Electronic copy available at: http://ssrn.com/abstract=1676272

 

Cowles Foundation 

 for Research in Economics 

 at Yale University

Cowles Foundation Discussion Paper No. 1770 

DATING THE TIMELINE OF FINANCIAL BUBBLES 

DURING THE SUBPRIME CRISIS 

Peter C. B. Phillips and Jun Yu 

September 2010

An author index to the working papers in the 

Cowles Foundation Discussion Paper Series is located at: 

http://cowles.econ.yale.edu/ 

This paper can be downloaded without charge from the 

Social Science Research Network Electronic Paper Collection: 

http://ssrn.com/abstract=1676272
Electronic copy available at: http://ssrn.com/abstract=1676272

Dating the Timeline of Financial Bubbles

During the Subprime Crisis1

Peter C. B. Phillips

Yale University, University of Auckland,

University of Southampton & Singapore Management University

Jun Yu

Singapore Management University

August 31, 2010

1Phillips acknowledges support from the NSF under Grant No. SES 06-47086. Yu acknowledges

support from the Singapore Ministry of Education AcRF Tier 2 fund under Grant No. T206B4301-

RS. Peter C. B. Phillips, Cowles Foundation for Research in Economics, Yale University, Box 208281,

Yale Station, New Haven, Connecticut 06520-8281. Email: peter.phillips@yale.edu. Jun Yu, School of

Economics and Sim Kee Boon Institute for Financial Economics, Singapore Management University, 90

Stamford Road, Singapore 178903. Email: yujun@smu.edu.sg.
Electronic copy available at: http://ssrn.com/abstract=1676272

Abstract

A new recursive regression methodology is introduced to analyze the bubble characteristics

of various Önancial time series during the subprime crisis. The methods modify a technique

proposed in Phillips, Wu and Yu (2010) and provide a technology for identifying bubble behavior and consistent dating of their origination and collapse. The tests also serve as an early

warning diagnostic of bubble activity. Seven relevant Önancial series are investigated, including three Önancial assets (the Nasdaq index, home price index and asset-backed commercial

paper), two commodities (the crude oil price and platinum price), one bond rate (Baa), and

one exchange rate (Pound/USD). Statistically signiÖcant bubble characteristics are found in

all of these series. The empirical estimates of the origination and collapse dates suggest an

interesting migration mechanism among the Önancial variables: a bubble Örst emerged in the

equity market during mid-1995 lasting to the end of 2000, followed by a bubble in the real

estate market between January 2001 and July 2007 and in the mortgage market between November 2005 and August 2007. After the subprime crisis erupted, the phenomenon migrated

selectively into the commodity market and the foreign exchange market, creating bubbles which

subsequently burst at the end of 2008, just as the e§ects on the real economy and economic

growth became manifest. Our empirical estimates of the origination and collapse dates match

well with the general datetimes of this crisis put forward in a recent study by Caballero, Farhi

and Gourinchas (2008).

Keywords: Financial bubbles, Crashes, Date stamping, Explosive behavior, Mildly explosive

process, Subprime crisis, Timeline.

JEL classiÖcation: C15, G12
There is a very real danger, fellow citizens, that the Icelandic economy in the

worst case could be sucked into the whirlpool, and the result could be national bankruptcy (Prime Minister Geir Haarde, televised address to Icelandic Nation, October

8, 2008)

Between 40 and 45 percent of the worldís wealth has been destroyed in little less

than a year and a half. (Stephen Schwarzman, March 11, 2009)

Federal Reserve policymakers should deepen their understanding about how to

combat speculative bubbles to reduce the chances of another Önancial crisis (Donald

Kohn, Federal Reserve Board Vice Chairman, March 24, 2010)

1 Introduction

Financial bubbles have been a longstanding topic of interest for economists, involving both

theorists and empirical researchers. Some of the main issues have focused on mechanisms for

modeling bubbles, reconciling bubble-like behavior in the context of rational expectations of

future earnings, mechanisms for detecting bubbles, and measuring their extent, exploring causes

and the psychology of investor behavior, and considering suitable policy responses. While there

is general agreement that Önancial bubbles give rise to misallocation of resources and can have

serious e§ects on real economic activity, as yet there has been little consensus among economists

and policy makers on how to address the many issues raised above.

The global Önancial turmoil over 2008-2009, triggered by the subprime crisis in the US

and its subsequent e§ect on commodity markets, exchange rates and real economic activity,

has led to renewed interest among economists in Önancial bubbles and their potential global

consequences. There is now widespread recognition among policy makers as well as economists

that changes in the global economy over the last decade, far from decoupling economic activity

as was earlier believed, have led to powerful latent Önancial linkages that have increased risks

in the event of a large common shock. The magnitude of the crisis is so large, the mechanism

so complex, and the consequences so important to the real economy that understanding the

phenomena, exploring its causes and mapping its evolution have presented major challenges to

the economics profession. As the headers that lead this article indicate, a substantial percentage

of the worldís accrued wealth has been destroyed within 18 months of the subprime crisis with

manifold e§ects ranging from the collapse of major Önancial institutions to the near bankruptcy

of national economies. There is also recognition that new empirical methods are needed to

improve understanding of speculative phenomena and to provide early warning diagnostics of

1
Figure 1: Time series plots of real prices for three Önancial assets: monthly observations on the

Nasdaq index from February 1973 to January 2009; monthly observations on the house price

index from January 1987 to January 2009; monthly observations on the outstanding value of

asset backed commercial paper from January 2001 to January 2009. All series are normalized

by the CPI. The estimated bubble origination and collapse dates are also shown on the Ögures.

Önancial bubbles.

The recent background of Önancial exuberance and collapse with concatenating e§ects

across markets and nations provides a rich new environment for empirical research. The most

urgent ongoing questions relate to matters of Öscal, monetary, and regulatory policies for securing Önancial stability and buttressing real economic activity. In this regard, there have been

serious disagreements among economists and policy makers about the e§ectiveness and consequences of various bailout and recovery plans proposed by North American, European and

Asian governments to deal with the Önancial crisis. Beyond these immediate policy issues are

underlying questions relating to the emergence of the phenomena and its evolutionary course

through the Önancial and economic systems. It is these latter issues that form the focus of

interest of the present paper.

The subprime crisis is not an isolated empirical event. In a recent article, Caballero, Farhi

and Gourinchas (2008a, CFG hereafter) argued that the Internet bubble in the 1990s, the asset

bubbles over 2005-2006, the subprime crisis in 2007, and the commodity bubbles of 2008 are

2
all closely related. Similar views of the interconnectedness of the crisis phenomena are held by

most economists and media commentators and this interpretation is also generally supported

by the timeline in which the various crisis events have unfolded. CFG go further and put

forward a sequential hypothesis concerning bubble creation and collapse that accounts for the

course of the Önancial turmoil in the U.S. economy using a simple general equilibrium model

without monetary factors but with goods that may be partially securitized. Date stamping

the timeline of the origination and collapse of the various bubbles is a critical element in the

validity of this sequential hypothesis.

The present paper uses new econometric methodology to test if and when bubbles emerged

and collapsed in the stock market, the real estate market, the mortgage market, the commodity

market, and the foreign exchange market over the period surrounding the subprime crisis.

Many series are studied. In particular, we investigate the bubble characteristics in the Nasdaq

index over February 1973 to January 2009, the U.S. house price index over January 1987

to January 2009, outstanding asset backed commercial paper (ABCP) over January 2001 to

January 2009, the price of crude oil over January 1999 to January 2009, platinum prices over

January 1999 to January 2009, Baa bond rates over January 3, 2006 to January 30, 2009, and

Pound/USD exchange rates over March 17, 2006 to March 20, 2009. Figs. 1-3 show the time

series plot of the Örst three, next two, and last two series, respectively. Our methods enable

us to determine whether a bubble emerged in each series, date stamp the origination in that

event, and correspondingly assess whether the bubble collapsed and the date of that collapse.

The empirical date stamps so determined are then matched against the hypothesized sequence

of events described in the model of CFG.

