TESTING FOR MULTIPLE BUBBLES: LIMIT THEORY OF REAL‐TIME DETECTORS

• Published date: 01 November 2015
• Date: 01 November 2015
• DOI: http://doi.org/10.1111/iere.12131

INTERNATIONAL ECONOMIC REVIEW

Vol. 56, No. 4, November 2015

TESTING FOR MULTIPLE BUBBLES: LIMIT THEORY

OF REAL-TIME DETECTORS∗

BYPETER C. B. PHILLIPS,SHUPING SHI,AND JUN YU

Yale University, U.S.A, University of Auckland, New Zealand, University of Southampton,

U.K., and Singapore Management University, Singapore; Macquarie University and CAMA,

Australia; Singapore Management University, Singapore

This article provides the limit theory of real-time dating algorithms for bubble detection that were suggested in

Phillips, Wu, and Yu (PWY; International Economic Review 52 [2011], 201–26) and in a companion paper by the

present authors (Phillips, Shi, and Yu, 2015; PSY; International Economic Review 56 [2015a], 1099–1134. Bubbles are

modeled using mildly explosive bubble episodes that are embedded within longer periods where the data evolve as a

stochastic trend, thereby capturing normal market behavior as well as exuberance and collapse. Both the PWY and

PSY estimates rely on recursive right-tailed unit root tests (each with a different recursive algorithm) that may be

used in real time to locate the origination and collapse dates of bubbles. Under certain explicit conditions, the moving

window detector of PSY is shown to be a consistent dating algorithm even in the presence of multiple bubbles. The

other algorithms are consistent detectors for bubbles early in the sample and, under stronger conditions, for subsequent

bubbles in some cases. These asymptotic results and accompanying simulations guide the practical implementation of

the procedures. They indicate that the PSY moving window detector is more reliable than the PWY strategy, sequential

application of the PWY procedure, and the CUSUM procedure.

1. INTRODUCTION

A recent article by Phillips, Wu, and Yu (2011; PWY) developed new econometric method-

ology for real-time bubble detection. When it was applied to NASDAQ data in the 1990s,

the algorithm revealed that evidence in the data supported Greenspan’s declaration of “irra-

tional exuberance” in December 1996 and that this evidence of market exuberance had existed

for some 16 months prior to that declaration. Greenspan’s remark therefore amounted to an

assertion that could have been evidence-based if the test had been conducted at the time.

Greenspan formulated his comment as a question: “How do we know when irrational ex-

uberance has unduly escalated asset values?” It was this very question that the recursive test

procedure in PWY was designed to address. Correspondingly, an element of the methodology

that is critical for empirical applications and policy assessment is the consistency of the test.

Ideally we want a test whose size goes to zero and whose power goes to unity as the sample

size passes to infinity. Then in very large samples there will be no false positive declarations of

exuberance and no false negative assessments where asset price bubbles are missed.

PWY gave heuristic arguments showing that their recursive methodology produced a con-

sistent test for exuberance, and they provided a real-time dating algorithm for finding the

∗Manuscript received July 2015.

1The current article and its empirical companion “Testing for Multiple Bubbles: Historical Episodes of Exuber-

ance and Collapse in the S&P 500” build on work originally circulated in the paper “Testing for Multiple Bub-

bles” and its technical supplement. We are grateful to the editor and referee for valuable comments on the origi-

nal version and thank many colleagues for helpful discussions. Phillips acknowledges support from the NSF under

Grant Nos. SES 09-56687 and SES 12-58258. Shi acknowledges the Financial Integrity Research Network (FIRN)

for funding support. Yu acknowledges the financial support from Singapore Ministry of Education Academic Re-

search Fund Tier 2 under the grant number MOE2011-T2-2-096. Please address correspondence to: Peter C.B. Phillips,

Cowles Foundation for Research in Economics, Yale University, Box 208281, New Haven, CT 06520-8281, U.S.A.

E-mail: peter.phillips@yale.edu.

1079

C

(2015) by the Economics Department of the University of Pennsylvania and the Osaka University Institute of Social

and Economic Research Association

1080 PHILLIPS,SHI,AND YU

bubble origination and termination dates that was used in analyzing the NASDAQ data. The

present article provides a rigorous limit theory showing formal test consistency of the PWY

bubble detection procedure and the consistency of its associated dating algorithm under certain

conditions, notably the existence of a single bubble period in the data.2This limit theory is part

of a much larger formal investigation undertaken here that examines the asymptotic properties

of bubble detection algorithms when there may be multiple episodes of exuberance in the data,

under which the PWY procedure does not perform nearly as well. As argued in the authors’

companion paper (Phillips, Shi, and Yu, 2015a, hereafter PSY), data over long historical peri-

ods often include several crises involving financial exuberance and collapse. Bubble detection

in this context of multiple episodes of exuberance and collapse is much more complex and is

the main subject of the PSY paper, which develops a new moving window bubble detector

that has some substantial advantages for long data series characterized by multiple financial

crises.

The dating algorithms of PWY and PSY are now being applied to a wide range of markets

that include energy, real estate, and commodities as well as financial assets.3This methodology

and its various applications have also attracted the attention of central bank economists, fiscal

regulators, and the financial press.4It is therefore important that the limit properties and

performance characteristics of these dating algorithms be well understood to assist in guiding

practitioners about the suitable choice of procedures for implementation in empirical work and

policy assessment exercises.

The PWY and PSY strategies for bubble detection involve the comparison of a sequence of

recursive test statistics with corresponding critical value sequences. Crossing times of these crit-

ical value lines provide the corresponding date estimates of bubble origination and termination.

