Section 3.2 gives the definitions and basic properties of stationary short memory, long memory and unit root processes, and explicitly
states the testing procedure. Section 3.3 gives the asymptotic results, using the standard
asymptotics. The asymptotic limits of the two component tests are derived and consistency of the two-part test is proved. Section 3.4 presents the fixed-b asymptotics. Section
3.5 presents the results of simulations which explore the finite sample properties of the
new test. Section 3.6 summarizes and concludes. Finally, an Appendix gives some proofs
and technical details.

3.2

Setup and Assumptions

The data is assumed to be generated by the DGP:

yt = µ + t , t = 1, 2, ..., T.

(2.1)

That is, non-zero level of the yt series is allowed, but not trend. Allowing for trend would
not change the basic principles of the research in this chapter, but it would change the
asymptotics.

3.2.1

Null Hypothesis

Under the null hypothesis { t }∞
t=1 is either a stationary short memory process or a unit
root process. That is, either { t }∞
t=1 itself is a stationary short memory process or it is a
cumulation of a short memory process.
t
Let {zt }∞
t=1 be a time series with zero mean, and let Zt = ∑ j=1 z j be its partial sum.

124

{zt }∞
t=1 is said to be a short-memory process if it satisfies the following two conditions.
Assumption N1
σ2 = lim T −1 E ZT2
T →∞

exists and is nonzero.

(2.2)

Assumption N2

∀r ∈ [0, 1], T −1/2 Z[rT ] ⇒ σW (r ),

(2.3)

where [rT ] denotes the integer part of rT, ⇒ means weak convergence, and W (r ) is the
standard Wiener process.
In addition to Assumptions N1 and N2, further regularity conditions are necessary
for the consistency of HAC (heteroskedasticity and autocorrelation consistent) estimators.
Examples of such conditions can be found in Andrews (1991), Newey and West (1987),
De Jong and Davidson (2000), Jansson (2002), and Hansen (1992). It is implicitly assumed
that one or more of these sets of conditions hold, so that the HAC estimators that appear
in our test statistics are consistent.
Unit root processes are the other class of DGP which belongs to the null hypothesis. A
time series is said to be a unit root process if its first difference is a short memory process.
Equivalently, a cumulation of a short memory process is a unit root process. That is, Zt is
a unit root process if

(1 − L) Zt ≡ zt ∼ short memory process.

125

(2.4)

3.2.2

Alternative Hypothesis

Under the alternative hypothesis, { t }∞
t=1 is a fractionally integrated process. Specifically,
consider the alternative that

t

follows an I (d) process with 0 < d < 1 :

(1 − L ) d

t

= ut , ut ∼ i.i.d Normal(0, σu2 ),

The class of I (d) processes with 0 < d <

1
2

(2.5)

has been widely used in econometrics to

represent long memory processes1 . More generally, a stationary process is said to have
long memory if
n

lim

n→∞

∑

γ j = ∞,

(2.6)

j=−n

where γ j is the autocovariance at lag j. Lo (1991) uses the following form of autocovariance function as a definition of a long-range dependent (long memory) process.




 j2d−1 L( j) for d ∈ (0, 1 ) or

2
γj ∼
as j → ∞ ,
 − j2d−1 L( j) for d ∈ (− 1 , 0)




(2.7)

2

where L( j) is a slowly varying function2 at infinity. This form of autocovariance function
includes the autocovariance function of the I (d) process with 0 < d <

1
2,

and is more

general in the sense that it would accommodate the case that ut in (2.5) is a short memory
but not necessarily an i.i.d. process. However, it does not accommodate the case of

1
2

≤

d < 1.
This chapter considers the I (d) process with i.i.d. innovations as in equation (2.5),
which was analyzed by Granger and Joyeux (1980) and Hosking (1981). When − 21 < d <
1
2,

d = 0, the process is a stationary long memory process, while it is a nonstationry long

memory process for

1
2

≤ d < 1. For d > − 21 , the process is invertible and has infinite

1 For

more comprehensive treatment of this topic, see Giraitis et al. (2012).
(1991, page 1286): A function L( x ) is said to be slowly varying at infinity if limt→∞ L(tx )/L(t) = 1.
An example is log x.
2 Lo

126

order moving average representation:

t

= (1 − L )

−d

∞

ut =

∑ b j ut− j ,

bj =

j =0

and when d <

1
2

Γ ( j + d)
Γ ( d ) Γ ( j + 1)

(2.8)

it has infinite order AR representation:

(1 − L )

d

∞

t

=

∑ aj

t− j

= ut , a0 = 1, a j =

j =0

Γ ( j − d)
for j ≥ 1.
Γ (1 − d ) Γ ( j + 1)

(2.9)

Also, for − 12 < d < 12 , d = 0 the process has a slowly decaying autocovariance function
given by
σu2 Γ (1 − 2d) Γ (k + d)
γk =
∼ ck2d−1 as k → ∞ for some constant c,
Γ ( d ) Γ (1 − d ) Γ ( k + 1 − d )

(2.10)

and this autocovariance function satisfies (2.6) and (2.7). Hosking (1981) provided further
results with ARMA(p, q) innovations.
To establish the asymptotic results in this chapter, an invariance principle under the
alternative hypothesis is needed. Davydov (1970) and Sowell (1990) provide an invariance principle for the fractionally integrated processes with i.i.d. innovations. Lee and
Schmidt (1996) use the result in Sowell (1990), replacing his rth moment-condition by a
normality assumption. Lo (1991) bases his asymptotic analysis upon the result of Taqqu
(1975), assuming stationarity and Gaussianity of the long-memory process. More recently,
Qiu and Lin (2011) proved an invariance principle for the fractionally integrated process
with strong near-epoch dependent innovations, which is the most general functional central limit theorem for fractionally integrated processes currently available. The analysis
in this chapter will focus on the I (d) process with normal i.i.d. innovations, following Lee
and Schmidt (1996). Specifically, this chapter will use the functional central limit theorem

127

appearing in Sowell (1990) which is restated in Lee and Schmidt (1996), as follows.3
1
Suppose {zt }∞
t=1 is generated by (2.5) with − 2 < d <

1
2

and let σT2 = var ( ZT ) . From

Sowell (1990),

σT2 = σu2 ·

Γ (1 + d + T ) Γ (1 + d )
Γ (1 − 2d)
×
−
Γ ( T − d)
Γ (−d)
(1 + 2d) Γ (1 + d) Γ (1 − d)

(2.11)

and as T → ∞,
σT2
T 1+2d

→ σu2 ·

Γ (1 − 2d)
≡ ωd2 .
(1 + 2d) Γ (1 + d) Γ (1 − d)

(2.12)

Also, Sowell (1990) gives the following invariance principle for the fractionally integrated processes with − 21 < d <

1
2

:
σT−1 Z[rT ] ⇒ Wd (r ),

(2.13)

T −(d+1/2) Z[rT ] ⇒ ωd Wd (r ),

(2.14)

or equivalently

where Wd (r ) is the fractional Brownian motion of Mandelbrot and Van Ness (1968), which
is defined as
Wd (r ) =

3.2.3

1
·
Γ (1 + d )

1
0

(r − s)d dW (s)

(2.15)

Test Statistics and the Rejection Rule

As above, the model is:
yt = µ + t , t = 1, 2, ..., T.
The null hypothesis is: H0 :

t

(2.16)

is a stationary short-memory process or a unit root

process with short-memory innovations, and the alternative hypothesis is: H1 :

t

is an

I (d) process with 0 < d < 1 and with normal i.i.d. innovations.
3 This result could presumably be generalized under the more general conditions in Qiu and Lin (2011).
The technical details involved would be orthogonal to the main points of this paper.

128

New rejection rule will be based on two statistics. The first statistic, denoted by K1 ,
should have a non-degenerate limit only when the innovation

t

is stationary short-

memory, and should diverge when the innovation has a unit root or is a long memory
process. Conversely, the second statistic, denoted by K2 , should have a non-degenerate
asymptotic distribution only when the innovation has a unit root, and should converge
to zero when the

t

is a stationary short memory or when it is a long memory (0 < d < 1)

process. If one can find two such statistics, K1 and K2 , the following rejection rule will
asymptotically have size of 5% :
Reject H0 if K1 > cv10.95 and K2 < cv20.05 ,
where cv10.95 is the upper 5% critical value from the asymptotic distribution of K1 when
the error term is a short memory process, and cv20.05 is the lower 5% critical value from
the asymptotic distribution of K2 when the error term is a unit root process. Now two
specific statistics (K1 and K2 ) are proposed to make this procedure operational.
The first statistic (K1 ) is the KPSS statistic (KPSS, 1992) which is defined as

ηµ =

T −2 ∑tT=1 St2
,
s2 ( l )

(2.17)

where s2 (l ) is a HAC estimator. This statistic is constructed using the OLS residuals
ej

T
j =1

from equation (2.16). More specifically,

et = yt − y = yt −

1
T

T

∑ y j , St =

j =1

t

∑ ej,

(2.18)

j =1

l

s2 (l ) = γ0 + 2 ∑ w(s, l )γs ,
s =1

where w(s, l ) = 1 −

s
l +1

(Bartlett kernel) , γs =

1
T

∑ Tj=s+1 e j e j−s . Note that with the Bartlett

kernel the number of lags, l, determines the maximum lag of the sample autocovariances

129

considered for estimating the long run variance of the error term. It is assumed that l → ∞
but l/T → 0 as T → ∞. This assumption characterizes the "traditional asymptotics" and
is made to ensure the consistency of the test. Later Section will make some comparisions
to the "fixed-b" asymptotics that arise when b ≡

l +1
T

has a fixed non-zero limit.

Section 3.3 collects existing asymptotic results for ηµ . The only case in which ηµ has a
non-degenerate limiting distribution occurs when the innovation follows a short-memory
process. Under all the other cases, i.e. a unit root process and the processes described by
our alternative hypothesis, ηµ diverges to infinity.
Also, a statistic K2 is needed that has a non-degenerate limit only under the unit root
innovation processes. Shin and Schmidt (1992) and Breitung (2002) considered a slightly
modified KPSS statistic,

l
T ηµ ,

which converges to a non-degenerate limiting distribution

under the unit root innovation process and goes to zero under short-memory error processes. But as shown by Lee and Amsler (1997), this statistic cannot distinguish I (1)
processes from I (d) processes with

1
2

< d < 1. This chapter therefore suggest an alter-

native statistic which can distinguish I (1) processes from I (d) processes with
(nonstationary long-memory processes) as well as from I (d) with 0 < d <

1
2

1
2

≤d<1

(stationary

long-memory processes) and stationary short-memory processes.
Consider the differenced model,

yt =

t

=

et .

(2.19)

The second statistic K2 will be the KPSS statistic based on the differenced data { yt }tT=2 .
The statistic is given by
ηµd =

T −2 ∑tT=2 St2
,
s2 ( l )

130

(2.20)

where
t

St =

t

∑

yj =

j =2

∑

j

=

t

−

1,

(2.21)

j =2
l

s2 (l ) = γ0 + 2 ∑ w(s, l )γs ,
s =1

with γs =

1
T −1

∑tT=s+2 ∆yt ∆yt−s =

1
T −1

∑tT=s+2 ∆ t ∆

t−s .

It will be shown that this statistic has a non-degenerate distribution in the case of a unit
root, while it converges to zero under stationary short memory or under I (d) with 0 <
d < 1. Note that this will be a lower-tail test. Other unit root test could have considered,
notably variants of the Dickey-Fuller test. The simulations will also compare the results
with different unit root tests. From a technical point of view, ηµd is attractive because the
proof of the consistency of the test is relatively straightforward.

3.3

Asymptotic Results

This Section discusses the asymptotic distributions of the ηµ and ηµd statistics when

t

is I (0), I (d) with 0 < d < 1, and I (1). The asymptotic theory is established under the
assumption that, as T → ∞, l → ∞ but l/T → 0, where l is the number of lags used in the
Bartlett kernel for estimation of the relevant long-run variance. These are the "traditional
asymptotics," as opposed to the "fixed-b asymptotics" which will be discussed in a later
Section of this chapter.

3.3.1

Asymptotic Results for ηµ

The existing results on the limit of ηµ are collected from Kwiatkowski et al. (1992), Shin
and Schmidt (1992), Lee and Schmidt (1996), and Lee and Amsler (1997). Table 3.1 shows
those results. Note that the case of d =

1
2

is missing in Table 3.1. Results for d =

1
2

similar

to those in Table 3.1 are not currently available, so this case will be treated separately in
131

this Section. The implications of these results for the asymptotic distribution of ηµ are
summarized in Theorem 7.
Theorem 7. Given the data generation process in (2.16), the KPSS statistic, ηµ defined in (2.17)
has following asymptotic limits.
A. When

t

is a short-memory process (Kwiatkowski et al. (1992)),
1

ηµ ⇒

B. When

t

0

B(r )2 dr.