The econometric methods used here are closely related to those proposed in Phillips, Wu

and Yu (2009, PWY hereafter). In particular, the methods rely on forward recursive regressions

coupled with sequential right-sided unit root tests. The sequential tests assess period by period

evidence for unit root behavior against mildly explosive alternatives. Mildly explosive behavior

may be modeled by an autoregressive process with a root () that exceeds unity but that is still

in the general vicinity of unity. Phillips and Magdalinos (2007a, 2007b, PM hereafter) show

that this ëmildly explosiveívicinity of unity can be successfully modeled in terms of deviations

of the form   1 = c=kn > 0, where c is a positive constant and kn is a sequence that passes

to inÖnity with, but more slowly than, the sample size n; so that  ! 1: These processes

therefore involve only mild departures from strict (rational) martingale behavior in markets.

They include submartingale processes of the type that have been used to model rational bubble

behavior in Önance (Evans, 1991; Campbell, Lo and McKinley, 1998). PM (2007a, 2007b) have

3
investigated this class of process, developed a large sample asymptotic theory, and shown that

these models are amenable to econometric inference, unlike purely explosive processes for which

no central limit theory is applicable. An important di§erence between the methods proposed in

the current paper and those in PWY lies in the manner in which the initialization is handled.

In PWY, the initial condition is Öxed to be the Örst observation in the full sample whereas

in the current paper the initial observation is selected based on an information criterion. The

use of information criteria in the selection of the the initial observation allows for sharper

identiÖcation of the bubble origination date.

PWY applied forward recursive regression methods to Nasdaq stock prices during the 1990s,

and using sequential tests against mildly explosive alternatives were able to date-stamp the

origination of Önancial exuberance in the Nasdaq market to mid-1995, prior to the famous

remark of Alan Greenspan in December 1996 about irrational exuberance in Önancial markets.

This test therefore revealed that there was anticipatory empirical evidence supporting mildly

explosive behavior in stock prices over a year prior to Greenspanís remarks. In ongoing work,

Phillips and Yu (2009) have developed a limit theory for this date stamping technology and

checked the Önite sample capability of this procedure to identify and date bubble behavior.

The date stamp estimators were shown to be consistent for the origination and collapse of

bubble behavior and the dating mechanism was shown to work well in Önite samples.

The present paper uses this methodology to explore the sequential pattern of events of the

current Önancial crisis. Dating helps to characterize the phenomena by identifying the individual events and by Öxing their extent and sequencing. It may be viewed as a Örst step in

understanding the phenomena and in searching for causes of the behavioral changes involved

in bubble origination and collapse. Date stamping also assists in evaluating hypotheses about

the concatenation of bubble activity over time and across markets, such as those developed in

CFG. The forward recursive regression approach used here enables early identiÖcation of the

appearance of mildly explosive behavior in asset prices, thereby providing anticipatory evidence

of a (local) move away from martingale behavior. This evidence may be used as an early warning diagnostic of (Önancial) exuberance and thereby assist policy makers in surveillance and

regulatory actions, as urged by Fed Vice Chairman Donald Kohn in the header of this article.

Similarly, the approach helps to identify a subsequent switch back to martingale behavior as

explosive sentiment collapses.

Empirical evidence of emergent mildly explosive behavior is found in many of the time series

studied here, and in all of the series (except for the Pound/USD exchange rate) manifesting

mildly explosive behavior there is further evidence of subsequent collapse. Figs. 1-3 show

4
Figure 2: Time series plots of real prices for two commodities: monthly observations of crude

oil prices and monthly observations of platinum prices, both from January 1999 to January

2009. Both series are normalized by the CPI. The estimated bubble origination and collapse

dates are also shown on the Ögures.

5
Figure 3: Time series plots of two Önancial variable: daily observations of Baa bond rates from

January 3, 2006 to January 30, 2009 and weekly observations of Pound/USD exchange rates

from March 17, 2006 to March 20, 2009. The estimated bubble origination and collapse dates

are also shown on the Ögures.

6
the origination and collapse dates for the bubbles identiÖed in the seven Önancial time series

mentioned earlier. For the three series depicted in Fig. 1, the bubbles emerged and collapsed

prior to the subprime crisis. For the two series depicted in Fig. 2 and the two series depicted

in Fig. 3, the bubbles all emerged after the subprime crisis. These Öndings reveal a sequence

of mildly explosive events each followed by a Önancial collapse that corroborates the sequential

hypothesis given in CFG. Consideration of a wider group of related Önancial series following

the eruption of the subprime crisis indicates that bubbles of the type found in the series in

Figs. 2-3 are not always evident in other commodities or currencies. Accordingly, the empirical

evidence supports a selective migration of the bubble activity through Önancial markets as the

subprime crisis evolved and liquid funds searched for safe havens.

The plan of the paper is as follows. Section 2 reviews the econometric methodology for

dating bubble characteristics, discusses rational bubble and variable discount rate sources of

Önancial exuberance, outlines some of the relevant facts concerning the subprime crisis, and

relates the timeline implications of the theoretical results obtained in CFG (2008a). Section 3

describes the data that is used in the present empirical study. Section 4 presents the empirical

Öndings and matches the estimates to the theory of CFG (2008a). Section 5 concludes.

2 Bubbles, the Subprime Crisis, and Econometric Dating

2.1 Bubbles and Crashes

In the popular press, the term ìÖnancial bubbleî refers to a situation where the price of a

Önancial asset rapidly increases and does so in a speculative manner that is distinct from what

is considered to be the assetís intrinsic value. The term carries the innuendo that the increase

is not justiÖed by economic fundamentals and that there is, accordingly, risk of a subsequent

collapse in which the asset price falls precipitously. In such cases, the bubble phenomenon is

typically conÖrmed in retrospect.

A common deÖnition that makes this usage precise is that bubble conditions arise when

the price of an asset signiÖcantly exceeds the fundamental value that is determined by the

discounted expected value of the cash áows that ownership of the asset can generate. However,

discount rates may be variable and, as demonstrated below, the time proÖle of the discount

rate can have important e§ects on the characteristics of the fundamental price and may even

propagate explosive price behavior.

An important secondary characteristic of the bubble phenomenon is that during both the

run up and run down periods the asset is subject to high volume trading in which the direction

7
of change is widely anticipated (and relied upon), as distinct from normal market conditions in

which the asset price follows a near martingale. It is this deviation from martingale behavior

that provides a mechanism for identifying both the emergence of the boom phase of a bubble

behavior and its subsequent crash.

This distinction is recognized in the rational bubble literature, which characterizes the boom

phase of a bubble in terms of explosive dynamics or submartingale behavior. This property

contrasts with the e¢ cient market martingale property, which implies unit root time series

dynamic behavior. To explain the di§erence in terms of the commonly used present value

model, let Pt be the stock price at time t before the dividend payout, Dt be the dividend

payo§ from the asset at time t, and r be the discount rate (r > 0). The standard no arbitrage

condition implies that

Pt =

1

1 + r

Et(Pt+1 + Dt+1); (1)

and recursive substitution yields

Pt = Ft + Bt; (2)

where Ft =

P1

i=1

(1 + r)

iEt(Dt+i) and

Et(Bt+1) = (1 + r)Bt: (3)

Hence, the asset price is decomposed into two components, a ìfundamentalî component, Ft;

that is determined by expected future dividends, and a supplementary solution corresponding

to the ìbubbleî component, Bt: In the absence of bubble conditions, Pt = Ft. Otherwise,

Pt = Ft +Bt and price embodies the explosive component Bt; which satisÖes the submartingale

property (3). Consequently, under bubble conditions, Pt will manifest the explosive behavior

inherent in Bt. This explosive property is very di§erent from the random wandering (or unit

root) behavior that is present in Ft when Dtis a martingale and that is commonly found for

asset prices in the empirical literature.

