ESSAYS ON TIME SERIES ECONOMETRICS WITH FINANCIAL APPLICATIONS

By

Taeyoon Hwang

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

Economics—Doctor of Philosophy

2024

ABSTRACT

This dissertation provides the developments and extensions of methodologies in time series econo-

metrics and their financial applications. Chapter 1 of the disseration introduces and summarizes the

following chapters. Chapter 2 develops an estimating equation approach to construct confidence

intervals for autocorrelation functions for time series with general stationary serial correlation

structures. Its empirical application using S&P 500 index returns shows that conclusions about

market efficiency and volatility clustering during pre and post-Covid periods using the estimating

equation approach contrast with conclusions using traditional (and often incorrectly used) methods.

Chapter 3 develops fixed-𝑏 asymptotics results for heteroskedasticity autocorrelation robust (HAR)

Wald tests for regressions for high frequency data using an existing continuous time framework. Its

empirical application suggests that the validity of the uncovered interest parity hypothesis depends

on whether normal or fixed-𝑏 critical values are used. Chapter 4 investigates the distribution of

realized US corporate bond return volatility using a compound poisson process setting and realized

volatility.

Copyright by
TAEYOON HWANG
2024

ACKNOWLEDGEMENTS

First and foremost, I would like to express my deepest gratitude to God for guiding me through my

Ph.D. journey in good health. I firmly believe that it was with His presence that I have reached this

point.

I am profoundly grateful to my advisor, Professor Tim Vogelsang, the person I am most thankful

to during my Ph.D. journey. His guidance and support enabled me to complete this dissertation

and reach the end of my Ph.D. journey. He was not only the best advisor but also a great friend. I

will never forget the valuable research activities and moments of daily life we shared at MSU.

I would also like to extend my sincere thanks to the members of my dissertation committee,

Professor Kyoo il Kim, Professor Antonio Galvao, and Professor Hao Jiang, for their constructive

feedback and support. I am also grateful to Michigan State University for allowing me to start my

Ph.D. and for the support provided throughout my studies. Additionally, I would like to thank the

staff of the MSU Econ department for their support, and my colleagues, the Econ graduate students,

for their camaraderie and shared experiences.

Special thanks go to Professor Sunku Hahn and Professor Jay Pil Choi, who guided me into

the field of economics. I would also like to dedicate this dissertation to the late President Chang

Young Jung, who is now with God and made great contributions to the academic field of Korean

economics.

I would like to express my gratitude to everyone who prayed for me, especially the members

of the Korean Church of Lansing.

I am also thankful to my friends Hyunki Kim, Kijoo Song,

and Doyeon Kim, who brought laughter into my life, and to Jaeho Choi and Jaehyeong Lim, who

discussed the future with me and provided support. I would also like to dedicate this dissertation

to my dearly beloved cousin, Mansoo Hwang, who was loved by God and went to be with Him far

too early.

Lastly, I am deeply indebted to my parents and partner for their unwavering support and

encouragement. To my parents, and my partner, Mira Choi, thank you for your constant love and

understanding.

iv

TABLE OF CONTENTS

CHAPTER 1

BIBLIOGRAPHY . .

. .

.

INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1
4

CHAPTER 2

AN ESTIMATING EQUATION APPROACH FOR ROBUST CONFIDENCE
INTERVALS FOR AUTOCORRELATIONS OF STATIONARY TIME
5
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
SERIES .
.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
TABLES AND FIGURES . . . . . . . . . . . . . . . . . . . . . 39

.
.

.
.

.
.

.

BIBLIOGRAPHY .
APPENDIX 2A

CHAPTER 3

SOME FIXED-𝑏 RESULTS FOR REGRESSIONS WITH HIGH FREQUENCY
DATA OVER LONG SPANS . . . . . . . . . . . . . . . . . . . . . . . 72
.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
PROOFS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
TABLES AND FIGURES . . . . . . . . . . . . . . . . . . . . . 114

.

.

.

BIBLIOGRAPHY .
APPENDIX 3A
APPENDIX 3B

CHAPTER 4

THE DISTRIBUTION OF REALIZED US CORPORATE BOND
RETURN VOLATILITY . . . . . . . . . . . . . . . . . . . . . . . . . . 123
.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
TABLES AND FIGURES . . . . . . . . . . . . . . . . . . . . . 136

.

.

.

BIBLIOGRAPHY .
APPENDIX 4A

v

CHAPTER 1

INTRODUCTION

This dissertation aims to develop and extend time series econometric methodologies with pro-

viding their financial applications. Therefore, the following chapters in this dissertation include

econometric methods and their empirical application using financial time series data.

Chapter 2 of this dissertation develops an estimating equation approach to obtain valid con-

fidence intervals for autocorrelations for stationary time series. The autocorrelation function is

a fundamental statistical analysis tool, widely used in empirical research across various scientific

domains. Yet, it is surprising to note that limited work has been done on providing easily im-

plementable methods for inference about the autocorrelation function of time series data under

empirically realistic assumptions (e.g., the relaxation of independent identically distributed (i.i.d.)

assumption for underlying innovations). Romano and Thombs (1996) pointed out that the Bartlett

formula, a primary approach for inference for the autocorrelation function, becomes invalid when

the assumption of i.i.d innovations is relaxed. This issue, coupled with the lack of a robust and

easily implementable method for such inference, poses challenges in the research domain: 1) The

previous literature might yield potentially misleading economic implications when utilizing the

Bartlett formula for inference, if the i.i.d assumption is violated. For example, Bollerslev and

Mikkelsen (1996) and Andersen et al. (2003) present figures with the confidence bands based on

the Bartlett formula to illustrate dependence properties within series of volatilities. 2) From a

practical standpoint, many statistical packages commonly used by researchers rely on the Bartlett

formula for inference.

Chapter 2 addresses the issue by developing a simple and easy to implement estimating equation

approach for robust inference for the autocorrelation function. The estimating equation is estimated

by ordinary least squares and inference is heteroskedasticity and autocorrelation robust (HAR).

The approach is robust in three ways: innovations can be weak white noise, innovations can have

asymmetric distributions, and inference does not require a specific model of serial correlation.

Extensive Monte Carlo simulations in Chapter 2 highlight the robustness of the approach. An

1

empirical application using S&P 500 index returns shows that, in the post-Covid period, conclusions

about market efficiency and volatility clustering using the approach contrast with conclusions using

the traditional approaches.

Chapter 3 of this dissertation develops fixed-𝑏 asymptotics results for HAR Wald tests for

high frequency data using the continuous time framework of Chang et al. (2023). In the fields

of finance and macroeconomics, high frequency data is increasingly being adopted for research.

Distinct asymptotics, rooted in a continuous time framework, have been established for regressions

involving such high frequency data. These asymptotics consider the time between observations

𝛿 → 0 and sample span 𝑇 → ∞ jointly, different than standard asymptotics of discrete time, as

suggested in Chang et al. (2023). In this context, Chapter 3 develops fixed-𝑏 asymptotic results for

HAR Wald tests for high frequency stationary regression and cointegrating regression under the

continuous time framework. Fixed-𝑏 asymptotics1 captures the impact of kernel and bandwidth

choices on the sampling distributions of HAR test statistics and typically provides more accurate

inference than traditional asymptotics. Chapter 3 shows that fixed-𝑏 limits of HAR Wald tests for

high frequency stationary regressions in the continuous time setting are the same as the standard

discrete time fixed-𝑏 limits. The simulation study in Chapter 3 shows that fixed-𝑏 critical values

provide rejection probabilities closer to nominal levels than traditional chi-square critical values

under data generated by Ornstein-Uhlenbeck processes, which are continuous-time analogues of

autoregregressive lag 1 (AR(1)) processes. As an empirical application, Chapter 3 provide some

basic results on the uncovered interest parity (UIP) puzzle by using Yen/US dollar exchange rate

returns and 2-year/10-year government bond yields of the US and Japan from 1991 to 2022,

providing evidence that validity of the UIP hypothesis depends on whether normal or fixed-𝑏

critical values are used.

Chapter 4 of this dissertation focuses on empirical research on the volatility of a financial

instrument, US corporate bonds. Given that corporate bond prices are illiquid and display irregular

trading patterns that differ from other assets like stocks, I model the price dynamics of bond prices

1For more about fixed-𝑏 asymptotic theory, please see Kiefer and Vogelsang (2005).

2

with discrete jumps using a compound Poisson process (CPP). Then, I investigate the distribution of

realized US corporate bond return volatility using realized volatility (RV) introduced in Andersen

et al. (2001). Monte Carlo simulations are designed to examine finite sample properties of RV under

CPP. These simulations consider various structures for the variance of the price jump including

the Heston model and allow a different mean value for the number of daily transactions (number

of price jumps) for the processes. The simulation results indicate that RV is a solid approximation

for integrated volatility when the mean of daily transactions is set to about 146, showing that

the mean absolute percentage error is around 10.2% for the case where the variance of the jump

follows the Heston model. For the empirical analysis, I build series of daily realized volatilities

for US corporate bonds from 2013 to 2018 by using high frequency corporate bond transaction

data (recorded every second) from the Financial Industry Regulatory Authority’s Trade Reporting

and Compliance Engine and link them with corporate bond characteristics (such as credit ratings,

issued amounts and yield) to examine the conditional distributions of the volatilities based on each

bond characteristic.

3

BIBLIOGRAPHY

Andersen, T. G., Bollerslev, T., Diebold, F. X., and Ebens, H. (2001). The distribution of realized

stock return volatility. Journal of financial economics, 61(1):43–76.

Andersen, T. G., Bollerslev, T., Diebold, F. X., and Labys, P. (2003). Modeling and forecasting

realized volatility. Econometrica, 71(2):579–625.

Bollerslev, T. and Mikkelsen, H. O. (1996). Modeling and pricing long memory in stock market

volatility. Journal of econometrics, 73(1):151–184.

Chang, Y., Lu, Y., and Park, J. Y. (2023). Understanding regressions with observations collected
at high frequency over long span. Working paper, School of Economics, University of Sydney.

Kiefer, N. M. and Vogelsang, T. J. (2005). A new asymptotic theory for heteroskedasticity-

autocorrelation robust tests. Econometric Theory, 21:1130–1164.

Romano, J. P. and Thombs, L. A. (1996). Inference for autocorrelations under weak assumptions.

Journal of the American Statistical Association, 91(434):590–600.

4

CHAPTER 2

AN ESTIMATING EQUATION APPROACH FOR ROBUST CONFIDENCE INTERVALS
FOR AUTOCORRELATIONS OF STATIONARY TIME SERIES
(CO-AUTHORED WITH TIM VOGELSANG)

2.1

Introduction

The autocorrelation function is a fundamental quantity in time series analysis with the sample

autocovariance routinely computed for observed time series. Approximating the sampling dis-

tribution of the estimated autocorrelation is a key tool in understanding the potential population

autocorrelation and the underlying dynamics of a time series. The seminal work by Bartlett (1946)

derived a formula, known as the ‘Bartlett formula’, for the asymptotic covariance matrix of sample

autocorrelations under the assumption that the underlying time series is covariance stationary with

independent, identically, distributed (i.i.d.) innovations. For a given parametric specification of the

autocorrelation function, the Bartlett formula enables one to compute feasible confidence intervals

and conduct hypothesis testing for autocorrelations. However, it has been pointed out in the liter-

ature that inference using the Bartlett formula is invalid when the i.i.d. innovation assumption is

relaxed. See Romano and Thombs (1996) and references.

Upon relaxing the i.i.d. assumption, Romano and Thombs (1996) derived the asymptotic

distribution of sample autocorrelations when the underlying innovations are only uncorrelated.

Allowing innovations to be uncorrelated but otherwise dependent permits many stationary nonlinear

processes frequently used in time series analysis. Another advantage of the approach of Romano

and Thombs (1996) is that it does not depend on any particular structure for generating the

stationary processes. However, to compute confidence intervals for sample autocorrelations they

suggest using the moving block bootstrap and subsampling schemes that may have been viewed

as computationally intensive at the time the Romano and Thombs (1996) paper was written. This

may be the reason their methods have not been adopted by widely used software packages.

In

The co-author has approved that the co-authored chapter is included. The co-author’s contact: Tim Vogelsang,
Department of Economics, 486 W. Circle Drive, 110 Marshall-Adams Hall, Michigan State University East Lansing,
MI 48824-1038. email: tjv@msu.edu

5

contrast to resampling methods, Lobato (2001) employed nonparametric kernel estimators of the

asymptotic variance of sample autocorrelations. A recent paper by Wang and Sun (2020) used a

similar approach but with orthonormal series variance estimators. Both of those papers focused

on tests of zero autocorrelation and not the construction of generally valid confidence intervals for

estimated autocorrelations.

There is a related strand of the literature that focuses on extending Bartlett’s asymptotic variance

formula that is valid for uncorrelated but potentially dependent innovations. Francq and Zakoïan

(2009) derive a generalized Bartlett formula for the case where innovations of the time series

process are weak white noise process. The formula obtained by Francq and Zakoïan (2009) can

be viewed as a closed-form version of the general asymptotic variance given by Romano and

Thombs (1996) that is represented in terms of the autocorrelation function of the time series, the

autocorrelation of the square of the innovations, and a kurtosis parameter. Their formula also

relies on a symmetry assumption for the fourth moments of the innovations. Implementation of

the generalized Bartlett formula is relatively straightforward for simple autocorrelation structures

like moving average models but is very complicated in general. It is likely for this reason that the

generalized Bartlett variance formula has not been implemented in standard software packages.

While the literature has highlighted the dependence of the original Bartlett formula on the

assumption of i.i.d. innovations, many modern statistical packages still rely on Bartlett formula for

deriving variance estimators of sample autocorrelations and for inference about autocorrelations.

Furthermore, even if the assumption of i.i.d.

innovations is valid, many software packages im-

plement a version of the Bartlett formula that is not valid for general stationary serial correlation

structures. For example, in Stata’s manual, the formula for the estimated variance of, (cid:98)
autocorrelation at lag 𝑘, is given by

𝜌𝑘 , the sample

𝑉

𝑎𝑟 ((cid:98)
(cid:98)

𝜌𝑘 ) =

1/𝑇

1
𝑇

(cid:8)1 + 2 (cid:205)𝑘−1

𝜌2
𝑖=1 (cid:98)
𝑖

(cid:9)

𝑘 = 1

𝑘 > 1,

(2.1)





where 𝑇 is the sample size. This formula assumes, for the purposes of computing an estimated

variance for (cid:98)

𝜌𝑘 and conducting inference about 𝜌𝑘 , that the true time series is a moving average

6

process with lag 𝑘 − 1, i.e. MA(𝑘 − 1). This is equivalent to carrying out a sequence of tests where
𝜌1 is used to test hypothesis about 𝜌1 conditional on the series being i.i.d. (MA(0)), (cid:98)
(cid:98)
test hypothesis about 𝜌2 conditional on the series being MA(1), ..., (cid:98)
about 𝜌𝑘 conditional on the series being is MA(𝑘 − 1). Suppose the series is MA(3). Then the

𝜌𝑘 is used to test hypothesis

𝜌2 is used to

𝜌1, (cid:98)

𝜌2 and (cid:98)

variance formulas for (cid:98)
𝜌𝑘
What is missing in the statistical packages is a method for computing confidence intervals for (cid:98)
(values of 𝜌𝑘 that cannot be rejected by a test), that are valid for general stationary serial correlation

𝜌3 are invalid along with corresponding confidence intervals.

structures and do not require the assumption of i.i.d. innovations.

In this paper we develop a simple estimating equation approach for computing confidence

intervals for estimated autocorrelations. The estimating equation approach extends the Lobato

(2001) and Wang and Sun (2020) approaches to the general stationary serial correlation case. Except

in narrow special cases, the asymptotic variances of the estimated autocorrelations take a sandwich

form and well known heteroskedasticity autocorrelation robust (HAR) variance estimators can be

used in a straightforward manner. We focus on kernel and orthonormal series HAR estimators and

use fixed-smoothing theory (Kiefer and Vogelsang (2005), Sun (2013)) to generate critical values

for computing confidence intervals. Following Lazarus et al. (2018) we consider HAR variance

estimators that impose the null leading to more reliable inference. Confidence intervals using

null-imposed HAR variance estimators are obtained using similar methods as used by Vogelsang

and Nawaz (2017). Our approach is easy to implement and can be viewed as a method for

operationalizing Romano and Thombs (1996) without needing resampling methods for valid first

order asymptotic inference.

The paper is organized as follows. Section 2 reviews estimation and inference of/for the

autocorrelation function of a stationary time series. In section 3 we develop a simple estimating

equation approach using HAR tests for inference. We show that fixed-smoothing asymptotics

applies to the test statistics. Our theory allows innovations of the time series to be white noise

driven by random variables whose distributions are potentially skewed. We show how to calculate

confidence intervals when the null is imposed on the variance estimator. Section 4 provides a

7

simulation study that documents finite sample null rejection probabilities and power for various

data generating processes (DGPs). Comparisons are made to existing approaches. Section 5

provides an empirical application using returns of the S&P 500 stock index. Some implications

about market efficiency and volatility clustering of the S&P 500 index during pre- and post-Covid

periods are obtained. Section 6 concludes the paper.

2.2 Preliminaries

Consider a real-valued covariance stationary time series, {𝑦𝑡 }, with mean 𝐸 (𝑦𝑡) = 𝜇. The

autocovariance and autocorrelation functions for 𝑦𝑡 are given as

𝛾𝑘 = 𝐸 [(𝑦𝑡 − 𝜇) (𝑦𝑡−𝑘 − 𝜇)] ,

𝑘 = 0, ±1, ±2, . . . ,

𝜌𝑘 = 𝛾𝑘 /𝛾0.

For a sample of 𝑇 observations {𝑦1, 𝑦2, ..., 𝑦𝑇 } define the sample autocovariance function as

𝛾𝑘 = 𝑇 −1
(cid:98)

𝑇
∑︁

𝑡=𝑘+1

(𝑦𝑡 − ¯𝑦) (𝑦𝑡−𝑘 − ¯𝑦) ,

𝑘 = 0, 1, 2, . . . , 𝑇 − 1,

where ¯𝑦 = 𝑇 −1 (cid:205)𝑇
𝑡=1

𝑦𝑡, and define the sample autocorrelation function as

𝜌𝑘 = (cid:98)
(cid:98)

𝛾0.
𝛾𝑘 /(cid:98)

(2.2)

The seminal work of Bartlett (1946) provided a formula, now known as Bartlett’s formula, for

the asymptotic variances and covariances of (cid:98)
i.i.d. innovations. Let 𝑦𝑡 be expressed by the Wold decomposition,

𝜌𝑘 when 𝑦𝑡 is a stationary linear time series driven by

𝑦𝑡 − 𝜇 =

∞
∑︁

𝜙𝑚𝜖𝑡−𝑚,

𝜌1, . . . ,

𝑚=−∞
where 𝜖𝑡 is an 𝑖.𝑖.𝑑.(0, 𝜎2) innovation. Then the vector of sample autocorrelations up to lag
𝑚, (cid:98)𝝆 = ((cid:98)
of corresponding population autocorrelations up to lag 𝑚. The asymptotic variance-covariance
𝑖, 𝑗 , the 𝑖 𝑗 𝑡ℎ element of the 𝑚 × 𝑚 matrix V𝐵, given by Bartlett’s
matrix of (cid:98)𝝆 is given by 𝑇 −1V𝐵 with 𝑣 𝐵
formula:

𝜌𝑚)′, asymptotically follows a normal distribution with mean 𝝆, the vector
(cid:98)

∞
∑︁

𝑣 𝐵
𝑖, 𝑗 =

(cid:8)𝜌ℓ+𝑖 𝜌ℓ+ 𝑗 + 𝜌ℓ−𝑖 𝜌ℓ+ 𝑗 + 2𝜌𝑖 𝜌 𝑗 𝜌2

ℓ − 2𝜌𝑖 𝜌ℓ 𝜌ℓ+ 𝑗 − 2𝜌 𝑗 𝜌ℓ 𝜌ℓ+𝑖}.

ℓ=−∞

8

Despite its wide usage in textbooks and statistical packages, Bartlett’s formula is only valid when

𝜖𝑡 is i.i.d. Use of Bartlett’s formula for inference is potentially invalid when 𝜖𝑡 is an uncorrelated

process (e.g. white noise process), but not i.i.d.. Specifically, using mixing conditions that allow

white noise innovations, Romano and Thombs (1996) derived an asymptotic normality result for
√
𝑇 ((cid:98)𝝆 − 𝝆) with asymptotic variance-covariance matrix V𝑅𝑇 with 𝑖 𝑗 𝑡ℎ elements given by

𝑣 𝑅𝑇
𝑖, 𝑗 = 𝛾−2
0

(cid:2)𝑐𝑖+1, 𝑗+1 − 𝜌𝑖𝑐1, 𝑗+1 − 𝜌 𝑗 𝑐1,𝑖+1 + 𝜌𝑖 𝜌 𝑗 𝑐1,1

(cid:3) ,

where 𝑐𝑖+1, 𝑗+1 = (cid:205)∞
𝑐𝑖+1, 𝑗+1 is the (𝑖 + 1, 𝑗 + 1)𝑡ℎ element of the asymptotic variance-covariance matrix of
where 𝜸 = (𝛾0, ..., 𝛾𝑚)′ and (cid:98)𝜸 = ((cid:98)
Thombs (1996) propose resampling methods for constructing confidence intervals for 𝜌𝑘 . For tests

𝑑=−∞ cov (cid:0)𝑦0𝑦𝑖, 𝑦𝑑 𝑦𝑑+ 𝑗 (cid:1). Note that Romano and Thombs (1996) showed that
√
𝑇 ((cid:98)𝜸 − 𝜸)
𝑖, 𝑗 , Romano and

𝛾𝑚)′. Given the complicated nature of 𝑣 𝑅𝑇
(cid:98)

𝛾0, ...,

of zero autocorrelation Lobato (2001) proposed nonparametric kernel estimators of 𝑐𝑖+1, 𝑗+1 and

Wang and Sun (2020) used series to estimate 𝑐𝑖+1, 𝑗+1. Neither study focused on confidence intervals

for 𝜌𝑘 when the time series has autocorrelation.

Closed form formulas for 𝑣 𝑅𝑇

𝑖, 𝑗 were obtained by Francq and Zakoïan (2009) for some models

of 𝑦𝑡 with 𝜖𝑡 being white noise with a symmetry condition imposed on the fourth moments of

𝜖𝑡. Francq and Zakoïan (2009) label these formulas ‘generalized Bartlett’ formulas. For example,

suppose that 𝑦𝑡 is a weak white noise process (i.e. 𝑦𝑡 = 𝜖𝑡 where 𝜖𝑡 is a weak white noise process).
𝑖, 𝑗 = 𝑣 𝐵

The generalized Bartlett formula is given by 𝑣𝐺 𝐵

𝑖, 𝑗 + 𝑣 𝐵∗

𝑖, 𝑗 where

𝑖,𝑖 = 1, 𝑣𝐵∗
𝑣 𝐵

𝑖,𝑖 =

𝛾𝜖 2 (𝑖)
[𝛾𝜖 (0)]2

,

(2.3)

and 𝑣 𝐵

𝑖, 𝑗 = 𝑣 𝐵∗

𝑖, 𝑗 = 0 if 𝑖 ≠ 𝑗 with 𝛾𝜖 2 (𝑖) being the autocovariance function of 𝜖 2

𝑡 at lag 𝑖 and 𝛾2

𝜖 (0)

being the variance of 𝜖𝑡. When the data generating process of 𝑦𝑡 is an MA(𝑞) model, Francq and

Zakoïan (2009) show that

𝑣 𝐵
𝑖,𝑖 =

𝑞
∑︁

ℓ=−𝑞

ℓ ,
𝜌2

𝑣𝐵∗
𝑖,𝑖 =

1
[𝛾𝜖 (0)]2

𝑞
∑︁

ℓ=−𝑞

ℓ ,
𝛾𝜖 2 (𝑖 − ℓ) 𝜌2

for all 𝑖 > 𝑞. Francq and Zakoïan (2009) do not provide formulas for 𝑖 ≤ 𝑞.

9

While the results of Lobato (2001), Wang and Sun (2020) and Francq and Zakoïan (2009) are

useful in specific contexts, they are not comprehensive enough to be used to construct confidence

intervals for 𝜌𝑘 . Therefore, we develop a systematic and simple approach to the construction of

confidence intervals that does not require resampling methods. Because our approach is based

on the inversion of 𝑡-statistics, resampling methods could be used to obtain critical values for the

construction of confidence intervals. We leave such an investigation to future research.

2.3 Theory

2.3.1 An Estimating Equation Approach For Autocorrelation Inference

In this section we develop an estimating equation approach that uses HAR 𝑡-statistics for

inference regarding autocorrelations where we relax the assumption that the innovations, 𝜖𝑡, are

i.i.d.. There are a few advantages of this approach. First, the HAR tests we use are well known and

easy to apply in practice. Second, we show that fixed-smoothing asymptotics can be used for the

test statistics providing critical values that depend on tuning parameters used to estimate variances.

Third, it is straightforward to construct confidence intervals for both the cases where the null

hypothesis about the autocorrelation is a) imposed and b) not imposed on the variance estimator.

As we show, imposing the null on the variance estimator can help reduce distortions in finite sample

rejections under the null similar to what was found for stationary time series regressions by Lazarus

et al. (2018) and Vogelsang (2018).

Consider the following estimation equation for a stationary time series 𝑦𝑡:

𝑦𝑡 = 𝑐 + 𝜌𝑘 𝑦𝑡−𝑘 + 𝜂(𝑘)

𝑡

,

(2.4)

where 𝑐 = 𝜇(1 − 𝜌𝑘 ) and 𝑡 = 𝑘 + 1, 𝑘 + 2, . . . , 𝑇. Regression (2.4) allows consistent estimation of

𝑐 and 𝜌𝑘 because

𝐸

(cid:17)

(cid:16)

𝜂(𝑘)
𝑡

= 0,

(cid:16)

𝐸

𝑦𝑡−𝑘 𝜂(𝑘)

𝑡

(cid:17)

= 0.

These conditions are easy to establish as follows. Taking the mean of both sides of (2.4) gives

𝐸 (𝑦𝑡) = 𝑐 + 𝜌𝑘 𝐸 (𝑦𝑡−𝑘 ) + 𝐸

(cid:16)

𝜂(𝑘)
𝑡

(cid:17)

.

𝜇 = 𝜇(1 − 𝜌𝑘 ) + 𝜌𝑘 𝜇 + 𝐸

(cid:16)

𝜂(𝑘)
𝑡

(cid:17)

,

10

Replacing 𝐸 (𝑦𝑡) and 𝐸 (𝑦𝑡−𝑘 ) with 𝜇 and because 𝑐 = 𝜇(1 − 𝜌𝑘 ), it follows that

𝜇 = 𝜇(1 − 𝜌𝑘 ) + 𝜌𝑘 𝜇 + 𝐸

(cid:17)

(cid:16)

𝜂(𝑘)
𝑡

= 𝜇 + 𝐸

(cid:16)

𝜂(𝑘)
𝑡

(cid:17)

,

in which case it follows that 𝐸

(cid:17)

(cid:16)

𝜂(𝑘)
𝑡

= 0. To show 𝐸

(cid:16)

𝑦𝑡−𝑘 𝜂(𝑘)

𝑡

(cid:17)

= 0, calculate 𝑐𝑜𝑣 (𝑦𝑡−𝑘 , 𝑦𝑡)

giving

𝑐𝑜𝑣 (𝑦𝑡−𝑘 , 𝑦𝑡) = 𝑐𝑜𝑣

(cid:16)

𝑦𝑡−𝑘 , 𝑐 + 𝜌𝑘 𝑦𝑡−𝑘 + 𝜂(𝑘)
(cid:16)

𝑡

(cid:17)

= 𝜌𝑘 𝑐𝑜𝑣 (𝑦𝑡−𝑘 , 𝑦𝑡−𝑘 ) + 𝑐𝑜𝑣

𝑦𝑡−𝑘 , 𝜂(𝑘)

𝑡

(cid:17)

,

or equivalently

𝛾𝑘 = 𝜌𝑘 𝛾0 + 𝑐𝑜𝑣

(cid:16)

It then directly follows that 𝑐𝑜𝑣
(cid:17)

(cid:16)

𝑦𝑡−𝑘 𝜂(𝑘)

𝑡

= 0.
Except in certain special cases, 𝜂(𝑘)

that 𝐸

𝑦𝑡−𝑘 , 𝜂(𝑘)

𝑡

(cid:17)

=

𝛾𝑘
𝛾0
(cid:17)
𝑦𝑡−𝑘 , 𝜂(𝑘)

𝑡

(cid:16)

𝛾0 + 𝑐𝑜𝑣

(cid:16)

𝑦𝑡−𝑘 , 𝜂(𝑘)

𝑡

(cid:17)

= 𝛾𝑘 + 𝑐𝑜𝑣

(cid:16)

𝑦𝑡−𝑘 , 𝜂(𝑘)

𝑡

(cid:17)

.

= 0. Because 𝐸

(cid:17)

(cid:16)

𝜂(𝑘)
𝑡

= 0, it must also be the case

𝑡 will have serial correlation. By construction 𝜂(𝑘)

𝑡

by

𝜂(𝑘)
𝑡 = 𝑦𝑡 − 𝑐 − 𝜌𝑘 𝑦𝑡−𝑘 = (𝑦𝑡 − 𝜇) − 𝜌𝑘 (𝑦𝑡−𝑘 − 𝜇) .

is given

(2.5)

Suppose 𝑦𝑡 is a finite order autoregressive moving average process (𝐴𝑅𝑀 𝐴( 𝑝, 𝑞)) given by

𝜙(𝐿) (𝑦𝑡 − 𝜇) = 𝜃 (𝐿)𝜖𝑡,

where 𝜙(𝐿) = 1 − 𝜙1𝐿 − 𝜙2𝐿2 − . . . − 𝜙 𝑝 𝐿 𝑝, 𝜃 (𝐿) = 1 + 𝜃1𝐿 + 𝜃2𝐿2 + . . . + 𝜃 𝑝 𝐿𝑞 and 𝐿 is the lag
operator. Applying the 𝜙(𝐿) lag polynomial to both sides of (2.5) gives

𝜙(𝐿)𝜂(𝑘)

𝑡 = 𝜙(𝐿) (𝑦𝑡 − 𝜇) − 𝜙(𝐿) 𝜌𝑘 (𝑦𝑡−𝑘 − 𝜇) = 𝜙(𝐿) (𝑦𝑡 − 𝜇) − 𝜌𝑘 𝜙(𝐿)𝐿 𝑘 (𝑦𝑡 − 𝜇)

= 𝜙(𝐿) (𝑦𝑡 − 𝜇) − 𝜌𝑘 𝐿 𝑘 𝜙(𝐿) (𝑦𝑡 − 𝜇) = 𝜃 (𝐿)𝜖𝑡 − 𝜌𝑘 𝐿 𝑘 𝜃 (𝐿)𝜖𝑡

(cid:16)

1 − 𝜌𝑘 𝐿 𝑘 (cid:17)

=

𝜃 (𝐿)𝜖𝑡 .

(2.6)

We see from (2.6) that 𝜂(𝑘)

𝑡

is an 𝐴𝑅𝑀 𝐴( 𝑝, 𝑞 + 𝑘) process.

Suppose that 𝑦𝑡 in uncorrelated. Then 𝑝 = 𝑞 = 0 and 𝜌𝑘 = 0, and it follows that (2.6) simplifies
is uncorrelated. Whether or not 𝑦𝑡−𝑘 𝜂(𝑘)

= 𝜖𝑡 in which case 𝜂(𝑘)

has serial correlation is

𝑡

𝑡

to 𝜂(𝑘)
𝑡

11

more complicated and depends on 𝑘, the serial correlation in 𝑦𝑡, and whether 𝜖𝑡 has dependence
in higher order moments. Cases where 𝑦𝑡−𝑘 𝜂(𝑘)

has no serial correlation should be viewed as

𝑡

exceptions rather than the rule, and inference based on estimation of (2.4) should be made robust

to serial correlation (and conditional heteroskedasticity).

It is convenient to rewrite the estimation equation (2.4) as

𝑦𝑡 = x′

𝑡−𝑘 𝛽+𝜂(𝑘)

𝑡

(2.7)

where x𝑡−𝑘 =

(cid:104)

1 𝑦𝑡−𝑘

(cid:105) ′

(cid:104)

and 𝛽 =

𝑐 𝜌𝑘

(cid:105) ′

. The ordinary least squares (OLS) estimator of 𝛽

from (2.7) is given by the usual formula

(cid:101)𝛽 =

𝑐
(cid:101)
𝜌𝑘
(cid:101)

















(cid:32) 𝑇

∑︁

=

𝑡=𝑘+1

x𝑡−𝑘 x′

𝑡−𝑘

(cid:33) −1

𝑇
∑︁

𝑡=𝑘+1

x𝑡−𝑘 𝑦𝑡 .

Using the Frisch-Waugh-Lovell Theorem, (cid:101)

𝜌𝑘 can be equivalently expressed as

(cid:205)𝑇

𝑡=𝑘+1

𝜌𝑘 =
(cid:101)

(cid:0)𝑦𝑡−𝑘 − ¯𝑦{1,𝑇−𝑘 }
(cid:205)𝑇

(cid:1) (cid:0)𝑦𝑡 − ¯𝑦{𝑘+1,𝑇 }
(cid:1) 2

(cid:0)𝑦𝑡−𝑘 − ¯𝑦{1,𝑇−𝑘 }

𝑡=𝑘+1

(cid:1)

,

where

¯𝑦{1,𝑇−𝑘 } =

1
𝑇 − 𝑘

𝑇−𝑘
∑︁

𝑡=1

𝑦𝑡,

¯𝑦{𝑘+1,𝑇 } =

1
𝑇 − 𝑘

𝑇
∑︁

𝑡=𝑘+1

𝑦𝑡 .

Define the 2 × 1 vector, v(𝑘)

𝑡

, as

and its partial sum process

v(𝑘)
𝑡 = x𝑡−𝑘 𝜂(𝑘)

𝑡 =

𝜂(𝑘)
𝑡

𝑦𝑡−𝑘 𝜂(𝑘)

𝑡









,









S(𝑘)
[𝑟𝑇] =

[𝑟𝑇]
∑︁

𝑡=𝑘+1

v(𝑘)
𝑡

,

where [𝑟𝑇] is the integer part of 𝑟𝑇 with 𝑟 ∈ [0, 1]. Using standard calculations,

𝑐 − 𝑐)

√
𝑇 ((cid:101)
√
𝜌𝑘 − 𝜌𝑘 )
𝑇 ((cid:101)









√

𝑇

(cid:16)

(cid:101)𝛽 − 𝛽

(cid:17)

=

=








(cid:32)
𝑇 −1

(cid:33) −1

x𝑡−𝑘 x′

𝑡−𝑘

𝑇 −1/2

𝑇
∑︁

𝑡=𝑘+1

(cid:32)
𝑇 −1

𝑇
∑︁

=

𝑡=𝑘+1

(cid:33) −1

x𝑡−𝑘 x′

𝑡−𝑘

𝑇 −1/2

𝑇
∑︁

𝑡=𝑘+1

x𝑡−𝑘 𝜂(𝑘)

𝑡

(cid:32)
𝑇 −1

𝑇
∑︁

v(𝑘)
𝑡 =

x𝑡−𝑘 x′

𝑡−𝑘

(cid:33) −1

𝑇 −1/2S(𝑘)
𝑇 .

𝑡=𝑘+1

𝑇
∑︁

𝑡=𝑘+1

12

The asymptotic variance of (cid:101)𝛽 depends on the probability limit of 𝑇 −1 (cid:205)𝑇
run variance of v(𝑘)

𝑡 which we denote by

𝑡=𝑘+1 x𝑡−𝑘 x′

𝑡−𝑘 and the long

𝛀(𝑘) = 𝚪(𝑘)
0

+

∞
∑︁

(cid:16)

𝑗=1

𝑗 + 𝚪(𝑘)′
𝚪(𝑘)

𝑗

(cid:17)

,

where 𝚪(𝑘)

𝑗 = 𝐸 (v(𝑘)

𝑡 v(𝑘)′
𝑡− 𝑗 ).

The following two assumptions are sufficient to obtain an asymptotic normality result for
(cid:16)

. We use the symbol ⇒ to denote weak convergence in distribution.

(cid:17)

(cid:101)𝛽 − 𝛽

√

𝑇

𝑡=𝑘+1 v(𝑘)

Assumption 2.1 𝑇 −1/2 (cid:205)[𝑟𝑇]
[𝑟𝑇] ⇒ 𝚲(𝑘)W2(𝑟), where 𝚲(𝑘) is the matrix square
root of 𝛀(𝑘), i.e. 𝛀(𝑘) = 𝚲(𝑘)𝚲(𝑘)′, 𝑟 ∈ [0, 1], and W2(𝑟) is a 2 × 1 vector of independent Wiener
processes (W2(𝑟) ∼ 𝑁 (0, 𝑟I2) where I2 is a 2 × 2 identity matrix).

= 𝑇 −1/2S(𝑘)

𝑡

Assumption 2.2 𝑇 −1 (cid:205)[𝑟𝑇]

𝑡=𝑘+1 x𝑡−𝑘 x′








Assumption 2.1 is a functional central limit theorem (FCLT) for the scaled partial sums of v(𝑘)
.
(cid:17)

, where 𝑟 ∈ [0, 1].

𝑝
→ 𝑟Q =𝑟

𝜇 𝛾0 + 𝜇2

Assumption 2.1 is stronger than what is needed for an asymptotic normality result for









𝑡−𝑘

√

𝑇

𝜇

1

(cid:16)

𝑡

(cid:101)𝛽 − 𝛽

but is used to obtain fixed-smoothing results for HAR test statistics. Inference is discussed in the next

section. A primitive condition for Assumption 2.1 to hold is that 𝑦𝑡 is near epoch dependence (𝐿2-

NED) with sufficient 𝛼-mixing. See Lobato (2001) for details for the case of zero autocovariance

tests. Additional details on sufficient conditions for FCLTs using NED and mixing can be found in
de Jong and Davidson (2000). Note that because i) v(𝑘)
ii) 𝜂(𝑘)
𝑡

is a filtered version of 𝑦𝑡−𝑘 , properties of transformations of NED processes play a role in

involves the product of 𝑦𝑡−𝑘 and 𝜂(𝑘)

, and

𝑡

𝑡

primitive conditions sufficient for Assumption 2.1; see Davidson (1994). Assumption 2.2 holds as

long as 𝑦𝑡−𝑘 is a second order stationary process. As long as 𝛾0 > 0 it follows that Q−1 exists.

We can directly derive the asymptotic distribution of

√

𝑇

(cid:16)

(cid:17)

(cid:101)𝛽 − 𝛽

under Assumptions 1 and 2

as

√

𝑇

(cid:16)

(cid:101)𝛽 − 𝛽

(cid:17)

=

𝑐 − 𝑐)

√
𝑇 ((cid:101)
√
𝜌𝑘 − 𝜌𝑘 )
𝑇 ((cid:101)

















⇒ Q−1𝚲W2(1) ∼ 𝑁

(cid:16)

0, Q−1𝛀(𝑘)Q−1(cid:17)

≡ 𝑁 (0, V(𝑘)).

13

𝜌𝑘 is V(𝑘)
The asymptotic variance of (cid:102)
calculations can be used to show that V(𝑘)

22 , which is the (2,2) element of V(𝑘). Straightforward
𝜌𝑘 obtained
22 is the same as the asymptotic variance for (cid:98)

by Romano and Thombs (1996) (see their equation (6)). Therefore, (cid:102)
to (cid:98)
out using well known estimators for V(𝑘) that are simple to implement in practice.

𝜌𝑘 . The advantage of using (cid:102)

𝜌𝑘 via the regression (2.4) is that inference about 𝜌𝑘 can be carried

𝜌𝑘 is asymptotically equivalent

The asymptotic variance, V(𝑘), is estimated as follows. The natural estimator of Q is given by

(cid:101)Q = (𝑇 − 𝑘)−1

𝑇
∑︁

𝑡=𝑘+1

x𝑡−𝑘 x′

𝑡−𝑘 .

Because the middle matrix of V(𝑘) is the long-run variance-covariance matrix of v(𝑘)

𝑡

, we can use

a nonparametric kernel estimator of the form

(cid:101)𝛀(𝑘) = (cid:101)𝚪(𝑘)

0

+

𝑇−𝑘−1
∑︁

𝑗=1

𝑘

(cid:18) 𝑗
𝑀

(cid:19) (cid:16)

𝑗 + (cid:101)𝚪(𝑘)′
(cid:101)𝚪(𝑘)

𝑗

(cid:17)

,

(cid:101)𝚪(𝑘)
𝑗 = (𝑇 − 𝑘)−1

𝑇
∑︁

𝑡=𝑘+ 𝑗+1

𝑡 (cid:101)v(𝑘)′
(cid:101)v(𝑘)
𝑡− 𝑗 ,

where

𝜂(𝑘)
(cid:101)v(𝑘)
𝑡 = x𝑡−𝑘 (cid:101)
𝑡

,

𝜂(𝑘)
𝑡 = 𝑦𝑡 − x′
(cid:101)

𝑡−𝑘 (cid:101)𝛽 = 𝑦𝑡 − (cid:101)

𝑐 − (cid:101)

𝜌𝑘 𝑦𝑡−𝑘 ,

(2.8)

𝑘 (𝑥) is a kernel function, and 𝑀 is a truncation lag or bandwidth. (cid:101)𝛀(𝑘) is the usual kernel HAR
. This leads to an estimator of V(𝑘) given by

𝜂(𝑘)
long run variance estimator using OLS residuals, (cid:101)
𝑡

(cid:101)V(𝑘) = (cid:101)Q−1

(cid:101)𝛀(𝑘)

(cid:101)Q−1.

We also consider a variant of (cid:101)𝛀(𝑘) that imposes the null hypothesis being tested about 𝜌𝑘 .

Suppose we are interested in testing the null hypothesis

𝐻0 : 𝜌𝑘 = 𝑎,

where 𝑎 is a given number in the (−1, 1) range. Define the null-imposed residuals for (2.4) as

𝜂(𝑘)∗
𝑡
(cid:101)

= 𝑦𝑡 − (cid:0) ¯𝑦{𝑘+1,𝑇 } − 𝑎 ¯𝑦{1,𝑇−𝑘 }

(cid:1) − 𝑎𝑦𝑡−𝑘 = (cid:0)𝑦𝑡 − ¯𝑦{𝑘+1,𝑇 }

(cid:1) − 𝑎 (cid:0)𝑦𝑡−𝑘 − ¯𝑦{1,𝑇−𝑘 }

(cid:1)

14

The null-imposed kernel estimator of 𝛀(𝑘) uses (cid:101)v(𝑘)∗
of (cid:101)v(𝑘)

and is given by

𝑡

𝑡

𝜂(𝑘)∗
= x𝑡−𝑘 (cid:101)
𝑡

− 1
𝑇−𝑘

(cid:205)𝑇

𝜂(𝑘)∗
𝑠=𝑘+1 x𝑠−𝑘 (cid:101)
𝑠

in place

(cid:101)𝛀(𝑘)∗ = (cid:101)𝚪(𝑘)∗

0

+

𝑇−𝑘−1
∑︁

𝑗=1

𝑘

(cid:18) 𝑗
𝑀

(cid:19) (cid:16)

(cid:101)𝚪(𝑘)∗

𝑗

+ (cid:101)𝚪(𝑘)∗′

𝑗

(cid:17)

,

(cid:101)𝚪(𝑘)∗

𝑗

= (𝑇 − 𝑘)−1

𝑇
∑︁

𝑡=𝑘+ 𝑗+1

𝑡 (cid:101)v(𝑘)∗′
(cid:101)v(𝑘)∗

𝑡− 𝑗

.