(2.22)

is a unit root process (Shin and Schmidt (1992)),
l
ηµ ⇒
T

2
r
0 W ( s ) ds dr
1
2
0 W ( s ) ds

1
0

(2.23)

implying ηµ → ∞ in probability.
C. When

t

is a fractionally integrated process with 0 < d < 1/2 (Lee and Schmidt (1996)),
l
T

2d

1

ηµ ⇒

0

Bd (r )2 dr

(2.24)

implying ηµ → ∞ in probability.
D. When

t

is a fractionally integrated process with 1/2 < d < 1 (Lee and Amsler (1997)),
l
T

ηµ ⇒

1
0

2
r
0 W d∗ ( s ) ds dr
1
2
0 W d∗ ( s ) ds

(2.25)

implying ηµ → ∞ in probability.
The above results cover all of the cases except d = 1/2. This case is covered by the
following Theorem.
Theorem 8. Given the data generation process in (2.16), the KPSS statistic, ηµ defined in (2.17)
diverges to infinity when the error term is an I ( 21 ) process.

132

Proof: See the Appendix.
Theorems 7 and 8 imply that the KPSS ηµ test is consistent against a unit root and also
against I (d) alternatives for all 0 < d < 1.

3.3.2

Asymptotic Results for ηµd

This Section considers the asymptotic behavior of ηµd under stationary short- or longmemory errors, under unit root errors, under nonstationary long-memory errors with
1
2

< d < 1, and under nonstationary long-memory errors with d = 12 . Recall
ηµd ≡

T −2 ∑tT=2 St2
, with St ≡
s2 ( l )

t

∑ ∆y j =

j =2

t

∑∆

j

=

t

−

1,

j =2

T
1
∆yt ∆yt−s ,
s (l ) = γ0 + 2 ∑ w(s, l )γs , and γs =
T − 1 t=∑
s +2
s =1
l

2

where ∆yt = yt − yt−1 , and ∆

t

=

t

−

t −1 .

Since ηµd is a unit root test statistic, its asymptotic distribution is established under
first, the unit root null. Then it will be shown that the limit of the statistic is zero for the
case of stationary short memory and for I (d) processes with 0 < d < 1.
Proposition 8. Under the data generation process in (2.16) with the error term
root process, and under the assumption that l → ∞ and

l
T

t

being a unit

→ 0 as T → ∞, the statistic, ηµd

defined in (2.20) weakly converges:
ηµd ⇒

1
0

W (r )2 dr,

(2.26)

where W (r ) is the standard Wiener process.
Proof: See the Appendix.
This is a lower tail test. The 1%, 5% and 10% lower tail critical values are 0.034, 0.056,
and 0.076, respectively. These are different from the critical values of the KPSS unit root

133

test in Shin and Schmidt (1992) and Breitung (2002) because the data is differenced instead
of demeaning the terms in St . As a consequence the result in (2.26) involves an ordinary
Wiener process as opposed to a demeaned Wiener process.
The next three results together prove that the limit of the statistic is zero for all cases
except the case of unit root errors.
Theorem 9. Under the data generation process in (2.16) with the error term
short- or long-memory process and under the assumption that l → ∞ and

l
T

t

being a stationary

→ 0 as T → ∞, the

statistic, ηµd defined in (2.20) has the following limiting distribution:
l
T
where γ0 = E

2
t

−1

d
ηµd →

γ0 + 12
,
2γ0

(2.27)

p

. Therefore, ηµd → 0.

Proof: See the Appendix.
The next Proposition shows the statistic ηµd can distinguish fractionally integrated processes with

1
2

< d < 1 from unit root processes. Recall that this is not the case for the

KPSS unit root test (Lee and Amsler (1997)).
Proposition 9. Under the data generation process in (2.16) with the error term
ally integrated with i.i.d. normal innovations and with

1
2

t

being fraction-

< d < 1 (so, a nonstationary long-

memory process), and under the assumption that l → ∞ and

l
T

→ 0 as T → ∞, the statistic, ηµd

defined in (2.20) has the following limiting distribution:
l
T

2d∗

ηµd ⇒

1
0

Wd∗ (r )2 dr,

(2.28)
p

where d∗ = d − 1 and Wd∗ (r ) is the fractional Brownian motion. Therefore, ηµd → 0.
Proof: See the Appendix.
Lastly, consider the case of I

1
2

.

134

Theorem 10. Under the data generation process in (2.16) with the error term
integrated with i.i.d. normal innovations and with d =
process) and under the assumption that l → ∞ and

l
T

1
2

t

being fractionally

(so, a nonstationary long-memory

→ 0 as T → ∞, the statistic, ηµd defined in

(2.20) converges to zero.
Proof: See the Appendix.

3.3.3

Correct Size and Consistency of the Double-KPSS Test

Now go back to the rejection rule in Section 3.2. The null of integer integration is rejected if the KPSS test rejects short memory and if the unit root test based on ηµd rejects
unit root. This two-part test has correct size asymptotically and is consistent against I (d)
alternatives with 0 < d < 1.
Proposition 10. Suppose the data generation process is given by (2.16). Under the null hypothesis of integer integration and under the assumption that l → ∞ and

l
T

→ 0 as T → ∞, the

rejection rule
Reject H0 if ηµ > cv10.95 and ηµd < cv20.05 ,
gives a test with asymptotic size of 5%, where cv10.95 is the upper 5% percentile of (2.22) and cv20.05
is the lower 5% percentile of (2.26). Also, the test is consistent againt the alternative hypothesis of
I (d) with 0 < d < 1.
Proof: The Proposition follows immediately from the results of Sections 3.3.1 and 3.3.2.
(1) If the series is I (0), asymptotically the KPSS test will reject with probability 0.05 and
the ηµd test will reject with probability one, so the Double-KPSS test will reject with probability 0.05. (2) If the series is I (1), the KPSS test will reject with probability one and the ηµd
test will reject with probability 0.05, so the Double-KPSS test will reject with probability
0.05. (3) If the series is I (d) with 0 < d < 1, asymptotically both tests will reject with
probability one, and so the Double-KPSS test will reject with probability one.

135

3.4

Fixed-b Asymptotic Results

Let b =

l +1
T ,

the ratio of the number of lags (plus one) to the sample size. The asymptotic

results of the previous Section were obtained under the assumption that b → 0 as T → ∞.
This Section discusses the asymptotic distribution of the ηµ and ηµd statistics under the
"fixed-b" assumption that b is held constant as T → ∞. The idea of fixed-b asymptotics
was proposed by Kiefer and Vogelsang (2005) and Hashimzade and Vogelsang (2008).
The fixed-b approach gives a random limit of the HAC estimator which depends on the
choice of kernel and the bandwidth ratio b. The fixed-b approach is known to produce
a better finite sample approximation to the distribution of test statistics in a variety of
settings. Amsler et al. (2009) derived the fixed-b asymptotic distribution of the KPSS ηµ
statistic under the I (0) null and under the I (1) alternative. Proposition 11 present their
results.
Proposition 11. Given the data generation process in (2.16), under the assumption that b =
l +1
T

∈ [0, 1] is held constant as T increases, the KPSS statistic, ηµ defined in (2.17) has the

following fixed-b asymptotic limits.
When

t

is a short-memory process,

ηµ ⇒

where Q0 (b) ≡
When

t

2
b

1
0

B(r )2 dr −

1− b
0

B(r )2 dr
,
Q0 ( b )

(2.29)

B(r ) B(r + b)dr with B(r ) ≡ W (r ) − rW (1).

is a unit root process,
ηµ ⇒

where Q1 (b) ≡
W (s) −

1
0

2
b

1
0

P(r )2 dr −

1− b
0

1
0

P(r )2 dr
,
Q1 ( b )

P(r ) P(r + b)dr with P(r ) ≡

1
0 W ( u ) du.

Proof: See Amsler et al. (2009).
136

(2.30)
r
0 W ( s ) ds

with W (s) ≡

As Proposition 11 shows, the KPSS statistic ηµ has a nondegenerate limit under both
I (0) and I (1) data generation processes. So the KPSS ηµ test does not give a consistent test
against the unit root under the fixed-b assumption. In the present context, the DoubleKPSS test would be conservative (undersized). If the DGP is I (1), the ηµd test would reject
with probability 0.05, but the KPSS ηµ test would reject with probability less than one,
under fixed-b asymptotics. So the probability of both tests rejecting would be less than
0.05. However, these issues should not be regarded as consequential. The assumption of
fixed-bandwidth ratio does not recommend any rules for selecting the number of lags.
But, no matter how one chooses the number of lags, it will be positive, and the fixed-b
critical values will usually give a test of more accurate size than the traditional (b = 0)
critical values. That is, fixed-b asymptotics is simply viewed as a way of generating a
more accurate approximation to the finite sample distribution of the statistic.
The next Proposition provides the fixed-b limits of ηµd under I (0) and I (1) data generation processes. To avoid confusion with the previous definition of b and l, let the number
of lags and the ratio be denoted by l and b =

l +1
T −1 .

(We have T − 1 instead of T because

one observation is used up in differencing.)
Proposition 12. Given the data generation process in (2.16), under the assumption that b =
l +1
T −1

∈ [0, 1] is held constant as T increases, the KPSS statistic, ηµd has the following fixed-b

asymptotic limits:
When

t

is a short-memory process,
ηµd

where γ0 = E
When

t

2
t

and

∞

⇒

γ0 +
2
b γ0

+

denotes the weak limit of

2
1

2
1

+
T

2
∞

,

(2.31)

as T → ∞.

is a unit root process,

ηµd ⇒

1
2
0 W (r ) dr
,
Q1d (b )

137

(2.32)

where W (r ) is the standard Wiener process and
Q1d (b ) ≡

2
b

1
0

W (r )2 dr −

1− b
0

W (r )W (r + b )dr −

1
1− b

W (r )W (1)dr + W (1)2 .

Proof: See the Appendix.
The fixed-b critical values for ηµ and ηµd are simulated using i.i.d. N(0,1) pseudo random numbers with T = 1, 000 and 50, 000 replications. Table 3.2 provides these fixed-b
critical values. The fixed-b critical values for ηµ are slightly different from those in Amsler
et al. (2009). This is partly due to randomness of the simulations, but it is also due to a
slight difference in the definitions of b. They had b =

l
T

whereas now we have b =

l +1
T .

Since the critical values were simulated using T = 1, 000, b = 0.02 in Amsler et al. (2009)
would correspond to b = 0.021 in this chapter, for example. For bigger T (i.e. asymptotically) this difference obviously disappears.
The simulation results in Amsler et al. (2009) showed that the size distortion associated with strong short-run persistence of the I (0) DGP can be fixed by using a relatively
large number of lags and the corresponding fixed-b critical values. In their results, with
the original KPSS critical value being used, as the short run persistence gets higher, the
overrejection of the KPSS test gets worse. One can reduce the rejection frequency by using
a higher number of lags but now the test becomes subject to an underrejection problem
which is translated into low power. However, by taking a relatively large number of lags
and using the fixed-b critical values, this problem can be partially fixed. Exactly the same
considerations apply to the ηµd test and the Double-KPSS test.

3.5

Monte Carlo Simulations

This Section reports the results of simulations designed to investigate the finite sample
size and power properties of the Double-KPSS test. Some comparisons of the ηµd test and
the ADF test will be made.

138

3.5.1

Design of the Experiment

The data generating processes to be considered in the simulations are as follows.
1. I (0) DGP:

yt = µ + t ,
t

with µ = 0,

0

= ρ

t −1

(2.33)

+ ut ,

= 0, ρ ∈ {0, 0.25, 0.5, 0.75, 0.95}, ut ∼ i.i.d. Normal (0, 1).

2. I (1) DGP:

yt = µ + t ,
t

=

t −1

(2.34)

+ ηt ,

ηt = ut − φut−1 ,

with µ = 0,

0

= u0 = 0, φ ∈ {0, 0.25, 0.5, 0.75, 0.95}, ut ∼i.i.d. Normal (0, 1).

