Introduction

Main Results

Numerical Experiments

Conclusion

High Quantile Regression for Tail Dependent Time Series1
Ting Zhang
tingzhang@uga.edu
Department of Statistics
University of Georgia

June 27, 2023

BU-Keio-Tsinghua Workshop 2023, Boston, MA

1

Research supported by NSF CAREER Award DMS-2131821.
Ting Zhang (tingzhang@uga.edu)

High Quantile Regression for Tail Dependent Time Series

Introduction

Main Results

Numerical Experiments

Conclusion

What is quantile regression?
Suppose we observe (yi , xi ), i = 1, . . . , n, where
I yi : response variable;
I xi : a set of explanatory variables.

Traditional linear regression:
I concerns the effect of xi on the mean of yi ;I regression coefficient can be estimated by least squares:nX2β̌ n = argmin(yi − x>i η) .
η∈Rp

i=1

Quantile regression (Koenker and Bassett, 1978, Econometrica):
I concerns the effect of xi on quantiles of yi ;I regression coefficient for the (1 − α)-th quantile can be estimated byβ̃ n = argmin
η∈Rp

n
X

φ1−α (yi − x>
i η),

i=1

+

where φ1−α (y) = (1 − α)y + α(−y)+ is the check function.
Ting Zhang (tingzhang@uga.edu)

High Quantile Regression for Tail Dependent Time Series

Introduction

Main Results

Numerical Experiments

Conclusion

Why high quantiles?
Traditional quantile regression:
I the quantile level 1 − α is assumed to be fixed.
I Examples:
median: 1 − α = 0.5;
first quartile: 1 − α = 0.25;
third quartile: 1 − α = 0.75;...

Recent applications require the study of tail phenomena:
I Abrevaya (2001, Empirical Economics): the analysis of low
percentiles of the birthweight distribution;
I Tsai (2012, J. Int. Financial Mark. Inst. Money): relationshipbetween stock price index and exchange rates when exchange rates
are extremely high or low;
I Elsner et al. (2008, Nature): understanding temporal trends ofstrong tropical cyclones;
I Rhines et al. (2017, Journal of Climate): trend analysis on
temperatures of the coldest days in North America;
I ...
Ting Zhang (tingzhang@uga.edu)

High Quantile Regression for Tail Dependent Time Series

Introduction

Main Results

Numerical Experiments

Conclusion

How to accommodate high quantiles?
Traditional quantile regression:n
Xβ̃ n = argmin
φ1−α (yi − x>
i η),
η∈Rp

i=1

where the quantile level 1 − α is treated as fixed.
High quantile regression:β̂ n = argmin
η∈Rp

n
X

φ1−αn (yi − x>
i η),

i=1

where the quantile level 1 − αn → 1 as n → ∞.
I allows the quantile level to approach the unit as the sample size
increases to capture the tail phenomena;
I theoretical properties can be more challenging due to the double
asymptotics;
I convergence rate may no longer be the universal n1/2 as in theconventional setting.
Ting Zhang (tingzhang@uga.edu)

High Quantile Regression for Tail Dependent Time Series

Introduction

Main Results

Numerical Experiments

Conclusion

What has been done on high quantile regression?
Wang, Li and He (2012, JASA) commented that
“To our knowledge, relatively little has been done for the estimationof high conditional quantiles.”Existing results were mainly developed for independent data:
I de Haan and Ferreira (2006);
I Belloni and Chernozhukov (2011, AoS);
I Wang, Li and He (2012, JASA);
I Wang and Li (2013, JASA);I He, Cheng and Tong (2016);
I Wang and Wang (2016, Statistica Sinica);
I D’Haultfœuille, Maurel and Zhang (2018, JoE);
I Zhang (2018, AoS);
I ...

What if the data exhibits dependence?
I For time series data, dependence is the rule rather than the exception.
Ting Zhang (tingzhang@uga.edu)

High Quantile Regression for Tail Dependent Time Series

Introduction

Main Results

Numerical Experiments

Conclusion

7
6
5

^
Quantiles of β2

2

3

4

5
4
2

3

^
Quantiles of β2

67What if we simply ignore the dependence?