Notice that (cid:101)v(𝑘)∗
important for power by Lazarus et al. (2018) and Vogelsang (2018) when imposing the null on the

𝜂(𝑘)∗
is the demeaned version of x𝑡−𝑘 (cid:101)
𝑡

. This simple demeaning was found to be

𝑡

variance estimator. The null-imposed estimator of V(𝑘) is given by

(cid:101)V(𝑘)∗ = (cid:101)Q−1

(cid:101)𝛀(𝑘)∗

(cid:101)Q−1.

Lastly, there is one thing to point out about the bandwidth 𝑀.

In practice data dependent
methods are often used to choose 𝑀. Those formulas are functions of the proxy used for v(𝑘)
when estimating 𝛀(𝑘). For (cid:101)𝛀(𝑘) data dependent bandwidths are functions of (cid:101)v(𝑘)
. For (cid:101)𝛀(𝑘)∗ data
dependent bandwidths would typically be functions of (cid:101)v(𝑘)∗
𝜂(𝑘)∗
𝑡
Having the bandwidth depend on the null value of 𝜌𝑘 complicates the computation of confidence
intervals. Things are much simpler when (cid:101)𝛀(𝑘)∗ uses the same data dependent bandwidth as (cid:101)𝛀(𝑘).

and would depend on 𝑎 through (cid:101)

.

𝑡

𝑡

𝑡

Details are provided in Section 3.3.

2.3.2

Inference about 𝜌𝑘

In this section we focus on simple tests of the autocorrelation for a given lag, 𝑘. We propose HAR
𝑡-tests using the variance estimators (cid:101)V(𝑘) and (cid:101)V∗(𝑘) and an additional variant of those estimators.
Our tests are valid for covariance stationary 𝑦𝑡 driven by weak white noise innovations. The case

of i.i.d. innovations is automatically handled.

For a given lag value, 𝑘, suppose we want to test the simple hypothesis

𝐻0 : 𝜌𝑘 = 𝑎,

where, because 𝜌𝑘 is a correlation parameter, 𝑎 is a given value in the range (−1, 1). The test

could be two-sided or one-sided using the appropriate rejection rule. We analyze the following two

15

𝑡-statistics:

where (cid:101)𝑉 (𝑘)

22 and (cid:101)𝑉 (𝑘)∗

22

𝑡 (𝑘) =
(cid:101)

𝜌𝑘 − 𝑎)
((cid:101)
√︃
𝑇−𝑘 (cid:101)𝑉 (𝑘)
1

22

,

𝑡 (𝑘)∗ =
(cid:101)

𝜌𝑘 − 𝑎)
((cid:101)
√︃
𝑇−𝑘 (cid:101)𝑉 (𝑘)∗
1

22

(2.9)

are the (2,2) elements of the respective variance matrix estimators.

Rather than seek sufficient conditions under which (cid:101)V(𝑘) and (cid:101)V(𝑘)∗ are consistent estimators,
we adopt the fixed-smoothing asymptotic approach (often called fixed-𝑏 asymptotics in the context

of kernel variance estimators). We do this to generate reference distributions for (cid:101)

𝑡 (𝑘) and (cid:101)

𝑡 (𝑘)∗

that depend on the choice of kernel and bandwidth and capture, to some extent, the impact of the

sampling distribution of the variance estimators on the 𝑡-statistics. As has been documented in the

time series econometrics literature (Kiefer and Vogelsang (2005), Sun et al. (2008), Gonçalves and

Vogelsang (2011), Zhang and Shao (2013), Lazarus et al. (2018) and Lazarus et al. (2021)), more

accurate inference is obtained using critical values from fixed-𝑏 reference distributions. Fixed-𝑏

asymptotic results are derived using an asymptotic nesting where the bandwidth to sample size

ratio, 𝑏 = 𝑀/𝑇 ∈ (0, 1], is held fixed as 𝑇 → ∞.

The following Theorem gives the fixed-𝑏 limits of the kernel variance estimators under As-

sumptions 1 and 2.

Theorem 2.1 Let 𝑀 = 𝑏𝑇 where 𝑏 ∈ (0, 1] is fixed. Under Assumptions 1 and 2, as 𝑇 → ∞, the
fixed-𝑏 limits of (cid:101)𝛀(𝑘), and (cid:101)𝛀(𝑘)∗are given by

(cid:101)𝛀(𝑘) ⇒ 𝚲(𝑘)

(cid:101)P2(𝑏)𝚲(𝑘)′, (cid:101)𝛀(𝑘)∗ ⇒ 𝚲(𝑘)

(cid:101)P2(𝑏)𝚲(𝑘)′,

where (cid:101)P2(𝑏) is a 2 × 2 stochastic matrix that is a function of the 2 × 1 vector of Brownian bridges,
(cid:102)W2(𝑟) = W2(𝑟) − 𝑟W2(1) and the form of (cid:101)P2(𝑏) depends on 𝑘 (𝑥).

Notice that the fixed-𝑏 limits of (cid:101)𝛀(𝑘) and (cid:101)𝛀(𝑘)∗ are the same1. Furthermore, the limits are the same

those obtained by Kiefer and Vogelsang (2005) in stationary time series regressions. Kiefer and

Vogelsang (2005) provide details on how the form of (cid:101)P2(𝑏) depends on the shape of the kernel. In

1It was first pointed out by Lazarus et al. (2018) that demeaning (cid:101)

null-imposed long run variance estimator as for the null-not-imposed long run variance estimator.

𝑣 (𝑘 )∗
𝑡

gives the same fixed-𝑏 limit for the

16

our simulations we use the Parzen kernel

𝑘 (𝑥) =

1 − 6𝑥2 + 6|𝑥|3 for |𝑥| ≤ 1
2

2(1 − |𝑥|)3 for 1

2 ≤ |𝑥| ≤ 1

0 for |𝑥| > 1,






giving

(cid:101)P2(𝑏) = −

∬

|𝑟−𝑠|<𝑏

1
𝑏2

𝑘′′ (cid:16) 𝑟 − 𝑠
𝑏

(cid:17)

(cid:102)W2(𝑟)(cid:102)W2(𝑟)′𝑑𝑟𝑑𝑠,

where 𝑘′′(𝑥) is the second derivative of 𝑘 (𝑥).

Using Theorem 2.1, the fixed-𝑏 limits of the 𝑡-statistics immediately follow from arguments in

Kiefer and Vogelsang (2005) and are given by

𝑡 (𝑘) ⇒
(cid:101)

𝑊1(1)
√︃

(cid:101)𝑃1(𝑏)

,

𝑡 (𝑘)∗ ⇒
(cid:101)

𝑊1(1)
√︃

(cid:101)𝑃1(𝑏)

,

where (cid:101)𝑃1(𝑏) is a scalar version of (cid:101)P2(𝑏) defined in terms of the scalar standard Wiener process
𝑊1(𝑟) in place of W2(𝑟). The fixed-𝑏 limiting distributions are nonstandard but the critical values

are easily tabulated using simulation methods. The following formula can be used to compute right

tail fixed-𝑏 critical values:

𝑐𝑣𝛼/2(𝑏) = 𝑧𝛼/2 + 𝜆1(𝑏 · 𝑧𝛼/2) + 𝜆2(𝑏 · 𝑧2

𝛼/2) + 𝜆3(𝑏 · 𝑧3

𝛼/2) + 𝜆4(𝑏2 · 𝑧𝛼/2) + 𝜆5(𝑏2 · 𝑧2

𝛼/2)

+ 𝜆6(𝑏2 · 𝑧3

𝛼/2) + 𝜆7(𝑏3 · 𝑧𝛼/2) + 𝜆8(𝑏3 · 𝑧2

𝛼/2) + 𝜆9(𝑏3 · 𝑧3

𝛼/2),

where 𝑧𝛼/2 is the right tail critical value from a standard normal distribution and the 𝜆 coefficients

depend on the kernel. Left tail critical values follow by symmetry around zero.2 Notice that the

critical values reduce to the 𝑁 (0, 1) distribution as 𝑏 → 0. This follows from the result, shown by

Kiefer and Vogelsang (2005), that 𝑝 lim𝑏→0 (cid:101)𝑃1(𝑏) = 1. Table 2A.1 gives the 𝜆 coefficients for the

Parzen kernel.

There are other methods for estimating long run variances. An alternative to the kernel approach

is the orthonormal series (OS) approach of Müller (2007) and Sun (2013) which has been applied

2Kiefer and Vogelsang (2005) show that 𝑊1 (1), which is distributed 𝑁 (0, 1), is independent of (cid:101)𝑃1 (𝑏) in which
(cid:101)𝑃1(𝑏) has a mixture normal distribution and therefore has a density symmetric around zero.

case 𝑊1 (1)/

√︃

17

to tests of zero autocorrelation tests by Wang and Sun (2020). The OS long run variance estimator

uses a finite set of orthonormal functions Φℓ (·), ℓ = 1, 2, ..., 𝐾 with the following properties

(Assumption 3.1.(b) of Sun (2013)):

Assumption 2.3 For ℓ = 1, 2, . . . , 𝐾, the basis functions Φℓ (·) are continuously differentiable and
orthonormal in 𝐿2 [0, 1] and satisfy ∫ 1

Φℓ (𝑥)𝑑𝑥 = 0.

0

Define (cid:101)𝚲ℓ =

(cid:101)v(𝑘)
𝑡
imposed and the null-imposed OS long run variance estimators of 𝛀(𝑘) are given by

𝑡=𝑘+1 Φℓ (cid:0) 𝑡
(cid:205)𝑇

𝑡=𝑘+1 Φℓ (cid:0) 𝑡
(cid:205)𝑇

and (cid:101)𝚲∗

(cid:101)v(𝑘)∗

1√
𝑇−𝑘

1√
𝑇−𝑘

ℓ =

𝑇

𝑇

(cid:1)

(cid:1)

𝑡

. The null-not-

(cid:101)𝛀(𝑘)
𝑂𝑆 =

1
𝐾

𝐾
∑︁

ℓ=1

giving the variance estimators

(cid:101)𝚲ℓ(cid:101)𝚲′

ℓ, (cid:101)𝛀(𝑘)∗

𝑂𝑆 =

1
𝐾

𝐾
∑︁

ℓ=1

(cid:101)𝚲∗

ℓ (cid:101)𝚲∗′
ℓ

(cid:101)V(𝑘)
𝑂𝑆 = (cid:101)Q−1

𝑂𝑆 (cid:101)Q−1, (cid:101)V(𝑘)∗
(cid:101)𝛀(𝑘)

𝑂𝑆 = (cid:101)Q−1

(cid:101)𝛀(𝑘)∗
𝑂𝑆 (cid:101)Q−1.

The corresponding 𝑡-statistics are given by

𝑡 (𝑘)
𝑂𝑆 =
(cid:101)

𝜌𝑘 − 𝑎)
((cid:101)
√︃ 1
𝑇−𝑘 (cid:101)𝑉 (𝑘)∗
𝑂𝑆,22
𝑡 (𝑘)∗
𝑂𝑆 where 𝐾 is held fixed as 𝑇 → ∞.
Following Sun (2013), we use asymptotic limits for (cid:101)
This is another example of fixed-smoothing asymptotics, called fixed-𝐾 asymptotics, that generates

𝜌𝑘 − 𝑎)
((cid:101)
√︃ 1
𝑇−𝑘 (cid:101)𝑉 (𝑘)
𝑂𝑆,22

𝑡 (𝑘)
𝑂𝑆 and (cid:101)

𝑡 (𝑘)∗
𝑂𝑆 =
(cid:101)

.

,

reference distributions that, in this case, capture the number of orthonormal series and the impact,

to some extent, of the sampling distribution of the variance estimators on the 𝑡-statistics. Our

assumptions allow direct application of results in Sun (2013) giving

𝑡 (𝑘)
𝑂𝑆 ⇒ 𝑡𝐾,
(cid:101)

𝑡 (𝑘)∗
𝑂𝑆 ⇒ 𝑡𝐾,
(cid:101)

where 𝑡𝐾 is a standard 𝑡-distribution with 𝐾 degrees of freedom. A nice feature of the OS approach

is that the fixed-𝐾 limit is a well known distribution and critical values are easily calculated using

standard statistical software. For a given set of orthonormal series, the value 𝐾 needs to be chosen
in practice. As in the kernel variance estimator case, we use data dependent methods based on (cid:101)v(𝑘)
𝑡 (𝑘)
the null-not-imposed proxy for v(𝑘)
𝑂𝑆 and (cid:101)

, for both (cid:101)

𝑡 (𝑘)∗
𝑂𝑆 .

,

𝑡

𝑡

18

2.3.3 Computation of Confidence Intervals

When the null is not imposed on the variance estimator, a (1 − 𝛼)% two-tail confidence interval

can be computed in the usual way as

√︂

𝜌𝑘 ± 𝑐𝑣𝛼/2 ·
(cid:101)

1

𝑇 − 𝑘 (cid:101)𝑉 (𝑘)

22

,

where 𝑐𝑣𝛼/2 is the critical value taken from the relevant reference distribution (standard normal

or fixed-𝑏).

In contrast, when the null is imposed on the variance estimator, computation of

confidence intervals is more complicated because the variance estimator depends on the null value

of 𝜌𝑘 . Fortunately, the end points of the confidence interval can be computed using the roots of a

second order polynomial. The calculation is very similar to the confidence intervals obtained by

Vogelsang and Nawaz (2017) for trend ratio parameters.

Recall the formula for the null-imposed 𝑡-statistic given by (2.9). A two tailed (1 − 𝛼)%

confidence interval is the collection of values of 𝑎 such that the null hypothesis is not rejected using

the inequality

(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)

𝜌𝑘 − 𝑎)
((cid:101)
√︃
𝑇−𝑘 (cid:101)𝑉 (𝑘)∗
1

22

(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)

≤ 𝑐𝑣𝛼/2.

What complicates the calculation is that (cid:101)𝑉 (𝑘)∗

22

depends on 𝑎 as we now show.

It is convenient to write (cid:101)𝑉 (𝑘)∗

22

in terms of quantities from the estimating equation (2.4) with the

intercept projected out using the Frisch-Waugh-Lovell Theorem. Let (cid:165)𝑦𝑡 and (cid:165)𝑦𝑡−𝑘 denote demeaned
values where (cid:165)𝑦𝑡 = 𝑦𝑡 − ¯𝑦{𝑘+1,𝑇 } and (cid:165)𝑦𝑡−𝑘 = 𝑦𝑡−𝑘 − ¯𝑦{1,𝑇−𝑘 }. Then (cid:101)

𝜌𝑘 can be written as

𝜌𝑘 =
(cid:101)

(cid:205)𝑇

𝑡=𝑘+1 (cid:165)𝑦𝑡−𝑘 (cid:165)𝑦𝑡
(cid:205)𝑇
𝑡=𝑘+1 (cid:165)𝑦2
𝑡−𝑘

,

𝜂(𝑘)
𝑡 = (cid:165)𝑦𝑡 − 𝑎 (cid:165)𝑦𝑡−𝑘 .
(cid:101)

𝜂(𝑘)
and (cid:101)
𝑡

can be written as

Define

(cid:165)𝑣 (𝑘)∗
𝑡

= (cid:165)𝑦𝑡−𝑘 ( (cid:165)𝑦𝑡 − 𝑎 (cid:165)𝑦𝑡−𝑘 ) = (cid:165)𝑦𝑡−𝑘 (cid:165)𝑦𝑡 − 𝑎 (cid:165)𝑦2

𝑡−𝑘 .

19

Then we rewrite (cid:101)𝑉 (𝑘)∗

22

equivalently as

(cid:101)𝑉 (𝑘)∗
22 = (cid:165)𝑄−1 (cid:165)Ω(𝑘)∗ (cid:165)𝑄−1,

(cid:205)𝑇

𝑡=𝑘+1 (cid:165)𝑦2

where (cid:165)𝑄 = 1
𝑇−𝑘
scalar process (cid:165)𝑣 (𝑘)∗
be equivalently written as a quadratic form. For (cid:165)Ω(𝑘)∗ the quadratic form is

𝑡−𝑘 and (cid:165)Ω(𝑘)∗ is the kernel long run variance estimator computed using the
. It is well known in the literature that kernel long run variance estimators can

𝑡

(cid:165)Ω(𝑘)∗ = (𝑇 − 𝑘)−1

= (𝑇 − 𝑘)−1

𝑇
∑︁

𝑇
∑︁

𝑡=𝑘+1
𝑇
∑︁

𝑠=𝑘+1
𝑇
∑︁

𝑡=𝑘+1

𝑠=𝑘+1

(cid:165)𝑣 (𝑘)∗
𝑡

𝑘𝑡𝑠 (cid:165)𝑣 (𝑘)∗
𝑠

( (cid:165)𝑦𝑡−𝑘 (cid:165)𝑦𝑡 − 𝑎 (cid:165)𝑦2

𝑡−𝑘 )𝑘𝑡𝑠 ( (cid:165)𝑦𝑠−𝑘 (cid:165)𝑦𝑠 − 𝑎 (cid:165)𝑦2

𝑠−𝑘 ),

where 𝑘𝑡𝑠 = 𝑘

(cid:17)

(cid:16) |𝑡−𝑠|
𝑀

. Rearranging (cid:165)Ω(𝑘)∗ gives

(cid:165)Ω(𝑘)∗ = (cid:165)Ω(𝑘)∗
11

− 2𝑎 (cid:165)Ω(𝑘)∗
12

+ 𝑎2 (cid:165)Ω(𝑘)∗
22

,

(2.10)

where

(cid:165)Ω(𝑘)∗
11 = (𝑇 − 𝑘)−1

(cid:165)Ω(𝑘)∗
12 = (𝑇 − 𝑘)−1

(cid:165)Ω(𝑘)∗
22 = (𝑇 − 𝑘)−1

𝑇
∑︁

𝑇
∑︁

𝑡=𝑘+1
𝑇
∑︁

𝑠=𝑘+1
𝑇
∑︁

𝑡=𝑘+1
𝑇
∑︁

𝑠=𝑘+1
𝑇
∑︁

𝑡=𝑘+1

𝑠=𝑘+1

(cid:165)𝑦𝑡−𝑘 (cid:165)𝑦𝑡 𝑘𝑡𝑠 (cid:165)𝑦𝑠−𝑘 (cid:165)𝑦𝑠,

(cid:165)𝑦𝑡−𝑘 (cid:165)𝑦𝑡 𝑘𝑡𝑠 (cid:165)𝑦2

𝑠−𝑘 ,

𝑡−𝑘 𝑘𝑡𝑠 (cid:165)𝑦2
(cid:165)𝑦2

𝑠−𝑘 .

Using these variance formulas, we obtain an equivalent formula for (cid:101)

𝑡 (𝑘)∗ given by

𝑡 (𝑘)∗ =
(cid:101)

√︂

1
𝑇−𝑘

𝜌𝑘 − 𝑎)
((cid:101)
(cid:165)𝑄−2 (cid:16) (cid:165)Ω(𝑘)∗

− 2𝑎 (cid:165)Ω(𝑘)∗
12

11

.

(cid:17)

+ 𝑎2 (cid:165)Ω(𝑘)∗
22

The confidence interval for 𝜌𝑘 is the values of 𝑎 such that

(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)

√︂

1
𝑇−𝑘

𝜌𝑘 − 𝑎)
((cid:101)
(cid:165)𝑄−2 (cid:16) (cid:165)Ω(𝑘)∗

− 2𝑎 (cid:165)Ω(𝑘)∗
12

11

(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)

≤ 𝑐𝑣𝛼/2,

+ 𝑎2 (cid:165)Ω(𝑘)∗
22

(cid:17)

20

or equivalently

𝜌𝑘 − 𝑎)
((cid:101)
(cid:165)𝑄−1 (cid:16) (cid:165)Ω(𝑘)∗

− 2𝑎 (cid:165)Ω(𝑘)∗
12

11

√︂

1
𝑇−𝑘

(cid:169)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:171)

The inequality (2.11) can be rewritten as

2

(cid:170)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:172)

+ 𝑎2 (cid:165)Ω(𝑘)∗
22

(cid:17)

≤ 𝑐𝑣2

𝛼/2

.

(2.11)

𝑐2𝑎2 + 2𝑐1𝑎 + 𝑐0 ≤ 0,

(2.12)

where

𝑐2 = 1 −

1
𝑇 − 𝑘

(cid:165)𝑄−2 (cid:165)Ω(𝑘)∗
22

· 𝑐𝑣2

𝛼/2

,

𝑐1 =

1
𝑇 − 𝑘

𝜌2
𝑐0 = (cid:101)
𝑘 −

· 𝑐𝑣2

(cid:165)𝑄−2 (cid:165)Ω(𝑘)∗
12
(cid:165)𝑄−2 (cid:165)Ω(𝑘)∗
11

1
𝑇 − 𝑘

𝜌𝑘 ,
𝛼/2 − (cid:101)

· 𝑐𝑣2

𝛼/2

.

Notice the importance of using a bandwidth rule for 𝑀 that does not depend on 𝑎. Otherwise (cid:165)Ω(𝑘)∗
11 ,
(cid:165)Ω(𝑘)∗
12

22 would depend on 𝑎 greatly complicating the solution to (2.12).

and (cid:165)Ω(𝑘)∗

The values of 𝑎 satisfying the inequality (2.12) are determined by the roots of the polynomial

𝑝(𝑎) = 𝑐2𝑎2 + 2𝑐1𝑎 + 𝑐0.

This polynomial has a similar form to the polynomial analyzed by Vogelsang and Nawaz (2017).

Let 𝑟1 and 𝑟2 be the roots of 𝑝(𝑎) and order them 𝑟1 ≤ 𝑟2 when they are real roots. The discriminant
1 − 𝑐2𝑐0, so the shape of the confidence interval for 𝑎

of the quadratic equation 𝑝(𝑎) is given by 𝑐2

depends on the signs of 𝑐2 and 𝑐2

1 − 𝑐2𝑐0.

There are four cases. Case 1 has 𝑐2 > 0 and 𝑐2

1 − 𝑐2𝑐0 ≥ 0 in which case the roots are real and
1 − 𝑐2𝑐0 < 0 in which case 𝑝(𝑎) opens upward and its vertex
is above zero giving roots that are complex numbers and an empty confidence interval. Case 3 has

𝑎 ∈ [𝑟1, 𝑟2]. Case 2 has 𝑐2 > 0 and 𝑐2

𝑐2 < 0 and 𝑐2

𝑝(𝑎) opens downward and its vertex is above zero. Case 4 has 𝑐2 < 0 and 𝑐2

1 − 𝑐2𝑐0 > 0 in which case the roots are real and 𝑎 ∈ (−1, 𝑟1] ∪ [𝑟2, 1) given that
1 − 𝑐2𝑐0 ≤ 0 in which
case 𝑎 ∈ (−1, 1). It is important to note that Case 2 is impossible because the confidence interval
cannot be empty given that it always contains the value 𝑎 = (cid:101)

𝑡 (𝑘)∗ = 0 in this case and a

𝜌𝑘 because (cid:101)

21

non-rejection is obtained. The other cases are possible although it is not easy to find intuition as to

the likelihood of each case.

First examine the sign of 𝑐2. One can show that

𝛼/2 will be
greater than 1 for commonly used significance levels. Therefore, the sign of 𝑐2 is inconclusive.

< 1, however, 𝑐𝑣2

1
𝑇−𝑘

(cid:165)𝑄−2 (cid:165)Ω(𝑘)∗
22

Whether or not 𝑐2 is positive depends on the kernel, bandwidth, the significance level and the data.

converges to zero in which case it is more likely that 𝑐2 is positive.

As 𝑇 increases,

(cid:165)𝑄−2 (cid:165)Ω(𝑘)∗
22
Next examine the sign of 𝑐2

1
𝑇−𝑘

1 − 𝑐2𝑐0. Algebra gives

1 − 𝑐2𝑐0 =
𝑐2

(cid:16) (cid:165)Ω(𝑘)∗

12

− (cid:165)Ω(𝑘)∗
11

(cid:165)Ω(𝑘)∗
22

(cid:17) (cid:18)

+

(cid:16) (cid:165)Ω(𝑘)∗

11

− 2(cid:101)

𝜌𝑘 (cid:165)Ω(𝑘)∗
12

𝜌2
+ (cid:101)
𝑘

1
𝑇 − 𝑘
(cid:165)Ω(𝑘)∗
22

(cid:19) 2

(cid:165)𝑄−2 · 𝑐𝑣2

𝛼/2

(cid:17) (cid:18)

1
𝑇 − 𝑘

(cid:165)𝑄−2 · 𝑐𝑣2

𝛼/2

(cid:19)

.

1 − 𝑐2𝑐0 is expressed as the sum of the two terms. The second term is the formula for
𝜌𝑘 in (2.10) and is scaled by a positive quantity. With appropriate choice of kernel,

We see that 𝑐2
(cid:165)Ω(𝑘)∗ with 𝑎 = (cid:101)
kernel long run variances like (cid:165)Ω(𝑘)∗ are non-negative as argued by Priestley (1981) and Newey and

West (1987). Therefore, the second term is non-negative. However, the first term is inconclusive
because (cid:165)Ω(𝑘)∗
12

can be positive or negative. Therefore, the sign of 𝑐2

1 − 𝑐2𝑐0 is also

− (cid:165)Ω(𝑘)∗
11

(cid:165)Ω(𝑘)∗
22

inconclusive.

Confidence intervals can be computed using the orthonormal series variance estimator analo-
replaced, respectively, by (cid:165)Ω(𝑘)∗
(cid:205)𝐾

ℓ,1, (cid:165)Ω(𝑘)∗
(cid:165)Λ∗
(cid:165)Λ∗
𝑂𝑆,12 =
ℓ=1
𝐾
ℓ,1
(cid:1) (cid:165)𝑦𝑡−𝑘 (cid:165)𝑦𝑡 and
𝑡=𝑘+1 Φℓ (cid:0) 𝑡
(cid:205)𝑇

ℓ,2 where (cid:165)Λ∗
(cid:165)Λ∗

𝑂𝑆,11 = 1
1√
𝑇−𝑘

ℓ,1 =

(cid:165)Λ∗
ℓ,2

(cid:205)𝐾

ℓ=1

𝐾

𝑇

(cid:205)𝐾

gously with (cid:165)Ω(𝑘)∗
(cid:165)Λ∗
1
ℓ=1
𝐾
ℓ,1
(cid:165)Λ∗
ℓ,2 = 1√
𝑇−𝑘

12 , and (cid:165)Ω(𝑘)∗
11 , (cid:165)Ω(𝑘)∗
ℓ,2, and (cid:165)Ω(𝑘)∗
(cid:165)Λ∗
𝑂𝑆,22 = 1
(cid:1) (cid:165)𝑦2
𝑡=𝑘+1 Φℓ (cid:0) 𝑡
(cid:205)𝑇
𝑡−𝑘 .

22

𝑇

2.4 Monte Carlo Simulations

In this section we study finite sample properties of the proposed 𝑡-statistics for testing

𝐻0 : 𝜌𝑘 = 𝑎

through extensive Monte Carlo simulations. We use 5,000 replications in all cases. We compare

our 𝑡-statistics, (cid:101)
critical values are used for (cid:101)

𝑡 (𝑘)
𝑡 (𝑘)∗, (cid:101)
𝑂𝑆 and (cid:101)
𝑡 (𝑘), (cid:101)

𝑡 (𝑘), (cid:101)

𝑡 (𝑘)∗
𝑂𝑆 with each other and with some existing approaches. Fixed-𝑏
𝑡 (𝑘)∗ and critical values from the 𝑡𝐾 distribution (fixed-𝐾 critical

22

𝑡 (𝑘)
𝑂𝑆 and (cid:101)

𝑡 (𝑘)∗
𝑂𝑆 . We also provide some results for (cid:101)

values) are used for (cid:101)
values to show the value of using fixed-𝑏 critical values. For (cid:101)
bandwidth, denoted by (cid:101)𝑀, proposed by Sun et al. (2008) that balances size distortions and power
of the tests, the ‘test-optimal-𝑀’. The weighting parameter that balances type 1 and type 2 errors
𝑡 (𝑘)∗, also uses (cid:101)𝑀 so that its bandwidth does not depend

is set to 10. The null-imposed statistic, (cid:101)

𝑡 (𝑘) we used the data dependent

𝑡 (𝑘)∗ using 𝑁 (0, 1) critical

𝑡 (𝑘)
𝑂𝑆 we used the data dependent smoothing parameter,
on the value of the null being tested. For (cid:101)
denoted by (cid:101)𝐾, proposed by Phillips (2005) that minimizes the mean square error of the variance
𝑡 (𝑘)∗
𝑂𝑆 , also uses (cid:101)𝐾 to avoid dependence
estimator, the ‘MSE-optimal-𝐾’. The null-imposed statistic, (cid:101)
on the value of the null being tested. For both (cid:101)𝑀 and (cid:101)𝐾 we use well known AR(1) plug-in methods
, the null-not-imposed proxy for v(𝑘)
(see ?) that are functions of (cid:101)v(𝑘)

given by equation (2.8).

𝑡

𝑡

Results are given for a broad set of data generating processes (DGPs) where 𝑦𝑡 follows the

𝐴𝑅𝑀 𝐴(1, 1) process

𝑦𝑡 = 𝜙𝑦𝑡−1 + 𝜖𝑡 + 𝜃𝜖𝑡−1,

(2.13)

where 𝜇 = 0 without loss of generality given that we include in a intercept in the estimating equation

(2.4). Special cases include uncorrelated 𝑦𝑡 (𝜙 = 0, 𝜃 = 0) and 𝐴𝑅(1) (𝜃 = 0) and 𝑀 𝐴(1) (𝜙 = 0)

processes. Results are given for nine DGPs of the innovation process, 𝜖𝑡, ranging from i.i.d.

to

cases with increasing dependence in higher moments.

DGP 1: IID : 𝜖𝑡 = 𝑢𝑡 ∼ 𝑖.𝑖.𝑑.𝑁 (0, 1).

DGP 2: MDS : 𝜖𝑡 = 𝑢𝑡𝑢𝑡−1, 𝑢𝑡 ∼ 𝑖.𝑖.𝑑.𝑁 (0, 1)..

DGP 3: GARCH : 𝜖𝑡 = ℎ𝑡𝑢𝑡 and ℎ2

𝑡 = 0.1 + 0.09𝜖 2

𝑡−1 + 0.9ℎ2

𝑡−1, 𝑢𝑡 ∼ 𝑖.𝑖.𝑑.𝑁 (0, 1).

DGP 4: WN-1 : 𝜖𝑡 = 𝑢𝑡 + 𝑢𝑡−1𝑢𝑡−2, 𝑢𝑡 ∼ 𝑖.𝑖.𝑑.𝑁 (0, 1).

DGP 5: WN-2: 𝜖𝑡 = 𝑢2

𝑡 𝑢𝑡−1, 𝑢𝑡 ∼ 𝑖.𝑖.𝑑.𝑁 (0, 1).

DGP 6: WN-NLMA: 𝜖𝑡 = 𝑢𝑡−2𝑢𝑡−1(𝑢𝑡−2 + 𝑢𝑡 + 1), 𝑢𝑡 ∼ 𝑖.𝑖.𝑑.𝑁 (0, 1).

DGP 7: WN-BILIN: 𝜖𝑡 = 𝑢𝑡 + 0.5𝑢𝑡−1𝜖𝑡−2, 𝑢𝑡 ∼ 𝑖.𝑖.𝑑.𝑁 (0, 1).

23

DGP 8: WN-GAM1 : 𝜖𝑡 = 𝑢𝑡 + 𝑢𝑡−1𝑢𝑡−2, 𝑢𝑡 = 𝜁𝑡 − 𝐸 [𝜁𝑡], 𝜁𝑡 ∼ 𝑖.𝑖.𝑑.𝐺𝑎𝑚𝑚𝑎(0.3, 0.4).

DGP 9: WN-GAM2 : 𝜖𝑡 = 𝑢𝑡 − 𝑢𝑡−1𝑢𝑡−2, 𝑢𝑡 = 𝜁𝑡 − 𝐸 [𝜁𝑡], 𝜁𝑡 ∼ 𝑖.𝑖.𝑑.𝐺𝑎𝑚𝑚𝑎(0.3, 0.4).

DGP 1 is an i.i.d. Gaussian innovation and serves as a benchmark given that all approaches are

valid for this case. DGP 2 relaxes the i.i.d. assumption and 𝜖𝑡 is a martingale difference sequence

(MDS) innovation that has been studied in the literature. See Romano and Thombs (1996) and

Francq and Zakoïan (2009). DGP 3 is a 𝐺 𝐴𝑅𝐶𝐻 (1, 1) innovation typical in financial time series.

DGPs 4-9 are white noise processes with stronger dependence than the MDS case. DGP 4 is

from Hansen (2022) and 𝜖𝑡 follows a white noise process that is a function of an underlying i.i.d.

Gaussian process. DGP 5 is a white noise process from Wang and Sun (2020). DGPs 6 and 7

are white noise processes from Lobato (2001). DGPs 8 and 9 build white noise process using

independent centered Gamma random variables generating some skewness in 𝑢𝑡.

2.4.1 Null Rejections for Uncorrelated Time Series

We first focus on the case where 𝑦𝑡 is uncorrelated, i.e. 𝜌𝑘 = 0 or equivalently 𝜙 = 0, 𝜃 = 0 in

(2.13). For this case we focus on the first order autocorrelation (𝑘 = 1) and examine tests of the

null hypothesis

𝐻0 : 𝜌1 = 0.

We consider the (original) Bartlett formula, the generalized Bartlett formula, and White standard

errors for constructing 𝑡-statistics that we compare to our proposed 𝑡-statistics. We carry out two-

tailed tests with a nominal significance level of 0.05. The original Bartlett formula always uses

𝑣 𝐵
1,1 = 1 whether or not 𝑦𝑡 is i.i.d. For the generalized Bartlett formula, we use the formula (2.3)
from Francq and Zakoïan (2009) for a white noise process. White standard errors are a special case
of (cid:101)Ω(𝑘) where only the (cid:101)Γ(𝑘)
the lag one autocorrelation, we also include the zero autocorrelation test of Taylor (1984) which

term is used. Because testing 𝜌1 = 0 is a zero autocorrelation test for

0

has recently been extended by Dalla et al. (2022). The Taylor (1984) (cid:101)
(cid:205)𝑛
(cid:16)(cid:205)𝑛
𝑡=2

𝑒𝑡1 = (𝑦𝑡 − ¯𝑦) (𝑦𝑡−1 − ¯𝑦) .

𝑒𝑡1
(cid:17) 1/2

𝜏1 =
(cid:101)

𝑡=2

,

𝑒2
𝑡1

𝜏1 𝑡-statistic is given by

24

Dalla et al. (2022) provide conditions under which (cid:101)
tributed. We also report results using the bootstrap method suggested by Romano and Thombs

𝜏1 is asymptotically standard normally dis-

(1996) where the bootstrapped version of (cid:98)
their equation (11) on page 594). We report results using the moving block bootstrap with block

𝜌𝑘 is centered around (cid:98)

𝜌𝑘 but is not standardized (see

length equal to

√

𝑇. For the case where the DGP for 𝜖𝑡 is i.i.d. we also report results using block

length equal to 1 (the i.i.d. bootstrap). We obtained results using the stationary bootstrap and the

circular bootstrap but exclude them from reporting because they give similar results and patterns

as the moving block bootstrap. We also obtained results using subsampling but found those results

less accurate than the bootstrap and those results are omitted.

Figures 1.1 through 1.9 plot empirical null rejection probabilities for each of the nine cases for

𝜖𝑡. Results are given for sample sizes 𝑇 = 100, 200, 500 and 2000. The labels Fixed-𝑏 (SPJ) and

Fixed-𝑏-𝐻0 (SPJ) correspond to (cid:101)
(SPJ) corresponds to (cid:101)
dependent bandwidth, (cid:101)𝑀, was used for all three tests. The labels OS (MSE) and OS-𝐻0 (MSE)

𝑡 (1)∗ using 𝑁 (0, 1) critical values. The (SPJ) label indicates that the same data

𝑡 (1)∗ respectively using fixed-𝑏 critical values. 𝑁 (0, 1)-𝐻0

𝑡 (1) and (cid:101)

correspond to (cid:101)

𝑡 (1)
𝑂𝑆 and (cid:101)

𝑡 (1)∗
𝑂𝑆 using the same (cid:101)𝐾 smoothing parameter.

To understand many of the patterns in Figures 1.1 - 1.9, it is useful to keep in mind that
= 𝜖𝑡−1𝜖𝑡 when 𝑦𝑡 is uncorrelated. For the IID, MDS and GARCH DGPs, 𝑣 (1)

is obviously

𝑡

𝑣 (1)
𝑡
uncorrelated. While not as obvious, 𝑣 (1)
WN-NLMA and WN-BILIN. In contrast, 𝑣 (1)

𝑡

because one can show that 𝐸

(cid:16)

𝑡 𝑣 (1)
𝑣 (1)
𝑡−1

(cid:17)

is uncorrelated for the white noise processes WN-1, WN-2,

𝑡
= 𝐸 (cid:0)𝑢3
𝑡

is positively autocorrelated for the WN-GAM1 DGP
(cid:1) > 0 for the Gamma

𝑡 ) > 0 given that 𝐸 (cid:0)𝑢3
𝑡

(cid:1) 𝐸 (𝑢2

parameters we use. The sign change in the WN-GAM2 DGP generates negative autocorrelation3
in 𝑣 (1)
𝑡

= −𝐸 (cid:0)𝑢3
𝑡

because 𝐸

𝑡 ) < 0.

(cid:1) 𝐸 (𝑢2

(cid:16)

(cid:17)

𝑡 𝑣 (1)
𝑣 (1)
𝑡−1

Figure 2A.1.1 depicts null rejection probabilities for the IID DGP (𝑦𝑡 = 𝜖𝑡 is i.i.d.). There are

slight over-rejections for (cid:101)
(red squares dash-dotted line) for 𝑇 = 100 because for this method there is variability in (cid:101)

𝑡 (1) (null-not-imposed kernel HAR statistic) with fixed-𝑏 critical values

𝑣 (1)
𝑡

3Notice that the WN-1 and WN-GAM1,WN-GAM2 DGPs take the same form. The reason that 𝑣 (1)
𝑡
𝑡 ) = 0 and it follows that 𝐸

for WN-1 is because 𝑢𝑡 is normally distributed. Normality implies that 𝐸 (𝑢3

is uncorrelated
(cid:16)
𝑡 𝑣 (1)
𝑣 (1)
= 0.
𝑡 −1

(cid:17)

25

from estimating 𝜌1 that matters when 𝑇 is relatively small. Imposing the null for the kernel HAR

approach reduces over-rejections as illustrated by (cid:101)

𝑡 (1)∗ using fixed-𝑏 critical values (purple up-

arrow dotted line). Using normal critical values for (cid:101)
𝑡 (1)
over-rejections and illustrates the benefits of using fixed-𝑏 critical values. The null rejections of(cid:101)
𝑂𝑆
𝑡 (1)∗
𝑂𝑆 (null-imposed, light green down-arrow dashed
(null-not-imposed, orange x’s solid line) and (cid:101)
𝑡 (1) and (cid:101)

𝑡 (1)∗. Rejections are close to 0.05 for all traditional

line) are similar to the rejections of (cid:101)

𝑡 (1)∗ (gold circle dash-dotted line) shows some

methods (Bartlett formula (blue dot solid lines ‘Bartlett(IID)’), generalized Bartlett (light blue star

dashed line ‘GB-WN’), Taylor (yellow rhombus dotted line ‘Taylor’) and White standard errors

(green right-arrow dotted line ‘White’)).

It is surprising to see that the i.i.d. bootstrap (black

down-arrow dashed line ‘IID-bootstrap’) does not work for the i.i.d. DGP. Null rejections for the

i.i.d. bootstrap are about 0.33 even when 𝑇 increases to 2000. Interestingly, the moving block

bootstrap (black circle dotted line ‘MBB’) performs better than the i.i.d. bootstrap even though the

data has no dependence. Even so, rejections with the moving block bootstrap range from 0.15 with

𝑇 = 100 to about 0.07 with 𝑇 = 2000 whereas all non-bootstrap tests have rejections close to 0.05

when 𝑇 = 100 and very close to 0.05 when 𝑇 = 2000.

Figures 1.2-1.7 relax the i.i.d. assumption and give results for 𝑦𝑡 being an MDS, GARCH and

the various white noise series that satisfy, with the exception of the original Bartlett variance, the

conditions of the traditional approaches. We see similar patterns as in the i.i.d. case, however more

size distortions occur for (cid:101)

𝑡 (1)
𝑡 (1) and (cid:101)
𝑂𝑆 (null-not-imposed) for smaller sample sizes. In contrast, (cid:101)

𝑡 (1)∗

and (cid:101)

𝑡 (1)∗
𝑂𝑆 (null-imposed) have rejections close to 0.05. This indicates potential size improvements

by imposing the null, consistent with the findings in Lazarus et al. (2018) and Vogelsang (2018)

in stationary regression settings. The traditional Bartlett formula shows over-rejections which

is expected with the i.i.d. assumption violated. The moving block bootstrap continues to have

substantial over-rejections especially for small sample sizes for all DGPs. The other traditional

methods work reasonably well as expected given that 𝑦𝑡 satisfies the required assumptions for those

methods.

Figures 1.8 and 1.9 give results for the white noise case with Gamma distributed innovations.

26

𝑡 (1)∗
𝑂𝑆

𝑡 (1)∗ and (cid:101)
𝑡 (1)
𝑡 (1) and (cid:101)
𝑂𝑆, have

For the WN-GAM1 DGP (Figure 2A.1.8) all tests show some over-rejections with (cid:101)

(null-imposed) having rejections closest to 0.05. The null-not-imposed tests, (cid:101)
substantial over-rejections for small 𝑇 but rejections approach 0.05 as 𝑇 increases. All of the

traditional methods have over-rejections even when 𝑇 is large because this DGP violates the
assumptions for those methods. In particular Taylor and White are designed for the case where 𝑣 (1)

𝑡

is uncorrelated and that fails here. The generalized Bartlett formula uses a symmetry assumption

for cross fourth moments of 𝜖𝑡 that is violated in the Gamma distribution case. Figure 2A.1.9 shows

that if we flip the sign on 𝑢𝑡−1𝑢𝑡−2, rejections change dramatically with all tests under-rejecting.
Under-rejections make sense because flipping the sign generates negative autocorrelation in 𝑣 (1)

𝑡

for the WN-GAM2 DGP. The traditional methods can have very low rejections close to zero.

As 𝑇 increases the rejections using the estimating equation approach tends towards 0.05 but the

traditional methods do not. The moving block bootstrap continues to over-reject and does not

perform as well as non-bootstrap methods.

It is a common misconception that 𝑦𝑡 = 𝜖𝑡 being uncorrelated implies that 𝑣 (1)

𝑡 = 𝜖𝑡−1𝜖𝑡 will

be uncorrelated. However, because it is possible for 𝜖𝑡−1𝜖𝑡 to have serial correlation when 𝜖𝑡 is

uncorrelated, the generalized-Bartlett, White, and Taylor approaches are not necessarily valid when

𝑦𝑡 is uncorrelated. One benefit of the estimating equation approach is that it automatically handles
white noise innovations including the case where 𝑣 (1)

has serial correlation.

𝑡

Finally, our simulation results for the bootstrap are puzzling especially in the i.i.d. case given

the relatively simple form of (cid:98)
expected is part of an ongoing research project that we will report in a follow-up paper.

𝜌1. An analytical analysis of why the bootstrap is not performing as

2.4.2 Null Rejections for Serially Correlated Time Series

Next we focus on cases of serially correlated time series where 𝜌𝑘 ≠ 0. We continue to focus

on tests of the first order autocorrelation (𝑘 = 1) and consider the null hypothesis

𝐻0 : 𝜌1 = 𝜌(0)
1

,

where 𝜌(0)
1

is the true value of 𝜌1, and 𝜌(0)

1 depends on the serial correlation structure of 𝑦𝑡. We
exclude the Taylor and White approaches because they are no longer valid when 𝜌𝑘 ≠ 0. We do not

27

report bootstrap results because of the bootstrap’s relatively poor performance with uncorrelated

data.