3. I (d) DGP:

yt = µ + t ,

(1 − L ) d

t

(2.35)

= ut ∼ i.i.d. Normal (0, 1),

with µ = 0, and d ∈ {0.1, 0.2, 0.3, 0.4, 0.45, 0.499, 0.5, 0.6, 0.7, 0.75, 0.8, 0.9}.
To generate I (d) processes with 0 < d <

1
2

Toeplitz matrix was used (formed from the

autocovariances, as in Diebold and Rudebusch (1991)). For the case of

1
2

≤ d < 1, first

generated I (d) processes with − 12 ≤ d < 0 (again using the Toeplitz matrix) and cumulated them to obtain the I (d) processes with

1
2

≤ d < 1. This is the same procedure as

in Lee and Schmidt (1996). The experiments considered T = 50, 100, 200, 500, 1, 000, and
139

2, 000 and the number of replications was 5, 000. The numbers of lags used for computing
the statistics were l0 (= 0), l4 , l12 , l25 and l50 , where
lk ≡ k ·

T
100

1/4

.

(2.36)

For the ADF test, with p lags, p4 , p12 , and p25 lags were considered, where pk is defined
in the same way as in equation (2.36).
As a matter of notation, ηµ × ηµd will denote the Double-KPSS test based on ηµ and ηµd .
Similarly ηµ × ADF will denote the double test but using the ADF test instead of ηµd .

3.5.2

Results with Standard Critical Values

This Section discusses the results for the ηµ , ηµd and ηµ × ηµd tests, using the "standard"
critical values that are valid asymptotically when l → ∞ and

l
T

→ 0 as T → ∞. These

results are given in Tables 3.3, 3.4 and 3.5. Each table contains the results for two sample
sizes (3.3: T = 50 and 100; 3.4: T = 200 and 500; 3.5: T = 1, 000 and 2, 000). The formatting
for each sample size is the same.
The results for the KPSS ηµ test are similar to those from previous simulations and
will be discussed only briefly. Size under the I (0) null with ρ = 0 is essentially correct
for l0 but the test is undersized with more lags. The test is oversized when ρ > 0 and
severely so for the largest values of ρ (like ρ = 0.95). Size improves very slowly as T
increases. Power against an I (1) alternative rises when T increases, falls as the number of
lags increases, and falls as φ increases (since the series approaches stationarity as φ → 1).
Power against I (d) alternatives grows with d, falls as the number of lags increases, and
grows (but slowly) as T increases.
The results for the ηµd unit root test show a pattern that is similar to what was seen
for the KPSS ηµ test, but reversed. Size under the I (1) null with φ = 0 is essentially
correct, but the test is undersized with more lags. The test is oversized when φ > 0

140

and severely so for the largest values of φ (like φ = 0.95). Size generally improves as T
increases. Power against I (0) alternatives rises when T increases, falls as the number of
lags increases, and falls as ρ increases (since the series approaches I (1) as ρ → 1). Power
against I (d) alternatives grows as d decreases, falls as the number of lags increases, and
grows as T increases.
All of these statements would also be true for the ADF test. Some comparisons of the
performance of the ηµd test and the ADF test will be given in Section 3.5.4.
Now turn to the issue of main interest, the performance of the Double-KPSS (ηµ × ηµd )
test. This test rejects the null of integer integration if both the ηµ short-memory test and
the ηµd unit root test reject their respective null hypotheses. As a result, the upward size
distortions caused by short run dynamics must be smaller for the Double-KPSS test than
for either of the individual tests. In many cases the rejection probability for the DoubleKPSS test is at least approximately equal to the product of the rejection probabilities for
the two component tests, but this is not always the case (The two tests are not independent).
Consider first the size of the Double-KPSS test under the I (0) null. In the most empirically relevant cases, like l4 lags with T = 100, or l12 lags with T = 200 or 500, its size
is reasonably accurate, except perhaps for the biggest values of ρ. For the largest sample
sizes (T = 1, 000 and 2, 000) the test has fairly accurate size, except for the case of ρ = 0.95,
if the test uses l12 × l12 or l25 × l25 lags. However, the size of the test does not improve
uniformly as T increases, because loosely speaking the power of the unit root test goes to
one faster than the size of the short-memory test goes to 0.05. But as a general statement
the size of the test is surprisingly good over a broad range of values of ρ.
Now consider the size of the test under the I (1) null. Once again the size is reasonably
accurate, except perhaps for the biggest values of φ, if reasonable numbers of lags, like
l4 lags with T = 100, or l12 lags with T = 200 or 500 are used. The test is if anything
undersized (due to the use of the standard critical values despite the positive number of

141

lags) for the smaller values of φ. For the largest values of T (1, 000 and 2, 000), size is quite
good with l12 × l12 or l25 × l25 lags, except when φ = 0.75 or 0.95.
Finally, consider the power of the test against I (d) alternatives. Now there is a potential problem, because if one uses the numbers of lags mentioned above as sufficient
to control size, power is low. For example, if l12 lags, with T = 200 is used, the highest
power is only 0.099 (against d = 0.4) and with T = 500 the highest power is 0.386 (against
d = 0.5). Of course, there is a trade-off between size and power. If one uses only l4 lags,
maximal power is 0.488 for T = 200 and 0.786 for T = 500. But with only l4 lags, there
are large size distortions under the null for the larger values of ρ (for the I (0) null) or
φ (for the I (1) null). It takes a very large sample size (like T = 1, 000 or 2, 000) to have
reasonable power with l12 lags.
So, what can one conclude from these simulations? In a view the main practical question is how large the sample size needs to be so that one can reasonably conclude that a
rejection from the test is due to its power against a fractional alternative, as opposed to
size distortions of one or both of the two component tests. This obviously will depend on
the values of d against which we require power, as well as the values of the nuisance parameters that we want the null hypothesis to encompass. As an extreme example, there
is no hope of success if we want to include in the I (0) null AR processes with local to
unity roots, or if we want to include in the I (1) null ARI MA(0, 1, 1) processes with local
to unity MA roots.
The simulations results seem to indicate that the test can in fact reasonably distinguish
fractional integration from non-extreme I (0) or I (1) processes, but that it will take a large
sample size to do so. For example, for T = 500 and for the tests using l12 lags, power
for d in the range [0.3, 0.7] is at least twice as large as the maximal size distortion for I (0)
processes with AR roots less than or equal to 0.75 or for I (1) processes with MA roots
less than or equal to 0.75. For smaller sample sizes, this statement would not be true, and
to make a similar statement that is true would require a smaller range of d and/or a more

142

restrictive range of AR or MA roots. Conversely, to make a similar statement that is true
for a larger range of d or of AR and MA parameters will require a larger sample size. For
example, with T = 2000, power for d in the range [0.2, 0.8] is at least twice as large as
the maximal size distortion for I (0) processes with ρ less than or equal to 0.75 or for I (1)
processes with φ less than or equal to 0.75. The obvious problem with these statements is
that, for economic time series data, T = 2000, or for that matter T = 500, is a very large
sample size.

3.5.3

Results with Fixed-b Critical Values

The fixed-b critical values, for the relevant values of b, are smaller than the traditional
critical values for the KPSS ηµ test (an upper tail test) and larger for the ηµd test (a lower tail
test). The fixed-b critical values will therefore lead to more rejections than the traditional
critical values, if the same number of lags is used in both cases. If the number of lags
increases with sample size but more slowly than sample size, the difference in the critical
values (and the rejection probabilities) will go to zero since b will go to zero.
The fixed-b critical values are very successful in removing the underrejection problem
that occurs for the KPSS ηµ and ηµd tests when there are no short-run dynamics and the
sample size is not large. For example, for KPSS ηµ with T = 50 and I (0) data with ρ = 0,
and with l12 lags, compare size of 0.014 with traditional critical values to 0.053 with fixedb critical values. Or, for the ηµd test with T = 50 and I (1) data with φ = 0, and with l12
lags, compare 0.000 with traditional critical values to 0.039 with fixed-b critical values.
For the Double-KPSS test there is also improvement in size in these cases from using the
fixed-b critical values, but the improvement is not so striking.
Upward size distortions in the presence of short run dynamics (I (0) data with large
positive ρ or I (1) data with large positive φ) are worse when the fixed-b critical values are
used. Also power against I (d) alternatives is higher when the fixed-b critical values are
used. However, these differences are not large when the sample size is big enough for us
143

to have reasonable power (e.g., T greater than or equal to 500). Of course, that is because
the rule for the choice of lags in the simulations implies that b goes to zero as T grows,
but that is a reasonable feature for such a rule to have. Using the fixed-b critical values is
recommended but recognize that if the number of lags is chosen reasonably this is likely
to not make much difference.

3.5.4

Comparison of the ηµd Test and the ADF Test

Although it is not the focus of this research, a new unit root test has been proposed in this
chapter and it is relevant to ask how it compares to other existing unit root tests. There
are of course a great many other existing unit root tests. The ADF τµ test will be taken
as a standard of comparison . The number of lagged differences included in the ADF
regression is denoted as p, and in making comparisons of size and power a value of p
will be matched to the same value of l, the number of lags used for long run variance
estimation in the ηµd test.
Table 3.9 gives size and power for the ADF test for T = 50, 100, 200 and 500, and these
results can be compared to the results previously given in Table 3.3 and 3.4.
In terms of the size of the test, the results are mixed. However, for the larger sample
sizes, the ADF test with p12 lags has smaller size distortions than the ηµd test with l12
lags for the larger values of φ. The ADF test generally has higher power against I (0)
alternatives, while the ηµd test has higher power against I (d) alternatives except when d is
very small (e.g. 0.1).
If one compares the Double-KPSS (ηµ × ηµd ) test to the ηµ × ADF test, similar statements apply, but the differences are much smaller. In fact, the similarities between these
two double tests far outweigh the differences. Unsurprisingly, perhaps, the precise choice
of unit root test to use is not the main issue here.

144

3.6

Conclusions

This chapter proposed a Double KPSS test to test the null of integer integration (I (0) or
I (1)) against the alternative of fractional integration (I (d) with d between zero and one).
The null of integer integration is rejected if the KPSS test rejects the null of short memory
and a unit root test rejects the null of a unit root. A new unit root test was suggested
for use in this testing procedure, but any other unit root test like the ADF test is also
possible. This would be a good preliminary test to use before estimating a fractional
model. An alternative, of course, is to just estimate the fractional model and see whether
the estimated d is significantly different from zero and from one. However, there appears
to be no clear consensus in the existing literature on how to allow for short-run dynamics
in estimating d and conducting inference about it.
The consistency of the test were proved. The main practical question is how large the
sample size needs to be so that one can reasonably conclude that a rejection from the test
is due to its power against a fractional alternative, as opposed to size distortions of the
two component tests. The simulations results seem to indicate that the test can in fact
distinguish fractional integration from non-extreme I (0) or I (1) processes, but that it will
take a very large sample size to do reliably. This is not a surprising result. It takes a lot
of data to distinguish I (0) from I (1) processes, if the range of short-run dynamics is not
severely restricted. Now we are trying to do more, for example, to distinguish a unit root
process from an I (d) process with d = 0.8. An important contribution of this chapter is to
try to quantify how much data that takes. The required sample sizes would be very large
indeed for macroeconomic applications, but perhaps not for applications in finance.