Over long periods of time, some asset prices like equities also tend to manifest empirical

evidence of a drift component. Unit root time series with a drift can generate periods of runup if the variance of the martingale component is small and the drift is strong enough. But

accumulated gains in such cases are at most of O(n) for sample size n: In practice, of course,

the drift component is usually small and is generally negligible over short periods, so the unit

root behavior is the dominant characteristic and clear evidence of gains only shows up over long

horizons. On the other hand, the run up rate in an explosive process is O((1 + r)

n

) for some

r > 0, as in (3), and is therefore much greater. This di§erence between linear and exponential

8
growth combined with the nonlinear curvature in an explosive process are testable properties

distinguishing the two processes. In terms of model (1) and its solution (2), both Bt and Pt

increase rapidly during the boom phase of the bubble according to Et(Bt+h) = (1 + r)

hBt

and the initialization B0 > 0: But when the bubble conditions collapse and the particular

solution disappears, Pt = Ft which corresponds to a sudden collapse in the asset price. If the

dividend process Dt follows a martingale, reáecting market conditions generating cash áows,

then Ftis similarly a martingale and is cointegrated with Dt. Under such conditions, the

presence of an additional ìrational bubbleîsubmartingale component Btin Pt can account for

an explosive-type run up in the asset price Pt.

Explosiveness in Bt and hence in Pt suggests that Ptis predictable during an explosive

period. While this may be at odds with the e¢ cient market hypothesis, the predictability in

stock returns at short horizons is consistent with what has been documented in the recent empirical literature ñsee, for example, Ang and Bekaert (2006). At longer horizons, explosiveness

is subject to collapse, generating a (long run) martingale like feature in the price and making

returns more di¢ cult to predict. This latter Önding is also empirically documented in Ang and

Bekaert (2006).

Importantly, making the discount factor rt either stationary or integrated of order one does

not change qualitatively our analysis because the implications for the statistical properties of

Ft; Bt and Pt are the same as with the constant r. For example, if rtis stationary, (3) becomes

Et(Bt+1) = (1 + rt)Bt: (4)

Then, if (3) is Ötted, r =



T

t=1(1 + rt)

1=T > 1, implying an explosive process for Bt and hence

Pt; even if Ftitself is not explosive.

2.2 The E§ects of a Time Varying Discount Rate

This paper interprets explosiveness in price as su¢ cient evidence for bubbles and this interpretation holds true under a varieties of assumptions on the discount rate. As indicated above,

certain time proÖles for the discount rate can have an important e§ect on the characteristics of

the fundamental price. The present section illustrates this possibility by developing a simple

propagating mechanism for explosive behavior in the fundamental price under a time varying

discount rate.

If dividends grow at a constant rate rD with rD < r in (1),1the fundamental value of the

1This assumption obviously violates the assumption we adopted earlier, namely, constancy, stationarity or

integration of order 1.

9
stock price

Ft =

Dt

r  rD

: (5)

This is the well-known Gordon growth model. It is evident that in this case the fundamental

value can be very sensitive to changes in r when r is close to rD. In fact, the fundamental value

diverges as r & rD; so that a price run-up is evidently possible under certain time proÖles for

the discount rate. This simple Gordon model reveals the potential impact of a time varying

discount rate, but it provides no price dynamics. The following argument provides an analytic

formulation that shows how an explosive time path in fundamental values can be generated by

time variation in the discount rate.

Consider a continuous time version of (5) with time varying discount rate rt, viz.,

Ft =

Z 1

0

exp(srt+s)EtDt+sds: (6)

Suppose dividends have a constant expected growth rate rD such that

EtDt+s = exp(rDs)Dt; (7)

and then Dtis a martingale when rD = 0. Combining (6) and (7)

Ft =

Z 1

0

exp(s(rt+s  rD))Dtds: (8)

Given some Öxed time point tb; constants ca > 0 and 1 > 2 > 0; let the time proÖle of the

discount rate rt+s for t 2 (0; tb] be as follows:

rt+s =



rD +

tbts

s

ca +

1

s

for 0  s < tb  t

rD + ca +

2

s

for s  tb  t

: (9)

Then, the discount rate decreases towards some level rD +

1

tbt

as t + s % tb and jumps to the

level rD + ca +

2

tbt

immediately thereafter, as shown in Fig. 4. Thus, the time proÖle of the

discount factor has a structural break at tb in which a higher rate of discounting occurs at tb:

The break itself widens asymptotically as t % tb:

10
Figure 4: Time Path of the Discount Rate rt+s in (9)

We then have

Ft=Dt =

Z 1

0

exp(s(rt+s  rD))ds

=

Z tbt

0

exp(ca(tb  t  s)  1)ds+

Z 1

tbt

exp(cas  2)ds

= e

1

"

e

ca(tbts)

ca

#tbt

0

+ e

2



e

cas

ca

1

tbt

=

e

1

ca

h

1  e

ca(tbt)

i

+

e

2

ca

e

ca(tbt)

=

e

1

ca

+



e

2  e1



ca

e

ca(tbt)

:= t;

and the time path of Ft=Dtis explosive over t 2 (0; tb]: Over this interval, Ft evolves according

to the di§erential equation

dFt =



e

2  e

1



e

ca(tbt)Dtdt + tdDt

:

Since caFt=Dt = e

1 +



e

2  e1



e

ca(tbt)

; we have

dFt =



e

2  e1



e

ca(tbt)

e1 + (e2  e1 ) eca(tbt)

caFtdt + tdDt; for t 2 (0; tb]:

11
For t close to tb the generating mechanism for Ft

is approximately

dFt =



e

2  e1



e1 + (e2  e1 )

caFtdt + tdDt

=

n

1  e

(12)

o

caFtdt + tdDt;

which is an explosive di§usion because

cb =

n

1  e

(12)

o

ca > 0;

since ca > 0 and e

(12) < 1: The discrete time path of Ft

in this neighbourhood is therefore

propagated by an explosive autoregressive process with coe¢ cient  = e

cb > 1:

The heuristic explanation of this behavior is as follows. As t % tb there is growing anticipation that the discount factor will soon increase. Under such conditions, investors anticipate

the present to become more important in valuing assets. This anticipation in turn leads to

an ináation of current valuations and price fundamentals Ft become explosive as this process

continues.

On the other hand, for t > tb we have

rt+s = rD + ca +

2

s

for s > 0 ;

and then

Ft=Dt =

Z 1

0

exp(s(rt+s  rD))ds

=

Z 1

0

exp(cas  2)ds

= e

2



e

cas

ca

1

0

=

e

2

ca

:

So, Ft =

e2

ca

Dt for t > tb and price fundamentals are collinear with Dt: When Dtis a Brownian

motion or an integrated process in discrete time, Ft and Dt are cointegrated. Thus, after time

tb; price fundamentals comove with Dt:

It follows that the time proÖle (9) for the discount rate rtinduces a subinterval of explosive

behavior in Ft before tb: In this deterministic setting, it is known as time tb approaches that

there will be an upwards shift in the discount factor that makes present valuations more important. A more realistic model might allow for uncertainty in this time proÖle and a stochastic

trajectory for rt that accommodated potential upwards shifts of this type.

Econometric dating procedures of the type described below may be used to assess evidence

for subperiods of explosive price behavior that are induced by such time variation in the discount

factor, just as for other potential sources of Önancial exuberance.

12
2.3 Subprime Crisis and Event Timeline

The subprime mortgage crisis is generally regarded as an important triggering element in the

ongoing global Önancial crisis. The subprime event began with a dramatic rise in mortgage

delinquencies and foreclosures starting in late 2006 in the US, as easy initial adjustment rate

mortgage terms began to expire and reÖnancing became more di¢ cult at the same time as

house prices were falling. The event had wider and, soon, global consequences because of the

huge scale of mortgage backed securities (MBS) in the Önancial system, extending the impact of

mortgage failure to the asset positions of investment and commercial banks. The crisis became

apparent in the last week of July 2007 when German bank regulators and government o¢ cials

organized a $5 billion bail out of IKB, a small bank in Germany. We may therefore treat the

beginning of August 2007 as the public onset date of the subprime crisis, although the realities

in terms of rising mortgage delinquencies commenced earlier.

