TESTING FOR MULTIPLE BUBBLES: HISTORICAL EPISODES OF EXUBERANCE AND COLLAPSE IN THE S&P 500

• Author: Shuping Shi,Jun Yu,Peter C. B. Phillips
• Date: 01 November 2015
• DOI: http://doi.org/10.1111/iere.12132
• Published date: 01 November 2015

INTERNATIONAL

ECONOMIC

REVIEW

November 2015

Vol. 56, No. 4

TESTING FOR MULTIPLE BUBBLES: HISTORICAL EPISODES OF EXUBERANCE

AND COLLAPSE IN THE S&P 500∗

BYPETER C. B. PHILLIPS,SHUPING SHI,AND JUN YU1

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

Recent work on econometric detection mechanisms has shown the effectiveness of recursive procedures in identifying

and dating financial bubbles in real time. These procedures are useful as warning alerts in surveillance strategies

conducted by central banks and fiscal regulators with real-time data. Use of these methods over long historical periods

presents a more serious econometric challenge due to the complexity of the nonlinear structure and break mechanisms

that are inherent in multiple-bubble phenomena within the same sample period. To meet this challenge, this article

develops a new recursive flexible window method that is better suited for practical implementation with long historical

time series. The method is a generalized version of the sup augmented Dickey–Fuller (ADF) test of Phillips et al.

(“Explosive behavior in the 1990s NASDAQ: When did exuberance escalate asset values?” International Economic

Review 52 (2011), 201–26; PWY) and delivers a consistent real-time date-stamping strategy for the origination and

termination of multiple bubbles. Simulations show that the test significantly improves discriminatory power and leads

to distinct power gains when multiple bubbles occur. An empirical application of the methodology is conducted on

S&P 500 stock market data over a long historical period from January 1871 to December 2010. The new approach

successfully identifies the well-known historical episodes of exuberance and collapses over this period, whereas the

strategy of PWY and a related cumulative sum (CUSUM) dating procedure locate far fewer episodes in the same

sample range.

Economists have taught us that it is unwise and unnecessary to combat asset price bubbles and

excessive credit creation. Even if we were unwise enough to wish to prick an asset price bubble,

we are told it is impossible to see the bubble while it is in its inflationary phase. (George Cooper,

2008)

1. INTRODUCTION

As financial historians have argued recently (Ferguson, 2008; Ahamed, 2009), financial crises

are often preceded by an asset market bubble or rampant credit growth. The global financial

crisis of 2007–2009 is no exception. In its aftermath, central bank economists and policymakers

∗Manuscript received July 2015.

1This article and its technical companion “Testing for Multiple Bubbles: Limit Theory of Real Time Detectors”

(Phillips et al., 2015a, this issue) build on work that was originally circulated in 2011 in a long paper entitled “Testing

for Multiple Bubbles” accompanied by a long supplement of technical results. We are grateful to the editor and

three referees for helpful comments, as well as many colleagues, seminar participants, and central bank economists

for valuable discussions. Phillips acknowledges support from the NSF under grant numbers SES 09-56687 and SES

12-58258. Shi acknowledges the Financial Integrity Research Network (FIRN) for funding support. Yu acknowledges

support from the Singapore Ministry of Education for Academic Research Fund under 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.

1043

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

and Economic Research Association

1044 PHILLIPS,SHI,AND YU

have been affirming the Basel III accord to work to stabilize the financial system by way of

guidelines on capital requirements and related measures to control “excessive credit creation.”

In this process of control, an important practical issue of market surveillance involves the

assessment of what is “excessive.” But as Cooper (2008) puts it in the header cited above from

his recent bestseller, many economists have declared the task to be impossible and that it is

imprudent to seek to combat asset price bubbles. How then can central banks and regulators

work to offset a speculative bubble when they are unable to assess whether one exists?

One contribution that econometric techniques can offer in this complex exercise of market

surveillance and policy action is the detection of exuberance in financial markets by explicit

quantitative measures. These measures are not simply ex post detection techniques but antici-

pative dating algorithms that use data only up to the point of analysis for ongoing assessment,

giving an early warning diagnostic that can assist regulators in market monitoring. If history has

a habit of repeating itself and human learning mechanisms do fail, as financial historians such

as Ferguson (2008)2assert, then quantitative warnings may serve as useful alert mechanisms to

both market participants and regulators in real time.

Several attempts to develop ex post econometric tests have been made in the literature going

back some decades (see Gurkaynak, 2008, for a recent review). Phillips, Wu, and Yu (2011, PWY

hereafter) recently proposed a recursive method that can detect exuberance in asset price series

during an inflationary phase. The approach is ex ante (or anticipative) as an early warning alert

system, so that it meets the needs of central bank surveillance teams and regulators, thereby

addressing one of the key concerns articulated by Cooper (2008). The method is especially

effective when there is a single-bubble episode in the sample data, as in the 1990s NASDAQ

episode analyzed in the PWY paper and in the 2000s U.S. house price bubble analyzed in

Phillips and Yu (2011).

Just as historical experience confirms the existence of many financial crises (Ahamed reports

60 different financial crises since the 17th century 3), when the sample period is long enough there

will often be evidence of multiple asset price bubbles in the data. The econometric identification

of multiple bubbles with periodically collapsing behavior over time is substantially more difficult

than identifying a single bubble. The difficulty arises from the complex nonlinear structure

involved in the multiple breaks that produce the bubble phenomena. Multiple breaks typically

diminish the discriminatory power of existing test mechanisms such as the recursive tests given

in PWY. These power reductions complicate attempts at econometric dating and enhance the

need for new approaches that do not suffer from this problem. If econometric methods are to be

useful in practical work conducted by surveillance teams, they need to be capable of dealing with

multiple-bubble phenomena. Of particular concern in financial surveillance is the reliability of

a warning alert system that points to inflationary upturns in the market. Such warning systems

ideally need to have a low false detection rate to avoid unnecessary policy measures and a high

positive detection rate that ensures early and effective policy implementation.