145

Table 3.1: Summary of the existing asymptotic results for ηµ
H0
t

∼

St = ∑tj=1 e j :

I (0)
√1 S[rT ]
T

I (1)
1
S
T 3/2 [rT ]

⇒ σB(r ),

where B(r ) = W (r ) − rW (1)
∑tT=1 St2 ⇒ σ2

∑ St2 :

1
T2

s2 ( l ) :

s2 ( l ) → σ 2

1
0

where W (s) =
1
T4

B(r )2 dr

p

r
0 W ( s ) ds,
1
W (s) − 0 W (u)du

⇒σ

∑tT=1 St2 ⇒ σ2
1 2
lT s ( l )

2
r
0 W ( s ) ds dr

1
0

1
2
0 W ( s ) ds

⇒ σ2

H1
t

∼

St = ∑tj=1 e j :

∑ St2 :
s2 ( l ) :

I (d), 0 < d < 1/2

I (d), 1/2 < d < 1

1
S
⇒ ωd Bd (r ),
T d+1/2 [rT ]
where Bd (r ) = Wd (r ) − rWd (1)

1
T 2( d +1)

∑tT=1 St2 ⇒ ωd2

1
0

Bd (r )2 dr

p

l −2d s2 (l ) → ωd2

⇒ ωd∗ 0 W d∗ (s)ds,
where d∗ = d − 1 and
1
W d∗ (r ) = Wd∗ (r ) − 0 Wd∗ (s)ds
1
T 2d∗ +4

∑tT=1 St2 ⇒ ωd2∗
1
s2 ( l )
lT 2d∗ +1

146

r

1
S
T d∗ +3/2 [rT ]

1
0

⇒ ωd2∗

2
r
0 W d∗ ( s ) ds dr
1
2
0 W d∗ ( s ) ds

Table 3.2: Fixed-b Critical Values for ηµ and ηµd , Bartlett kernel, l = bT − 1, l = b ( T − 1) − 1

b
0
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
0.20
0.22
0.24
0.26
0.28
0.30
0.32
0.34
0.36
0.38
0.40
0.42
0.44
0.46
0.48
0.50

ηµ
upper tail
5%
1%
0.463 0.739
0.453 0.705
0.446 0.673
0.439 0.639
0.434 0.609
0.428 0.582
0.421 0.561
0.416 0.541
0.409 0.522
0.403 0.504
0.398 0.489
0.394 0.476
0.391 0.464
0.388 0.455
0.385 0.447
0.383 0.441
0.381 0.435
0.380 0.430
0.381 0.426
0.380 0.424
0.381 0.422
0.383 0.421
0.386 0.422
0.390 0.425
0.394 0.430
0.400 0.437

ηµd
lower tail
1%
5%
0.034 0.056
0.038 0.060
0.042 0.063
0.045 0.067
0.049 0.071
0.053 0.075
0.057 0.078
0.062 0.082
0.066 0.086
0.069 0.090
0.073 0.095
0.077 0.099
0.080 0.103
0.083 0.107
0.086 0.111
0.088 0.115
0.090 0.119
0.092 0.122
0.093 0.126
0.095 0.129
0.095 0.132
0.096 0.135
0.098 0.137
0.098 0.139
0.099 0.141
0.100 0.142

b
0.52
0.54
0.56
0.58
0.60
0.62
0.64
0.66
0.68
0.70
0.72
0.74
0.76
0.78
0.80
0.82
0.84
0.86
0.88
0.90
0.92
0.94
0.96
0.98
1

147

ηµ
upper tail
5%
1%
0.405 0.445
0.410 0.453
0.414 0.461
0.419 0.468
0.424 0.473
0.429 0.479
0.432 0.485
0.436 0.490
0.439 0.493
0.442 0.496
0.445 0.499
0.448 0.499
0.449 0.501
0.452 0.500
0.454 0.498
0.456 0.498
0.458 0.496
0.462 0.494
0.465 0.490
0.468 0.488
0.473 0.487
0.478 0.487
0.484 0.488
0.491 0.492
NA
NA

ηµd
lower tail
1%
5%
0.100 0.144
0.101 0.145
0.101 0.147
0.102 0.148
0.102 0.150
0.103 0.151
0.104 0.151
0.104 0.152
0.104 0.153
0.105 0.154
0.106 0.155
0.106 0.156
0.106 0.157
0.107 0.158
0.107 0.159
0.107 0.160
0.108 0.160
0.108 0.161
0.108 0.162
0.109 0.163
0.109 0.163
0.109 0.164
0.110 0.165
0.110 0.166
0.110 0.166

Table 3.3: Size and Power Using Standard 5% Critical Values with Traditional Lag Choices,T =
50 and 100
T=50
lag

l0

ηµ
l4

l12

l0

ηµd
l4

l12

l0 × l0

ηµ × ηµd
l4 × l4

l12 × l12

0.014
0.016
0.021
0.035
0.118

0.959
0.923
0.867
0.641
0.108

(Power)
0.478
0.484
0.478
0.324
0.048

0.001
0.001
0.001
0.001
0.001

0.050
0.134
0.280
0.379
0.082

(Size)
0.015
0.021
0.031
0.030
0.003

0.000
0.000
0.000
0.000
0.000

∼ I (0)
ρ=0
0.25
0.50
0.75
0.95

(Size)
0.053 0.045
0.149 0.062
0.336 0.102
0.653 0.214
0.924 0.530

∼ I (1)
φ= 0
0.25
0.50
0.75
0.95

0.961
0.944
0.895
0.709
0.126

(Power)
0.711
0.703
0.679
0.571
0.100

0.353
0.348
0.331
0.262
0.035

0.042
0.141
0.400
0.786
0.955

(Size)
0.021
0.028
0.064
0.208
0.466

0.000
0.000
0.000
0.000
0.001

0.030
0.112
0.321
0.518
0.117

(Size)
0.002
0.003
0.011
0.062
0.036

0.000
0.000
0.000
0.000
0.000

∼ I (d)
d =0.1
0.2
0.3
0.4
0.45
0.499
0.5
0.6
0.7
0.75
0.8
0.9

0.133
0.252
0.399
0.546
0.616
0.674
0.679
0.775
0.851
0.879
0.907
0.939

(Power)
0.082
0.138
0.198
0.276
0.322
0.372
0.369
0.456
0.533
0.571
0.603
0.662

0.021
0.036
0.053
0.076
0.084
0.100
0.100
0.131
0.174
0.200
0.229
0.283

0.946
0.925
0.890
0.820
0.774
0.723
0.720
0.572
0.389
0.294
0.215
0.102

(Power)
0.464
0.430
0.379
0.322
0.282
0.251
0.239
0.167
0.110
0.085
0.066
0.034

0.001
0.001
0.001
0.000
0.000
0.001
0.000
0.000
0.000
0.000
0.000
0.000

0.124
0.227
0.340
0.427
0.450
0.458
0.457
0.402
0.290
0.222
0.164
0.077

(Power)
0.029
0.043
0.045
0.050
0.047
0.042
0.043
0.027
0.017
0.012
0.009
0.003

0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000

l0

ηµ
l4

l12

l0

ηµd
l4

l12

l0 × l0

ηµ × ηµd
l4 × l4

l12 × l12

0.027
0.033
0.041
0.068
0.250

0.999
0.993
0.983
0.944
0.262

(Power)
0.751
0.764
0.797
0.784
0.172

0.040
0.046
0.064
0.100
0.029

0.043
0.142
0.354
0.673
0.247

(Size)
0.027
0.036
0.063
0.137
0.055

0.001
0.001
0.000
0.001
0.000

0.006
0.006
0.006
0.008
0.032

0.044
0.149
0.431
0.769
0.299

(Size)
0.009
0.015
0.040
0.188
0.178

0.000
0.000
0.000
0.001
0.002

0.041
0.037
0.034
0.031
0.026
0.029
0.028
0.023
0.018
0.015
0.013
0.010

0.155
0.343
0.528
0.681
0.726
0.744
0.742
0.690
0.533
0.427
0.311
0.130

(Power)
0.061
0.113
0.164
0.201
0.207
0.202
0.201
0.151
0.091
0.069
0.047
0.022

0.001
0.001
0.001
0.000
0.001
0.000
0.000
0.000
0.000
0.000
0.000
0.000

t

t

t

T=100
lag

∼ I (0)
ρ=0
0.25
0.50
0.75
0.95

(Size)
0.043 0.040
0.143 0.052
0.361 0.088
0.718 0.194
0.978 0.591

∼ I (1)
φ=0
0.25
0.50
0.75
0.95

0.993
0.989
0.978
0.905
0.300

(Power)
0.821
0.818
0.809
0.747
0.265

0.578
0.577
0.571
0.541
0.186

(Size)
0.046 0.029
0.157 0.041
0.451 0.091
0.863 0.332
0.998 0.718

∼ I (d)
d =0.1
0.2
0.3
0.4
0.45
0.499
0.5
0.6
0.7
0.75
0.8
0.9

0.156
0.346
0.539
0.715
0.783
0.830
0.830
0.908
0.954
0.967
0.975
0.985

(Power)
0.091
0.173
0.272
0.373
0.424
0.473
0.474
0.561
0.640
0.678
0.710
0.774

0.052
0.086
0.131
0.181
0.214
0.238
0.241
0.310
0.380
0.410
0.440
0.513

0.997
0.992
0.983
0.958
0.936
0.907
0.904
0.775
0.572
0.454
0.330
0.139

t

t

t

(Power)
0.738
0.713
0.670
0.609
0.564
0.509
0.509
0.370
0.235
0.175
0.128
0.062

148

Table 3.4: Size and Power Using Standard 5% Critical Values with Traditional Lag Choices, T =
200 and 500
T=200
lag

l4

ηµ
l12

l25

l4

ηµd
l12

l25

l4 × l4

ηµ × ηµd
l12 × l12

l25 × l25

∼ I (0)
ρ=0
0.25
0.50
0.75
0.95

(Size)
0.044 0.040
0.061 0.047
0.098 0.055
0.223 0.079
0.711 0.311

0.029
0.031
0.033
0.042
0.127

0.945
0.951
0.965
0.975
0.574

(Power)
0.555
0.569
0.615
0.706
0.333

0.008
0.009
0.012
0.021
0.032

0.041
0.057
0.093
0.215
0.358

(Size)
0.022
0.026
0.033
0.050
0.046

0.000
0.000
0.001
0.000
0.000

∼ I (1)
φ=0
0.25
0.50
0.75
0.95

(Power)
0.948 0.720
0.948 0.721
0.945 0.720
0.919 0.707
0.557 0.466

0.524
0.525
0.523
0.514
0.340

0.037
0.058
0.127
0.445
0.917

(Size)
0.022
0.023
0.031
0.094
0.499

0.004
0.003
0.004
0.003
0.006

0.026
0.041
0.100
0.385
0.501

(Size)
0.004
0.004
0.006
0.028
0.211

0.000
0.000
0.000
0.000
0.001

∼ I (d)
d =0.1
0.2
0.3
0.4
0.45
0.499
0.5
0.6
0.7
0.75
0.8
0.9

0.132
0.264
0.403
0.540
0.597
0.646
0.647
0.742
0.819
0.852
0.879
0.923

(Power)
0.084
0.143
0.219
0.292
0.334
0.379
0.381
0.457
0.526
0.558
0.594
0.658

0.051
0.082
0.117
0.161
0.183
0.214
0.215
0.271
0.333
0.364
0.391
0.457

0.940
0.926
0.904
0.863
0.831
0.785
0.785
0.646
0.442
0.341
0.246
0.105

(Power)
0.550
0.534
0.500
0.439
0.391
0.348
0.350
0.237
0.145
0.111
0.079
0.042

0.009
0.008
0.008
0.008
0.009
0.008
0.007
0.007
0.006
0.006
0.006
0.005

0.121
0.241
0.358
0.457
0.483
0.486
0.488
0.447
0.321
0.253
0.184
0.074

(Power)
0.046
0.074
0.096
0.099
0.094
0.084
0.091
0.057
0.029
0.022
0.016
0.006

0.001
0.001
0.001
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000

l4

ηµ
l12

l25

l4

ηµd
l12

l25

t

t

t

T=500
lag

l4 × l4

ηµ × ηµd
l12 × l12

l25 × l25

∼ I (0)
ρ=0
0.25
0.50
0.75
0.95

(Size)
0.054 0.049
0.067 0.054
0.094 0.060
0.192 0.084
0.728 0.322

0.043
0.044
0.047
0.056
0.143

0.997
0.997
0.999
1.000
0.995

(Power)
0.873
0.883
0.918
0.971
0.969

0.556
0.570
0.616
0.728
0.784

0.053
0.067
0.094
0.192
0.724

(Size)
0.042
0.046
0.054
0.080
0.305

0.023
0.025
0.028
0.039
0.081

∼ I (1)
φ=0
0.25
0.50
0.75
0.95

(Power)
0.992 0.901
0.992 0.901
0.992 0.901
0.990 0.897
0.888 0.791

0.730
0.729
0.728
0.728
0.661

0.045
0.059
0.123
0.440
0.981

(Size)
0.038
0.042
0.054
0.160
0.778

0.023
0.025
0.027
0.046
0.400

0.041
0.054
0.117
0.431
0.870

(Size)
0.020
0.022
0.030
0.114
0.601

0.003
0.003
0.004
0.012
0.231

∼ I (d)
d =0.1
0.2
0.3
0.4
0.45
0.499
0.5
0.6
0.7
0.75
0.8
0.9

0.173
0.365
0.555
0.721
0.785
0.832
0.830
0.906
0.950
0.964
0.974
0.987

(Power)
0.119
0.225
0.341
0.461
0.515
0.566
0.568
0.662
0.746
0.778
0.808
0.861

0.084
0.145
0.220
0.297
0.339
0.379
0.381
0.461
0.530
0.571
0.609
0.672

0.996
0.995
0.991
0.980
0.967
0.950
0.947
0.844
0.642
0.507
0.366
0.141

(Power)
0.868
0.860
0.839
0.799
0.760
0.715
0.715
0.565
0.361
0.272
0.186
0.083

0.555
0.546
0.515
0.460
0.418
0.363
0.364
0.242
0.139
0.102
0.076
0.042

0.173
0.362
0.549
0.705
0.755
0.786
0.782
0.756
0.600
0.477
0.345
0.134

(Power)
0.101
0.192
0.281
0.359
0.380
0.384
0.386
0.339
0.221
0.163
0.108
0.045

0.047
0.075
0.101
0.113
0.109
0.095
0.099
0.063
0.029
0.020
0.014
0.008

t

t

t

149

Table 3.5: Size and Power Using Standard 5% Critical Values with Traditional Lag Choices, T =
1,000 and 2,000
T=1,000
lag

