Forecasting Using Functional Coefficients Autoregressive Models

Munich Personal RePEc Archive

Forecasting Using Functional Coefficients Autoregressive Models

Bruno, Giancarlo

ISAE (Institute for studies and economic analisys) Rome (Italy)

June 2008

Online at https://mpra.ub.uni-muenchen.de/42335/

MPRA Paper No. 42335, posted 01 Nov 2012 05:38 UTC

ISTITUTO DI STUDI E ANALISI ECONOMICA

Forecasting Using Functional Coefficients Autoregressive Models

by

Giancarlo Bruno

ISAE, Institute for Studies and Economic Analyses, P.zza dell’Indipendenza, 4 00185 Rome, Italy - Tel: +39-06-4448 2719, fax: +39-06-4448 2249

email: g.bruno@isae.it

Working paper n. 98 June 2008

The Series “Documenti di Lavoro” of the Istituto di Studi e Analisi Economica – Institute for Studies and Economic Analyses (ISAE) hosts the preliminary results of the research projects carried out within ISAE. The diffusion of the papers is subject to the favourable opinion of an anonymous referee, whom we would like to thank. The opinions expressed are merely the Authors’ own and in no way involve the ISAE responsability.

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ABSTRACT

The use of linear parametric models for forecasting economic time series is widespread among practitioners, in spite of the fact that there is a large evidence of the presence of non-linearities in many of such time series.

However, the empirical results stemming from the use of non-linear models are not always as good as expected. This has been sometimes associated to the difficulty in correctly specifying a non-linear parametric model. I this paper I cope with this issue by using a more general non-parametric approach, which can be used both as a preliminary tool for aiding in specifying a suitable parametric model and as an autonomous modelling strategy. The results are promising, in that the non-parametric approach achieve a good forecasting record for a considerable number of series.

Keywords: Non-linear time-series models, non-parametric models.

JEL Classification: C22, C53.

CONTENTS

1 INTRODUCTION ... 5

2 THE FUNCTIONAL COEFFICIENTS AUTOREGRESSIVE (FCAR) MODEL...7

2.1 Estimation ... 8

2.2 Model identification ... 9

3 DATA ... 10

4 EMPIRICAL RESULTS... 12

5 CONCLUDING REMARKS ... 16

References ... 18

APPENDIX ... 20

✶ ■◆❚❘❖❉❯❈❚■❖◆

■♥ t❤✐s ♣❛♣❡r ■ ❡①♣❧♦r❡ t❤❡ ✉s❡❢✉❧♥❡ss ♦❢ ❛ ❝❧❛ss ♦❢ ♥♦♥✲❧✐♥❡❛r ♥♦♥✲

♣❛r❛♠❡tr✐❝ t✐♠❡ s❡r✐❡s ♠♦❞❡❧s ✐♥ ♣r♦❞✉❝✐♥❣ ♠✉❧t✐✲st❡♣ ❛❤❡❛❞ ❢♦r❡❝❛sts

❢♦r ❛ s❡t ♦❢ ❡❝♦♥♦♠✐❝ t✐♠❡ s❡r✐❡s✳ ❚❤❡ r❡s✉❧ts ❛r❡ ❝♦♠♣❛r❡❞ ✇✐t❤ t❤♦s❡

❞❡r✐✈❡❞ ❢r♦♠ ❛♣♣❧②✐♥❣ ❛ ❧✐♥❡❛r ♣❛r❛♠❡tr✐❝ ♠♦❞❡❧ ❜❡❧♦♥❣✐♥❣ t♦ t❤❡ ✇❡❧❧

❦♥♦✇♥ ❝❧❛ss ♦❢ t❤❡ ❛✉t♦r❡❣r❡ss✐✈❡✱ ♠♦✈✐♥❣ ❛✈❡r❛❣❡ ✭❆❘▼❆✮ ♠♦❞❡❧s✳

❚❤❡ ♠❛✐♥ ✜♥❞✐♥❣ ✐s t❤❛t t❤❡ ✉s❡ ♦❢ ♥♦♥✲❧✐♥❡❛r ♥♦♥✲♣❛r❛♠❡tr✐❝ ♠♦❞❡❧s

❝❛♥ ✐♠♣r♦✈❡ t❤❡ ❢♦r❡❝❛st✐♥❣ ♣❡r❢♦r♠❛♥❝❡✱ ✐rr❡s♣❡❝t✐✈❡ ♦❢ t❤❡ ❧✐♥❡❛r✐t②

♦❢ t❤❡ s❡r✐❡s ❡①❛♠✐♥❡❞ ✭❛s ♠❡❛s✉r❡❞ ❜② s♦♠❡ ✉s✉❛❧ t❡sts✮✳ ❙✉❝❤ ❛♥

✐♠♣r♦✈❡♠❡♥t r❡s✉❧ts t♦ ❜❡ s✉❜st❛♥t✐❛❧ ✐♥ ♦✉r ❞❛t❛s❡t ✐❢ ♦♥❡ ♠❛✐♥ ❢♦❝✉s

✐s t❤❡ ♠❡❞✐❛♥ ❛❜s♦❧✉t❡ ❡rr♦r ♦r t❤❡ ❞✐r❡❝t✐♦♥❛❧ ❡rr♦r r❛t❤❡r t❤❛♥ t❤❡

♠❡❛♥ ❛❜s♦❧✉t❡ ♦r t❤❡ ♠❡❛♥ sq✉❛r❡ ❡rr♦r✳

❚❤❡ ✇✐❞❡s♣r❡❛❞ ✉s❡ ♦❢ ❧✐♥❡❛r ♣❛r❛♠❡tr✐❝ ♠♦❞❡❧s✱ ✐♥ ♣❛rt✐❝✉❧❛r

❆❘▼❆ ♠♦❞❡❧s✱ ✐s ❜❛s❡❞ ♦♥ t❤❡ ❛ss✉♠♣t✐♦♥ t❤❛t ❛ t✐♠❡ s❡r✐❡s ❝❛♥

❜❡ ❡①♣r❡ss❡❞ ❛s t❤❡ r❡❛❧✐s❛t✐♦♥ ♦❢ ❛ st♦❝❤❛st✐❝ ♣r♦❝❡ss ❛s t❤❡ ❢♦❧❧♦✇✲

✐♥❣✿

X_t =

∞

X

j=0

ψ_jZ_t−j Z_t ∼ IID(0, σ²) ✭✶✮

✇✐t❤ P_∞

j=0|ψ_j| < ∞✳

■♥ t❤✐s ❝❛s❡ t❤❡ ❜❡st ♠❡❛♥ sq✉❛r❡ ♣r❡❞✐❝t♦r ✐s ❡q✉❛❧ t♦ t❤❡ ❜❡st

❧✐♥❡❛r ♣r❡❞✐❝t♦r ❛♥❞ ♣r♦❝❡ss ✭✶✮ ✐s ✉s✉❛❧❧② ❛♣♣r♦①✐♠❛t❡❞ ❜② ❛ s♦ ❝❛❧❧❡❞

ARMA(p, q) ♠♦❞❡❧✿

(1−ϕ₁L−. . . ϕ_pL^p)X_t = (1 +θ₁L +· · ·+θ_qL^q)ε_t ✭✷✮

✇❤❡r❡ L ✐s t❤❡ ❧❛❣ ♦♣❡r❛t♦r s✉❝❤ t❤❛t L^kX_t = X_t−k ❛♥❞ p ❛♥❞ q ❛r❡

t②♣✐❝❛❧❧② ♦❢ ❧♦✇ ♦r❞❡r✳

■t ✐s ♥❡❝❡ss❛r② t♦ str❡ss ✭❇r♦❝❦✇❡❧❧ ❛♥❞ ❉❛✈✐s✱ ✶✾✾✶✮ t❤❛t t❤❡ ❲♦❧❞

❞❡❝♦♠♣♦s✐t✐♦♥ ♦♥❧② ✐♥s✉r❡s t❤❛t ❛♥② ③❡r♦✲♠❡❛♥ ❝♦✈❛r✐❛♥❝❡ st❛t✐♦♥❛r②

♣r♦❝❡ss ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ✐♥ ❛ s✐♠✐❧❛r ✇❛② ❛s ✐♥ ✭✶✮ ❜✉t ✇✐t❤ Z_t ∼ W N(0, σ²)❀ ✐♥ t❤✐s ❝❛s❡ ♦♥❡ ❤❛s t♦ ❛❞❞ t❤❡ ❤②♣♦t❤❡s✐s ♦❢ ❣❛✉ss✐❛♥✐t② t♦ t❤❡ s❡q✉❡♥❝❡ Z_t ✐♥ ♦r❞❡r t♦ ❤❛✈❡ t❤❛t t❤❡ ❜❡st ❧✐♥❡❛r ♣r❡❞✐❝t♦r ✐s t❤❡ ❜❡st ♣r❡❞✐❝t♦r ✐♥ ♠❡❛♥ sq✉❛r❡ s❡♥s❡✱ ♦t❤❡r✇✐s❡ ❛ ♥♦♥✲❧✐♥❡❛r ♠♦❞❡❧

s❤♦✉❧❞ ❜❡ ✉s❡❞✳

■♥❞❡❡❞✱ ♠❛♥② t✐♠❡ s❡r✐❡s ❡①❤✐❜✐t ♥♦♥✲❧✐♥❡❛r ❢❡❛t✉r❡s✱ s✉❝❤ ❛s ♥♦♥✲

♥♦r♠❛❧✐t②✱ ❛s②♠♠❡tr✐❝ ❝②❝❧❡s✱ ❜✐✲♠♦❞❛❧✐t②✱ t✐♠❡ ✐rr❡✈❡rs✐❜✐❧✐t②✱ ♣r❡❞✐❝✲

t✐♦♥ ♣❡r❢♦r♠❛♥❝❡ ❞❡♣❡♥❞✐♥❣ ♦♥ t❤❡ st❛rt✐♥❣ ♣♦✐♥t✱ ❡t❝✳✱ t❤✉s ♠❛❦✐♥❣

t❤❡ ❧✐♥❡❛r ❤②♣♦t❤❡s✐s ❤❛r❞ t♦ ♠❛✐♥t❛✐♥✳ ❚❤❡r❡❢♦r❡✱ ✐♥ s✉❝❤ ❝❛s❡s ✐t

✺

❝♦✉❧❞ ❜❡ ❛♣♣r♦♣r✐❛t❡ t♦ ✉s❡ ♠♦r❡ ❣❡♥❡r❛❧ ✭♥♦♥✲❧✐♥❡❛r✮ ♠♦❞❡❧s t♦ ❞❡✲

s❝r✐❜❡ t❤♦s❡ s❡r✐❡s ❛s ✇❡❧❧ ❛s t♦ ❢♦r❡❝❛st t❤❡♠✳

❆♥ ✐ss✉❡ ✇✐t❤ t❤✐s ❛♣♣r♦❛❝❤ ✐s r❡♣r❡s❡♥t❡❞ ❜② t❤❡ ❧❛r❣❡ ♥✉♠❜❡r ♦❢

♥♦♥✲❧✐♥❡❛r ♣❛r❛♠❡tr✐❝ ♠♦❞❡❧s ✇❤✐❝❤ ❝❛♥ ❜❡ ❝♦♥str✉❝t❡❞❀ ♠♦r❡♦✈❡r✱ ✐♥

♣r❛❝t✐❝❛❧ ❛♣♣❧✐❝❛t✐♦♥s✱ t❤❡ s✉♣❡r✐♦r ❢♦r❡❝❛st✐♥❣ ♣❡r❢♦r♠❛♥❝❡ ♦❢ ♥♦♥✲

❧✐♥❡❛r ♠♦❞❡❧s ❤❛s ❜❡❡♥ ❤❛r❞❧② ♦❜s❡r✈❡❞✳ ❆♥ ❡①t❡♥s✐✈❡ ❛♥❛❧②s✐s ♦❢ t❤❡

❢♦r❡❝❛st✐♥❣ ♣❡r❢♦r♠❛♥❝❡ ♦❢ ❧✐♥❡❛r ✈s✳ ♥♦♥✲❧✐♥❡❛r ♣❛r❛♠❡tr✐❝ ♠♦❞❡❧s

✇❛s ❝❛rr✐❡❞ ♦✉t ❜② ❙t♦❝❦ ❛♥❞ ❲❛ts♦♥ ✭✶✾✾✽✮✿ t❤❡ ❛✉t❤♦rs ✜♥❞ t❤❛t

❧✐♥❡❛r ❛✉t♦r❡❣r❡ss✐✈❡ ♠♦❞❡❧s ✇✐t❤ ✉♥✐t r♦♦t ♣r❡t❡st✐♥❣ ♦✉t♣❡r❢♦r♠ t❤❡

♥♦♥✲❧✐♥❡❛r ♣❛r❛♠❡tr✐❝ ♠♦❞❡❧s ❝♦♥s✐❞❡r❡❞✳

■♥ ❛ r❡❝❡♥t ❝♦♥tr✐❜✉t✐♦♥✱ ❚❡räs✈✐rt❛ ✭✷✵✵✻✮ ❛r❣✉❡s t❤❛t ❛ ❝♦♠❜✐♥❛✲

t✐♦♥ ♦❢ ❛ ❧❛r❣❡ ♥✉♠❜❡r ♦❢ ♥♦♥✲❧✐♥❡❛r ♠♦❞❡❧s ❝❛♥ ♦❜t❛✐♥ ♣♦✐♥t ❢♦r❡❝❛sts s✉♣❡r✐♦r t♦ ❧✐♥❡❛r ♦♥❡s✳ ❚❤✐s ❝❛♥ ❜❡ ❡①♣❧❛✐♥❡❞ ❜② t❤❡ ❢❛❝t t❤❛t t❤❡