This article responds to this need by providing a new framework for testing and dating bubble

phenomena when there may be multiple bubbles in the data. The mechanisms developed here

extend those of PWY by allowing for flexible window widths in the recursive regressions on

which the test procedures are based. The approach adopted in PWY uses a sup ADF (SADF)

to test for the presence of a bubble based on sequence of forward recursive right-tailed ADF

unit root tests. PWY then proposed a dating strategy, which identifies points of origination

and termination of a bubble based on a backward regression technique. When there is a single

bubble in the data, it is known that this dating strategy is consistent, as was first shown in an

unpublished working paper by Phillips and Yu (2009) whose results are subsumed as a special

case within this work. Other break testing procedures such as Chow tests, model selection, and

CUSUM tests may also be applied as dating mechanisms. Extensive simulations conducted

2“Nothing illustrates more clearly how hard human beings find it to learn from history than the repetitive history of

stock market bubbles” (Ferguson, 2008).

3“Financial booms and busts were, and continue to be, a feature of the economic landscape. These bubbles and

crises seem to be deep-rooted in human nature and inherent to the capitalist system. By one count there have been 60

different crises since the 17th century” (Ahamed, 2009).

TESTING FOR MULTIPLE BUBBLES 1045

recently by Homm and Breitung (2012) indicate that the PWY procedure works satisfactorily

against other recursive (as distinct from full sample) procedures for structural breaks and is

particularly effective as a real-time bubble detection algorithm. Importantly, the procedure can

detect market exuberance arising from a variety of sources, including mildly explosive behavior

that may be induced by changing fundamentals such as a time-varying discount factor.

As shown here, when the sample period includes multiple episodes of exuberance and col-

lapse, the PWY procedures may suffer from reduced power and can be inconsistent, thereby

failing to reveal the existence of bubbles. This weakness is a particular drawback in ana-

lyzing long time series or rapidly changing market data where more than one episode of

exuberance is suspected. To overcome this weakness and deal with multiple breaks of exu-

berance and collapse, this article proposes a generalized sup ADF (GSADF) method to test for

the presence of bubbles as well as a recursive backward regression technique to time-stamp

the bubble origination and termination dates. Like PWY, the new procedures rely on recursive

right-tailed ADF tests but use flexible window widths in their implementation. Instead of fixing

the starting point of the recursion on the first observation, the GSADF test extends the sample

coverage by changing both the start point and the endpoint of the recursion over a feasible

range of flexible windows. This test is therefore a right-sided double recursive test for a unit

root and is analogous to double recursive left-sided ex post tests of persistence such as that

considered in Leybourne et al. (2007).

The new dating strategy is an ex ante procedure and extends the dating strategy of PWY

by changing the start point in the real-time analysis. Since the new procedures cover more

subsamples of the data and have greater window flexibility, they are designed to outperform the

PWY procedures in detecting explosive behavior when multiple episodes occur in the data. This

expected enhancement in performance by the new procedures is demonstrated here in simu-

lations, which compare the two methods in terms of their size and power in bubble detection.

Moreover, the new procedure delivers a consistent dating mechanism when multiple bubbles

occur, in contrast to the original version of the PWY dating strategy, which can be inconsistent

when multiple bubbles occur. The technique is therefore well suited to analyzing long historical

time series. Throughout the article consistency refers to consistency in determining the relevant

sample fraction of the break point instead of the sample observation, as is usual in structural

break asymptotic theory.

In addition to the GSADF test and ex ante dating algorithm, a modified version of the original

PWY algorithm is developed in which the detection procedure is repeated sequentially with

re-initialization after the detection of each bubble. This sequential PWY algorithm works with

subsamples of the data with different initializations in the recursions and therefore in theory

is capable of detecting multiple bubbles. We also consider a detection mechanism based on a

recursive CUSUM test suggested recently in Homm and Breitung (2012).

An empirical application of these methodologies is conducted to S&P 500 stock market data

over the period January 1871 to December 2010. The new approach successfully identifies all

the well-known historical episodes of exuberance over this period, including the great crash, the

postwar boom in 1954, Black Monday in October 1987, and the dot-com bubble. The strategy

of PWY is much more conservative and locates only a single episode over the same historical

period, catching the 1990s stock bubble. The sequential PWY algorithm is similarly conservative

in detecting bubbles in this data set, as is the CUSUM procedure.

The organization of the article is as follows: Section 2 discusses reduced-form model specifi-

cation issues for bubble testing, describes the new rolling window recursive test, and gives its

limit theory. Section 3 proposes date-stamping strategies and outlines their properties in single,

multiple, and no-bubble scenarios. Section 4 reports the results of simulations investigating size,

power, and performance characteristics of the various tests and dating strategies. In Section 5,

the new procedures, the original PWY test, the sequential PWY test, and the CUSUM test

are all applied to the S&P 500 price–dividend ratio data over 1871–2010. Section 6 concludes.

Proofs of the main results under the null are given in the Appendix.

A companion paper in this journal (Phillips et al., 2015a) develops the limit theory and

consistency properties of the dating procedures of this article covering both single and