l4

ηµ
l12

l25

l4

ηµd
l12

l25

l4 × l4

ηµ × ηµd
l12 × l12

l25 × l25

∼ I (0)
ρ=0
0.25
0.50
0.75
0.95

(Size)
0.047 0.045
0.057 0.049
0.076 0.056
0.147 0.073
0.641 0.272

0.042
0.044
0.046
0.055
0.133

1.000
1.000
1.000
1.000
1.000

(Power)
0.966
0.971
0.985
0.999
1.000

0.804
0.817
0.866
0.946
0.996

0.047
0.057
0.076
0.147
0.641

(Size)
0.043
0.047
0.055
0.073
0.272

0.034
0.037
0.040
0.052
0.132

∼ I (1)
φ=0
0.25
0.50
0.75
0.95

(Power)
0.998 0.959
0.998 0.959
0.998 0.959
0.997 0.958
0.975 0.931

0.849
0.848
0.848
0.848
0.819

0.049
0.061
0.117
0.390
0.988

(Size)
0.042
0.046
0.064
0.174
0.851

0.034
0.037
0.042
0.085
0.599

0.048
0.060
0.116
0.387
0.963

(Size)
0.030
0.034
0.050
0.148
0.786

0.013
0.014
0.016
0.041
0.467

∼ I (d)
d =0.1
0.2
0.3
0.4
0.45
0.499
0.5
0.6
0.7
0.75
0.8
0.9

0.197
0.414
0.630
0.799
0.860
0.899
0.899
0.953
0.978
0.987
0.991
0.995

(Power)
0.139
0.273
0.422
0.576
0.637
0.695
0.696
0.792
0.859
0.885
0.906
0.939

0.106
0.192
0.299
0.406
0.459
0.507
0.509
0.600
0.681
0.713
0.747
0.800

1.000
1.000
0.999
0.997
0.994
0.986
0.984
0.925
0.740
0.603
0.446
0.186

(Power)
0.964
0.962
0.950
0.923
0.903
0.866
0.870
0.729
0.513
0.395
0.291
0.126

0.802
0.795
0.780
0.748
0.706
0.652
0.645
0.504
0.322
0.239
0.170
0.081

0.197
0.414
0.630
0.797
0.853
0.886
0.884
0.879
0.718
0.591
0.438
0.182

(Power)
0.133
0.262
0.398
0.524
0.567
0.591
0.593
0.555
0.410
0.319
0.234
0.099

0.085
0.150
0.227
0.293
0.311
0.305
0.300
0.257
0.159
0.112
0.076
0.032

l4

ηµ
l12

l25

l4

ηµd
l12

l25

t

t

t

T=2,000
lag

l4 × l4

ηµ × ηµd
l12 × l12

l25 × l25

∼ I (0)
ρ=0
0.25
0.50
0.75
0.95

(Size)
0.048 0.046
0.058 0.050
0.074 0.054
0.132 0.067
0.599 0.242

0.044
0.045
0.047
0.054
0.122

1.000
1.000
1.000
1.000
1.000

(Power)
0.995
0.996
0.999
1.000
1.000

0.940
0.949
0.971
0.994
1.000

0.048
0.058
0.074
0.132
0.599

(Size)
0.046
0.050
0.054
0.067
0.242

0.041
0.043
0.046
0.054
0.122

∼ I (1)
φ=0
0.25
0.50
0.75
0.95

(Power)
1.000 0.989
1.000 0.989
1.000 0.989
1.000 0.989
0.998 0.982

0.940
0.940
0.941
0.941
0.927

0.051
0.062
0.111
0.375
0.995

(Size)
0.045
0.049
0.066
0.165
0.878

0.040
0.041
0.048
0.092
0.677

0.051
0.062
0.110
0.375
0.993

(Size)
0.042
0.046
0.062
0.159
0.861

0.027
0.028
0.033
0.069
0.614

∼ I (d)
d =0.1
0.2
0.3
0.4
0.45
0.499
0.5
0.6
0.7
0.75
0.8
0.9

0.223
0.496
0.745
0.890
0.932
0.958
0.959
0.987
0.995
0.996
0.998
1.000

(Power)
0.163
0.345
0.533
0.693
0.764
0.815
0.817
0.892
0.938
0.952
0.966
0.981

0.125
0.254
0.397
0.523
0.588
0.651
0.651
0.753
0.820
0.848
0.873
0.910

1.000
1.000
1.000
1.000
1.000
0.999
0.998
0.977
0.849
0.707
0.552
0.217

(Power)
0.995
0.994
0.993
0.985
0.976
0.957
0.955
0.858
0.651
0.527
0.385
0.157

0.938
0.937
0.925
0.895
0.869
0.834
0.829
0.692
0.489
0.369
0.258
0.115

0.223
0.496
0.745
0.890
0.932
0.957
0.958
0.965
0.844
0.704
0.550
0.216

(Power)
0.163
0.344
0.529
0.683
0.745
0.777
0.779
0.755
0.596
0.487
0.359
0.146

0.117
0.237
0.367
0.467
0.505
0.528
0.525
0.493
0.361
0.271
0.185
0.080

t

t

t

150

Table 3.6: Size and Power Using 5% Fixed-b Critical Values with Traditional Lag Choices, T = 50
and 100
T=50
lag
t

t

t

t

t

l0

l4

l12

l25

l0

l4

l12

l25

l0 × l0

ηµ × ηµd
l4 × l4 l12 × l12

l25 × l12

∼ I (0)
ρ=0
0.25
0.5
0.75
0.95

0.053
0.149
0.336
0.653
0.924

(Size)
0.052 0.053
0.073 0.061
0.114 0.071
0.236 0.102
0.551 0.247

0.057
0.061
0.065
0.066
0.079

0.959
0.923
0.867
0.641
0.108

(Power)
0.597 0.085
0.608 0.096
0.613 0.121
0.487 0.168
0.103 0.083

0.031
0.034
0.042
0.058
0.057

0.050
0.134
0.280
0.379
0.082

(Size)
0.025
0.005
0.035
0.005
0.056
0.007
0.073
0.010
0.016
0.006

0.003
0.002
0.002
0.002
0.000

∼ I (1)
φ=0
0.25
0.5
0.75
0.95

0.961
0.944
0.895
0.709
0.126

(Power)
0.727 0.495
0.717 0.492
0.696 0.481
0.589 0.414
0.114 0.097

0.168
0.170
0.168
0.154
0.079

0.042
0.141
0.400
0.786
0.955

(Size)
0.041 0.039
0.059 0.039
0.113 0.037
0.321 0.044
0.586 0.077

0.039
0.037
0.030
0.026
0.033

0.030
0.112
0.321
0.518
0.117

(Size)
0.007
0.003
0.012
0.003
0.031
0.003
0.129
0.003
0.058
0.005

0.001
0.001
0.001
0.005
0.004

∼ I (d)
d =0.1
0.2
0.3
0.4
0.45
0.499
0.5
0.6
0.7
0.75
0.8
0.9

0.133
0.252
0.399
0.546
0.616
0.674
0.679
0.775
0.851
0.879
0.907
0.939

(Power)
0.095 0.072
0.156 0.097
0.222 0.128
0.301 0.162
0.348 0.185
0.397 0.208
0.396 0.210
0.484 0.257
0.556 0.316
0.593 0.343
0.623 0.374
0.681 0.436

0.065
0.073
0.083
0.089
0.094
0.101
0.101
0.113
0.126
0.132
0.137
0.155

0.946
0.925
0.890
0.820
0.774
0.723
0.720
0.572
0.389
0.294
0.215
0.102

(Power)
0.578 0.085
0.556 0.082
0.508 0.081
0.443 0.083
0.409 0.081
0.373 0.078
0.359 0.075
0.265 0.072
0.181 0.064
0.145 0.061
0.119 0.056
0.073 0.045

0.034
0.035
0.037
0.033
0.031
0.032
0.030
0.034
0.036
0.036
0.036
0.039

0.124
0.227
0.340
0.427
0.450
0.458
0.457
0.402
0.290
0.222
0.164
0.077

(Power)
0.047
0.005
0.075
0.005
0.089
0.005
0.093
0.005
0.093
0.005
0.092
0.006
0.091
0.006
0.069
0.005
0.044
0.005
0.033
0.005
0.027
0.004
0.014
0.003

0.003
0.003
0.002
0.002
0.002
0.002
0.002
0.002
0.002
0.002
0.001
0.001

l50

l4

l50

l4 × l4

ηµ × ηµd
l12 × l12 l25 × l25

l50 × l25

T=100
lag
t

ηµd

ηµ

ηµd

ηµ
l4

l12

l25

l12

l25

∼ I (0)
ρ=0
0.25
0.5
0.75
0.95

0.045
0.060
0.095
0.208
0.608

(Size)
0.045 0.048
0.052 0.052
0.061 0.057
0.093 0.068
0.305 0.137

0.051
0.053
0.058
0.065
0.054

0.801
0.813
0.841
0.839
0.244

(Power)
0.391 0.054
0.410 0.057
0.447 0.072
0.508 0.109
0.209 0.132

0.028
0.028
0.032
0.042
0.072

0.032
0.044
0.073
0.158
0.092

(Size)
0.013
0.002
0.015
0.003
0.016
0.004
0.023
0.010
0.015
0.012

0.002
0.002
0.002
0.001
0.002

∼ I (1)
φ=0
0.25
0.5
0.75
0.95

0.832
0.830
0.820
0.758
0.277

(Power)
0.621 0.445
0.617 0.444
0.615 0.442
0.588 0.422
0.236 0.182

0.086
0.085
0.084
0.081
0.079

0.042
0.059
0.123
0.389
0.777

(Size)
0.043 0.044
0.046 0.042
0.063 0.040
0.123 0.038
0.372 0.051

0.044
0.040
0.038
0.036
0.030

0.017
0.026
0.060
0.238
0.203

(Size)
0.004
0.005
0.005
0.005
0.008
0.004
0.027
0.003
0.072
0.003

0.002
0.002
0.002
0.004
0.007

∼ I (d)
d =0.1
0.2
0.3
0.4
0.45
0.499
0.5
0.6
0.7
0.75
0.8
0.9

0.100
0.186
0.288
0.390
0.439
0.489
0.489
0.575
0.655
0.691
0.728
0.786

(Power)
0.077 0.065
0.121 0.085
0.171 0.110
0.231 0.146
0.259 0.163
0.292 0.182
0.292 0.184
0.360 0.220
0.428 0.266
0.459 0.290
0.494 0.316
0.562 0.383

0.057
0.062
0.062
0.074
0.077
0.078
0.076
0.080
0.078
0.080
0.081
0.083

0.788
0.767
0.726
0.667
0.623
0.576
0.577
0.434
0.294
0.230
0.172
0.091

(Power)
0.393 0.054
0.379 0.055
0.362 0.059
0.314 0.060
0.285 0.060
0.246 0.062
0.254 0.057
0.193 0.060
0.143 0.060
0.120 0.057
0.098 0.057
0.065 0.050

0.028
0.029
0.030
0.032
0.033
0.036
0.034
0.038
0.038
0.039
0.039
0.041

0.073
0.134
0.197
0.235
0.243
0.249
0.246
0.196
0.131
0.102
0.076
0.037

(Power)
0.023
0.020
0.030
0.025
0.035
0.026
0.032
0.023
0.035
0.023
0.030
0.017
0.034
0.018
0.022
0.014
0.015
0.010
0.010
0.008
0.008
0.006
0.006
0.004