♣r❡s❡♥❝❡ ♦❢ ♥♦♥✲❧✐♥❡❛r ❢❡❛t✉r❡s ✐♥ ❛❝t✉❛❧ t✐♠❡ s❡r✐❡s ✐s ♦❢t❡♥ ❝♦✉♣❧❡❞

✇✐t❤ t❤❡ ❞✐✣❝✉❧t② ✐♥ s♣❡❝✐❢②✐♥❣ ❛ ❝♦rr❡❝t ♣❛r❛♠❡tr✐❝ ♥♦♥✲❧✐♥❡❛r ♠♦❞❡❧✳

❚❤✐s ❧❛st ♦❜s❡r✈❛t✐♦♥ ❝❛♥ ❧❡❛❞ t♦ t❤❡ ❛❧t❡r♥❛t✐✈❡ ❛♣♣r♦❛❝❤ ♦❢ ❧❡t✲

t✐♥❣ t❤❡ ❞❛t❛ s♣❡❝✐❢② t❤❡ ✉♥❦♥♦✇♥ ♥♦♥✲❧✐♥❡❛r ❢✉♥❝t✐♦♥❛❧ ❢♦r♠✱ ✐✳❡✳

✉s✐♥❣ ❛ ♥♦♥✲♣❛r❛♠❡tr✐❝ ❛♣♣r♦❛❝❤✳ ■♥ t❤✐s ♣❛♣❡r ■ ✇✐❧❧ ❛♥❛❧②s❡ t❤❡

❢♦r❡❝❛st✐♥❣ ♣❡r❢♦r♠❛♥❝❡✱ ❢♦r ❛ ❧❛r❣❡ ♥✉♠❜❡r ♦❢ t✐♠❡ s❡r✐❡s✱ ♦❢ t❤❡

❛♣♣❧✐❝❛t✐♦♥ ♦❢ ❛ ♥♦♥✲♣❛r❛♠❡tr✐❝✱ ♥♦♥✲❧✐♥❡❛r ❢♦r❡❝❛st✐♥❣ ♠♦❞❡❧✳

❚❤❡ ♥♦♥✲❧✐♥❡❛r ♥♦♥✲♣❛r❛♠❡tr✐❝ ♠♦❞❡❧ ■ ❝♦♥s✐❞❡r ❤❡r❡ ✐s t❤❡ s♦

❝❛❧❧❡❞ ❢✉♥❝t✐♦♥❛❧ ❝♦❡✣❝✐❡♥ts ❛✉t♦r❡❣r❡ss✐✈❡ ♠♦❞❡❧ ✭❋❈❆❘✮✱ ✇❤✐❝❤ ✐♥

♣r❛❝t✐❝❡ ✐s ❛♥ ❛✉t♦r❡❣r❡ss✐✈❡ ♠♦❞❡❧ ✇❤❡r❡ t❤❡ ❝♦❡✣❝✐❡♥ts ❛r❡ ❛❧❧♦✇❡❞

t♦ ✈❛r② ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ ❛ ❧❛❣ ♦❢ t❤❡ ♠♦❞❡❧❧❡❞ ✈❛r✐❛❜❧❡✳ ❚❤❡ ♥♦♥✲

♣❛r❛♠❡tr✐❝ ♥❛t✉r❡ ♦❢ t❤❡ ♠♦❞❡❧ ❧✐❡s ✐♥ t❤❡ ❢❛❝t t❤❛t t❤❡ ❢✉♥❝t✐♦♥❛❧

❢♦r♠ ♦❢ t❤❡ ❝♦❡✣❝✐❡♥ts ✐s ❧❡❢t ✉♥s♣❡❝✐✜❡❞✳ ❲❤✐❧❡ ♦t❤❡r ♣❛♣❡rs ❤❛✈❡

❡①❛♠✐♥❡❞ s♦♠❡ ❛s♣❡❝ts ♦❢ t❤❡ ❢♦r❡❝❛st✐♥❣ ♣❡r❢♦r♠❛♥❝❡ ♦❢ ❋❈❆❘ ♠♦❞✲

❡❧s✱ s✉❝❤ ❛s ✐♥ ❈❤❡♥ ❛♥❞ ❚s❛② ✭✶✾✾✸✮ ❛♥❞ t❤❡ r❡❝❡♥t ✇♦r❦ ♦❢ ❍❛r✈✐❧❧

❛♥❞ ❘❛② ✭✷✵✵✺✮✱ ♥♦♥❡t❤❡❧❡ss✱ t♦ ♠② ❦♥♦✇❧❡❞❣❡✱ ✐t ❧❛❝❦s ❛ ♠♦r❡ ❡①✲

t❡♥s✐✈❡ st✉❞② ♦❢ t❤❡ ❢♦r❡❝❛st✐♥❣ ✉s❡❢✉❧♥❡ss ♦❢ s✉❝❤ ♠♦❞❡❧s ✇✐t❤ r❡❛❧

❞❛t❛ s❡ts✳

❚❤❡ ♣❛♣❡r ✐s ♦r❣❛♥✐s❡❞ ❛s ❢♦❧❧♦✇s✿ s❡❝t✐♦♥ ✷ ♣r❡s❡♥ts t❤❡ ♠♦❞❡❧

❛❞♦♣t❡❞❀ s❡❝t✐♦♥ ✸ r❡✈✐❡✇s t❤❡ ❞❛t❛ ✉s❡❞❀ s❡❝t✐♦♥ ✹ ♣r❡s❡♥ts t❤❡ s❡t✲

✉♣ ♦❢ t❤❡ ❡♠♣✐r✐❝❛❧ ❡①❡r❝✐s❡ ❛s ✇❡❧❧ ❛s t❤❡ ♠❛✐♥ r❡s✉❧ts✱ ✇❤✐❧❡ t❤❡ ✜♥❛❧

s❡❝t✐♦♥ ❝♦♥❝❧✉❞❡s ❛♥❞ ♣r❡s❡♥ts s♦♠❡ ✐ss✉❡s ❢♦r ❢✉t✉r❡ r❡s❡❛r❝❤✳

✻

✷ ❚❍❊ ❋❯◆❈❚■❖◆❆▲ ❈❖❊❋❋■❈■❊◆❚❙ ❆❯❚❖❘❊●❘❊❙❙■❱❊

✭❋❈❆❘✮ ▼❖❉❊▲

❆ ✈❡r② ❣❡♥❡r❛❧ ♥♦♥✲♣❛r❛♠❡tr✐❝ s❡tt✐♥❣ ❢♦r t✐♠❡ s❡r✐❡s ♠♦❞❡❧❧✐♥❣ ❝❛♥

❜❡ s♣❡❝✐✜❡❞ ❛s ❛ ♥♦♥✲♣❛r❛♠❡tr✐❝ ❛✉t♦r❡❣r❡ss✐✈❡ ✭◆❆❘✮ ♠♦❞❡❧✿

X_t = g(X_t−1, X_t−2, . . . , X_t−p) +ε_t, t = p + 1, . . . , T ✭✸✮

✇❤❡r❡ ε_t ✐s ❛ ♠❛rt✐♥❣❛❧❡ ❞✐✛❡r❡♥❝❡ ♣r♦❝❡ss ❛♥❞ {X_t, . . . , X_t−p} ✐s ❛ str✐❝t❧② st❛t✐♦♥❛r② β✲♠✐①✐♥❣ ♣r♦❝❡ss✳

❊st✐♠❛t✐♦♥ ♦❢ t❤❡ ✉♥❦♥♦✇♥ ❢✉♥❝t✐♦♥ g(·) ❝❛♥ ❜❡ ❝❛rr✐❡❞ ♦✉t ❡✳❣✳

❜② ♠❡❛♥s ♦❢ ❛ ❦❡r♥❡❧ ❡st✐♠❛t♦r✳ ▼♦r❡ ❣❡♥❡r❛❧❧② ❛ ❧♦❝❛❧ ♣♦❧②♥♦♠✐❛❧

❛♣♣r♦❛❝❤ ❝❛♥ ❜❡ ✉s❡❞✿ ✐♥ t❤❛t ❝❛s❡ ❛ ❦❡r♥❡❧ ❡st✐♠❛t♦r ❝❛♥ ❜❡ s❡❡♥ t♦

❜❡ ❡q✉✐✈❛❧❡♥t t♦ ❛ ❧♦❝❛❧ ❝♦♥st❛♥t ❡st✐♠❛t♦r✳ ❊✣❝✐❡♥❝② r❡❛s♦♥s s❤♦✇

t❤❛t ✐♥ t❤✐s s❡tt✐♥❣ ❛ ❧♦❝❛❧ ❧✐♥❡❛r ❡st✐♠❛t♦r s❤♦✉❧❞ ❜❡ ♣r❡❢❡rr❡❞ ✭❋❛♥

❛♥❞ ●✐❥❜❡❧s✱ ✶✾✾✻✮✳

◆❡✈❡rt❤❡❧❡ss✱ t❤❡ ❣❡♥❡r❛❧✐t② ♦❢ ♠♦❞❡❧ ✭✸✮ ❝♦♠❡s ✇✐t❤ ❛ ❝♦st✿ t❤❡

s♦ ❝❛❧❧❡❞ ❝✉rs❡ ♦❢ ❞✐♠❡♥s✐♦♥❛❧✐t②✱ t❤❛t ✐s t❤❡ s❛♠♣❧❡ s✐③❡ r❡q✉✐r❡❞ ❢♦r

❤❛✈✐♥❣ ❛ ♣❡r❢♦r♠❛♥❝❡ ❝♦♠♣❛r❛❜❧❡ t♦ t❤❡ ❝❛s❡ ✇❤❡r❡ p = 1 ❣r♦✇s

❡①♣♦♥❡♥t✐❛❧❧② ❢❛st ✭❋❛♥ ❛♥❞ ❨❛♦✱ ✷✵✵✸✱ ♣❛❣❡ ✸✶✼✮✱ ✇❤✐❝❤ ✐♥ ♣r❛❝t✐❝❡

♠❡❛♥s t❤❛t✱ ❢♦r t❤❡ ✉s✉❛❧ s❛♠♣❧❡ s✐③❡s ♦❜s❡r✈❡❞ ✐♥ ❡❝♦♥♦♠✐❝ t✐♠❡

s❡r✐❡s✱ p ❝❛♥ ❜❡ ❛t ♠♦st ♦♥❡ ♦r t✇♦✳

❉✐✛❡r❡♥t ♠❡❛♥s ❤❛✈❡ ❜❡❡♥ ♣r♦♣♦s❡❞ t♦ ♦✈❡r❝❛♠❡ t❤✐s ♣r♦❜❧❡♠✱

r❡str✐❝t✐♥❣ ✐♥ s♦♠❡ ✇❛② t❤❡ ❜❡❤❛✈✐♦✉r ♦❢ t❤❡ ❢✉♥❝t✐♦♥ g(·) ✐♥ ♠♦❞❡❧

✭✸✮✳ ❆ ✉s✉❛❧ ♠❛♥♥❡r t♦ ❛❝❝♦♠♣❧✐s❤ t❤✐s✱ ❢♦r ❡①❛♠♣❧❡✱ ✐s ❜② ✉s✐♥❣ s♦

❝❛❧❧❡❞ ❛❞❞✐t✐✈❡ ♠♦❞❡❧s✱ t❤❛t ✐s ♠♦❞❡❧s ❧✐❦❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿

X_t = a₁(X_t−1) + . . .+a_p(X_t−p) +ε_t t = p+ 1, . . . , T. ✭✹✮

■♥ t❤❡ s❛♠❡ s♣✐r✐t ♦t❤❡r s♦❧✉t✐♦♥s ❤❛✈❡ ❜❡❡♥ ♣r♦♣♦s❡❞✱ ❛♠♦♥❣

t❤❡♠ t❤❡ ❢✉♥❝t✐♦♥❛❧ ❝♦❡✣❝✐❡♥t ❛✉t♦r❡❣r❡ss✐✈❡ ✭❋❈❆❘✮ ♠♦❞❡❧✿

X_t =a₁(X_t−d)X_t−1+. . .+a_p(X_t−d)X_t−p+ε_t t = p+1, . . . , T ✭✺✮

✇❤✐❝❤ ❤❛s ❜❡❡♥ ✐♥tr♦❞✉❝❡❞ ❜② ❈❤❡♥ ❛♥❞ ❚s❛② ✭✶✾✾✸✮✱ ✇❤✐❧❡ ❈❤❡♥

❛♥❞ ▲✐✉ ✭✷✵✵✶✮ ❛♥❞ ❈❛✐ ❡t ❛❧✳ ✭✷✵✵✵✮ ❢✉rt❤❡r ❛❞❞r❡ss t❤❡ ✐ss✉❡s ♦❢

❡st✐♠❛t✐♦♥✱ t❡st✐♥❣ ❛♥❞ s♠♦♦t❤✐♥❣ ♣❛r❛♠❡t❡r s❡❧❡❝t✐♦♥✳ ❚❤✐s ❦✐♥❞ ♦❢

♠♦❞❡❧ ❤❛s s♦♠❡ ❛♣♣❡❛❧✐♥❣ ❢❡❛t✉r❡s✱ ✐♥ t❤❛t ✐t ♥❡sts t❤❡ ✉s✉❛❧ ❧✐♥❡❛r ❆❘

♠♦❞❡❧✱ ❛s ✇❡❧❧ ❛s s♦♠❡ ♣♦♣✉❧❛r ♥♦♥✲❧✐♥❡❛r ♣❛r❛♠❡tr✐❝ ♠♦❞❡❧s✱ s✉❝❤ ❛s t❤r❡s❤♦❧❞ ❛✉t♦r❡❣r❡ss✐✈❡ ✭❚❆❘✮ ❛♥❞ ❡①♣♦♥❡♥t✐❛❧ ❛✉t♦r❡❣r❡ss✐✈❡ ✭❊❳✲

