Quantitative Economics 2 (2011), 455–491 1759-7331/20110455

Dating the timeline of financial bubbles during the

subprime crisis
Peter C. B. Phillips
Yale University, University of Auckland, University of Southampton, and
Singapore Management University

Jun Yu
Singapore Management University

A new recursive regression methodology is introduced to analyze the bubble char-

acteristics of various financial time series during the subprime crisis. The meth-
ods modify a technique proposed in Phillips, Wu, and Yu (2011) and provide a
technology for identifying bubble behavior with consistent dating of their origi-
nation and collapse. The tests serve as an early warning diagnostic of bubble ac-
tivity and a new procedure is introduced for testing bubble migration across mar-
kets. Three relevant financial series are investigated, including a financial asset
price (a house price index), a commodity price (the crude oil price), and one bond
price (the spread between Baa and Aaa). Statistically significant bubble character-
istics are found in all of these series. The empirical estimates of the origination
and collapse dates suggest a migration mechanism among the financial variables.
A bubble emerged in the real estate market in February 2002. After the subprime
crisis erupted in 2007, the phenomenon migrated selectively into the commodity
market and the bond market, creating bubbles which subsequently burst at the
end of 2008, just as the effects on the real economy and economic growth became
manifest. Our empirical estimates of the origination and collapse dates and tests
of migration across markets match well with the general dateline of the crisis put
forward in the recent study by Caballero, Farhi, and Gourinchas (2008a).
Keywords. Financial bubbles, crashes, date stamping, explosive behavior, migra-
tion, mildly explosive process, subprime crisis, timeline.
JEL classification. C15, G01, G12.

Peter C. B. Phillips: peter.phillips@yale.edu

Jun Yu: yujun@smu.edu.sg
Phillips acknowledges support from the NSF under Grants SES 06-47086 and SES 09-56687. Yu acknowl-
edges support from the Singapore Ministry of Education AcRF Tier 2 fund under Grant T206B4301-RS. We
wish to thank a co-editor, three anonymous referees, seminar participants in many universities, the 2010
International Symposium on Econometric Theory and Applications held at Singapore Management Uni-
versity, the World Congress of the Econometric Society in Shanghai, the second Singapore Conference on
Quantitative Finance at Saw Centre for Quantitative Finance, the Workshop on Econometric and Financial
Studies After Crisis at Academia Sinica, the 2011 Shanghai Econometrics Workshop, and the Frontiers in
Financial Econometrics Workshop for helpful comments.

Copyright © 2011 Peter C. B. Phillips and Jun Yu. Licensed under the Creative Commons Attribution-
NonCommercial License 3.0. Available at http://www.qeconomics.org.
DOI: 10.3982/QE82
456 Phillips and Yu Quantitative Economics 2 (2011)

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 world’s 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 financial crisis (Donald Kohn, Fed-
eral Reserve Board Vice Chairman, March 24, 2010).

1. Introduction
Financial bubbles have been a longstanding topic of interest for economists, involv-
ing 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 mea-
suring their extent, exploring causes and the psychology of investor behavior, and con-
sidering suitable policy responses. While there is general agreement that financial bub-
bles give rise to misallocation of resources and can have serious effects 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 financial turmoil over 2008–2009, triggered by the subprime crisis in the
United States and its subsequent effect on commodity markets, exchange rates, and real
economic activity, has led to renewed interest among economists in financial 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 pow-
erful latent financial 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 con-
sequences so important to the real economy that understanding the phenomenon, ex-
ploring its causes, and mapping its evolution have presented major challenges to the
economics profession. As the quotations that preface this article indicate, a substan-
tial percentage of the world’s accrued wealth was destroyed within 18 months of the
subprime crisis, with manifold effects ranging from the collapse of major financial in-
stitutions to the near bankruptcy of national economies. There is also recognition that
new empirical methods are needed to improve understanding of speculative phenom-
ena and to provide early warning diagnostics of financial bubbles.
The recent background of financial exuberance and collapse with concatenating
effects across markets and nations provides a rich new environment for empirical re-
search. The most urgent ongoing questions relate to matters of fiscal, monetary, and
regulatory policies for securing financial stability and buttressing real economic activ-
ity. Beyond these immediate policy issues are underlying questions relating to the emer-
gence of the phenomenon and its evolutionary course through the financial and eco-
nomic systems. It is these latter issues that form the focus of interest of the present pa-
per.
Quantitative Economics 2 (2011) Timeline of financial bubbles 457

The subprime crisis is not an isolated empirical event. In a recent article, Caballero,
Farhi, and Gourinchas (CFG) (2008a) 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 all closely related. CFG go further and put forward a sequential hypothesis
concerning bubble creation and collapse that accounts for the course of the financial
turmoil in the U.S. economy using a simple general equilibrium model without mone-
tary factors, but with goods that may be partially securitized. Date-stamping the time-
line of the origination and collapse of the various bubbles is a critical element in the
validity of this sequential hypothesis. Empirical evaluation further requires some econo-
metric technology for testing the presence of bubble migration across markets.
The present paper uses econometric methodology to test if and when bubbles
emerged and collapsed in the real estate market, the commodity market, and the bond
market over the period surrounding the subprime crisis. New econometric methods are
introduced for testing bubble migration across markets. Several series are studied. In
particular, we investigate the bubble characteristics in the U.S. house price index from
January 1990 to January 2009, the price of crude oil from January 1999 to January 2009,
and the spread between Baa and Aaa bond rates from January 3, 2006 to July 2, 2009.
Figure 1 shows time series plots of the three series. Our methods enable us to determine
whether a bubble emerged in each series, to date-stamp the origination in that event,
and correspondingly to assess whether the bubble collapsed and the date of that col-
lapse. The empirical date stamps so determined are then matched against the hypothe-
sized 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 (2011; PWY hereafter). In particular, the methods rely on forward recursive
regressions coupled with sequential right-sided unit root tests. The sequential tests as-
sess period by period evidence for unit root behavior against mildly explosive alterna-
tives. Mildly explosive behavior can be modeled by an autoregressive process with a root
(ρ) that exceeds unity but that is still in the general vicinity of unity. Phillips and Mag-
dalinos (PM) (2007a, 2007b) showed that this “mildly explosive” vicinity of unity can be
successfully modeled in terms of deviations of the form ρ − 1 = c/kn > 0, where c is a
positive constant and kn is a sequence that passes to infinity with, but more slowly than,
the sample size n, so that ρ → 1. These processes therefore involve only mild depar-
tures from strict (rational) martingale behavior in markets. They include submartingale
processes of the type that have been used to model rational bubble behavior in finance
(Evans (1991), Campbell, Lo, and McKinley (1997)). PM (2007a, 2007b) have 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.
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 financial exuberance in the Nasdaq market to mid-1995,
prior to the famous remark of Alan Greenspan in December 1996 about irrational ex-
uberance in financial markets. This test therefore revealed that there was anticipatory
empirical evidence supporting mildly explosive behavior in stock prices over a year prior
458 Phillips and Yu Quantitative Economics 2 (2011)