0.003
0.003
0.003
0.001
0.002
0.002
0.002
0.002
0.003
0.002
0.002
0.002

151

Table 3.7: Size and Power Using 5% Fixed-b Critical Values with Traditional Lag Choices, T =
200 and 500
T=200
lag
t

t

t

t

t

l4

l12

l25

l50

l4

l12

l25

l50

l4 × l4

ηµ × ηµd
l12 × l12 l25 × l25

l50 × l25

∼ I (0)
ρ=0
0.25
0.5
0.75
0.95

0.047
0.066
0.102
0.233
0.720

(Size)
0.053 0.053
0.057 0.055
0.065 0.060
0.094 0.074
0.340 0.186

0.053
0.056
0.057
0.064
0.091

0.956
0.963
0.974
0.983
0.641

(Power)
0.663 0.346
0.677 0.356
0.727 0.395
0.826 0.475
0.540 0.407

0.046
0.049
0.060
0.084
0.150

0.045
0.063
0.100
0.227
0.414

(Size)
0.034
0.017
0.038
0.016
0.044
0.019
0.074
0.023
0.119
0.020

0.017
0.018
0.021
0.025
0.017

∼ I (1)
φ=0
0.25
0.50
0.75
0.95

0.951
0.950
0.948
0.923
0.564

(Power)
0.741 0.587
0.739 0.587
0.738 0.583
0.725 0.576
0.487 0.414

0.377
0.378
0.378
0.376
0.283

0.048
0.073
0.150
0.478
0.930

(Size)
0.047 0.048
0.051 0.051
0.069 0.053
0.180 0.083
0.616 0.286

0.046
0.046
0.046
0.041
0.046

0.034
0.054
0.122
0.419
0.516

(Size)
0.011
0.002
0.013
0.003
0.022
0.004
0.078
0.010
0.287
0.095

0.002
0.002
0.003
0.006
0.060

∼ I (d)
d =0.1
0.2
0.3
0.4
0.45
0.499
0.5
0.6
0.7
0.75
0.8
0.9

0.137
0.271
0.415
0.550
0.608
0.655
0.659
0.749
0.826
0.860
0.884
0.925

(Power)
0.099 0.084
0.167 0.123
0.241 0.173
0.318 0.221
0.364 0.246
0.406 0.281
0.408 0.281
0.481 0.343
0.550 0.398
0.584 0.428
0.617 0.459
0.684 0.518

0.072
0.091
0.115
0.141
0.152
0.174
0.175
0.204
0.238
0.257
0.278
0.323

0.952
0.941
0.920
0.884
0.852
0.811
0.811
0.677
0.479
0.377
0.280
0.126

(Power)
0.661 0.344
0.652 0.334
0.624 0.318
0.575 0.285
0.531 0.262
0.492 0.239
0.492 0.237
0.375 0.188
0.258 0.142
0.201 0.122
0.158 0.101
0.086 0.071

0.048
0.049
0.051
0.053
0.055
0.057
0.053
0.058
0.060
0.059
0.055
0.053

0.128
0.252
0.375
0.476
0.504
0.513
0.516
0.478
0.357
0.289
0.216
0.094

(Power)
0.065
0.054
0.109
0.079
0.143
0.102
0.161
0.107
0.165
0.104
0.158
0.102
0.163
0.104
0.126
0.078
0.083
0.047
0.060
0.034
0.045
0.025
0.021
0.013

0.027
0.035
0.038
0.034
0.028
0.024
0.030
0.018
0.011
0.009
0.007
0.004

l50

l4

l50

l4 × l4

ηµ × ηµd
l12 × l12 l25 × l25

l50 × l25

T=500
lag
t

ηµd

ηµ

ηµd

ηµ
l4

l12

l25

l12

l25

∼ I (0)
ρ=0
0.25
0.5
0.75
0.95

0.054
0.070
0.097
0.197
0.733

(Size)
0.055 0.054
0.058 0.056
0.064 0.060
0.090 0.068
0.335 0.166

0.052
0.054
0.056
0.063
0.099

0.998
0.998
1.000
1.000
0.997

(Power)
0.901 0.672
0.908 0.687
0.941 0.738
0.983 0.848
0.984 0.927

0.365
0.377
0.415
0.507
0.639

0.054
0.069
0.097
0.197
0.730

(Size)
0.049
0.035
0.052
0.037
0.059
0.042
0.088
0.055
0.325
0.139

0.033
0.035
0.039
0.049
0.084

∼ I (1)
φ=0
0.25
0.5
0.75
0.95

0.992
0.992
0.992
0.990
0.890

(Power)
0.908 0.749
0.908 0.748
0.907 0.748
0.906 0.747
0.802 0.685

0.593
0.593
0.592
0.593
0.552

0.048
0.067
0.131
0.454
0.984

(Size)
0.051 0.047
0.055 0.049
0.070 0.057
0.193 0.101
0.815 0.545

0.049
0.048
0.050
0.060
0.227

0.044
0.062
0.126
0.445
0.874

(Size)
0.028
0.013
0.031
0.014
0.043
0.017
0.146
0.037
0.641
0.347

0.007
0.008
0.009
0.021
0.269

∼ I (d)
d =0.1
0.2
0.3
0.4
0.45
0.499
0.5
0.6
0.7
0.75
0.8
0.9

0.178
0.371
0.559
0.725
0.790
0.836
0.834
0.909
0.952
0.965
0.975
0.988

(Power)
0.129 0.102
0.239 0.164
0.356 0.244
0.478 0.324
0.530 0.365
0.583 0.408
0.582 0.409
0.677 0.489
0.758 0.559
0.790 0.596
0.819 0.636
0.871 0.693

0.085
0.123
0.172
0.220
0.244
0.278
0.278
0.339
0.405
0.436
0.464
0.530

0.997
0.996
0.992
0.983
0.970
0.954
0.952
0.855
0.658
0.523
0.387
0.153

(Power)
0.897 0.669
0.888 0.662
0.871 0.647
0.835 0.598
0.803 0.564
0.755 0.524
0.757 0.516
0.619 0.390
0.419 0.253
0.322 0.196
0.233 0.146
0.107 0.084

0.363
0.354
0.336
0.290
0.274
0.242
0.245
0.181
0.134
0.112
0.096
0.069

0.178
0.369
0.554
0.712
0.762
0.794
0.790
0.770
0.617
0.493
0.367
0.146

(Power)
0.114
0.090
0.210
0.144
0.304
0.207
0.389
0.264
0.415
0.280
0.422
0.288
0.421
0.287
0.386
0.264
0.269
0.178
0.207
0.135
0.148
0.094
0.064
0.036

0.066
0.104
0.147
0.178
0.182
0.186
0.176
0.138
0.085
0.065
0.046
0.022

152

Table 3.8: Size and Power Using 5% Fixed-b Critical Values with Traditional Lag Choices, T =
1,000 and 2,000
T=1,000
lag
t

t

t

t

t

l4

l12

l25

l50

l4

l12

l25

l50

l4 × l4

ηµ × ηµd
l12 × l12 l25 × l25

l50 × l25

∼ I (0)
ρ=0
0.25
0.5
0.75
0.95

0.049
0.059
0.079
0.151
0.647

(Size)
0.049 0.051
0.054 0.052
0.059 0.056
0.077 0.063
0.286 0.145

0.049
0.050
0.052
0.055
0.092

1.000
1.000
1.000
1.000
1.000

(Power)
0.973 0.851
0.978 0.866
0.989 0.901
0.999 0.966
1.000 0.998

0.621
0.635
0.685
0.807
0.964

0.049
0.059
0.079
0.151
0.647

(Size)
0.048
0.044
0.052
0.045
0.059
0.051
0.077
0.061
0.286
0.144

0.042
0.043
0.046
0.053
0.091

∼ I (1)
φ=0
0.25
0.5
0.75
0.95

0.998
0.998
0.998
0.997
0.976

(Power)
0.963 0.861
0.962 0.860
0.962 0.860
0.961 0.860
0.934 0.832

0.707
0.707
0.707
0.707
0.686

0.052
0.065
0.123
0.396
0.988

(Size)
0.053 0.052
0.057 0.053
0.076 0.064
0.195 0.117
0.867 0.659

0.054
0.055
0.058
0.079
0.413

0.051
0.063
0.121
0.393
0.965

(Size)
0.040
0.023
0.044
0.023
0.059
0.029
0.170
0.066
0.805
0.528

0.012
0.012
0.015
0.036
0.419

∼ I (d)
d =0.1
0.2
0.3
0.4
0.45
0.499
0.5
0.6
0.7
0.75
0.8
0.9

0.201
0.418
0.635
0.803
0.862
0.901
0.902
0.955
0.978
0.987
0.991
0.996

(Power)
0.146 0.116
0.281 0.207
0.432 0.316
0.584 0.425
0.647 0.476
0.703 0.527
0.707 0.528
0.799 0.619
0.862 0.693
0.890 0.728
0.911 0.761
0.943 0.815

0.094
0.150
0.221
0.296
0.334
0.378
0.378
0.456
0.529
0.563
0.592
0.647

1.000
1.000
0.999
0.998
0.994
0.987
0.986
0.929
0.749
0.615
0.455
0.194

(Power)
0.972 0.848
0.968 0.844
0.960 0.829
0.935 0.800
0.916 0.764
0.886 0.716
0.885 0.711
0.754 0.573
0.547 0.392
0.431 0.307
0.319 0.227
0.143 0.115

0.620
0.616
0.601
0.570
0.531
0.486
0.485
0.366
0.248
0.200
0.153
0.093

0.201
0.418
0.635
0.802
0.857
0.889
0.888
0.884
0.728
0.603
0.448
0.190

(Power)
0.140
0.099
0.271
0.172
0.413
0.257
0.541
0.330
0.584
0.350
0.613
0.357
0.614
0.351
0.581
0.313
0.443
0.213
0.356
0.163
0.263
0.121
0.115
0.055

0.080
0.126
0.180
0.227
0.240
0.253
0.246
0.217
0.146
0.108
0.075
0.030

l50

l4

l50

l4 × l4

ηµ × ηµd
l12 × l12 l25 × l25

l50 × l25

T=2,000
lag
t

ηµd

ηµ

ηµd

ηµ
l4

l12

l25

l12

l25

∼ I (0)
ρ=0
0.25
0.5
0.75
0.95

0.049
0.060
0.074
0.134
0.604

(Size)
0.049 0.047
0.051 0.049
0.056 0.052
0.070 0.057
0.245 0.133

0.047
0.048
0.049
0.052
0.080

1.000
1.000
1.000
1.000
1.000

(Power)
0.996 0.954
0.997 0.960
0.999 0.979
1.000 0.996
1.000 1.000

0.802
0.818
0.866
0.950
1.000

0.049
0.060
0.074
0.134
0.604

(Size)
0.049
0.045
0.051
0.047
0.056
0.051
0.070
0.057
0.245
0.133

0.045
0.046
0.049
0.052
0.080

∼ I (1)
φ=0
0.25
0.5
0.75
0.95

1.000
1.000
1.000
1.000
0.998

(Power)
0.990 0.944
0.990 0.944
0.990 0.943
0.989 0.943
0.983 0.932

0.815
0.815
0.815
0.815
0.810

0.052
0.065
0.114
0.380
0.995

(Size)
0.051 0.052
0.058 0.054
0.073 0.061
0.176 0.108
0.886 0.708

0.053
0.054
0.058
0.078
0.482

0.051
0.064
0.114
0.380
0.993

(Size)
0.048
0.037
0.054
0.038
0.069
0.044
0.171
0.084
0.870
0.648

0.019
0.020
0.024
0.051
0.546

∼ I (d)
d =0.1
0.2
0.3
0.4
0.45
0.499
0.5
0.6
0.7
0.75
0.8
0.9

0.226
0.499
0.749
0.892
0.935
0.959
0.960
0.988
0.995
0.996
0.998
1.000

(Power)
0.167 0.131
0.350 0.267
0.538 0.407
0.700 0.535
0.770 0.599
0.819 0.662
0.824 0.663
0.893 0.759
0.942 0.826
0.954 0.855
0.966 0.878
0.982 0.916

0.107
0.191
0.287
0.393
0.434
0.487
0.489
0.570
0.651
0.688
0.718
0.769

1.000
1.000
1.000
1.000
1.000
0.999
0.999
0.978
0.853
0.712
0.557
0.222

(Power)
0.996 0.953
0.995 0.950
0.993 0.940
0.988 0.912
0.979 0.892
0.961 0.856
0.961 0.852
0.869 0.723
0.669 0.526
0.543 0.408
0.408 0.295
0.167 0.136