P❆❘✮ ♠♦❞❡❧s❀ ❛❧s♦ ❙❊❚❆❘ ♠♦❞❡❧s ❝❛♥ ❜❡ ❝♦♥s✐❞❡r❡❞ ❛s ♥❡st❡❞ ✐♥ t❤✐s

✼

❢r❛♠❡✇♦r❦✳ ▼♦r❡♦✈❡r✱ ✐t ❤❛s ❛ ♥✐❝❡ ✐♥t❡r♣r❡t❛t✐♦♥✱ ❛s t❤❡ ❝♦❡✣❝✐❡♥ts

❞❡♣❡♥❞ ♦♥ t❤❡ ✏st❛t❡✑ ♦❢ t❤❡ ✈❛r✐❛❜❧❡ X_t−d ✐♥ ❛ s♠♦♦t❤ ✇❛②✱ ❞✐✛❡r✲

❡♥t❧② ❢r♦♠ ✇❤❛t ❤❛♣♣❡♥s ✐♥ t❤❡ ❚❆❘ ♠♦❞❡❧✱ ✇❤❡r❡ t❤❡ ❛✉t♦r❡❣r❡s✲

s✐✈❡ ♣❛r❛♠❡t❡rs s❤✐❢t ❞✐s❝♦♥t✐♥✉♦✉s❧② ❢♦❧❧♦✇✐♥❣ t❤❡ ❞✐s❝r❡t❡ ♥✉♠❜❡r ♦❢

st❛t❡s ❛ss♦❝✐❛t❡❞ t♦ t❤❡ ✈❛r✐❛❜❧❡ X_t−d✳ ❙✉❝❤ ❛ ♠♦❞❡❧ r❡♠❛✐♥s s✉✣✲

❝✐❡♥t❧② ❣❡♥❡r❛❧ t♦ ❤❛♥❞❧❡ ♠❛♥② ❦✐♥❞s ♦❢ ♥♦♥✲❧✐♥❡❛r✐t✐❡s ♦❢t❡♥ ❢♦✉♥❞ ✐♥

♠❛❝r♦❡❝♦♥♦♠✐❝ t✐♠❡ s❡r✐❡s✱ ✇❤✐❧❡ r❡❞✉❝✐♥❣ ❝♦♥s✐❞❡r❛❜❧② t❤❡ ♣r♦❜❧❡♠

♦❢ ♠♦❞❡❧ ❝♦♠♣❧❡①✐t②✿ t❤❡ ✉♥❦♥♦✇♥ ❢✉♥❝t✐♦♥s✱ ✐♥ ❢❛❝t✱ ❞❡♣❡♥❞ ♦♥❧② ♦♥

♦♥❡ ✈❛r✐❛❜❧❡ ✐♥ t❤✐s s❡t✲✉♣✳

✷✳✶ ❊st✐♠❛t✐♦♥

❊st✐♠❛t✐♦♥ ♦❢ ♠♦❞❡❧ ✭✺✮ ❝♦♥s✐sts ✐♥ t❤❡ ❡st✐♠❛t✐♦♥ ♦❢ t❤❡ ✉♥❦♥♦✇♥

❢✉♥❝t✐♦♥s a_i(·)✳ Pr♦✈✐❞❡❞ s✉✐t❛❜❧❡ ❝♦♥❞✐t✐♦♥s ♦♥ t❤❡✐r s♠♦♦t❤♥❡ss✱

t❤✐s ❝❛♥ ❜❡ ❝❛rr✐❡❞ ♦✉t ❜② ❧♦❝❛❧ ❛✈❡r❛❣✐♥❣ t❡❝❤♥✐q✉❡s✱ s✉❝❤ ❛s ❦❡r✲

♥❡❧ ❡st✐♠❛t✐♦♥ ♦r ❧♦❝❛❧ ♣♦❧②♥♦♠✐❛❧ ❡st✐♠❛t✐♦♥❀ ❢♦❧❧♦✇✐♥❣ t❤❡ ❡✣❝✐❡♥❝② r❡❛s♦♥s s❤♦✇❡❞ ✐♥ ❋❛♥ ❛♥❞ ●✐❥❜❡❧s ✭✶✾✾✻✮ ✇❡ ✇✐❧❧ ✉s❡ ❛ ❧♦❝❛❧ ❧✐♥❡❛r

❡st✐♠❛t♦r ✇❤✐❝❤ ❝❛♥ ❜❡ s❤♦✇♥ t♦ ❜❡ ✉♥✐❢♦r♠❧② ❜❡tt❡r t❤❛♥ t❤❡ ❧♦❝❛❧

❝♦♥st❛♥t ✭❦❡r♥❡❧✮ ❡st✐♠❛t♦r✳

▲❡t U_t = X_t−d✳ ❚❤✐s ✐♠♣❧✐❡s t❤❛t t❤❡ ❢♦❧❧♦✇✐♥❣ ❢✉♥❝t✐♦♥ ♠✉st ❜❡

♠✐♥✐♠✐③❡❞✱ ✇✐t❤ r❡s♣❡❝t t♦ {a_i, b_i}✿

T

X

t=p+1

( X_t −

p

X

i=1

[a_i +b_i(U_t −u)]X_t−i )2

1 hK

U_t −u h

✭✻✮

✇❤❡r❡ K(·) ✐s ❛ ❦❡r♥❡❧ ❢✉♥❝t✐♦♥✱ h ✐s ❛ s♠♦♦t❤✐♥❣ ♣❛r❛♠❡t❡r ✭❜❛♥❞✲

✇✐❞t❤✮ ❛♥❞ u ✐s t❤❡ ♣♦✐♥t ✇❤❡r❡ t❤❡ r❡❣r❡ss✐♦♥ ❢✉♥❝t✐♦♥ ✐s ❡✈❛❧✉❛t❡❞✳

❚❤❡ ❧♦❝❛❧ ❧✐♥❡❛r r❡❣r❡ss✐♦♥ ❡st✐♠❛t❡ ♦❢ a_i(·)✐♥ ✭✺✮ ✐s t❤❡♥ s✐♠♣❧②aˆ_i(u)✳

❘❡s♦rt✐♥❣ t♦ ♠❛tr✐① ♥♦t❛t✐♦♥ ❛♥❞ ❞❡♥♦t✐♥❣ ✇✐t❤ XXX˜ t❤❡ 2 × np

♠❛tr✐① ✇✐t❤ t❤❡ t✲t❤ r♦✇ ❣✐✈❡♥ ❜②✿

{X_t−1, . . . , X_t−p, X_t−1K_h(X_t−d −u), . . . , X_t−pK_h(X_t−d −u)},

✇❤❡r❡ K_h = ¹_hK(·/h)✳ ▲❡tt✐♥❣ YYY = (X_1+p, . . . , X_T) ❛♥❞

WWW = di ag{(K_h(X_p+1−d), . . . , K_h(X_T−d)}✱ t❤❡♥ t❤❡ ♣r♦❜❧❡♠ ❝❛♥ ✇r✐t✲

t❡♥ ❛s✿

argmin_β(YYY −XXXβ˜ββ)^′WWW(YYY −XXXβ˜ββ) ✭✼✮

s♦ t❤❛t t❤❡ s♦❧✉t✐♦♥ ✈❡❝t♦r ✐s✿

βˆ

ββ = ( ˜XXX^′WWWXXX)˜ ⁻¹XXX˜^′WWW YYY ✭✽✮

✽

✇❤❡r❡ βββˆ = (ˆa₁, . . . ,aˆ_p,ˆb₁, . . . ,ˆb_p)

❚❤❡ ❛♣♣r♦❛❝❤ ❥✉st ❞❡s❝r✐❜❡❞ tr❡❛ts t❤❡ s♠♦♦t❤✐♥❣ ♣❛r❛♠❡t❡r h

❛s ❛ ❝♦♥st❛♥t ♦✈❡r t❤❡ ❞♦♠❛✐♥ ♦❢ u✳ ❆♥ ❛❧t❡r♥❛t✐✈❡ ✐s r❡♣r❡s❡♥t❡❞

❜② t❤❡ ❦✲♥❡❛r❡st ♥❡✐❣❤❜♦✉r ✭❦✲◆◆✮ ♠❡t❤♦❞✱ ✇❤❡r❡ ❢♦r ❡❛❝❤ ✈❛❧✉❡ u

✇❤❡r❡ t❤❡ ❢✉♥❝t✐♦♥ ✐s ❡✈❛❧✉❛t❡❞✱ ♦♥❧② t❤❡ k ♥❡❛r❡st ♦❜s❡r✈❛t✐♦♥s ❛r❡

✉s❡❞✱ ♣♦ss✐❜❧② ✇❡✐❣❤t❡❞ ❜② ❛ ❦❡r♥❡❧ ❢✉♥❝t✐♦♥✳ ❚❤✐s ✐s ❡q✉✐✈❛❧❡♥t t♦

s♣❡❝✐❢②✐♥❣ ❛ ✈❛r✐❛❜❧❡ ❜❛♥❞✇✐❞t❤✱ ✇❤✐❝❤ ❞❡♣❡♥❞s ♦♥ t❤❡ ♣♦✐♥t u ✇❤❡r❡

t❤❡ ❢✉♥❝t✐♦♥ ✐s ❡✈❛❧✉❛t❡❞✳ ■♥ ♣❛rt✐❝✉❧❛r✱ t❤✐s ❛♠♦✉♥ts t♦ ❤❛✈❡ ❛ ❧❛r❣❡r

❜❛♥❞✇✐❞t❤ ❢♦r t❤❡ ✐♥t❡r✈❛❧s ♦❢ t❤❡ u ❞♦♠❛✐♥ ✇❤❡r❡ ♦❜s❡r✈❛t✐♦♥s ❛r❡

❧❡ss ❢r❡q✉❡♥t ❛♥❞ ✈✐❝❡ ✈❡rs❛✳

❲❤✐❧❡ t❤❡ ❦✲♥❡❛r❡st ♥❡✐❣❤❜♦✉r ❛♣♣r♦❛❝❤ ❝♦✉❧❞ ❜❡ ✐♥ t❤❡♦r② ♠♦r❡

s✉✐t❡❞ t♦ t❤❡ ♣r♦❜❧❡♠ ♦❢ ❢♦r❡❝❛st✐♥❣✱ ✇❤✐❝❤ ✐s ❛ ✏❧♦❝❛❧✑ ♣r♦❜❧❡♠✱ ✐ts

❛❝t✉❛❧ ❡✛❡❝t✐✈❡♥❡ss ♠✉st ❜❡ ❝♦♥✜r♠❡❞ ✐♥ ♣r❛❝t✐❝❡✳

✷✳✷ ▼♦❞❡❧ ✐❞❡♥t✐✜❝❛t✐♦♥

■♥ ♦r❞❡r t♦ ❡st✐♠❛t❡ t❤❡ ♠♦❞❡❧ ✭✺✮ ✐t ✐s ♥❡❝❡ss❛r② t♦ s❡t ✉♣ ❛ ♣r♦❝❡❞✉r❡

t♦ ✐❞❡♥t✐❢② t❤❡ ❞✐✛❡r❡♥t ❡❧❡♠❡♥ts ✇❤✐❝❤ ♣❡rt❛✐♥ t♦ t❤❡ ❡st✐♠❛t✐♦♥

♣r♦❝❡ss ✐ts❡❧❢✳ ■♥ ♣❛rt✐❝✉❧❛r s✉✐t❛❜❧❡ ✈❛❧✉❡s ❢♦r p✱ d ❛♥❞ h ✭♦r k ❢♦r t❤❡

❦✲◆◆ ♠❡t❤♦❞✮ ♠✉st ❜❡ s♣❡❝✐✜❡❞✳ ■♥ ♦r❞❡r t♦ ❛❝❝♦♠♣❧✐s❤ t❤✐s t❛s❦ ■ s❧✐❣❤t❧② ♠♦❞✐❢② t❤❡ ♣r♦❝❡❞✉r❡ ♣r♦♣♦s❡❞ ❜② ❈❛✐ ❡t ❛❧✳ ✭✷✵✵✵✮✱ ❛❧❧♦✇✐♥❣

t♦ s❡❧❡❝t ❛ s✉❜s❡t ♦❢ ❧❛❣s ❜❡t✇❡❡♥ 1 ❛♥❞ p✳ ❙✉❝❤ ❛ ♣r♦❝❡❞✉r❡ ❢♦r

♠♦❞❡❧ ✐❞❡♥t✐✜❝❛t✐♦♥ ❧♦♦❦s ❧✐❦❡ ❛s ❢♦❧❧♦✇s✿

✶✳ ❋✐rst✱ ❛ ♠❛①✐♠✉♠ ✈❛❧✉❡ ❢♦r p ✐s ❣✐✈❡♥✱ ❞❡♣❡♥❞✐♥❣ ♦♥ t❤❡ ❢r❡✲

q✉❡♥❝② ♦❢ t❤❡ t✐♠❡ s❡r✐❡s✿ ❞❛✐❧② ❛♥❞ ♠♦♥t❤❧② s❡r✐❡s ❛r❡ ❣✐✈❡♥ ❛

✈❛❧✉❡ ♦❢ p = 13✱ ✇❤✐❧❡ ❢♦r q✉❛rt❡r❧② ❛♥❞ ❛♥♥✉❛❧ s❡r✐❡s ■ ❝♦♥s✐❞❡r p = 5✳

✷✳ ❆ s✉❜s❡t ♦❢ s✐❣♥✐✜❝❛♥t ❧❛❣s ❢r♦♠ t❤❡ s❡t {1, . . . , p} ✐s t❤❡♥ s❡✲

❧❡❝t❡❞✳ ❚❤✐s ✐s ❞♦♥❡ ✇✐t❤ ❛ ♥♦♥✲♣❛r❛♠❡tr✐❝ ✈❡rs✐♦♥ ♦❢ t❤❡ ✜♥❛❧

♣r❡❞✐❝t✐♦♥ ❡rr♦r ✭❋P❊✮ ❝r✐t❡r✐♦♥ ✇❤✐❝❤ ❤❛s ❜❡❡♥ ♣r♦♣♦s❡❞ ❜②

❚s❝❤❡r♥✐❣ ❛♥❞ ❨❛♥❣ ✭✷✵✵✵✮ ✐♥ t❤❡ ❝❛s❡ ♦❢ t❤❡ ❣❡♥❡r❛❧ ♠♦❞❡❧

✭✸✮ ✭■♥❞❡❡❞✱ t❤❡ ♣r♦♣♦s❡❞ ♠❡t❤♦❞ ✐s ✈❛❧✐❞ ❛❧s♦ ✐♥ ❝❛s❡ ♦❢ ❤❡t✲

❡r♦s❦❡❞❛st✐❝✐t②✮✳

✸✳ ❖♥❝❡ ❛ s✉❜s❡t ♦❢ {1, . . . , p} ❤❛s ❜❡❡♥ s❡❧❡❝t❡❞✱ ❛ ❢♦r♠ ♦❢ ♠✉❧t✐✲

❢♦❧❞ ❝r♦ss✲✈❛❧✐❞❛t✐♦♥✱ ❛s ♣r♦♣♦s❡❞ ❜② ❈❛✐ ❡t ❛❧✳ ✭✷✵✵✵✮✱ ✐s ✉s❡❞

t♦ s❡❧❡❝t ❜♦t❤ t❤❡ ❧❛❣ d ♦❢ t❤❡ st❛t❡ ✈❛r✐❛❜❧❡ X_t−d ❛♥❞ t❤❡

❜❛♥❞✇✐❞t❤h ✭♦r✱ ✐♥ t❤❡ ❝❛s❡ ♦❢ ❦✲♥❡❛r❡st ♥❡✐❣❤❜♦✉r✱ t❤❡ ♦♣t✐♠❛❧

✈❛❧✉❡ ♦❢ k✮✱ ❛s s♣❡❝✐✜❡❞ ✐♥ t❤❡ ♥❡①t s✉❜s❡❝t✐♦♥✳

✾

✷✳✷✳✶ ❙❡❧❡❝t✐♦♥ ♦❢ t❤❡ ❜❛♥❞✇✐❞t❤ ❛♥❞ ♦❢ t❤❡ d ♣❛r❛♠❡t❡r

❆ ♠✉❧t✐✲❢♦❧❞ ❝r♦ss✲✈❛❧✐❞❛t✐♦♥ ♣r♦❝❡❞✉r❡ ✇❛s ♣r♦♣♦s❡❞ ❜② ❈❛✐ ❡t ❛❧✳

✭✷✵✵✵✮ t♦ ❛❧❧♦✇ t❤❡ s✐♠✉❧t❛♥❡♦✉s ❝❤♦✐❝❡ ♦❢ p✱ d ❛♥❞ h✳ ❍❡r❡ ■ ✉s❡ ✐t

♦♥❧② ❢♦r h ❛♥❞ d✱ ❤❛✈✐♥❣ ❛❧r❡❛❞② s❡❧❡❝t❡❞ ❛ s✉✐t❛❜❧❡ s✉❜s❡t ♦❢ s✐❣♥✐✜✲

❝❛♥t ❧❛❣s✳ ❚❤❡ ♣r♦❝❡❞✉r❡ ✇♦r❦s ❛s ❢♦❧❧♦✇s✳

▲❡t ✉s t❛❦❡ t✇♦ ♣♦s✐t✐✈❡ ✐♥t❡❣❡rs m ❛♥❞ Qs✉❝❤ t❤❛t T > mQ❀ t❤❡

✐❞❡❛ ✐s t♦ ✉s❡ q s✉❜✲s❡r✐❡s ♦❢ ❧❡♥❣t❤ T −qm✱ ✇✐t❤ q = 1,2, . . . , Q✱ t♦ ❡st✐♠❛t❡ t❤❡ ✉♥❦♥♦✇♥ ❢✉♥❝t✐♦♥❛❧ ❝♦❡✣❝✐❡♥ts✱ t❤❡♥ ✉s❡ t❤❡s❡ ❡st✐✲