Two versions of the generalized Bartlett approach are included. One assumes that 𝑦𝑡 is white

noise (GB-WN) and the other assumes 𝑦𝑡 follows an MA(1) process (GB-MA)4. The formula for

GB-MA is given by

𝑣 𝐵∗
1,1 =

𝛾𝜖 2 (0)
[𝛾𝜖 (0)]2

(cid:2)𝜌𝜖 2 (1) (1 − 4𝜌1 + 4𝜌4

1) + 𝜌𝜖 2 (2) 𝜌2
1

(cid:3) .

We derived this formula using the general expression in Francq and Zakoïan (2009). The corre-

sponding estimator is obtained by plugging in estimators of the parameters. We estimate 𝜌1 using
(2.2). We estimate 𝛾𝜖 (0) using the sample variance of (cid:98)
MA(1) model to 𝑦𝑡 − 𝑦. The parameters 𝛾𝜖 2 (0), 𝜌𝜖 2 (1), 𝜌𝜖 2 (2) are estimated using sample analogs

𝜖𝑡 are the residuals from fitting an

𝜖𝑡 where (cid:98)

𝜖 2
𝑡 .
computed with (cid:98)

Results are given for the MA(1) case in Figures 2-6 and the AR(1) case in Figures 6-11. Results

for ARMA(1,1) specifications are similar and are omitted. We exclude DGPs WN-2,WN-NLMA

and WN-BILIN for 𝜖𝑡 given the similarity in patterns to WN-1. We also exclude WN-GAM2. We

continue to use two-tailed tests with 0.05 nominal level. Each figure has four panels corresponding

to the sample sizes 𝑇 = 100, 200, 500, and 1000. The 𝑥-axis indicates the value of either 𝜃 or 𝜙.
For the MA(1) case, 𝜌(0)

1 = 𝜃/(cid:0)1 + 𝜃2(cid:1) and for the AR(1) case 𝜌(0)

1 = 𝜙.

Figure 2A.2 gives results for MA(1) case with 𝜖𝑡 i.i.d. Not surprisingly, all approaches work

reasonably well except for GB-WN which under-rejects unless 𝜃 = 0. This is expected given

that GB-WN is invalid except when 𝜃 = 0. Figure 2A.3 gives MA(1) results where 𝜖𝑡 follows

the MDS DGP. The traditional Bartlett approach (MA(1)) over-rejects because 𝜖𝑡 is not i.i.d. For

𝑇 = 100, GB-MA (green star dashed line) tends to over-reject. Rejections become closer to 0.05 as

𝑇 increases. The small sample distortions are likely caused by the need to estimate 𝜃. Similar to

MA(1) with 𝜖𝑡 i.i.d, GB-WN continues to under-reject. The null-imposed kernel HAR test, (cid:101)
works well whether normal critical values (N(0,1)-𝐻0) or fixed-𝑏 critical values (Fixed-b-𝐻0) are
4We do not implement versions of the generalized Bartlett approach designed for the case when 𝑦𝑡 has the AR(1)
component because the form of the generalized Bartlett variance formula for the AR(1) case is complicated and is very
difficult to implement.

𝑡 (1)∗,

28

used. The null-imposed orthonormal series test, (cid:101)

imposing the null leads to over-rejections for both (cid:101)

𝑡 (1)∗
𝑂𝑆 (OS-𝐻0) has similar performance to(cid:101)
𝑡 (1)
𝑂𝑆 (OS) when 𝑇 is relatively
𝑡 (1) (Fixed-b) and (cid:101)

𝑡 (1)∗. Not

small. This again illustrates that more reliable inference under the null is obtained by imposing

the null on the kernel and orthonormal series variance estimators. When 𝜖𝑡 is a GARCH process,

Figure 2A.4 shows that all methods works well except for the traditional Bartlett and GB-WN as

one would expect. Figure 2A.5 gives results for the case of 𝜖𝑡 being white noise (WN-1 DGP) and

we see that patterns are similar to the MDS case. In contrast, patterns are clearly different when 𝜖𝑡

is the white noise driven by Gamma errors (WN-GAM1 DGP) as seen in Figure 2A.6. None of the

Bartlett approaches are valid in this case and rejections are either well above or well below 0.05.

The null-imposed HAR approaches, (cid:101)

𝑡 (1)∗ and (cid:101)

𝑡 (1)∗
𝑂𝑆 , perform best especially with fixed-𝑏 critical

values. Not imposing the null can lead to nontrivial over-rejections. While rejections of the HAR

tests get closer to 0.05 with larger sample sizes, there are still some size distortions even with

𝑇 = 2000. Our conjecture is that the CLT and FCLT ‘kick in’ more slowly as 𝑇 increases in the

Gamma distribution case.

We now turn to Figures 7-11 for the AR(1) results. Keep in mind that both GB-MA and GB-WN

use formulas based on a misspecified model and are not expected to perform well. Figure 2A.7

gives results for 𝜖𝑡 i.i.d. We can see that the misspecified GB approaches have size distortions that

persist with larger 𝑇. The Bartlett (AR(1)) and HAR tests perform reasonably well with small 𝑇

with some slight over-rejections. Rejections are close to 0.05 with 𝑇 = 2000. Figures 8, 9 and

10 give AR(1) results for 𝜖𝑡 MDS, GARCH and WN-1 respectively. When the errors are MDS

and GARCH (Figures 8 and 9), we can see that the null-imposed HAR tests (cid:101)
𝑡 (1)∗
𝑂𝑆 (OS-𝐻0) perform well with null rejections reasonably close to 0.05. When 𝜖𝑡 is white noise
(cid:101)
(Figures 10 and 11), all approaches exhibit over-rejections when 𝜙 > 0 especially as 𝜙 approaches

𝑡 (1)∗ (Fixed-b-𝐻0) and

1. Increasing 𝑇 improves the performance of the HAR approaches.

2.4.3 Power Analysis

In this subsection we study finite sample power of the test statistics. We use size-adjusted power

to account for the size distortions of the tests. This allows power comparisons with the same null

29

rejections. Because we use size-adjusted finite sample critical values, there is no need to distinguish

between using 𝑁 (0, 1) and fixed-𝑏 critical values for (cid:101)
‘Kernel’ and (cid:101)
AR(1) and 𝜖𝑡 is IID (DGP 1), WN-NLMA (DGP 6) and WN-GAM1 (DGP 8). The null hypothesis

𝑡 (1)∗ as ‘Kernel-𝐻0’. We report three sets of power results for the case where 𝑦𝑡 is

𝑡 (1)∗. In the power figures we label (cid:101)

𝑡 (1) as

in all cases is 𝐻0 : 𝜌1 = 0 with the alternative given by 𝐻1 : 𝜌1 = 𝜙. We use a 0.1 grid for 𝜙 on the

interval [−0.5, 0.5]. Results are reported for 𝑇 = 100, 200, 300, and 500.

Figure 2A.12 gives results for 𝜖𝑡 IID. Size-adjusted power is essentially the same across all

tests. Figure 2A.13 gives results for 𝜖𝑡 WN-NLMA. Size-adjusted power is similar across tests

although one can see that the null-imposed HAR tests have slightly lower power for negative values

of 𝜌1. This is more apparent in Figure 2A.14 where results for 𝜖𝑡 WN-GAM1 are given. With

𝑇 = 100, power is lower for the null-imposed tests for negative values of 𝜌1. Interestingly, these

power differences disappear when 𝑇 = 500. There are also some asymmetries in power around 𝜌1

in the white noise cases, especially WN-GAM1, that do not occur with 𝜖𝑡 IID.

While the null-imposed HAR tests can have lower power than the null-not-imposed HAR tests,

the power differences are relatively small and disappear as 𝑇 increases. Given the superior null

rejections of the null-imposed-tests and their respectable power, we can recommended them in

practice.

2.4.4 Null Rejection Probabilities Across Lags

The finite sample results to this point have focused on the case of 𝑘 = 1. In this subsection we

provide results for other values of 𝑘. We report results for the 𝐴𝑅(1) case for 𝜙 = 0 and 𝜙 = 0.5

with 𝑘 ranging from 1 to 10. The null hypothesis is

𝐻0 : 𝜌𝑘 = 𝜙𝑘 ,

given the AR(1) structure. Results are reported for the HAR tests and the recursive MA approach

used by the software Stata given by equation (2.1). We report results for 𝑇 = 50, 100, 250 and

1000. We continue to focus on two-sided tests with a nominal level of 0.05. Results for 𝜙 = 0

are given in Figures 15-19 for 𝜖𝑡 IID, MDS, GARCH, WN-1 and WN-GAM1. For 𝜖𝑡 IID (Figure

2A.15) the HAR tests, especially the null-imposed versions, work well for all 𝑘 with rejections very

30

close to 0.05 as 𝑇 increases. The Stata procedure (blue dot solid line labeled ‘Software’) works

well with large 𝑇 but under-rejects for small 𝑇 and larger values of 𝑘. This makes sense because the

estimated variance used by Stata increases mechanically as 𝑘 increases. Figure 2A.16 shows that

when 𝜖𝑡 is an MDS, the null-imposed HAR tests continue to perform well but the null-not-imposed

HAR tests have some over-rejections with small values of 𝑇. The Stata procedure relies on the i.i.d.

assumption for 𝜖𝑡 and breaks down for 𝑘 = 1. In the case of GARCH innovations, Figure 2A.17

shows that the HAR tests perform well, again imposing the null works best. The Stata procedure

completely breaks down. When 𝜖𝑡 is white noise, Figures 18 and 19 show that the null-imposed

HAR tests continue to work well for all 𝑘 including the 𝑇 = 50 case. Not imposing the null results

in HAR tests that can have substantial over-rejections for small values of 𝑘 especially when 𝑇 is

not large. The Stata procedure breaks down for 𝑘 = 1, 2 but works reasonably well for 𝑘 ≥ 3.

These results show that when 𝑦𝑡 is uncorrelated, the Stata procedure only works when 𝑦𝑡 is i.i.d. In

contrast, the HAR tests with the null-imposed work quite well including the case of 𝑦𝑡 being white

noise.

The results with 𝜙 = 0.5 are given in Figures 20-24 for the same cases for 𝜖𝑡. The Stata procedure

is not valid for any of these cases given the AR(1) structure. The null-imposed HAR tests work

well overall but do have some relatively minor size distortions when 𝑇 = 50. The null-not-imposed

HAR tests can have substantial over-rejections with small values of 𝑇 and small values of 𝑘. An

interesting contrast can also be seen in these figures for (cid:101)

some over-rejections, they are less pronounced for (cid:101)

𝑡 (𝑘)∗ than for (cid:101)
𝑡 (𝑘)∗
𝑂𝑆 and uses series to estimate the long run variance. The reason is that the MSE

a kernel and (cid:101)

𝑡 (𝑘)∗ and (cid:101)
𝑡 (𝑘)∗
𝑂𝑆 . This is not because (cid:101)

𝑡 (𝑘)∗
𝑂𝑆 . When these two tests have
𝑡 (𝑘)∗ uses

criteria for smoothing parameters of long run variances leads to less smoothing than the test based

criteria. Less smoothing (e.g. smaller bandwidths for kernel estimators) is well known to lead to

tests with a greater tendency to over-reject in finite samples when fixed-smoothing critical values

are used (see the simulations in Kiefer and Vogelsang (2005) for the kernel case). The reason that
𝑡 (𝑘)∗
𝑂𝑆 tends to over-reject more than (cid:101)
(cid:101)

𝑡 (𝑘)∗ is because (cid:101)𝐾 leads to less smoothing than (cid:101)𝑀.

31

2.4.5 Shape of Confidence Intervals

In section 3.3, we showed that confidence intervals computed with the null-imposed HAR

statistics can take three forms. In this section we investigate the likelihood of the forms for some

representative DGPs from our simulation design. We provide results for confidence intervals using
𝑡 (𝑘)∗
𝑡 (𝑘)∗ with fixed-𝑏 critical values. Results with (cid:101)
𝑂𝑆 using 𝑡𝐾 critical values are similar and are not
(cid:101)
reported. Tables 2 and 3 give results for the AR(1) case with 𝜖𝑡 IID and 𝜖𝑡 WN-NLMA (DGP 6).

These results nicely show the range of possibilities. Results are given for 𝑇 = 50, 100, 250, 500

and AR(1) values 𝜙 = 0, 0.25, 0.7, −0.7. We use 10,000 replications.

Tables 2 and 3 are organized as follows. For each pair of values for 𝜙 and 𝑇, we report the

empirical probabilities of each confidence interval type (Prob), the empirical coverage probability

of the confidence interval (ECP), and the average confidence interval length (𝐶 𝐼) conditional on

the confidence interval type and overall. The AR-IID results in Table 2A.2 serve as a benchmark.

The first panel of the table (𝜙 = 0) gives result for when 𝑦𝑡 is i.i.d. We can see that for all sample

sizes the probability of obtaining the typical [𝑟1, 𝑟2] confidence interval is 1.0. As 𝜙 moves away

from 0 and for smaller values of 𝑇, there are very small, but non-zero, probabilities of obtaining

the confidence intervals (−1, 𝑟1] ∪ [𝑟2, 1) and (−1, 1).

Table 2A.3 shows very different patterns from Table 2A.2. With no autocorrelation or relatively

weak autocorrelation (𝜙 = 0.25), there is about a 50% chance of shapes (−1, 𝑟1] ∪ [𝑟2, 1) and (−1, 1)

with 𝑇 small.

In these cases, the empirical coverages and confidence lengths are larger than the

[𝑟1, 𝑟2] case (this is obviously true by construction when the confidence interval is (−1, 1)). As 𝑇

increases or 𝜙 moves farther away from zero, the probability of [𝑟1, 𝑟2] confidence interval shape

increases. As one expect, average confidence interval lengths shrink as 𝑇 increases.

These results show that for smaller sample sizes and more complex dependence in 𝑦𝑡 and

its innovations, 𝜖𝑡, disjoint and possibly very wide confidence intervals can occur. While some

empirical practitioners may be bothered by disjoint or wide confidence intervals, we view these cases

as providing the practitioner with a signal that 𝑦𝑡 has potentially complex serial correlation structure

with innovations that have complex dependence in higher moments that matter for inference about

32

the autocorrelations of 𝑦𝑡. In other words, disjoint or wide confidence intervals are signals of data

that has limited information about autocorrelation structure.

2.5 Empirical Application

The autocorrelation function is widely used as a preliminary step in analyzing financial time series.

The Bartlett formula is commonly used as part of graphical evidence of autocorrelation structure.

For example, Bollerslev and Mikkelsen (1996) provides a figure of sample autocorrelations for

absolute daily returns of the S&P 500 index with the 95% confidence bands5 implied by the Bartlett

formula for i.i.d. data to illustrate volatility clustering and its long-term dependence. Andersen

et al. (2003) provides figures of sample autocorrelations for daily exchange rate realized volatilities

before and after fractional differencing along with the i.i.d. Bartlett confidence bands to graphically

confirm evidence of long memory.

While the i.i.d. Bartlett confidence bands are routinely reported in practice, it is important to

keep in mind the limitations of these confidence bands. First, the confidence bands are only valid

if the data is i.i.d. If the data is uncorrelated but not i.i.d. (martingale difference, white noise), then

the bands are no longer valid. Second, the bands can only be used to test the null hypothesis that

the series is i.i.d. Once it is determined that the series has dependence, the bands cannot be used

to assess significance of autocorrelations at specific lags because the bands are not generally valid

when there is serial correlation.

A more informative approach is to report confidence intervals using (cid:101)

𝑡 (𝑘)∗ or (cid:101)

𝑡 (𝑘)∗
𝑂𝑆 allowing

inference about autocorrelations that is valid for general serial correlation structures and innovations

that are not necessarily i.i.d. As an illustration we provide some empirical results for S&P 500 index

returns and absolute returns for two sets of time periods (before Covid and during/after Covid) that

have the same number of observations (913 observations for each) but exhibit different estimated

autocorrelation patterns and confidence intervals. Figure 2A.25 provides plots of the returns and

the absolute returns for the full time span of the observations from June 28, 2016 to September 28,

2023. Figure 2A.26 plots estimated autocorrelations for S&P 500 returns for daily data from June

5A confidence band is used to test the null hypothesis of zero autocorrelation and is not a confidence interval. An

estimated value outside the band is a rejection of the null.

33

28, 2016 to February 12, 2020 (Panel (a)) and February 13, 2020 to September 28, 2023 (Panel

(b)). Red circles are the sample autocorrelations given by (2.2) and blue dots are autocorrelations

estimated by OLS using (2.4). The dashed red lines are i.i.d. Bartlett confidence bands. The

gray area is the Stata confidence bands using equation (2.1). The black lines with bars are 95%

confidence intervals computed using (cid:101)

𝑡 (𝑘)∗ with fixed-𝑏 critical values. The dash-dot green lines are

95% confidence bands using (cid:101)

𝑡 (𝑘)∗ that can be used to test a given autocorrelation is zero. One can

equivalently test an autocorrelation is zero by checking that the confidence interval contains zero.

Figure 2A.26 gives results for returns which provides information about market efficiency. Panel

(a) shows that estimated autocorrelations of returns are close to zero and, in nearly every case, not

statistically significant. If one used the Bartlett or Stata confidence bands, one would conclude there

is no evidence to reject the null that returns are uncorrelated (equity market is efficient). However,

that conclusion is subject to the caveat that the bands are only valid if the innovations are i.i.d. In

contrast, the confidence intervals using (cid:101)

𝑡 (𝑘)∗ allow more robust inference. Because nearly all the

confidence intervals contain zero, we cannot reject the null returns are uncorrelated whether or not

innovations are i.i.d. or are simply uncorrelated.

Panel (b) of Figure 2A.26 is distinctly different and interesting because conclusions depend

critically on the method used and its assumptions. Using the Bartlett or Stata confidence bands,

one would conclude there is evidence to reject the null hypothesis that returns are uncorrelated

in the Covid/Post-Covid period given that many sample autocorrelations are outside the bands.

This conclusion is only valid if innovations are i.i.d. Furthermore, these bands cannot be used

to conclude anything further about the autocorrelation structure because the confidence bands are

not confidence intervals. In contrast, the (cid:101)

𝑡 (𝑘)∗ confidence intervals tell a different story. While the

estimated autocorrelations are larger in magnitude compared to the pre-Covid period, nearly all the

confidence intervals contain zero. Therefore, using robust confidence intervals, one cannot reject

that returns are uncorrelated in the Covid/Post-Covid period. The fact that confidence intervals

are wider in this period is an indication that the innovations have potentially more complex higher

order dependence and/or GARCH effects than the pre-Covid period.

34

Figure 2A.27 gives results for absolute returns which provides information about volatility clus-

tering and the dependence structure of volatility (Bollerslev and Mikkelsen (1996)). In Panel (a)

we see positive estimated autocorrelations with tight confidence intervals. While the estimated au-

tocorrelations are not large in magnitude, they persistent at long lags and are statistically significant

(all confidence intervals do not contain zero). This evidence implies volatility clustering during the

pre-Covid period. Panel (b) is an interesting contrast. While estimated autocorrelations are larger,

confidence intervals are substantially wider. Notice that we cannot reject that the first six lags have

zero autocorrelation. While it may be tempting to argue that there is stronger evidence for volatility

clustering and higher persistence during the Covid/Post-Covid period, the wide confidence intervals

suggest something else may be happening in this period that warrants further investigation. Here,

if one only looked at the Bartlett or Stata confidence bands, a potentially misleading conclusion

might be reached.

2.6 Conclusion

This paper develops an estimating equation approach for robust confidence intervals for the

autocorrelation function of a stationary time series. Our approach is applicable to general stationary

time series with uncorrelated innovations that can have dependence in higher order moments

(innovations do not have to be i.i.d.). Except for narrow exceptions, the asymptotic variance of

estimated autocorrelations take a sandwich form. The asymptotic variance can be directly estimated

by well known HAR variance estimators allowing 𝑡-statistics and confidence intervals to be easily

constructed. We consider HAR variance estimators that impose the null leading to more reliable

inference. We provide conditions under which fixed-smoothing critical values can be used for

𝑡-tests and confidence intervals and recommend those critical values be used in practice.

Our extensive simulation study shows that the tests based on the null-imposed variance estimator

in conjunction with fixed-smoothing critical values leads to inference about the autocorrelation

function that works well in practice both in terms of controlling null rejection probabilities and

having good power. Our approach can be used to report generally valid confidence intervals for

covariance stationary time series under weak assumptions for the innovations. In contrast existing

35

software packages typically report confidence bands based on strong assumptions that can only be

used to test narrow hypotheses (and are often misused in practice). Our approach is an improvement

and allows the testing of significantly broader hypothesis about the autocorrelation function in a

highly robust manner.

Our simulation results also reveal a puzzle regarding the use of the bootstrap for inference about

the autocorrelation function. For the case of uncorrelated data (including the case of i.i.d. data) we

find that the block bootstrap and related bootstrap approaches do not perform as well as expected

even in the case where the data is i.i.d. and the i.i.d. bootstrap is used. An analysis of the bootstrap

applied to inference about the autocorrelation function is a topic of ongoing research that we plan

to report in a follow-up paper.

36

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38

APPENDIX 2A

TABLES AND FIGURES

Table 2A.1 𝑐𝑣𝛼/2(𝑏) Polynomial Coefficients, Parzen Kernel
𝜆5
𝜆2
-.5787
.1191

𝜆8
-.0237

𝜆7
.0254

𝜆6
.4326

𝜆4
.4962

𝜆3
.0863

𝜆9
-.0237

𝜆1
.4375

Table 2A.2 Shape of Confidence Intervals using (cid:101)

𝑡 (𝑘)∗, AR-IID

Case
[𝑟1, 𝑟2]
(−1, 𝑟1] ∪ [𝑟2, 1)
(−1, 1)
Total

Case
[𝑟1, 𝑟2]
(−1, 𝑟1] ∪ [𝑟2, 1)
(−1, 1)
Total

(cid:91)𝑃𝑟𝑜𝑏

1.000
0.000
0.000
1.000

(cid:91)𝑃𝑟𝑜𝑏

1.000
0.000
0.000
1.000

Case
[𝑟1, 𝑟2]
(−1, 𝑟1] ∪ [𝑟2, 1)
(−1, 1)
Total

(cid:91)𝑃𝑟𝑜𝑏

0.996
0.004
<0.001
1.000

Case
[𝑟1, 𝑟2]
(−1, 𝑟1] ∪ [𝑟2, 1)
(−1, 1)
Total

(cid:91)𝑃𝑟𝑜𝑏

0.987
0.012
0.002
1.000

T=50

ECP

0.955
-
-
0.955

T=50

ECP

0.942
-
-
0.942

T=50

ECP

0.898
0.895
1.000
0.898

T=50

ECP

0.947
1.000
1.000
0.948

𝜙 = 0.0
T=100

𝐶 𝐼 (cid:91)𝑃𝑟𝑜𝑏

ECP

𝐶 𝐼 (cid:91)𝑃𝑟𝑜𝑏

0.618
-
-
0.618

1.000
0.000
0.000
1.000

0.408
-
-
0.408

1.000
0.000
0.000
1.000

0.951
-
-
0.951
𝜙 = 0.25
T=100

𝐶 𝐼 (cid:91)𝑃𝑟𝑜𝑏

ECP

𝐶 𝐼 (cid:91)𝑃𝑟𝑜𝑏

0.607
-
-
0.607

1.000
0.000
0.000
1.000

0.399
-
-
0.399

1.000
0.000
0.000
1.000

0.941
-
-
0.941
𝜙 = 0.7
T=100

𝐶 𝐼 (cid:91)𝑃𝑟𝑜𝑏

ECP

𝐶 𝐼 (cid:91)𝑃𝑟𝑜𝑏

0.532
1.000
1.875 <0.001
0.000
2.000
1.000
0.538

0.326
1.885
-
0.327

1.000
0.000
0.000
1.000

0.909
1.000
-
0.909
𝜙 = −0.7
T=100

𝐶 𝐼 (cid:91)𝑃𝑟𝑜𝑏

1.000
0.536
1.935 <0.001
0.000
2.000
1.000
0.555

ECP

0.939
1.000
-
0.939

𝐶 𝐼 (cid:91)𝑃𝑟𝑜𝑏

0.329
1.945
-
0.330

1.000
0.000
0.000
1.000

T=250

ECP

0.953
-
-
0.953

T=250

ECP

0.948
-
-
0.948

T=250

ECP

0.930
-
-
0.930

T=250

ECP

0.941
-
-
0.941

T=1000

𝐶 𝐼 (cid:91)𝑃𝑟𝑜𝑏

0.251
-
-
0.251

1.000
0.000
0.000
1.000

ECP

0.948
-
-
0.948

𝐶 𝐼

0.125
-
-
0.125

T=1000

𝐶 𝐼 (cid:91)𝑃𝑟𝑜𝑏

0.245
-
-
0.245

1.000
0.000
0.000
1.000

ECP

0.952
-
-
0.952

𝐶 𝐼

0.121
-
-
0.121

T=1000

𝐶 𝐼 (cid:91)𝑃𝑟𝑜𝑏

0.188
-
-
0.188

1.000
0.000
0.000
1.000

ECP

0.943
-
-
0.943

𝐶 𝐼

0.090
-
-
0.090

T=1000

𝐶 𝐼 (cid:91)𝑃𝑟𝑜𝑏

0.187
-
-
0.187

1.000
0.000
0.000
1.000

ECP

0.944
-
-
0.944

𝐶 𝐼

0.090
-
-
0.090

39

Table 2A.3 Shape of Confidence Intervals using (cid:101)

𝑡 (𝑘)∗, AR-WN-NLMA

Case
[𝑟1, 𝑟2]
(−1, 𝑟1] ∪ [𝑟2, 1)
(−1, 1)
Total

Case
[𝑟1, 𝑟2]
(−1, 𝑟1] ∪ [𝑟2, 1)
(−1, 1)
Total

Case
[𝑟1, 𝑟2]
(−1, 𝑟1] ∪ [𝑟2, 1)
(−1, 1)
Total

Case
[𝑟1, 𝑟2]
(−1, 𝑟1] ∪ [𝑟2, 1)
(−1, 1)
Total

(cid:91)𝑃𝑟𝑜𝑏

0.488
0.134
0.378
1.000

(cid:91)𝑃𝑟𝑜𝑏

0.510
0.121
0.369
1.000

(cid:91)𝑃𝑟𝑜𝑏

0.662
0.093
0.245
1.000

(cid:91)𝑃𝑟𝑜𝑏

0.611
0.109
0.280
1.000

𝜙 = 0.0

T=100

𝐶 𝐼 (cid:91)𝑃𝑟𝑜𝑏

1.125
1.592
2.000
1.519

0.747
0.109
0.144
1.000

ECP

0.966
0.987
1.000
0.973

𝐶 𝐼 (cid:91)𝑃𝑟𝑜𝑏

0.883
1.549
2.000
1.117

0.944
0.036
0.019
1.000

𝜙 = 0.25

T=100

𝐶 𝐼 (cid:91)𝑃𝑟𝑜𝑏

1.098
1.610
2.000
1.493

0.750
0.107
0.142
1.000

ECP

0.957
0.983
1.000
0.966

𝐶 𝐼 (cid:91)𝑃𝑟𝑜𝑏

0.864
1.595
2.000
1.104

0.939
0.042
0.019
1.000

𝜙 = 0.7

T=100

𝐶 𝐼 (cid:91)𝑃𝑟𝑜𝑏

0.791
1.671
2.000
1.169

0.812
0.076
0.112
1.000

ECP

0.942
0.984
1.000
0.952

𝐶 𝐼 (cid:91)𝑃𝑟𝑜𝑏

0.608
1.769
2.000
0.852

0.930
0.049
0.021
1.000

𝜙 = −0.7

T=100

𝐶 𝐼 (cid:91)𝑃𝑟𝑜𝑏

0.769
1.711
2.000
1.216

0.796
0.080
0.125
1.000

ECP

0.975
0.995
1.000
0.980

𝐶 𝐼 (cid:91)𝑃𝑟𝑜𝑏

0.610
1.771
2.000
0.876

0.948
0.034
0.018
1.000

T=50

ECP

0.968
0.985
1.000
0.983

T=50

ECP

0.961
0.977
1.000
0.977

T=50

ECP

0.957
0.972
1.000
0.969

T=50

ECP

0.976
0.987
1.000
0.984

T=250

ECP

0.955
1.000
1.000
0.958

T=250

ECP

0.946
1.000
1.000
0.950

T=250

ECP

0.925
0.992
1.000
0.930

T=250

ECP

0.968
0.988
1.000
0.969

T=1000

𝐶 𝐼 (cid:91)𝑃𝑟𝑜𝑏

1.000
0.587
1.487 <0.001
2.000 <0.001
1.000
0.648

ECP

0.958
1.000
1.000
0.958

𝐶 𝐼

0.304
1.572
2.000
0.304

T=1000

𝐶 𝐼 (cid:91)𝑃𝑟𝑜𝑏

0.572
1.562
2.000
0.640

0.999
0.001
0.000
1.000

ECP

0.945
1.000
-
0.945

𝐶 𝐼

0.293
1.559
-
0.295

T=1000

𝐶 𝐼 (cid:91)𝑃𝑟𝑜𝑏

0.388
1.854
2.000
0.494

0.996
0.004
0.000
1.000

ECP

0.929
1.000
-
0.929

𝐶 𝐼

0.193
1.859
-
0.199

T=1000

𝐶 𝐼 (cid:91)𝑃𝑟𝑜𝑏

0.393
1.801
2.000
0.470

0.998
0.002
0.001
1.000

ECP

0.966
1.000
1.000
0.966

𝐶 𝐼

0.184
1.868
2.000
0.188

40

Figure 2A.1.1 Graphs of null rejection probabilities, 𝐻0 : 𝜌1 = 0

Figure 2A.1.2 Graphs of null rejection probabilities, 𝐻0 : 𝜌1 = 0

41

100200300400500600700Sample Size (T)0.000.050.100.150.200.250.300.35Null Rejection ProbDGP: IID : yt=†t=ut and ut∼iidNormal(0,1)1400160018002000Bartlett (IID)GB-WNTaylorWhiteN(0,1)-H0 (SPJ)Fixed-b (SPJ)Fixed-b-H0 (SPJ)OS (MSE)OS-H0 (MSE)IID-bootstrapMBB100200300400500600700Sample Size (T)0.000.050.100.150.200.250.30Null Rejection ProbDGP: MDS : yt=†t=utut−1 and ut∼iidNormal(0,1)1400160018002000Bartlett (IID)GB-WNTaylorWhiteN(0,1)-H0 (SPJ)Fixed-b (SPJ)Fixed-b-H0 (SPJ)OS (MSE)OS-H0 (MSE)MBBFigure 2A.1.3 Graphs of null rejection probabilities, 𝐻0 : 𝜌1 = 0

Figure 2A.1.3: Graphs of null rejection probabilities, 𝐻0 : 𝜌1 = 0

42

100200300400500600700Sample Size (T)0.0000.0250.0500.0750.1000.1250.1500.1750.200Null Rejection ProbDGP: GARCH : yt=†t=htut and h2t=0.1+0.09†2t−1+0.9h2t−1, ut∼iidN(0,1)1400160018002000Bartlett (IID)GB-WNTaylorWhiteN(0,1)-H0 (SPJ)Fixed-b (SPJ)Fixed-b-H0 (SPJ)OS (MSE)OS-H0 (MSE)MBB100200300400500600700Sample Size (T)0.000.050.100.150.200.25Null Rejection ProbDGP: WN-1 : yt=†t=ut+ut−1ut−2 and ut∼iidNormal(0,1)1400160018002000Bartlett (IID)GB-WNTaylorWhiteN(0,1)-H0 (SPJ)Fixed-b (SPJ)Fixed-b-H0 (SPJ)OS (MSE)OS-H0 (MSE)MBBFigure 2A.1.5: Graphs of null rejection probabilities, 𝐻0 : 𝜌1 = 0

Figure 2A.1.6: Graphs of null rejection probabilities, 𝐻0 : 𝜌1 = 0

43

100200300400500600700Sample Size (T)0.000.050.100.150.200.250.300.35Null Rejection ProbDGP: WN-2 : yt=†t=u2tut−1 and ut∼iidNormal(0,1)1400160018002000Bartlett (IID)GB-WNTaylorWhiteN(0,1)-H0 (SPJ)Fixed-b (SPJ)Fixed-b-H0 (SPJ)OS (MSE)OS-H0 (MSE)MBB100200300400500600700Sample Size (T)0.000.050.100.150.200.250.300.350.40Null Rejection ProbDGP: WN-NLMA: yt=†t=ut−2ut−1(ut−2+ut+1) and ut∼iidNormal(0,1)1400160018002000Bartlett (IID)GB-WNTaylorWhiteN(0,1)-H0 (SPJ)Fixed-b (SPJ)Fixed-b-H0 (SPJ)OS (MSE)OS-H0 (MSE)MBBFigure 2A.1.7: Graphs of null rejection probabilities, 𝐻0 : 𝜌1 = 0

Figure 2A.1.8: Graphs of null rejection probabilities, 𝐻0 : 𝜌1 = 0

44

100200300400500600700Sample Size (T)0.000.050.100.150.200.25Null Rejection ProbDGP: WN-BILIN: yt=†t=ut+0.5ut−1†t−2 and ut∼iidNormal(0,1)1400160018002000Bartlett (IID)GB-WNTaylorWhiteN(0,1)-H0 (SPJ)Fixed-b (SPJ)Fixed-b-H0 (SPJ)OS (MSE)OS-H0 (MSE)MBB100200300400500600700Sample Size (T)0.000.050.100.150.200.250.300.350.40Null Rejection ProbDGP: WN-GAM1 : yt=†t=ut+ut−1ut−2 and ut=ζt−E[ζt], ζt∼iidGamma(0.3,0.4)1400160018002000Bartlett (IID)GB-WNTaylorWhiteN(0,1)-H0 (SPJ)Fixed-b (SPJ)Fixed-b-H0 (SPJ)OS (MSE)OS-H0 (MSE)MBBFigure 2A.1.9: Graphs of null rejection probabilities, 𝐻0 : 𝜌1 = 0

45

100200300400500600700Sample Size (T)0.000.020.040.060.080.10Null Rejection ProbDGP: WN-GAM2 : yt=†t=ut−ut−1ut−2 and ut=ζt−E[ζt], ζt∼iidGamma(0.3,0.4)1400160018002000Bartlett (IID)GB-WNTaylorWhiteN(0,1)-H0 (SPJ)Fixed-b (SPJ)Fixed-b-H0 (SPJ)OS (MSE)OS-H0 (MSE)MBBFigure 2A.2 Null rejection probabilities, 𝐻0 : 𝜌1 =

𝜃

(1+𝜃2) , MA-IID

46

0.10.30.50.70.90.10.30.50.70.9θ0.0000.0250.0500.0750.1000.1250.150Null Rejection ProbT=100Bartlett (MA)GB-MAGB-WNN(0,1)-H0 (SPJ)Fixed-b-H0 (SPJ)Fixed-b (SPJ)OS-H0 (MSE)OS (MSE)0.10.30.50.70.90.10.30.50.70.9θ0.0000.0250.0500.0750.1000.1250.150Null Rejection ProbT=200Bartlett (MA)GB-MAGB-WNN(0,1)-H0 (SPJ)Fixed-b-H0 (SPJ)Fixed-b (SPJ)OS-H0 (MSE)OS (MSE)0.10.30.50.70.90.10.30.50.70.9θ0.0000.0250.0500.0750.1000.1250.150Null Rejection ProbT=500Bartlett (MA)GB-MAGB-WNN(0,1)-H0 (SPJ)Fixed-b-H0 (SPJ)Fixed-b (SPJ)OS-H0 (MSE)OS (MSE)0.10.30.50.70.90.10.30.50.70.9θ0.0000.0250.0500.0750.1000.1250.150Null Rejection ProbT=2000Bartlett (MA)GB-MAGB-WNN(0,1)-H0 (SPJ)Fixed-b-H0 (SPJ)Fixed-b (SPJ)OS-H0 (MSE)OS (MSE)DGP: MA-IID : yt=†t+θ†t−1, where †t∼iidNormal(0,1)Figure 2A.3: Null rejection probabilities, 𝐻0 : 𝜌1 =

𝜃

(1+𝜃2) , MA-MDS

47

0.10.30.50.70.90.10.30.50.70.9θ0.000.050.100.150.200.25Null Rejection ProbT=100Bartlett (MA)GB-MAGB-WNN(0,1)-H0 (SPJ)Fixed-b-H0 (SPJ)Fixed-b (SPJ)OS-H0 (MSE)OS (MSE)0.10.30.50.70.90.10.30.50.70.9θ0.000.050.100.150.200.25Null Rejection ProbT=200Bartlett (MA)GB-MAGB-WNN(0,1)-H0 (SPJ)Fixed-b-H0 (SPJ)Fixed-b (SPJ)OS-H0 (MSE)OS (MSE)0.10.30.50.70.90.10.30.50.70.9θ0.000.050.100.150.200.25Null Rejection ProbT=500Bartlett (MA)GB-MAGB-WNN(0,1)-H0 (SPJ)Fixed-b-H0 (SPJ)Fixed-b (SPJ)OS-H0 (MSE)OS (MSE)0.10.30.50.70.90.10.30.50.70.9θ0.000.050.100.150.200.25Null Rejection ProbT=2000Bartlett (MA)GB-MAGB-WNN(0,1)-H0 (SPJ)Fixed-b-H0 (SPJ)Fixed-b (SPJ)OS-H0 (MSE)OS (MSE)DGP: MA-MDS : yt=†t+θ†t−1, where †t=utut−1, ut∼iidNormal(0,1)Figure 2A.4: Null rejection probabilities, 𝐻0 : 𝜌1 =

𝜃

(1+𝜃2) , MA-GRACH

48

0.10.30.50.70.90.10.30.50.70.9θ0.0000.0250.0500.0750.1000.1250.1500.1750.200Null Rejection ProbT=100Bartlett (MA)GB-MAGB-WNN(0,1)-H0 (SPJ)Fixed-b-H0 (SPJ)Fixed-b (SPJ)OS-H0 (MSE)OS (MSE)0.10.30.50.70.90.10.30.50.70.9θ0.0000.0250.0500.0750.1000.1250.1500.1750.200Null Rejection ProbT=200Bartlett (MA)GB-MAGB-WNN(0,1)-H0 (SPJ)Fixed-b-H0 (SPJ)Fixed-b (SPJ)OS-H0 (MSE)OS (MSE)0.10.30.50.70.90.10.30.50.70.9θ0.0000.0250.0500.0750.1000.1250.1500.1750.200Null Rejection ProbT=500Bartlett (MA)GB-MAGB-WNN(0,1)-H0 (SPJ)Fixed-b-H0 (SPJ)Fixed-b (SPJ)OS-H0 (MSE)OS (MSE)0.10.30.50.70.90.10.30.50.70.9θ0.0000.0250.0500.0750.1000.1250.1500.1750.200Null Rejection ProbT=2000Bartlett (MA)GB-MAGB-WNN(0,1)-H0 (SPJ)Fixed-b-H0 (SPJ)Fixed-b (SPJ)OS-H0 (MSE)OS (MSE)DGP: MA-GARCH : yt=†t+θ†t−1, where †t=htut and h2t=0.1+0.09†2t−1+0.9h2t−1, ut∼iidN(0,1)Figure 2A.5 Null rejection probabilities, 𝐻0 : 𝜌1 =

𝜃

(1+𝜃2) , MA-WN-1

49

0.10.30.50.70.90.10.30.50.70.9θ0.0000.0250.0500.0750.1000.1250.1500.1750.200Null Rejection ProbT=100Bartlett (MA)GB-MAGB-WNN(0,1)-H0 (SPJ)Fixed-b-H0 (SPJ)Fixed-b (SPJ)OS-H0 (MSE)OS (MSE)0.10.30.50.70.90.10.30.50.70.9θ0.0000.0250.0500.0750.1000.1250.1500.1750.200Null Rejection ProbT=200Bartlett (MA)GB-MAGB-WNN(0,1)-H0 (SPJ)Fixed-b-H0 (SPJ)Fixed-b (SPJ)OS-H0 (MSE)OS (MSE)0.10.30.50.70.90.10.30.50.70.9θ0.0000.0250.0500.0750.1000.1250.1500.1750.200Null Rejection ProbT=500Bartlett (MA)GB-MAGB-WNN(0,1)-H0 (SPJ)Fixed-b-H0 (SPJ)Fixed-b (SPJ)OS-H0 (MSE)OS (MSE)0.10.30.50.70.90.10.30.50.70.9θ0.0000.0250.0500.0750.1000.1250.1500.1750.200Null Rejection ProbT=2000Bartlett (MA)GB-MAGB-WNN(0,1)-H0 (SPJ)Fixed-b-H0 (SPJ)Fixed-b (SPJ)OS-H0 (MSE)OS (MSE)DGP: MA-WN-1 : yt=†t+θ†t−1, where †t=ut+ut−1ut−2, ut∼iidNormal(0,1)Figure 2A.6 Null rejection probabilities, 𝐻0 : 𝜌1 =

𝜃

(1+𝜃2) , MA-WN-Gamma

50

0.10.30.50.70.90.10.30.50.70.9θ0.0000.0250.0500.0750.1000.1250.1500.1750.200Null Rejection ProbT=100Bartlett (MA)GB-MAGB-WNN(0,1)-H0 (SPJ)Fixed-b-H0 (SPJ)Fixed-b (SPJ)OS-H0 (MSE)OS (MSE)0.10.30.50.70.90.10.30.50.70.9θ0.0000.0250.0500.0750.1000.1250.1500.1750.200Null Rejection ProbT=200Bartlett (MA)GB-MAGB-WNN(0,1)-H0 (SPJ)Fixed-b-H0 (SPJ)Fixed-b (SPJ)OS-H0 (MSE)OS (MSE)0.10.30.50.70.90.10.30.50.70.9θ0.0000.0250.0500.0750.1000.1250.1500.1750.200Null Rejection ProbT=500Bartlett (MA)GB-MAGB-WNN(0,1)-H0 (SPJ)Fixed-b-H0 (SPJ)Fixed-b (SPJ)OS-H0 (MSE)OS (MSE)0.10.30.50.70.90.10.30.50.70.9θ0.0000.0250.0500.0750.1000.1250.1500.1750.200Null Rejection ProbT=2000Bartlett (MA)GB-MAGB-WNN(0,1)-H0 (SPJ)Fixed-b-H0 (SPJ)Fixed-b (SPJ)OS-H0 (MSE)OS (MSE)DGP: MA-WN-GAM1 : yt=†t+θ†t−1, where †t=ut+ut−1ut−2 and ut=ζt−E[ζt], ζt∼iidGamma(0.3,0.4)Figure 2A.7 Null rejection probabilities, 𝐻0 : 𝜌1 = 𝜙, AR-IID