0.800
0.799
0.786
0.755
0.723
0.677
0.675
0.538
0.369
0.279
0.211
0.110

0.226
0.499
0.749
0.892
0.934
0.958
0.959
0.966
0.848
0.709
0.556
0.221

(Power)
0.166
0.124
0.349
0.254
0.534
0.382
0.691
0.484
0.754
0.529
0.785
0.554
0.790
0.551
0.767
0.523
0.617
0.397
0.504
0.308
0.381
0.222
0.158
0.099

0.103
0.182
0.271
0.354
0.381
0.405
0.403
0.380
0.293
0.224
0.155
0.061

153

Table 3.9: Size and Power of ADF and ηµ × ADF Tests Using Standard Critical Values with
Traditional Lag Choices, T = 50, 100, 200, and 500
T=50
lag
t

t

t

∼ I (0)
ρ=0
0.25
0.50
0.75
0.95
∼ I (1)
φ=0
0.25
0.50
0.75
0.95
∼ I (d)
d =0.1
0.2
0.3
0.4
0.45
0.499
0.5
0.6
0.7
0.75
0.8
0.9

p0

ADF
p4

0.041
0.157
0.515
0.956
1.000

(Power)
0.789
0.683
0.495
0.220
0.056
(Size)
0.044
0.041
0.052
0.182
0.716

1.000
1.000
0.999
0.978
0.941
0.882
0.880
0.686
0.422
0.307
0.219
0.094

(Power)
0.672
0.536
0.406
0.294
0.249
0.209
0.207
0.143
0.102
0.084
0.070
0.053

p4

ADF
p12

1.000
1.000
0.978
0.474
0.064

T=100
ηµ × ADF
l4 × p4 l12 × p12

p12

l0 × p 0

0.072
0.069
0.061
0.047
0.038

0.053
0.149
0.320
0.238
0.043

0.038
0.034
0.031
0.030
0.065

0.032
0.119
0.413
0.665
0.126

(Size)
0.000
0.000
0.001
0.002
0.012
(Size)
0.020
0.018
0.014
0.010
0.001

0.063
0.057
0.050
0.043
0.043
0.042
0.042
0.040
0.037
0.039
0.038
0.037

0.133
0.252
0.398
0.525
0.559
0.561
0.564
0.476
0.298
0.217
0.159
0.068

(Power)
0.001
0.001
0.002
0.004
0.004
0.007
0.007
0.010
0.013
0.015
0.017
0.017

p0

ADF
p4

0.000
0.000
0.000
0.000
0.000

1.000
1.000
1.000
0.974
0.116

0.000
0.000
0.000
0.000
0.000

0.054
0.190
0.585
0.978
1.000

(Power)
0.994
0.983
0.932
0.623
0.091
(Size)
0.051
0.050
0.058
0.182
0.947

0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000

1.000
1.000
1.000
1.000
1.000
0.998
0.998
0.930
0.692
0.516
0.362
0.145

(Power)
0.970
0.901
0.761
0.573
0.479
0.401
0.397
0.255
0.155
0.128
0.102
0.068

p4

ADF
p12

l0 × p 0

0.346
0.315
0.270
0.185
0.060

0.043
0.143
0.361
0.693
0.105

0.043
0.044
0.041
0.041
0.206

0.051
0.179
0.563
0.883
0.300

(Size)
0.002
0.001
0.003
0.008
0.013
(Size)
0.023
0.024
0.022
0.016
0.008

0.274
0.205
0.158
0.123
0.111
0.101
0.100
0.082
0.067
0.059
0.055
0.048

0.156
0.346
0.539
0.715
0.782
0.828
0.828
0.838
0.645
0.484
0.338
0.133

(Power)
0.002
0.005
0.009
0.012
0.013
0.017
0.017
0.019
0.019
0.020
0.021
0.024

T=200
lag
t

t

t

∼ I (0)
ρ=0
0.25
0.50
0.75
0.95
∼ I (1)
φ=0
0.25
0.50
0.75
0.95
∼ I (d)
d =0.1
0.2
0.3
0.4
0.45
0.499
0.5
0.6
0.7
0.75
0.8
0.9

0.046
0.046
0.052
0.165
0.997

(Power)
1.000
1.000
1.000
0.999
0.661
(Size)
0.047
0.045
0.045
0.047
0.480

1.000
1.000
0.997
0.956
0.892
0.791
0.788
0.543
0.317
0.231
0.169
0.096

(Power)
0.751
0.595
0.444
0.313
0.265
0.225
0.224
0.153
0.110
0.094
0.079
0.060

1.000
1.000
1.000
1.000
0.899

p25

l4 × p 4

ηµ × ADF
l4 × p4 l12 × p12

p12

0.007
0.008
0.010
0.011
0.002
0.004
0.011
0.056
0.170
0.049

0.011
0.017
0.025
0.034
0.040
0.048
0.048
0.053
0.038
0.028
0.019
0.006

T=500
ηµ × ADF
l12 × p12 l25 × p25

0.875
0.867
0.846
0.778
0.359

0.054
0.067
0.094
0.192
0.422

0.046
0.047
0.048
0.047
0.098

0.043
0.042
0.041
0.043
0.371

(Size)
0.043
0.044
0.047
0.056
0.135
(Size)
0.018
0.073
0.359
0.719
0.661

0.193
0.149
0.116
0.089
0.081
0.077
0.076
0.060
0.052
0.049
0.046
0.044

0.045
0.066
0.077
0.064
0.064
0.063
0.062
0.051
0.044
0.043
0.040
0.039

(Power)
0.051
0.082
0.117
0.161
0.183
0.214
0.215
0.267
0.249
0.184
0.110
0.034

0.028
0.028
0.030
0.033
0.030

1.000
1.000
1.000
1.000
0.899

0.011
0.011
0.010
0.011
0.121

0.046
0.046
0.052
0.165
0.997

(Power)
1.000
1.000
1.000
0.999
0.661
(Size)
0.047
0.045
0.045
0.047
0.480

0.004
0.005
0.005
0.004
0.003
0.003
0.003
0.002
0.002
0.002
0.001
0.002

1.000
1.000
1.000
0.999
0.999
0.991
0.990
0.870
0.552
0.398
0.262
0.106

(Power)
1.000
0.993
0.944
0.798
0.695
0.575
0.572
0.356
0.203
0.152
0.115
0.071

154

p25

l4 × p 4

ηµ × ADF
l12 × p12 l25 × p25

0.875
0.867
0.846
0.778
0.359

0.054
0.067
0.094
0.192
0.422

0.046
0.047
0.048
0.047
0.098

0.043
0.042
0.041
0.043
0.371

(Size)
0.043
0.044
0.047
0.056
0.135
(Size)
0.018
0.073
0.359
0.719
0.661

0.749
0.592
0.432
0.300
0.252
0.206
0.205
0.139
0.094
0.083
0.068
0.055

0.173
0.355
0.484
0.498
0.459
0.384
0.378
0.253
0.154
0.115
0.094
0.062

(Power)
0.084
0.145
0.220
0.297
0.339
0.379
0.381
0.461
0.516
0.492
0.367
0.095

0.028
0.028
0.030
0.033
0.030
0.011
0.011
0.010
0.011
0.121

0.050
0.077
0.094
0.091
0.083
0.072
0.069
0.040
0.022
0.018
0.013
0.010

APPENDIX

155

Appendix for Chapter 3
Proof of Theorem 8
Liu (1998), Theorem 3.4, shows that ∑ St2 = O p T 3 when

t

1
2

is I

. It also shows

that s2 (0) = O p (ln T ) . However, he does not establish the order in probability of s2 (l )
when l → ∞ as T → ∞. The proof here will establish the limiting behavior of s2 (l ) when
l → ∞ using results of Tanaka (1999). Tanaka provides the invariance principle for I

1
2

processes having i.i.d. innovations. He defines the process
ts2T − s2j 1
1
XT (t) = y j + 2
y j − y j −1 ,
sT
s j − s2j−1 s T

s2j−1
s2T

≤t≤

s2j
s2T

,

(2.37)

where (1 − L)1/2 yt = ut ∼ I ID (0, σu2 ) and s2j = Var y j . Lemma 2.1 of Liu (1998) shows
that
s2j =
with K =

2σu2
π

4σu2
π

j

1

∑ 2k − 1 = K · L( j)

(2.38)

k =1

and
L( T )
= 1.
T →∞ log T

(2.39)

lim

Theorem 2.1 in Tanaka (1999) states XT = { XT (t)} weakly converges to the standard
Wiener process defined on [0, 1] . Note that in our terminology,

t

replaces yt in Tanaka

(1999). Rewrite s2 (l ) as
s2 ( l ) =

1
T

T

l

∑ e2t + 2 ∑

t =1

1−

s =1

156

s
l+1

1
T

T

∑

t = s +1

et et−s .

Multiplying s2 (l ) by

1
l ln T

· ln1T yields

1
1
1 1 1
·
· s2 ( l ) =
l ln T ln T
l ln T T

+

T

∑ (ln T )−1/2 et · (ln T )−1/2 et

(2.40)

t =1

1 l
s
1−
∑
l s =1
l+1

1 1
ln T T

T

∑

(ln T )−1/2 et · (ln T )−1/2 et−s .

t = s +1

Consider the absolute value of the second sum.
1 l
s
1−
∑
l s =1
l+1

≤

1 l
s
1−
∑
l s =1
l+1

=

≤

1 1
ln T T

T

∑

(ln T )−1/2 et · (ln T )−1/2 et−s

1 1
· max
ln T T 1≤s≤l

1 1 1
· max
2 ln T T 1≤s≤l

T

∑

(2.41)

t = s +1
T

∑

(ln T )−1/2 et · (ln T )−1/2 et−s

t = s +1

(ln T )−1/2 et · (ln T )−1/2 et−s

t = s +1

T
1 1 1
· max ∑ (ln T )−1/2 et · (ln T )−1/2 et−s .
2 ln T T 1≤s≤l t=s+1

It turns out that it is enough to show this last expression is o p (1).

max

1≤ s ≤ l

1 1
ln T T

T

∑

(ln T )−1/2 et · (ln T )−1/2 et−s

t = s +1

1 1
≤ max
( T − s) max (ln T )−1/2 et · (ln T )−1/2 et−s
ln
T
T
s +1≤ t ≤ T
1≤ s ≤ l
1
≤ max
max (ln T )−1/2 et · max (ln T )−1/2 et
1≤ t ≤ T
1≤s≤l ln T 1≤t≤ T
1
1
=
max (ln T )−1/2 et · max (ln T )−1/2 et =
· O p (1) = o p (1).
ln T 1≤t≤T
ln T
1≤ t ≤ T

157

(2.42)

The second to last equality in (2.42) comes from the following.
max (ln T )−1/2 et

1≤ t ≤ T

≤

max

1≤ t ≤ T

(ln T )−1/2

≤ 2 · max (ln T )

+ (ln T )−1/2

t

−1/2

K1/2
2 · max
·
sT
1≤ t ≤ T

t

1≤ t ≤ T

(2.43)
t

with large T

= 2K1/2 · max | XT (r )| ⇒ 2K1/2 · max |W (r )| = O p (1).
0≤r ≤1

0≤r ≤1

The weak convergence result in the last line follows from Tanaka (1999).
Similarly, one can show

1 1 1
l ln T T

∑tT=1

√ 1 et
ln T

·

√ 1 et
ln T

= o p (1). Hence

1
l (ln T )2

· s2 ( l ) =

o p (1). Now rewrite ηµ as
ηµ =
and recall that
above p lim

1
T3

1
T3

∑tT=1 St2

1
s2 ( l )
l (ln T )2

×

T
l (ln T )2

,

(2.44)

∑tT=1 St2 = O p (1) and its weak limit is not zero (Liu 1998). Also from the

1
s2 ( l )
l (ln T )2

= 0. Therefore, since

T
l (ln T )2

goes to infinity under the traditional

choice of the number of the lags, ηµ diverges to infinity if l → ∞ and

l
T

→ 0 as T → ∞.

Proof of Proposition 8
In this case, ∆

t

is a short memory process with zero mean. So the limiting behavior of

s2 (l ) and St should be the same as that of s2 (l ) and St from the model with short memory
error and no intercept. Hence the followings are immediate:
p
1
s2 (l ) → σ2 , √ S[rT ] ⇒ σW (r ),
T
1
1 T 2
2
S
⇒
σ
W (r )2 dr, and
∑
t
2
T t =2
0

ηµd ⇒

1
0

(2.45)

W (r )2 dr.