Much has already been written about the causes of this crisis and a host of factors have been

suggested, including poor appreciation of the risks associated with MBS, weak underwriting

standards and risk assessment practices in general, increasingly complex Önancial products,

high levels of Önancial leverage with associated vulnerabilities, shortfalls in understanding the

impact of large common shocks on the Önancial system, and inadequate monitoring by policy

makers and regulators of the accumulating risk exposure in the Önancial markets. We refer

readers to Brunnermeier (2008), Greenlaw, Hatzius, Kashyap and Shin (2008) and Hull (2008)

for detailed discussions of the subprime crisis and its manifold implications. Our concern in the

present paper is with the crisis timeline and, more speciÖcally, the issues of empirically dating

the origination and collapse of the various Önancial bubbles that occurred as the crisis events

unfolded.

Prior to the subprime crisis and following the collapse in dot.com stocks in 2000-2001, the

housing market in many states of the US sustained rapid increases in valuations fueled by a

period of low interest rates, large foreign capital ináows, and high-risk lending practices of

Önancial institutions. In the resulting boom, home ownership in the US increased to 69.2% in

2004 from 64% in 1994 (Callis and Cavanaugh, 2007) and nominal house prices increased by

more than 180% over the period 1997-2006 (Panel 2 in Fig. 1). Household debt, as a percentage of disposable income, increased from 77% to 127% over the period 1990-2007 (Economist,

November 22, 2008). At the same time, the MBS market, derived from residential mortgages,

mushroomed, and major banks and Önancial institutions around the world invested in securities that were ultimately founded on the U.S. housing market. For example, the nominal

13
outstanding amount of asset backed commercial paper (ABCP) increased by more than 80%

over the period July 2004 to July 2007 (See Panel 3 of Fig. 1).

The concatenation of events that occured after the housing market peaked in 2005 and went

into decline, followed by the subprime mortgage crisis and subsequent repercussions on Önancial

institutions over 2007-2008 and Önally the impact on world trade and real economic activity, is

now well known. Securities backed by subprime mortgages lost most of their value, investors lost

conÖdence, and liquidity dried up as money áowed to assets which appeared to have inherently

lower risk, such as Treasury bonds, and to other assets like commodities, and currencies such

as the U.S. dollar and the Japanese Yen (mainly through the unwinding of the carry trade

industry), generating a so-called áight-to-quality. In consequence, commodity prices soared,

some currencies like the U.S. dollar appreciated, while others like the British pound rapidly

declined. As the crisis deepened, stock markets around the world fell, and commercial banks,

mortgage lenders and insurance companies failed. Consumption and investment expenditures

dropped, many OECD economies went into serious recession, export driven economies in Asia

sustained double digit percentage declines in exports, growth slowed signiÖcantly in China,

and world trade declined. Concomitant with these real economic e§ects, global demand for

commodities declined and commodity prices fell.

In a recent study, CFG (2008a) proposed a model which seeks to explain the main features

of this sequence of complex interlinked Önancial crises. The CFG model links together global

Önancial asset scarcity, global imbalances, the real estate bubble, the subprime crisis, and the

commodity bubble in a general equilibrium macroeconomic environment without monetary

factors. The model is based on CFG (2008b) and assumes that the economy has two countries

(U and M) and features two goods (X and Z). A key part of the CFG framework is a sequence

of hypotheses involving successive bubble creations and collapses, which we brieáy review as

follows.

Country U is interpreted as the U.S. and country M as the emerging market economies and

commodity producers. Good X is a non-storable good, a fraction of which can be capitalized,

and is produced by both countries. Good Z is a storable commodity and is produced only

by country M. A presumption in the model is that there exists a global imbalance at period

t0. The imbalance can be interpreted as arising from continuing capital áows from emerging

markets to the U.S. as the U.S. runs a growing trade deÖcit with emergent economies, which

in turn rely more heavily on export driven growth.

In order to allow country U to have both a large current account deÖcit and low interest

rates, a fundamental assumption that CFG makes is that a bubble developed initially in country

14
U: In practical terms, this may be viewed as a bubble in the equity, housing and mortgage

markets in the U.S., the latter providing Önancial assets that o§er su¢ cient rewards to be

attractive to the rest of the world. Another fundamental assumption is that the bubble bursts

at t = 0, leaving investors (both locals and foreigners) to look for alternative stores of value. In

the Örst stage, a áight-to-quality reaction migrates the bubble to ìgoodîassets and so the price

of commodities (notably, Z) jumps, which results in a signiÖcant wealth transfer from U to M.

In the second stage, under the assumption that the Önancial asset crisis and wealth transfer

precipitates a severe growth slowdown, the excess demand for the ìgoodî asset is destroyed,

leading to a decrease in inventory of the good Z, and the bubble in commodity prices collapses.

Accordingly, this model can describe events in which asset bubbles emerged and subsequently collapsed creating a sequence of bubble e§ects in one market after another. When the

real estate bubble crashed and the value of MBS securities fell substantially, liquidity áowed

into other markets creating bubbles in commodities and oil markets as investors transferred

Önancial assets. The deepening Önancial crisis then sharply slowed down economic growth,

which in turn destroyed the commodity bubbles. Obviously, this story makes strong predictions concerning the timing of the origination and the collapse of various bubble phenomena

in di§erent markets. To evaluate the evidence in support of such interpretations of the events,

consistent date stamping of those events is critical.

2.4 Econometric Dating of the Timeline

Bubbles can be deÖnitively identiÖed only in hindsight after a market correction

(Economist, June 18, 2005)

The time path of Ptin the rational bubble model (with bubble component Bt) is explosive.

Similarly, in the run-up phase of a Önancial bubble, a pattern of stochastically explosive or

mildly explosive behavior is a characteristic feature. The econometric determination of bubble

behavior therefore relies on a test procedure having power to discriminate between unit root

(or martingale like) local behavior in a process and mildly explosive stochastic alternatives.

The same distinction in reverse is required during a bubble collapse. Phillips and Magdalinos

(2007a, 2007b, hereafter PM) analyzed the properties of mildly explosive stochastic processes

and developed a limit theory for autoregressive coe¢ cient estimation and inference in that

context.

PWY (2009) used forward recursive regression techniques and PM asymptotics to test for

the presence of mildly explosive behavior in 1990s Nasdaq data and to date stamp the orig15
ination and collapse of the Nasdaq bubble. It was shown that a sup unit root test against a

mildly explosive alternative obtained from forward recursive regressions has power to detect

periodically collapsing bubbles. To improve the power and sharpen date detection, this paper

modiÖes the sup test of PWY by selecting the initial condition based on an information criterion. The new methods are used in combination with the limit theory in Phillips and Yu

(2009) which establishes consistency of the dating estimators.

The key idea of PWY is simple to implement and relies on recursively calculated rightsided unit root tests to assess evidence for mildly explosive behavior in the data. In particular,

for time series fXtg

n

t=1, we apply standard unit root tests (such as the coe¢ cient test or the

Dickey-Fuller t test) with usual unit root asymptotics under the null against the alternative of

an explosive or mildly explosive root. The test is a right-sided test and therefore di§ers from

the usual left-sided tests for stationarity. Contrary to the quotation that heads this section, it

is possible by means of these tests to identify the emergence of mildly explosive behavior as it

occurs, thereby presaging bubble conditions. It is not necessary to wait for a market correction

to identify bubble conditions in hindsight.