(a) House Prices (b) Oil Price

(c) Bond Price

Figure 1. Time series plots of real prices for three financial assets: (a) monthly observations of
the house price index from January 1990 to January 2009 adjusted by rental; (b) monthly obser-
vations of crude oil prices adjusted by supply; and (c) daily observations of the spread between
Baa bond rates and Aaa bond rates from January 3, 2006 to July 2, 2009. The estimated bubble
origination and collapse dates are also shown on the figures.

to Greenspan’s remarks. In ongoing work, Phillips and Yu (2009) and Phillips, Shi, and
Yu (2011) developed a limit theory for this date stamping technology, explored multiple
bubble detection, and checked the finite sample capability of the 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 finite samples.
We use this methodology to explore the sequential pattern of events of the current
financial crisis. Dating helps to characterize the phenomenon by identifying the individ-
ual events and by fixing their extent and sequencing. It may be viewed as a first step in
understanding the phenomenon and in searching for causes of the behavioral changes
involved in bubble origination and collapse. Date stamping in conjunction with migra-
tion analysis assists in evaluating hypotheses about the concatenation of bubble activity
over time and across markets, such as those developed by CFG. The forward recursive
regression approach used here enables early identification of the appearance of mildly
explosive behavior in asset prices, thereby providing anticipatory evidence of a (local)
Quantitative Economics 2 (2011) Timeline of financial bubbles 459

move away from martingale behavior. This evidence can be used as an early warning di-
agnostic of (financial) exuberance, and thereby can assist policy makers in surveillance
and regulatory actions, as urged by Fed Vice Chairman Donald Kohn in one of the open-
ing quotes 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 all of the time
series studied here, and in all of them that manifest mildly explosive behavior, there is
further evidence of subsequent collapse. Figure 1 shows the origination and collapse
dates for the bubbles identified in the three financial time series mentioned earlier.
For the real estate market, the bubbles emerged prior to the subprime crisis. For the
other series, the bubbles all emerged after the subprime crisis. These findings reveal a
sequence of mildly explosive events, each followed by a financial collapse that corrobo-
rates the sequential hypothesis given by CFG. Consideration of a wider group of related
financial series following the eruption of the subprime crisis indicates that bubbles of
the type found in the series in Figure 1 are not always evident in other commodities. Ac-
cordingly, the empirical evidence supports a selective migration of the bubble activity
through financial markets as the subprime crisis evolved and liquid funds searched for
safe havens.
The present paper differs from PWY in three aspects. The first difference involves the
treatment of initialization. In PWY, the initial condition is fixed to be the first observa-
tion in the full sample, whereas in this paper, the initial observation is selected based
on an information criterion. The use of information criteria in the selection of the initial
observation allows for sharper identification of the bubble origination date. As a result,
when a long series is available, the new method may not necessarily use all the obser-
vations to identify the most recent bubble episode. Second, in this paper, a method for
testing bubble migration is developed, a limit theory for the new procedure is obtained,
and the migration test is implemented in the empirical application. Finally, the empiri-
cal focus of this paper is the subprime crisis and some of the events unfolding over the
period 2002–2009.
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 financial exuberance, outlines some of the relevant facts concerning the sub-
prime crisis, and relates the timeline implications of the theoretical results obtained in
CFG (2008a). Section 3 describes the data that are used in the present empirical study.
Section 4 presents the empirical findings and matches the estimates to the theory of
CFG (2008a). Section 5 concludes. New limit theory for the migration test is developed
in the Appendix.

2. Bubbles, the subprime crisis, econometric dating, and bubble migration

2.1 Bubbles and crashes
In the popular press, the term “financial bubble” refers to a situation where the price of a
financial asset rapidly increases and does so in a speculative manner that is distinct from
what is considered to be the asset’s intrinsic value. The term carries the innuendo that
460 Phillips and Yu Quantitative Economics 2 (2011)

the increase is not justified 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 confirmed in retrospect.
A common definition that makes this usage precise is that bubble conditions arise
when the price of an asset significantly exceeds the fundamental value that is deter-
mined by the discounted expected value of the cash flows that ownership of the asset
can generate. However, discount rates may be variable and, as demonstrated below, the
time profile of the discount rate can have important effects 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 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 behav-
ior. This property contrasts with the efficient market martingale property, which implies
unit root time series dynamic behavior. To explain the difference in terms of the com-
monly used present value model, let Pt be the stock price at time t before the dividend
payout, let Dt be the dividend payoff from the asset at time t, and let r be the discount
rate (r > 0). The standard no arbitrage condition implies that

1
Pt = Et (Pt+1 + Dt+1 ) (1)
1+r
and recursive substitution yields

Pt = Ft + Bt  (2)
∞ −i E
where Ft = i=1 (1 + r) t (Dt+i ) and

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

Hence, the asset price is decomposed into two components: a “fundamental” compo-
nent, Ft , that is determined by expected future dividends, and a supplementary solution
that corresponds to the “bubble” component, Bt .1 In the absence of bubble conditions,
Pt = Ft . Otherwise, Pt = Ft + Bt and price embodies the explosive component Bt , which
satisfies the submartingale property (3). Consequently, under bubble conditions, Pt will
manifest the explosive behavior inherent in Bt . This explosive property is very different
from the random wandering (or unit root) behavior that is present in Ft when Dt is 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 em-
pirical evidence of a drift component. Unit root time series with a drift can generate
1 Extensions of the framework (1)–(3) to log linear approximations such as those in Campbell and Shiller

(1988) and the validity of these approximations are considered in Lee and Phillips (2011).
Quantitative Economics 2 (2011) Timeline of financial bubbles 461

periods of run-up 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 ev-
idence 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 difference between linear and exponential growth combined with the non-
linear curvature in an explosive process are testable properties that distinguish 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)h Bt and the ini-
tialization B0 > 0. But when the bubble conditions collapse and the particular solution
disappears, then Pt = Ft , which corresponds to a sudden collapse in the asset price. If
the dividend process Dt follows a martingale, reflecting market conditions that gener-
ate cash flows, then Ft is similarly a martingale and is cointegrated with Dt . Under such
conditions, the presence of an additional “rational bubble” submartingale component
Bt in Pt can account for an explosive run-up in the asset price Pt .
Importantly, making the discount factor rt either stationary or integrated of order 1
does not change our analysis qualitatively because the implications for the statistical
properties of Ft , Bt , and Pt are the same as with the constant r. For example, if rt is
stationary, (3) becomes

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


Then if (3) is fitted, r = ( Tt=1 (1 + rt ))1/T > 1, implying an explosive process for Bt and
hence Pt , even if Ft itself is not explosive.