♠❛t❡s t♦ ♣r♦❞✉❝❡ ✜tt❡❞ ✈❛❧✉❡s ❢♦r t❤❡ ♥❡①t m ♦❜s❡r✈❛t✐♦♥s✳

▲❡t ✉s ❞❡♥♦t❡ ✇✐t❤ aˆ_j,q t❤❡ ❡st✐♠❛t❡ ♦❢ a(·)_j ✉s✐♥❣ T −qm ❞❛t❛

♣♦✐♥ts✱ ✇❡ ❤❛✈❡ t❤❛t ❢♦r ❡❛❝❤ q t❤❡ ❛✈❡r❛❣❡ ♣r❡❞✐❝t✐♦♥ ❡rr♦r ✐s ❣✐✈❡♥

❜②✿

AP E_q(h, d) = 1 m

T−qm+m

X

t=T−qm+1

"

X_t −

p

X

j=1

ˆ

a_j,q(X_t−d)X_t−j

#2

. ✭✾✮

▼♦r❡♦✈❡r✱ ❢♦r ❣✐✈❡♥ h ❛♥❞ d ❞❡✜♥❡ t❤❡ ❛✈❡r❛❣❡ ❢♦r❡❝❛st✐♥❣ ❡rr♦r✿

AP E(h, d) =Q⁻¹

Q

X

q=1

AP E_q(h, d) . ✭✶✵✮

❚❤❡ ✈❛❧✉❡ ♦❢h❛♥❞d ❛r❡ t❤❡♥ s❡❧❡❝t❡❞ s✉❝❤ t❤❛t ✭✶✵✮ ✐s ♠✐♥✐♠✐③❡❞✳

■♥ t❤❡ ❡♠♣✐r✐❝❛❧ ❡①❡r❝✐s❡ ■ ✉s❡ t❤❡ ✈❛❧✉❡s Q = 4 ❛♥❞ m = 0.1T✳

❆❧❧ t❤❡ ♣r❡✈✐♦✉s st❡♣s ❝❛♥ ❜❡ r❡♣❡❛t❡❞ ✐♥ ♠✉❝❤ t❤❡ s❛♠❡ ✇❛② ❢♦r t❤❡ ❦✲◆◆ ♠❡t❤♦❞✱ s✉❜st✐t✉t✐♥❣ k ❢♦r h ✐♥ ❡q✉❛t✐♦♥s ✭✾✮ ❛♥❞ ✭✶✵✮✳

✸ ❉❆❚❆

❚❤❡ ❛✐♠ ♦❢ t❤✐s ❡①❡r❝✐s❡ ✐s t♦ t❡st t❤❡ ❢♦r❡❝❛st✐♥❣ ♣❡r❢♦r♠❛♥❝❡ ♦❢ t❤❡

❋❈❆❘ ♠♦❞❡❧ ✇✐t❤ r❡s♣❡❝t t♦ ❛❝t✉❛❧ ❡❝♦♥♦♠✐❝ t✐♠❡ s❡r✐❡s✳ ❋♦r t❤✐s r❡❛s♦♥ ■ ❞♦ ♥♦t r❡❧② ♦♥ s✐♠✉❧❛t❡❞ ❡①❛♠♣❧❡s✱ ❡✈❡♥ t❤♦✉❣❤ t❤❡s❡ ❝♦✉❧❞

❜❡ ✐♠♣♦rt❛♥t ❢♦r ❛ss❡ss✐♥❣ t❤❡ ❢♦r❡❝❛st✐♥❣ ❜❡❤❛✈✐♦✉r ♦❢ s✉❝❤ ♠♦❞❡❧

❢♦r ❛ ❣✐✈❡♥ ❞❛t❛ ❣❡♥❡r❛t✐♥❣ ♣r♦❝❡ss ✭❉●P✮✳

❖❜✈✐♦✉s❧② t❤❡ ❝❤♦✐❝❡ ♦❢ t❤❡ t❡st✐♥❣ ❞❛t❛s❡t ❧✐♠✐ts ✐♥ s♦♠❡ r❡s♣❡❝t t❤❡ ❣❡♥❡r❛❧✐t② ♦❢ t❤❡ r❡s✉❧ts ♦❜t❛✐♥❡❞✱ ❜✉t t❤✐s ✐s ❛♥ ✉♥❛✈♦✐❞❛❜❧❡ ❧✐♠✲

✐t❛t✐♦♥ ✐♥ ❛♥ ❡♠♣✐r✐❝❛❧ ❡①❡r❝✐s❡ ❧✐❦❡ t❤✐s ♦♥❡✳ ■ tr✐❡❞ t♦ ❝♦♣❡ ✇✐t❤

t❤✐s ❝r✐t✐❝❛❧ ❛s♣❡❝t ❜② t❛❦✐♥❣ ❛ s❡t ♦❢ s❡r✐❡s ✇❤✐❝❤ ❛r❡ ✇✐❞❡❧② ✉s❡❞ ✐♥

t❤❡ t✐♠❡ s❡r✐❡s ❛♥❞ ❢♦r❡❝❛st✐♥❣ ❧✐t❡r❛t✉r❡ ❛♥❞ t❤❛t s❤♦✇ ❛ ❝❡rt❛✐♥ ❞❡✲

❣r❡❡ ♦❢ ❤❡t❡r♦❣❡♥❡✐t② ❛s r❡❣❛r❞s t♦ ❢r❡q✉❡♥❝②✱ ❧✐♥❡❛r✐t②✱ ❛♥❞ st♦❝❤❛st✐❝

❢❡❛t✉r❡s ✐♥ ❣❡♥❡r❛❧✳

✶✵

❚❤❡ ❞❛t❛ ✉s❡❞ ✐♥ t❤✐s ♣❛♣❡r ❝♦♠❡ ♠❛✐♥❧② ❢r♦♠ t❤❡ ❞❛t❛s❡ts ❝♦♥✲

t❛✐♥❡❞ ✐♥ t❤❡ s♦❢t✇❛r❡ ❘ ✭✷✵✵✼✮ ❛♥❞ ✐♥ ♣❛rt✐❝✉❧❛r ✐♥ t❤❡ ♣❛❝❦❛❣❡ ❢♠❛

✭❍②♥❞♠❛♥✱ ✷✵✵✼❛✮❀ s♦♠❡ s❡r✐❡s ❝♦♠❡ ♦✉t ❛❧s♦ ❢r♦♠ ❞❛t❛ ❝♦♥t❛✐♥❡❞

✐♥ t❤❡ ♣❛❝❦❛❣❡s ts❡r✐❡s ✭❚r❛♣❧❡tt✐ ❛♥❞ ❍♦r♥✐❦✱ ✷✵✵✻✮✱ ❢♦r❡❝❛st

✭❍②♥❞♠❛♥✱ ✷✵✵✼❜✮ ❛♥❞ ♠❋✐❧t❡r ✭❇❛❧❝✐❧❛r✱ ✷✵✵✼✮✳ ❉❡t❛✐❧❡❞ ✐♥❢♦r♠❛✲

t✐♦♥ ❛❜♦✉t t❤❡ t✐♠❡ s❡r✐❡s ❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥ t❤❡ ❆♣♣❡♥❞✐① ❆✳