Note that the weak limit of ηµd is a functional of a standard Wiener process instead of a
Brownian bridge process (for the KPSS test of short memory) or a demeaned Brownian
motion (for the KPSS unit root test). This is because we difference the data instead of

158

demeaning the terms in St .
Proof of Theorem 9
Rewrite the numerator as T −2 ∑tT=2 St2 = T −2 ∑tT=2 (
T −2 ∑tT=2

t

−1

T

∑

St2

t =2

∑tT=2

1
=
T
1
T

=
1
T

−

1)

t

+

2
1

2

= T −2 ∑tT=2

2
t

−2

1

·

+ T −1 12 . Multiplying by T gives
T

since

t

T

∑

2
t

2
t

+

t =2

T

∑

T

1
−2 1
T

2
1

∑

(2.46)

t =2

+ o p (1),

t =2

p

t

→ 0. Therefore,

T

−1

T

d

∑ St2 → γ0 +

2
1.

(2.47)

t =2

Second, to figure out the limiting behavior of s2 (l ), rewrite γs as below.
1
T

γs =

=

1
T

T

∑

T

∑

( t−

t −1 ) ( t − s

T

t t−s

t = s +2

t − s −1 )

T

∑

−

−

t = s +2

t t − s −1

−

t = s +2

∑

T

t −1 t − s

+

t = s +2

∑

t −1 t − s −1

.

t = s +2

Plugging this into s2 (l ) yields:

−

−

1
T
1
T

T

1
T

s2 ( l ) =

∑

t =2

T

∑

t =2

∑

t

s =1

l

t

1
T

T

∑

T

t = s +2

l

T

t −1

t + 2 ∑ w ( s, l )
s =1

159

1
T

t t−s

t = s +2

1
t−1 + 2 ∑ w ( s, l )
T
s =1

T

t =2

l

t + 2 ∑ w ( s, l )

∑

∑

t = s +2

t t − s −1

t −1 t − s

(2.48)

1
T

+

T

l

∑

t −1 t −1

t =2

+ 2 ∑ w(s, l )
s =1

1
T

T

∑

.

t −1 t − s −1

t = s +2

Then the equation (2.48) can be rewritten by collecting the terms according to the time
lags of cross products of

m n ’s

s2 ( l ) =

+

+

1
T

∑ [2 − 2w(1, l )]

+

t t

T T

t =2

+
T

1 1

(2.49)

T −1

∑ 2 [2w(1, l ) − w(0, l ) − w(2, l )]

t t −1

t =3

T T −1

+ 2 [w(1, l ) − 1]

2 1

T

+

T −1

∑ 2 [2w(2, l ) − w(1, l ) − w(3, l )]

t t −2

t =4

2 [w(2, l ) − w(1, l )] · [
T

T T −2

+

3 1]

+···

T −1

∑

2 [2w(l − 1, l ) − w(l − 2, l ) − w(l, l )]

t t − l +1

t = l +1

+

2 [w(l − 1, l ) − w(l − 2, l )] · [
T
1
+
T

+

T −1

2 [w(1, l ) − w(0, l )] ·
1
+
T

1
+
T

1
T

(i.e. the value of m − n).

T T − l +1

+

l 1]

T −1

∑

2 [2w(l, l ) − w(l − 1, l )]

t t−l

t = l +2

2 [w(l, l ) − w(l − 1, l )] · [
T

T T −l

+

l +1 1 ]

−

1
T

T

∑

2w(l, l )

t t − l −1 .

t = l +2

Now, fix l and let T increase to infinity. Because max1≤s,t≤T |

t s|

= O p (T )

4

one can

obtain the following as T increases:
p

s2 (l ) → [2 − 2w(1, l )] γ0 + 2 [2w(1, l ) − w(0, l ) − w(2, l )] γ1
4 Notice that max

2
t
1≤s,t≤ T | t s | = max1≤t≤ T t and recall that for t ∼ I (1), max1≤t≤ T √ T
equivalently max1≤t≤T t2 = O p ( T ) . Hence we conclude that max1≤t≤T t2 = O p ( T ) when t

short or long memory process.

160

(2.50)
2

= O p (1), or

is a stationary

+ · · · + 2 [2w(l − 1, l ) − w(l − 2, l ) − w(l, l )] γl −1 + 2 [2w(l, l ) − w(l − 1, l )] γl − 2w(l, l )γl +1 .
Note that with the Bartlett kernel w( j, l ) = 1 −

j
l +1 ,

(2.49) can be simplified because

2w( j, l ) − w( j − 1, l ) − w( j + 1, l ) = 0

for j = 1, 2, ..., l. Hence
p

s2 (l ) → [2 − 2w(1, l )] γ0 − 2w(l, l )γl +1 =

2
(γ0 − γl +1 ) .
l+1

(2.51)

p

Therefore, l · s2 (l ) → 2γ0 as l increases since γl +1 → 0 under the assumption of either
stationary short- or stationary long-memory process.
Now it is straightforward to see that
T d
T
ηµ =
l
l

T −2 ∑tT=2 St2
s2 ( l )

=

T −1 ∑tT=2 St2
,
ls2 (l )

(2.52)

and therefore, using (2.47) and (2.51),
T d d γ0 + 12
η →
.
l µ
2γ0
p

This implies that ηµd → 0 as l → ∞, T → ∞,

l
T

(2.53)

→ 0.

Proof of Proposition 9
In this case, ∆

t

∼ I (d∗ ) where d∗ = d − 1 with − 21 < d∗ < 0. This means ∆

anti-persistent process. From Table 3.1 in Section 3.3.1,
1
T 2( d ∗

S
+1) [rT ]

≡
=

1
T d∗ +1/2
1

[rT ]

∑∆

[rT ]

∑∆
T d∗ +1/2
t =1

⇒ ωd∗ Wd∗ (r ),

161

t

t =2

t

−

1
T d∗ +1/2

∆

1

t

is an

so
T

1
T 2( d ∗ +1)

∑ St2 ⇒ ωd2∗

t =2

1
0

Wd∗ (r )2 dr,

and
p

l −2d∗ s2 (l ) → ωd2∗ .
Therefore,
l
T
That is, ηµd = O p

T 2d∗
l

2d∗

1

ηµd ⇒

0

Wd∗ (r )2 dr.

and ηµd goes to zero as l → ∞, T → ∞,

l
T

→ 0.

Proof of Theorem 10
Since

t

∼ I (1/2), it it true that ∆yt = ∆

t

∼ I (−1/2). Fix l and increase T to get

s2 (l ) → σ2 (l ) = γ0∗ + 2 ∑ls=1 Ws,l γs∗ , where γs∗ = E (∆ t ∆

t−s ) .

As in Lee and Schmidt

(1996, p.291), one can show

2

( l + 1) σ ( l ) = ( l

+ 1) γ0∗

l

+ 2 ∑ (l + 1 − s) γs∗

(2.54)

s =1

l +2

= var

∑∆

= var (

j

l +2

−

1)

j =2

= var (

l +2 ) + var ( 1 ) − 2ρ

where ρ is the correlation between

l +2

and

var ( t ) =
where L(t) = 4 ∑tj=1

1
2j−1

and

L(t)
ln t

1.

var (

l +2 )

var ( 1 ),

Recall from equation (2.38) that

2σu2
L ( t ),
π

→ 1 as t increases. Divide equation (2.54) by ln(l + 1)

and let l increase to yield

( l + 1) σ 2 ( l )
2σ2
→ u as l grows.
ln(l + 1)
π

162

This is due to the facts that var ( 1 )/ ln(l + 1) → 0 as l → ∞ and |ρ| ≤ 1. Hence

p lim

l · s2 ( l )
( l + 1) s2 ( l )
2σ2
= p lim
= u ≡ K,
ln l
ln(l + 1)
π
ln l
l

which implies s2 (l ) = O p

when T and l grow but l/T goes to zero.

Next, consider the sum of St2 in the numerator of ηµd . Look at the absolute value of the
appropriately scaled sum and see
1
T ln T

T

1
T

∑ St2 =

t =2

≤ 4 · max √
1≤ t ≤ T

T

1

∑ √ln T (

√

K1/2
·
4 · max
sT
1≤ t ≤ T

t

−

t =2
2

1
ln T

1) ·

t

t

1
ln T

( t−

1)

with large T

= 4K1/2 · max | XT (r )| ⇒ 4K1/2 · max |W (r )| = O p (1).
0≤r ≤1

0≤r ≤1

Therefore, one can obtain

ηµd

T −2 ∑tT=2 St2
=
=
s2 ( l )

=

l ln T
ln l T

l ln T
ln l T · O p (1)
2σu2
π + o p (1)

T
1
2
T ln T ∑t=2 St
l 2
ln l s ( l )

= o p (1).

Proof of Proposition 12
Unlike the case for ηµ , the calculation of ηµd does not involve demeaning so that the
full sum of the data ST is not zero. So, the correct representation of the HAC estimator
with the Bartlett kernel is (see Hashimzade and Vogelsang (2008), page 161)

s (l ) =

T −1

2

2

b ( T − 1)2

−

∑

St2

t =2

2
b ( T − 1)2

−

T −b ( T −1)−1

2
b ( T − 1)2

T −1

∑

t = T − b ( T −1)

163

St S T +

∑

S t S t + b ( T −1)

t =2

1
S2 ,
T−1 T

(2.55)

with b =

l +1
T −1 .

Now suppose

s (l ) =

T −1

2

2

∑(

b ( T − 1)2

=

2

T −1

∑

2

t + b ( T −1)

−

1

(2.56)

( t−

−

1) ( T

1)

+

2
t

T −1

−2

∑

1

t =2

t

+ ( T − 2)

t

+

2
1

1
(
T−1

−

t =2

T

−

1)

2
b ( T − 1)2

2

×

T −b ( T −1)−1
t

t + b ( T −1) −

∑

1

2
1

+ T − b ( T − 1) − 1 ·

t + b ( T −1)

t =2



T −1

2
b ( T − 1)

1)

t = T − b ( T −1)

2
b ( T − 1)

( t−



∑



t =2

−

∑

−

T −1

T −b ( T −1)−1

∑

1)

t =2

2
b ( T − 1)

−

t

in (2.55) yields

T −b ( T −1)−1

2

t =2



−

t− 1

follows a short-memory process. Plugging St =

t

2

∑





T −1
t T

∑

−

t = T − b ( T −1)

1( t

ηµd

=

T) + b

( T − 1)

2
1 +

t = T − b ( T −1)

≡ H (b ) +
Combining this with

+

1
(
T−1

T

−

1)

2

∑tT=2 St =

1
T2

∑tT=2

2
t

−2

T
1 ∑ t =2 t

+ ( T − 1)

2
1

1
T2

∑tT=2 St
=
s2 ( l )

1
T

∑tT=2

2
t

−2

T
1 ∑ t =2 t

+ ( T − 1)

2
1

Denote the weak limit of

T

as T grows as

∞

T
T −1

T

−

1)

.

1
T2

T · H (b ) +

1
(
T−1

(

T

−

1)

2

gives

.

and apply the functional central limit theo-

rem and continuous mapping theorem to obtain

2
b

(γ0 + b 1
γ0 + 12
2
2
b γ0 + 1 +

=

Secondly, suppose that

2
1

γ0 +

ηµd ⇒

t

∞) + ( ∞
2
∞

−

1)

2

γ0 + 12
<
2γ0 + 12 +

2
∞

<1

follows a unit root process. Rearranging equation (2.56)

164

2

gives
2
( T − 1)2 2
s (l ) =
2
bT
T

−

2
bT

T −1

∑

T −1

∑

T −1/2

2
t

−

t =2

T −1/2

t

· T −1/2

2
bT

T+

t = T − b ( T −1)

T −b ( T −1)−1

∑

T −1/2

t

· T −1/2

t + b ( T −1)

t =2

T−1
T −1/2 (
T

T−

1)

2

+ o p (1),

where the remaining terms are negligible since
T −1/2

1

= o p (1),

and
1
T2

T −1

∑

t 1

= T −1/2

1·

t =2

1
T

T −1

∑ T −1/2

t

= o p (1)O p (1) = o p (1).

t =2

Therefore by applying the functional CLT and continuous mapping theorem one can
show
ηµd

⇒

1
2
0 W (r ) dr
,
Q1d (b )

where W (r ) is the standard Wiener process and
Q1d (b ) ≡

2
b

1
0

W (r )2 dr −

1− b
0

W (r )W (r + b )dr −

Note that this limit does not degenerate to
limit of ηµ in Proposition 11, which is

1
2

1
2

1
1− b

W (r )W (1)dr + W (1)2 .

for b = 1. This is in contrast with the fixed-b

under both I (0) and I (1) DGPs.

165

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