2.2 The effects of a time varying discount rate

This paper interprets explosiveness in price as sufficient evidence for bubbles and this
interpretation holds true under a variety of assumptions on the discount rate. As in-
dicated above, certain time profiles for the discount rate can have an important effect
on the characteristics of the fundamental price. The present section illustrates this pos-
sibility 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),2 the fundamental value of
the stock price is

Dt
Ft =  (5)
r − rD

This is the well known Gordon growth model. It is evident that in this case the funda-
mental 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
2 This assumption obviously violates the assumption we adopted earlier, namely, constancy, stationarity,

or integration of order 1.
462 Phillips and Yu Quantitative Economics 2 (2011)

Figure 2. Time path of the discount rate rt+s in (9).

certain time profiles for the discount rate. This simple Gordon model reveals the po-
tential 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 , namely
 ∞
Ft = exp(−srt+s )Et Dt+s ds (6)
0

Suppose dividends have a constant expected growth rate rD such that

Et Dt+s = exp(r D s)Dt  (7)

Then Dt is a martingale when rD = 0. Combining (6) and (7) yield

 ∞
Ft = exp(−s(rt+s − rD ))Dt ds (8)
0

Given some fixed time point tb , constants ca > 0, and λ1 > λ2 > 0, let the time profile of
the discount rate rt+s for t ∈ (0 tb ] be
⎧
⎪ t −t −s λ1
⎨ rD + b ca +  for 0 ≤ s < tb − t,
rt+s = s s (9)
⎪
⎩ rD + ca + λ2  for s ≥ tb − t.
s
λ1
Then the discount rate decreases toward some level rD + tb −t as t + s  tb and jumps to
the level rD +ca + t λ−t
2
immediately thereafter, as shown in Figure 2. Thus, the time profile
b
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 .
We then have
 ∞
Ft /Dt = exp(−s(rt+s − rD )) ds
0
Quantitative Economics 2 (2011) Timeline of financial bubbles 463
 tb −t  ∞
= exp(−ca (tb − t − s) − λ1 ) ds + exp(−ca s − λ2 ) ds
0 tb −t

tb −t
∞
e−ca (tb −t−s) e−ca s
= e−λ1 + e−λ2
ca 0 −ca tb −t

e−λ1   e−λ2
= 1 − e−ca (tb −t) + e−ca (tb −t)
ca ca
e−λ1 (e−λ2 − e−λ1 ) −ca (tb −t)
= + e := σt
ca ca
and the time path of Ft /Dt is explosive over t ∈ (0 tb ]. Over this interval, Ft evolves ac-
cording to the differential equation

dFt = (e−λ2 − e−λ1 )e−ca (tb −t) Dt dt + σt dDt 

Since ca Ft /Dt = e−λ1 + (e−λ2 − e−λ1 )e−ca (tb −t) , we have

(e−λ2 − e−λ1 )e−ca (tb −t)

dFt = ca Ft dt + σt dDt for t ∈ (0 tb ]
e−λ1 + (e−λ2 − e−λ1 )e−ca (tb −t)
For t close to tb , the generating mechanism for Ft is approximately

(e−λ2 − e−λ1 )
dFt = ca Ft dt + σt dDt
e−λ1
+ (e−λ2 − e−λ1 )

= 1 − e−(λ1 −λ2 ) ca Ft dt + σt dDt 

which is an explosive diffusion because


cb = 1 − e−(λ1 −λ2 ) ca > 0

since ca > 0 and e−(λ1 −λ2 ) < 1. The discrete time path of Ft in this neighborhood is there-
fore propagated by an explosive autoregressive process with coefficient ρ = ecb > 1.
The heuristic explanation of this behavior is as follows. As t  tb there is growing an-
ticipation 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 inflation of current valuations and price fundamentals Ft become
explosive as this process continues.
On the other hand, for t > tb , we have
λ2
rt+s = rD + ca + for s > 0
s
and then
 ∞
Ft
= exp(−s(rt+s − rD )) ds
Dt 0
 ∞
= exp(−ca s − λ2 ) ds
0
464 Phillips and Yu Quantitative Economics 2 (2011)

∞
−λ2 e−ca s e−λ2
=e = 
−ca 0 ca
−λ
So Ft = e ca 2 Dt for t > tb and price fundamentals are collinear with Dt . When Dt is 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 profile (9) for the discount rate rt induces 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 upward shift in the discount factor that makes present
valuations more important. A more realistic model might allow for uncertainty in this
time profile and for a stochastic trajectory for rt that accommodates potential upward
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 vari-
ation in the discount factor, just as for other potential sources of financial exuberance.

2.3 Subprime crisis and event timeline

The subprime mortgage crisis is generally regarded as an important triggering element
in the ongoing global financial crisis. The subprime event began with a dramatic rise in
mortgage delinquencies and foreclosures that started in late 2006 in the United States,
as easy initial adjustment rate mortgage terms began to expire and refinancing became
more difficult at the same time that house prices were falling. The event had wider
and, soon, global consequences because of the huge scale of mortgage backed securi-
ties (MBS) in the financial 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 officials organized a
$5 billion bailout 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 financial products, high levels of financial leverage with associated vulnera-
bilities, shortfalls in understanding the impact of large common shocks on the finan-
cial system, and inadequate monitoring by policy makers and regulators of the accumu-
lating risk exposure in the financial markets. We refer readers to Brunnermeier (2009),
Greenlaw, Hatzius, Kashyap, and Shin (2008), and Hull (2008) for detailed discussions of
the subprime crisis and its manifold implications. The concern of the present work is
the crisis timeline and, more specifically, the issues of empirically dating the origination
and collapse of the various financial 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 United States sustained rapid increases
in valuations fueled by a period of low interest rates, large foreign capital inflows, and
Quantitative Economics 2 (2011) Timeline of financial bubbles 465