51

0.10.30.50.70.90.10.30.50.70.9φ0.0000.0250.0500.0750.1000.1250.150Null Rejection ProbT=100Bartlett (AR)GB-MAGB-WNN(0,1)-H0 (SPJ)Fixed-b-H0 (SPJ)Fixed-b (SPJ)OS-H0 (MSE)OS (MSE)0.10.30.50.70.90.10.30.50.70.9φ0.0000.0250.0500.0750.1000.1250.150Null Rejection ProbT=200Bartlett (AR)GB-MAGB-WNN(0,1)-H0 (SPJ)Fixed-b-H0 (SPJ)Fixed-b (SPJ)OS-H0 (MSE)OS (MSE)0.10.30.50.70.90.10.30.50.70.9φ0.0000.0250.0500.0750.1000.1250.150Null Rejection ProbT=500Bartlett (AR)GB-MAGB-WNN(0,1)-H0 (SPJ)Fixed-b-H0 (SPJ)Fixed-b (SPJ)OS-H0 (MSE)OS (MSE)0.10.30.50.70.90.10.30.50.70.9φ0.0000.0250.0500.0750.1000.1250.150Null Rejection ProbT=2000Bartlett (AR)GB-MAGB-WNN(0,1)-H0 (SPJ)Fixed-b-H0 (SPJ)Fixed-b (SPJ)OS-H0 (MSE)OS (MSE)DGP: AR-IID : yt=φyt−1+†t, where †t∼iidNormal(0,1)Figure 2A.8 Null rejection probabilities, 𝐻0 : 𝜌1 = 𝜙, AR-MDS

52

0.10.30.50.70.90.10.30.50.70.9φ0.000.050.100.150.200.25Null Rejection ProbT=100Bartlett (AR)GB-MAGB-WNN(0,1)-H0 (SPJ)Fixed-b-H0 (SPJ)Fixed-b (SPJ)OS-H0 (MSE)OS (MSE)0.10.30.50.70.90.10.30.50.70.9φ0.000.050.100.150.200.25Null Rejection ProbT=200Bartlett (AR)GB-MAGB-WNN(0,1)-H0 (SPJ)Fixed-b-H0 (SPJ)Fixed-b (SPJ)OS-H0 (MSE)OS (MSE)0.10.30.50.70.90.10.30.50.70.9φ0.000.050.100.150.200.25Null Rejection ProbT=500Bartlett (AR)GB-MAGB-WNN(0,1)-H0 (SPJ)Fixed-b-H0 (SPJ)Fixed-b (SPJ)OS-H0 (MSE)OS (MSE)0.10.30.50.70.90.10.30.50.70.9φ0.000.050.100.150.200.25Null Rejection ProbT=2000Bartlett (AR)GB-MAGB-WNN(0,1)-H0 (SPJ)Fixed-b-H0 (SPJ)Fixed-b (SPJ)OS-H0 (MSE)OS (MSE)DGP: AR-MDS : yt=φyt−1+†t, where †t=utut−1, ut∼iidNormal(0,1)Figure 2A.9: Null rejection probabilities, 𝐻0 : 𝜌1 = 𝜙, AR-GRACH

53

0.10.30.50.70.90.10.30.50.70.9φ0.0000.0250.0500.0750.1000.1250.1500.1750.200Null Rejection ProbT=100Bartlett (AR)GB-MAGB-WNN(0,1)-H0 (SPJ)Fixed-b-H0 (SPJ)Fixed-b (SPJ)OS-H0 (MSE)OS (MSE)0.10.30.50.70.90.10.30.50.70.9φ0.0000.0250.0500.0750.1000.1250.1500.1750.200Null Rejection ProbT=200Bartlett (AR)GB-MAGB-WNN(0,1)-H0 (SPJ)Fixed-b-H0 (SPJ)Fixed-b (SPJ)OS-H0 (MSE)OS (MSE)0.10.30.50.70.90.10.30.50.70.9φ0.0000.0250.0500.0750.1000.1250.1500.1750.200Null Rejection ProbT=500Bartlett (AR)GB-MAGB-WNN(0,1)-H0 (SPJ)Fixed-b-H0 (SPJ)Fixed-b (SPJ)OS-H0 (MSE)OS (MSE)0.10.30.50.70.90.10.30.50.70.9φ0.0000.0250.0500.0750.1000.1250.1500.1750.200Null Rejection ProbT=2000Bartlett (AR)GB-MAGB-WNN(0,1)-H0 (SPJ)Fixed-b-H0 (SPJ)Fixed-b (SPJ)OS-H0 (MSE)OS (MSE)DGP: AR-GARCH : yt=φyt−1+†t, where †t=htut and h2t=0.1+0.09†2t−1+0.9h2t−1, ut∼iidN(0,1)Figure 2A.10: Null rejection probabilities, 𝐻0 : 𝜌1 = 𝜙, AR-WN-1

54

0.10.30.50.70.90.10.30.50.70.9φ0.0000.0250.0500.0750.1000.1250.1500.1750.200Null Rejection ProbT=100Bartlett (AR)GB-MAGB-WNN(0,1)-H0 (SPJ)Fixed-b-H0 (SPJ)Fixed-b (SPJ)OS-H0 (MSE)OS (MSE)0.10.30.50.70.90.10.30.50.70.9φ0.0000.0250.0500.0750.1000.1250.1500.1750.200Null Rejection ProbT=200Bartlett (AR)GB-MAGB-WNN(0,1)-H0 (SPJ)Fixed-b-H0 (SPJ)Fixed-b (SPJ)OS-H0 (MSE)OS (MSE)0.10.30.50.70.90.10.30.50.70.9φ0.0000.0250.0500.0750.1000.1250.1500.1750.200Null Rejection ProbT=500Bartlett (AR)GB-MAGB-WNN(0,1)-H0 (SPJ)Fixed-b-H0 (SPJ)Fixed-b (SPJ)OS-H0 (MSE)OS (MSE)0.10.30.50.70.90.10.30.50.70.9φ0.0000.0250.0500.0750.1000.1250.1500.1750.200Null Rejection ProbT=2000Bartlett (AR)GB-MAGB-WNN(0,1)-H0 (SPJ)Fixed-b-H0 (SPJ)Fixed-b (SPJ)OS-H0 (MSE)OS (MSE)DGP: AR-WN-1 : yt=φyt−1+†t, where †t=ut+ut−1ut−2, ut∼iidNormal(0,1)Figure 2A.11: Null rejection probabilities, 𝐻0 : 𝜌1 = 𝜙, AR-WN-Gamma

55

0.10.30.50.70.90.10.30.50.70.9φ0.0000.0250.0500.0750.1000.1250.1500.1750.200Null Rejection ProbT=100Bartlett (AR)GB-MAGB-WNN(0,1)-H0 (SPJ)Fixed-b-H0 (SPJ)Fixed-b (SPJ)OS-H0 (MSE)OS (MSE)0.10.30.50.70.90.10.30.50.70.9φ0.0000.0250.0500.0750.1000.1250.1500.1750.200Null Rejection ProbT=200Bartlett (AR)GB-MAGB-WNN(0,1)-H0 (SPJ)Fixed-b-H0 (SPJ)Fixed-b (SPJ)OS-H0 (MSE)OS (MSE)0.10.30.50.70.90.10.30.50.70.9φ0.0000.0250.0500.0750.1000.1250.1500.1750.200Null Rejection ProbT=500Bartlett (AR)GB-MAGB-WNN(0,1)-H0 (SPJ)Fixed-b-H0 (SPJ)Fixed-b (SPJ)OS-H0 (MSE)OS (MSE)0.10.30.50.70.90.10.30.50.70.9φ0.0000.0250.0500.0750.1000.1250.1500.1750.200Null Rejection ProbT=2000Bartlett (AR)GB-MAGB-WNN(0,1)-H0 (SPJ)Fixed-b-H0 (SPJ)Fixed-b (SPJ)OS-H0 (MSE)OS (MSE)DGP: AR-WN-GAM1 : yt=φyt−1+†t, where †t=ut+ut−1ut−2 and ut=ζt−E[ζt], ζt∼iidGamma(0.3,0.4)Figure 2A.12: Size adjusted power, 𝐻0 : 𝜌1 = 0, 𝐻1 : 𝜌1 = 𝜙, AR-IID

56

0.40.30.20.10.00.10.20.30.4ρ10.00.10.20.30.40.50.60.70.80.91.0Size Adjusted PowerT=100Bartlett (IID)GB-MAGB-WNKernel-H0 (SPJ)Kernel (SPJ)OS-H0 (MSE)OS (MSE)0.40.30.20.10.00.10.20.30.4ρ10.00.10.20.30.40.50.60.70.80.91.0Size Adjusted PowerT=200Bartlett (IID)GB-MAGB-WNKernel-H0 (SPJ)Kernel (SPJ)OS-H0 (MSE)OS (MSE)0.40.30.20.10.00.10.20.30.4ρ10.00.10.20.30.40.50.60.70.80.91.0Size Adjusted PowerT=300Bartlett (IID)GB-MAGB-WNKernel-H0 (SPJ)Kernel (SPJ)OS-H0 (MSE)OS (MSE)0.40.30.20.10.00.10.20.30.4ρ10.00.10.20.30.40.50.60.70.80.91.0Size Adjusted PowerT=500Bartlett (IID)GB-MAGB-WNKernel-H0 (SPJ)Kernel (SPJ)OS-H0 (MSE)OS (MSE)DGP: AR-IID : yt=φyt−1+†t, where †t∼iidNormal(0,1)Figure 2A.13: Size adjusted power, 𝐻0 : 𝜌1 = 0, 𝐻1 : 𝜌1 = 𝜙, AR-WN-NLMA

57

0.40.30.20.10.00.10.20.30.4ρ10.00.10.20.30.40.50.60.70.80.91.0Size Adjusted PowerT=100Bartlett (IID)GB-MAGB-WNKernel-H0 (SPJ)Kernel (SPJ)OS-H0 (MSE)OS (MSE)0.40.30.20.10.00.10.20.30.4ρ10.00.10.20.30.40.50.60.70.80.91.0Size Adjusted PowerT=200Bartlett (IID)GB-MAGB-WNKernel-H0 (SPJ)Kernel (SPJ)OS-H0 (MSE)OS (MSE)0.40.30.20.10.00.10.20.30.4ρ10.00.10.20.30.40.50.60.70.80.91.0Size Adjusted PowerT=300Bartlett (IID)GB-MAGB-WNKernel-H0 (SPJ)Kernel (SPJ)OS-H0 (MSE)OS (MSE)0.40.30.20.10.00.10.20.30.4ρ10.00.10.20.30.40.50.60.70.80.91.0Size Adjusted PowerT=500Bartlett (IID)GB-MAGB-WNKernel-H0 (SPJ)Kernel (SPJ)OS-H0 (MSE)OS (MSE)DGP: AR-WN-NLMA : yt=φyt−1+†t, where †t=ut−2ut−1(ut−2+ut+1), ut∼iidNormal(0,1)Figure 2A.14: Size adjusted power, 𝐻0 : 𝜌1 = 0, 𝐻1 : 𝜌1 = 𝜙, AR-WN-Gamma

58

0.40.30.20.10.00.10.20.30.4ρ10.00.10.20.30.40.50.60.70.80.91.0Size Adjusted PowerT=100Bartlett (IID)GB-MAGB-WNKernel-H0 (SPJ)Kernel (SPJ)OS-H0 (MSE)OS (MSE)0.40.30.20.10.00.10.20.30.4ρ10.00.10.20.30.40.50.60.70.80.91.0Size Adjusted PowerT=200Bartlett (IID)GB-MAGB-WNKernel-H0 (SPJ)Kernel (SPJ)OS-H0 (MSE)OS (MSE)0.40.30.20.10.00.10.20.30.4ρ10.00.10.20.30.40.50.60.70.80.91.0Size Adjusted PowerT=300Bartlett (IID)GB-MAGB-WNKernel-H0 (SPJ)Kernel (SPJ)OS-H0 (MSE)OS (MSE)0.40.30.20.10.00.10.20.30.4ρ10.00.10.20.30.40.50.60.70.80.91.0Size Adjusted PowerT=500Bartlett (IID)GB-MAGB-WNKernel-H0 (SPJ)Kernel (SPJ)OS-H0 (MSE)OS (MSE)DGP: AR-WN-GAM1 : yt=φyt−1+†t, where †t=ut+ut−1ut−2 and ut=ζt−E[ζt], ζt∼iidGamma(0.3,0.4)Figure 2A.15: Null rejection probabilities, 𝐻0 : 𝜌𝑘 = 0, IID

59

12345678910lag k0.0000.0250.0500.0750.1000.1250.1500.1750.200Null Rejection ProbT=50SoftwareFixed-b-H0 (SPJ)Fixed-b (SPJ)OS-H0 (MSE)OS (MSE)12345678910lag k0.0000.0250.0500.0750.1000.1250.1500.1750.200Null Rejection ProbT=100SoftwareFixed-b-H0 (SPJ)Fixed-b (SPJ)OS-H0 (MSE)OS (MSE)12345678910lag k0.0000.0250.0500.0750.1000.1250.1500.1750.200Null Rejection ProbT=250SoftwareFixed-b-H0 (SPJ)Fixed-b (SPJ)OS-H0 (MSE)OS (MSE)12345678910lag k0.0000.0250.0500.0750.1000.1250.1500.1750.200Null Rejection ProbT=1000SoftwareFixed-b-H0 (SPJ)Fixed-b (SPJ)OS-H0 (MSE)OS (MSE)DGP: IID : where †t∼iidNormal(0,1)Figure 2A.16: Null rejection probabilities, 𝐻0 : 𝜌𝑘 = 0, MDS

60

12345678910lag k0.000.050.100.150.200.250.30Null Rejection ProbT=50SoftwareFixed-b-H0 (SPJ)Fixed-b (SPJ)OS-H0 (MSE)OS (MSE)12345678910lag k0.000.050.100.150.200.250.30Null Rejection ProbT=100SoftwareFixed-b-H0 (SPJ)Fixed-b (SPJ)OS-H0 (MSE)OS (MSE)12345678910lag k0.000.050.100.150.200.250.30Null Rejection ProbT=250SoftwareFixed-b-H0 (SPJ)Fixed-b (SPJ)OS-H0 (MSE)OS (MSE)12345678910lag k0.000.050.100.150.200.250.30Null Rejection ProbT=1000SoftwareFixed-b-H0 (SPJ)Fixed-b (SPJ)OS-H0 (MSE)OS (MSE)DGP: MDS : where †t=utut−1, ut∼iidNormal(0,1)Figure 2A.17: Null rejection probabilities, 𝐻0 : 𝜌𝑘 = 0, GRACH

61

12345678910lag k0.0000.0250.0500.0750.1000.1250.1500.1750.200Null Rejection ProbT=50SoftwareFixed-b-H0 (SPJ)Fixed-b (SPJ)OS-H0 (MSE)OS (MSE)12345678910lag k0.0000.0250.0500.0750.1000.1250.1500.1750.200Null Rejection ProbT=100SoftwareFixed-b-H0 (SPJ)Fixed-b (SPJ)OS-H0 (MSE)OS (MSE)12345678910lag k0.0000.0250.0500.0750.1000.1250.1500.1750.200Null Rejection ProbT=250SoftwareFixed-b-H0 (SPJ)Fixed-b (SPJ)OS-H0 (MSE)OS (MSE)12345678910lag k0.0000.0250.0500.0750.1000.1250.1500.1750.200Null Rejection ProbT=1000SoftwareFixed-b-H0 (SPJ)Fixed-b (SPJ)OS-H0 (MSE)OS (MSE)DGP: GARCH : where †t=htut and h2t=0.1+0.09†2t−1+0.9h2t−1, ut∼iidN(0,1)Figure 2A.18: Null rejection probabilities, 𝐻0 : 𝜌𝑘 = 0, WN-1

62

12345678910lag k0.0000.0250.0500.0750.1000.1250.1500.1750.200Null Rejection ProbT=50SoftwareFixed-b-H0 (SPJ)Fixed-b (SPJ)OS-H0 (MSE)OS (MSE)12345678910lag k0.0000.0250.0500.0750.1000.1250.1500.1750.200Null Rejection ProbT=100SoftwareFixed-b-H0 (SPJ)Fixed-b (SPJ)OS-H0 (MSE)OS (MSE)12345678910lag k0.0000.0250.0500.0750.1000.1250.1500.1750.200Null Rejection ProbT=250SoftwareFixed-b-H0 (SPJ)Fixed-b (SPJ)OS-H0 (MSE)OS (MSE)12345678910lag k0.0000.0250.0500.0750.1000.1250.1500.1750.200Null Rejection ProbT=1000SoftwareFixed-b-H0 (SPJ)Fixed-b (SPJ)OS-H0 (MSE)OS (MSE)DGP: WN-1 : where †t=ut+ut−1ut−2, ut∼iidNormal(0,1)Figure 2A.19: Null rejection probabilities, 𝐻0 : 𝜌𝑘 = 0, WN-Gamma

63

12345678910lag k0.000.050.100.150.200.250.30Null Rejection ProbT=50SoftwareFixed-b-H0 (SPJ)Fixed-b (SPJ)OS-H0 (MSE)OS (MSE)12345678910lag k0.000.050.100.150.200.250.30Null Rejection ProbT=100SoftwareFixed-b-H0 (SPJ)Fixed-b (SPJ)OS-H0 (MSE)OS (MSE)12345678910lag k0.000.050.100.150.200.250.30Null Rejection ProbT=250SoftwareFixed-b-H0 (SPJ)Fixed-b (SPJ)OS-H0 (MSE)OS (MSE)12345678910lag k0.000.050.100.150.200.250.30Null Rejection ProbT=1000SoftwareFixed-b-H0 (SPJ)Fixed-b (SPJ)OS-H0 (MSE)OS (MSE)DGP: WN-GAM1 : where †t=ut+ut−1ut−2 and ut=ζt−E[ζt], ζt∼iidGamma(0.3,0.4)Figure 2A.20: Null rejection probabilities, 𝐻0 : 𝜌𝑘 = 𝜙𝑘 , 𝜙 = 0.5, AR-IID

64

12345678910lag k0.0000.0250.0500.0750.1000.1250.1500.1750.200Null Rejection ProbT=50SoftwareFixed-b-H0 (SPJ)Fixed-b (SPJ)OS-H0 (MSE)OS (MSE)12345678910lag k0.0000.0250.0500.0750.1000.1250.1500.1750.200Null Rejection ProbT=100SoftwareFixed-b-H0 (SPJ)Fixed-b (SPJ)OS-H0 (MSE)OS (MSE)12345678910lag k0.0000.0250.0500.0750.1000.1250.1500.1750.200Null Rejection ProbT=250SoftwareFixed-b-H0 (SPJ)Fixed-b (SPJ)OS-H0 (MSE)OS (MSE)12345678910lag k0.0000.0250.0500.0750.1000.1250.1500.1750.200Null Rejection ProbT=1000SoftwareFixed-b-H0 (SPJ)Fixed-b (SPJ)OS-H0 (MSE)OS (MSE)DGP: AR-IID : yt=φyt−1+†t, where †t∼iidNormal(0,1)Figure 2A.21: Null rejection probabilities, 𝐻0 : 𝜌𝑘 = 𝜙𝑘 , 𝜙 = 0.5, AR-MDS

65

12345678910lag k0.0000.0250.0500.0750.1000.1250.1500.1750.200Null Rejection ProbT=50SoftwareFixed-b-H0 (SPJ)Fixed-b (SPJ)OS-H0 (MSE)OS (MSE)12345678910lag k0.0000.0250.0500.0750.1000.1250.1500.1750.200Null Rejection ProbT=100SoftwareFixed-b-H0 (SPJ)Fixed-b (SPJ)OS-H0 (MSE)OS (MSE)12345678910lag k0.0000.0250.0500.0750.1000.1250.1500.1750.200Null Rejection ProbT=250SoftwareFixed-b-H0 (SPJ)Fixed-b (SPJ)OS-H0 (MSE)OS (MSE)12345678910lag k0.0000.0250.0500.0750.1000.1250.1500.1750.200Null Rejection ProbT=1000SoftwareFixed-b-H0 (SPJ)Fixed-b (SPJ)OS-H0 (MSE)OS (MSE)DGP: AR-MDS : yt=φyt−1+†t, where †t=utut−1, ut∼iidNormal(0,1)Figure 2A.22: Null rejection probabilities, 𝐻0 : 𝜌𝑘 = 𝜙𝑘 , 𝜙 = 0.5, AR-GRACH

66

12345678910lag k0.0000.0250.0500.0750.1000.1250.1500.1750.200Null Rejection ProbT=50SoftwareFixed-b-H0 (SPJ)Fixed-b (SPJ)OS-H0 (MSE)OS (MSE)12345678910lag k0.0000.0250.0500.0750.1000.1250.1500.1750.200Null Rejection ProbT=100SoftwareFixed-b-H0 (SPJ)Fixed-b (SPJ)OS-H0 (MSE)OS (MSE)12345678910lag k0.0000.0250.0500.0750.1000.1250.1500.1750.200Null Rejection ProbT=250SoftwareFixed-b-H0 (SPJ)Fixed-b (SPJ)OS-H0 (MSE)OS (MSE)12345678910lag k0.0000.0250.0500.0750.1000.1250.1500.1750.200Null Rejection ProbT=1000SoftwareFixed-b-H0 (SPJ)Fixed-b (SPJ)OS-H0 (MSE)OS (MSE)DGP: AR-GARCH : yt=φyt−1+†t, where †t=htut and h2t=0.1+0.09†2t−1+0.9h2t−1, ut∼iidN(0,1)Figure 2A.23: Null rejection probabilities, 𝐻0 : 𝜌𝑘 = 𝜙𝑘 , 𝜙 = 0.5, AR-WN-1

67

12345678910lag k0.0000.0250.0500.0750.1000.1250.1500.1750.200Null Rejection ProbT=50SoftwareFixed-b-H0 (SPJ)Fixed-b (SPJ)OS-H0 (MSE)OS (MSE)12345678910lag k0.0000.0250.0500.0750.1000.1250.1500.1750.200Null Rejection ProbT=100SoftwareFixed-b-H0 (SPJ)Fixed-b (SPJ)OS-H0 (MSE)OS (MSE)12345678910lag k0.0000.0250.0500.0750.1000.1250.1500.1750.200Null Rejection ProbT=250SoftwareFixed-b-H0 (SPJ)Fixed-b (SPJ)OS-H0 (MSE)OS (MSE)12345678910lag k0.0000.0250.0500.0750.1000.1250.1500.1750.200Null Rejection ProbT=1000SoftwareFixed-b-H0 (SPJ)Fixed-b (SPJ)OS-H0 (MSE)OS (MSE)DGP: AR-WN-1 : yt=φyt−1+†t, where †t=ut+ut−1ut−2, ut∼iidNormal(0,1)Figure 2A.24: Null rejection probabilities, 𝐻0 : 𝜌𝑘 = 𝜙𝑘 , 𝜙 = 0.5, AR-WN-Gamma

68

12345678910lag k0.000.050.100.150.200.250.30Null Rejection ProbT=50SoftwareFixed-b-H0 (SPJ)Fixed-b (SPJ)OS-H0 (MSE)OS (MSE)12345678910lag k0.000.050.100.150.200.250.30Null Rejection ProbT=100SoftwareFixed-b-H0 (SPJ)Fixed-b (SPJ)OS-H0 (MSE)OS (MSE)12345678910lag k0.000.050.100.150.200.250.30Null Rejection ProbT=250SoftwareFixed-b-H0 (SPJ)Fixed-b (SPJ)OS-H0 (MSE)OS (MSE)12345678910lag k0.000.050.100.150.200.250.30Null Rejection ProbT=1000SoftwareFixed-b-H0 (SPJ)Fixed-b (SPJ)OS-H0 (MSE)OS (MSE)DGP: AR-WN-GAM1 : yt=φyt−1+†t, where †t=ut+ut−1ut−2 and ut=ζt−E[ζt], ζt∼iidGamma(0.3,0.4)Figure 2A.25: Graphs of S&P 500 index daily returns and absolute returns

69

20172018201920202021202220232024Date0.100.050.000.050.10ReturnS&P 500 Index Daily Returns, June 28, 2016 to September 28, 2023ReturnFebruary 13, 202020172018201920202021202220232024Date0.000.020.040.060.080.100.12Absolute ReturnS&P 500 Index Daily Absolute Returns, June 28, 2016 to September 28, 2023Absolute ReturnFebruary 13, 2020Figure 2A.26: Estimated autocorrelations for S&P 500 index returns during pre- and post-Covid

70

0.02.55.07.510.012.515.017.520.0k (Lag)0.50.40.30.20.10.00.10.20.30.40.5Estimated Autocorrelation(a) S&P 500 Index Returns, June 28, 2016 to February 12, 2020 (pre-Covid)Estimated AutocorrelationConfidence Interval - HARConfidence Band - HARSample AutocorrelationConfidence Band - Stata FormulaConfidence Band - Bartlett Formula0.02.55.07.510.012.515.017.520.0k (Lag)0.50.40.30.20.10.00.10.20.30.40.5Estimated Autocorrelation(b) S&P 500 Index Returns, February 13, 2020 to September 28, 2023 (post-Covid)Estimated AutocorrelationConfidence Interval - HARConfidence Band - HARSample AutocorrelationConfidence Band - Stata FormulaConfidence Band - Bartlett FormulaFigure 2A.27: Estimated autocorrelations for S&P 500 index absolute returns during pre- and
post-Covid

71

0.02.55.07.510.012.515.017.520.0k (Lag)1.000.750.500.250.000.250.500.751.00Estimated Autocorrelation(a) S&P 500 Index Absolute Returns, June 28, 2016 to February 12, 2020 (pre-Covid)Estimated AutocorrelationConfidence Interval - HARConfidence Band - HARSample AutocorrelationConfidence Band - Stata FormulaConfidence Band - Bartlett Formula0.02.55.07.510.012.515.017.520.0k (Lag)1.000.750.500.250.000.250.500.751.00Estimated Autocorrelation(b) S&P 500 Index Absolute Returns, February 13, 2020 to September 28, 2023 (post-Covid)Estimated AutocorrelationConfidence Interval - HARConfidence Band - HARSample AutocorrelationConfidence Band - Stata FormulaConfidence Band - Bartlett FormulaCHAPTER 3

SOME FIXED-𝑏 RESULTS FOR REGRESSIONS WITH HIGH FREQUENCY DATA
OVER LONG SPANS
(CO-AUTHORED WITH TIM VOGELSANG)

3.1

Introduction

This paper develops fixed-𝑏 asymptotic results for heteroskedasticity autocorrelation robust

(HAR) Wald statistics for regressions with high frequency data in a continuous time framework. Our

results are obtained within the theoretical framework developed by Chang et al. (2023) (hereafter

CLP). Our results complement and extend the analysis in CLP. Our results are related to, and

complement, recent work by Pellatt and Sun (2023) who focus on orthonormal series estimators

of long run variances and develop fixed-smoothing asymptotic results for corresponding HAR

statistics.

Motivated by high frequency data, CLP investigate the asymptotic properties of HAR Wald

tests in a regression model where the observed discrete time series data is generated by an un-

derlying continuous time model. Focusing on consistency/inconsistency of kernel based long run

variance estimators, CLP show that HAR Wald statistics can diverge to infinity under some high

frequency conditions, but this spuriousness can disappear when using data-dependent bandwidth

selection methods compatible to high frequency data. In particular, CLP conclude that the An-

drews (1991) data dependent approach works more reliably with high frequency data than the

Newey and West (1994) data dependent approach. While suggestive of finite sample properties,

consistency/inconsistency of a long-run variance estimator only partially reflects the impact of the

bandwidth/kernel on the sampling distribution of the HAR test statistic. In contrast, the fixed-𝑏

approach of Kiefer and Vogelsang (2005) more fully captures the impact of the bandwidth/kernel

on the first order asymptotic distribution of the HAR test statistic.

This chapter is based on the published paper: Hwang, Taeyoon and Vogelsang, Timothy J, "Some fixed-b results
for regressions with high frequency data over long spans" Journal of Econometrics, 2024, forthcoming. DOI Link:
https://doi.org/10.1016/j.jeconom.2024.105773. The co-author has approved that the co-authored chapter is included.
The co-author’s contact: Tim Vogelsang, Department of Economics, 486 W. Circle Drive, 110 Marshall-Adams Hall,
Michigan State University East Lansing, MI 48824-1038. email: tjv@msu.edu

72

In this paper we obtain fixed-𝑏 asymptotic results for the statistics analyzed by CLP using

the same continuous time framework. We find that the fixed-𝑏 limits of the HAR Wald tests

in stationary high frequency regressions estimated by ordinary least squares (OLS) are the same

as the standard fixed-𝑏 limits in Kiefer and Vogelsang (2005). For cointegrating high frequency

regressions the fixed-𝑏 limits generally have non-pivotal limits. However, for the special case

where the stochastic processes in the continuous time regression follow Brownian motions and the

regressors are independent of the errors, the fixed-𝑏 limits are pivotal and are the same as those

obtained by Bunzel (2006) in discrete time settings. For the case of cointegration with endogeneity,

we analyze the integrated modified OLS (IM-OLS) estimator of Vogelsang and Wagner (2014) and

an associated test that is asymptotically pivotal under fixed-𝑏 asymptotics. We find that the fixed-𝑏

limit in the CLP high frequency setting is the same as that obtained by Vogelsang and Wagner

(2014). Using the language of CLP, we can say that fixed-𝑏 critical values are high frequency

compatible.

When fixed-𝑏 limits are pivotal with respect to serial correlation nuisance parameters but depend

on the bandwidth and kernel, the use of fixed-𝑏 critical values rather than chi-square critical values

is expected to improve inference regardless of the method used to obtain the bandwidth. See,

for example, Kiefer and Vogelsang (2005) and Lazarus et al. (2018) for simulation evidence and

Gonçalves and Vogelsang (2011), Lazarus et al. (2021), and Sun et al. (2008) for theoretical and

simulation evidence.

We assess the performance of fixed-𝑏 critical values using a simulation study using the same

data generating process (DGP) as CLP. Consistent with the existing fixed-𝑏 literature, we find the

use of fixed-𝑏 critical values systematically performs better than chi-square critical values regardless

of the method used to choose the bandwidth. We extend the simulation results of CLP by reporting

results for additional persistence parameters for the Ornstein-Uhlenbeck process (OU process) used

to generate the data. As in CLP we compare/contrast the Andrews (1991) and Newey and West

(1994) data dependent methods. We also include the data dependent method proposed by Sun

et al. (2008) where the bandwidth choice minimizes a weighted average of type I and type II errors.

73

Similar to CLP we find that the Andrews (1991) bandwidth performs reliably with respect to the

frequency of observations especially when fixed-𝑏 critical values are used. The performance of the

Newey and West (1994) bandwidth depends critically on the choice of pre-tuning parameters with

some choices leading to severe over-rejections with high frequency data while other choices leading

to better, but not fully satisfactory, performance. The Sun et al. (2008) bandwidth performance is

similar to the Andrews (1991) bandwidth with less over-rejections when fixed-𝑏 critical values are

used. The continuous rule of thumb (CRT) bandwidth rule suggested by CLP tends to over-reject

more substantially than the Andrews (1991) and Sun et al. (2008) especially when the data has

strong persistence.

While the Andrews (1991), Sun et al. (2008) and CRT bandwidths tend to perform well at high

frequencies (assuming the persistence in the data is not too strong relative to the span), we find

that null rejections are remarkably stable across sampling frequencies ranging from high to low.

This stability holds for data with strong and mild persistence. We show that the source of this

stability in null rejections is stability in bandwidth sample size ratios (𝑏-values) across sampling

frequencies. This stability holds by construction for the CRT bandwidth. It is more surprising this

stability holds for the Andrews (1991) and Sun et al. (2008) bandwidths, and we provide a simple

theoretical explanation.

We also report some power results in our simulations and find that for persistent series, power

is stable across sampling frequencies. In contrast, power falls as sampling frequency decreases

for mildly persistent series. Therefore, we can recommend that practitioners use data sampled at

higher frequencies.

It is important to compare and contrast our analysis with Pellatt and Sun (2023). Both analyses

obtain fixed-smoothing results for HAR tests in high frequency settings and provide reference

distributions for critical values that improve finite sample inference relative to using standard

critical values. In the stationary case we focus on kernel based tests whereas Pellatt and Sun (2023)

focus on orthonormal series based tests.

In cointegration settings we obtain results for kernel

based tests for both OLS and IM-OLS estimators whereas Pellatt and Sun (2023) obtain results

74

for orthonormal series based tests for estimators based on the orthonormal series transformation

proposed by Hwang and Sun (2018). Because we focus on kernel long run variance estimators, we

are able to provide some useful and interesting results on bandwidth choice that further refine CLP’s

results on high-frequency compatible bandwidths. Taken together, our analysis and that of Pellatt

and Sun (2023) provide a useful set of fixed-smoothing results for inference in the high frequency

regression setting of CLP. That we and Pellatt and Sun (2023) find that methods originally proposed

for discrete settings can be applied in the CLP high frequency setting in exactly the same way with

existing fixed-smoothing reference distributions is a positive contribution for empirical practice

and allows empirical researchers to use high or low frequency data for HAR inference with one set

of methods.

The rest of the paper is organized as follows.

In section 3.2, the model is given, and the

continuous time framework of CLP is described. Section 3.3 reviews standard fixed-𝑏 asymptotic

theory for HAR tests in stationary regressions and then provides fixed-𝑏 results for high frequency

asymptotics using the continuous time framework of CLP. Results are provided for kernel based

tests using OLS and, in the case of cointegration, IM-OLS. Because of the nonstandard form of

fixed-𝑏 asymptotic distributions, Section 3.4 describes numerical methods based on simulations

that are used to obtain critical value functions used for the finite sample simulations. Section 3.5

provides some finite sample simulation results. Section 3.6 has an illustrative empirical application

for simple regressions used to test the uncovered interest parity condition. Section 3.7 gives some

concluding remarks. Proofs are given in the Appendix.

3.2 Model

We focus on the model and setup used by CLP. Consider the continuous time regression model

𝑌𝑡 = 𝑋′

𝑡 𝛽 + 𝑈𝑡,

(3.1)

where 0 ≤ 𝑡 ≤ 𝑇, 𝑇 is the span (e.g. number of years), 𝛽 is a 𝑘 × 1 vector of parameters, 𝑋𝑡 is a

𝑘 × 1 vector of continuous time processes, and 𝑈𝑡 is a scalar continuous time process.

Following CLP the continuous time model can be discretized as follows. Suppose data is

sampled at discrete time periods with 𝛿 denoting the time interval between discrete observations.

75

Letting 𝑖 = 1, 2, . . . , 𝑛 index the discrete observations, the link between the discrete observations

and the underlying continuous time processes is given by

𝑦𝑖 = 𝑌𝑖𝛿 and 𝑥𝑖 = 𝑋𝑖𝛿,

where 𝑖𝛿 is the time, 𝑡, at which discrete observation, 𝑖, is observed. Because 𝑡 ∈ [0, 𝑇], it follows

that 𝑛𝛿 = 𝑇. Thus, 𝑦𝑖 and 𝑥𝑖 are discrete sample paths observed at time intervals 𝛿 from 𝑋𝑡 and 𝑌𝑡

respectively. The sampling frequency is inversely related to 𝛿.

We can write the discrete time regression analogous to (3.1) using 𝑥𝑖 and 𝑦𝑖 as

𝑦𝑖 = 𝑥′

𝑖 𝛽 + 𝑢𝑖.

(3.2)

where 𝑢𝑖 = 𝑈𝑖𝛿. Suppose (3.2) is estimated by ordinary least squares (OLS):

(cid:98)𝛽 =

(cid:32) 𝑛
∑︁

𝑖=1

(cid:33) −1 𝑛
∑︁

𝑥𝑖𝑥′
𝑖

𝑥𝑖 𝑦𝑖,

𝑖=1

and we are interested in testing linear hypotheses about 𝛽 of the form

𝐻0 : 𝑅𝛽 = 𝑟, 𝐻1 : 𝑅𝛽 ≠ 𝑟,

where 𝑅 is a known 𝑞 × 𝑘 matrix with rank 𝑞 and 𝑟 is a known 𝑞 × 1 vector. Following CLP we focus

on two heteroskedasticity autocorrelation robust (HAR) Wald statistics. The first Wald statistic is

appropriate for certain cointegration regressions and is given by

𝐺 ( (cid:98)𝛽) =

(cid:16)

𝑅 (cid:98)𝛽 − 𝑟

(cid:33) −1

𝑥𝑖𝑥′
𝑖

𝑅′

𝑛𝑅

(cid:17)′ 

𝜔2

(cid:98)




(cid:32) 𝑛
∑︁

𝑖=1

−1








(cid:16)

𝑅 (cid:98)𝛽 − 𝑟

(cid:17)

,

𝑛 is an estimator of the long run variance of 𝑢𝑖. The second Wald statistic is appropriate

𝜔2
where (cid:98)
for stationary regressions and is given by

𝐻 ( (cid:98)𝛽) = (𝑅 (cid:98)𝛽 − 𝑟)′

(cid:32) 𝑛
∑︁

𝑅

𝑖=1








(cid:33) −1

𝑥𝑖𝑥′
𝑖

𝑛(cid:98)Ω𝑛

(cid:33) −1

𝑥𝑖𝑥′
𝑖

𝑅′

(cid:32) 𝑛
∑︁

𝑖=1

−1








(𝑅 (cid:98)𝛽 − 𝑟),

where (cid:98)Ω𝑛 is an estimator of long run variance of 𝑥𝑖𝑢𝑖. For the case of data sampled at a given
frequency, the asymptotic properties (as 𝑇, 𝑛 → ∞, 𝛿 fixed) of these Wald statistics are well studied

76

𝜔2
in the literature. A key contribution of CLP is the analysis of these Wald statistics when (cid:98)
(cid:98)Ω𝑛 are kernel long run variance estimators where the time interval between observations shrinks

𝑛 and

with the sample size, i.e. 𝛿 → 0 as 𝑇, 𝑛 → ∞.

In this “high frequency” asymptotic setting,

𝜔2
𝜔2
𝑛 and (cid:98)Ω𝑛 under which (cid:98)
CLP establish conditions for the bandwidths of (cid:98)
estimators leading to asymptotically valid inference using the Wald statistics. In particular, CLP

𝑛 and (cid:98)Ω𝑛 are consistent

𝜔2
show that the parametric plug-in data dependent bandwidth rule of Andrews (1991) ensures (cid:98)
(cid:98)Ω𝑛 are consistent in the high frequency asymptotics case. In contrast, the non-parametric plug-in

𝑛 and

𝜔2
data dependent bandwidth rule of Newey and West (1994) results in (cid:98)
because the bandwidths are too small in the high frequency asymptotics case.

𝑛 and (cid:98)Ω𝑛 being inconsistent

Here we explore a related but different question.

If the bandwidths are modeled as a fixed

𝜔2
proportion of the sample size (i.e. the fixed-𝑏 asymptotics nesting is used for (cid:98)
fixed-𝑏 limits in the CLP high frequency asymptotic setting the same as the well known limits for

𝑛 and (cid:98)Ω𝑛), are the

the fixed sampling frequency case (𝛿 fixed)? As will be shown, the answer is yes if the assumptions

in the CLP framework are slightly strengthened to be analogous to assumptions used in discrete

(𝛿 fixed) settings. This suggests that fixed-𝑏 critical values can be used to improve inference for

sampling frequencies that range from low to high.

3.3 Theory

3.3.1 Fixed-b Theory for Discrete Stationary Regressions

Our starting point is a review of fixed-𝑏 theory for the 𝐻 ( (cid:98)𝛽) statistic in stationary regressions
for a given sampling frequency (𝛿 fixed) as developed by Kiefer and Vogelsang (2005). Let Ω𝑣

denote the long run variance of 𝑣𝑖 = 𝑥𝑖𝑢𝑖 defined as

Ω𝑣 = Γ𝑣 (0) +

∞
∑︁

𝑗=1

(cid:0)Γ𝑣 ( 𝑗) + Γ𝑣 ( 𝑗)′(cid:1) ,

where Γ𝑣 ( 𝑗) = 𝐸 (𝑣𝑖𝑣′

𝑖− 𝑗 ). Let Λ𝑣 denote the matrix square root of Ω𝑣, i.e. Ω𝑣 = Λ𝑣Λ𝑣′. The

kernel based nonparametric estimator of Ω𝑣 is given by

𝑛 = (cid:98)Γ𝑣
(cid:98)Ω𝑣

𝑛 (0) +

𝑛−1
∑︁

𝑗=1

(cid:19) (cid:16)

𝑘

(cid:18) 𝑗
𝑀𝑛

𝑛 ( 𝑗) + (cid:98)Γ𝑣
(cid:98)Γ𝑣

𝑛 ( 𝑗)′(cid:17)

,

77

where (cid:98)Γ𝑣

𝑛 ( 𝑗) = 𝑛−1 (cid:205)𝑛

𝑣′
𝑣𝑖(cid:98)
𝑖− 𝑗 and (cid:98)
𝑖= 𝑗+1 (cid:98)

𝑣𝑖 = 𝑥𝑖(cid:98)

𝑢𝑖 = 𝑥𝑖

(cid:16)

𝑦𝑖 − 𝑥′

(cid:17)
𝑖 (cid:98)𝛽

.

Fixed-𝑏 asymptotic results are obtained using an asymptotic nesting for the bandwidth, 𝑀𝑛, such

that 𝑀𝑛 = 𝑏𝑛 where 𝑏 ∈ (0, 1] is held constant as the sample size, 𝑛, grows. With the frequency

of observation held fixed (equivalent to 𝛿 held fixed), standard fixed-𝑏 theory applies under two

sufficient assumptions. We use the symbol ⇒ to denote weak convergence in distribution.

Assumption 3.1 (a) 𝑛−1 (cid:205)[𝑟𝑛]
𝑖=1
𝑛−1/2 (cid:205)[𝑟𝑛]
𝑡=1

𝑥𝑖𝑥′
𝑖

𝑝
→ 𝑟𝑄, where 𝑟 ∈ [0, 1] and 𝑄−1 exists, and (b) 𝑛−1/2 (cid:205)[𝑟𝑛]
𝑡=1

𝑥𝑖𝑢𝑖 =

𝑣𝑖 ⇒ Λ𝑣𝑊𝑘 (𝑟), where 𝑟 ∈ [0, 1], and 𝑊𝑘 (𝑟) is a 𝑘 × 1 vector of independent Wiener

processes, 𝑊𝑘 (𝑟) ∼ 𝑁 (0, 𝑟 𝐼𝑘 ).

Kiefer and Vogelsang (2005) show that under Assumption 3.1 the fixed-𝑏 limit of (cid:98)Ω𝑣

𝑛 is given

by

𝑛 ⇒ Λ𝑣 𝑃𝑘 (𝑏)Λ𝑣′,
(cid:98)Ω𝑣

where 𝑃𝑘 (𝑏) is a stochastic process that is a function of the Brownian bridge, 𝐵𝑘 (𝑟) = 𝑊𝑘 (𝑟) −

𝑟𝑊𝑘 (1), where the form of 𝑃𝑘 (𝑏) depends on the kernel, 𝑘 (𝑥). Relevant to our simulations is the

case where 𝑘 (𝑥) is the Parzen kernel and 𝑃𝑘 (𝑏) is given by

𝑃𝑘 (𝑏) = −

∬

|𝑟−𝑠|<𝑏

1
𝑏2

𝑘′′ (cid:16) 𝑟 − 𝑠
𝑏

(cid:17)

𝐵𝑘 (𝑟)𝐵𝑘 (𝑠)′𝑑𝑟𝑑𝑠,

(3.3)

where 𝑘′′(𝑥) is the second derivative of

𝑘 (𝑥) =






1 − 6𝑥2 + 6 |𝑥|3

for

|𝑥| ≤ 1
2

2 (1 − |𝑥|)3

for

1

2 ≤ |𝑥| ≤ 1

0 for

|𝑥| > 1.

For the case of the widely used Bartlett kernel

𝑃𝑘 (𝑏) =

2
𝑏

∫ 1

0

𝐵𝑘 (𝑟)𝐵𝑘 (𝑟)′𝑑𝑟 −

1
𝑏

∫ 1−𝑏

0

𝐵𝑘 (𝑟)𝐵𝑘 (𝑟 + 𝑏)′𝑑𝑟 −

1
𝑏

∫ 1−𝑏

0

𝐵𝑘 (𝑟 + 𝑏)𝐵𝑘 (𝑟)′𝑑𝑟. (3.4)

See Kiefer and Vogelsang (2005) for details. The fixed-𝑏 limit of 𝐻 ( (cid:98)𝛽) is given by

𝐻 ( (cid:98)𝛽) ⇒ 𝑊𝑞 (1)′𝑃𝑞 (𝑏)−1𝑊𝑞 (1).