More speciÖcally, we estimate the following autoregressive speciÖcation by recursive least

squares

Xt =  + Xt1 + "t; "t  iid (0; 2); (10)

allowing for the fact that the iid assumption may be relaxed with the usual (possibly semiparametric) adjustments to the tests. The null hypothesis is H0 :  = 1 and the right-tailed

alternative hypothesis is H1 :  > 1; which allows for mildly explosive autoregressions with

 = 1 + c=kn; where kn ! 1 and kn=n ! 0:

The regression in the Örst recursion uses  0 = [nr0] observations, for some fraction r0 of

the total sample where [] denotes the integer part of its argument. Subsequent regressions

employ this originating data set supplemented by successive observations giving a sample of

size  = [nr] for r0  r  1: Denote the corresponding coe¢ cient test statistic and the DickeyFuller t statistic by DF

r and DFtr

, namely,

DF

r

:= 



^ ( )  1



; DFt

r

:= P

j=1 X~ 2

j1

^

2



!1=2



^ ( )  1



; (11)

where ^ is the least squares estimate of  based on the Örst  = [nr] observations, ^

2



is

the corresponding estimate of 

2

; and X~

j1 = Xj1  

1 P

j=1 Xj1: Obviously, DF

1

and

DFt

1

correspond to the full sample test statistics. Under the null hypothesis of pure unit root

dynamics and using standard weak convergence methods (Phillips, 1987), we have the following

16
limit theory as  = [nr] ! 1 for all r 2 [r0; 1]

DF

r )

R 1

0 W dW f

R 1

0 Wf2

; DFt

r )

R 1

0 W dW f

R 1

0 Wf2

1=2

; (12)

where W is standard Brownian motion and Wf (r) = W (r) 

R 1

0 W is demeaned Brownian

motion.

If Model (10) is the true data generating process for all t, then recursive regressions are

unnecessary. In this case, a right-sided unit root test based on the full sample is able to distinguish a unit root null from an explosive alternative. In practice, of course, empirical bubble

characteristics are much more complicated than model (10) and involve some regime change(s)

between unit root (martingale) behavior with  = 1 and mildly explosive behavior with  > 1

and potential re-initialization as market temperature shifts from normal to exuberant sentiment and back again. A distinguishing empirical feature of bubble behavior is that market

correction typically occurs as sentiment reverts back and mildly explosive behavior collapses.

A model to capture this type of reversion was Örst constructed by Evans (1991) who argued

that conventional unit root tests had little power in detecting periodically collapsing bubbles

generated in this manner. As shown in Phillips and Yu (2009), such a model which mixes a

unit root process with a collapsed explosive process actually behaves like a unit root process

over the full sample (in fact, with some bias toward stationarity as explained below), thereby

invalidating the standard unit root test as a discriminating criterion when it is applied to the

full sample.

To Önd evidence for the presence of a bubble in the full sample, PWY (2009) suggest using

a sup statistic based on the recursive regression. This involves comparing supr DFt

r with the

right tailed critical values from the limit distribution based on supr2[r0;1] R r

0 W dW= f

R r

0 Wf2

1=2

.

Similarly, for the coe¢ cient test, one can compare the sup statistic supr DF

r with the right

tailed critical values from the limit distribution based on supr2[r0;1] R r

0 W dW= f

R r

0 Wf2

.

Our approach to Önding the timeline of the bubble dynamics also makes use of forward

recursive regressions. We date the origination of the bubble by the estimate ^e = [nr^e] ; where

r^e = inf

sr0

n

s : DF

s > cv

n

o

; or r^e = inf

sr0

n

s : DFt

s > cvdf

n

o

; (13)

and cv

n

(cv

df

n

) is the right-side 100n% critical value of the limit distribution of the DF

r

(DFt

r

) statistic based on  s = [ns] observations, and nis the size of the one-sided test.

Conditional on Önding some originating date r^e for (mildly) explosive behavior, we date the

17
collapse of the bubble by ^f = [nr^f ] ; where

r^f = inf

sr^e+

log(n)

n

n

s : DF

s < cv

n

o

; or r^f = inf

sr^e+

log(n)

n

n

s : DFt

s < cvdf

n

o

: (14)

This dating rule for ^f requires that the duration of the bubble is nonnegligible ñat least a small

inÖnity as measured by log n, so that episodes of smaller order than log n are not considered

signiÖcant in the dating algorithm for  f : This requirement helps to reduce the type I error in

the unit root test without a§ecting the consistency property of the estimator.

The consistent estimation of re and rf requires a slow divergence rate of critical values.

For practical implementation, we set the critical value sequencesncv

n

; cv

df

n

o

according to an

expansion rule such as cv

n

= (log log2[nr])=2 and cv

df

n

= (log log2[nr])=4: Both these critical

values diverge at a slowly varying rate with cv

df

n

< cv

n

: For practically reasonable sample

sizes, these critical values are close to the 1% critical values for DF

1

and DFt

1

. For example,

when n = 100, cv

n

= (log log2 n)=2 = 1:17 and cv

df

n

= (log log2 n)=4 = 0:58. The 1% critical

values for DF

1

and DFt

1

are 1.14 and 0.63, respectively. These critical value expansion rates

have been trialed in extensive simulations in Phillips and Yu (2009) and found to give very

satisfactory results in terms of small size and high discriminatory power.

Under the mildly explosive bubble model,

Xt = Xt11 ft <  eg + nXt11 f e  t   f g (15)

+

0

@

X

t

k= f +1

"k + X

 f

1

A 1 ft >  f g + "t 1 ft   f g

n = 1 +

c

n

; c > 0;  2 (0; 1);

Phillips and Yu (2009) showed that r^e

p! re and r^fp! rf under some general regularity

conditions. Model (15) mixes together two processes, a unit root process and a mildly explosive

process with a root above 1 taking the form n = 1 + c

n . This type of mildly explosive process

over  e  t   f was originally proposed and analyzed by PM (2007a, 2007b). However, the

above system is more complex because it involves regime switches from unit root to mildly

explosive behavior at  e and from the mildly explosive root back to a unit root at  f . At  f ,

the switch also involves a re-initialization of the process and Xt collapses to X

 f

; corresponding

to a bubble collapse back to fundamental values prevailing prior to the emergence of the bubble.

We may, for instance, set X

 f = X e + X

for some Op (1) random quantity X; so that X

 f

is within an Op (1) realization of the pre-bubble value of Xt:

18
Under this model speciÖcation (15), Phillips and Yu (2009) showed that when  = [nr] 2

[ e;  f ),

DF

r = 



^n ( )  1



= n

1

rc + op (1) ! +1;

and

DFt

r =

 P

j=1 Xe2

j1

^

2



!1=2



^n ( )  1



= n

1=2

c

3=2

r

3=2

2

1=2r

1=2

e

f1 + op (1)g ! +1:

Hence, provided cv

n

goes to inÖnity at a slower rate than n

1 and cv

df

n

(r) goes to inÖnity

at a slower rate than n

1=2

, DF

r and DFtr both consistently estimate re. Moreover, when

 = [nr] >  f ,

DF

r = 



^n ( )  1



= n

1

rc ! 1; (16)

and

DFt

r =

 P

j=1 Xe2

j1

^

2



!1=2



^n ( )  1



= n

(1+)=2

c

1=2

r

1=2

2

1=2

f1 + op (1)g ! 1: (17)

Hence, DF

r and DFtr both consistently estimate rf . Importantly, (16) diverges to negative

inÖnity, so it is apparent that in the post bubble period  >  f the autoregressive coe¢ cient

^n ( ) is biased downwards, which in this case means biased towards stationarity. This bias is

explained by the fact that the collapse of the bubble produces a mean reverting e§ect in the

data, which manifests in the limit theory as a slight bias towards stationarity in the estimated

unit root.

We now provide some heuristic discussion about the capacity of these forward recursive

regression tests to capture the timeline of bubble activity. The tests have discriminatory power

because they are sensitive to the changes that occur when a process undergoes a change from

a unit root to a mildly explosive root or vice versa. This sensitivity is much greater than in

left-sided unit root tests against stationary alternatives, due to the downward bias and long left

tail in the distribution of the autoregressive coe¢ cient in unit root and near stationary cases.

By contrast, as is apparent ex post in the data when there has been a bubble, the trajectories

implied by unit root and mildly explosive processes di§er in important ways. Although a unit

root process can generate successive upward movements, these movements still have a random

wandering quality unlike those of a stochastically explosive process where there is a distinct

nonlinearity in movement and little bias in the estimation of the autoregressive coe¢ cient.