high-risk lending practices of financial institutions. In the resulting boom, home own-
ership in the United States increased to 69.2% in 2004 from 64% in 1994 (Callis and Ca-
vanaugh (2007)) and nominal house prices increased by more than 180% over the period
1997–2006. 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 financial institutions around the world invested in securities that were ultimately
founded on the U.S. housing market. For example, the nominal outstanding amount of
asset backed commercial paper (ABCP) increased by more than 80% over the period July
2004 to July 2007.
The concatenation of events that occurred after the housing market peaked in 2005
and went into decline, followed by the subprime mortgage crisis and subsequent reper-
cussions on financial institutions over 2007–2008 and finally the impact on world trade
and real economic activity, is now well known. Securities backed by subprime mort-
gages lost most of their value, investors lost confidence, and liquidity dried up as money
flowed 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 flight to quality. In consequence, commodity prices soared. As the crisis deep-
ened, stock markets around the world fell, and commercial banks, mortgage lenders,
and insurance companies failed. Consumption and investment expenditures dropped,
many Organization for Economic Cooperation and Development (OECD) economies
went into serious recession, export driven economies in Asia sustained double digit per-
centage declines in exports, growth slowed significantly in China, and world trade de-
clined. Concomitant with these real economic effects, 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 financial crises. The CFG model links
together global financial 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); it 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 cre-
ations and collapses, which we briefly review as follows.
Country U is interpreted as the United States and country M as the emerging mar-
ket economies and commodity producers. Good X is a nonstorable good, a fraction of
which can be capitalized, and is produced by both countries. Good Z is a storable com-
modity 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 flows from emerging markets to the United States as the United States
runs a growing trade deficit with emergent economies, which in turn rely more heavily
on export driven growth.
To allow country U to have both a large current account deficit and low interest rates,
a fundamental assumption that CFG made is that a bubble developed initially in coun-
try U. In practical terms, this may be viewed as a bubble in the equity, housing, and
466 Phillips and Yu Quantitative Economics 2 (2011)

mortgage markets in the United States, the latter providing financial assets that offer
sufficient rewards to be attractive to the rest of the world. Another fundamental assump-
tion is that the bubble bursts at t = 0, leaving investors (both local and foreign) to look for
alternative stores of value. In the first stage, a flight-to-quality reaction migrates the bub-
ble to “good” assets and so the price of commodities (notably Z) jumps, which results
in a significant wealth transfer from U to M. In the second stage, under the assumption
that the financial asset crisis and wealth transfer precipitates a severe growth slowdown,
the excess demand for the good 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 effects in one market after an-
other. When the real estate bubble crashed and the value of MBS securities fell substan-
tially, liquidity flowed into other markets, creating bubbles in commodities and oil mar-
kets as investors transferred financial assets. The deepening financial crisis then sharply
slowed down economic growth, which in turn destroyed the commodity bubbles. Ob-
viously, this story makes strong predictions concerning the timing of the origination
and the collapse of various bubble phenomena in different markets. To evaluate the evi-
dence 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 definitively identified only in hindsight after a market correction (Economist,
June 18, 2005).

The time path of Pt in the rational bubble model (with bubble component Bt ) is ex-
plosive. Similarly, in the run-up phase of a financial bubble, a pattern of stochastically
explosive or mildly explosive behavior is a characteristic feature. The econometric de-
termination of bubble behavior therefore relies on a test procedure having power to dis-
criminate 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 (PM) (2007a, 2007b) analyzed the properties
of mildly explosive stochastic processes and developed a limit theory for autoregressive
coefficient estimation and inference in that context.
PWY (2011) 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 origination and collapse of the Nasdaq bubble. It was shown that a sup unit
root test against a mildly explosive alternative obtained from forward recursive regres-
sions has the power to detect periodically collapsing bubbles. To improve the power and
sharpen date detection, this paper modifies 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) and Phillips, Shi, and Yu (2011), which es-
tablishes consistency of the dating estimators.
The key idea of PWY is simple to implement and relies on recursively calculated
right-sided unit root tests to assess evidence for mildly explosive behavior in the data.
Quantitative Economics 2 (2011) Timeline of financial bubbles 467

In particular, for time series {Xt }nt=1 , we apply standard unit root tests (such as the co-
efficient 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 differs 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 condi-
tions. It is not necessary to wait for a market correction to identify bubble conditions in
hindsight.
More specifically, by recursive least squares, we estimate the autoregressive specifi-
cation

Xt = μ + δXt−1 + εt  εt ∼ iid(0 σ 2 ) (10)

allowing for the fact that the independent and identically distributed (iid) assumption
may be relaxed to serially dependent errors with martingale difference primitives mak-
ing the usual (possibly semiparametric) adjustments to the tests that are now standard
practice in left-sided unit root tests. The null hypothesis is H0 : δ = 1 and the right-tailed
alternative hypothesis is H1 : δ > 1, which allows for mildly explosive autoregressions
with δ = 1 + c/kn , where kn → ∞ and kn /n → 0. The latter requirement ensures that the
process Xt is mildly explosive in the sense of PM (2007a). If kn = O(n) and δ = 1 + c/n,
then the alternative is local to unity, and Xt has random wandering behavior and is no
longer mildly explosive. In that event, consistent dating of periods of exuberance (c > 0)
is not possible.
The regression in the first recursion uses τ0 =
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 τ =
nr for r0 ≤ r ≤ 1. Denote the corresponding coefficient test
statistic and the Dickey–Fuller t statistic by DF δr and DF tr , namely
⎛ ⎞1/2