78

This limiting random variable is a function of a 𝑞 × 1 vector of standard Wiener processes and

depends on the kernel (through 𝑃𝑞 (𝑏)) and the bandwidth (through 𝑏) but is otherwise pivotal.

Similar, although different, results are obtained for 𝐺 ( (cid:98)𝛽) in cointegrated regressions. See Bunzel

(2006) for details.

It is not obvious whether the fixed-𝑏 results for 𝐺 ( (cid:98)𝛽) and 𝐻 ( (cid:98)𝛽) in the fixed sampling frequency

case continue to hold in the high frequency asymptotic framework of CLP. Using the theoretical

tools of CLP, we obtain fixed-𝑏 results in the high frequency setting.

3.3.2 Fixed-b High Frequency Asymptotics for Stationary Regressions

In this section we obtain fixed-𝑏 asymptotic results for the 𝐻 ( (cid:98)𝛽) statistic in the high frequency

framework of CLP using slightly strengthened assumptions from CLP appropriate for stationary

regressions. Following CLP, let 𝑍 (equivalently 𝑍𝑡) denote a continuous time stochastic process

and assume that 𝑍 = 𝑍 𝑐 + 𝑍 𝑑 such that 𝑍 𝑐 is the continuous component and 𝑍 𝑑 is a jump component

defined as 𝑍 𝑑

𝑡 = (cid:205)

0≤𝑠≤𝑡 Δ𝑍𝑠 where Δ𝑍𝑡 = 𝑍𝑡 − 𝑍𝑡−. We assume a version of Assumption D1 from

CLP holds, and we assume that a version of Lemma 3.1 from CLP holds for partial sums:

Assumption 3.2 Defining 𝑍 = 𝑋 𝑋′ or 𝑋𝑈 and 𝑧𝑖 = 𝑍𝑖𝛿 for 𝑖 = 1, . . . , 𝑛, suppose that for 𝑟 ∈ (0, 1]

1
𝑛

[𝑟𝑛]
∑︁

𝑖=1

𝑧𝑖 =

1
𝑇

∫ 𝑟𝑇

0

𝑍𝑡 𝑑𝑡 + 𝑂 𝑝 (Δ𝛿,𝑟𝑇 (∥𝑍 ∥)) ,

for all small 𝛿 and large 𝑇 where Δ𝛿,𝑟𝑇 (∥𝑍 ∥) = sup0≤𝑠,𝑡≤𝑟𝑇 sup|𝑡−𝑠|≤𝛿

(cid:13)
(cid:13)𝑍 𝑐

𝑡 − 𝑍 𝑐
𝑠

(cid:13)
(cid:13) and ∥·∥ is the

Euclidean norm.

Assumption 3.3 Δ𝛿,𝑇 (∥ 𝑋 𝑋′∥) → 0 and

√

𝑇Δ𝛿,𝑇 (∥ 𝑋𝑈 ∥) → 0 as 𝛿 → 0 and 𝑇 → ∞.

Assumptions 3.2 and 3.3 allow sample moments to approximate continuous time analogs. CLP

argue that Assumption 3.3 is not particularly strong, nor is Assumption 3.6 given below (equivalent

to Assumption D2 in CLP)1. The next assumption is sufficient to obtain continuous time fixed-𝑏

results and is equivalent to Assumption C1 of CLP strengthened to hold for partial sums.

1In the discussion after introducing Assumptions D1 and D2, CLP point out that these assumptions allow the
continuous time processes to be Brownian motions but also allow the processes to have more local volatility and be
more explosive globally. CLP argue that 𝛿 generally needs to go to 0 faster than 𝑇 goes to ∞ and that the relative rate
would depend on how locally volatile and explosive the processes are.

79

Assumption 3.4 We assume that for 𝑟 ∈ (0, 1]: (a)

∫ 𝑟𝑇

𝑇 −1

0

𝑋𝑡 𝑋′

𝑡 𝑑𝑡

𝑝
→ 𝑟𝑄,

as 𝑇 → ∞ for some nonrandom matrix 𝑄 > 0, and (b)

𝑇 −1/2

∫ 𝑟𝑇

0

𝑋𝑡𝑈𝑡 𝑑𝑡 ⇒ Λ𝑊𝑘 (𝑟) ∼ 𝑁 (0, 𝑟Ω),

as 𝑇 → ∞ where Ω is the long run variance of 𝑋𝑡𝑈𝑡.

Assumption 3.4 is effectively a continuous time analog to Assumption 3.1 and is slightly stronger

than Assumption C1 of CLP. It rules out certain nonstationary behavior for 𝑋𝑡 and 𝑋𝑡𝑈𝑡. Note that

Assumption 3.4(b) is a continuous time functional central limit theorem (invariance principle) for

𝑋𝑡𝑈𝑡 whereas Assumption C1(b) of CLP is a continuous time central limit theorem. An analogous

condition to Assumption 3.4(b) was used by Lu and Park (2019) (their equation (2)) to obtain

a continuous time fixed-𝑏 result for kernel long run variance estimators applied to a vector of

continuous time processes known to be mean zero. The fixed-𝑏 results of Lu and Park (2019)

cannot be directly applied to regression settings because of need to estimate 𝛽 when constructing
𝑥𝑖(cid:98)
𝑢𝑖. This is equivalent to having to estimate an unknown mean before estimating a long run
variance. This changes the fixed-𝑏 limit compared to the known mean case - see Hashimzade and

Vogelsang (2008) for details.

Using Assumptions 3.2, 3.3, and 3.4, the following theorem holds for the 𝐻 ( (cid:98)𝛽) statistic.

Theorem 3.1 Let 𝑀𝑛 = 𝑏𝑛 where 𝑏 ∈ (0, 1] is fixed. Assume 𝐻0 : 𝑅𝛽 = 𝑟 holds. Then, under
Assumptions 3.2, 3.3, and 3.4, as 𝛿 → ∞ and 𝑇 → ∞, 𝐻 ( (cid:98)𝛽) ⇒ 𝑊𝑞 (1)′𝑃𝑞 (𝑏)−1𝑊𝑞 (1).

The proof is given in the appendix. Theorem 3.1 shows that the fixed-𝑏 limit of 𝐻 ( (cid:98)𝛽) in the

high frequency asymptotic framework of CLP is the same as in the fixed sampling frequency case

as long as the assumptions used by CLP are strengthened to hold for partial sums. Critical values

of the limiting distribution depends on the kernel and bandwidth sample size ratio, 𝑏, but otherwise

are pivotal. Therefore, critical values are easily obtained using simulation methods.

80

Next we analyze both of the 𝐺 ( (cid:98)𝛽) and 𝐻 ( (cid:98)𝛽) statistics under assumptions in CLP suited for

cointegrating regressions.

3.3.3 Fixed-b High Frequency Asymptotics for Cointegrating Regressions

As is well known in the literature, fixed-𝑏 limits of HAR statistics depend on the stationarity

properties of 𝑋𝑡, 𝑈𝑡 and 𝑋𝑡𝑈𝑡. Cointegrating regression corresponds to the case where 𝑋𝑡 is a

Brownian motion (continuous time unit root process) and 𝑈𝑡 is stationary. We consider the case
where 𝑋𝑡 includes an intercept and write 𝑋𝑡 = [1 (cid:101)𝑋𝑡]′, and its discretized version 𝑥𝑖 as 𝑥𝑖 = [1 (cid:101)
where (cid:101)𝑋𝑡 and (cid:101)
to obtain fixed-𝑏 results for cointegrating regression. The space of cadlag functions is denoted by

𝑥𝑖 are (𝑘 − 1) × 1 vectors. Two versions of assumptions used by CLP are sufficient

𝑥𝑖]′,

𝐷 [0, 1].

Assumption 3.5 (CLP Assumption C2) Assume that (a) for 𝑋𝑇 (𝑟) defined as 𝑋𝑇 (𝑟) = Λ−1

𝑇 𝑋𝑟𝑇 on

[0,1] with an appropriate nonsingular normalizing sequence (Λ𝑇 ) of matrices, it follows that

𝑋𝑇 (𝑟) ⇒ 𝑋 ◦(𝑟),

in the product space of 𝐷 [0, 1] as 𝑇 → ∞ with linearly independent limit process 𝑋 ◦(𝑟), and (b)
if we define 𝑆𝑇 (𝑟) on [0, 1] as 𝑆𝑇 (𝑟) = 𝑇 −1/2 ∫ 𝑟𝑇

𝑈𝑠𝑑𝑠 then

0

𝑆𝑇 (𝑟) ⇒ 𝑈◦(𝑟),

in 𝐷 [0, 1] jointly with 𝑋𝑇 (𝑟) ⇒ 𝑋 ◦(𝑟) in the product space of 𝐷 [0, 1] as 𝑇 → ∞, where 𝑈◦(𝑟)

is a Brownian motion with 𝑈◦(𝑟) = 𝜆𝑢𝑤𝑢 (𝑟) where 𝜆2

𝑢 = lim𝑇→∞ 𝑇 −1𝐸 (∫ 𝑇

0

𝑈𝑡 𝑑𝑡)2 > 0, which is

assumed to exist. 𝑤𝑢 (𝑟) is a standard Wiener process.

Assumption 3.6 (CLP Assumption D2 (modified)) Assume (a) ∥Λ𝑇 ∥2Δ𝛿,𝑇 (∥ 𝑋 𝑋′∥) → 0,
√

√

𝑇 ∥Λ𝑇 ∥Δ𝛿,𝑇 (∥ 𝑋𝑈 ∥) → 0 and (b)

𝑇Δ𝛿,𝑇 (∥𝑈 ∥) → 0, ∥Λ𝑇 ∥Δ𝛿,𝑇 (∥ 𝑋 ∥) → 0 as 𝛿 → 0 and

𝑇 → ∞.

Because we assume the first element of 𝑋𝑡 is an intercept variable, the first element of 𝑋 ◦ is the

identity function. As pointed out in CLP in the discussion of their Assumptions C1 and C2, the

81

random components of 𝑋 ◦ can be general diffusion processes; see also Kim and Park (2017). The

classic cointegration case is obtained when the random components of 𝑋 ◦ are Brownian motions.

Assumption 3.5(b) assumes a continuous time functional central limit theorem holds for the partial

integrals of 𝑈𝑡. The limiting Brownian motion, 𝑈◦, can be correlated with 𝑋 ◦. A continuous time

version of cointegrating regression without endogeneity between the regressors and error holds for

the special case where 𝑋 ◦ is a Brownian motion that is independent of 𝑈◦. We also extend the list

of 𝑍 processes in Assumption 3.2 (labeled 3.2∗) to include the processes in Assumption 3.6 (b).

The next theorem gives fixed-𝑏 results for the 𝐺 ( (cid:98)𝛽) and 𝐻 ( (cid:98)𝛽) statistics under Assumptions 3.5
and 3.6 (CLP Assumptions C2,D2). The limits depend on a 𝑞 × 𝑘 matrix 𝑅∗ that depends on the

form of Λ𝑇 that is defined as follows. Suppose there exists a 𝑞 × 𝑞 nonsingular scaling matrix, Λ𝑅
𝑇 ,

such that lim𝑇→∞ Λ𝑅

𝑇 𝑇 −1/2𝑅Λ−1

𝑇 exists and is a matrix with rank equal to 𝑞. Then define

𝑅∗ = lim
𝑇→∞

Λ𝑅

𝑇 𝑇 −1/2𝑅Λ−1
𝑇 .

(3.5)

When the null hypothesis depends on estimated parameters that converge at the same rate, it will be

the case that 𝑅∗ = 𝑅. However, for a row of 𝑅 that corresponds to a null hypothesis that is a linear

combination of estimated parameters that converge at different rates, the corresponding row of 𝑅∗

will have nonzero elements corresponding to the estimated parameters in that linear combination

that converge the slowest.

Theorem 3.2 Let 𝑀𝑛 = 𝑏𝑛 where 𝑏 ∈ (0, 1] is fixed. Assume 𝐻0 : 𝑅𝛽 = 𝑟 holds. Under

Assumptions 3.2∗, 3.5 and 3.6, as 𝛿 → 0 and 𝑇 → ∞,

𝐺 ( (cid:98)𝛽) ⇒ (𝑅∗𝐶)′ (cid:2)𝑃𝐺 (𝑏)𝑅∗𝑄−1

◦ 𝑅∗′(cid:3) −1 𝑅∗𝐶,

𝐻 ( (cid:98)𝛽) ⇒ (𝑅∗𝐶)′ (cid:2)𝑅∗𝑄−1

◦ 𝑃𝐻 (𝑏)𝑄−1

◦ 𝑅∗′(cid:3) −1 𝑅∗𝐶,

where

𝑄◦ =

∫ 1

0

𝑋 ◦(𝑠) 𝑋 ◦(𝑠)′𝑑𝑠, 𝐶 = 𝑄−1
◦

∫ 1

0

𝑋 ◦ (𝑠) 𝑑𝑤𝑢 (𝑠),

𝑃𝐺 (𝑏) is a function of

𝐵𝐺 (𝑟) = 𝑤𝑢 (𝑟) −

(cid:18)∫ 𝑟

𝑋 ◦(𝑠)′𝑑𝑠

(cid:19)

𝐶,

0

82

and 𝑃𝐻 (𝑏) is a function of

𝐵𝐻 (𝑟) =

∫ 𝑟

0

𝑋 ◦(𝑠)𝑑𝑤𝑢 (𝑠) −

𝑋 ◦(𝑠) 𝑋 ◦(𝑠)′𝑑𝑠

(cid:19)

𝐶,

(cid:18)∫ 𝑟

0

where the forms of 𝑃𝐺 (𝑏) and 𝑃𝐻 (𝑏) are the same as 𝑃𝑘 (𝑏) with 𝐵𝐺 (𝑟) and 𝐵𝐻 (𝑟) in place of

𝐵𝑘 (𝑟).

The proof is given in the appendix.

In general, the fixed-𝑏 limits of 𝐺 ( (cid:98)𝛽) and 𝐻 ( (cid:98)𝛽) given
by Theorem 3.2 are nonpivotal and depend on nuisance parameters related to the structure of 𝑋 ◦

and dependence between 𝑋 ◦ and 𝑈◦.

In the special case where 𝑋 ◦ is a Brownian motion that

is independent of 𝑈◦, the limits in Theorem 3.2 simplify, and are identical to, the fixed-𝑏 limits

obtained by Bunzel (2006) for cointegrating regressions. These limits depend on 𝑏 and the kernel

along with the number of stochastic regressors and the presence of the intercept regressor.

To be concrete, suppose that





𝑥 (𝑟)
𝑥𝑊

Λ

(cid:101)
(cid:101)

𝑥 (𝑟) is a (𝑘 − 1) × 1 vector of independent Wiener processes that are independent of 𝑤𝑢 (𝑟)
where 𝑊
(cid:101)

𝑋 ◦ (𝑟) =









(3.6)

1

,

(𝑈◦). In this case the random part of 𝑋 ◦ is a Brownian motion with long run variance Ω
𝑥Λ′
𝑥 = Λ
𝑥.
(cid:101)
(cid:101)
(cid:101)

Note that in this case the scaling matrix, Λ𝑇 , has the form

Λ𝑇 =

1

01×(𝑘−1)

0(𝑘−1)×1 𝑇 1/2𝐼𝑘−1









.









The following Lemma gives the fixed-𝑏 limits for this special case.

Lemma 3.1 Define

𝑔 (𝑟) =

.

1





𝑥 (𝑟)
𝑊


(cid:101)










For the case where 𝑋 ◦ is given by (3.6), the limits in Theorem 3.2 become

𝑄◦ =

(cid:18)∫ 𝑟

0

∫ 1

0

𝑔(𝑠)𝑔(𝑠)′𝑑𝑠, 𝐶 = 𝑄−1
◦

∫ 1

0

𝑔(𝑠)𝑑𝑤𝑢 (𝑠),

(cid:19)

𝑔(𝑠)′𝑑𝑠

𝐶, 𝐵𝐻 (𝑟) =

∫ 𝑟

0

𝑔(𝑠)𝑑𝑤𝑢 (𝑠) −

𝑔(𝑠)𝑔(𝑠)′𝑑𝑠

(cid:19)

𝐶.

(cid:18)∫ 𝑟

0

𝐵𝐺 (𝑟) = 𝑤𝑢 (𝑟) −

83

Careful examination of 𝐶, 𝐵𝐺 (𝑟), and 𝐵𝐻 (𝑟) in the Lemma reveals that these terms, and the

limits of 𝐺 ( (cid:98)𝛽) and 𝐻 ( (cid:98)𝛽), are pivotal and only depend on 𝑤𝑢 (𝑟) and 𝑔(𝑟) in addition to 𝑏 and the
kernel. For a given 𝑏 and kernel, asymptotic critical values are easily simulated but depend on the

𝑥 (𝑟), the presence of the intercept in 𝑔(𝑟), and 𝑅∗.
dimension of 𝑊
(cid:101)

3.3.4 Fixed-b High Frequency Asymptotics for Cointegrating Regressions with Endogenous

Regressors

For the case of discrete time cointegrating regressions with endogenous regressors, there are

many methods proposed in the literature to obtain asymptotically pivotal test statistics.

In the

continuous time framework, Pellatt and Sun (2023) analyze the discrete time approach of Hwang

and Sun (2018) that uses a transformation of the regression using orthonormal basis functions.

Pellatt and Sun (2023) show that the test statistics proposed by Hwang and Sun (2018) have

the same asymptotic limits in the high frequency setting of CLP. Another approach that delivers

asymptotic pivotal test statistics in discrete time is the integrated modified OLS (IM-OLS) approach

of Vogelsang and Wagner (2014) where kernels are used to estimate relevant long run variance

parameters. Therefore, it is natural to analyze IM-OLS tests in the CLP high frequency setting.

The IM-OLS approach is based on a simple transformation and augmentation of the discrete

time regression (3.2) where we continue to focus on the case where 𝑋𝑡 = [1 (cid:101)𝑋𝑡]′ and 𝑥𝑖 = [1 (cid:101)
Partial summing both sides of (3.2) and including (cid:101)

𝑥𝑖 after partial summing gives

𝑥𝑖]′.

𝑆𝑦
𝑖 = 𝑆𝑥′

𝑖 𝛾 + 𝑆𝑢
𝑥′
𝑖 𝛽 + (cid:101)
𝑖 ,

(3.7)

𝑖 = (cid:205)𝑖

where 𝑆𝑦
𝑥 𝑗 , (cid:101)
parameters. It is convenient to stack 𝑆𝑥

𝑖 = (cid:205)𝑖

𝑦 𝑗 , 𝑆𝑥

𝑗=1

𝑗=1

𝑥𝑖 = (cid:101)
𝑖 and (cid:101)

𝑖 , 𝑆𝑢
𝑥
𝑥𝑖−1 + 𝑣(cid:101)

𝑖 = (cid:205)𝑖
𝑥𝑖 into a single vector 𝑥∗

𝑗=1

𝑢 𝑗 , and 𝛾 is a (𝑘 − 1) × 1 vector of

𝑖 and write (3.7) more compactly

as

where

𝑆𝑦
𝑖 = 𝑥∗′

𝑖 𝜃 + 𝑆𝑢
𝑖 ,

(3.8)

𝑥∗
𝑖 =

𝑆𝑥
𝑖

𝑥𝑖
(cid:101)

















,

𝜃 =

𝛽

𝛾









.









84

The IM-OLS estimator is given by OLS applied to (3.8):

(cid:101)𝜃 =

(cid:32) 𝑛
∑︁

𝑖=1

𝑖 𝑥∗′
𝑥∗
𝑖

(cid:33) −1 𝑛
∑︁

𝑖=1

𝑖 𝑆𝑦
𝑥∗
𝑖 .

The corresponding continuous time regression is given by

where 𝑆𝑌

𝑡 = ∫ 𝑡

0

𝑌𝑠𝑑𝑠, 𝑆𝑈

𝑡 = ∫ 𝑡

0

Wald statistic given by

(cid:101)𝑊 ∗ = (𝑅(cid:101)𝜃 − 𝑟)′

where 𝑐𝑖 = (cid:205)𝑛

𝑗=1

𝑗 − (cid:205)𝑖
𝑥∗

𝑗=1

𝑆𝑌
𝑡 = 𝑋 ∗′

𝑡 𝜃 + 𝑆𝑈
𝑡 ,

𝑈𝑠𝑑𝑠, and 𝑋 ∗

𝑡 =

∫ 𝑡
0

𝑋𝑠𝑑𝑠

(cid:101)𝑋𝑡

















where (cid:101)𝑋𝑡 = ∫ 𝑡

0

𝑥
𝑠 𝑑𝑠. We focus on the
𝑉(cid:101)

(cid:32) 𝑛
∑︁

(cid:33) (cid:32) 𝑛
∑︁

(cid:33) −1 (cid:32) 𝑛
∑︁

𝑖 𝑥∗′
𝑥∗
𝑖



(cid:101)𝜆2∗
𝑥 𝑅

𝑢·(cid:101)



𝑗 . The dimension of the 𝑅 matrix and 𝑟 vector are adjusted with zeros
𝑥∗

(𝑅(cid:101)𝜃 − 𝑟),

𝑖 𝑥∗′
𝑥∗
𝑖

𝑐𝑖𝑐′
𝑖

(3.9)








𝑅′

𝑖=1

𝑖=1

𝑖=1

(cid:33) −1

−1

to accommodate 𝛾 in the model but the restrictions being tested about 𝛽 remain the same. The long

run variance estimator is given by

(cid:101)𝜆2∗
𝑥 = 𝑛−1
𝑢·(cid:101)

𝑛
∑︁

𝑛
∑︁

𝑖=2

𝑗=2

𝑘

(cid:19)

(cid:18) |𝑖 − 𝑗 |
𝑀𝑛

Δ(cid:101)𝑆𝑢∗

𝑖 Δ(cid:101)𝑆𝑢∗
𝑗 ,

where Δ(cid:101)𝑆𝑢∗

𝑖 = (cid:101)𝑆𝑢∗

𝑖 − (cid:101)𝑆𝑢∗

𝑖−1 are the first differences of the OLS residuals, (cid:101)𝑆𝑢∗

𝑖

, from the regression

𝑆𝑦
𝑖 = 𝑆𝑥′

𝑖 𝛾 + 𝑧′
𝑥′
𝑖 𝛽 + (cid:101)

𝑖𝛿 + 𝑆𝑢∗
𝑖

,

(3.10)

where 𝑧𝑖 = 𝑖 (cid:205)𝑛

𝑗=1

𝑗 − (cid:205)𝑖−1
𝑥∗
𝑗=1

(cid:205) 𝑗

𝑚=1

𝑚. The extra 𝑧𝑖 regressors are included to ensure that the fixed-𝑏
𝑥∗

limit of (cid:101)𝑊 ∗ is pivotal. See Vogelsang and Wagner (2014) for details.

To obtain the high frequency fixed-𝑏 limit of (cid:101)𝑊 ∗ we extend Assumptions 3.5 and 3.6 to
accommodate the partials sums and augmented regressors. We focus on the case where 𝑋 ◦ (𝑟) is

given by (3.6) and we write the assumptions in terms of

𝑋 ∗∗

𝑡 =









𝑡 ∫ 𝑇
0

𝑋 ∗
𝑡
𝑠 𝑑𝑠 − ∫ 𝑡
𝑋 ∗

0

(cid:16)∫ 𝑠
0

(cid:17)

𝑑𝑠

𝑋 ∗

𝜈 𝑑𝜈

,









which collects the continuous time variables corresponding to the regressors in (3.10).

85

Assumption 3.7 Λ−1

𝑇 𝑋 ∗∗

𝑟𝑇 ⇒ 𝑋 ◦

∗∗(𝑟) on [0,1] with an appropriate nonsingular normalizing se-

quence (Λ𝑇 ) of matrices where

(cid:16)∫ 𝑠
0

𝑋 𝑜
∗ (𝜈)𝑑𝜈

(cid:17)

𝑑𝑠

, 𝑋 ◦

∗ (𝑟) = Π𝑔∗(𝑟),









𝑋 ◦
∗∗(𝑟) =

Π =

0

1

0

0

𝑟 ∫ 1
0

𝑋 𝑜
∗ (𝑟)
∗ (𝑠)𝑑𝑠 − ∫ 𝑟
𝑋 𝑜


















𝑈𝑠𝑑𝑠 ⇒ 𝑈◦(𝑟) = 𝜆𝑢·(cid:101)

0 Λ
𝑥
(cid:101)

0 Λ
𝑥
(cid:101)












0

0

, 𝑔∗(𝑟) =

𝑟

∫ 𝑟
0

𝑥 (𝑠)𝑑𝑠
𝑊
(cid:101)
𝑥 (𝑟)
𝑊
(cid:101)












,











(cid:17)′

0

and 𝑆𝑇 (𝑟) = 𝑇 −1/2 ∫ 𝑟𝑇

𝑥𝑤𝑢·(cid:101)
𝑥 (𝑟) + Ω𝑢
𝑥
(cid:101)
𝑥 (𝑟), 𝜆2
a scalar standard Wiener process independent of 𝑊
(cid:101)

Λ−1
𝑥
(cid:101)
𝑥 = 𝜆2
𝑢·(cid:101)
𝑥 (𝑟).
𝑥𝑊
𝑥 is the covariance between 𝑈◦(𝑟) and Λ
of 𝑈◦(𝑟), and Ω𝑢
(cid:101)
(cid:101)
(cid:101)

(cid:16)

𝑊
𝑥 (𝑟) = 𝐵𝑢 (𝑟) where 𝑤𝑢·(cid:101)
(cid:101)
𝑥, 𝜆2
𝑥 Ω′
𝑥Ω−1
𝑢 − Ω𝑢
𝑢
(cid:101)
(cid:101)
(cid:101)

𝑢 is the variance

𝑥 (𝑟) is

√
𝑇Δ𝛿,𝑇 (∥𝑈 ∥) → 0, Δ𝛿,𝑇 (∥ (cid:101)𝑋 ∥) → 0,

√

𝑇Δ𝛿,𝑇 (∥𝑉(cid:101)

𝑥 ∥) → 0, ∥Λ𝑇 ∥2Δ𝛿,𝑇 (∥ 𝑋 ∗∗𝑋 ∗∗′ ∥) →

Assumption 3.8
√

0 and

𝑇 ∥Λ𝑇 ∥Δ𝛿,𝑇 (∥ 𝑋 ∗∗𝑆𝑈 ∥) → 0 as 𝛿 → 0 and 𝑇 → ∞.

We also extend the list of 𝑍 processes in Assumption 3.2 (labeled 3.2∗∗) to include the processes

in Assumption 3.8. We can now state the fixed-𝑏 limiting result for the IM-OLS (cid:101)𝑊 ∗ statistic in the

following theorem.

Theorem 3.3 Let 𝑀𝑛 = 𝑏𝑛 where 𝑏 ∈ (0, 1] is fixed. Assume 𝐻0 : 𝑅𝜃 = 𝑟 holds. Under

Assumptions 3.2∗∗, 3.7 and 3.8, as 𝛿 → 0 and 𝑇 → ∞,
𝜒2
𝑞
(𝑏)

(cid:101)𝑊 ∗ ⇒

𝑃∗∗
1

where 𝜒2

𝑞 is a chi-square random variable with 𝑞 degrees freedom independent of 𝑃∗∗

1 (𝑏) where

1 (𝑏) takes the same form as 𝑃1(𝑏) in (3.3) or (3.4) with
𝑃∗∗
(cid:19) −1 ∫ 1

(cid:18)∫ 1

∫ 𝑟

𝐵∗∗
1 (𝑟) =

𝑑𝑤𝑢·(cid:101)

𝑥 (𝑠)−𝑔∗∗ (𝑟)′

𝑔∗∗ (𝑠) 𝑔∗∗ (𝑠)′

0
in place of 𝐵1(𝑟) with

0

0

0

𝑔∗∗(𝑟) =









𝑔∗(𝑟)

𝑟 ∫ 1
0

𝑔∗(𝑠)𝑑𝑠 − ∫ 𝑟

0

(cid:16)∫ 𝑠
0

𝑔∗(𝜈)𝑑𝜈

(cid:17)

𝑑𝑠

.









86

(cid:18)∫ 1

𝑔∗∗ (𝜈) 𝑑𝜈 −

(cid:19)

𝑔∗∗ (𝜈) 𝑑𝜈

∫ 𝑠

0

𝑑𝑤𝑢·(cid:101)

𝑥 (𝑠),

The limit given by Theorem 3.3 is the same as the limit in the discrete time case as obtained by

Vogelsang and Wagner (2014).

3.4 Fixed-b Critical Values

Because fixed-𝑏 asymptotic limits are nonstandard, numerical methods are used to compute

critical values. Here we focus on critical values using the Parzen kernel because the finite sample

simulations use the Parzen kernel to make direct comparisons with CLP. Appendix B of Vogelsang

(2011), the working paper version of Vogelsang (2012), provides a numerical method for compu-

tation of fixed-𝑏 critical values for the Bartlett kernel for stationary regressions. We use the same

method here for the Parzen kernel for the following cases: (i) H-statistic stationary regression,

(ii) G-statistic cointegration regression (without endogeneity) and (iii) H-statistic cointegration

regression (without endogeneity). To align with our simulation results and empirical illustration,

we report critical value functions for a simple regression with an intercept and one regressor for

two hypotheses. The first is a test of the joint null hypothesis that the intercept parameter is zero

(𝛽1 = 0) and the slope parameter is 1 (𝛽2 = 1). The second is a test that the slope parameter is 1. In

the stationary case, the fixed-𝑏 critical values only depend on the number of restrictions, 𝑞 = 2 and

1 respectively. In cointegration regressions, as shown by Propositions 1 and 2 of Bunzel (2006)

and our Lemma 3.1, the fixed-𝑏 critical values also depend on the number of stochastic regressors

in the model, the form of deterministic regressors (the intercept), and 𝑅∗.

Following Vogelsang (2011) let 𝑐𝑣𝛼 (𝑏) denote the critical value for a given statistic for sig-

nificance level 𝛼 using a bandwidth sample size ratio 𝑏. Using 50,000 replications, 𝑐𝑣𝛼 (𝑏) was

simulated using normalized partial sums of independent, identically distributed (i.i.d.) 𝑁 (0, 1)

random variables using 1,000 steps to approximate the Wiener processes in the asymptotic distri-

butions. These simulations were carried out for the values of 𝑏 = 0.02, 0.04, . . . , 0.98, 1.0. Using

87

the simulated critical values, we fit critical value functions of the form:

𝑐𝑣𝛼 (𝑏) = 𝑧2

𝑞,𝛼 + 𝜆1(𝑏 · 𝑧2

𝑞,𝛼) + 𝜆2(𝑏 ·

(cid:17) 2

(cid:16)

𝑧2
𝑞,𝛼

) + 𝜆3(𝑏 ·

(cid:17) 3

(cid:16)

𝑧2
𝑞,𝛼

) + 𝜆4(𝑏2 · 𝑧2

𝑞,𝛼) + 𝜆5(𝑏2 ·

(cid:16)

(cid:17) 2

)

𝑧2
𝑞,𝛼

(3.11)

+ 𝜆6(𝑏2 ·

(cid:17) 3

(cid:16)

𝑧2
𝑞,𝛼

) + 𝜆7(𝑏3 · 𝑧2

𝑞,𝛼) + 𝜆8(𝑏3 ·

(cid:17) 2

(cid:16)

𝑧2
𝑞,𝛼

) + 𝜆9(𝑏3 ·

(cid:16)

(cid:17) 3

),

𝑧2
𝑞,𝛼

where 𝑧2

𝑞,𝛼 is the critical value from a 𝜒2

𝑞 (chi-square with 𝑞 degrees of freedom) random variable.

For 𝐻0 : 𝛽1 = 0, 𝛽2 = 1 we have 𝑞 = 2, and for 𝐻0 : 𝛽2 = 1 we have 𝑞 = 1. Notice that, by

construction, 𝑐𝑣𝛼 (0) = 𝑧2

𝑞,𝛼 so that when 𝑏 = 0 the critical values are chi-square. For a given

statistic, the values of the 𝜆𝑖 coefficients were obtained using least squares. The fits, as measured

by the regression 𝑅2, are excellent in all cases (no smaller than 0.995). Table 3B.1 gives the 𝜆𝑖

coefficients.

As shown by Vogelsang and Wagner (2014) fixed-𝑏 critical values for (cid:101)𝑊 ∗ depend on the number

of integrated regressors, the form of the deterministic regressors and the hypothesis being tested

in addition to the kernel and bandwidth. To test the joint null hypothesis, 𝛽1 = 0 and 𝛽2 = 1,
we simulated fixed-𝑏 critical values for (cid:101)𝑊 ∗ based on the Parzen kernel for testing the joint null

hypothesis in a cointegrating regression with an intercept and one integrated regressor. Then using

the fixed-𝑏 critical values, we fit a critical value function for 𝛼 = 0.05 for (cid:101)𝑊 ∗. The critical value

function is given by

𝑐𝑣0.05(𝑏) =






5.96 + 8.73·𝑏 + 551.46·𝑏2 − 1950.49·𝑏3 + 7145.52·𝑏4, when 𝑏 ≤ 0.2

−770.1 + 16182.2·𝑏 − 144138·𝑏2 + 727975.7·𝑏3 − 2283734.7·𝑏4

+4650336.4·𝑏5 − 6114285.0·𝑏6 + 4986435.8·𝑏7 − 2287834.8·𝑏8 + 450664.3·𝑏9,

when 𝑏 > 0.2

(3.12)

where the fit as measured by the regression 𝑅2 was larger than 0.999.

For testing the slope parameter only, 𝛽2 = 1, using simulated critical values for the case of an

intercept and one integrated regressor provided by the supplementary material of Vogelsang and

88

Wagner (2014), we fit a critical value function for 𝛼 = 0.05 for (cid:101)𝑊 ∗. The critical value function is

given by

𝑐𝑣0.05(𝑏) =






3.84 − 6.61·𝑏 + 930.69·𝑏2 − 12634.81·𝑏3 + 102402.08·𝑏4

−396004.20·𝑏5 + 605085.78·𝑏6, when 𝑏 ≤ 0.2

−41.98 + 1353.9·𝑏 − 15398.9·𝑏2 + 96187.7·𝑏3 − 354315.7·𝑏4

+821676.6·𝑏5 − 1182990.3·𝑏6 + 1018299.0·𝑏7 − 478903.5·𝑏8 + 94608.1·𝑏9,

when 𝑏 > 0.2

(3.13)

where the fit as measured by the regression 𝑅2 was larger than 0.999.

More generally, asymptotic critical values for (cid:101)𝑊 ∗ can be simulated using a simple Monte Carlo
𝑖 𝛽 + 𝑢𝑖, with

simulation procedure2. Suppose an empirical application uses regression (3.2), 𝑦𝑖 = 𝑥′
𝑖 = 1, 2, . . . , 𝑛 where 𝑥𝑖 = [1 (cid:101)
processes. Critical values for (cid:101)𝑊 ∗ for testing a given hypothesis about 𝛽 using a given kernel and
value of 𝑏 can be computed as follows:

𝑥𝑖 is a (𝑘 − 1) × 1 vector of unit root

𝑥𝑖]′ is a 𝑘 × 1 vector and (cid:101)

Step (1) Generate realizations of (cid:101)
𝑥
𝑥
𝑖 where {𝜖(cid:101)
𝑥𝑖−1 + 𝜖(cid:101)

𝑥𝑖 and 𝑢𝑖 using a large value of 𝑛 such as 𝑛 = 1000. The vector
𝑥0
𝑖 } is a sequence of i.i.d. 𝑁 (0, 𝐼𝑘−1) vectors with (cid:101)
𝑥𝑖 is generated as (cid:101)
(cid:101)
equal to a zero vector. Generate a realization of {𝑢𝑖} as a sequence of i.i.d. 𝑁 (0, 1) random

𝑥𝑖 = (cid:101)

𝑥
𝑖 }. Let 𝛽𝑟 denote the 𝛽 vector with the null hypothesis
variables that are independent of {𝜖(cid:101)

imposed. Elements of 𝛽𝑟 that do not involve the null hypothesis can be set to zero without

loss of generality because (cid:101)𝑊 ∗ is exactly invariant to those elements. Using the realizations
𝑥𝑖 = [1 (cid:101)

𝑥𝑖]′ and 𝑢𝑖, a realization of 𝑦𝑖 is calculated as 𝑦𝑖 = 𝑥′

𝑖 𝛽𝑟 + 𝑢𝑖.

Step (2) Using the realized {𝑦𝑖} and {𝑥𝑖} data from Step 1, compute the IM-OLS statistic (cid:101)𝑊 ∗ using
(3.9) with the given kernel and 𝑀𝑛 = 𝑏†𝑛 where 𝑏† is the bandwidth sample size ratio from

the empirical application.

2We thank a referee for suggesting that we include this algorithm in the paper.

89

Step (3) Repeat Steps (1) and (2) many times (for example 10,000 or more), and obtain the (1 − 𝛼)

quantile of the realizations of (cid:101)𝑊 ∗. Use that quantile as the critical value for (cid:101)𝑊 ∗ computed

from the empirical application.

As an example, recall the case of 𝑘 = 2 for testing 𝛽1 = 0 and 𝛽2 = 1 as previously discussed.
𝑥𝑖 is a scalar process generated as (cid:101)
Here (cid:101)
𝑥
𝑖 . The vector 𝛽𝑟 is given by 𝛽𝑟 = [0 1]′. For cases
𝑢𝑖 ∼ 𝑖.𝑖.𝑑.𝑁 (0, 1) that is independent of 𝜖(cid:101)

𝑖 ∼ 𝑖.𝑖.𝑑.𝑁 (0, 1) and (cid:101)

𝑥
𝑥
𝑖 where 𝜖(cid:101)
𝑥𝑖−1 + 𝜖(cid:101)

𝑥0 = 0, and

𝑥𝑖 = (cid:101)

where more complicated deterministic regressors are included in 𝑥𝑖, those regressors would simply

be included in the simulated 𝑥𝑖 vector.

3.5 Simulations

3.5.1 Finite Sample Simulation Environment

In this section we use simulations to explore finite sample properties of 𝐻 ( (cid:98)𝛽) for the stationary
regression case and (cid:101)𝑊 ∗ for the cointegration case with and without endogeneity. Patterns are
similar for the 𝐺 ( (cid:98)𝛽) and 𝐻 ( (cid:98)𝛽) statistics in the cointegration case without endogeneity and are

not reported. We focus on performance of the kernel based tests across sampling frequencies to

explore empirical null rejections, bandwidth behavior, and power across frequencies. We do not

make comparisons across long run variance estimators given that Pellatt and Sun (2023) provide

extensive comparisons between kernel long run variance estimators and orthonormal series long

run variance estimators where overall performance between the two was found to be similar.

We use the same data generating process (DGP) as CLP to facilitate comparisons. The long run

variances of 𝐻 ( (cid:98)𝛽) and (cid:101)𝑊 ∗ are implemented with the Parzen kernel in all cases. We consider five

bandwidth rules. These rules include three of the bandwidth rules used by CLP in their simulations:

the Andrews (1991) AR(1) plug-in rule (AD), the Newey and West (1994) nonparametric plug-in

rule (NW) using the pre-tuning parameters suggested by Newey and West (1994) and the CRT

bandwidth rule proposed by CLP. The other two bandwidth rules are: the Sun et al. (2008) AR(1)

plug-in rule that balances size distortions and power (SPJ), and a variant of the NW rule that uses

different pre-tuning parameters which we label NW-Tune.

Following CLP, we focus on a continuous time regression model with an intercept and one

90

regressor given by

𝑌𝑡 = 𝛽1 + 𝛽2𝑋𝑡 + 𝑈𝑡,

where 𝑋𝑡 now denotes a univariate stochastic regressor. Both 𝑋𝑡 and 𝑈𝑡 are Ornstein–Uhlenbeck

processes given by

𝑑𝑋𝑡 = −𝜅𝑥 𝑋𝑡 𝑑𝑡 + 𝜎𝑥 𝑑𝑉𝑡, 𝑑𝑈𝑡 = −𝜅𝑢𝑈𝑡 𝑑𝑡 + 𝜎𝑢𝑑𝑊𝑡,

𝑉𝑡

𝑊𝑡

















=









1 0
√

𝜋

1 − 𝜋2

𝜉1𝑡

𝜉2𝑡

















,









where 𝜉1𝑡 and 𝜉2𝑡 are standard Brownian motions independent each other. The DGP for 𝑉𝑡 and 𝑊𝑡

is similar to the DGP used by Pellatt and Sun (2023). The variances of 𝑉𝑡 and 𝑊𝑡 are normalized to

be one and 𝜋 is the correlation between 𝑉𝑡 and 𝑊𝑡. In the stationary simulations both 𝜅𝑥 and 𝜅𝑢 are

strictly positive and 𝜋 = 0. In the cointegration case 𝜅𝑥 = 0 and 𝜋 can be non-zero. Using a span of

𝑇 = 30 for a given replication, we generate 7,560 daily (252 weekday observations per year) sample

paths for each OU process. Lower frequency series (such as monthly/quarterly) are constructed

from the generated daily series. We focus on testing the joint null hypothesis, 𝐻0 : 𝛽1 = 0, 𝛽2 = 1,

using a nominal significance level of 0.05. When used, fixed-𝑏 critical values are computed using

the critical value function given by (3.11) using coefficients from the H-stationary line of Table

3B.1 for the stationary regression case. The critical value function given by (3.12) is used for the

cointegrating regression case. We used 2,000 replications in all cases.

We also carried out simulations for the case where 𝑋𝑡 follows Feller’s Square Root (SR) process

as in CLP and Pellatt and Sun (2023) which is given by 𝑑𝑋𝑡 = 𝜅𝑥 (𝜇𝑥 − 𝑋𝑡) 𝑑𝑡 + 𝜎𝑥

√

𝑋𝑡 𝑑𝑉𝑡. The

patterns in the results are very similar to the OU process case and are not reported.

3.5.2 Bandwidth Formulas

To help with interpreting some of the finite sample patterns it is useful to examine some of the

formulas for the bandwidth rules we used. The Andrews (1991) formula for the Parzen kernel is

given by

(cid:98)𝑀 𝐴𝐷

𝑛 = 2.6614

(cid:18)

𝜌2
4(cid:98)
𝜌)4
(1 − (cid:98)

𝑛

(cid:19) 1/5

,

91

where 𝑛 is the sample size of the discretized series and (cid:98)
to (cid:98)
residuals. For the stationary regression we follow Andrews (1991) and place zero weight on the

𝑢𝑖 (cointegrating regression) where (cid:98)

𝑢𝑖 (stationary regression) or (cid:98)

𝜌 is the estimated AR(1) parameter fit

𝑢𝑖 are the OLS

𝑣𝑖 = 𝑥𝑖(cid:98)

𝑣𝑖 = (cid:98)

intercept component of the long-run variance to ensure that the bandwidth rule is invariant to the

scale of the data.