■♥ t❛❜❧❡ ✶ ✇❡ s❤♦✇ t❤❡ t✐♠❡ s❡r✐❡s ❝❧❛ss✐✜❡❞ ❛❝❝♦r❞✐♥❣ t♦ t❤❡✐r

❧❡♥❣t❤ ❛♥❞ ❢r❡q✉❡♥❝②❀ ♠♦st ♦❢ t❤❡ s❡r✐❡s ❤❛✈❡ ❜❡t✇❡❡♥ ✶✵✵ ❛♥❞ ✷✵✵

♦❜s❡r✈❛t✐♦♥s ❛♥❞ ❛r❡ r❡❝♦r❞❡❞ ❛t ♠♦♥t❤❧② ❢r❡q✉❡♥❝②✳

❚❛❜❧❡ ✶✿ ❚✐♠❡ s❡r✐❡s ✉s❡❞ ❜② ❢r❡q✉❡♥❝② ❛♥❞ ❧❡♥❣t❤

❳❳

❳

❳❳

❳

❳

❳❳

❳

❳

❳❳

❢r❡q✉❡♥❝② ❧❡♥❣t❤❳ < ✶✵✵ ✶✵✶✲✷✵✵ ✷✵✶✲✸✵✵ > ✸✵✵

❛♥♥✉❛❧ ✶ ✸ ✶

q✉❛rt❡r❧② ✻ ✶

♠♦♥t❤❧② ✶✵ ✸ ✹

❞❛✐❧② ✶ ✶

❆❧❧ t❤❡ s❡r✐❡s ✇❡r❡ ♠❛❞❡ st❛t✐♦♥❛r② ❜② ❞✐✛❡r❡♥❝✐♥❣ ✭♣♦ss✐❜❧② ❛❢t❡r

❛ ❧♦❣ ♦r sq✉❛r❡ r♦♦t tr❛♥s❢♦r♠❛t✐♦♥✮❀ s❡❛s♦♥❛❧ ❞✐✛❡r❡♥❝❡s ✇❡r❡ ❛❧✇❛②s

✐♠♣♦s❡❞ ♦♥ s❡❛s♦♥❛❧ s❡r✐❡s❀ t❤❡ ♥❡❡❞ ❢♦r ✜rst ❞✐✛❡r❡♥❝❡ ✇❛s t❡st❡❞

❜② ♠❡❛♥s ♦❢ ❆❉❋✱ PP ❛♥❞ ❑P❙❙ t❡sts✿ ❛ ✜rst ❞✐✛❡r❡♥❝❡ ✐s ✐♠♣♦s❡❞

✇❤❡♥ ❛t ❧❡❛st t✇♦ ♦❢ t❤❡ ❛❢♦r❡♠❡♥t✐♦♥❡❞ t❡sts ❣✐✈❡ ❛♥ ✐♥❞✐❝❛t✐♦♥ ❛t

✾✺✪ ❝♦♥✜❞❡♥❝❡ ❧❡✈❡❧ ✐♥ ❢❛✈♦✉r ♦❢ t❤❡ ♣r❡s❡♥❝❡ ♦❢ ❛ ✉♥✐t r♦♦t✳

❚r❛♥s❢♦r♠❡❞ s❡r✐❡s ✇❡r❡ t❡st❡❞ ❢♦r ❧✐♥❡❛r✐t②✿ t❤❡ t❡sts ♣r♦♣♦s❡❞ ❜②

❍✐♥✐❝❤ ✭✶✾✽✷✮✱ ❑❡❡♥❛♥ ✭✶✾✽✺✮✱ ▲❡❡ ❡t ❛❧✳ ✭✶✾✾✸✮ ❛♥❞ ❚❡räs✈✐rt❛ ❡t ❛❧✳

✭✶✾✾✸✮ ✇❡r❡ ✉s❡❞✳ ◆♦♥❡ ♦❢ t❤❡s❡ t❡sts ♣r♦♣♦s❡ ❛ s♣❡❝✐✜❝ ❛❧t❡r♥❛t✐✈❡✳

■♥ ❛❞❞✐t✐♦♥✱ ▲❥✉♥❣✲❇♦① t❡st ♦♥ sq✉❛r❡❞ r❡s✐❞✉❛❧s ♦❢ t❤❡ ✜tt❡❞ ❆❘▼❆

♠♦❞❡❧ ✇❡r❡ ❝❛❧❝✉❧❛t❡❞✿ ✐♥ ❝❛s❡ ♦❢ ❧✐♥❡❛r✐t② t❤❡ sq✉❛r❡❞ r❡s✐❞✉❛❧s s❤♦✉❧❞

✐♥ ❢❛❝t ❜❡ ✇❤✐t❡ ♥♦✐s❡ ❛♥❞ ❞❡♣❛rt✉r❡ ❢r♦♠ t❤✐s ❜❡❤❛✈✐♦✉r ❝❛♥ ❜❡ t❛❦❡♥

❛s ❡✈✐❞❡♥❝❡ ♦❢ t❤❡ ♣r❡s❡♥❝❡ ♦❢ ♥♦♥✲❧✐♥❡❛r✐t✐❡s✳ ❉❡t❛✐❧❡❞ r❡s✉❧ts ♦♥ t❤❡

tr❛♥s❢♦r♠❛t✐♦♥ ✉s❡❞ ❛♥❞ ♦♥ t❤❡ r❡s✉❧ts ♦❢ ✉♥✐t r♦♦t ❛♥❞ ❧✐♥❡❛r✐t② t❡sts

❢♦r ❡❛❝❤ s❡r✐❡s ❛r❡ ♣r❡s❡♥t❡❞ ✐♥ ❆♣♣❡♥❞✐① ❇

❚❤❡ ♥❛t✉r❛❧ ❜❡♥❝❤♠❛r❦ ❛❣❛✐♥st ✇❤✐❝❤ t♦ ❝♦♠♣❛r❡ t❤❡ ❢♦r❡❝❛st✐♥❣

♣❡r❢♦r♠❛♥❝❡ ♦❢ t❤❡ ♠♦❞❡❧ ❝♦♥s✐❞❡r❡❞ ✐s t❤❡ ✇❡❧❧ ❦♥♦✇♥ ❆❘▼❆ ♠♦❞❡❧

❛s ✐♥ ✭✷✮✳ ❚❤❡ s✉❜s❡t ♦❢ s✐❣♥✐✜❝❛♥t ❧❛❣s ♦❢ t❤❡ ❛✉t♦r❡❣r❡ss✐✈❡ ❛♥❞

♠♦✈✐♥❣ ❛✈❡r❛❣❡ ♣♦❧②♥♦♠✐❛❧s ❤❛✈❡ ❜❡❡♥ s❡❧❡❝t❡❞ ♦♥ t❤❡ ❜❛s✐s ♦❢ t❤❡

❆■❈❈ ❝r✐t❡r✐♦♥ ✭s❡❡ ❇r♦❝❦✇❡❧❧ ❛♥❞ ❉❛✈✐s✱ ✶✾✾✶✱ ♣✳ ✸✵✷✮✱ s❡❛r❝❤✐♥❣

✶✶

❚❛❜❧❡ ✷✿ ❘❡s✉❧ts ♦❢ ❧✐♥❡❛r✐t② t❡sts

♥✉♠❜❡r ♦❢

t❡sts s✐❣♥❢✳ ❛t

✺✪ s❡r✐❡s

✵ ✺

✶ ✼

✷ ✾

✸ ✸

✹ ✻

✺ ✶

✇✐t❤✐♥ ❛ ♠❛①✐♠✉♠ ❧❛❣ ♦❢ ✶✸ ❜♦t❤ ❢♦r t❤❡ ❛✉t♦r❡❣r❡ss✐✈❡ ❛♥❞ t❤❡

♠♦✈✐♥❣ ❛✈❡r❛❣❡ ❧❛❣ ♣♦❧②♥♦♠✐❛❧✳ ❆ ❢✉rt❤❡r ❜❡♥❝❤♠❛r❦ ✐s r❡♣r❡s❡♥t❡❞

❜② ❛ s✐♠♣❧❡ r❛♥❞♦♠ ✇❛❧❦ ♠♦❞❡❧✳

✹ ❊▼P■❘■❈❆▲ ❘❊❙❯▲❚❙

❊❛❝❤ ♠♦❞❡❧ ❝♦♥s✐❞❡r❡❞ ✇❛s ✐❞❡♥t✐✜❡❞ ✉s✐♥❣ t❤❡ ✜rst t✇♦ t❤✐r❞s ♦❢ ♦❜✲

s❡r✈❛t✐♦♥s ❢♦r ❡❛❝❤ s❡r✐❡s✱ ✇❤✐❧❡ t❤❡ r❡♠❛✐♥✐♥❣ t❤✐r❞ ✇❛s ❧❡❢t ✐♥ ♦r❞❡r t♦ ❝❛rr② ♦✉t ❛ tr✉❡ ♦✉t✲♦❢✲s❛♠♣❧❡ ❢♦r❡❝❛st✐♥❣ ❡①❡r❝✐s❡✳ ❋♦r ❛❧❧ ❡st✐✲

♠❛t✐♦♥s ■ ✉s❡❞ ❛ ●❛✉ss✐❛♥ ❦❡r♥❡❧✳ ❆♣♣❡♥❞✐① ❈ ❝♦♥t❛✐♥s s♦♠❡ ❞❡t❛✐❧❡❞

✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t t❤❡ ♠♦❞❡❧s s❡❧❡❝t❡❞ ❢♦r ❡❛❝❤ s❡r✐❡s✳ ❆ r❡❝✉rs✐✈❡

s❝❤❡♠❡ ✇❛s ✉s❡❞ t♦ ❣❡t ❢♦r❡❝❛sts ✉♣ t♦ ✶✷ st❡♣✲❛❤❡❛❞ ❢♦r t❤❡ ❡✈❛❧✉✲

❛t✐♦♥ ♣❡r✐♦❞✳ ■♥ t❤✐s ♣❛♣❡r ✇❤❡♥ ■ r❡❢❡r t♦ ♠✉❧t✐✲st❡♣ ❢♦r❡❝❛sts ■ ♠❡❛♥

t❤❛t t❤❡ s✲st❡♣✲❛❤❡❛❞ ❢♦r❡❝❛st ✐s ♦❜t❛✐♥❡❞ ✐t❡r❛t✐✈❡❧②✱ ❝♦♥s✐❞❡r✐♥❣ ❛s tr✉❡ ✈❛❧✉❡s t❤❡ ❢♦r❡❝❛sts ❢♦r t❤❡ 1,2, . . . , s−1 st❡♣✲❛❤❡❛❞ ♦❜t❛✐♥❡❞ ✐♥

t❤❡ ♣r❡✈✐♦✉s r♦✉♥❞s✳ ■♥❞❡❡❞✱ ■ ❛♠ ❛✇❛r❡ t❤✐s ✐s ♥♦t t❤❡ ♦♥❧② ♣♦ss✐❜❧❡

♣r♦❝❡❞✉r❡ ✐♥ t❤❡ ❢r❛♠❡✇♦r❦ ♦❢ ♥♦♥✲❧✐♥❡❛r ♠♦❞❡❧s ✭❍❛r✈✐❧❧ ❛♥❞ ❘❛②✱

✷✵✵✺✱ ❡✳❣✳✮✳ ❆♥②✇❛②✱ t❤❡ r❡s✉❧ts ❣✐✈❡♥ ✐♥ t❤❡ ♣r❡✈✐♦✉s r❡❢❡r❡♥❝❡ ❢♦r

❋❈❆❘ ♠♦❞❡❧s ❞♦ ♥♦t s❡❡♠ t♦ ❝❧❡❛r❧② s✉♣♣♦rt ❛ ♣❛rt✐❝✉❧❛r ♠❡t❤♦❞✱ s♦

■ ✉s❡ t❤❡ ♠♦st ✇✐❞❡s♣r❡❛❞ ♦♥❡ ❛♠♦♥❣ ♣r❛❝t✐t✐♦♥❡rs✳ ▼♦r❡♦✈❡r✱ ♦t❤❡r

❛♣♣r♦❛❝❤❡s✱ s✉❝❤ ❛s ❛ ❞✐r❡❝t ♠✉❧t✐✲st❡♣ ❛♣♣r♦❛❝❤✱ ✐♠♣❧② t❤❡ ✐❞❡♥t✐✜✲

❝❛t✐♦♥ ♦❢ ❞✐✛❡r❡♥t ♠♦❞❡❧s ❢♦r ❞✐✛❡r❡♥t ❢♦r❡❝❛st✐♥❣ ❤♦r✐③♦♥s ❛♥❞ t❤✐s

❝♦✉❧❞ ❜❡ ❛♥ ✐ss✉❡ ✐♥ t❤❡ ♣r❡s❡♥t ❝♦♥t❡①t ❢♦r ❛t ❧❡❛st t✇♦ r❡❛s♦♥s✿

✜rst✱ t❤❡ ❝♦♠♣✉t❛t✐♦♥❛❧ ❜✉r❞❡♥ ✐s ♠✉❝❤ ❤❡❛✈✐❡r t❤❛♥ t❤❡ ♣r❡s❡♥t ❛♣✲

♣r♦❛❝❤❀ s❡❝♦♥❞✱ t❤❡ ❝❤♦✐❝❡ ♦❢ t❤❡ st❛t❡ ❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡ ❜❡❝♦♠❡s

✶✷

♠♦r❡ q✉❡st✐♦♥❛❜❧❡✳

❚❤❡ ♣❛r❛♠❡t❡rs ♦❢ t❤❡ ❆❘▼❆ ♠♦❞❡❧s ❛s ✇❡❧❧ ❛s t❤❡ ❜❛♥❞✇✐❞t❤ h

✭♦r t❤❡ k ♣❛r❛♠❡t❡r✮ ✇❡r❡ r❡✲❡st✐♠❛t❡❞ ❛t ❡❛❝❤ ♣❡r✐♦❞ t✳ ▼♦r❡♦✈❡r✱

❛ tr✐♠♠✐♥❣ ✇❛s ❛❞♦♣t❡❞ ❛s ✐♥ ❙t♦❝❦ ❛♥❞ ❲❛ts♦♥ ✭✶✾✾✽✮✱ t❤❛t ✐s

❢♦r❡❝❛sts ✇✐t❤ ❡①❝❡♣t✐♦♥❛❧ ✈❛❧✉❡s ✇❡r❡ ❡①❝❧✉❞❡❞ ❛♥❞ r❡♣❧❛❝❡❞ ❜② ❛

♥♦✲❝❤❛♥❣❡ ❢♦r❡❝❛st s♦ ❛s t♦ s✐♠✉❧❛t❡ ❛ ❤✉♠❛♥ ✐♥t❡r✈❡♥t✐♦♥ ♦♥ t❤❡

❛✉t♦♠❛t✐❝ ❣❡♥❡r❛t❡❞ ❢♦r❡❝❛sts^✶✳ ◗✉❛♥t✐t❛t✐✈❡❧② t❤✐s ✇❛s ❝♦♥✜♥❡❞ t♦

✼✻ ❝❛s❡s ♦✉t ♦❢ ✷✷✾✷ ❢♦r❡❝❛sts ❣❡♥❡r❛t❡❞ ✭✸✳✸✪✮❀ t❤✐s ❝♦♥❝❡r♥❡❞

❡ss❡♥t✐❛❧❧② ❢♦✉r s❡r✐❡s ✇❤✐❝❤ ❝♦♥t❛✐♥ ✻✵ ♦✉t ♦❢ t❤❡ ✼✻ ❝❛s❡s ❝♦♥s✐❞❡r❡❞✳

❋♦r❡❝❛st✐♥❣ ♣❡r❢♦r♠❛♥❝❡ ✇❛s ❡✈❛❧✉❛t❡❞ ✇✐t❤ r❡❢❡r❡♥❝❡ t♦ s♦♠❡

✉s✉❛❧ ✐♥❞✐❝❛t♦rs✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ❞❡♥♦t✐♥❣ ✇✐t❤ y_t t❤❡ tr✉❡ ♦❜s❡r✈❛t✐♦♥

♦❢ ✈❛r✐❛❜❧❡ y ❛t t✐♠❡ t ❛♥❞ ✇✐t❤ yˆ_st t❤❡ s✲st❡♣ ❛❤❡❛❞ ❢♦r❡❝❛st ❢♦r

✈❛r✐❛❜❧❡ y ❛t t✐♠❡ t✱ ❛♥❞ ✇✐t❤ 1, . . . , τ t❤❡ ✐♥t❡r✈❛❧ ♦❢ ❡✈❛❧✉❛t✐♦♥✱ ■

❝❛❧❝✉❧❛t❡❞ t❤❡ ❢♦❧❧♦✇✐♥❣ ♠❡❛s✉r❡s✿

• ♠❡❛♥ ❡rr♦r ✭▼❊✮✿ ¹_τ Pτ

t=1(y_t −yˆ_ts)❀

• ♠❡❛♥ ❛❜s♦❧✉t❡ ❡rr♦r ✭▼❆❊✮✿ _τ¹P_τ

t=1|y_t −yˆ_ts|❀

• r♦♦t ♠❡❛♥ sq✉❛r❡s❡rr♦r ✭❘▼❙❊✮✿ q

1 τ

Pτ

t=1(y_t −yˆ_ts)²❀

• ♠❡❞✐❛♥ ❡rr♦r ✭▼❡❞❊✮✿ Med {y_t −yˆ_ts}_t=1,...,τ❀

• ♠❡❞✐❛♥ ❛❜s♦❧✉t❡ ❡rr♦r ✭▼❡❞❆❊✮✿ Med{|y_t −yˆ_ts|}_t=1,...,τ✳

■♥ ❛❞❞✐t✐♦♥ t♦ t❤❡ ♣r❡✈✐♦✉s ♠❡❛s✉r❡s✱ ✇❤✐❝❤ ♠❛✐♥❧② ❛❞❞r❡ss t❤❡

q✉❡st✐♦♥ ♦❢ ❤♦✇ ❝❧♦s❡ ✐s t❤❡ ❢♦r❡❝❛st ✈❛❧✉❡ t♦ t❤❡ r❡❛❧✐s❡❞ ♦♥❡✱ ■ ✉s❡

❛ ❢✉rt❤❡r ❡✈❛❧✉❛t✐♦♥ ❝r✐t❡r✐♦♥✱ ✇❤✐❝❤ ✐s ❣✐✈❡♥ ❜② t❤❡ ♣❡r❢♦r♠❛♥❝❡ ♦❢ ❛

❣✐✈❡♥ ♠♦❞❡❧ ✐♥ ❝♦rr❡❝t❧② ♣r❡❞✐❝t✐♥❣ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ ❝❤❛♥❣❡ ✐♥ t❤❡ t✐♠❡

s❡r✐❡s t♦ ❜❡ ❢♦r❡❝❛st✳ ■♥ ❢❛❝t✱ ✐t ❝♦✉❧❞ ✇❡❧❧ ❜❡ t❤❡ ❝❛s❡ t❤❛t ❛ ❢♦r❡✲

❝❛st✐♥❣ ♠♦❞❡❧ ✐s ✈❡r② ❣♦♦❞ ❛t ❢♦r❡❝❛st✐♥❣ ❛ ✈❛r✐❛❜❧❡ ♣r♦❞✉❝✐♥❣ s♠❛❧❧

❡rr♦rs✱ ✇❤✐❧❡ ❜❡✐♥❣ ✐♥❛❝❝✉r❛t❡ ❛t ❢♦r❡❝❛st✐♥❣ t❤❡ s✐❣♥ ♦❢ ✐ts ❝❤❛♥❣❡

✭❛♥❞ ✈✐❝❡ ✈❡rs❛✮✳ ■♥❞❡❡❞✱ ✐♥ s♦♠❡ ❝♦♥t❡①ts✱ ❛ ❝♦rr❡❝t s✐❣♥ ❢♦r❡❝❛st

❝♦✉❧❞ ❜❡ ❛ ✈❛❧✉❛❜❧❡ ❛ss❡t ✐♥ ❡✈❛❧✉❛t✐♥❣ t❤❡ ♣r❡❞✐❝t✐♦♥ ❛❜✐❧✐t②✳ ❍❛✈✐♥❣

s❛✐❞ t❤❛t✱ ■ ✉s❡ ❛❧s♦ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝r✐t❡r✐♦♥✿