τ
2
⎜ X̃j−1 ⎟
⎜ j=1 ⎟
⎜ ⎟
DF δr := τ(δ̂τ − 1) t
DF r := ⎜ ⎟ (δ̂τ − 1) (11)
⎜ σ̂τ2 ⎟
⎝ ⎠

where δ̂τ is the least squares estimate of δ based on the first τ =

nr observations, σ̂τ2

is the corresponding estimate of σ 2 , and X̃j−1 = Xj−1 − τ−1 τj=1 Xj−1 . Obviously, DF δ1
and DF t1 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, as τ =
nr → ∞ for all r ∈ [r0  1], the limit theory
 r  r
r W̃r (s) dW (s) W̃r (s) dW (s)
DF δr ⇒ 0  r  DF tr ⇒ 0
1/2  (12)
r
W̃r2 (s) 2
W̃ (s)
0 r
0
468 Phillips and Yu Quantitative Economics 2 (2011)
r
where W is standard Brownian motion and W̃r (s) = W (s) − 1r 0 W is demeaned Brown-
ian motion.3
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, em-
pirical bubble characteristics are much more complicated than model (10) and involve
some regime change(s) between unit root (martingale) behavior with δ = 1 and mildly
explosive behavior with δ > 1, and potential reinitialization 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
first constructed by Evans (1991), who argued that conventional unit root tests had little
power to detect 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 stan-
dard unit root test as a discriminating criterion when it is applied to the full sample.
To find evidence for the presence of a bubble in the full sample, PWY (2011) sug-
gested using a sup statistic based on the recursive regression. This involves compar-
ing supr DF tr with the right-tailed critical values from the limit distribution based on
r r
supr∈[r0 1] 0 W̃ dW /( 0 W̃ 2 )1/2 . Similarly, for the coefficient test, one can compare the
sup statistic supr DF δr with the right tailed critical values from the limit distribution
r r
based on supr∈[r0 1] r 0 W̃ dW / 0 W̃ 2 .
Our approach to finding the timeline of the bubble dynamics also makes use of
forward recursive regressions. We date the origination of the bubble by the estimate
τ̂e =
nr̂e , where
 df 
r̂e = inf s : DF δs > cvβ
δ
n
or r̂e = inf s : DF ts > cvβn (13)
s≥r0 s≥r0

df
δ (cv ) is the right-side 100β % critical value of the limit distribution of the DF δ
cvβn βn n r
(DF tr ) statistic based on τs =
ns observations, and βn is the size of the one-sided test.
Conditional on finding some originating date r̂e for (mildly) explosive behavior, we date
the collapse of the bubble by τ̂f =
nr̂f , where
 df 
r̂f = inf s : DF δs < cvβ
δ
n
or r̂f = inf s : DF ts < cvβn  (14)
s≥r̂e +γ ln(n)/n s≥r̂e +γ ln(n)/n

This dating rule for τ̂f requires that the duration of the bubble is nonnegligible — at
least a small infinity as measured by the quantity γ ln n so that episodes of smaller order
than γ ln n are not considered significant in the dating algorithm for τf . The parameter
r r 1 1 r r
3 Note that rW̃r (s) dW (s)/ 0 W̃r (s)2 =d 0 W̃1 (s) dW (s)/ 0 W̃1 (s)2 and 0 W̃r (s) dW (s)/( 0 W̃r (s)2 )1/2 =d
0
1 1 2 1/2 so that the recursive limit distributions in (12) are all equivalent to those
0 W̃1 (s) dW (s)/( 0 W̃1 (s) )
based on a full sample of size n.
Quantitative Economics 2 (2011) Timeline of financial bubbles 469

γ can be set so that the minimum duration is tuned to the sampling interval. This min-
imal duration requirement helps to reduce the type I error in the unit root test, so that
false detections are controlled, without affecting the consistency property of the estima-
tor.
The consistent estimation of re and rf requires a slow divergence rate of critical
values so that test size tends to zero as n → ∞. For practical implementation, we set
the critical value sequences {cvβδ  cvdf } according to an expansion rule such as cvδ =
n βn βn
df
−044 + ln(
nr)/C and cvβn = −008 + ln(
nr)/C. Both these critical values diverge at a
df δ . For practically reasonable sample sizes, these criti-
slowly varying rate with cvβn < cvβ n
cal values are close to the 5% critical values for DF δ1 and DF t1 if the constant C is chosen
to be large, say 100. For example, when n = 100, cvβ δ = −044 + ln(
nr)/C = −0394 and
n
df
cvβn = −008 + ln(
nr)/C = −0034. The 5% critical values for DF δ1 and DF t1 are −0.44
and −0.08, respectively. Similar critical value expansion rates have been trialed in ex-
tensive simulations in Phillips and Yu (2009) and found to give very satisfactory results
in terms of small size and high discriminatory power. More conversative rules for these
critical values are obtained by choosing smaller values of the constant C, as in the appli-
cation reported later in the paper.
Under the mildly explosive bubble model,

Xt = Xt−1 1{t < τe } + δn Xt−1 1{τe ≤ t ≤ τf }

 t 

∗
+ εk + Xτf 1{t > τf } + εt 1{t ≤ τf } (15)
k=τf +1

c
δn = 1 +  c > 0 α ∈ (0 1)
nα
p p
Phillips and Yu (2009) showed that r̂e → re and r̂f → 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 δn = 1 + ncα . This type of mildly
explosive process over τe ≤ t ≤ τf was originally proposed and analyzed by PM (2007a,
2007b). The above system is more complex because it involves regime switches from unit
root to mildly explosive behavior at τe and from the mildly explosive root back to a unit
root at τf . At τf , the switch also involves a reinitialization of the process and Xt collapses
to Xτ∗f , corresponding to a bubble collapse back to fundamental values prevailing prior
to the emergence of the bubble. We may, for instance, set Xτ∗f = Xτe + X ∗ for some Op (1)
random quantity X ∗ , so that Xτ∗f is within an Op (1) realization of the pre-bubble value
of Xt .
Under this model specification (15), Phillips and Yu (2009) showed that when τ =
[nr] ∈ [τe  τf ),

DF δr = τ(δ̂n (τ) − 1) = n1−α rc + op (1) → +∞

470 Phillips and Yu Quantitative Economics 2 (2011)

and
⎛ ⎞1/2

τ
2
⎜ X̃j−1 ⎟
⎜ j=1 ⎟ c 3/2 r 3/2
⎜ ⎟
DF tr = ⎜ ⎟ (δ̂n (τ) − 1) = n1−α/2 {1 + op (1)} → +∞
⎜ σ̂τ2 ⎟ 21/2 re
1/2
⎝ ⎠

δ goes to infinity at a slower rate than n1−α and that cv (r) df

Hence, provided that cvβ n βn
goes to infinity at a slower rate than n1−α/2 , DF δr and DF tr both consistently estimate re .
Moreover, when τ = [nr] > τf ,

DF δr = τ(δ̂n (τ) − 1) = −n1−α rc → −∞ (16)

and
⎛ ⎞1/2

τ
2
⎜ X̃j−1 ⎟
⎜ j=1 ⎟ c 1/2 r 1/2
⎜ ⎟
DF tr =⎜ ⎟ (δ̂n (τ) − 1) = −n(1+α)/2 {1 + op (1)} → −∞ (17)
⎜ σ̂τ2 ⎟ 21/2
⎝ ⎠