The Newey and West (1994) bandwidth takes the same form as (cid:98)𝑀 𝐴𝐷
𝜌 estimated nonparametrically using kernel estimators that require bandwidths of their own (pre-

on (cid:98)
tuning parameters). Formulas can be found in Newey and West (1994). Newey and West (1994)

𝑛 with the term that depends

recommend the deterministic pre-tuning rule 4(𝑛/100)4/25 and this gives the NW data dependent

bandwidth. We also consider the pre-tuning rule 8.5(𝑛/100)21/25 which gives the NW-Tune data

dependent bandwidth. The formula for the Sun et al. (2008) bandwidth is

(cid:98)𝑀 𝑆𝑃𝐽

𝑛

=





(cid:17) 1/3

𝑛

(cid:16) 2(cid:98)
𝜌𝑐
𝜌)2
(1−(cid:98)
log(𝑛)

𝑤𝐺′

1,0(𝑧2
0.539𝑧2

1,𝛿 (𝑧2
1,𝛼) − 𝐺′
𝛼𝐾𝜏 (𝑧2
𝛼)

1,𝛼)

if

𝜌𝑐
2(cid:98)
𝜌)2
(1−(cid:98)

> 0

otherwise

(cid:17)

, 𝐾𝜏 (𝑥) =

𝛿2
2𝑥

𝐺 ′

3,𝜏 (𝑥),

where

(cid:16)

12

𝑐 =

𝜌 is the same AR(1) estimator used for (cid:98)𝑀 𝐴𝐷

and (cid:98)
(non)central chi-square random variable with 𝑗 degrees of freedom and noncentrality parameter

𝑗,𝜏 (·) is the probability density function of a

, 𝐺′

𝑛

𝜏2, and 𝑧2

1,𝛼 is the critical value from a chi-square random variable with one degree of freedom.
The parameters 𝑤 and 𝜏 control the trade-off between size distortions and power. We use 𝑤 = 10

and 𝜏 = 2.

Finally, the CRT bandwidth of CLP is given by the formula

𝑀𝐶 𝑅𝑇
𝑛

= 𝑛1/5𝛿−4/5 = 𝑛1/5

(cid:19) −4/5

(cid:18)𝑇
𝑛

= 𝑛𝑇 −4/5.

3.5.3 Stationary Regression Results

For the stationary regressions we set 𝜋 = 0 and use two pairs of persistence parameters:

(𝜅𝑥, 𝜅𝑢) = (0.1, 6.9) and (𝜅𝑥, 𝜅𝑢) = (0.5, 0.5) where the first pair is from CLP and has relatively

92

low persistence in 𝑈𝑡. The second pair has similar persistence in 𝑋𝑡 but much more persistence in

𝑈𝑡. We use the same standard deviation parameter values as in CLP: (𝜎𝑥, 𝜎𝑢) = (1.5514, 2.7566).

3.5.3.1 Empirical Null Rejections

Figure 3B.1 plots empirical null rejection probabilities for 𝐻 ( (cid:98)𝛽). Panels (a) and (c) give results
for (𝜅𝑥, 𝜅𝑢) = (0.1, 6.9) (the CLP case) whereas panels (b) and (d) give results for (𝜅𝑥, 𝜅𝑢) =

(0.5, 0.5) (the more persistent case). The top panels give results using chi-square critical values

whereas the bottom panels give results using fixed-𝑏 critical values. The 𝑥-axis is 𝛿 which ranges

from 1/252 (daily frequency) to 1/4 (quarterly frequency). Figure 3B.1(a) essentially replicates

some finite sample results from CLP although rejections using the AD bandwidth are just above

0.1 in contrast to the rejections in CLP where rejections were close to 0.07. Rejections using

the SPJ bandwidth are slightly higher than those with AD, and rejections with CRT are between

them. Rejections with NW are similar for low to medium frequencies and over-rejections occur at

the daily (high) frequency. The alternative version of NW, NW-Tune, tends to over-reject across

all sampling frequencies but does not show the big jump in over-rejection at the daily frequency.

Figure 3B.1(c) shows that, except for NW, rejections are improved (closer to 0.05) when fixed-𝑏

critical values are used. Rejections are below 0.1 in all cases.

Figures 1(b,d) show what happens when the persistence is stronger for the given span. In Figure

3B.1(b) we see that all bandwidths lead to substantial over-rejections when the chi-square critical

value is used. The NW bandwidth continues to over-reject more substantially at high frequencies

consistent with the CLP finding that the NW bandwidth is not high frequency compatible. As

Figure 3B.1(d) shows, using fixed-𝑏 critical values substantially reduces over-rejections for AD and

especially SPJ. Modest improvements are seen for CRT and NW-Tune. Even with the improvements

that the fixed-𝑏 critical values provide, over-rejections remain because the persistence is strong

relative to the span (magnitude of 𝑇).

3.5.3.2 Bandwidth Patterns Across Sampling Frequencies

One pattern that is interesting in all four panels of Figure 3B.1 is that rejections for AD, SPJ

and CRT are stable across frequency of observation especially when fixed-𝑏 critical values are

93

used. To help explain this stability, we computed average bandwidth to sample size ratios (across

replications) denoted by

(cid:98)𝑏 𝐴𝐷 = (cid:98)𝑀 𝐴𝐷

𝑛
𝑛

, (cid:98)𝑏𝑆𝑃𝐽 = (cid:98)𝑀 𝑆𝑃𝐽

𝑛
𝑛

, 𝑏𝐶 𝑅𝑇 =

𝑀𝐶 𝑅𝑇
𝑛
𝑛

,

(and similarly for NW and NW-Tune). Plots of these ratios are given in the two panels of Figure

3B.2. We see that (cid:98)𝑏 𝐴𝐷 and (cid:98)𝑏𝑆𝑃𝐽 are nearly flat across sampling frequencies with slight decreases
at lower frequencies (larger 𝛿). By construction, 𝑏𝐶 𝑅𝑇 is flat across sampling frequencies because

𝑏𝐶 𝑅𝑇 =

𝑀𝐶 𝑅𝑇
𝑛
𝑛

𝑛𝑇 −4/5
𝑛

=

= 𝑇 −4/5,

(3.14)

which is the same for all frequencies for a given value of 𝑇. The bandwidth ratio for NW decreases

as the sampling frequency increases and NW-tune has a similar pattern that is shifted up. The

fact that (cid:98)𝑏 𝐴𝐷, (cid:98)𝑏𝑆𝑃𝐽 and 𝑏𝐶 𝑅𝑇 are stable across frequencies explains why rejections are similar

across frequencies. It is well known from the fixed-smoothing literature that the extent to which

over-rejections occur depends on the bandwidth sample size ratio and not the bandwidth itself.

Another interesting pattern in Figure 3B.1(c,d) is that rejections using SPJ are lower than AD or

CRT when fixed-𝑏 critical values are used. This is obvious in Figure 3B.1(d). Why does SPJ give

rejections closer to the nominal level? From Figure 3B.2(b) we see that (cid:98)𝑏𝑆𝑃𝐽 > (cid:98)𝑏 𝐴𝐷 on average and
both are larger than 𝑏𝐶 𝑅𝑇 . This makes sense because the SPJ bandwidth rule is known to give larger

bandwidths than AD given that SPJ balances size distortions and power rather than minimizing the

mean square error of the variance estimator.

What is not obvious from Figure 3B.2(a,b) is why (cid:98)𝑏 𝐴𝐷 and (cid:98)𝑏𝑆𝑃𝐽 are stable across sampling

frequencies. In the next section we provide some simple theoretical arguments that can help explain

this finite sample pattern.

3.5.3.3 Stability of Bandwidth Sample Size Ratios Across Sampling Frequencies

To understand Figure 3B.2(a,b), some simple theoretical calculations holding the span, 𝑇, fixed

are useful. We hold 𝑇 fixed because the patterns observed in those figures are for a given value of

the span (𝑇 = 30). Recall that the AD and SPJ bandwidths are functions of the estimated AR(1)
parameter of 𝑥𝑖(cid:98)

𝑢𝑖 which is a proxy for 𝑥𝑖𝑢𝑖. Because 𝑥𝑖𝑢𝑖 is the product of two independent discretized

94

OU processes, each of which are AR(1) processes with AR(1) parameters 𝜌𝑥 = exp (−𝜅𝑥 (𝑇/𝑛))

and 𝜌𝑢 = exp (−𝜅𝑢 (𝑇/𝑛)) respectively, 𝑥𝑖𝑢𝑖 has an AR(1) structure with AR(1) parameter

𝜌 = 𝜌𝑥 𝜌𝑢 = exp

(cid:18)

− (𝜅𝑥 + 𝜅𝑢)

(cid:19)

.

𝑇
𝑛

(3.15)

Letting 𝜅𝑥𝑢 = 𝜅𝑥 + 𝜅𝑢 and using (3.15), we can write the AD and SPJ bandwidths as

𝑀 𝐴𝐷

𝑛 = 2.6614

(cid:19) 1/5

(cid:18)

4𝜌2
(1 − 𝜌)4

𝑛

(cid:32)

= 2.6614

𝑀 𝑆𝑃𝐽
𝑛

=

(cid:19) 1/3

(cid:18)

2𝜌𝑐
(1 − 𝜌)2

𝑛

(cid:32)

=

2 exp (cid:0)−𝜅𝑥𝑢
𝑇
𝑛
(1 − exp (cid:0)−𝜅𝑥𝑢

4 exp (cid:0)−2𝜅𝑥𝑢
(1 − exp (cid:0)−𝜅𝑥𝑢
(cid:33) 1/3
(cid:1) 𝑐
(cid:1))2
𝑇
𝑛

𝑛

,

(cid:33) 1/5

,

(cid:1)
(cid:1))4

𝑇
𝑛
𝑇
𝑛

𝑛

giving

𝑏 𝐴𝐷 =

𝑀 𝐴𝐷
𝑛
𝑛

= 2.6614

(cid:32)

4 exp (cid:0)−2𝜅𝑥𝑢
(1 − exp (cid:0)−𝜅𝑥𝑢
2 exp (cid:0)−𝜅𝑥𝑢
𝑇
𝑛
(1 − exp (cid:0)−𝜅𝑥𝑢

(cid:1) 𝑐
(cid:1))2
𝑇
𝑛

1
𝑛2

(cid:1)
𝑇
𝑛
(cid:1))4
𝑇
𝑛
(cid:33) 1/3

(cid:33) 1/5

,

1
𝑛4

,

𝑏𝑆𝑃𝐽 =

(cid:32)

𝑀 𝑆𝑃𝐽
𝑛
𝑛

=

for the bandwidth ratios. Using the expansion

exp

(cid:18)

−𝜅𝑥𝑢

(cid:19)

𝑇
𝑛

= 1 − 𝜅𝑥𝑢 (

𝑇
𝑛

) +

𝑛 ))2

(𝜅𝑥𝑢 ( 𝑇
2!

−

𝑛 ))3

(𝜅𝑥𝑢 ( 𝑇
3!

+ · · ·,

we can easily show (for 𝑇 fixed) that

𝑏 𝐴𝐷 = 2.6614

lim
𝑛→∞

(cid:18)

4
(𝜅𝑥𝑢𝑇)4

(cid:19) 1/5

,

𝑏𝑆𝑃𝐽 =

lim
𝑛→∞

(cid:18)

𝑐
(𝜅𝑥𝑢𝑇)2

(cid:19) 1/3

.

(3.16)

(3.17)

(3.18)

which suggests that, for data sampled at high frequencies, 𝑏 𝐴𝐷 and 𝑏𝑆𝑃𝐽 are positive. The more

persistent the data, i.e. the closer 𝜅𝑥𝑢 is to zero, the larger 𝑏 𝐴𝐷 and 𝑏𝑆𝑃𝐽 will be.

The stability of 𝑏𝐶 𝑅𝑇 across sampling frequencies in Figure 3B.2(a,b) is obvious and expected

given (3.14). Because the formulas for (3.16) and (3.17) are not constant functions with respect

to 𝑛, the large 𝑛 limits of 𝑏 𝐴𝐷 and 𝑏𝑆𝑃𝐽 are only useful in understanding what happens for very

high frequency cases (large 𝑛). To provide a more complete picture, in Figure 3B.3 we plot the

theoretical 𝑏 𝐴𝐷 and 𝑏𝑆𝑃𝐽 functions (3.16) and (3.17) for the case of 𝑇 = 30 for 𝑛 ranging from high

frequencies (daily, on the left) to low frequencies (yearly, on the right). For the high persistence

95

case (𝜅𝑥, 𝜅𝑢) = (0.5, 0.5), 𝑏 𝐴𝐷 and 𝑏𝑆𝑃𝐽 are nearly flat across sampling frequencies especially in

the daily (𝛿 = 1/252) to quarterly (𝛿 = 1/4) range. Interestingly, the stability of 𝑏 𝐴𝐷 and 𝑏𝑆𝑃𝐽 also

holds for lower frequency cases with a slight decline at the annual frequency (𝛿 = 1). In the lower

persistent case (𝜅𝑥, 𝜅𝑢) = (0.1, 6.9) we see that 𝑏 𝐴𝐷 and 𝑏𝑆𝑃𝐽 continue to be flat in the daily to

quarterly range but show noticeable decline at the annual frequency.

The surprising finding that 𝑏 𝐴𝐷 and 𝑏𝑆𝑃𝐽 are stable in the simulations is explained by the relative

flatness of 𝑏 𝐴𝐷 and 𝑏𝑆𝑃𝐽 as functions of the sampling frequency at least for the AR(1) plug-in case.

Because 𝑏 𝐴𝐷 and 𝑏𝑆𝑃𝐽 do not converge to zero as the sampling frequency becomes very high (𝑛

becomes large), 𝑏 𝐴𝐷 and 𝑏𝑆𝑃𝐽 remain “high frequency stable” - a complementary finding to the

high frequency compatible finding of 𝑀 𝐴𝐷

𝑛

by CLP.

3.5.3.4 Finite Sample Power

In this section we report finite sample power (not size-adjusted) of 𝐻 ( (cid:98)𝛽) for testing 𝐻0 : 𝛽1 = 0,
𝛽2 = 1 using fixed-𝑏 critical values in all cases. Results are given in the four panels of Figure 3B.4.

Panels (a) and (c) give results for (𝜅𝑥, 𝜅𝑢) = (0.1, 6.9) with alternatives (𝛽1, 𝛽2) = (0.02, 1.02),

(0.04, 1.06). Panels (b) and (d) give results for (𝜅𝑥, 𝜅𝑢) = (0.5, 0.5) with alternatives (𝛽1, 𝛽2) =

(0.3, 1.3), (0.95, 1.95). The format of the figures is the same as for the null simulations.

First, notice that power increases as the alternatives move farther away from the null (going from

top panels to bottom panels). This is not surprising and is expected. The more interesting patterns

are how power depends on the sampling frequency for a given alternative. In the high persistence

case (Figure 3B.4(b,d)), power is nearly the same across sampling frequencies for the AD, SPJ and

CRT bandwidths. For the less persistence case (Figure 3B.4(a,c)), power noticeably decreases as

the sample frequency decreases (as 𝛿 increases). This suggests that using high frequency data when

the data is not too persistent gives higher power. A similar finding was reported by Pellatt and Sun

(2023).

This decline in power as the sampling frequency decreases cannot be explained by the band-

widths because the bandwidth sample size ratios are stable across sampling frequencies. To see

why power decreases as the sampling frequency decreases, we can calculate the signal to noise

96

ratio for the slope estimator analogous to the calculations (3.16) and (3.17) for the bandwidths. The

signal to noise ratio is simply the inverse of the approximate variance of (cid:98)𝛽2 given by

(cid:104)

𝑣𝑎𝑟

(cid:16)

(cid:98)𝛽2

(cid:17) (cid:105) −1

≈ 𝑛 · 𝑣𝑎𝑟 (𝑥𝑖)Ω−1𝑣𝑎𝑟 (𝑥𝑖),

where Ω is the long run variance of 𝑥𝑖𝑢𝑖. Straightforward calculations give

𝑛 · 𝑣𝑎𝑟 (𝑥𝑖)Ω−1𝑣𝑎𝑟 (𝑥𝑖) =

𝑛 (cid:0)1 − exp (cid:0)−𝜅𝑥𝑢
(cid:0)1 − exp (cid:0)−2𝜅𝑥𝑢

𝑇
𝑛
𝑇
𝑛

(cid:1)(cid:1) 2 𝜎2
(cid:1)(cid:1) 𝜎2

𝑥 𝜅𝑢
𝑢 𝜅𝑥

=

(cid:2)𝑛 (cid:0)1 − exp (cid:0)−𝜅𝑥𝑢
𝑇
𝑛
𝑛 (cid:0)1 − exp (cid:0)−2𝜅𝑥𝑢

(cid:1)(cid:1)(cid:3) 2 𝜎2
(cid:1)(cid:1) 𝜎2
𝑇
𝑛

𝑥 𝜅𝑢
𝑢 𝜅𝑥

.

For 𝑇 fixed it is easy to show, using (3.18), that as 𝑛 → ∞,

𝑛 · 𝑣𝑎𝑟 (𝑥𝑖)Ω−1𝑣𝑎𝑟 (𝑥𝑖) =

lim
𝑛→∞

𝜅𝑥𝑢𝜎2
𝑥 𝜅𝑢
𝑢 𝜅𝑥
2𝜎2

𝑇 .

In contrast, it is easy to see that as 𝑛 → 0,

𝑛 · 𝑣𝑎𝑟 (𝑥𝑖)Ω−1𝑣𝑎𝑟 (𝑥𝑖) = 0.

lim
𝑛→0

We see that, for a given span, the signal to noise ratio is finite for high frequencies and decreases

to zero as the sampling frequency decreases.

3.5.4 Cointegrating Regression Results

We report a set of simulation results for the IM-OLS statistic, (cid:101)𝑊 ∗, for cointegrating regressions

in Figures 4-7 using a format that is analogous to Figures 1,2 and 4. The data dependent bandwidths

𝑥 were computed using the OLS residuals, not Δ(cid:101)𝑆𝑢∗
for (cid:101)𝜆2∗
𝑢·(cid:101)
on Δ(cid:101)𝑆𝑢∗
reported for 𝜅𝑢 = 0.393 (highly persistent 𝑈𝑡) and 𝜅𝑢 = 6.287 (low persistent 𝑈𝑡). Results were

are severely biased. In all cases 𝜅𝑥 = 0. Following Pellatt and Sun (2023), results are

, given that estimated values of 𝜌 based

𝑖

𝑖

obtained for 𝜅𝑢 = 1.572 but are not reported as those results consistently fall between results for the

other two 𝜅𝑢 values. Following CLP we used 𝜎𝑢 = 0.0097 and 𝜎𝑥 = 0.0998. Results are given for

𝜋 = 0, 0.75. Results for 𝜋 = 0.25, 0.5 are very similar and are not reported. Fixed-𝑏 critical values

based on (3.12) are used in all cases.

Overall, the patterns in Figures 5-7 are similar to the patterns in Figures 1-3 giving similar

findings in cointegrating regressions, including the endogenous case, as in the stationary regression

case. There can be substantial over-rejections in the high persistence case. As persistence decreases

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(for the given value of 𝑇), empirical null rejections approach the nominal level. Figure 3B.6

shows that the AD, SPJ bandwidth ratios are stable across sampling frequencies for both levels

of persistence. Figure 3B.7 shows that power in the high persistence case is not sensitive to the

sampling frequency. Like for stationary regressions, power noticeably declines as the sampling

frequency decreases in the lower persistence case.

3.5.5 Summary of Simulations

The simulation results for 𝐻 ( (cid:98)𝛽) (stationary regression) and (cid:101)𝑊 ∗ (cointegrated regression) gen-

erated by OU processes can be summarized as follows.

If the bandwidth sample size ratio is

constant across sampling frequencies and fixed-𝑏 critical values are used, null rejections are sim-

ilar. Therefore, with regard to size distortions, the sampling frequency does not matter. While

higher frequency data has stronger autocorrelation for given values of 𝜅𝑥 and 𝜅𝑢, it also has a larger

sample size. Intuitively, the null rejections are stable across sampling frequencies when the same

bandwidth sample size ratio is used because what matters is strength of autocorrelation relative to

the number of observations. This balance between autocorrelation and sample size is stable across

sampling frequencies.

Power, on the other hand, can depend on the sampling frequency if the persistence is not strong.

Power increases as the sampling frequency increases. Because the 𝑏 𝐴𝐷, 𝑏𝑆𝑃𝐽, and 𝑏𝐶 𝑅𝑇 are stable

across sampling frequencies and provide stability in null rejections across sampling frequencies,

using high frequency data can lead to higher power without sacrificing additional size distortions

(over what is already present given the underlying persistence as measured by 𝜅𝑥 and 𝜅𝑢 relative to

the span, 𝑇).

For given persistence in the data (given 𝜅𝑥 and 𝜅𝑢), when fixed-𝑏 critical values are used, SPJ

gives the least size distorted tests followed by AD followed by CRT. This happens because 𝑏𝑆𝑃𝐽

tends to be larger than 𝑏 𝐴𝐷 which tends to be larger than 𝑏𝐶 𝑅𝑇 . Of course, given the well known

trade-off between size distortions and power as 𝑏 increases when fixed-𝑏 critical values are used, the

power rankings are the opposite. Which bandwidth to use in practice depends on the implications

of this trade-off to the practitioner.

98

3.6 Empirical Application

In this section we provide some basic empirical results on the uncovered interest parity (UIP)

puzzle as an illustration. UIP is the well studied hypothesis that interest rate differentials between

two countries should be equal to the expected return of the exchange rate. There is a large literature3

that tests UIP using the following simple regression that fits within our analysis:

𝑠𝑖+Δ − 𝑠𝑖 = 𝛽1 + 𝛽2(𝑖𝑛𝑡𝑖 − 𝑖𝑛𝑡∗

𝑖 ) + 𝑢𝑖,

where 𝑠𝑖 is the logarithm of the exchange rate at time 𝑖, 𝑖𝑛𝑡𝑖 is the interest rate on a domestic bond

of maturity Δ and 𝑖𝑛𝑡∗
𝑖

is the interest rate on a foreign country bond of the same maturity. The

null hypothesis of UIP is 𝐻0 : 𝛽1 = 0 and 𝛽2 = 1 and the null hypothesis of a milder version of

UIP is 𝐻0 : 𝛽2 = 1 which is our focus here. Diez de los Rios and Sentana (2011) examine tests of

(mild) UIP in a continuous time framework and focus on using high sampling frequencies in the

data which we do here.

For our empirical illustration, we examine the UIP hypothesis for the US-Japan case, where

the domestic country is the US and the foreign country is Japan. The data for the exchange rate

and interest rates are obtained from Refinitiv Workplace (formerly Thomson-Reuters) and are daily

observations. The sample period is from 1991/01/02 to 2022/11/01 giving a span of up to 30 years

depending on the bond maturity horizon. For the interest rates 𝑖𝑛𝑡𝑡 and 𝑖𝑛𝑡∗

𝑡 , we use the yields on

the benchmark government bonds of the domestic (US) and the foreign country (Japan) with two

different maturities. The first is a 2-year bond. The second is a 10-year bond following Chinn and

Meredith (2004).

Because exchange rates and interest rates are highly persistent, we use a 𝑡-statistic based on

𝐺 ( (cid:98)𝛽) for testing 𝐻0 : 𝛽2 = 1 which is appropriate for the case of a cointegration regression. We
use fixed-𝑏 critical values calculated with (3.11) using coefficients from the G-cointegration line of

Panel B of Table 3B.1. To accommodate possible endogeneity, we also use a 𝑡-statistics based on

(cid:101)𝑊 ∗ of IM-OLS using critical values given by (3.13). We report results for four bandwidth selection

3See Engel et al. (2022) for a recent empirical paper and Engel (2014) for a broader survey of empirical work

testing UIP.

99

rules (AD, SPJ, NW, CRT). For each bandwidth method, we calculate standard errors, the 𝑏 ratio

and the 𝑡-statistic for both the OLS and IM-OLS cases. We also provide 95% confidence intervals

for 𝛽2 using fixed-𝑏 critical values and 𝑁 (0, 1) critical values.

Table 3B.2 gives results for the 2-year bond maturity and Table 3B.3 gives results for the 10-year

bond maturity. In each table results are given for daily, weekly, monthly and quarterly sampling

frequencies. Both OLS and IM-OLS are considered. Looking at the daily frequency results in

Table 3B.2, we see that the 𝑡-statistics based on 𝐺 ( (cid:98)𝛽) using AD and SPJ are -2.189 and -2.473

respectively, rejecting the UIP hypothesis at the 5% level when using the normal critical value.

However, the fixed-𝑏 critical values for the 𝑡-statistic using (cid:98)𝑏 𝐴𝐷 and (cid:98)𝑏𝑆𝑃𝐽 are ±4.053 and ±5.213
respectively. Therefore, the UIP hypothesis is not rejected when using fixed-𝑏 critical values. The

same finding is made with the 𝑡-statistics based on (cid:101)𝑊 ∗ of IM-OLS. Interestingly (cid:98)𝛽2 and (cid:101)𝛽2 are
stable across the frequencies but have opposite signs with (cid:98)𝛽2 negative and (cid:101)𝛽2 close to 0.5.

Recall that 𝐺 ( (cid:98)𝛽) and (cid:101)𝑊 ∗ use the same (cid:98)𝑏 values for each bandwidth method. Notice that (cid:98)𝑏 𝐴𝐷
and (cid:98)𝑏𝑆𝑃𝐽 yield wider confidence intervals with fixed-𝑏 critical values compared to normal critical
values. These patterns hold across sampling frequencies. The values of the (cid:98)𝑏 𝐴𝐷 and (cid:98)𝑏𝑆𝑃𝐽 are large

(0.479 to 0.777) and each are roughly stable across sampling frequencies as expected given the

finite sample simulations. Also, as expected, (cid:98)𝑏𝑆𝑃𝐽 is larger than (cid:98)𝑏 𝐴𝐷. Large values of (cid:98)𝑏 𝐴𝐷 and (cid:98)𝑏𝑆𝑃𝐽

are an indication of high persistence in the regression error. Using normal critical values with large

bandwidth ratios would lead to misleading inference (type 1 error well above the nominal level). The

NW bandwidths are substantially smaller than AD and SPJ. Table 3B.2 shows that (cid:98)𝑏𝑁𝑊 is very small

for high sampling frequencies (0.007) and is increasing to 0.11 at the quarterly frequency, the same

pattern that we observe in the finite sample simulations. This illustrates that the NW bandwidth is

not high frequency compatible as argued by CLP, and confidence intervals based on NW can be

misleadingly tight. By construction 𝑏𝐶 𝑅𝑇 is the same across sampling frequencies. However, the

value of 𝑏𝐶 𝑅𝑇 = 0.066 is substantially smaller than (cid:98)𝑏 𝐴𝐷 and (cid:98)𝑏𝑆𝑃𝐽 suggesting confidence intervals
using 𝑏𝐶 𝑅𝑇 can be too tight relative to the persistence in the regression errors.

Table 3B.3 gives results for the 10-year benchmark government bond. While the point estimates

100

of 𝛽2 are now larger than 1 (about 1.4) for both OLS and IM-OLS, bandwidth patterns and inference

conclusions are similar to Table 3B.2. The (cid:98)𝑏 values continue to be large and roughly stable across

sampling frequencies. UIP cannot be rejected for either test.

3.7 Conclusion

In this paper we develop fixed-𝑏 asymptotic results for HAR Wald statistics for regressions

with high frequency data in the continuous time framework of Chang et al. (2023). We find that

the fixed-𝑏 limits of the HAR Wald tests in stationary high frequency regressions are the same

as the standard fixed-𝑏 limits in Kiefer and Vogelsang (2005). For cointegrating high frequency

regressions the fixed-𝑏 limits generally have non-pivotal limits. For the special case where the

stochastic processes in the continuous time regression follow Brownian motions and the regressors

are independent of the errors, the fixed-𝑏 limits are pivotal and are the same as those obtained by

Bunzel (2006) in discrete time settings. We also analyzed a Wald statistic from Vogelsang and

Wagner (2014) using their IM-OLS estimator and obtained fixed-𝑏 limits that are same as the limits

in Vogelsang and Wagner (2014). Our results in conjunction with the results in Pellatt and Sun

(2023) for orthonormal series approaches (including the cointegration estimator of Hwang and Sun

(2018)) establish that fixed-𝑏 (more generally fixed-smoothing) critical values are high frequency

compatible.

In a simulation study where data is generated according to OU processes we find that the

Andrews (1991), Sun et al. (2008) and CLP’s CRT bandwidths tend to perform well not only at

high frequencies (assuming the persistence in the data is not too strong relative to the span), but give

null rejections that are remarkably stable across sampling ranging from high to low frequencies.

This stability holds for data with strong and mild persistence. We show that the source of this

stability in null rejections is stability in bandwidth sample size ratios (𝑏-values) across sampling

frequencies. This stability holds by construction for the CRT bandwidth. It is more surprising

this stability holds for the Andrews (1991), Sun et al. (2008) bandwidths and we provide a simple

theoretical explanation.

We also report some power results in our simulations and find that for persistent series, power

101

is stable across sampling frequencies. In contrast, power falls as sampling frequency decreases

for mildly persistent series. Therefore, we can recommend that practitioners use data sampled at

higher frequencies.

102

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104

APPENDIX 3A

PROOFS

We provide proofs for the case of the Bartlett kernel given the simple form of the corresponding

long run variance estimators. Algebra is similar for other kernels. When writing formulas for 𝐺 ( (cid:98)𝛽)
and 𝐻 ( (cid:98)𝛽) note that, when the null hypothesis 𝐻0 : 𝑅𝛽 = 𝑟 is true, we can write

𝑅 (cid:98)𝛽 − 𝑟 = 𝑅 (cid:98)𝛽 − 𝑅𝛽 = 𝑅

(cid:16)

(cid:17)

(cid:98)𝛽 − 𝛽

so that 𝑅 (cid:98)𝛽 − 𝑟 can be replaced with 𝑅
IM-OLS, we can replace 𝑅(cid:98)𝜃 − 𝑟 (with suitably augmented 𝑅 and 𝑟) with 𝑅

(cid:98)𝛽 − 𝛽

in the formulas for 𝐺 ( (cid:98)𝛽) and 𝐻 ( (cid:98)𝛽). Similarly for

(cid:16)

(cid:98)𝜃 − 𝜃

(cid:17)

.

(cid:16)

(cid:17)

Proof of Theorem 3.1: Under Assumptions 3.2, 3.3 and 3.4 as 𝛿 → 0 and 𝑇 → ∞, we have the

following results.

1
𝑛

[𝑟𝑛]
∑︁

𝑥𝑖𝑥′

𝑖 = 𝑇 −1

∫ 𝑟𝑇

0

𝑋𝑡 𝑋′

𝑡 𝑑𝑡 + 𝑜 𝑝 (1)

𝑝
→ 𝑟𝑄,

𝑖=1
√

𝑥𝑖𝑢𝑖 =

√

√
𝑇 1
𝑛

[𝑟𝑛]
∑︁

𝑖=1

𝛿
𝑛

[𝑟𝑛]
∑︁

𝑖=1

𝑥𝑖𝑢𝑖 = 𝑇 −1/2

∫ 𝑟𝑇

0

𝑋𝑡𝑈𝑡 𝑑𝑡 + 𝑜 𝑝 (1) ⇒ Λ𝑊𝑘 (𝑟).

Using these limits gives

√

𝑇

(cid:16)

(cid:98)𝛽 − 𝛽

(cid:17)

=

(cid:32)

1
𝑛

𝑛
∑︁

𝑖=1

𝑥𝑖𝑥′
𝑖

(cid:33) −1 √

𝑇 1
𝑛

𝑛
∑︁

𝑖=1

𝑥𝑖𝑢𝑖 ⇒ 𝑄−1Λ𝑊𝑘 (1).

Next, we derive the fixed-𝑏 limit of (cid:98)Ω upon appropriate scaling. For the Bartlett kernel, we can

write

where

(cid:98)Ω𝑛 =

2
𝑀𝑛𝑛

𝑛−1
∑︁

𝑖=1

𝑖 ((cid:98)𝑆𝑣
(cid:98)𝑆𝑣

𝑖 )′ −

1
𝑀𝑛𝑛

𝑛−𝑀𝑛−1
∑︁

(cid:16)

𝑖 ((cid:98)𝑆𝑣
(cid:98)𝑆𝑣

𝑖+𝑀𝑛)′ + (cid:98)𝑆𝑣

𝑖+𝑀𝑛 ((cid:98)𝑆𝑣

𝑖 )′(cid:17)

,

𝑖=1

[𝑟𝑛]
∑︁

𝑗=1

𝑢 𝑗 .
𝑥 𝑗 (cid:98)

(cid:98)𝑆𝑣
[𝑟𝑛] =

105

Scaling (cid:98)𝑆𝑣

[𝑟𝑛] by 𝑇 1/2/𝑛 gives

𝑇 1/2 1

𝑛 (cid:98)𝑆𝑣

[𝑟𝑛] = 𝑇 1/2 1
𝑛

= 𝑇 1/2 1
𝑛

= 𝑇 1/2 1
𝑛

[𝑟𝑛]
∑︁

𝑗=1
[𝑟𝑛]
∑︁

𝑗=1
[𝑟𝑛]
∑︁

𝑗=1

𝑢 𝑗 = 𝑇 1/2 1
𝑛

𝑥 𝑗 (cid:98)

[𝑟𝑛]
∑︁

𝑗=1

𝑥 𝑗 (𝑦 𝑗 − 𝑥′

𝑗 (cid:98)𝛽)

𝑥 𝑗 (𝑥′

𝑗 𝛽 + 𝑢 𝑗 − 𝑥′

𝑗 (cid:98)𝛽) = 𝑇 1/2 1
𝑛

[𝑟𝑛]
∑︁

𝑗=1

𝑥 𝑗 (𝑢 𝑗 − 𝑥′

𝑗 ( (cid:98)𝛽 − 𝛽))

𝑥 𝑗 𝑢 𝑗 − 𝑇 1/2 1
𝑛

[𝑟𝑛]
∑︁

𝑗=1

𝑥 𝑗 𝑥′

𝑗 ( (cid:98)𝛽 − 𝛽)

[𝑟𝑛]
∑︁

𝑥 𝑗 𝑢 𝑗 −

1
𝑛

[𝑟𝑛]
∑︁

𝑗=1

𝑥 𝑗 𝑥′
𝑗

√
𝑇 ( (cid:98)𝛽 − 𝛽)

= 𝑇 1/2 1
𝑛

= 𝑇 −1/2

𝑗=1
∫ 𝑟𝑇

𝑋𝑡𝑈𝑡 𝑑𝑡 + 𝑜 𝑝 (1) −

∫ 𝑟𝑇

(cid:18)
𝑇 −1

𝑋𝑡 𝑋′

𝑡 𝑑𝑡 + 𝑜 𝑝 (1)

(cid:19) √

𝑇 ( (cid:98)𝛽 − 𝛽)

0

0
⇒ Λ𝑊𝑘 (𝑟) − 𝑟𝑄𝑄−1Λ𝑊𝑘 (1) ≡ Λ𝐵𝑘 (𝑟),

as 𝛿 → 0 and 𝑇 → ∞. Scaling (cid:98)Ω𝑛 by 𝛿 = 𝑇/𝑛 gives

𝛿(cid:98)Ω𝑛 = 𝑇 1

𝑛 (cid:98)Ω𝑛 =

2
𝑀𝑛𝑛

𝑇 1
𝑛

𝑛−1
∑︁

𝑖 ((cid:98)𝑆𝑣
(cid:98)𝑆𝑣

𝑖 )′ −

𝑛−𝑀𝑛−1
∑︁

[(cid:98)𝑆𝑣

𝑖 ((cid:98)𝑆𝑣

𝑖+𝑀𝑛)′ + (cid:98)𝑆𝑣

𝑖+𝑀𝑛 ((cid:98)𝑆𝑣

𝑖 )′]

𝑖=1
𝑖 (𝑇 1/2 1

𝑇 1/2 1

𝑛 (cid:98)𝑆𝑣

𝑛 (cid:98)𝑆𝑣

𝑖 )′ −

1
𝑀𝑛

𝑖=1
[𝑇 1/2 1

𝑛 (cid:98)𝑆𝑣

𝑖 𝑇 1/2 1
𝑛

𝑖+𝑀𝑛)′ + 𝑇 1/2 1
((cid:98)𝑆𝑣

𝑖+𝑀𝑛 (𝑇 1/2 1

𝑛 (cid:98)𝑆𝑣

𝑛 (cid:98)𝑆𝑣

𝑖 )′]

𝑇 1/2 1

𝑖 (𝑇 1/2 1

𝑛 (cid:98)𝑆𝑣

𝑛 (cid:98)𝑆𝑣

𝑖 )′ −

1
𝑏𝑛

[𝑇 1/2 1

𝑛 (cid:98)𝑆𝑣

𝑖 (𝑇 1/2 1

𝑛 (cid:98)𝑆𝑣

𝑖+𝑏𝑛)′ + 𝑇 1/2 1

𝑛 (cid:98)𝑆𝑣

𝑖+𝑏𝑛 (𝑇 1/2 1

𝑛 (cid:98)𝑆𝑣

𝑖 )′]

𝐵𝑘 (𝑟)𝐵𝑘 (𝑟)′𝑑𝑟 −

1
𝑏

𝐵𝑘 (𝑟)𝐵𝑘 (𝑟 + 𝑏)′𝑑𝑟 −

1
𝑏

∫ 1−𝑏

0

𝐵𝑘 (𝑟 + 𝑏)𝐵𝑘 (𝑟)′𝑑𝑟

(cid:19)

Λ′

1
𝑀𝑛𝑛

𝑇 1
𝑛

𝑛−𝑀𝑛−1
∑︁

𝑖=1
𝑛−𝑏𝑛−1
∑︁

𝑖=1
∫ 1−𝑏

0

𝑛−1
∑︁

𝑖=1
𝑛−1
∑︁

=

=

2
𝑀𝑛

2
𝑏𝑛

⇒ Λ

𝑖=1
(cid:18) 2
𝑏

∫ 1

0

≡ Λ𝑃𝑘 (𝑏)Λ′.

The result for 𝐻 ( (cid:98)𝛽) as 𝛿 → 0 and 𝑇 → ∞ is straightforward to obtain as follows:

𝐻 ( (cid:98)𝛽) = 𝑅

(cid:16)

(cid:98)𝛽 − 𝛽

(cid:33) −1

𝑥𝑖𝑥′
𝑖

𝑛(cid:98)Ω𝑛

(cid:32) 𝑛
∑︁

𝑅

𝑖=1

(cid:17)′ 






(cid:33) −1

𝑥𝑖𝑥′
𝑖

𝑅′

(cid:32) 𝑛
∑︁

𝑖=1








√

𝑇

(cid:16)

= 𝑅

(cid:98)𝛽 − 𝛽

(cid:32)

1
𝑛

𝑛
∑︁

𝑖=1

𝑅

(cid:17)′ 






(cid:33) −1

𝑥𝑖𝑥′
𝑖

𝑇 1
𝑛 (cid:98)Ω𝑛

(cid:32)

1
𝑛

𝑛
∑︁

𝑖=1

(cid:33) −1

𝑥𝑖𝑥′
𝑖

𝑅′








−1

−1

(cid:16)

𝑅

(cid:17)

(cid:98)𝛽 − 𝛽

√

𝑇

𝑅

(cid:16)

(cid:17)

(cid:98)𝛽 − 𝛽

106

√

𝑇

(cid:16)

= 𝑅

(cid:98)𝛽 − 𝛽

(cid:17)′ (cid:34)

(cid:18)
𝑇 −1

𝑅

∫ 𝑇

0

𝑋𝑡 𝑋′

𝑡 𝑑𝑡 + 𝑜 𝑝 (1)

(cid:19) −1

𝑇 1
𝑛 (cid:98)Ω𝑛

∫ 𝑇

(cid:18)
𝑇 −1

0

𝑋𝑡 𝑋′

𝑡 𝑑𝑡 + 𝑜 𝑝 (1)

(cid:35) −1

(cid:19) −1

𝑅′

√

𝑇

𝑅

(cid:16)

(cid:17)

(cid:98)𝛽 − 𝛽

⇒ [𝑅𝑄−1Λ𝑊𝑘 (1)]′−1 [𝑅𝑄−1Λ𝑃𝑘 (𝑏)Λ′𝑄−1𝑅′]−1 [𝑅𝑄−1Λ𝑊𝑘 (1)]

= [Λ∗𝑊𝑞 (1)]′[Λ∗𝑃𝑞 (𝑏)Λ∗′]−1 [Λ∗𝑊𝑞 (1)] = 𝑊𝑞 (1)′𝑃𝑞 (𝑏)−1𝑊𝑞 (1).

Note that Λ∗ is the square root matrix of 𝑅𝑄−1ΛΛ′𝑄−1𝑅′.

Proof of Theorem 3.2: We first give the derivation for 𝐻 ( (cid:98)𝛽) followed by the derivation for 𝐺 ( (cid:98)𝛽).
𝑥𝑖] and 𝑈◦(𝑠) = 𝜆𝑢𝑤𝑢 (𝑠). Under Assumptions 3.2, 3.5 and 3.6
Recall that 𝑋′

𝑡 = [1 (cid:101)𝑋𝑡], 𝑥′

𝑖 = [1 (cid:101)

we have these results for the partial sums:

1
𝑛

[𝑟𝑛]
∑︁

𝑖=1

Λ−1

𝑇 𝑥𝑖𝑥′

𝑖Λ−1′

𝑇 =

1
𝑇

∫ 𝑟𝑇

0

Λ−1

𝑇 𝑋𝑡 𝑋′

𝑡 Λ−1′

𝑇 𝑑𝑡 + 𝑜 𝑝 (1) ⇒

∫ 𝑟

0

𝑋 ◦(𝑠) 𝑋 ◦(𝑠)′𝑑𝑠,

[𝑟𝑛]
∑︁

Λ−1

𝑇 𝑥𝑖𝑢𝑖 =

𝑇 1/2 1
𝑛

Scaling

𝑖=1
(cid:16)

(cid:98)𝛽 − 𝛽

(cid:17)

by 𝑇 1/2Λ′

𝑇 gives

1
√
𝑇

∫ 𝑟𝑇

0

Λ−1

𝑇 𝑋𝑡𝑈𝑡 𝑑𝑡 + 𝑜 𝑝 (1) ⇒

∫ 𝑟

0

𝑋 ◦(𝑠)𝑑𝑈◦(𝑠) = 𝜆𝑢

∫ 𝑟

0

𝑋 ◦(𝑠)𝑑𝑤𝑢 (𝑠).

𝑇 1/2Λ′
𝑇

(cid:16)

(cid:98)𝛽 − 𝛽

(cid:17)

=

(cid:32)

⇒

1
𝑛

𝑖=1
(cid:18)∫ 1

𝑛
∑︁

Λ−1

𝑇 𝑥𝑖𝑥′

𝑖Λ−1′
𝑇

(cid:33) −1 (cid:32)

𝑇 1/2 1
𝑛

(cid:33)

Λ−1

𝑇 𝑥𝑖𝑢𝑖

𝑛
∑︁

𝑖=1

𝑋 ◦(𝑠) 𝑋 ◦(𝑠)′𝑑𝑠

(cid:19) −1

𝜆𝑢

∫ 1

𝑋 ◦(𝑠)𝑑𝑤𝑢 (𝑠)

0
(cid:18)∫ 1

0

= 𝜆𝑢

𝑋 ◦(𝑠) 𝑋 ◦(𝑠)′𝑑𝑠

0
(cid:19) −1 ∫ 1

0

𝑋 ◦(𝑠)𝑑𝑤𝑢 (𝑠) ≡ 𝜆𝑢𝐶.