• ❢r❛❝t✐♦♥ ♦❢ ❝♦rr❡❝t❡❞ ❞✐r❡❝t✐♦♥❛❧ ❢♦r❡❝❛sts✿ ¹_τ Pτ t=1I_(y

t−yt−1)(ˆyt s−yt−1)=1✳

❚❤❡ ❉✐❡❜♦❧❞✲▼❛r✐❛♥♦ t❡st ✇❛s ✉s❡❞ t♦ ❛ss❡ss t❤❡ s✐❣♥✐✜❝❛♥❝❡ ♦❢

❞✐✛❡r❡♥❝❡s ✐♥ ❢♦r❡❝❛st✐♥❣ ❛❝❝✉r❛❝② ❛♠♦♥❣ t❤❡ ✈❛r✐♦✉s ♠♦❞❡❧s✳ ■♥

✶❋♦r❡❝❛sts ✇❤✐❝❤ ♣r♦❞✉❝❡❞ ❛ ❝❤❛♥❣❡ ❡①❝❡❡❞✐♥❣ t❤❡ ♠❛①✐♠✉♠ ♦❜s❡r✈❡❞ ✐♥ t❤❡ ♣❛st ♦❢ t❤❡ s❡r✐❡s

✇❡r❡ ❡①❝❧✉❞❡❞✳

✶✸

♣❛rt✐❝✉❧❛r t❤❡ ✈❛r✐❛♥t ♣r♦♣♦s❡❞ ❜② ❍❛r✈❡② ❡t ❛❧✳ ✭✶✾✾✽✮ ✇❛s ✉s❡❞✳

▲❡t ✉s ❞❡♥♦t❡ ✇✐t❤ e_{i t} t❤❡ ❢♦r❡❝❛st✐♥❣ ❡rr♦rs st❡♠♠✐♥❣ ❢r♦♠ ♠♦❞❡❧

i ❛t t✐♠❡ t✱ t❤❡♥ ✇❤❡♥ ❝♦♠♣❛r✐♥❣ τ ❢♦r❡❝❛sts st❡♠♠✐♥❣ ❢r♦♠ t✇♦

❝♦♠♣❡t✐♥❣ ♠♦❞❡❧s i ❛♥❞ j t❤❡ ❉✐❡❜♦❧❞✲▼❛r✐❛♥♦ st❛t✐st✐❝s ✐s✿

DM = τ⁻¹Pτ

t=1[g(e_{i t})−g(e_jt)]

q

τ⁻¹2πf_d(0)ˆ

✭✶✶✮

✇❤❡r❡f_d(0)✐s ❛ ❝♦♥s✐st❡♥t ❡st✐♠❛t❡ ♦❢ t❤❡ s♣❡❝tr❛❧ ❞❡♥s✐t② ♦❢τ⁻¹Pτ

t=1[g(e_{i t})−

g(e_jt)] ❛t ❢r❡q✉❡♥❝② ✵✳ ❚❤❡ ❉✐❡❜♦❧❞✲▼❛r✐❛♥♦ st❛t✐st✐❝s s❤♦✉❧❞ ❜❡ ❝♦♥✲

❢r♦♥t❡❞ ✇✐t❤ ❛ st❛♥❞❛r❞ ♥♦r♠❛❧ ❞✐str✐❜✉t✐♦♥✳ ■♥ t❤✐s ♣❛♣❡r ■ ❝♦♥s✐❞❡r t❤❡ ❢✉♥❝t✐♦♥ g(·) = | · |✳ ■♥ ♣❛rt✐❝✉❧❛r ■ ✉s❡❞ t❤❡ ✈❛r✐❛♥t ♦❢ t❤❡ t❡st

♣r♦♣♦s❡❞ ❜② ❍❛r✈❡② ❡t ❛❧✳ ✭✶✾✾✽✮ ✇❤❡r❡ t❤❡ ❢♦r❡❝❛st✐♥❣ ❤♦r✐③♦♥ s ✐s

❛❧s♦ t❛❦❡♥ ✐♥t♦ ❛❝❝♦✉♥t✿

DM^∗ =

τ + 1−2s +τ⁻¹s(s −1) τ

1/2

DM. ✭✶✷✮

❚❤❡ ❛✉t❤♦rs ♣r♦♣♦s❡ t♦ ❝♦♠♣❛r❡ s✉❝❤ ❛ st❛t✐st✐❝ ✇✐t❤ t❤❡ ❙t✉❞❡♥t t

❞✐str✐❜✉t✐♦♥ ✇✐t❤ τ −1 ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠✳

■♥ ✇❤❛t ❢♦❧❧♦✇s ■ tr② t♦ s✉♠♠❛r✐③❡ t❤❡ ♠❛✐♥ r❡s✉❧ts ♦❢ t❤❡ ❢♦r❡✲

❝❛st✐♥❣ ❡①❡r❝✐s❡✳ ❋✐rst ♦❢ ❛❧❧✱ t❤❡ r❡s✉❧ts ♣r❡s❡♥t❡❞ ❛r❡ r❡❧❛t✐✈❡ t♦ t❤❡

❋❈❆❘ ♠♦❞❡❧ ✇✐t❤ ✜①❡❞ ❜❛♥❞✇✐❞t❤ ❛♥❞ t♦ t❤❡ ❆❘▼❆ ♦♥❡✳ ❆❝t✉❛❧❧② t❤❡ ❋❈❆❘ ♠♦❞❡❧ ✇✐t❤ ❛ ✈❛r✐❛❜❧❡ ❜❛♥❞✇✐❞t❤ ✭❦✲◆◆ ❡st✐♠❛t♦r✮ r❡s✉❧t❡❞

❛❧✇❛②s ✐♥ ❛ ♣♦♦r❡r ❢♦r❡❝❛st✐♥❣ ♣❡r❢♦r♠❛♥❝❡ t❤❛♥ t❤❡ ♦♥❡ ✇✐t❤ ✜①❡❞

❜❛♥❞✇✐❞t❤✳ ▼♦r❡♦✈❡r t❤❡ ♥❛✐✈❡ ✭r❛♥❞♦♠ ✇❛❧❦✮ ❢♦r❡❝❛st r❡s✉❧ts ❛r❡

❛❧s♦ ❞✐s❝❛r❞❡❞ ❜❡❝❛✉s❡ t❤❡② ❛r❡ ❛❧♠♦st ❛❧✇❛②s s✐❣♥✐✜❝❛♥t❧② ♦✉t♣❡r✲

❢♦r♠❡❞ ❜② ❛❧❧ t❤❡ ♦t❤❡r ♠❡t❤♦❞s^✷✳

❚❛❜❧❡ ✸ ♣r❡s❡♥ts t❤❡ ❛❣❣r❡❣❛t❡ r❡s✉❧ts ❝♦♥❝❡r♥✐♥❣ ❢♦r❡❝❛sts ✇✐t❤

❤♦r✐③♦♥ ❢r♦♠ ✶ t♦ ✻ st❡♣✲❛❤❡❛❞❀ ✐♥ ♣❛rt✐❝✉❧❛r t❤❡ ♣❡r❝❡♥t❛❣❡ ♦❢ ❝❛s❡s

✇❤❡r❡ ❋❈❆❘ ♠♦❞❡❧ ♦✉t♣❡r❢♦r♠s ❆❘▼❆ ♠♦❞❡❧ ❛r❡ s❤♦✇♥✳ ❈♦♥s✐❞❡r✲

✐♥❣ ✸✶ t✐♠❡ s❡r✐❡s ✇✐t❤ ✶✷ s❡t ♦❢ ❢♦r❡❝❛sts ❢♦r ❡❛❝❤ t✐♠❡ s❡r✐❡s✱ ✇❡

❤❛✈❡ ❛ t♦t❛❧ ♦❢ ✶✽✻ ♣♦ss✐❜❧❡ ❝♦♠♣❛r✐s♦♥s ❢♦r ❤♦r✐③♦♥s ✶ t♦ ✻ ❛♥❞ ✶✽✻

❢♦r ❤♦r✐③♦♥s ✼ t♦ ✶✷✳ ❚❤❡ r❡s✉❧ts ❛r❡ ❜r♦❦❡♥ ❞♦✇♥ ❜② t❤❡ ❞❡❣r❡❡ ♦❢

♥♦♥✲❧✐♥❡❛r✐t② ♦❢ t❤❡ s❡r✐❡s✱ ❝♦♥s✐❞❡r✐♥❣ s❡♣❛r❛t❡❧② t❤❡ s❡r✐❡s ❢♦r ✇❤✐❝❤

♦♥❧② ✵ ♦r ✶ t❡sts r❡❥❡❝t❡❞ ❧✐♥❡❛r✐t② ✭❛t ✾✺✪ ❝♦♥✜❞❡♥❝❡ ❧❡✈❡❧✮✱ s❡r✐❡s

❢♦r ✇❤✐❝❤ t❤✐s ✇❛s tr✉❡ ❢♦r ✷ ♦r ✸ t❡sts✱ ❛♥❞ ✜♥❛❧❧② s❡r✐❡s ✇❡r❡ ❛❧♠♦st

❛❧❧ t❡sts ✭✹✕✺✮ r❡❥❡❝t❡❞ ❧✐♥❡❛r✐t②✳

✷❖❜✈✐♦✉s❧②✱ t❤❡ ❝♦♠♣❧❡t❡ r❡s✉❧ts ❛r❡ ❛✈❛✐❧❛❜❧❡ ❢r♦♠ t❤❡ ❛✉t❤♦r✳

✶✹

❚❛❜❧❡ ✸✿ ❋♦r❡❝❛st✐♥❣ ♣❡r❢♦r♠❛♥❝❡ ❛t ✶ t♦ ✻ st❡♣✲❛❤❡❛❞✿ ♣❡r❝❡♥t❛❣❡

♦❢ ❝❛s❡s t❤❡ ❋❈❆❘ ♠♦❞❡❧ ♣❡r❢♦r♠s ❜❡tt❡r t❤❛♥ t❤❡ ❆❘▼❆ ♠♦❞❡❧

♠♦❞❡❧✳ ❇r❡❛❦❞♦✇♥ ❜② ♥♦♥✲❧✐♥❡❛r✐t② s❝♦r❡✳

★ s❡r✐❡s ♥♦♥✲

❧✐♥❡❛r✐t②^❛ ▼❆❊ ❘▼❙❊ ▼❡❞❆❊ ❞✐r❡❝t✐♦♥❛❧❡rr♦r

✶✷ ✵✕✶ ✹✼✳✷✪ ✸✹✳✼✪ ✹✽✳✻✪ ✼✹✳✻✪

✶✷ ✷✕✸ ✹✽✳✻✪ ✺✵✳✵✪ ✺✶✳✹✪ ✾✹✳✷✪

✼ ✹✕✺ ✷✸✳✽✪ ✶✹✳✸✪ ✸✺✳✼✪ ✽✷✳✾✪

✸✶ ❛❧❧ ✹✷✳✺✪ ✸✻✳✵✪ ✹✻✳✽✪ ✽✹✳✻✪

❛✏♥♦♥✲❧✐♥❡❛r✐t②✑ st❛♥❞s ❢♦r ❞❡❣r❡❡ ♦❢ ♥♦♥✲❧✐♥❡❛r✐t② ❛s ♠❡❛s✉r❡❞ ❜② t❤❡ ♥✉♠❜❡r ♦❢ t❡sts ✇❤✐❝❤

r❡❥❡❝t❡❞ ❧✐♥❡❛r✐t②✳ ❙♦✱ ✜rst r♦✇ r❡❢❡r t♦ t❤❡ ✶✷ s❡r✐❡s ✇❤✐❝❤ ✇❡r❡ ♠❛✐♥❧② ❥✉❞❣❡❞ ❧✐♥❡❛r ❜❡❝❛✉s❡ ♦♥❧②

❛t ♠♦st ✶ t❡st r❡❢✉s❡❞ t❤❡ ❧✐♥❡❛r✐t② ❤②♣♦t❤❡s✐s✱ ❡t❝✳

❲❤❛t ❡♠❡r❣❡s ✐s t❤❛t ❛ ♣❡r❝❡♥t❛❣❡ ❜❡t✇❡❡♥ ✸✻✪ ❛♥❞ ✹✼✪ ♦❢

t❤❡ ❝❛s❡s ❝♦♥s✐❞❡r❡❞✱ ❞❡♣❡♥❞✐♥❣ ✉♣♦♥ t❤❡ ❝r✐t❡r✐♦♥ ❝❤♦s❡♥✱ s❡❡ ❛♥

✐♠♣r♦✈❡♠❡♥t ✐♥ t❤❡ ❢♦r❡❝❛st✐♥❣ ♣❡r❢♦r♠❛♥❝❡ ✇✐t❤ t❤❡ ✉s❡ ♦❢ ❋❈❆❘

♠♦❞❡❧✳ ❚❤❡ ❝r✐t❡r✐♦♥ ❝❤♦s❡♥ ✐♥✢✉❡♥❝❡s t❤❡ r❡s✉❧ts✱ ✇✐t❤ t❤❡ ❘▼❙❊

❢❛✈♦✉r✐♥❣ ♠♦r❡ t❤❡ ❆❘▼❆ ♠♦❞❡❧✿ t❤✐s ✐s ❧✐❦❡❧② t♦ ❜❡ ❛ss♦❝✐❛t❡❞ t♦ t❤❡

♣r❡s❡♥❝❡ ♦❢ ❛ ❢❡✇ ❧❛r❣❡ ❡rr♦rs ✐♥ s♦♠❡ ♦❢ t❤❡ ❢♦r❡❝❛sts ♣r♦❞✉❝❡❞ ✇✐t❤

t❤❡ ❋❈❆❘ ♠♦❞❡❧✱ ❛♥❞ ✐s ❝♦♥s✐st❡♥t ✇✐t❤ t❤❡ ❜❡tt❡r r❡s✉❧ts ♦❜t❛✐♥❡❞

❜② t❤❡ ❧❛tt❡r ❝♦♠♣❛r✐♥❣ t❤❡ ♠❡❞✐❛♥ ❛❜s♦❧✉t❡ ❡rr♦r✳

❆ ✈❡r② ❣♦♦❞ r❡s✉❧t ❢♦r t❤❡ ♥♦♥✲❧✐♥❡❛r ♠♦❞❡❧ ❝♦♠❡s ♦✉t ❝❤❡❝❦✲

✐♥❣ t❤❡ ❞✐r❡❝t✐♦♥❛❧ ❡rr♦r✱ ✇❤✐❝❤ ❛❧♠♦st ✐♥✈❛r✐❛❜❧② ♣✐❝❦s ✉♣ t❤❡ ❋❈❆❘

♠♦❞❡❧ ❛s t❤❡ ❜❡st ♣❡r❢♦r♠✐♥❣ ♠♦❞❡❧✳

❋♦r ❛❧❧ t❤❡ ❝r✐t❡r✐❛ ❝♦♥s✐❞❡r❡❞ t❤❡r❡ ✐s ❛♥ ✐♠♣r♦✈❡♠❡♥t ✐♥ t❤❡ ❢♦r❡✲

❝❛st✐♥❣ ♣❡r❢♦r♠❛♥❝❡ ♦❢ t❤❡ ❋❈❆❘ ♠♦❞❡❧ ♣❛ss✐♥❣ ❢r♦♠ t❤❡ ♠♦r❡ ✏❧✐♥✲

❡❛r✑ s❡r✐❡s ✭✜rst r♦✇ ♦❢ t❛❜❧❡ ✸✮ t♦ t❤❡ ✐♥t❡r♠❡❞✐❛t❡ ♦♥❡s ✭s❡❝♦♥❞

r♦✇✮✳ ❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ t❤❡r❡ ✐s ❛ ❞r♦♣ ✐♥ t❤❡ ♣❡r❢♦r♠❛♥❝❡ ✇✐t❤ t❤❡