Hence, DF δr and DF tr both consistently estimate rf . Importantly, (16) diverges to nega-

tive infinity, so it is apparent that in the post-bubble period τ > τf the autoregressive
coefficient δ̂n (τ) is biased downward, which in this case means biased toward stationar-
ity. This bias is explained by the fact that the collapse of the bubble is sharp following τf
in model (15) and produces a mean reverting effect in the data, which manifests in the
limit theory as a slight bias toward stationarity in the estimated unit root.
We now provide some heuristic discussion about the capacity of these forward re-
cursive regression tests to capture the timeline of bubble activity. The tests have dis-
criminatory 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 sen-
sitivity 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 co-
efficient 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 differ 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 coefficient. Forward re-
cursive regressions are sensitive to the changes implied by this nonlinearity. When data
from the explosive (bubble) period are included in estimating the autoregressive coeffi-
cient, these observations quickly influence the estimate and its asymptotic behavior due
to the dominating effect of the signal from mildly explosive data. This difference in sig-
nal between the two periods provides identifying information and explains why the two
Quantitative Economics 2 (2011) Timeline of financial bubbles 471

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 con-
tinues to dominate that of the unit root period. This domination, which at this point is
effectively 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 PM (2009).
More than this, the crash and reinitialization give the appearance in the data of a form
of mean reversion to an earlier state, so that the estimated autoregressive coefficient is
smaller than unity and the classical unit root test statistics diverge to minus infinity, 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 ini-
tial condition based on the Schwarz (1978) Bayesian information criterion (BIC). In PWY,
the initial observation in each recursive regression was fixed to the first observation of
the full sample. While this choice is convenient, when time series mix a nonexplosive
regime with an explosive regime, a more powerful test is obtained if the recursive statis-
tics are calculated using sample data from a single regime for bubble detection. This ob-
servation motivates us to use the data to choose the initialization. The method follows
an approach to endogenous initialization in time series regression that was suggested in
Phillips (1996).
Suppose an origination date τ̂e has been identified by the procedure of PWY.4 Let
nmin be the number of observations in a base sample of the observations {Xτe −nmin +1     
Xτe }. The base sample may be constructed by taking some percentage of the sample
before τ̂e . In our applications below, we use 10%. For the base sample, we compare the
BIC value of two competing models: a unit root model and an autoregressive model. If
the BIC value of the unit root model is smaller and the point estimate of δ is larger than 1,
we reset the initial condition to τe − nmin + 1. Otherwise, we expand the base sample to
{Xτe −nmin      Xτe }, so that 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 δ is larger
than 1, we reset the initial condition to τe − nmin . This exercise is repeated until the BIC
value of the unit root model is smaller. If the sample eventually becomes {X1      Xτe }
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 τ̂0 , then the recursive testing methodology of PWY is applied from τ̂0 . With
this initialization, denote the estimate of the origination date τ̂e (τ̂0 ) and the estimate
of the collapse date τ̂f (τ̂0 ). Obviously, τ̂e (1) = τ̂e and τ̂f (1) = τ̂f . However, if τ̂0 > 1, it is
possible that τ̂e (τ̂0 ) = τ̂e and τ̂f (τ̂0 ) = τ̂f . In general, it is expected that τ̂e (τ̂0 ) ≤ τ̂e since
the backward recursion to locate the initialization τ̂0 begins from τ̂e .
4 In case no bubble is found, no change in the PWY procedure is required. However, a flexible moving

window recursive approach is also possible, which allows for variable initializations and may be more ef-
fective in assessing evidence for multiple bubbles. See Phillips, Shi, and Yu (2011).
472 Phillips and Yu Quantitative Economics 2 (2011)

Assume the sample is {Xτe −nmin −nk +1      Xτe }. The BIC value of the unit root model
is
⎛ ⎞

τe
2
⎜ ( Xt − X) ⎟
⎜ t=τ −n −n ⎟ ln(n + n )
⎜ e min k ⎟ k min
ln ⎜ ⎟+ 
⎜ nk + nmin ⎟ nk + nmin
⎝ ⎠

whereas the BIC value of the autoregression is

⎛ ⎞
τe
2
⎜ (Xt − μ̂ − δ̂Xt−1 ) ⎟
⎜ t=τ −n −n ⎟ 2 ln(n + n )
⎜ e min k ⎟ k min
ln ⎜ ⎟+ 
⎜ nk + nmin ⎟ nk + nmin
⎝ ⎠

τe
where X = n 1+n t=τe −nmin −nk +1 Xt , and δ̂ and μ̂ are the ordinary least squares (OLS)
min k
estimators of δ and μ from the autoregressive model

Xt = μ + δXt−1 + εt 

It is known that when the criterion is applied in this way, BIC can consistently (i.e.,
almost surely as n → ∞) distinguish a unit root model from a stationary model with-
out 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 ex-
plosive 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 specify
transient behavior.

2.6 Testing bubble migration

As discussed earlier, the CFG model links together bubbles from different markets in a
migration mechanism. Using the econometric dating algorithm, bubble periods in dif-
ferent markets can be formally dated as we described above. Then, with recursive statis-
tics that measure the existence and intensity of bubble phenomena in different markets,
we may empirically test whether a bubble migrates from one market to another market.
This section outlines a new reduced form procedure for testing such bubble migration
from one series Xt to another series Yt .
To fix ideas, let θX (τ) be the coefficient in an autoregression characterizing the
τ=
nr
time series {Xt }t=1 , which may be recursively estimated by least squares regression
as θ̂X (τ). We explicitly allow for the coefficient θX (τ) to be time dependent so that it
captures any structural changes in the coefficient arising from exuberance and collapse.
The goal is to explain potential migrationary effects of these changes on the behavior of
a second time series Yt .
Quantitative Economics 2 (2011) Timeline of financial bubbles 473

Suppose our dating mechanism identifies a bubble in Xt at τeX =

nreX  and the au-
toregressive estimate θ̂X (τ) peaks at τpX =
nrpX . In a similar way, define θY (τ), θ̂Y (τ),
τ=
nr
reY , and rpY for the time series {Yt }t=1 . It is assumed that rpY > rpX and that this in-
equality is confirmed by the dating algorithm. In practice, we will be working with date
estimators obtained by recursive regression.
Let m = τpY − τpX =
nrpY  −
nrpX  be the number of observations in the interval
(τpX  τpY ]. We consider formulating over the interval (τpX  τpY ] an empirical model in
which the null generating mechanism of Yt involves an autoregressive coefficient θY
that transitions from a unit root to a mildly explosive root at τeY =
nreY  for some reY ∈
(rpX  rpY ), namely