2

3
4
5
6
7
Quantiles Ignoring Tail Dependence

2

3
4
5
6
7
Quantiles Based on the Developed Results“Ignoring the dependence can lead to misleading p-values!”
Ignoring the dependence
Developed results
Ting Zhang (tingzhang@uga.edu)

0.1
0.224
0.087

0.05
0.150
0.043

0.01
0.059
0.007

High Quantile Regression for Tail Dependent Time Series

Introduction

Main Results

Numerical Experiments

Conclusion

What about high quantile regression under dependence?
Chernozhukov (2005, AoS):
I An influential work that considered the possibility of allowing timeseries data by using the strong mixing framework of Rosenblatt(1956), and the derived asymptotic distribution is the same fordependent and independent cases.
“No effect if ignoring the dependence! Why?”

Ting Zhang (tingzhang@uga.edu)

High Quantile Regression for Tail Dependent Time Series

Introduction

Main Results

Numerical Experiments

Conclusion

What about high quantile regression under dependence?
Chernozhukov (2005, AoS):
I An influential work that considered the possibility of allowing timeseries data by using the strong mixing framework of Rosenblatt(1956), and the derived asymptotic distribution is the same fordependent and independent cases.
“No effect if ignoring the dependence! Why?”
I To handle high quantiles, however, besides the mixing condition, an
additional condition was required to control the joint probability ofnearby tail events; see also Chernozhukov and Fernández-Val (2011).
Such a condition can be interpreted as a negligibility condition on taildependence.
“Dependence is only allowed to exist in non-tail regions!”

What I contribute:
1

2

Propose a new framework based on tail adversarial stability forasymptotic theory of tail dependent time series;
Develop an asymptotic theory for high quantile regression estimatorsfor data with potentially non-negligible tail dependence.
“Tail dependence indeed affects high quantile regression estimators!”
Ting Zhang (tingzhang@uga.edu)

High Quantile Regression for Tail Dependent Time Series

Introduction

Main Results

Numerical Experiments

Conclusion

Tail Adversarial Stability
A New Framework for Asymptotic Theory Allowing Tail Dependence

Ting Zhang (tingzhang@uga.edu)

High Quantile Regression for Tail Dependent Time Series

Introduction

Main Results

Numerical Experiments

Conclusion

Background and Motivation
A fundamental problem in statistics is to develop limit theorems:
I important for understanding parameter estimators;
I useful for guiding statistical inference procedures.

The development of such limit theorems, however, often requires
some fundamental assumptions or beliefs about the observed data.
“Assumption of Independence: most commonly used but notsuitable for time series data”
Rosenblatt (1956) proposed the influential strong mixing condition:
I coordinates well with a big blocks small blocks argument and leads tothe development of various limit theorems;
I involves a supremum over two sigma algebras and is in generaldifficult to calculate (Wu, 2005);
I does not seem to be ideal for high quantile regression as
Chernozhukov (2005, AoS) imposed an additional negligibility
condition to control the degree of tail dependence.
Ting Zhang (tingzhang@uga.edu)

High Quantile Regression for Tail Dependent Time Series

Introduction

Main Results

Numerical Experiments

Conclusion

Wu (2005) proposed an influential alternative based on thefunctional dependence measure:
I coordinates well with the projection operator of Hannan (1979) andhas been successful in obtaining various limit theorems;
I easy to calculate, especially for linear processes;
I tends to summarize the dependence over the whole support and may
not be suitable for investigating tail dependence.

“Requires significant innovations over the correlation”
I propose a new framework:
I convenient and mathematically rigorous for developing limit theoremsof tail dependent time series;
I based on a previously undescribed tail adversarial stability notion;
I coordinates well with an m-dependent martingale approximation
scheme that leads to desired limit theorems;
I can be easily calculated for the moving maximum process (Hall, Pengand Yao, 2002).
Ting Zhang (tingzhang@uga.edu)

High Quantile Regression for Tail Dependent Time Series

Introduction

Main Results

Numerical Experiments

Conclusion

Suppose we observe a stationary time series Y1 , . . . , Yn according to
Yi = G(Fi ),Fi = (. . . , i−1 , i ),I G is a measurable function representing the physical system;I j , j ∈ Z, are iid innovations (system inputs);
I covers a wide range of processes (Wiener, 1958 and Wu, 2005).