♠♦r❡ ♥♦♥✲❧✐♥❡❛r s❡r✐❡s

P❛ss✐♥❣ t♦ t❤❡ ❧♦♥❣❡st ❢♦r❡❝❛st✐♥❣ ❤♦r✐③♦♥s ✭✼ t♦ ✶✷ st❡♣✲❛❤❡❛❞✮

t❤❡ ♣❡r❢♦r♠❛♥❝❡ ♦❢ t❤❡ ♥♦♥✲❧✐♥❡❛r ♠♦❞❡❧ ❛♣♣❡❛rs t♦ ❜❡ ❡✈❡♥ ❜❡tt❡r

♦♥ ❛✈❡r❛❣❡✱ ❛s s❤♦✇♥ ✐♥ t❛❜❧❡ ✹✿ ✐♥ ♦✈❡r ✹✹✪ ♦❢ ❝❛s❡s t❤❡ ❋❈❆❘

♠♦❞❡❧ ♦✉t♣❡r❢♦r♠s t❤❡ ❆❘▼❆ ♦♥❡✱ ❝♦♥s✐❞❡r✐♥❣ t❤❡ ▼❆❊ ❛♥❞ ❘▼❙❊

❝r✐t❡r✐❛✱ ✇❤✐❧❡ t❤✐s ♣❡r❝❡♥t❛❣❡ r✐s❡s t♦ ✻✵✪ ✇❤❡♥ ▼❡❞❆❊ ✐s ❝♦♥s✐❞✲

❡r❡❞✳ ❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ t❤❡r❡ ✐s ❛ ❞❡t❡r✐♦r❛t✐♦♥ ♦❢ t❤❡ ♣❡r❢♦r♠❛♥❝❡

❢♦r t❤❡ ❞✐r❡❝t✐♦♥❛❧ ❡rr♦r✿ ♥❡✈❡rt❤❡❧❡ss✱ ❛❝❝♦r❞✐♥❣ t♦ t❤✐s ❝r✐t❡r✐♦♥ st✐❧❧

♠♦r❡ t❤❛♥ ✻✵✪ ♦❢ t❤❡ s❡r✐❡s ❛r❡ ❜❡st ❢♦r❡❝❛st ✇✐t❤ t❤❡ ❋❈❆❘ ♠♦❞❡❧✱

✐rr❡s♣❡❝t✐✈❡ ♦❢ t❤❡✐r ❧✐♥❡❛r✐t②✳

❚❤❡ s✐❣♥✐✜❝❛♥❝❡ ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ ✐♥ ❢♦r❡❝❛st✐♥❣ ♣❡r❢♦r♠❛♥❝❡ ✐s

✶✺

❚❛❜❧❡ ✹✿ ❋♦r❡❝❛st✐♥❣ ♣❡r❢♦r♠❛♥❝❡ ❛t ✼ t♦ ✶✷ st❡♣✲❛❤❡❛❞✿ ♣❡r❝❡♥t❛❣❡

♦❢ ❝❛s❡s t❤❡ ❋❈❆❘ ♠♦❞❡❧ ♣❡r❢♦r♠s ❜❡tt❡r t❤❛♥ t❤❡ ❆❘▼❆ ♠♦❞❡❧

♠♦❞❡❧✳ ❇r❡❛❦❞♦✇♥ ❜② ♥♦♥✲❧✐♥❡❛r✐t② s❝♦r❡✳

★ s❡r✐❡s ♥♦♥✲

❧✐♥❡❛r✐t②^❛ ▼❆❊ ❘▼❙❊ ▼❡❞❆❊ ❞✐r❡❝t✐♦♥❛❧❡rr♦r

✶✷ ✵✕✶ ✺✶✳✹✪ ✺✺✳✻✪ ✻✷✳✺✪ ✻✷✳✷✪

✶✷ ✷✕✸ ✺✵✳✵✪ ✹✸✳✶✪ ✻✽✳✶✪ ✻✶✳✹✪

✼ ✹✕✺ ✸✶✳✵✪ ✷✽✳✻✪ ✹✷✳✾✪ ✻✵✳✻✪

✸✶ ❛❧❧ ✹✻✳✷✪ ✹✹✳✻✪ ✻✵✳✷✪ ✻✶✳✺✪

❛s❡❡ ❢♦♦t♥♦t❡ ✐♥ t❛❜❧❡ ✸

❝❛rr✐❡❞ ♦✉t ❜② ♠❡❛♥s ♦❢ t❤❡ ❉▼ t❡st ✭✶✷✮✱ ❛♥❞ ❛ s✉♠♠❛r② ♦❢ t❤❡

r❡s✉❧ts ✐s ♣r❡s❡♥t❡❞ ✐♥ t❛❜❧❡s ✺ ❛♥❞ ✻✳

❚❤❡ r❡s✉❧ts ❞♦ ♥♦t s❤♦✇ ❛ ❝❧❡❛r ❝✉t ♣❛tt❡r♥✱ ♥♦r ✇✐t❤ r❡❢❡r❡♥❝❡

t♦ t❤❡ ❧✐♥❡❛r✐t② ♦❢ t❤❡ s❡r✐❡s✱ ♥❡✐t❤❡r ✇✐t❤ r❡❢❡r❡♥❝❡ t♦ t❤❡ ❢♦r❡❝❛st✐♥❣

❤♦r✐③♦♥❀ ❛❝t✉❛❧❧②✱ ✇❤❡♥ t❤❡ ❢♦r❡❝❛st✐♥❣ ❤♦r✐③♦♥s ✶✕✻ ❛r❡ ❝♦♥s✐❞❡r❡❞

✐♥ ❤❛❧❢ t❤❡ s❡r✐❡s ✭✶✻ ♦✉t ♦❢ ✸✶✮ t❤❡r❡ ❛r❡ ♥♦ s✐❣♥✐✜❝❛♥t ❞✐✛❡r❡♥❝❡s

❛♠♦♥❣ ❆❘▼❆ ❛♥❞ ❋❈❆❘ ❢♦r❡❝❛sts✱ ✇❤✐❧❡ ❢♦r ✼✕✶✷ ❤♦r✐③♦♥s t❤✐s ♥✉♠✲

❜❡r r✐s❡s t♦ ✷✶✳ ❚❤❡r❡❢♦r❡✱ t❤❡ ✐♥❢♦r♠❛t✐✈❡ ❝♦♥t❡♥t ♦❢ t❤❡ t❡st ❛♣♣❡❛rs

♥♦t t♦ ❜❡ ✈❡r② ❤✐❣❤ ✐♥ t❤✐s ❝♦♥t❡①t✱ ❛♥❞ ✐t s✉r❡❧② ❞❡s❡r✈❡s s♦♠❡ ❞❡❡♣❡r

❛♥❛❧②s✐s✳

❚❛❜❧❡ ✺✿ ❚❡st ♦❢ ❢♦r❡❝❛st✐♥❣ ❛❝❝✉r❛❝② ❛t ✶✕✻ st❡♣ ❛❤❡❛❞✳ P❡r❝❡♥t❛❣❡

♦❢ ❝❛s❡s t❤❡ t❡st ✐s s✐❣♥✐✜❝❛♥t ❛t ✾✺✪✳ ❇r❡❛❦❞♦✇♥ ❜② ♥♦♥✲❧✐♥❡❛r✐t② s❝♦r❡✳

★ s❡r✐❡s ♥♦♥✲

❧✐♥❡❛r✐t②^❛ ❋❈❆❘ ❜❡tt❡r

t❤❛♥ ❆❘▼❆ ❆❘▼❆ ❜❡tt❡r t❤❛♥ ❋❈❆❘

✶✷ ✵✕✶ ✷✳✽✪ ✾✳✼✪

✶✷ ✷✕✸ ✷✵✳✽✪ ✽✳✸✪

✼ ✹✕✺ ✹✳✽✪ ✶✶✳✶✪

✸✶ ❛❧❧ ✶✵✳✷✪ ✾✳✺✪

❛s❡❡ ❢♦♦t♥♦t❡ ✐♥ t❛❜❧❡ ✸✳

✺ ❈❖◆❈▲❯❉■◆● ❘❊▼❆❘❑❙

❚❤❡ ♣✉r♣♦s❡ ♦❢ t❤✐s ♣❛♣❡r ✐s t♦ ❡✈❛❧✉❛t❡ t❤❡ ❢♦r❡❝❛st✐♥❣ ❛❜✐❧✐t② ❢♦r r❡❛❧✱

♠❛✐♥❧② ❡❝♦♥♦♠✐❝✱ t✐♠❡ s❡r✐❡s ♦❢ ❛ ♥♦♥✲❧✐♥❡❛r ♥♦♥✲♣❛r❛♠❡tr✐❝ ♠♦❞❡❧

✶✻

❚❛❜❧❡ ✻✿ ❚❡st ♦❢ ❢♦r❡❝❛st✐♥❣ ❛❝❝✉r❛❝② ❛t ✼✕✶✷ st❡♣ ❛❤❡❛❞✳ P❡r❝❡♥t❛❣❡

♦❢ ❝❛s❡s t❤❡ t❡st ✐s s✐❣♥✐✜❝❛♥t ❛t ✾✺✪✳ ❇r❡❛❦❞♦✇♥ ❜② ♥♦♥✲❧✐♥❡❛r✐t② s❝♦r❡✳

★ s❡r✐❡s ♥♦♥✲

❧✐♥❡❛r✐t②^❛ ❋❈❆❘ ❜❡tt❡r

t❤❛♥ ❆❘▼❆ ❆❘▼❆ ❜❡tt❡r t❤❛♥ ❋❈❆❘

✶✷ ✵✕✶ ✶✶✳✶✪ ✶✳✹✪

✶✷ ✷✕✸ ✾✳✼✪ ✶✶✳✶✪

✼ ✹✕✺ ✷✳✹✪ ✶✾✳✵✪

✸✶ ❛❧❧ ✽✳✻✪ ✾✳✶✪

❛s❡❡ ❢♦♦t♥♦t❡ ✐♥ t❛❜❧❡ ✸✳

✭❢✉♥❝t✐♦♥❛❧ ❝♦❡✣❝✐❡♥t ❛✉t♦r❡❣r❡ss✐✈❡✮ ✇✐t❤ t❤❛t ♦❢ ❛ ❝❧❛ss✐❝❛❧ ❧✐♥❡❛r✱

♣❛r❛♠❡tr✐❝ ♦♥❡ ✭❛✉t♦r❡❣r❡ss✐✈❡✱ ♠♦✈✐♥❣ ❛✈❡r❛❣❡✮✳ ❚❤❡ ❝♦♠♣❛r✐s♦♥

✇❛s ❝❛rr✐❡❞ ♦✉t ❜② tr②✐♥❣ t♦ ❜❡ ❛s ❝❧♦s❡ ❛s ♣♦ss✐❜❧❡ t♦ ❛ r❡❛❧ ❡①❡r✲

❝✐s❡✿ ♠♦❞❡❧s ✇❡r❡ ✐❞❡♥t✐✜❡❞ ✉s✐♥❣ ❥✉st ❛ s✉❜✲s❛♠♣❧❡ ♦❢ t❤❡ ❛✈❛✐❧❛❜❧❡

♦❜s❡r✈❛t✐♦♥s✱ ✇❤✐❧❡ t❤❡ r❡♠❛✐♥✐♥❣ ✇❡r❡ ✉s❡❞ t♦ ❣❡♥❡r❛t❡ ❢♦r❡❝❛sts✳

❚❤❡ ❝♦♠♣❛r✐s♦♥ ✇❛s ❝❛rr✐❡❞ ♦✉t ♦✈❡r ❛ ✈❛r✐❡t② ♦❢ ❡✈❛❧✉❛t✐♦♥ ❝r✐✲

t❡r✐❛✳ ❚❤❡ r❡s✉❧ts ❛r❡ ❡♥❝♦✉r❛❣✐♥❣✱ ✐♥ t❤❡ s❡♥s❡ t❤❛t t❤❡ ❢♦r❡❝❛st✐♥❣

♣❡r❢♦r♠❛♥❝❡ ♦❢ t❤❡ ❋❈❆❘ ♠♦❞❡❧ ✐s s✉♣❡r✐♦r t♦ t❤❛t ♦❢ t❤❡ ❆❘▼❆

♦♥❡ ✐♥ ❛ ♥♦♥✲♥❡❣❧✐❣✐❜❧❡ ♥✉♠❜❡r ♦❢ ❝❛s❡s✳ ❆ s♦♠❡✇❤❛t ❜❛❞ ♥❡✇ ✐s r❡♣r❡s❡♥t❡❞ ❜② t❤❡ ❢❛❝t t❤❛t✱ ✇❤✐❧❡ t❤❡ ♠❛✐♥ ♠♦t✐✈❛t✐♦♥ ❢♦r ✉s✐♥❣ ❛

❋❈❆❘ ♠♦❞❡❧ ❧✐❡s ✐♥ t❤❡ ♥♦♥✲❧✐♥❡❛r ♥❛t✉r❡ ♦❢ t❤❡ s❡r✐❡s ❛t ❤❛♥❞✱ ♥❡✈✲

❡rt❤❡❧❡ss✱ t❤❡r❡ ✐s ♥♦ ❝❧❡❛r ❝♦♥♥❡❝t✐♦♥ ❜❡t✇❡❡♥ t❤❡ r❡s✉❧ts ♦❢ s♦♠❡

❧✐♥❡❛r✐t② t❡sts ❛♥❞ t❤❡ ❢♦r❡❝❛st✐♥❣ ✐♠♣r♦✈❡♠❡♥t ♦❜t❛✐♥❛❜❧❡ ❢r♦♠ t❤❡

❋❈❆❘ ♠♦❞❡❧✳

❋✉rt❤❡r r❡s❡❛r❝❤ ✐s ♣❧❛♥♥❡❞ t♦ s❤❡❞ s♦♠❡ ♠♦r❡ ❧✐❣❤t ✇✐t❤ r❡❣❛r❞

t♦ ❞✐✛❡r❡♥t ❛s♣❡❝ts✳ ❆♠♦♥❣ ♦t❤❡rs✿ t❤❡ ❧✐♥❦ ❜❡t✇❡❡♥ ❧✐♥❡❛r✐t② t❡st

❞✐❛❣♥♦st✐❝ ❛♥❞ t❤❡ ❢♦r❡❝❛st✐♥❣ ♣❡r❢♦r♠❛♥❝❡ ♦❢ ♥♦♥✲❧✐♥❡❛r ♠♦❞❡❧s❀ t❤❡

✉s❡ ♦❢ ❛ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ ❧❛❣❣❡❞ ✈❛❧✉❡s ♦❢ t❤❡ ✈❛r✐❛❜❧❡ ♦❢ ✐♥t❡r❡st