1 τ < τeY =
nreY ,
θY = θY (τ) = cY
1 + α  τ ≥ τeY =
nreY .
n
In this model, θY (τ) has a structural break that produces exuberance at τ = τeY and the
localizing coefficient cY of the mildly explosive root is constant. The alternative hypoth-
esis of interest is that the autoregressive coefficient θY transitions to a mildly explosive
root whose recursive value, θY (τ), depends on the corresponding recursive autoregres-
sive coefficient, θX (τ) for the series Xt . In this event, the value of θY is in part deter-
mined by changes in θX , so that as the bubble in the series Xt collapses and θX (τ) falls,
the bubble migrates to the series Yt and manifests in an increasing coefficient θY (τ) that
exceeds unity. In effect, the bubble collapse in Xt influences and possibly augments ex-
uberance in Yt .
We allow the autoregressive coefficient for Xt to be local to unity upon the collapse
τ−τ
of the bubble in Xt , namely θX (τ) = 1 + cX ( mpX )/n for τ > τpX , which introduces a
nonzero and possibly negative localizing coefficient function so that cX (·) < 0 to deliver
a mean reverting effect during the collapse in Xt over τ ∈ (τpX  τpY ). Note that for τ =

nr, we have
     
τ − τpX τ − τpX n r − rpX −1
cX = cX = cX {1 + O(n )}
m n m rpY − rpX
 
r − rpX
∼ cX 
rpY − rpX

The dependence of θY on the changing behavior of θX (τ) can be captured through the
localizing coefficients as
 
τ − τpX
cY + dcX
m
θY (τ) = 1 + α
 τ =
nr r ≥ reY  (18)
n
so that exuberance in Yt , which is measured by the second term of (18), is influenced
by the evolving pattern of the autoregressive behavior in Xt , represented by the lo-
calizing coefficient function cX (·) of Xt . Thus, as the bubble in Xt collapses and Xt
returns to near martingale behavior, the localizing coefficient cX impacts the autore-
gressive parameter θY that determines the behavior of Yt . The simplest version of (18)
474 Phillips and Yu Quantitative Economics 2 (2011)

involves a linear relation for θX in which cX is constant and negative (cX < 0). Then
τ−τ
θY (τ) = 1 + cnYα + d cnXα ( mpX )2 and this specification is useful in what follows.
The null hypothesis is no migration (d = 0) and the alternative hypothesis is bub-
ble migration (d = 0). When interest focuses, as here, on the possibility of a collapse in
Xt inducing or increasing subsequent exuberance in Yt , the alternative may be signed
(d < 0) so that dcX > 0. Based on (18), we formulate the interactive model
τ − τpX
θY (τ) − 1 = β0n + β1n (θX (τ) − 1) + error
m
(19)
τ =
nrpX  + 1    
nrpY 

for data over the time interval (

nrpY 
nrpY ) of length m =
nrpX  −
nrpY  which
covers the period of collapse of Xt and the emergence of exuberance in Yt . The null and
alternative hypotheses in (19) are

H0 : β1n = 0 H1 : β1n < 0 (20)

The coefficients (β0n  β1n ) in (19) may be fitted by linear regression (using recursive es-
timates (θ̂Y (τ) θ̂X (τ)) of the autoregressive coefficients), giving (β̂0n  β̂1n ). The fitted
coefficient β̂1n is then tested for significance. The limit behavior of β̂1n under the null
can be used to construct a suitable test.
The empirical regression form of (19) involves recursive estimates of the variables
θY (τ) − 1 and θX (τ) − 1. The fitted slope coefficient is then

nrpY 

Z̃Y (τ)Z̃X (τ)
τ=
nrpX 
β̂1n = 

nrpY 

Z̃X (τ)2
τ=
nrpX 

where
τ − τpX
ZY (τ) = θ̂Y (τ) − 1 ZX (τ) = (θ̂X (τ) − 1) 
m
and

nrpY 
1 
Z̃a (τ) = Za (τ) − Za (s) for a = X Y
m
s=
nrpX 

The limit theory for β̂1n can be obtained using methods similar to those in Phillips, Shi,
and Yu (2011). Under H0 as n → ∞, we find that β̂1n = Op (1). An explicit expression for
the limiting form of β̂1n has been obtained and the limit depends on cX and the form
of the generating mechanism for Xt during the collapse period. Under H1 as n → ∞,
we find that β̂1n = Op (n1−α ) and is divergent. Details of these results are provided in the
Appendix.
Quantitative Economics 2 (2011) Timeline of financial bubbles 475

Table 1. Values of L(m).

a = 1/3 a=1 a=3

m = 20 0.43 1.30 3.90

m = 40 0.53 1.60 4.81

Given the limit behavior under the null and alternative, we construct an asymptot-
ically conservative and consistent test of H0 : β1n = 0 against H1 : β1n < 0 based on the
standardized statistic
1 L(m)
Zβ = β̂1n /L(m) where + → 0 as n → ∞ for any ε > 0
L(m) nε
for some slowly varying function L(m), such as a log(m) with a > 0 and where m =

nrpY  −
nrpX  = O(n). The test uses critical values from a standard N(0 1) distribu-
tion and rejects H0 (no bubble migration from X to Y ) in favor of H1 (bubble migration
from X to Y ) if |Zβ | > cvα where cvα is the 100α% critical value of the standard normal.
This test has asymptotically zero size (because β̂1n /L(m) →p 0 under the null) and unit
power (because Zβ = Op (n1−α /L(m)) diverges under the alternative). The test relies on
standard normal distribution critical values cvα and all the usual settings of α (i.e., 10%,
5%, and 1%) will control size to zero asymptotically as n → ∞. Values of Zβ may be re-
ported for a range of values of L(m) such as {a log m : 13 ≤ a ≤ 3}, reflecting the impact
of different normalization factors. Table 1 reports values of L(m) when m = 20 40 and
a = 1/3 1, and 3.
The test has the advantage that it does not require knowledge of the precise model
for Xt following the collapse of the bubble in Xt . In particular, knowledge of the localiz-
ing coefficient cX (or its precise functional time dependence in the more general case of
cX (·)) is not required to mount this test. The test is robust to any model for Xt that has
an autoregressive root that is local to unity, even with a localizing coefficient function
cX (·) that is nonconstant, and further allows for a wide range of potential (conditional
and unconditional) volatility in the error process. Moreover, this test does not require
standard error computations for implementation as it is a coefficient based test.
One weakness of the test is that it is consistent against alternatives in which Yt dis-
plays changing levels of exuberance that might be proxied by the regressor (θ̂X (τ) − 1) ×
τ−τpX
m in the empirical regression (21). This is a form of trend determined spurious re-
gression phenomena that arises because both dependent variable and regressor are
computed recursively and have limits that trend in the recursive argument (r) of the
regression sample size τ =
nr. In this event, of course, rejection of the null signals that
exuberance in Yt is evolving and that the pattern of the evolution is such that it can be
partly explained by the course of the collapse in the time series Xt following τpX . Rejec-
tion of H0 therefore establishes an association but does not confirm a causal link.