Denote the distribution function by F , then for small choices of a,
Yi > F −1 (1 − a) represents a tail event.
Let
Yi? = G(Fi? ),Fi? = (. . . , −1 , ?0 , 1 , . . . , i ),we considerθα (i) = sup pr{Yi? ≤ F −1 (1 − a) | Yi > F −1 (1 − a)}.
a∈(0,α]“Measures the tail adversarial effect of current innovation on Yi ”
I Example: If Yi does not depend on 0 , then θα (i) = 0.
Ting Zhang (tingzhang@uga.edu)

High Quantile Regression for Tail Dependent Time Series

Introduction

Main Results

Numerical Experiments

Recall that
Yi = G(Fi ),Fi = (. . . , −1 , 0 , 1 , . . . , i );Yi? = G(Fi? ),Fi? = (. . . , −1 , ?0 , 1 , . . . , i ),andConclusionθα (i) = sup pr{Yi? ≤ F −1 (1 − a) | Yi > F −1 (1 − a)}.
a∈(0,α]“Measures the tail adversarial effect of current innovation on Yi ”
For q ≥ 1, defineΘα,q (0) =∞X{θα (i)}1/q ,k ≥ 0,

i=0

that measures the cumulative tail adversarial effect on future
observations, then a process is said to be tail adversarial q-stable or(Yi ) ∈ TASq if
lim Θα,q (0) < ∞.
α↓0“Finite cumulative tail adversarial effect on future observations”
Ting Zhang (tingzhang@uga.edu)

High Quantile Regression for Tail Dependent Time Series

Introduction

Main Results

Numerical Experiments

Conclusion

Example: Moving maximum process (Hall, Peng and Yao, 2002):
Yi = max al i−l ,0≤l<∞

i = 1, . . . , n,I j : independent
with F (z) = exp(−z −γ ), γ > 0;
PFréchet∞γ
I well defined if l=0 al < ∞;I proven to be dense in the class of stationary processes whose
finite-dimensional distributions are extreme-value of a given type.

By elementary calculation, one can show that for α small enough,
P
γ/q
2aγi
21/q ∞ai
i=0θα (i) ≤ P∞ γ , Θα,q (0) ≤ P∞ γ 1/q ,( l=0 al )l=0 aland thus (Yi ) ∈ TASq if∞
Xγ/qai < ∞.
i=0“Calculating the strong mixing coefficient can be nontrivial and may
possibly lead to stronger conditions.”
Ting Zhang (tingzhang@uga.edu)

High Quantile Regression for Tail Dependent Time Series

Introduction

Main Results

Numerical Experiments

Conclusion

High Quantile Regression under Tail Dependence
Taking Advantage of the Newly Proposed Framework

Ting Zhang (tingzhang@uga.edu)

High Quantile Regression for Tail Dependent Time Series

Introduction

Main Results

Numerical Experiments

Conclusion

Consider the high quantile regression estimator:β̂ n = argmin
η∈Rp

n
X

φ1−αn (yi,n − x>i,n η),i=1

I φ1−αn (y) = (1 − αn )y + + αn (−y)+ is the check function;
I 1 − αn → 1 as n → ∞ to capture the tail phenomena;
I allows the observed data to be a triangular array.

Ting Zhang (tingzhang@uga.edu)

High Quantile Regression for Tail Dependent Time Series

Introduction

Main Results

Numerical Experiments

Conclusion

Theorem (Zhang, 2021)
Assume certain regularity conditions including the TASq , we haveβ̂ n − β n = Op (τn−1 ).
“τn : explicit expression available, depend on n, αn and tail behavior”
Examples:τn−1/2

Uniformn1/2 αnNormal(nαn )1/2 {log(αn−1 )}1/2

Exponential(nαn )1/2

Pareto(nαn1+2λ )1/2

“Convergence rate is no longer the universal n1/2 , and in certain
cases can even exceed the parametric rate”
Ting Zhang (tingzhang@uga.edu)

High Quantile Regression for Tail Dependent Time Series

Introduction

Main Results

Numerical Experiments

Conclusion

Theorem (Zhang, 2021)
Assume certain regularity conditions including the TASq , we haveτn (β̂ n − β n ) →d N (0, Γ),
where
Γ=

X

−1
ρk Υ−10 Υk Υ0 .