Next, we need to determine the scaling for (cid:98)Ω𝑛. First, scale (cid:98)𝑆𝑣

[𝑟𝑛] by 𝑇 1/2 1

𝑛 Λ−1

𝑇 to give

𝑇 1/2 1
𝑛

𝑇 (cid:98)𝑆𝑣
Λ−1

[𝑟𝑛] = 𝑇 1/2 1
𝑛

Λ−1
𝑇

= 𝑇 1/2 1
𝑛

Λ−1
𝑇

= 𝑇 1/2 1
𝑛

Λ−1
𝑇

[𝑟𝑛]
∑︁

𝑖=1
[𝑟𝑛]
∑︁

𝑖=1
[𝑟𝑛]
∑︁

𝑖=1

𝑢𝑖 = 𝑇 1/2 1
𝑥𝑖(cid:98)
𝑛

Λ−1
𝑇

[𝑟𝑛]
∑︁

𝑖=1

𝑥𝑖 (𝑦𝑖 − 𝑥′

𝑖 (cid:98)𝛽)

𝑥𝑖 (𝑥′

𝑖 𝛽 + 𝑢𝑖 − 𝑥′

𝑖 (cid:98)𝛽) = 𝑇 1/2 1
𝑛

Λ−1
𝑇

[𝑟𝑛]
∑︁

𝑖=1

𝑥𝑖 (𝑢𝑖 − 𝑥′

𝑖 ( (cid:98)𝛽 − 𝛽))

𝑥𝑖𝑢𝑖 − 𝑇 1/2 1
𝑛

Λ−1
𝑇

[𝑟𝑛]
∑︁

𝑖=1

𝑥𝑖𝑥′

𝑖 ( (cid:98)𝛽 − 𝛽)

107

[𝑟𝑛]
∑︁

Λ−1

𝑇 𝑥𝑖𝑥′

𝑖Λ−1′

𝑇 𝑇 1/2Λ′

𝑇 ( (cid:98)𝛽 − 𝛽)

= 𝑇 1/2 1
𝑛

[𝑟𝑛]
∑︁

𝑖=1

Λ−1

𝑇 𝑥𝑖𝑢𝑖 −

1
𝑛

∫ 𝑟

⇒

𝑋 ◦(𝑠)𝜆𝑢𝑑𝑤𝑢 (𝑠) −

𝑖=1
(cid:18)∫ 𝑟

0
(cid:32)∫ 𝑟

0

= 𝜆𝑢

𝑋 ◦(𝑠)𝑑𝑤𝑢 (𝑠) −

0
∫ 𝑟

0

𝑋 ◦(𝑠) 𝑋 ◦(𝑠)′𝑑𝑠

𝑋 ◦(𝑠) 𝑋 ◦(𝑠)′𝑑𝑠

(cid:19) (cid:34)

𝜆𝑢

(cid:18)∫ 1

0

(cid:18)∫ 1

0

𝑋 ◦(𝑠) 𝑋 ◦(𝑠)′𝑑𝑠

(cid:19) −1 ∫ 1

0

(cid:35)

𝑋 ◦(𝑠)𝑑𝑤𝑢 (𝑠)

𝑋 ◦(𝑠) 𝑋 ◦(𝑠)′𝑑𝑠

(cid:19) −1 ∫ 1

0

(cid:33)

𝑋 ◦(𝑠)𝑑𝑤𝑢 (𝑠)

≡ 𝜆𝑢 𝐵𝐻 (𝑟).

Next, scale (cid:98)Ω𝑛 as 𝛿Λ−1

𝑇 (cid:98)Ω𝑛Λ−1′

𝑇 = (𝑇/𝑛) Λ−1

𝑇 (cid:98)Ω𝑛Λ−1′

𝑇

to give

𝑇 1
𝑛

𝑇 (cid:98)Ω𝑛Λ−1′
Λ−1

𝑇 =

2
𝑀𝑛𝑛

𝑇 1
𝑛

Λ−1
𝑇

𝑛−1
∑︁

𝑖=1

𝑖 ((cid:98)𝑆𝑣
(cid:98)𝑆𝑣

𝑖 )′Λ−1′

𝑇 −

1
𝑀𝑛𝑛

𝑇 1
𝑛

=

2
𝑏𝑛

𝑛−1
∑︁

𝑖=1

𝑇 1/2 1
𝑛

𝑇 (cid:98)𝑆𝑣
Λ−1

𝑖 )′Λ−1′

𝑇 −

((cid:98)𝑆𝑣

𝑖 𝑇 1/2 1
𝑛
𝑖+𝑏𝑛𝑇 1/2 1
𝑛

+ 𝑇 1/2 1
𝑛

𝑇 (cid:98)𝑆𝑣
Λ−1

=

2
𝑏𝑛

𝑛−1
∑︁

𝑖=1

𝑇 1/2 1
𝑛

𝑇 (cid:98)𝑆𝑣
Λ−1

𝑖 )′Λ−1′

𝑇 −

((cid:98)𝑆𝑣

𝑖 𝑇 1/2 1
𝑛
𝑖+𝑏𝑛𝑇 1/2 1
𝑛

((cid:98)𝑆𝑣

𝑖 )′Λ−1′
𝑇 ]

+ 𝑇 1/2 1
𝑛

𝑇 (cid:98)𝑆𝑣
Λ−1

𝑛−𝑀𝑛−1
∑︁

Λ−1
𝑇

[(cid:98)𝑆𝑣

𝑖 ((cid:98)𝑆𝑣

𝑖+𝑀𝑛)′ + (cid:98)𝑆𝑣

𝑖+𝑀𝑛 ((cid:98)𝑆𝑣

𝑖 )′]Λ−1′

𝑇

𝑖=1
𝑛−𝑏𝑛−1
∑︁

𝑡=1

[𝑇 1/2 1
𝑛

𝑇 (cid:98)𝑆𝑣
Λ−1

𝑖 𝑇 1/2 1
𝑛

((cid:98)𝑆𝑣

𝑖+𝑏𝑛)′Λ−1′
𝑇

1
𝑏𝑛

((cid:98)𝑆𝑣

𝑖 )′Λ−1′
𝑇 ]

1
𝑏𝑛

𝑛−𝑏𝑛−1
∑︁

𝑡=1

[𝑇 1/2 1
𝑛

𝑇 (cid:98)𝑆𝑣
Λ−1

𝑖 𝑇 1/2 1
𝑛

((cid:98)𝑆𝑣

𝑖+𝑏𝑛)′Λ−1′
𝑇

⇒

2
𝑏

∫ 1

0

𝜆𝑢 𝐵𝐻 (𝑟)𝜆𝑢 𝐵𝐻 (𝑟)′𝑑𝑟 −

1
𝑏

∫ 1−𝑏

0

𝜆𝑢 𝐵𝐻 (𝑟)𝜆𝑢 𝐵𝐻 (𝑟 + 𝑏)′𝑑𝑟

∫ 1−𝑏

−

1
𝑏

𝜆𝑢 𝐵𝐻 (𝑟 + 𝑏)𝜆𝑢 𝐵𝐻 (𝑟)′𝑑𝑟

= 𝜆2
𝑢

(cid:18) 2
𝑏

∫ 1

0

𝐵𝐻 (𝑟)𝐵𝐻 (𝑟)′𝑑𝑟 −

0

1
𝑏

∫ 1−𝑏

0

𝐵𝐻 (𝑟)𝐵𝐻 (𝑟 + 𝑏)′𝑑𝑟 −

1
𝑏

∫ 1−𝑏

0

𝐵𝐻 (𝑟 + 𝑏)𝐵𝐻 (𝑟)′𝑑𝑟

(cid:19)

≡ 𝜆2

𝑢𝑃𝐻 (𝑏).

The result for 𝐻 ( (cid:98)𝛽) as 𝛿 → 0 and 𝑇 → ∞ is straightforward to obtain as follows:

𝐻 ( (cid:98)𝛽) =

(cid:16)

(cid:16)

𝑅

(cid:98)𝛽 − 𝛽

(cid:32) 𝑛
∑︁

𝑅

𝑖=1

(cid:17)(cid:17)′ 






(cid:33) −1

𝑥𝑖𝑥′
𝑖

𝑛(cid:98)Ω𝑛

(cid:33) −1

𝑥𝑖𝑥′
𝑖

(cid:32) 𝑛
∑︁

𝑖=1

(cid:16)

=

𝑅Λ−1′

𝑇 Λ′
𝑇

(cid:16)

(cid:98)𝛽 − 𝛽

𝑅Λ−1′
𝑇

(cid:17)(cid:17)′ 





𝑇 ( (cid:98)𝛽 − 𝛽)

× 𝑅Λ−1′

𝑇 Λ′

(cid:32) 𝑛
∑︁

𝑖=1

Λ−1

𝑇 𝑥𝑖𝑥′

𝑖Λ−1′
𝑇

108

𝑅( (cid:98)𝛽 − 𝛽)

−1

𝑅′







(cid:33) −1

𝑛Λ−1

𝑇 (cid:98)Ω𝑛Λ−1′
𝑇

(cid:32) 𝑛
∑︁

𝑖=1

Λ−1

𝑇 𝑥𝑖𝑥′

𝑖Λ−1′
𝑇

(cid:33) −1

Λ−1

𝑇 𝑅′

−1








(cid:16)

=

Λ𝑅

𝑇 𝑅Λ−1′

𝑇 Λ′
𝑇

(cid:17)(cid:17)′

(cid:16)

(cid:98)𝛽 − 𝛽

×

𝑇 𝑅Λ−1′
𝑇



Λ𝑅





(cid:32) 𝑛
∑︁

𝑖=1

Λ−1

𝑇 𝑥𝑖𝑥′

𝑖Λ−1′
𝑇

(cid:33) −1

𝑛Λ−1

𝑇 (cid:98)Ω𝑛Λ−1′
𝑇

(cid:32) 𝑛
∑︁

𝑖=1

Λ−1

𝑇 𝑥𝑖𝑥′

𝑖Λ−1′
𝑇

× Λ𝑅

𝑇 𝑅Λ−1′

𝑇 Λ′
𝑇

(cid:16)

(cid:17)

(cid:98)𝛽 − 𝛽

(cid:33) −1

𝑇 𝑅′Λ𝑅′
Λ−1
𝑇

−1








(cid:16)

=

Λ𝑅

𝑇 𝑇 −1/2𝑅Λ−1′

𝑇 𝑇 1/2Λ′
𝑇

(cid:17)(cid:17)′

(cid:16)

(cid:98)𝛽 − 𝛽

Λ𝑅

𝑇 𝑇 −1/2𝑅Λ−1′

𝑇

×








(cid:32)

1
𝑛

𝑛
∑︁

𝑖=1

Λ−1

𝑇 𝑥𝑖𝑥′

𝑖Λ−1′
𝑇

(cid:33) −1

𝑇 1
𝑛

𝑇 (cid:98)Ω𝑛Λ−1′
Λ−1
𝑇

(cid:32)

1
𝑛

𝑛
∑︁

𝑖=1

Λ−1

𝑇 𝑥𝑖𝑥′

𝑖Λ−1′
𝑇

(cid:33) −1

𝑇 𝑅′𝑇 −1/2Λ𝑅′
Λ−1
𝑇

−1








× Λ𝑅

𝑇 𝑇 −1/2𝑅Λ−1′

𝑇 𝑇 1/2Λ′

𝑇 ( (cid:98)𝛽 − 𝛽)

(cid:34)

⇒ (𝑅∗𝜆𝑢𝐶)′

𝑅∗

(cid:18)∫ 1

0

𝑋 ◦(𝑠) 𝑋 ◦(𝑠)′𝑑𝑠

(cid:19) −1

𝑢𝑃𝐻 (𝑏)
𝜆2

(cid:18)∫ 1

0

𝑋 ◦(𝑠) 𝑋 ◦(𝑠)′𝑑𝑠

(cid:19) −1

(cid:35) −1

𝑅∗′

𝑅∗𝜆𝑢𝐶

= (𝑅∗𝐶)′ (cid:2)𝑅∗𝑄−1

◦ 𝑃𝐻 (𝑏)𝑄−1

◦ 𝑅∗′(cid:3) −1 𝑅∗𝐶.

Note that we use (3.5) for the limit of Λ𝑅

𝑇 𝑇 −1/2𝑅Λ−1′
𝑇 .

𝜔2
The derivation for 𝐺 ( (cid:98)𝛽) is similar and only requires the fixed-𝑏 limit of (cid:98)

𝑛. Similar to (cid:98)Ω𝑛, for

𝑛 can be written as a function of (cid:98)𝑆𝑢
𝜔2
the case of the Bartlett kernel (cid:98)

𝑖 = (cid:205)[𝑟𝑛]

𝑗=1 (cid:98)

𝑢 𝑗 as

𝜔2
(cid:98)

𝑛 =

2
𝑀𝑛𝑛

𝑛−1
∑︁

(cid:16)

𝑖=1

(cid:17) 2

(cid:98)𝑆𝑢

𝑖

−

2
𝑀𝑛𝑛

𝑛−𝑀𝑛−1
∑︁

𝑖=1

𝑖 (cid:98)𝑆𝑢
(cid:98)𝑆𝑢

𝑖+𝑀𝑛

.

𝑛, we scale (cid:98)𝑆𝑢
𝜔2
To determine the scaling needed for (cid:98)

𝑖 by 𝑇 1/2/𝑛 to give

𝑇 1/2 1

𝑛 (cid:98)𝑆𝑢

[𝑟𝑇] = 𝑇 1/2 1
𝑛

= 𝑇 1/2 1
𝑛

= 𝑇 1/2 1
𝑛

[𝑟𝑛]
∑︁

𝑗=1
[𝑟𝑛]
∑︁

𝑗=1
[𝑟𝑛]
∑︁

𝑗=1

𝑢 𝑗 = 𝑇 1/2 1
(cid:98)
𝑛

[𝑟𝑛]
∑︁

𝑗=1

(𝑦 𝑗 − 𝑥′

𝑗 (cid:98)𝛽)

(𝑥′

𝑗 𝛽 + 𝑢 𝑗 − 𝑥′

𝑗 (cid:98)𝛽) = 𝑇 1/2 1
𝑛

[𝑟𝑛]
∑︁

𝑗=1

(𝑢 𝑗 − 𝑥′

𝑗 ( (cid:98)𝛽 − 𝛽))

𝑢 𝑗 − 𝑇 1/2 1
𝑛

[𝑟𝑛]
∑︁

𝑗=1

𝑗 ( (cid:98)𝛽 − 𝛽) = 𝑇 1/2 1
𝑥′
𝑛

[𝑟𝑛]
∑︁

𝑗=1

𝑢 𝑗 −

1
𝑛

[𝑟𝑛]
∑︁

𝑗=1

109

𝑗 Λ−1′
𝑥′

𝑇 𝑇 1/2Λ′

𝑇 ( (cid:98)𝛽 − 𝛽)

= 𝑇 −1/2

∫ 𝑟𝑇

0

𝑈𝑡 𝑑𝑡 + 𝑜 𝑝 (1) −

(cid:18) 1
𝑇

∫ 𝑟𝑇

𝑡 Λ−1′
𝑋′

𝑇 𝑑𝑡 + 𝑜 𝑝 (1)

(cid:19)

𝑇 1/2Λ′

𝑇 ( (cid:98)𝛽 − 𝛽)

0
(cid:18)∫ 1

𝑋 ◦(𝑠) 𝑋 ◦(𝑠)′𝑑𝑠

(cid:19) −1 ∫ 1

𝑋 ◦(𝑠)𝑑𝑤𝑢 (𝑠)

⇒ 𝜆𝑢𝑤𝑢 (𝑟) − 𝜆𝑢

𝑋 ◦(𝑠)′𝑑𝑠

∫ 𝑟

0

(cid:32)

= 𝜆𝑢

𝑤𝑢 (𝑟) −

∫ 𝑟

0

𝑋 ◦(𝑠)′𝑑𝑠

0
(cid:18)∫ 1

𝑋 ◦(𝑠) 𝑋 ◦(𝑠)′𝑑𝑠

0
(cid:19) −1 ∫ 1

𝑋 ◦(𝑠)𝑑𝑤𝑢 (𝑠)

(cid:33)

≡ 𝜆𝑢 𝐵𝐺 (𝑟).

0

0

𝜔2
Next, scaling (cid:98)

𝑛 by 𝛿 = 𝑇/𝑛 gives

𝛿

𝑛 = 𝑇 1
𝜔2
(cid:98)

𝜔2
𝑛 (cid:98)

𝑛 =

2
𝑀𝑛𝑛

𝑇 1
𝑛

𝑛−1
∑︁

(cid:16)

(cid:17) 2

(cid:98)𝑆𝑢

𝑖

−

2
𝑀𝑛𝑛

𝑇 1
𝑛

𝑛−𝑀𝑛−1
∑︁

𝑖 (cid:98)𝑆𝑢
(cid:98)𝑆𝑢

𝑖+𝑀𝑛

𝑖=1
(𝑇 1/2 1

𝑛 (cid:98)𝑆𝑢

𝑖 ) −

2
𝑏𝑛

𝑛−𝑏𝑛−1
∑︁

𝑖=1
(cid:18)
𝑇 1/2 1

𝑖=1

(cid:19) (cid:18)

𝑇 1/2 1

𝑛 (cid:98)𝑆𝑢

𝑖+𝑏𝑛

(cid:19)

𝑛 (cid:98)𝑆𝑢

𝑖

=

2
𝑏𝑛

𝑛−1
∑︁

𝑖=1

(cid:19)

(cid:18)
𝑇 1/2 1

𝑛 (cid:98)𝑆𝑢

𝑖

𝜆𝑢 𝐵𝐺 (𝑟)𝜆𝑢 𝐵𝐺 (𝑟)𝑑𝑟 −

𝜆𝑢 𝐵𝐺 (𝑟)𝜆𝑢 𝐵𝐺 (𝑟 + 𝑏)𝑑𝑟

∫ 1

⇒

2
𝑏

= 𝜆2
𝑢

0
(cid:18) 2
𝑏

∫ 1

0

𝐵𝐺 (𝑟)2𝑑𝑟 −

2
𝑏

0

∫ 1−𝑏

0

2
𝑏
∫ 1−𝑏

𝐵𝐺 (𝑟)𝐵𝐺 (𝑟 + 𝑏)𝑑𝑟

(cid:19)

≡ 𝜆2

𝑢𝑃𝐺 (𝑏).

The result for 𝐺 ( (cid:98)𝛽) follows using similar arguments as for 𝐻 ( (cid:98)𝛽) as 𝛿 → 0 and 𝑇 → ∞:

(cid:16)

(cid:16)

𝐺 ( (cid:98)𝛽) =

=

𝑅( (cid:98)𝛽 − 𝛽)

𝑛𝑅

𝜔2
(cid:98)

(cid:17)′ 






(cid:33) −1

𝑥𝑖𝑥′
𝑖

(cid:32) 𝑇
∑︁

𝑡=1

𝑅Λ−1′

𝑇 Λ′
𝑇

(cid:16)

(cid:98)𝛽 − 𝛽

(cid:17)(cid:17)′ 

𝑛𝑅Λ−1′
𝜔2

𝑇
(cid:98)




−1

𝑅′







(cid:32) 𝑇
∑︁

𝑡=1

(cid:16)

𝑅

(cid:17)

(cid:98)𝛽 − 𝛽

Λ−1

𝑇 𝑥𝑖𝑥′

𝑖Λ−1′
𝑇

(cid:33) −1

Λ−1

𝑇 𝑅′

−1








𝑅Λ−1′

𝑇 Λ′
𝑇

(cid:16)

(cid:17)

(cid:98)𝛽 − 𝛽

(cid:16)

=

Λ𝑅

𝑇 𝑅Λ−1′

𝑇 Λ′
𝑇

(cid:16)

(cid:98)𝛽 − 𝛽

(cid:17)(cid:17)′ 






𝑛Λ𝑅
𝜔2
(cid:98)

𝑇 𝑅Λ−1′
𝑇

(cid:32) 𝑇
∑︁

𝑡=1

Λ−1

𝑇 𝑥𝑖𝑥′

𝑖Λ−1′
𝑇

(cid:33) −1

𝑇 𝑅′Λ𝑅′
Λ−1
𝑇

−1








Λ𝑅

𝑇 𝑅Λ−1′

𝑇 Λ′
𝑇

(cid:16)

(cid:17)

(cid:98)𝛽 − 𝛽

(cid:16)

=

Λ𝑅

𝑇 𝑇 −1/2𝑅Λ−1′

𝑇 𝑇 1/2Λ′
𝑇

(cid:16)

(cid:98)𝛽 − 𝛽

𝑛Λ𝑅

𝑇 𝑇 −1/2𝑅Λ−1′

𝑇

𝜔2
𝑛 (cid:98)

(cid:17)(cid:17)′ 
𝑇 1





(cid:17)
(cid:98)𝛽 − 𝛽

× Λ𝑅

𝑇 𝑇 −1/2𝑅Λ−1′

𝑇 𝑇 1/2Λ′
𝑇

(cid:16)

(cid:32)

1
𝑛

𝑇
∑︁

𝑡=1

Λ−1

𝑇 𝑥𝑖𝑥′

𝑖Λ−1′
𝑇

(cid:33) −1

𝑇 𝑅′𝑇 −1/2Λ𝑅′
Λ−1
𝑇

−1








(cid:34)

⇒ (𝑅∗𝜆𝑢𝐶)′

𝑢𝑃𝐺 (𝑏)𝑅∗
𝜆2

𝑋 ◦(𝑠) 𝑋 ◦(𝑠)′𝑑𝑠

(cid:35) −1

(cid:19) −1

𝑅∗′

(cid:18)∫ 1

0

𝑅∗𝜆𝑢𝐶 = (𝑅∗𝐶)′ (cid:2)𝑃𝐺 (𝑏)𝑅∗𝑄−1

◦ 𝑅∗′(cid:3) −1 𝑅∗𝐶.

110

Proof of Theorem 3.3: Define the scaling matrix

𝑇 −1/2











Following Vogelsang and Wagner (2014) let 𝐴𝑅 be a 𝑞 × 𝑞 matrix such that 𝐴−1

𝑇 −1𝐼𝑘−1

𝐴𝐼 𝑀 =












𝐼𝑘−1

0

0

0

0

0

0

.

lim𝑇→∞ 𝑅∗

𝑇 = 𝑅∗ where 𝑅∗

𝑇 = 𝐴−1

𝑅 𝑅 𝐴𝐼 𝑀. Replacing 𝑅(cid:101)𝜃 − 𝑟 with 𝑅

(cid:17)

(cid:16)

(cid:101)𝜃 − 𝜃

𝐴𝐼 𝑀 and 𝐴−1
𝑅 :

(cid:101)𝑊 ∗ = [ 𝐴−1

𝑅 𝑅 𝐴𝐼 𝑀 𝐴−1

𝐼 𝑀 ((cid:101)𝜃 − 𝜃)]′

𝑅 exists and satisfies
we rewrite (cid:101)𝑊 ∗ using

𝑅 𝑅 𝐴𝐼 𝑀

×



𝑥 𝐴−1
(cid:101)𝜆2∗

𝑢·(cid:101)



× [ 𝐴−1

𝑅 𝑅 𝐴𝐼 𝑀 𝐴−1

𝐼 𝑀 ((cid:101)𝜃 − 𝜃)].

(cid:32)

𝐴𝐼 𝑀

𝑛
∑︁

𝑖=1

(cid:33) −1 (cid:32)

𝑖 𝑥∗′
𝑥∗

𝑖 𝐴𝐼 𝑀

𝐴𝐼 𝑀

(cid:33) (cid:32)

𝑐𝑖𝑐′

𝑖 𝐴𝐼 𝑀

𝐴𝐼 𝑀

𝑛
∑︁

𝑖=1

𝑛
∑︁

𝑖=1

(cid:33) −1

𝑖 𝑥∗′
𝑥∗

𝑖 𝐴𝐼 𝑀

𝐴𝐼 𝑀 𝑅′𝐴−1′

𝑅

−1








Noting that 𝐴−1

𝑅 𝑅 𝐴𝐼 𝑀 is 𝑅∗

𝑇 by definition and letting 𝐵𝐼 𝑀 = 𝑇 1/2 𝐴𝐼 𝑀, we have upon scaling inside

the middle inverse term:

(cid:101)𝑊 ∗ = [𝑅∗

𝑇 𝐴−1

𝐼 𝑀 ((cid:101)𝜃 − 𝜃)]′
(cid:32)
𝑛
∑︁

𝑇
𝑥 𝑅∗
𝑛 (cid:101)𝜆2∗
𝑇
𝑢·(cid:101)

1
𝑛

𝑖=1







(cid:32)

×

×

𝐵𝐼 𝑀

1
𝑛

𝑖 𝑥∗′
𝑥∗
𝑖

1
𝑛

𝐵𝐼 𝑀

(cid:33) −1 (cid:32)

1
𝑛

𝑛
∑︁

(cid:18)

𝑖=1

𝐵𝐼 𝑀

(cid:19) 1
𝑛

1
𝑛

𝑐𝑖𝑐′
𝑖

(cid:18) 1
𝑛

1
𝑛

𝐵𝐼 𝑀

(cid:19) (cid:33)

1
𝑛

𝑛
∑︁

𝑖=1

𝐵𝐼 𝑀

1
𝑛

𝑖 𝑥∗′
𝑥∗
𝑖

1
𝑛

𝐵𝐼 𝑀

(cid:33) −1

−1

𝑅∗′
𝑇








[𝑅∗

𝑇 𝐴−1

𝐼 𝑀 ((cid:101)𝜃 − 𝜃)].

Under Assumption 3.2∗∗ and 3.8, we have the following results for partial sums:

1
𝑛

[𝑟𝑛]
∑︁

𝑖=1

𝑢𝑖 =

1
𝑇

∫ 𝑟𝑇

0

𝑈𝑡 𝑑𝑡 + 𝑜 𝑝 (𝑇 −1/2),

1
𝑛

[𝑟𝑛]
∑︁

𝑖=1

𝑥𝑖 =
(cid:101)

1
𝑇

∫ 𝑟𝑇

0

(cid:101)𝑋𝑡 𝑑𝑡 + 𝑜 𝑝 (1),

1
𝑥 [𝑟𝑛] =
𝑛(cid:101)

1
𝑛

[𝑟𝑛]
∑︁

𝑖=1

𝑥
𝑣(cid:101)
𝑖 =

1
𝑇

∫ 𝑟𝑇

0

𝑥
𝑡 𝑑𝑡 + 𝑜 𝑝 (𝑇 −1/2) =
𝑉(cid:101)

1
𝑇 (cid:101)𝑋𝑟𝑇 + 𝑜 𝑝 (𝑇 −1/2),

111

(3A.1)

(3A.2)

(3A.3)

. Scaling 𝑥∗

[𝑟𝑛] by 𝐵𝐼 𝑀/𝑛 gives,

Recall that 𝑥∗

[𝑟𝑛] =












[𝑟𝑛]

(cid:205)[𝑟𝑛]

𝑥𝑖
𝑖=1 (cid:101)
𝑥 [𝑟𝑛]
(cid:101)

𝐵𝐼 𝑀 ·

1
𝑛

𝑥∗
[𝑟𝑛] = 𝐵𝐼 𝑀

1
𝑛






















𝑛 [𝑟𝑛]
1
(cid:205)[𝑟𝑛]

𝑥𝑖
𝑖=1 (cid:101)
𝑥 [𝑟𝑛]
1
𝑛(cid:101)

using (3A.2), (3A.3) and 𝑋 ∗

𝑟𝑇 =












∫ 𝑟𝑇
0 (cid:101)𝑋𝑠𝑑𝑠
(cid:101)𝑋𝑟𝑇












= 𝐵𝐼 𝑀











𝑟𝑇












𝑟 + 𝑜(1)

∫ 𝑟𝑇
0 (cid:101)𝑋𝑠𝑑𝑠 + 𝑜 𝑝 (1)
1
𝑇
𝑇 (cid:101)𝑋𝑟𝑇 + 𝑜 𝑝 (𝑇 −1/2)
1












= 𝐵𝐼 𝑀 ·

1
𝑇

𝑋 ∗
𝑟𝑇 + 𝑜 𝑝 (1),

. The limit of 𝐵𝐼 𝑀 · 1

𝑇 𝑋 ∗

𝑟𝑇 follows as

𝐵𝐼 𝑀 ·

1
𝑇

𝑋 ∗
𝑟𝑇 = 𝐵𝐼 𝑀












𝑟

1
𝑇

∫ 𝑟𝑇
0 (cid:101)𝑋𝑠𝑑𝑠
𝑇 (cid:101)𝑋𝑟𝑇
1

=























𝑟

1
𝑇

𝑇 −1/2 (cid:101)𝑋𝑠𝑑𝑠

∫ 𝑟𝑇
0
𝑇 −1/2 (cid:101)𝑋𝑟𝑇

⇒























𝑟

Λ
𝑥
(cid:101)

∫ 𝑟
𝑥 (𝑠)𝑑𝑠
𝑊
0
(cid:101)
𝑥 (𝑟)
𝑥𝑊
Λ
(cid:101)
(cid:101)












= Π𝑔∗(𝑟).

(3A.4)

Therefore, we have the following results:

1
𝑛

𝑛
∑︁

𝑖=1

𝐵𝐼 𝑀

1
𝑛

𝑥∗
𝑖 𝑥∗′
𝑖

1
𝑛

𝐵𝐼 𝑀 =

1
𝑇

∫ 𝑇

0

𝐵𝐼 𝑀

1
𝑇

𝑡 𝑋 ∗′
𝑋 ∗
𝑡

1
𝑇

𝐵𝐼 𝑀 𝑑𝑡 + 𝑜 𝑝 (1) ⇒ Π

1
𝑛

𝑛
∑︁

𝑖=1

𝐵𝐼 𝑀

1
𝑛

𝑖 𝑇 1/2 1
𝑥∗
𝑛

𝑆𝑢
𝑖 =

1
𝑇

∫ 𝑇

0

𝐵𝐼 𝑀

1
𝑇

𝑡 𝑇 −1/2𝑆𝑈
𝑋 ∗

𝑡 𝑑𝑡 + 𝑜 𝑝 (1) ⇒ Π

∫ 1

0

∫ 1

0

𝑔∗(𝑠)𝑔∗(𝑠)′𝑑𝑠Π′,

𝑔∗(𝑠)𝐵𝑢 (𝑠)𝑑𝑠.

We now derive the limit of 𝐴−1

𝐼 𝑀 ((cid:101)𝜃 − 𝜃) as 𝑇 → ∞ and 𝛿 → 0. Using algebra from Vogelsang and

Wagner (2014) and straightforward scaling gives

𝐴−1
𝐼 𝑀 ((cid:101)𝜃 − 𝜃) =

=

(cid:32)

(cid:32)

1
𝑛

1
𝑛

𝑛
∑︁

𝑖=1
𝑛
∑︁

𝑖=1

𝑇 1/2 𝐴𝐼 𝑀

1
𝑛

𝑖 𝑥∗′
𝑥∗
𝑖

1
𝑛

𝑇 1/2 𝐴𝐼 𝑀

(cid:33) −1 (cid:32)

1
𝑛

𝐵𝐼 𝑀

1
𝑛

𝑖 𝑥∗′
𝑥∗
𝑖

1
𝑛

𝐵𝐼 𝑀

(cid:33) −1 (cid:32)

1
𝑛

𝑛
∑︁

𝑖=1

𝐵𝐼 𝑀

𝑛
∑︁

𝑇 1/2 𝐴𝐼 𝑀

1
𝑛

𝑖 𝑇 1/2 1
𝑥∗
𝑛

𝑆𝑢
𝑖

(cid:33)

(cid:16)

−

(cid:17)′

𝑥Ω−1
0, 0, Ω𝑢
𝑥
(cid:101)
(cid:101)

𝑖=1
𝑖 𝑇 1/2 1
𝑥∗
𝑛

1
𝑛

(cid:33)

𝑆𝑢
𝑖

(cid:16)

−

𝑥Ω−1
0, 0, Ω𝑢
𝑥
(cid:101)
(cid:101)

(cid:17)′

.

Together these results give

𝐴−1
𝐼 𝑀 ((cid:101)𝜃 − 𝜃) ⇒

∫ 1

(cid:18)

Π

0

𝑔∗(𝑠)𝑔∗(𝑠)′𝑑𝑠Π′

(cid:19) −1 (cid:18)

∫ 1

Π

𝑔∗(𝑠)𝐵𝑢 (𝑠)𝑑𝑠

(cid:19)

(cid:16)

−

(cid:17)′

𝑥Ω−1
0, 0, Ω𝑢
𝑥
(cid:101)
(cid:101)

= 𝜆𝑢·(cid:101)

𝑥 (Π′)−1

(cid:18)∫ 1

0

𝑔∗(𝑠)𝑔∗(𝑠)′𝑑𝑠

112

0
(cid:19) −1 ∫ 1

0

𝑔∗(𝑠)𝑤𝑢·(cid:101)

𝑥 (𝑠)𝑑𝑠,

where the last equality holds using arguments from the proof of Theorem 3.2 from Vogelsang and

Wagner (2014) and the limit is identical to the limit obtained in Vogelsang and Wagner (2014).

Now consider the terms inside the inverse of (cid:101)𝑊 ∗. The argument for the limit of

(1/𝑛) (cid:205)𝑛

𝑖=1 (𝐵𝐼 𝑀/𝑛) (𝑐𝑖/𝑛) (cid:0)𝑐′

𝑖/𝑛(cid:1) (𝐵𝐼 𝑀/𝑛) is similar to that of (1/𝑛) (cid:205)𝑛

𝑖=1 (𝐵𝐼 𝑀/𝑛) 𝑥∗

𝑖 𝑥∗′

𝑖 (𝐵𝐼 𝑀/𝑛)

and is given by

1
𝑛

𝑛
∑︁

(cid:18)

𝑖=1

𝐵𝐼 𝑀

(cid:19) 1
𝑛

1
𝑛

𝑐𝑖𝑐′
𝑖

(cid:18) 1
𝑛

1
𝑛

(cid:19)

𝐵𝐼 𝑀

∫ 𝑡

0
(cid:19)′

(cid:19)′

𝐵𝐼 𝑀

1
𝑇

𝑋 ∗

𝑠 𝑑𝑠

𝑑𝑡+𝑜 𝑝 (1)

𝑑𝑟Π′.

=

1
𝑛

𝑛
∑︁

(cid:18)

𝑖=1

𝐵𝐼 𝑀

(cid:19)

1
𝑛

=

1
𝑇

∫ 𝑇

0

(cid:18) 1
𝑇

∫ 𝑇

0

𝐵𝐼 𝑀

1
𝑇

𝑋 ∗

𝑠 𝑑𝑠 −

1
𝑛

(cid:169)
(cid:173)
(cid:171)

1
𝑇

𝑛
∑︁

𝑗=1

𝑥∗

𝑗 −

1
𝑛

𝑖
∑︁

𝑗=1

1
𝑛

𝑛
∑︁

𝑗=1

𝑥∗

𝑗 −

1
𝑛

𝑖
∑︁

𝑗=1

′

𝑥∗
𝑗 (cid:170)
(cid:174)
(cid:172)

(cid:19)

𝐵𝐼 𝑀

(cid:18) 1
𝑛

∫ 𝑡

0

𝐵𝐼 𝑀

1
𝑇

𝑋 ∗

𝑠 𝑑𝑠

∫ 𝑇

0

𝐵𝐼 𝑀

1
𝑇

𝑋 ∗

𝑠 𝑑𝑠 −

1
𝑇

𝑥∗
(cid:169)
𝑗 (cid:170)
(cid:173)
(cid:174)
(cid:171)
(cid:172)
(cid:19) (cid:18) 1
𝑇

∫ 1

(cid:18)∫ 1

𝑔∗(𝑠)𝑑𝑠 −

∫ 𝑟

𝑔∗(𝑠)𝑑𝑠

(cid:19) (cid:18)∫ 1

𝑔∗(𝑠)𝑑𝑠 −

⇒ Π

0
This limit together with the limits of (1/𝑛) (cid:205)𝑛

0

0

0
𝑖=1 (𝐵𝐼 𝑀/𝑛) 𝑥∗

∫ 𝑟

0

𝑔∗(𝑠)𝑑𝑠

𝑖 𝑥∗′

𝑖 (𝐵𝐼 𝑀/𝑛) and 𝑅∗

𝑇 give an expres-

sion identical to that obtained by Vogelsang and Wagner (2014) for the inverse term apart from

𝑞 random variable using arguments in Vogelsang and Wagner (2014). The last step is to show

𝑥. Therefore, the limit of the parts of (cid:101)𝑊 ∗ that do not depend on (𝑇/𝑛) (cid:101)𝜆2∗
(𝑇/𝑛) (cid:101)𝜆2∗
𝑥 follow a
𝑢·(cid:101)
𝑢·(cid:101)
𝜒2
𝜆2
𝑢·(cid:101)
𝑥
𝑥 ⇒ 𝜆2
that (𝑇/𝑛) (cid:101)𝜆2∗
𝑥
𝑢·(cid:101)
𝑢·(cid:101)
𝜔2
derivation of the limit of (𝑇/𝑛) (cid:98)

1 (𝑏). While more tedious, the derivation follows the same steps as the
𝑃∗∗
𝑛 in Theorem 3.2 using (3A.1) and (3A.4). Details are omitted.

113

APPENDIX 3B

TABLES AND FIGURES

Table 3B.1 Fixed-𝑏 Right Tail Critical Value Polynomial Coefficients, Parzen Kernel

Panel A. Joint Null Hypothesis That Intercept is Zero (𝛽0 = 0) and Slope is One (𝛽1 = 1).
𝑅2

𝜆3

𝜆4

𝜆5
-.3118

𝜆6
.1336

𝜆7
.8846

𝜆8
-.1857

𝜆9
.1841

𝜆1
.7289

𝜆1
.6333

-.0421 2.6426

.9956

G-cointegration

H-stationary

.2688

.9880

-.0490 5.1346 -2.3769

.1518

-2.1792 3.0649

.2219

.9998

H-cointegration 1.3250 1.4466 -.0802 6.0381 -3.2097

.3246

-.1194

2.8672

.3661

.9987

G-cointegration

H-stationary

𝜆1
.8130

.9475

Panel B. Null Hypothesis That Slope is One (𝛽1 = 1).
𝜆7
-.1950

𝜆5
-1.1113

𝜆3
-.0079

𝜆1
.2969

𝜆4
.7675

𝜆6
.7940

𝜆8
.4250

𝜆9

𝑅2

-.2409 .9258

.3250

-.0078

.7082

.5386

-.01370

1.4406

-.7790

.2669

.9993

H-cointegration 1.3752

.3003

.1324

.1909

-1.9013

.8501

-.0859

.9864

-.3429 .9388

114

Table 3B.2 UIP Tests (US-Japan, 2-year government bond, Δ = 504 days)

√︃

𝐺 ( (cid:98)𝛽)

Daily,
𝑛 = 7678
(cid:98)𝛽1 = 0.011

(cid:101)𝑊 ∗

√︁
Daily,
𝑛 = 7678
(cid:101)𝛽1 = −0.004

√︃

𝐺 ( (cid:98)𝛽)

Weekly,
𝑛 = 1545
(cid:98)𝛽1 = 0.011

(cid:101)𝑊 ∗

√︁
Weekly,
𝑛 = 1545
(cid:101)𝛽1 = −0.005

√︃

𝐺 ( (cid:98)𝛽)

Monthly,
𝑛 = 364
(cid:98)𝛽1 = 0.01

√︁

(cid:101)𝑊 ∗

Monthly,
𝑛 = 364
(cid:101)𝛽1 = −0.005

√︃

𝐺 ( (cid:98)𝛽)

Quarterly,
𝑛 = 124
(cid:98)𝛽1 = 0.013

√︁

(cid:101)𝑊 ∗

Quarterly,
𝑛 = 124
(cid:101)𝛽1 = −0.004

AD

S.E.
𝑡𝛽2=1
Fixed-𝑏 CV
CI 𝑁 (0, 1)
CI Fixed-𝑏

0.611
-2.189
±4.053
(-1.53, 0.86)
(-2.81, 2.14)

S.E.
𝑡𝛽2=1
Fixed-𝑏 CV
CI Fixed-𝑏
(cid:98)𝑏-ratio

0.363
-1.34
±11.377
(-3.62, 4.64)
0.504

S.E.
𝑡𝛽2=1
Fixed-𝑏 CV
CI 𝑁 (0, 1)
CI Fixed-𝑏

0.616
-2.147
±3.917
(-1.53, 0.89)
(-2.73, 2.09)

S.E.
𝑡𝛽2=1
Fixed-𝑏 CV
CI Fixed-𝑏
(cid:98)𝑏-ratio

0.392
-1.222
±10.628
(-3.64, 4.68)
0.479

S.E.
𝑡𝛽2=1
Fixed-𝑏 CV
CI 𝑁 (0, 1)
CI Fixed-𝑏

0.601
-2.209
±4.435
(-1.5, 0.85)
(-2.99, 2.34)

S.E.
𝑡𝛽2=1
Fixed-𝑏 CV
CI Fixed-𝑏
(cid:98)𝑏-ratio

0.288
-1.621
±13.351
(-3.31, 4.38)
0.572

S.E.
𝑡𝛽2=1
Fixed-𝑏 CV
CI 𝑁 (0, 1)
CI Fixed-𝑏

0.618
-2.214
±3.946
(-1.58, 0.84)
(-2.8, 2.07)

S.E.
𝑡𝛽2=1
Fixed-𝑏 CV
CI Fixed-𝑏
(cid:98)𝑏-ratio

0.378
-1.271
±10.789
(-3.56, 4.6)
0.484

Bandwidth Rules

SPJ
NW
OLS (cid:98)𝛽2 = −0.337
0.308
0.54
-4.341
-2.473
±1.98
±5.213
(-0.94, 0.267)
(-1.4, 0.72)
(-0.95, 0.273)
(-3.15, 2.48)

IM-OLS (cid:101)𝛽2 = 0.514
0.462
0.179
-1.053
-2.711
±1.959
±16.557
(-0.39, 1.418)
(-2.46, 3.48)
0.7
0.007
OLS (cid:98)𝛽2 = −0.322
0.509
-2.596
±2.019
(-1.32, 0.676)
(-1.35, 0.706)

0.541
-2.445
±5.03
(-1.38, 0.74)
(-3.04, 2.4)

IM-OLS (cid:101)𝛽2 = 0.521
0.651
0.199
-0.735
-2.405
±2.005
±15.901
(-0.78, 1.827)
(-2.64, 3.69)
0.021
0.671
OLS (cid:98)𝛽2 = −0.327
0.733
-1.812
±2.1
(-1.76, 1.109)
(-1.87, 1.212)

0.541
-2.452
±5.717
(-1.39, 0.73)
(-3.42, 2.77)

IM-OLS (cid:101)𝛽2 = 0.533
0.88
0.134
-0.531
-3.492
±2.184
±18.12
(-1.39, 2.454)
(-1.89, 2.96)
0.05
0.777
OLS (cid:98)𝛽2 = −0.367
0.924
-1.48
±2.292
(-2.18, 1.443)
(-2.48, 1.75)

0.539
-2.537
±5.069
(-1.42, 0.69)
(-3.1, 2.36)

IM-OLS (cid:101)𝛽2 = 0.52
1.056
0.19
-0.455
-2.531
±2.692
±16.046
(-2.32, 3.361)
(-2.52, 3.56)
0.11
0.677

CRT

0.806
-1.658
±2.15
(-1.92, 1.24)
(-2.07, 1.4)

0.959
-0.507
±2.305
(-1.7, 2.72)
0.066

0.807
-1.638
±2.15
(-1.9, 1.26)
(-2.06, 1.41)

0.963
-0.497
±2.305
(-1.7, 2.74)
0.066

0.811
-1.635
±2.15
(-1.92, 1.26)
(-2.07, 1.42)

0.963
-0.485
±2.305
(-1.69, 2.75)
0.066

0.821
-1.666
±2.15
(-1.98, 1.24)
(-2.13, 1.4)

0.977
-0.491
±2.305
(-1.73, 2.77)
0.066

Notes: The null hypothesis is 𝐻0 : 𝛽2 = 1. The rows S.E., 𝑡𝛽2=1, Fixed-𝑏 CV, CI, and (cid:98)𝑏-ratio are reported for (cid:98)𝛽2
and (cid:101)𝛽2, the OLS and IM-OLS estimators respectively. The 𝑡-statistics are computed as
(cid:101)𝑊 ∗ with
the signs equal to the signs of (cid:98)𝛽2 − 1 and (cid:101)𝛽2 − 1 respectively. The value of (cid:98)𝑏 is the same for both test statistics.