❛s st❛t❡ ✈❛r✐❛❜❧❡❀ t❤❡ ❝♦♠♣❛r✐s♦♥ ♦❢ ♣r❡❞✐❝t✐♦♥ ✐♥t❡r✈❛❧s ✐♥st❡❛❞ ♦❢ ❥✉st t❤❡ ♣♦✐♥t ♦♥❡s✳

✶✼

❘❊❋❊❘❊◆❈❊❙

▼❡❤♠❡t ❇❛❧❝✐❧❛r✳ ♠❋✐❧t❡r✿ ▼✐s❝❡❧❧❛♥❡♦✉s t✐♠❡ s❡r✐❡s ✜❧✲

t❡rs✱ ✷✵✵✼✳ ❯❘▲ ❤tt♣✿✴✴✇✇✇✳♠❜❛❧❝✐❧❛r✳♥❡t✴♠❋✐❧t❡r✱

❤tt♣✿✴✴✇✇✇✳r✲♣r♦❥❡❝t✳♦r❣✳ ❘ ♣❛❝❦❛❣❡ ✈❡rs✐♦♥ ✵✳✶✲✸✳

P❡t❡r ❏✳ ❇r♦❝❦✇❡❧❧ ❛♥❞ ❘✐❝❤❛r❞ ❆✳ ❉❛✈✐s✳ ❚✐♠❡ ❙❡r✐❡s✿ ❚❤❡♦r② ❛♥❞

▼❡t❤♦❞s✳ ❙♣r✐♥❣❡r✱ ◆❡✇ ❨♦r❦✱ ✶✾✾✶✳

❩♦♥❣✇ ❈❛✐✱ ❏✐❛♥q✐♥❣ ❋❛♥✱ ❛♥❞ ◗✐✇❡✐ ❨❛♦✳ ❋✉♥❝t✐♦♥❛❧✲❝♦❡✣❝✐❡♥t r❡✲

❣r❡ss✐♦♥ ♠♦❞❡❧s ❢♦r ♥♦♥❧✐♥❡❛r t✐♠❡ s❡r✐❡s✳ ❏♦✉r♥❛❧ ♦❢ t❤❡ ❆♠❡r✐❝❛♥

❙t❛t✐st✐❝❛❧ ❆ss♦❝✐❛t✐♦♥✱ ✾✺✭✹✺✶✮✿✾✹✶✕✾✺✻✱ ❙❡♣ ✷✵✵✵✳

❘♦♥❣ ❈❤❡♥ ❛♥❞ ▲♦♥✲▼✉ ▲✐✉✳ ❋✉♥❝t✐♦♥❛❧ ❝♦❡✣❝✐❡♥t ❛✉t♦r❡❣r❡ss✐✈❡

♠♦❞❡❧s✿ ❊st✐♠❛t✐♦♥ ❛♥❞ t❡sts ♦❢ ❤②♣♦t❤❡s❡s✳ ❏♦✉r♥❛❧ ♦❢ ❚✐♠❡

❙❡r✐❡s ❆♥❛❧②s✐s✱ ✷✷✭✷✮✿✶✺✶✕✶✼✸✱ ✷✵✵✶✳

❘♦♥❣ ❈❤❡♥ ❛♥❞ ❘✉❡② ❙✳ ❚s❛②✳ ❋✉♥❝t✐♦♥❛❧✲❝♦❡✣❝✐❡♥t ❛✉t♦r❡❣r❡ss✐✈❡

♠♦❞❡❧s✳ ❏♦✉r♥❛❧ ♦❢ t❤❡ ❆♠❡r✐❝❛♥ ❙t❛t✐st✐❝❛❧ ❆ss♦❝✐❛t✐♦♥✱ ✽✽✭✹✷✶✮✿

✷✾✽✕✸✵✽✱ ▼❛r ✶✾✾✸✳

❏✳ ❋❛♥ ❛♥❞ ■✳ ●✐❥❜❡❧s✳ ▲♦❝❛❧ P♦❧②♥♦♠✐❛❧ ▼♦❞❡❧✐♥❣ ❛♥❞ ■ts ❆♣♣❧✐❝❛t✐♦♥s✳

❈❤❛♣♠❛♥ ❛♥❞ ❍❛❧❧✱ ▲♦♥❞♦♥✱ ✶✾✾✻✳

❏✐❛♥q✐♥❣ ❋❛♥ ❛♥❞ ◗✐✇❡✐ ❨❛♦✳ ◆♦♥❧✐♥❡❛r ❚✐♠❡ ❙❡r✐❡s✿ ◆♦♥♣❛r❛♠❡tr✐❝

❛♥❞ P❛r❛♠❡tr✐❝ ▼❡t❤♦❞s✳ ❙♣r✐♥❣❡r✱ ◆❡✇ ❨♦r❦✱ ✷✵✵✸✳

❉✳ ■✳ ❍❛r✈❡②✱ ❙✳ ❏✳ ▲❡②❜♦✉r♥❡✱ ❛♥❞ P✳ ◆❡✇❜♦❧❞✳ ❚❡sts ❢♦r ❢♦r❡❝❛st

❡♥❝♦♠♣❛ss✐♥❣✳ ❏♦✉r♥❛❧ ♦❢ ❇✉s✐♥❡ss ❛♥❞ ❊❝♦♥♦♠✐❝ ❙t❛t✐st✐❝s✱ ✶✻✿

✷✺✹✕✷✺✾✱ ✶✾✾✽✳

❏❛♥❡ ▲✳ ❍❛r✈✐❧❧ ❛♥❞ ❇♦♥♥♥✐❡ ❑✳ ❘❛②✳ ❆ ♥♦t❡ ♦♥ ♠✉❧t✐✲st❡♣ ❢♦r❡❝❛st✲

✐♥❣ ✇✐t❤ ❢✉♥❝t✐♦♥❛❧ ❝♦❡✣❝✐❡♥t ❛✉t♦r❡❣r❡ss✐✈❡ ♠♦❞❡❧s✳ ■♥t❡r♥❛t✐♦♥❛❧

❏♦✉r♥❛❧ ♦❢ ❋♦r❡❝❛st✐♥❣✱ ✷✶✿✼✶✼✕✼✷✼✱ ✷✵✵✺✳

▼✳ ❏✳ ❍✐♥✐❝❤✳ t❡st✐♥❣ ❢♦r ❣❛✉ss✐❛♥✐t② ❛♥❞ ❧✐♥❡❛r✐t② ♦❢ ❛ st❛t✐♦♥❛r② t✐♠❡✲s❡r✐❡s✳ ❏♦✉r♥❛❧ ♦❢ ❚✐♠❡ ❙❡r✐❡s ❆♥❛❧②s✐s✱ ✸✿✶✻✾✕✶✼✻✱ ✶✾✽✷✳

❘♦❜ ❏✳ ❍②♥❞♠❛♥✳ ❢♠❛✿ ❉❛t❛ s❡ts ❢r♦♠ ✧❋♦r❡✲

❝❛st✐♥❣✿ ♠❡t❤♦❞s ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s✧ ❜② ▼❛❦r✐❞❛❦✐s✱

❲❤❡❡❧✇r✐❣❤t ✫ ❍②♥❞♠❛♥ ✭✶✾✾✽✮✱ ✷✵✵✼❛✳ ❯❘▲

❤tt♣✿✴✴✇✇✇✳r♦❜❤②♥❞♠❛♥✳✐♥❢♦✴❘❧✐❜r❛r②✴❢♦r❡❝❛st✴✳ ❘

♣❛❝❦❛❣❡ ✈❡rs✐♦♥ ✶✳✵✾✳

✶✽

❘♦❜ ❏✳ ❍②♥❞♠❛♥✳ ❢♦r❡❝❛st✿ ❋♦r❡❝❛st✲

✐♥❣ ❢✉♥❝t✐♦♥s ❢♦r t✐♠❡ s❡r✐❡s✱ ✷✵✵✼❜✳ ❯❘▲

❤tt♣✿✴✴✇✇✇✳r♦❜❤②♥❞♠❛♥✳✐♥❢♦✴❘❧✐❜r❛r②✴❢♦r❡❝❛st✴✳ ❘

♣❛❝❦❛❣❡ ✈❡rs✐♦♥ ✶✳✵✾✳

❉✳ ▼✳ ❑❡❡♥❛♥✳ ❆ t✉❦❡② ♥♦♥❛❞❞✐t✐✈✐t②✲t②♣❡ t❡st ❢♦r t✐♠❡✲s❡r✐❡s ♥♦♥✲

❧✐♥❡❛r✐t②✳ ❇✐♦♠❡tr✐❦❛✱ ✼✷✿✸✾✕✹✹✱ ✶✾✽✺✳

❚✳❍✳ ▲❡❡✱ ❍✳ ❲❤✐t❡✱ ❛♥❞ ❈✳❲✳❏✳ ●r❛♥❣❡r✳ ❚❡st✐♥❣ ❢♦r ♥❡❣❧❡❝t❡❞

♥♦♥❧✐♥❡❛r✐t② ✐♥ t✐♠❡ s❡r✐❡s ♠♦❞❡❧s✳ ❏♦✉r♥❛❧ ♦❢ ❊❝♦♥♦♠❡tr✐❝s✱ ✺✻✿

✷✻✾✕✷✾✵✱ ✶✾✾✸✳

❏❛♠❡s ❍✳ ❙t♦❝❦ ❛♥❞ ▼❛r❦ ❲✳ ❲❛ts♦♥✳ ❆ ❝♦♠♣❛r✐s♦♥ ♦❢ ❧✐♥❡❛r ❛♥❞

♥♦♥❧✐♥❡❛r ✉♥✐✈❛r✐❛t❡ ♠♦❞❡❧s ❢♦r ❢♦r❡❝❛st✐♥❣ ♠❛❝r♦❡❝♦♥♦♠✐❝ t✐♠❡

s❡r✐❡s✳ ◆❇❊❘ ❲♦r❦✐♥❣ P❛♣❡rs ✻✻✵✼✱ ◆❛t✐♦♥❛❧ ❇✉r❡❛✉ ♦❢ ❊❝♦♥♦♠✐❝

❘❡s❡❛r❝❤✱ ■♥❝✱ ❏✉♥ ✶✾✾✽✳

❘ ❉❡✈❡❧♦♣♠❡♥t ❈♦r❡ ❚❡❛♠✳ ❘✿ ❆ ▲❛♥❣✉❛❣❡ ❛♥❞ ❊♥✈✐r♦♥♠❡♥t ❢♦r

❙t❛t✐st✐❝❛❧ ❈♦♠♣✉t✐♥❣✳ ❘ ❋♦✉♥❞❛t✐♦♥ ❢♦r ❙t❛t✐st✐❝❛❧ ❈♦♠♣✉t✐♥❣✱

❱✐❡♥♥❛✱ ❆✉str✐❛✱ ✷✵✵✼✳ ❯❘▲ ❤tt♣✿✴✴✇✇✇✳❘✲♣r♦❥❡❝t✳♦r❣✳ ■❙❇◆

✸✲✾✵✵✵✺✶✲✵✼✲✵✳

❚✳ ❚❡räs✈✐rt❛✱ ❈✳❋✳ ▲✐♥✱ ❛♥❞ ❈✳❲✳❏✳ ●r❛♥❣❡r✳ P♦✇❡r ♦❢ t❤❡ ♥❡✉r❛❧

♥❡t✇♦r❦ ❧✐♥❡❛r✐t② t❡st✳ ❏♦✉r♥❛❧ ♦❢ ❚✐♠❡ ❙❡r✐❡s ❆♥❛❧②s✐s✱ ✶✹✿✷✵✾✕

✷✷✵✱ ✶✾✾✸✳

❚✐♠♦ ❚❡räs✈✐rt❛✳ ❋♦r❡❝❛st✐♥❣ ❡❝♦♥♦♠✐❝ ✈❛r✐❛❜❧❡s ✇✐t❤ ♥♦♥❧✐♥❡❛r ♠♦❞✲

❡❧s✳ ■♥ ●r❛❤❛♠ ❊❧❧✐♦t✱ ❈❧✐✈❡ ❲✳❏✳ ●r❛♥❣❡r✱ ❛♥❞ ❆❧❧❛♥ ❚✐♠♠❡r♠❛♥♥✱

❡❞✐t♦rs✱ ❍❛♥❞❜♦♦❦ ♦❢ ❊❝♦♥♦♠✐❝ ❋♦r❡❝❛st✐♥❣✱ ✈♦❧✉♠❡ ■✱ ❝❤❛♣t❡r ✽✱

♣❛❣❡s ✹✶✸✕✺✼✳ ❊❧s❡✈✐❡r✱ ✷✵✵✻✳

❆❞r✐❛♥ ❚r❛♣❧❡tt✐ ❛♥❞ ❑✉rt ❍♦r♥✐❦✳ ts❡r✐❡s✿ ❚✐♠❡ ❙❡✲

r✐❡s ❆♥❛❧②s✐s ❛♥❞ ❈♦♠♣✉t❛t✐♦♥❛❧ ❋✐♥❛♥❝❡✱ ✷✵✵✻✳ ❯❘▲

❤tt♣✿✴✴❈❘❆◆✳❘✲♣r♦❥❡❝t✳♦r❣✴✳ ❘ ♣❛❝❦❛❣❡ ✈❡rs✐♦♥ ✵✳✶✵✲✹✳

❘♦❧❢ ❚s❝❤❡r♥✐❣ ❛♥❞ ▲✐❥✐❛♥ ❨❛♥❣✳ ◆♦♥♣❛r❛♠❡tr✐❝ ❧❛❣ s❡❧❡❝t✐♦♥ ❢♦r t✐♠❡

s❡r✐❡s✳ ❏♦✉r♥❛❧ ♦❢ ❚✐♠❡ ❙❡r✐❡s ❆♥❛❧②s✐s✱ ✷✶✭✹✮✿✹✺✼✕✹✽✼✱ ✷✵✵✵✳

✶✾