k∈ZI Υk : limit of n−1

Pn−|k|
i=1

xi,n x>i+|k|,n as n → ∞;I ρk : standardized lag-k tail dependence index of Zhang (2005).
Example: For independent data, the above central limit theorem
reduces toτn (β̂ n − β n ) →d N (0, Υ−1
0 ).
“Ignoring the dependence can lead to misleading p-values!”
Ting Zhang (tingzhang@uga.edu)

High Quantile Regression for Tail Dependent Time Series

Introduction

Main Results

Numerical Experiments

Conclusion

Simulation Studies and Data Analysis
Numerical Experiments

Ting Zhang (tingzhang@uga.edu)

High Quantile Regression for Tail Dependent Time Series

Introduction

Main Results

Numerical Experiments

Conclusion

A Simulation Study
Consider:765^
Quantiles of β2

2

3

4

5
4
2

3

^
Quantiles of β2

67I Trend analysis design: xi,n = (1, ti,n )> , ti,n = i/n;
I Moving maximum dependence: Ui,n = max(i , i−1 ) − an ;I Focus on the distribution of β̂2 , the coefficient of ti,n .

2

3
4
5
6
7
Quantiles Ignoring Tail Dependence

2

3
4
5
6
7
Quantiles Based on the Developed Results“Ignoring the dependence can lead to misleading p-values!”
Ignoring the dependence
Developed results
Ting Zhang (tingzhang@uga.edu)

0.1
0.224
0.087

0.05
0.150
0.043

0.01
0.059
0.007

High Quantile Regression for Tail Dependent Time Series

Introduction

Main Results

Numerical Experiments

Conclusion

Global temperature anoamalies (Celsius)

Application to a Temperature Data
1.0

0.5

0.0

−0.5

−1.0
1850

1900

1950
Time (year)

2000

Figure: Monthly global temperature anomalies in Celsius from 1850 to 2019

Temperature is an important factor in economics (Dell, Jones and
Olken, 2012).
Rust (2003) and Wu and Zhao (2007, JRSSB):I quadratic trend for the mean function.

Q: Is the quadratic trend still enough for high quantiles or if ahigher order polynomial (e.g. cubic) is needed?
Ting Zhang (tingzhang@uga.edu)

High Quantile Regression for Tail Dependent Time Series

Introduction

Main Results

Numerical Experiments

Conclusion

Application to a Temperature Data (Continued)
Testing if a cubic trend can be reduced to a quadratic one
Tail dependence
Cubic Coefficient Estimate
p-value

95% quantile
Adjusted
Ignored
1.211
1.211
0.270
0.008

99% quantile
Adjusted
Ignored
3.421
3.421
0.002
0.000

Table: High quantile regression estimators for the cubic coefficient and their associatedp-values for testing a zero null hypothesis against a two-sided alternative.

Findings:
I Ignoring the dependence ⇒ underestimated uncertainty ⇒distortedly narrower confidence intervals“can be a serious problem in risk management”
I the quadratic form continues to be adequate for the 95% quantile,
while a higher order polynomial is needed for the 99% quantile.
“the trend in 99% quantile can be more complicated”
Ting Zhang (tingzhang@uga.edu)

High Quantile Regression for Tail Dependent Time Series

Introduction

Main Results

Numerical Experiments

Conclusion

Zhang, T. (2021).
High-quantile regression for tail-dependent time series.
Biometrika, 108, 113–126.
To summarize, in this work I make two major contributions.
1

I propose a new framework:
I convenient and mathematically rigorous foundation for developing
limit theorems of tail dependent time series;
I based on a previously undescribed tail adversarial stability notion;
I coordinates well with an m-dependent martingale approximation
scheme that leads to desired limit theorems;
I can be easily calculated for the moving maximum process (Hall, Pengand Yao, 2002).

2

Taking advantage of the newly proposed framework:
I I consider the problem of high quantile regression for time series data;
I unlike the result of Chernozhukov (2005, AoS), dependence isallowed in both tail and non-tail regions;
I the associated convergence rate and a central limit theorem areestablished for a general class of tail dependent processes.
Ting Zhang (tingzhang@uga.edu)

High Quantile Regression for Tail Dependent Time Series

Introduction

Main Results

Numerical Experiments

Conclusion

Questions?

Thank You!
Zhang’s Statistics Lab: tingzhang@uga.edu
Ting Zhang (tingzhang@uga.edu)

High Quantile Regression for Tail Dependent Time Series