𝐺 ( (cid:98)𝛽) and

√︃

√︁

115

Table 3B.3 UIP Tests (US-Japan, 10-year government bond, Δ = 2520 days)

√︃

𝐺 ( (cid:98)𝛽)

Daily,
𝑛 = 5706
(cid:98)𝛽1 = −0.036

(cid:101)𝑊 ∗

√︁
Daily,
𝑛 = 5706
(cid:101)𝛽1 = −0.038

√︃

𝐺 ( (cid:98)𝛽)

Weekly,
𝑛 = 1148
(cid:98)𝛽1 = −0.036

(cid:101)𝑊 ∗

√︁
Weekly,
𝑛 = 1148
(cid:101)𝛽1 = −0.039

√︃

𝐺 ( (cid:98)𝛽)

Monthly,
𝑛 = 271
(cid:98)𝛽1 = −0.036

√︁

(cid:101)𝑊 ∗

Monthly,
𝑛 = 271
(cid:101)𝛽1 = −0.038

√︃

𝐺 ( (cid:98)𝛽)

Quarterly,
𝑛 = 92
(cid:98)𝛽1 = −0.035

√︁

(cid:101)𝑊 ∗

Quarterly,
𝑛 = 92
(cid:101)𝛽1 = −0.038

AD

S.E.
𝑡𝛽2=1
Fixed-𝑏 CV
CI 𝑁 (0, 1)
CI Fixed-𝑏

0.501
0.71
±3.333
(0.37, 2.34)
(-0.31, 3.02)

S.E.
𝑡𝛽2=1
Fixed-𝑏 CV
CI Fixed-𝑏
(cid:98)𝑏-ratio

0.247
2.098
±7.325
(-0.29, 3.33)
0.364

S.E.
𝑡𝛽2=1
Fixed-𝑏 CV
CI 𝑁 (0, 1)
CI Fixed-𝑏

0.491
0.743
±3.407
(0.4, 2.33)
(-0.31, 3.04)

S.E.
𝑡𝛽2=1
Fixed-𝑏 CV
CI Fixed-𝑏
(cid:98)𝑏-ratio

0.243
2.211
±7.741
(-0.34, 3.42)
0.38

S.E.
𝑡𝛽2=1
Fixed-𝑏 CV
CI 𝑁 (0, 1)
CI Fixed-𝑏

0.456
0.786
±3.984
(0.47, 2.25)
(-0.46, 3.17)

S.E.
𝑡𝛽2=1
Fixed-𝑏 CV
CI Fixed-𝑏
(cid:98)𝑏-ratio

0.201
2.493
±10.998
(-0.71, 3.71)
0.491

S.E.
𝑡𝛽2=1
Fixed-𝑏 CV
CI 𝑁 (0, 1)
CI Fixed-𝑏

0.448
0.74
±4.121
(0.45, 2.21)
(-0.51, 3.18)

S.E.
𝑡𝛽2=1
Fixed-𝑏 CV
CI Fixed-𝑏
(cid:98)𝑏-ratio

0.199
2.537
±11.744
(-0.83, 3.85)
0.516

SPJ

Bandwidth Rules
NW
OLS (cid:98)𝛽2 = 1.356
0.139
2.558
±1.983
(1.08, 1.628)
(1.08, 1.631)

0.437
0.814
±4.217
(0.5, 2.21)
(-0.49, 3.2)

0.186
2.789
±12.254
(-0.76, 3.79)
0.534

0.423
0.863
±4.324
(0.54, 2.19)
(-0.46, 3.19)

0.178
3.008
±12.799
(-0.75, 3.82)
0.552

0.363
0.987
±5.12
(0.65, 2.07)
(-0.5, 3.22)

0.143
3.517
±16.231
(-0.81, 3.81)
0.685

0.355
0.934
±5.303
(0.64, 2.03)
(-0.55, 3.21)

IM-OLS (cid:101)𝛽2 = 1.518
0.241
2.146
±1.96
(1.04, 1.991)
0.008
OLS (cid:98)𝛽2 = 1.365
0.226
1.617
±2.025
(0.92, 1.807)
(0.91, 1.822)

IM-OLS (cid:101)𝛽2 = 1.537
0.242
2.216
±2.016
(1.05, 2.025)
0.024
OLS (cid:98)𝛽2 = 1.358
0.349
1.028
±2.141
(0.67, 2.042)
(0.61, 2.105)

IM-OLS (cid:101)𝛽2 = 1.501
0.255
1.968
±2.283
(0.92, 2.083)
0.063
OLS (cid:98)𝛽2 = 1.331
0.447
0.742
±2.318
(0.46, 2.207)
(0.3, 2.367)
IM-OLS (cid:101)𝛽2 = 1.506
0.295
1.712
±2.769
(0.69, 2.323)
0.118

0.142
3.558
±16.863
(-0.89, 3.9)
0.714

See the notes to Table 2.

116

CRT

0.386
0.922
±2.199
(0.6, 2.11)
(0.51, 2.2)

0.27
1.92
±2.431
(0.86, 2.17)
0.081

0.383
0.953
±2.199
(0.61, 2.12)
(0.52, 2.21)

0.271
1.984
±2.431
(0.88, 2.19)
0.081

0.386
0.929
±2.199
(0.6, 2.11)
(0.51, 2.21)

0.266
1.885
±2.431
(0.85, 2.15)
0.081

0.394
0.841
±2.199
(0.56, 2.1)
(0.46, 2.2)

0.282
1.792
±2.431
(0.82, 2.19)
0.081

Figure 3B.1 Null rejection probabilities of 𝐻

(cid:16)

(cid:17)
(cid:98)𝛽

under OU processes, stationary regression

117

0.000.040.080.120.160.200.240.00.10.20.30.40.50.60.70.80.9Null Rejection Probability(a) u=6.9, x=0.1, chi-square cvADSPJNWNW_TuneCRT0.05 level0.000.040.080.120.160.200.240.00.10.20.30.40.50.60.70.80.9Null Rejection Probability(b) u=0.5, x=0.5, chi-square cvADSPJNWNW_TuneCRT0.05 level0.000.040.080.120.160.200.240.00.10.20.30.40.50.60.70.80.9Null Rejection Probability(c) u=6.9, x=0.1, fixed-b cvADSPJNWNW_TuneCRT0.05 level0.000.040.080.120.160.200.240.00.10.20.30.40.50.60.70.80.9Null Rejection Probability(d) u=0.5, x=0.5, fixed-b cvADSPJNWNW_TuneCRT0.05 levelFigure 3B.2 Average bandwidth ratios under OU processes, stationary regression

Figure 3B.3 Theoretical bandwidth ratios

118

0.000.040.080.120.160.200.240.00.10.20.30.40.50.60.70.80.9Average bandwidth ratio(a) u=6.9, x=0.1ADSPJNWNW_TuneCRT0.000.040.080.120.160.200.240.00.10.20.30.40.50.60.70.80.9Average bandwidth ratio(b) u=0.5, x=0.5ADSPJNWNW_TuneCRT0.00.10.20.30.40.50.60.70.80.90.00.10.20.30.40.50.60.7Ratio of Bandwidth to Sample SizeAD (x+u=1)SPJ (x+u=1)AD (x+u=7)SPJ (x+u=7)Figure 3B.4 Finite sample power of 𝐻

(cid:16)

(cid:17)
(cid:98)𝛽

under OU processes, stationary regression

119

0.000.040.080.120.160.200.240.00.10.20.30.40.50.60.70.80.9Power(a) u=6.9, x=0.1, 0=0.02, 1=1.02 ADSPJNWNW_TuneCRT0.000.040.080.120.160.200.240.00.10.20.30.40.50.60.70.80.9Power(b) u=0.5, x=0.5, 0=0.3, 1=1.3 ADSPJNWNW_TuneCRT0.000.040.080.120.160.200.240.00.10.20.30.40.50.60.70.80.9Power(c) u=6.9, x=0.1, 0=0.04, 1=1.06 ADSPJNWNW_TuneCRT0.000.040.080.120.160.200.240.00.10.20.30.40.50.60.70.80.9Power(d) u=0.5, x=0.5, 0=0.95, 1=1.95 ADSPJNWNW_TuneCRTFigure 3B.5 Null rejection probabilities of (cid:101)𝑊 ∗ of IM-OLS under OU processes, cointegrating
regression

120

0.000.040.080.120.160.200.240.00.10.20.30.40.50.60.70.80.91.0Null Rejection Probability(a) u=6.287, =0SPJADNWNW_TuneCRT0.000.040.080.120.160.200.240.00.10.20.30.40.50.60.70.80.91.0Null Rejection Probability(b) u=0.393, =0SPJADNWNW_TuneCRT0.000.040.080.120.160.200.240.00.10.20.30.40.50.60.70.80.91.0Null Rejection Probability(c) u=6.287, =0.75SPJADNWNW_TuneCRT0.000.040.080.120.160.200.240.00.10.20.30.40.50.60.70.80.91.0Null Rejection Probability(d) u=0.393, =0.75SPJADNWNW_TuneCRTFigure 3B.6 Average bandwidth ratios under OU processes, cointegrating regression

121

0.000.040.080.120.160.200.240.00.10.20.30.40.50.60.70.80.91.0Average bandwidth ratio(a) u=6.287, =0SPJADNWNW_TuneCRT0.000.040.080.120.160.200.240.00.10.20.30.40.50.60.70.80.91.0Average bandwidth ratio(b) u=0.393, =0SPJADNWNW_TuneCRT0.000.040.080.120.160.200.240.00.10.20.30.40.50.60.70.80.91.0Average bandwidth ratio(c) u=6.287, =0.75SPJADNWNW_TuneCRT0.000.040.080.120.160.200.240.00.10.20.30.40.50.60.70.80.91.0Average bandwidth ratio(d) u=0.393, =0.75SPJADNWNW_TuneCRTFigure 3B.7 Finite sample power of (cid:101)𝑊 ∗ of IM-OLS under OU processes, cointegrating regression

122

0.000.040.080.120.160.200.240.00.10.20.30.40.50.60.70.80.91.0Power(a) u=6.287, =0.75, 0 = 0.0002, 1 = 1.002SPJADNWNW_TuneCRT0.000.040.080.120.160.200.240.00.10.20.30.40.50.60.70.80.91.0Power(b) u=0.393, =0.75, 0 = 0.001, 1 = 1.01SPJADNWNW_TuneCRT0.000.040.080.120.160.200.240.00.10.20.30.40.50.60.70.80.91.0Power(c) u=6.287, =0.75, 0 = 0.0003, 1 = 1.004SPJADNWNW_TuneCRT0.000.040.080.120.160.200.240.00.10.20.30.40.50.60.70.80.91.0Power(d) u=0.393, =0.75, 0 = 0.002, 1 = 1.03SPJADNWNW_TuneCRTCHAPTER 4

THE DISTRIBUTION OF REALIZED US CORPORATE BOND RETURN VOLATILITY

4.1

Introduction

Volatility in the US corporate bond market (especially, daily volatility) has been less frequently

studied despite of its huge importance for the financial sector, macroeconomic conditions, and

portfolio management. The reasons could be that (1) US corporate bonds are illiquid, and (2) their

transactions are irregularly spaced.

Due to these characteristics and the high-frequency nature (recorded every second) of corporate

bond transaction data, the distribution of the volatility of corporate bonds has not been explored

extensively. Campbell and Taksler (2003) studied the effect of equity volatility on corporate bond

yields, but it is not about the volatility of corporate bonds.

To address the irregular price movements and illiquid transaction behavior of corporate bonds,

I utilize a Compound Poisson Process (CPP) to model the price dynamics of US corporate bonds.

CPP can well describe the irregular price movements and illiquid transaction behavior of corporate

bonds. Although the current continuous time stochastic diffusion model can take the irregularity

of US corporate bond transactions into account, it does not effectively describe the illiquidity of

US corporate bonds. CPP is designed to describe the behavior of a continuous stochastic process

which has random discrete jumps over time. Since its main focus is on the discrete change of the

process by the arrival of the random jump, it is adequate for modeling the prices of illiquid assets.

Unlike US stock trades, the US corporate bonds are not actively traded during a day, mostly showing

random transactions less than hundreds during a day according to US corporate bond transaction

data. Therefore, CPP is an appropriate process to describe the price dynamics of US corporate

bonds.

To estimate the volatility of corporate bonds, I utilize realized (daily) volatility suggested by

Andersen et al. (2001a). The realized volatility uses intraday returns measured at a fixed regular

interval (e.g. 5 minutes, 10 minutes) to estimate daily volatility of an asset. I slightly modify the

calculation of the realized volatility to adjust it to accommodate a CPP setting.

123

In Section 4.2, I review the RV and related literature. In Section 4.3, I explain the CPP setting

for US corporate bonds and slightly modify the RV under the setting. Then, I conduct Monte Carlo

simulations for the RV in Section 4.4.

For empirical analysis, I utilize the richness of Trade Reporting and Compliance Engine

(TRACE) of Financial Industry Regulatory Authority (FINRA). Section 4.5 explains the data

used for empirical analysis. The data allows one to obtain 99% of every bond transaction of US

corporate bond market which is recorded every second. Additionally, bond characteristics data for

TRACE can be obtained using Refinitiv Workspace. With the datasets, in Section 4.6, I analyze the

distributions of the RV of US corporate bonds. I investigate not only the unconditional distribution

of the RV of US corporate bonds, but also the conditional distribution of the RV using the bond

characteristics. I also provide an analysis for linkages between the corporate bond volatility and

returns of other financial instruments in Section 4.6. Then I conclude in Section 4.7.

4.2 Preliminaries

Andersen et al. (2001a,b, 2003) proposed realized volatility (RV), which is an ex-post, model-

free volatility measure. The method starts from the continuous time diffusion model given by

𝑑𝑃𝑡 = 𝜇𝑑𝑡 + 𝜎𝑡 𝑑𝑊𝑡,

(4.1)

where 𝑃 is a logarithmic price of the asset and 𝑊 is a standard Brownian motion. Under this model,

the integrated volatility (IV) is defined as

𝐼𝑉 =

∫ 𝑡

0

𝜎2

𝑠 𝑑𝑊𝑠.

They proposed the realized volatility as an estimator for IV, given by

𝑅𝑉 =

𝑚
∑︁

𝜏=1

𝜏,
𝑟 2

(4.2)

(4.3)

where 𝑟𝜏 = 𝑝𝜏 − 𝑝𝜏−1 and {𝑝𝜏}𝑚

𝜏=1 is a sequence of intraday log-prices with a fixed interval, such

as 5 minutes. In Andersen et al. (2003), they showed that RV is a consistent estimator for IV using

semi-martingale theory in Protter (1990). Using their seminal work about RV, many subsequent

124

studies extended the continuous time diffusion model by adding a jump component giving

𝑑𝑃𝑡 = 𝜇𝑑𝑡 + 𝜎𝑡 𝑑𝑊𝑡 + 𝑑𝐽𝑡,

(4.4)

where 𝐽 is a pure jump Lèvy process. As shown in subsequent studies, this can be expressed as a

Brownian semi-martingale with jumps given by

𝑑𝑃𝑡 = 𝜇𝑑𝑡 + 𝜎𝑡 𝑑𝑊𝑡 + 𝜅𝑡 𝑑𝑞𝑡,

(4.5)

where 𝜅𝑡 is size of the jump and 𝑑𝑞𝑡 is 1 when there is a jump and 0 if there is no jump at time 𝑡.

Thus, the return of an asset on interval [0, 𝑡] is expressed as

𝑟𝑡,0 = 𝜇𝑡 +

∫ 𝑡

0

𝜎𝑠𝑑𝑊𝑠 +

𝑁𝑡
∑︁

𝑗=1

𝑘 𝑗 ,

(4.6)

where the return of an asset is computed as 𝑟𝑡,0 = 𝑃𝑡 − 𝑃0. Here 𝑁𝑡 represents a number of jumps

on the interval (0, 𝑡], following a poisson process with intensity 𝜆 with jump size of 𝑘 𝑗 . Recall that

IV of the continuous time diffusion process is

𝐼𝑉 =

∫ 𝑡

0

𝜎2

𝑠 𝑑𝑊𝑠.

(4.7)

Additionally, it has been pointed out by the previous literature that RV converges to the following

expression under the model with a jump component,

𝑅𝑉

𝑝
→

∫ 𝑡

0

𝜎2

𝑠 𝑑𝑊𝑠 +

𝑁𝑡
∑︁

𝑗=1

𝜅2
𝑗

(4.8)

as 𝑚 → ∞. Thus, in order to estimate the integrated volatility, Barndorff-Nielsen and Shephard

(2004) devise bipower variation (BV) as a consistent estimator for integrated volatility given by

𝐵𝑉 =

𝜋

2

𝑚
∑︁

𝜏=2

𝑟𝜏𝑟𝜏−1

𝑝
→

∫ 𝑡

0

𝜎2

𝑠 𝑑𝑊𝑠

(4.9)

as 𝑚 → ∞. Hence, BV can be a consistent estimator when the jumps exist in the price dynamics

of the asset.

125

4.3 Model

Stochastic volatility models for assets like stocks and exchange rate typically start with a

continuous time diffusion process for the logarithmic price (𝑃𝑡) of an asset given by

𝑑𝑃𝑡 = 𝜇𝑑𝑡 + 𝜎𝑡 𝑑𝑊𝑡 .

(4.10)

As I discussed in the previous section, one can extend the process to include jumps given by

𝑑𝑃𝑡 = 𝜇𝑑𝑡 + 𝜎𝑡 𝑑𝑊𝑡 + 𝜅𝑡 𝑑𝑞𝑡 .

(4.11)

The processes are based on continuous time diffusion models valid for assets like stocks and

exchange rates which are heavily traded within time intervals. In contrast, US corporate bonds

are illiquid, and thus, their transactions are irregularly spaced. As they are illiquid, the price

of a bond is constant for some time intervals and shows sudden discrete jumps at irregular time

points. Therefore, the model for the price of a corporate bond should exclude continuous part for

its diffusion model, including only the jump part in its diffusion model for the price. To model this

illiquidity and irregularity of the corporate bond price, I adopt a Compound Poisson Process (CPP)

for the bond price diffusion model. The process 𝑃𝑡 is a Compound Poisson Process and is defined

as

𝑁𝑡
∑︁

𝜅 𝑗 ,

𝑃𝑡 =

(4.12)

𝑗=1
where 𝑁𝑡 follows a poisson process (which is the number of jumps) with an intensity parameter 𝜆

and each 𝜅 𝑗 is an 𝑖.𝑖.𝑑. random variable (which represents a jump size). Hence, utilizing the CPP

setting, my model for the (logarithmic) price of a US corporate bond at time 𝑡, 𝑃𝑡, is suggested as

𝑃𝑡 = 𝑃0 +

𝑁𝑡
∑︁

𝑗=1

𝜅 𝑗 ,

(4.13)

where I can think of the discrete jump size 𝜅 𝑗 as the percentage change (return) of the bond price

at that time point. The return of a US corporate bond during the interval (0, 𝑡] is expressed as

𝑟𝑡,0 =

𝑁𝑡
∑︁

𝑗=1

𝜅 𝑗 .

126

(4.14)

Next, I slightly modify RV for the CPP setting to estimate the volatility of US corporate bonds.

First, I define integrated volatility (IV) for US corporate bond price using the CPP setting. I start

by modifying the CPP of equation 4.13 by adding some assumptions. For the poisson process

𝑁𝑡 in equation 4.13, I assume an inhomogeneous poisson process with intensity 𝜆(𝑡) which is

depending on time, rather than a homogeneous poisson process with a constant intensity rate 𝜆,

since the transaction of financial assets shows the diurnal pattern that the frequency of the trade is

dependent on time. For the size of the jump of the log price (discrete change), I assume that the

jump size 𝜅 𝑗 follows normal distribution, thus 𝜅 𝑗 ∼ 𝑁 (0, 𝜙2

𝑗 (𝑡)), where 𝜙2

𝑗 (𝑡) of the CPP model for

the bond price corresponds with the instantaneous variance of the continuous time diffusion model.

Therefore, the integrated volatility of US corporate bond price on interval (0, 𝑡] is defined as

𝐼𝑉 𝑏𝑜𝑛𝑑

(0,𝑡] =

𝑁𝑡
∑︁

𝑗=1

𝜙2
𝑗 .

(4.15)

Then, I modify the RV to estimate the volatility of US corporate bonds under the CPP setting as

follows.

𝑅𝑉 =

𝑁𝑡
∑︁

𝑗=1

𝑗 .
𝜅2

(4.16)

I use this RV for Monte Carlo simulations and empirical analysis for US corporate bonds throughout

the paper. Notice that the suggested RV can be interpreted as the sum of the squares of returns (the

sum of the squares of discrete jumps in (logarithmic) bond price) at some time points during the

time interval (0, 𝑡].

4.4 Monte Carlo Simulation

To investigate the properties of the RV estimator under the CPP setting, I conduct Monte Carlo

simulations. I generate data using the below CPP setting with three different models for volatility

modeling. For the first model for the volatility (Case (1)), I assume that the variance of the jump

size, 𝜙2

𝑗 (𝑡), does not change over time within a day. Thus, the volatility is a constant within a

day for Case(1). In the second model (Case (2)), I assume that 𝜙2

𝑗 (𝑡) is randomly generated from

uniform distribution, which means that the instantaneous volatility of US corporate bond changes

every second within a day, following uniform distribution. In the third model (Case (3)), 𝜙2

𝑗 (𝑡)

127

changes every second similar to the second model, but it follows Heston model.

𝑃𝑡 = 𝑃0 +

𝑁𝑡
∑︁

𝑗=1

𝜅 𝑗 ,

𝑁𝑡 is a poisson process with intensity 𝜆(𝑡),

𝑗 (𝑡)),
𝜅 𝑗 ∼ 𝑁 (0, 𝜙2
𝑁𝑡
∑︁

𝑅𝑉 =

𝑗 ,
𝜅2

𝑗=1
𝑁𝑡
∑︁

𝑗=1

𝑗 ,
𝜙2

𝐼𝑉 =

(4.17)

(4.18)

(4.19)

(4.20)

(4.21)

where 𝑁𝑡 is the number of jumps (trades) within a day. As discussed, I introduce three different

models for volatility modeling for the simulations. Case (1) for volatility modeling is given by

𝜙2

𝑗 (𝑡) = 𝜙2

𝑑. Case (2) is modeled as 𝜙2

𝑗 (𝑡) ∼ 𝑢𝑛𝑖 𝑓 [0, 1]. Case (3) follows the Heston model and is

given by 𝑑𝜙2

𝑡 = 𝜅(𝜃 − 𝜙2

𝑡 ) + 𝜂𝜙𝑡 𝑑𝑊𝑡, where 𝑊𝑡 is standard Brownian motion.

I use the “Thinning Algorithm" for generating poisson process where 𝑡 is measured in seconds.

I conduct 5000 times (days) of simulations for each case. The closed form of 𝜆(𝑡) will not affect the

results, but it is set as a quadratic equation to accommodate the diurnal pattern of financial assets,

called the “Volatility Smile." I evaluate the finite sample performance of the RV as an estimator

of IV using mean absolute percentage error (MAPE) = 1
𝑛

(cid:205)𝑛

𝑡=1 | 𝑅𝑉−𝐼𝑉

𝐼𝑉

|. I also have three different

cases for the mean of intraday observations (the mean number of jump occurrences in a day) for

the simulations. I call it ‘Illiquid Bond Market’ when the mean of intraday observations is set to

26.6, and ‘Liquid Bond Market’ when the mean is set to about 60.3. When the mean is set to about

146.7, I call it ‘Thick Bond Market’.

Table 4A.1 reports the results of the Monte Carlo simulations. Based on the simulations, the

RV obtains comparably low MAPE under Case (1) than under other cases across the three different

averages of intraday observations. The MAPEs for all cases decline as the average of the intraday

observations grows, which is expected. The RV under Case (2) shows 19% of MAPE if the mean

of the observations is about 60 for a day. The MAPE for Case (2) is around 12% when having 146

128

jumps. The RV under Case (3) represents 16% of MAPE when the average of the observations is

about 60 for a day. The MAPE for Case (3) declines around 10% when having 146 jumps.

4.5 Data

The data for the empirical analysis of realized US corporate bond return volatility is obtained

from the Trade Reporting and Compliance Engine (TRACE) of the Financial Industry Regulatory

Authority (FINRA). I use two different databases to gather the data. The first database is the Wharton

Research Data Services (WRDS) where I acquire high-frequency transaction data of US corporate

bonds in TRACE. The transactions are recorded every second. The data includes the price, date,

and time of over-the-counter secondary market corporate bond transactions, which covers more

than 99% of US corporate bond transactions. The second database is Refinitiv Workspace, where

I obtain the data for the corporate bond characteristics such as credit ratings and original issued

amounts. The credit ratings data includes ratings from both Moody’s and Fitch Ratings. The

two datasets are easily linked together using the unique 9 digit number given to US corporate

bonds called the Committee on Uniform Securities Identification Procedures (CUSIP). My analysis

is from January 1, 2013, to December 31, 2018. For simple statistics of the characteristics of

interest, the means of the yield rate and the issued amount for the bonds during the period are

6.41 and 2,120,885,360.13, respectively. The standard deviations of each variable are 6.31 and

2,035,185,824.37.

I clean the TRACE dataset following Dick-Nielsen (2009). I remove cancellation transactions

and their original transaction, corrections and their original transaction, reversals and their original

transaction, and lastly, agency transactions (double counting problem). About 20% of 100 million

transactions in the data is removed by the steps.

To ensure more than the average daily observation of 60 for each bond (which could correspond

to the liquid market in the Monte Carlo simulations), I select bonds with more than 15,000

transactions per year, as 15,000 divided by 250 trading days in a year equals 60. I also remove

bonds with trading days less than 200 days.

129

4.6 Empirical Results

4.6.1 The Unconditional Distribution of Corporate Bond Volatility

I construct the unconditional distribution of the daily realized US corporate bond return volatil-

ity. Similar to the way Andersen et al. (2001b) constructed the unconditional distribution for 30

DJIA stocks, I report the distribution of the daily realized US corporate bond return volatility. First,

I calculate the daily realized volatility from the chosen corporate bonds which satisfy the standards

discussed in the previous section. Next, I calculate and report the mean, standard deviation, skew-

ness, and kurtosis of the daily realized volatilities for each combination of corporate bond and year.

Then, I construct the unconditional distribution of the mean of the daily RV. Table 4A.4 shows

the numbers (percentiles) describing the mean, standard deviation, skewness and kurtosis of the

daily realized volatilities. As shown in Table 4A.4, the median for the mean of the daily realized

volatilities is about 0.0042 and the mean value of it is about 0.0052. I also report the percentiles for

statistics from the logarithm of the standard deviation. The standard deviation is calculated as the

root of realized volatility. Its median is about -2.931 and the mean is -3.052. Figure 4A.1 shows the

unconditional distribution of the series of the mean of daily RVs for each combination of corporate

bond and year. The distributions is left-skewed.

In figure 4A.2, I compare the unconditional

distribution of the mean of log standard deviation of RV with standard normal distribution with

mean of -3.

4.6.2 The Conditional Distribution of Corporate Bond Volatility by Bond Characteristics

In this section, I construct the conditional distribution of the mean of daily RV by using bond

characteristics from the bond characteristics dataset. The characteristics that I use are credit rating,

(original) issued amount, and yield rate. For credit rating, I divide the volatility series into two

groups, ‘Investment Grade’ bonds and ‘High Yield’ bonds.

I classify a bond as a ‘Investment

Grade’ bond, if either a credit rating from Moody’s for the bond is above (or equal to) Baa3 or a

credit rating from Fitch for the bond is above (or equal to) BBB-. If either a credit rating from

Moody’s for the bond is below Baa3 or a credit rating from Fitch for the bond is below BBB-, then

I classify the bond as a ‘High Yield’ bond. If I have both credit ratings (from Moody’s and Fitch)

130

for the bond and the credit ratings conflict, I follow the rating from Moody’s for the classification.

Issued amount is the size of bond issued. I divide the daily volatilities as two groups by the median

value of the issued amount of the sample. Finally, I used yield rate as the one of bond characteristics

to divide the volatilities as two group. The criteria for the grouping is 5%. The bonds with a yield

rate higher than 5% are in one group, and the bonds with a yield rate less than 5% are in another

group.

As one can see in Table 4A.5, the conditional distributions of the mean of daily RV by bond

characteristics vary between the groups. The series of the mean of daily bond volatilities with

credit rating of investment grade has less mean and median than the series of high yield, which

implies that investment grade bonds are less volatile than high yield corporate bonds. Surprisingly,

the group of average daily volatility for less issued bonds have 3 times higher median than that of

the largely issued bonds group. The group of bonds with yield rate higher than 5% also have higher

value for the median than the group with less than 5%.

I also compare the conditional distributions of different bond characteristics using the graphs.

The graphs shows significant difference between the distributions with different characteristics as

I confirmed in the table as well. In Figure 4A.3, I draw two different distributions by credit rating

criteria, where the distribution with blue color represents the conditional distribution of mean of

daily RV of investment grade bonds and orange colored distribution represents that of high yield

bond. One can see that the conditional distribution of the high yield bonds is slightly shifted to the

right relative to that of the investment grade bonds, which means that high yield bonds are more

volatile than investment grade bonds. The difference of the distributions by bond characteristics

seems much more obvious in Figure 4A.4 and Figure 4A.5. The conditional distributions of small

size issued bonds in Figure 4A.4 and high yield rate bonds in Figure 4A.5 not only have bigger

mean and median, but also exhibit the different shapes with fatter and longer tails compared to

those of large-size issued bonds and low yield rate bonds. These figures can provide evidence that

bond characteristics are the main factors determining the volatility of corporate bonds.

131

4.6.3 The Conditional Distribution of Corporate Bond Volatility by Linkages with Other

Financial Markets

In this section, I examine the conditional distribution of the averaged daily realized volatility

of corporate bonds on a given day, considering the market conditions of other financial markets on

the same day. I consider three financial assets. The first one is the returns of the S&P 500 index,

which is representative market index for the US stock market. Secondly, I choose the returns of the

CBOE Volatility Index (VIX.), which is an implied volatility measure of the US stock market and

is often called as a fear gauge of the financial markets. Lastly, I use returns of US 30 year treasury

bond yield as a leading indicator for other types of bonds. The data for three financial instruments

is from Center for Research in Security Prices (CRSP). The data period is the same as the period

mentioned for the corporate bond data in Section 4.5.

I derive a time series index for the US corporate bond market volatility by averaging the RV of

corporate bonds for each day, which represents the degree of intraday volatility of the US corporate

bond market for that day. Then, I investigate how the (averaged) realized volatility index of the

US corporate bond market behaves under different conditions in other financial markets. One can

see that the volatility of the corporate bond market for that day shows different patterns depending

on the conditions of the other financial markets.

In Table 4A.6, when there are sizable price

movements in the other financial instruments, the means for the index of the corporate bond market

shows larger values compared to the means on days with relatively smaller price movements. When

there is more than a 1% price change in the S&P 500 index, whether it is a negative or a positive

shock, the mean for the volatility index of corporate bond market is larger compared to the mean

on days when the S&P 500 price change is within 1%. For CBOE VIX and 30 year T-Bill, one

observes similar patterns where sizable price changes over the particular thresholds (7% for VIX

and 0.025% for T-Bill) in these instruments yield a higher volatility index for the corporate bond

market. This implies that the conditional distributions of the index for the corporate bond market

by the size of price movements in other financial markets can vary.

Figure 4A.6 shows the conditional distribution of the z-scores of (averaged) realized volatility

132

index of the US corporate bonds for the days when the absolute value of the (z-scored) daily return

of the S&P 500 is less than 1, meaning a small price change. Figure 4A.7 provides the conditional

distribution of the (z-scored) index for the days when the absolute value of the (z-scored) daily

return of the S&P 500 is over 1, meaning that there are sizable shocks in the US stock market. One

can see that the values of the distribution are centered around zero with thin tails. That means when

there is a small price change in the S&P 500, the volatility index of the US corporate market tends

to be around the average level, not an extreme value. But, as one can find in Figure 4A.7, when

the absolute value of the z-scored daily return of the S&P 500 is larger than 1, which indicates the

US stock market has a sizable positive or negative shock on that day, the conditional distribution

of the volatility index for the US corporate bond market displays fat tails. Especially, in Figure

4A.7, when the z-scored S&P 500 daily return is less than -1 (a big negative shock), the mass of

distribution is not centered around the average, and the tail of the conditional distribution of the

volatility index is fatter compared to the distribution in Figure 4A.6. This implies that when there

is a substantial shock to the US stock market, the US corporate bond market tends to show larger

volatility on the same day, as it tends to have extreme values, as the fat tail shows. This could

provide evidence of the linkage between the US corporate bond market and the US stock market,

as Campbell and Taksler (2003) explored similar market linkage with analyzing the effect of equity

volatility on corporate bond yields.

4.7 Conclusion

Realized volatility has been an important measure for the volatility of financial assets since its

development in Andersen et al. (2001a,b, 2003). I investigate the distributions of the daily realized

volatility of US corporate bonds using the RV method, slightly modified for a CPP setting. To

describe the price dynamics of US corporate bonds, I employ a CPP setting to accommodate their

irregularity and illiquidity, and then I slightly modify the RV to fit the CPP setting for US corporate

bonds.

I empirically analyze the distribution of the RV of US corporate bonds using the high frequency

transaction data from TRACE of FINRA. Utilizing the advantage of millions of corporate bond

133

transactions recorded every second, I calculate the daily RV of bonds by using all transactions

during a day and construct their unconditional distribution. Linking the transaction data with the

bond characteristics data from TRACE via Refinitiv Workspace, I build the conditional distributions

of daily RV by bond characteristics. I find that there are significant differences in the shapes of

the conditional distributions constructed for the groups with different characteristics. The group

of bonds with high yield ratings in credit rating shows a higher mean and median for the mean of

the daily RV than the group with investment grades. The group of bonds with a small-size issued

amount and the group with a high yield rate display higher mean and median values, less centered

distributions, and fatter tails than the group with a large-size issued amount and the group with a

low yield rate, respectively.

I also examine linkages between the US corporate bond market and other financial markets. I

choose three representative instruments in the US stock market and the US treasury bond market.

I calculate the cross sectional mean of daily RV of corporate bonds for the realized volatility index

for each day. Then, I find that the group of days with greater price changes in those financial

instruments has higher realized volatility for the US corporate bond market than the group of

days with relatively small price movements in the financial instruments. I also discover that the

conditional distribution of the RV of the corporate bond market for the days with greater shocks in

the S&P 500 displays a less centered distribution and fatter tails, especially when the type of the

shocks is negative.

134

BIBLIOGRAPHY

Andersen, T. G., Bollerslev, T., Diebold, F. X., and Ebens, H. (2001a). The distribution of realized

stock return volatility. Journal of financial economics, 61(1):43–76.

Andersen, T. G., Bollerslev, T., Diebold, F. X., and Ebens, H. (2001b). The distribution of realized

stock return volatility. Journal of financial economics, 61(1):43–76.

Andersen, T. G., Bollerslev, T., Diebold, F. X., and Labys, P. (2003). Modeling and forecasting

realized volatility. Econometrica, 71(2):579–625.

Barndorff-Nielsen, O. E. and Shephard, N. (2004). Power and bipower variation with stochastic

volatility and jumps. Journal of financial econometrics, 2(1):1–37.

Campbell, J. Y. and Taksler, G. B. (2003). Equity volatility and corporate bond yields. The Journal

of finance, 58(6):2321–2350.

Dick-Nielsen, J. (2009). Liquidity biases in trace. The Journal of Fixed Income, 19(2):43.

Protter, P. (1990). Stochastic differential equations. Stochastic Integration and Differential Equa-

tions: A New Approach, pages 187–284.

135

APPENDIX 4A

TABLES AND FIGURES

Table 4A.1 Monte Carlo Simulation Result

Avg Intraday Obs ≈ 26.6
Illiquid Bond Market

Avg Intraday Obs ≈ 60.3
Liquid Bond Market

Avg Intraday Obs ≈ 146.7
Thick Bond Market

MC

Case(1) Case(2) Case(3) Case(1) Case(2) Case(3) Case(1) Case(2) Case(3)

MAPE 0.0403

0.2865

0.2432

0.0187

0.1935

0.1582

0.0058

0.1255

0.1021

Notes. The simulation is done 5,000 times (days). For the illiquid bond market, the mean value for jumps (intraday
transaction observations) is set at 26.6 per day. (The number of jumps in the simulation corresponds to the number
of intraday transaction observations in real data). For the liquid bond market, the mean value for jumps is set at
60.3 per day. In the thick bond market, the mean value for jumps is set at 146.7 per day.

Table 4A.2 Example of Cancellation of bond transaction

CUSIP

Company Symbol

Date

Time

Price Yield Trade Status Message No.

17275RAX0

17275RAX0

17275RAX0

CSCO

CSCO

CSCO

20190401 12:47:00 99.833 2.5906

20190401 12:47:00 99.833 2.5906

20190401 13:33:39 99.789 2.6281

T

X

T

49568

49568

57659

Table 4A.3 Example of Correction of bond transaction

CUSIP

Company Symbol

Date

Time

Price Volume Trade Status Message No. Orig Msg No.

023770AA8

023770AA8

023770AA8

AAL

AAL

AAL

20190402 12:57:52 98.27 500000

20190402 12:57:52 98.27 500000

20190402 12:57:52 98.17 414041.7

T

C

R

57113

57113

58249

57113

136

Table 4A.4 The unconditional daily corporate bond volatility distribution

Volatility

Log St.dev

Bond Mean

Std

Skew

Kurt

Mean

Std

Skew

Kurt

Min

0.10

0.25

0.50

0.75

0.90

0.0001 0.0002

0.3008

-0.9639

-4.8942 0.2401 -8.2086

-1.3260

0.0008 0.0007

0.8641

0.8597

-3.8394 0.3523 -3.2892

0.2771

0.0020 0.0018

1.2330

1.9862

-3.3500 0.4109 -1.7987

1.0867

0.0042 0.0034

1.8355

4.9924

-2.9313 0.4787 -1.1285

2.9454

0.0068 0.0054

3.2182

16.2460

-2.6569 0.5547 -0.6820

8.1248

0.0100 0.0105

5.2471

41.1743

-2.4399 0.6715 -0.2868

21.7149

Max

0.0283 0.0341 14.8560 230.3933 -1.9055 1.5826 0.9421 100.0411

Mean 0.0052 0.0047

2.6434

16.4441

-3.0517 0.5036 -1.4866

7.9331

Std

0.0047 0.0050

2.2749

30.6782

0.5592 0.1582 1.3769

14.0918

Notes. Volatility is computed by the RV method. The log of the standard deviation is the log of the square root of
the volatility calculated by the RV method.

Table 4A.5 Daily Corporate Bond Volatility Distribution by Bond Characteristics

Credit Rating

Issue Size

Yield Rate

Invest Grade

High Yield

Large Size

Small Size

Higher than 5% Lower than 5%

Bond Mean

Std Mean

Std Mean

Std Mean

Std Mean

Std Mean

Std

Min

0.10

0.25

0.50

0.75

0.90

0.0001 0.0002 0.0008 0.0012 0.0001 0.0002 0.0001 0.0004 0.0003 0.0003 0.0001 0.0000

0.0005 0.0005 0.0023 0.0022 0.0004 0.0005 0.0023 0.0021 0.0029 0.0023 0.0004 0.0005

0.0015 0.0012 0.0032 0.0029 0.0010 0.0010 0.0036 0.0032 0.0042 0.0033 0.0008 0.0008

0.0037 0.0030 0.0048 0.0041 0.0024 0.0020 0.0059 0.0043 0.0062 0.0046 0.0019 0.0016

0.0066 0.0051 0.0076 0.0061 0.0050 0.0038 0.0086 0.0073 0.0095 0.0077 0.0035 0.0024

0.0100 0.0085 0.0108 0.0123 0.0073 0.0056 0.0128 0.0134 0.0159 0.0143 0.0057 0.0036

Max

0.0253 0.0297 0.0283 0.0341 0.0200 0.0303 0.0283 0.0341 0.0283 0.0341 0.0129 0.0108

Mean 0.0049 0.0041 0.0062 0.0062 0.0035 0.0029 0.0070 0.0064 0.0079 0.0068 0.0025 0.0019

Std

0.0047 0.0043 0.0047 0.0063 0.0033 0.0035 0.0052 0.0057 0.0055 0.0061 0.0023 0.0017

Notes. The criteria for dividing the two groups based on credit rating are outlined in Section 4.6.2. Large size
bonds are defined as bonds with an issued amount higher than the median. There are also two groups by yield rate.
One is a group of bonds with a yield rate higher than 5%. The other group has a yield rate less than 5%.

137

Table 4A.6 Daily Bond Market Volatility by Different Market Regime

S&P 500

CBOE VIX

T-Bill 30Yrs

>1% <-1% Btw

>7% <-7% Btw >0.025% <-0.025% Btw

Mean

Std

0.0063 0.0063 0.0055 0.0061 0.0060 0.0055

0.0059

0.0060

0.0054

0.0036 0.0044 0.0031 0.0040 0.0035 0.0030

0.0031

0.0038

0.0032

Obs (days)

143

125

1242

219

207

1084

378

418

704

Notes. The statistics in the table (mean, standard deviation, number of observations) are for the realized volatility
index, which represents the average daily RV of US corporate bonds on each day, for the US corporate bond market
under different conditions of other financial instruments. Each column shows the statistics for different conditions
of price changes (daily returns) of the S&P 500, VIX, and T-Bill. For example, the first column reports the statistics
for the volatility index on the days when the S&P 500 has changed by more than 1%, and the second column shows
the statistics for the volatility index on the days when the S&P 500 has changed by less than -1%. The third column
shows the statistics for the days when the return of the S&P 500 is between -1% and 1%.

138

Figure 4A.1 The unconditional distribution of the mean of the daily RV of corporate bonds

Figure 4A.2 The unconditional distribution of the mean of log of standard deviation

139

í0HDQRI'DLO\5HDOL]HG9RODWLOLWLHVRI&RUSRUDWH%RQG'HQVLW\íííííí0HDQRI/RJ6WDQGDUG'HYLDWLRQVRI&RUSRUDWH%RQG'HQVLW\6WDQGDUG1RUPDOμ=−30HDQRI/RJ6WGFigure 4A.3 The conditional distributions of the mean of the daily RV by credit rating

Figure 4A.4 The conditional distributions of the mean of the daily RV by size of issued amount

140

í0HDQRI'DLO\5HDOL]HG9RODWLOLWLHVRI&RUSRUDWH%RQG'HQVLW\,QYHVWPHQW*UDGH+LJK<LHOGí0HDQRI'DLO\5HDOL]HG9RODWLOLWLHVRI&RUSRUDWH%RQG'HQVLW\/DUJH6L]H,VVXHG6PDOO6L]H,VVXHGFigure 4A.5 The conditional distributions of the mean of the daily RV by yield rate

Figure 4A.6 The conditional distributions of the mean of the daily RV of corporate bonds when the
return of the S&P 500 is small

141

í0HDQRI'DLO\5HDOL]HG9RODWLOLWLHVRI&RUSRUDWH%RQG'HQVLW\/RZ<LHOG5DWH+LJK<LHOG5DWHRealized Volatility of bonds 1.5 1.0 0.50.00.51.01.52.0S&P 500 Return 1.5 1.0 0.50.00.51.01.52.0Frequency020406080100120140Figure 4A.7 The conditional distributions of the mean of the daily RV of corporate bonds when the
return of the S&P 500 is big

142

Realized Volatility of bonds−1.5−1.0−0.50.00.51.01.52.0S&P 500 Return−1.5−1.0−0.50.00.51.01.52.0Frequency051015202530