Forward recursive regressions are sensitive to the changes implied by this nonlinearity. When

data from the explosive (bubble) period are included in estimating the autoregressive coe¢ -

cient, these observations quickly ináuence the estimate and its asymptotic behavior due to the

19
dominating e§ect of the signal from mildly explosive data. This di§erence in signal between

the two periods provides identifying information and explains why the two test procedures

consistently estimate the origination date. When the bubble bursts and the system switches

back to unit root behavior, the signal from the explosive period continues to dominate that of

unit root period. This domination, which at this point is e§ectively a domination by initial

conditions, is analogous to the domination by distant initializations that can occur in unit root

limit theory, as shown recently by Phillips and Magdalinos (2009). More than this, the crash

and re-initialization give the appearance in the data of a form of mean reversion to an earlier

state, so that the estimated autoregressive coe¢ cient is smaller than unity and the classical

unit root test statistics diverge to minus inÖnity, as shown in (16) and (17) above.

2.5 Initialization

To improve the power of the PWY procedure, we modify the methods by selecting the initial

condition based on the Schwarz (1978) BIC criterion. In PWY, the initial observation in each

recursive regression was Öxed to the Örst observation of the full sample. While this choice

is convenient, when time series mix a non-explosive regime with an explosive regime, a more

powerful test is obtained if the recursive statistics are calculated using sample data from a

single regime for bubble detection. This observation motivates us to use the data to choose

the initialization. The method follows an approach to endogenize initialization in time series

regression that was suggested in Phillips (1996).

Suppose an origination date ^e has been identiÖed by the procedure of PWY2. Let nmin

be the number of observations in a base sample of the observations fX enmin+1; : : : ; X e g:

The base sample may be constructed by taking some percentage of the sample before ^e: In

our applications below we use 10%: For the base sample, we compare the BIC value of two

competing models ñ a unit root model and an autoregressive model. If the BIC value of the

unit root model is smaller and the point estimate of  is larger than 1, we reset the initial

condition to  e  nmin + 1. Otherwise, we expand the base sample to fX enmin; : : : ; X e g,

that is another observation is added to the beginning of the sample. Based on the new sample,

we again compare the BIC value of the competing models. If the BIC value of the unit root

model is smaller and the point estimate of  is larger than 1, we reset the initial condition

to  e  nmin. This exercise is repeated until the BIC value of the unit root model is smaller.

2

In case no bubble is found, no change in the PWY procedure is required. However, a áexible moving

window recursive approach is also possible, which allows for variable initializations, and may be more e§ective

in assessing evidence for multiple bubbles.

20
If the sample eventually becomes fX1; : : : ; X e g and the BIC value of the unit root model is

still larger, we set the initial condition to t = 1; which is the same as that used in PWY. If

the initialization emerging from this procedure is ^0; then the recursive testing methodology of

PYW is applied from ^0. With this initialization, denote the estimate of the origination date

^e(^ 0) and the estimate of the collapse date ^f (^ 0). Obviously, ^e(1) = ^ e and ^f (1) = ^ f .

However, if ^0 > 1, it is possible that ^e(^ 0) 6= ^ e and ^f (^ 0) 6= ^ f . In general it is expected

that ^e(^ 0)  ^e since the backward recursion to locate the initialization ^0 begins from ^e:

Assume the sample is X enminnk+1; : : : ; X e. The BIC value of the unit root model is

ln P e

t= enminnk



Xt  X

2

nk + nmin !

+

ln(nk + nmin)

nk + nmin

;

whereas the BIC value of the autogression is

ln

0

B@

P e

t= enminnk



Xt  b  bXt1

2

nk + nmin

1

CA +

2 ln(nk + nmin)

nk + nmin

;

where X =

1

nmin+nk

P e

t= enminnk+1 Xt

, b and b are the OLS estimators of  and  from the

autoregressive model

Xt =  + Xt1 + "t:

It is known that when the criterion is applied in this way BIC can consistently (i.e. almost

surely as n ! 1) distinguish a unit root model from a stationary model without specifying

transient dynamics (see Phillips, 2008). Using similar methods it can be shown that BIC

consistently distinguishes a unit root model from a model with an explosive root. In essence,

the use of BIC to select the initialization is equivalent to the use of BIC to choose a break

point, although in the present case it is not necessary to speciÖcy transient behavior.

3 Data

Two datasets are studied in the empirical work reported here. The primary data constitute

seven Önancial time series: the monthly Nasdaq composite price index (without dividends)

over January 1990 to January 2009; the monthly U.S. house price index over January 1987 to

January 2009; the monthly outstanding asset backed commercial paper (ABCP) in the U.S.

over January 2001 to January 2009; monthly crude oil prices (in US dollars) over January

1999 to January 2009; monthly platinum prices (in US dollars) over January 1999 to January

21
2009; the daily Baa bond rates from January 3, 2006 to January 30, 2009; and the weekly

Pound/USD exchange rates from March 17, 2006 to March 20, 2009.

A secondary dataset is studied to check whether the empirical bubble characteristics found

in the primary series apply to other commodities and exchange rates. The secondary data

include some commodity prices such as monthly heating oil, co§ee, cotton, cocoa, sugar, feeder

cattle prices, all measured in USD and over January 1999 to January 2009, and some exchange

rates, such as the weekly Euro/USD exchange rates, the Yen/USD exchange rates and the

Cnd/USD exchange rates, all observed over March 17, 2006 to March 20, 2009.

The choice of the sampling periods is judiciously guided by CFG (2008a) because we aim

to match the empirical analysis with the predictions made in CFG. The CFG story begins with

the internet bubble in the Nasdaq in the 1990s ñsee page 7 in CFG ñand ends with the collapse

of all Önancial bubbles when the economy goes seriously into recession. For the Baa bond rates,

it is well known that a relevant event that signaled the e§ects of the credit crunch is the failure

of Lehman Brothers on September 15, 2008. The sampling period is chosen so that we have

enough observations before September 15 for the bubble test to have good power. Similar

auguments apply to the choice of the sampling period for the exchange rates. However, other

sampling intervals, all covering the subprime crisis period, have been used and the empirical

Öndings reported here are reasonably robust to the choice of the sample period.

The Nasdaq composite price index is obtained from Önance.yahoo.com. It extends the

sample used in PWY by including more recent observations from June 2005 to January 2009.

PWY found strong evidence of a bubble in the Nasdaq during the 1990s, associated with the

dom.com episode. We extend the sample period in order to check whether there are any subsequent bubbles prior to the subprime crisis. The house price index is the seasonally adjusted

S&P Case Shiller composite-10 index obtained from Robert Shillerís website, and represents

the maximum time span of this data. The outstanding commercial paper data is for asset

backed commercial paper (ABCP) obtained from the Federal Reserve Board. The ABCP time

series is a crude indicator of the size of the mortgage and subprime market. The crude oil price

series is based on WTI - Cushing, Oklahoma spot prices obtained from the Energy Information

Administration website. The platinum price series is obtained from the kitco website. The

Baa bond rates are averages of Baa industrial bond rates and are obtained from the Federal

Reserve Board. This variable measures the credit risk level and is particularly relevant because,

as the crisis unfolded, the sharp drop in the prices and market liquidity of all mortgage-backed

securities led a sharp increase in the price of risk and in spreads. Not surprisingly, mutual

mistrust amongst counterparties surged and bond rates jumped. Finally, the Pound/USD ex22
change rates are obtained from the Federal Reserve Board. For the secondary dataset, all the

commodity prices are downloaded from EconStats (http://www.econstats.com/index.htm) and

all the exchange rates are downloaded from the Federal Reserve Board. All time series, except

for the exchange rates and the Baa bond rates, are deáated using the Consumer Price Index

(CPI), which is obtained from the Department of Labor. Figs. 1-3 plot all of seven series in

the primary dataset. Table 1 reports some summary descriptive statistics for these seven time

series, including sample size, sample frequency, sample minimum, date of the minimum, sample

maximum, date of the maximum, as well as the coe¢ cient statistic (DF

1

) and DF-t statistic

(DFt

1

) based on the entire sample.