The PWY procedure uses recursively calculated right-sided unit root test statistics based on an

expanding window of observations up to the current data point, whereas PSY use a moving

window recursion of sup statistics based on a sequence of right-sided unit root tests calculated

over flexible windows of varying length taken up to the current data point. Inferences from the

PWY and PSY strategies about the presence of exuberance in the data, including the dating of

any exuberance or collapse, are drawn from these test sequences and the corresponding critical

value sequences. The goals of the present article are to explore the asymptotic and finite sample

properties of these two procedures for bubble dating and to build a methodology for analyzing

real-time detector asymptotics in this context.

Our findings can be summarized as follows. First, under some general conditions both the

PWY and PSY detectors are consistent when there is a single bubble in the sample period.

Second, when there are two bubbles in the sample period, the PWY detector for the first bubble

is consistent, whereas the PWY estimates associated with the second bubble are duration-

dependent. Specifically, the PWY strategy fails to detect the existence of the second bubble

(and hence cannot provide consistent date estimates for the timing of that bubble) when the

first bubble has longer duration than the second. But when the duration of the second bubble

exceeds the first, the PWY strategy can detect the second bubble but only with some delay.

Third, the PSY strategy and (under additional conditions) a sequential implementation of the

PWY strategy (to each individual bubble in turn) do provide consistent detectors for both

2The present article therefore subsumes the results contained in the unpublished working paper of Phillips and Yu

(2009) which is referenced in PWY and which first analyzed the asymptotic properties of the PWY procedure.

3See Phillips and Yu (2011b), Das et al. (2011), Homm and Breitung (2012), Gutierrez (2013), Bohl et al. (2013),

Etienne et al. (2013), Greenaway-McGrevy and Phillips (forthcoming), and Jiang et al. (2015), among others.

4For example, a Financial Times article (Meyer, 2013) reports the work of Etienne et al. (2013), which employs

the PSY dating algorithm to identify agricultural commodity bubbles. Recent working papers from the Hong Kong

Monetary Authority (Yiu et al., 2013) and the Central Bank of Colombia (Ojeda-Joya et al., 2013), use PSY in studying

real estate bubbles in Hong Kong and Columbia. Work for UNCTAD by Gilbert (2010) applies PWY to date bubbles

in commodity prices and test congressional testimony reasoning by Masters (2008), and recent financial press articles

(Phillips and Yu, 2011a, 2013) use PWY to assess current real estate and world stock market data for evidence of

bubbles using these methods.

TESTING FOR MULTIPLE BUBBLES 1081

bubbles, and these results hold irrespective of bubble duration. Thus, the PSY dating algorithm

and sequential application of the PWY procedure have desirable asymptotic properties in a

multiple bubbles scenario. One disadvantage of sequentially applying the PWY procedure is

that sufficient data are needed between bubbles to implement the procedure, and therefore

some origination dates may not be consistently estimated if the origination date is excluded

from the PWY sample recursion.

The article also reports simulations to evaluate the finite sample performance of these de-

tectors and date estimators, along with an alternative procedure based on CUSUM tests, as

proposed in recent work by Homm and Breitung (2012). The simulation results strongly cor-

roborate the asymptotic theory, indicating that the PSY detector is much more reliable than

PWY. On the other hand and with some exceptions that will be discussed in detail below, se-

quential application of the PWY procedure may perform nearly as well as the PSY algorithm.

The performance characteristics of the CUSUM procedure are found to be similar to those of

PWY. Overall, the results suggest that the PSY detector is a preferred procedure for practical

implementation, especially with long data series involving more than one bubble/crisis episode.

The rest of the article is organized as follows. Section 2 introduces the date stamping pro-

cedures that use recursive regressions and right-tailed unit root tests of the type considered in

PWY and PSY. This section also describes the models used to capture mildly explosive bubble

behavior when there are single and multiple bubble episodes in the data. Section 3 derives the

limit theory for the dating procedures under both single bubble and multiple bubble alternatives.

Finite sample performance is studied in Section 4, and Section 5 concludes. Two appendices

contain supporting lemmas and derivations for the limit theory presented in the article cover-

ing both single and multiple bubble scenarios. A technical supplement to the article (Phillips

et al., 2015b) provides a complete set of additional mathematical derivations that are needed

for the limit theory presented here. Computer code and Eviews software are now available for

implementation of the methods in the article.5

2. BUBBLE DATING ALGORITHMS

This section introduces three different dating algorithms—the original PWY detector, the

PSY detector, and a sequential version of the PWY detector. The approach in all of these

algorithms is to use recursive right-tailed unit root tests to assess evidence for mildly explosive

bubble behavior. In what follows we use the same models, tests, and notation as PSY to assist

in cross referencing between the two papers.

The null hypothesis is specified as suggested in Phillips et al. (2014): a random walk (or more

generally a martingale) process with an asymptotically negligible drift that we write in the form

Xt=kT −η+Xt−1+εt,with constant kand η>1/2,(1)

where Tis the sample size, εt

i.i.d.

∼(0,σ

2), and X0=Op(1).6Under these simple conditions,

partial sums of εtsatisfy the functional law

T−1/2T·



t=1

εt⇒B(·):=σW(·),(2)

where Wis standard Brownian motion. The framework can be extended to allow for martingale

difference sequence and more general weakly dependent innovations under conditions that

5Gauss and Matlab codes are available online at https://sites.google.com/site/shupingshi/PrgGSADF.zip?

attredirects=0. An Add-In for the Eviews software package is available in Caspi (2013).

6See Phillips and Magdalinos (2009) for the impact of alternative initializations on the limit theory.