Table 1: Summary statistics

Data Sample Freq Min Date Max Date DF

1 DFt1

Size (min) (max)

Nasdaq 432 M 1.1002 Sep 1974 27.66 Jan 2000 -4.6476 -1.5614

House 265 M .4924 Oct 1996 1.1340 Feb 2006 -0.3124 -0.5272

ABCP 97 M 3314.8 Sep 2008 5817.1 July 2007 -2.3631 -1.1098

Oil 121 M .0730 Feb 1999 .6118 June 2008 -3.6298 -1.5713

Platinum 121 M 2.0879 Aug 1999 9.5841 Mar 2008 -3.4287 -1.5088

Baa 772 D 6.08 Dec/21/06 9.54 Oct/31/08 -1.0436 -0.4544

Pound/USD 158 W .4775 Nov/9/07 .7240 Jan/23/09 1.9719 1.0594

The real Nasdaq index reached its maximum of 27.66 in January 2000 growing from a

minimum of 1.1 in September 1974. The house price index troughed in October 1996 and

peaked in February 2006. Interestingly, the minimum value for the ABCP was reached in

September 2008, barely a year after its maximum (July, 2007). This timing suggests a strong

decline in the index over the period August 2007 to September 2008 when the subprime crisis

swept through the mortgage market. The crude oil price and platinum price series follow the

same pattern, having their minima in the early part of the sample and reaching the maxima

in mid-2008. The rate for Baa is lowest (6.81) on December 21, 2006 and highest (9.54)

on October 31, 2008, shortly after the failure of Lehman Brothers on September 15. The

Pound/USD exchange rate series is volatile, moving rapidly from 0.4775 on November 9, 2007

to 0.7240 on Jan 23, 2009. At the 5% level, for only one series (namely the Pound/USD

exchange rate) is the unit root null rejected in favor of an explosive alternative for the full

sample (the 5% asymptotic critical values are, respectively, -0.13 and -0.07 for the two unit

root test statistics DF

1

and DFt

1

).

23
4 Empirical Results

Three phases have been identiÖed in connection with the subprime crisis. According to CFG

(2008a), each phase involves a speciÖc hypothesis that concerns related bubble activity. In the

Örst phase (A), before the subprime crisis publicly erupted, bubbles had emerged and burst in

the stock market, the housing market, and mortgage market. These bubbles all played a role

in global imbalances. In particular, three hypotheses are in order.

Hypothesis A1: A bubble originated and collapsed in the stock market prior to the

emergence of the subprime crisis.

Hypothesis A2: A bubble originated in the housing market following the Nasdaq crash

in late 2000 and burst when the subprime crisis emerged in August 2007.

Hypothesis A3: A bubble originated and collapsed in mortgage market securities, the

collapse coinciding with the public eruption of the subprime crisis in August 2007.

During the second phase (B), the subprime crisis broke and funds áowed selectively to

assets in other markets with lower perceived risk. In consequence, bubbles emerged in certain

commodity and foreign exchange markets and credit risk perceptions shot up, leading to the

following hypotheses:

Hypothesis B1: Bubbles originated in certain commodity price markets and exchange

rates following the eruption of the subprime crisis.

Hypothesis B2: Bubbles originated in the bond market as the subprime crisis unfolded.

In the third phrase (C), as perceptions increased that there would be a potentially serious

impact of the Önancial crisis on real economic activity in the U.S and globally, the Önancial

bubbles in commodity prices and the bond market collapsed. Correspondingly, we have another

hypothesis:

Hypothesis C: Bubbles that had arisen in commodity prices and the bond market collapsed.

We now report and discuss the empirical results. First, we check for statistical evidence

of the presence of bubble(s) in each of the time series based on the recursively calculated

sup statistics max DF

r and max DFtr

. Table 2 reports critical values for these two statistics

obtained by simulation for the two sample sizes, 100 and 500. The critical values for max DFt

r

are nearly identical to those reported in PWY. The critical values for max DF

r are about twice

as large as those for max DFt

r

. This is not surprising as the critical values for the conventional

unit root statistic DF

r are about twice as large as those for DFtr

(see, for example, Fuller,

24
2000).

Table 2: Critical values of max DF

r and max DFtr obtained in simulations

Sample Size Test Statistic 10% 5% 1%

500 max DF

r 2.3525 2.8791 3.9619

500 max DFt

r 1.1800 1.4603 2.0043

100 max DF

r 2.3221 2.9470 4.3412

100 max DFt

r 1.1914 1.5073 2.1899

The Örst two rows in Table 3 report values for the two statistics based on the seven time

series with  0 = 0:1. All cases show overwhelming evidence for the presence of bubbles. The

p-values are all less than 1% for data other than the crude oil price. For the crude oil price, the

p-value is between 1% and 5%. Judging from the magnitude of the two statistics, the bubble

characteristics are strongest in the Nasdaq, House prices and Baa.

Table 3: Testing the Presence of Bubbles and Date Stamping3

Nasdaq Home Price ABCP Oil Platinum Baa Pnd/USD

max DF

r 18.026 5.7668 4.5963 4.2131 5.063 19.156 7.5524

max DFt

r 8.2106 14.625 4.9612 2.3652 2.565 5.1876 2.6286

b e(^ 0) June/95 Sep/00 Aug/05 Mar/08 Jan/08 Oct/8/08 Oct/24/08

b f (^ 0) Nov/00 June/07 July/07 Aug/08 July/08 Dec/4/08 NA

Next we estimate the origination and collapse dates,  e and  f and report these in the last

two rows in Table 3. Table 4 shows the di§erence between b e(^ 0) and b f (^ 0), the estimates

based on the endogenized initialization, and b e(1) and b f (1), the estimates based on the initialization proposed by PWY. The two sets of estimates turn out to be the same for the Nasdaq,

Oil, Plantinum and Baa series. But they di§er by 1-4 periods for Home prices, ABCP and

Pound/USD (shown in bold face in Table 4). When they are di§erent, b e(^ 0) is always smaller,

suggesting that the endogenized initialization indeed gives an earlier warning.

Table 4: Estimates of the Origination and Collapse Dates Based on Endogenized

Initializations4

Nasdaq Home Price ABCP Oil Platinum Baa Pnd/USD

b e(1) June/95 Jan/01 Nov/05 Mar/08 Jan/08 Oct/8/08 Oct/31/08

b e(^ 0) June/95 Sep/00 Aug/05 Mar/08 Jan/08 Oct/8/08 Oct/24/08

b f (1) Nov/00 Aug/07 Aug/07 Aug/08 July/08 Dec/4/08 NA

b f (^ 0) Nov/00 June/07 July/07 Aug/08 July/08 Dec/4/08 NA

3The reported estimated  e and  f are based on DF

t

.

4The reported estimated  e and  f are based on DF

t

.

25
Time series plots of the recursively calculated statistics DF

r and DFtr are shown in Figs.

5-11. Superposed on these plots are the critical value paths, log log2[nr] =2; log log2[nr] =4, the

estimated dates b e(^ 0), b f (^ 0), and the onset date for the subprime crisis. Recall that Figs.

1-3 plot b e(^ 0) and b f (^ 0), together with the time series data. In all of these cases, we clearly

identify an explosive subperiod in the data.

Some general conclusions can be drawn from the estimates and Figs. 1ñ3 and 5-11. First,

the estimated origination and collapse dates seem to cover a subperiod of signiÖcant price

run-up in each of the time series. Second, the estimated origination dates are not the same

as the apparent beginning of these run-up periods. This may be because a unit root process

as well as processes with very mildly explosive roots that are closer to unity than an O



n

1



neighborhood can also generate mild run-ups but the latter are indistinguishable from a unit

root root process. The present tests have substantial discriminatory power for mildly explosive

roots beyond those of O



n

1



neighborhoods and are consistent against such alternatives.

Third, both statistics DF

r and DFtr

lead to the same or very similar dating estimates, except

for house prices, which are discussed below.

Some speciÖc conclusions can be drawn for the individual time series and these are summarized below.

1. For the Nasdaq series, a very signiÖcant bubble is found by both DF

r and DFtr

statistics

over June 1995 to Oct 2000. The estimates b e and b f are identical to those found in

PWY, although a di§erent statistic (the augmented Dickey-Fuller test) was employed

there, the data sources (both for the index and the deáator) di§er, and the sample period

is shorter in PWY. Interestingly, no bubble is found over the more recent period even

though there was some run up in prices before the subprime crisis. Note that the Nasdaq

bubble collapsed several years prior to the subprime crisis. So the results are entirely

consistent with hypothesis A1.

2. For the House Price series, again a very signiÖcant bubble is found by both DF

r and

DFt

r

; but this time during the 2000s. Compared with DFt

r

, the statistic DF

r

is three

months later in identifying the bubble but nine months earlier identifying the collapse of

the bubble. In both cases, our estimates of the bubble origination date in the early 2000s

strongly support the argument in Baker (2002), who claimed that there was a housing

bubble at the time. According to DF

r

, the bubble collapsed in June 2007 just before the

subprime crisis erupted, consistent with hypothesis A2.

3. For the ABCP series, a signiÖcant bubble is found by both DF

r and DFtr over August

26
2005 to July 2007. Note that in this case the origination date b e comes several years

later than that of the House Price series, reáecting the lag in packaging mortgages into

Önancial derivatives and related products. The bubble collapsed in July 2007 when the

subprime crisis became apparent, consistent with hypothesis A3.

4. For Crude Oil prices, neither DF

r nor DFtr

identiÖes a bubble before the subprime crisis

broke out. However, a signiÖcant bubble is found by both DF

r and DFtr over March

2008 to July 2008. The bubble emerged in March 2008 after the subprime crisis broke,

consistent with hypothesis B1. The bubble collapsed in August 2008, consistent with

hypothesis C.

5. For Platinum prices, the recursions of DF

r and DFtr both include two periods (one

at the beginning of the sample and the other in the middle of the sample) where the

statistics exceed the critical values. However, the durations are so short so that the log(n)

separating rule for minimum bubble duration suggests that these should be interpreted

as short-lived run-ups not bubbles. However, a signiÖcant bubble is found by both DF

r

and DFt

r over January 2008 to June 2008. The bubble emerged in January 2008 after

the subprime crisis broke, consistent with hypothesis B1. The bubble collapsed in July

2008, consistent with hypothesis C.

6. For the Baa bond rates, while both DF

r and DFtr

suggest random wandering behavior

for much of the period, both also indicate a short but signiÖcant bubble over the period

from October 18, 2008 to December 3, 2008. This period corresponds with the period of

the rapid acceleration of Önancial distress, soon after the Lehman Brothers bankruptcy.

The bubble emerged in the bond market on October 8, 2008 after the subprime crisis

erupted, consistent with hypotheses B2 and C.

7. Finally, for the Pound/USD exchange rate, a signiÖcant bubble is found by both DF

r

and DFt

r

: The bubble emerged on October 24, 2008 after the subprime crisis, consistent

with hypothesis C. However, the bubble persists until the end of our sample, March 20,

2009 and remains an ongoing characteristic in the data.

In sum, all these tests provide empirical support for hypotheses A - C, showing bubble

characteristics in the data that are consistent with the hypotheses. The empirical estimates on

the timeline of the crisis also broadly support the predictions made in the CFG (2008a) model.

Fig. 12 shows the complete timeline of the bubble process. The timeline shows how bubbles

27
Figure 5: Recursive calculation of the coe¢ cient test and t statistic for the real Nasdaq index

from February 1973 to January 2009, obtained from forward recursive regressions.

migrated from the equity market (in particular the Nasdaq index), Örst to the housing market,

and next to the mortgage market before the subprime crisis. After the subprime crisis, the

bubbles selectively moved to certain goods in commodity markets and certain currencies in the

foreign exchange market.

To assess whether or not bubble characteristics were a generic or speciÖc feature in commodity and foreign exchange markets during the Önancial crisis, we applied the methods more

broadly to many series in a secondary dataset. To preserve space, we present only the summary

empirical results in Table 4 without plotting the recursive test statistics.

Although it is clear from the empirical results obtained earlier that funds moved across

markets during the crisis period for áight-to-quality and áight-to-liquidity reasons, the results

in Table 4 suggest that investors were selective in transferring assets. For example, in the

28
Figure 6: Recursive calculation of the coe¢ cient test and t statistic for the real seasonally

adjusted home price composite 10 index from January 1987 to January 2009, obtained from

forward recursive regressions.

29
Figure 7: Recursive calculation of the coe¢ cient test and t statistic for the real outstanding

values for asset-backed commercial paper from January 2001 to January 2009, obtained from

forward recursive regressions.

30
Figure 8: Recursive calculation of the coe¢ cient test and t statistic for the real crude oil price

from January 1999 to January 2009, obtained from forward recursive regressions.

31
Figure 9: Recursive calculation of the coe¢ cient test and t statistic for the real platinum price

from January 1999 to January 2009, obtained from forward recursive regressions.

32
Figure 10: Recursive calculation of the coe¢ cient test and t statistic for the Baa bond rates

from January 3, 2006 to January 30, 2009, obtained from forward recursive regressions.

33
Figure 11: Recursive calculation of the coe¢ cient test and t statistic for the Pound/USD

exchange rates March 17, 2006 to March 20, 2009, obtained from forward recursive regressions.

34
commodity market, we identify a bubble in heating oil prices, with similar origination and

collapsing dates as those for crude oil prices. However, we Önd no evidence of bubbles in

co§ee, cotton, sugar, and feeder cattle prices. In the foreign exchange market, we locate a

bubble in the Cnd/USD exchange rate, which originated on September 21, 2007 and burst on

November 23, 2007. Interestingly, the origination date is about one year earlier than that

in the Pound/USD exchange rate. However, no bubble is found in the Euro/USD exchange

rate. Although the value of the max DF

r

statistic is marginally higher than the corresponding

critical value for Cocoa and the Yen/USD rate, detailed analysis of the recursive calculations

of the test statistic shows that the run-ups only lasted for a couple of periods and therefore do

not survive the log(n) separating rule for minimum bubble duration.

Table 4: Test Results for the Presence of Bubbles and Date Stamps

max DF

r max DFtr b e b f

Heating oil 6.9092 2.2416 March/08 August/08

Co§ee -1.6035 -0.7002 NA NA

Cotton -0.2466 -0.0866 NA NA

Cocoa 2.4876 0.9872 NA NA

Sugar -0.7408 -0.2220 NA NA

Feeder cattle 1.0336 0.4327 NA NA

Euro/USD 0.4091 0.3311 NA NA

Yen/USD 3.8949 1.4247 NA NA

Cnd/USD 4.0494 2.6956 Sep/21/07 Nov/23/07

5 Conclusions

This paper provides an empirical study of the bubble characteristics in several key Önancial

variables over an historical time period that includes the subprime crisis and its sequel, including

global e§ects. The econometric methods employed are based on recursive regression, rightsided unit root tests and a newly developed dating technology and associated limit theory from

Phillips and Yu (2009). These methods enable us to track the timeline of the crisis in terms of

the individual series by empirically dating the origination and collapse of each of the bubbles.

The dates are matched against the onset date for the subprime crisis as well as a speciÖc

sequential hypothesis concerning bubble migrations that are predicted in the theoretical model

proposed by CFG (2008a). Our estimates suggest that bubbles migrated from the equity market

to the housing market and on to the subprime mortgage derivative market before the crisis

broke. After the crisis erupted into the public arena, the pricing bubbles migrated to selected

commodity markets and, in some cases, the foreign exchange market, suggesting a áight-to35
quality or perceived safe haven phenomena. All these bubbles collapsed as the Önancial crisis

impacted real economic activity. The estimated sequence of the bubble migration phenomenon

is broadly consistent with the predictions of CFG (2008a).

The methods used here may also be used to provide early warning diagnostics for market

exuberance as they provide consistent tests for mildly explosive behavior. Such diagnostics may

assist policy makers in framing early monetary policy responses or other regulatory actions or

interventions to combat speculative bubbles in Önancial markets.

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37
Figure 12: Timeline of Önancial bubbles in the stock, real estate, mortgate, commodity, bond,

and foreign exchange markets. The panels show recursive calculations of the coe¢ cient statistic

and critical values highlighting the successive bubble episodes.

38