3. Data
Two data sets are studied in the empirical work reported here. The primary data consti-
tute three financial time series: the monthly U.S. house price index divided by the rental
476 Phillips and Yu Quantitative Economics 2 (2011)

measure from January 1990 to January 2009; monthly crude oil prices (in U.S. dollars)
normalized by the oil supply that is approximated by the U.S. inventory from January
1999 to January 2009; and the daily spread between the Baa and Aaa bond rates from
January 3, 2006 to July 2, 2009.
A secondary data set is studied to check whether the empirical bubble characteris-
tics found in the primary series apply to other commodities. The secondary data include
some commodity prices such as monthly heating oil, coffee, cotton, cocoa, sugar, and
feeder cattle prices, all measured in U.S. dollars, from January 1999 to January 2009.
The choice of the sampling periods is guided by CFG (2008a) because we aim to
match the empirical analysis with predictions they made. The CFG story begins with the
Internet bubble in the Nasdaq in the 1990s (see p. 7 in CFG (2008a)) and ends with the
collapse of all financial bubbles when the economy goes seriously into recession. For the
Baa bond rates, it is well known that a relevant event that signaled the effects 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 arguments apply to the choice of the sampling period for
the exchange rates. Longer sampling intervals, all covering the subprime crisis period,
have been used and the empirical findings reported here are robust to the choice of
the sample period. This is because earlier observations are discarded by the proposed
procedure of selecting the initial observation.
The house price index is the S&P Case–Shiller Composite-10 index obtained from
Shiller’s website. We measure fundamental values by standardizing the use of rental
data. The quarterly rental data are imputed using the method of Davis, Lehnert, and
Martin (2008) and are linearly interpolated to a monthly frequency.5 The ratio between
the two series is the first time series we use. The crude oil price series is based on WTI–
Cushing, Oklahoma spot prices obtained from the Energy Information Administration
website. We measure fundamental values using a measure of the oil supply based on
the inventory of crude oil in the United States.6 The ratio between the two series is the
second time series we use. The Baa (Aaa) bond rates are averages of Baa (Aaa) industrial
bond rates and are obtained from the Federal Reserve Board. The Baa bond rates mea-
sure the credit risk level and are particularly relevant because, as the crisis unfolded,
the drop in the prices and market liquidity of all mortgage-backed securities led a sharp
increases in the price of risk and in spreads. Unsurprisingly, mistrust among financial
counterparties surged and bond rates jumped. While the same argument may apply to
the Aaa bond rate, the effect is believed to be much less serious for this bond grade. As
a result, the spread between the Baa and Aaa bond rates is a measure of the credit risk
level.
For the secondary data set, all the commodity prices are downloaded from EconStats
at http://www.econstats.com/index.htm and deflated using the Consumer Price Index
5 The rental data can be downloaded from http://www.lincolninst.edu/subcenters/land-values/rent-
price-ratio.asp.
6 The inventory data can be downloaded from the Department of Energy, Monthly Energy Review, U.S.

crude oil ending stocks non-SPR (thousands of barrels).

Quantitative Economics 2 (2011) Timeline of financial bubbles 477

Table 2. Summary statistics.

Sample Date Date

Data Size Freq Min (min) Max (max) DF t1

House 229 M 05405 Sep 1996 11418 Feb 2006 −03024

Oil 121 M 12086 Feb 1999 146188 June 2008 −15152
Baa/Aaa 878 D 075 May/2/06 350 Dec/3/08 −06694

(CPI) obtained from the Department of Labor. Figure 1 plots the three series in the pri-
mary data set. Table 2 reports some summary descriptive statistics for these three time
series, including sample size, sample frequency, sample minimum, date of the mini-
mum, sample maximum, date of the maximum, as well as the DF t statistic (DF t1 ) based
on the entire sample.
The rental-adjusted house price index troughed in September 1996 and peaked
in February 2006. The supply-adjusted crude oil price has its minimum and wanders
around in the early part of the sample, reaching its maximum in mid-2008. The spread
between Baa and Aaa moves within a narrow range (between 0.75 and 1.0) in the first
half of the sample and reaches its highest point (3.5) on December 3, 2008, shortly after
the failure of Lehman Brothers on September 15. At the 5% level, in all cases the unit
root null cannot be rejected in favor of an explosive alternative for the full sample (the
5% asymptotic critical value is −0.08 for the unit root test statistic DF t1 ). That is, the full
sample analysis indicates that no bubble can be found in any of the three time series.

4. Empirical findings
Three phases have been identified in connection with the subprime crisis. According
to CFG (2008a), each phase involves a specific hypothesis that concerns related bubble
activity. In the first phase (A), before the subprime crisis publicly erupted, bubbles had
emerged and burst in the stock market, the housing market, and the mortgage market.
These bubbles all played a role in global imbalances. The following hypothesis is central.

Hypothesis A. A price bubble arises in the housing market. The property price bubble
originated before the subprime crisis surfaced in August 2007 and collapsed as the sub-
prime crisis broke.

During the second phase (B), the subprime crisis erupted and funds flowed selec-
tively to assets in other markets with lower perceived risk or greater opportunity. In con-
sequence, bubbles emerged in certain commodity markets and credit risk perceptions
rapidly elevated, leading to the following hypothesis:

Hypothesis B. Following the public eruption of the subprime crisis, new bubbles
emerged in (i) selected commodity price markets and (ii) bond markets.

In the third phase (C), perceptions increased that the financial crisis and associated
credit crunch might seriously impact real economic activity both in the United States
478 Phillips and Yu Quantitative Economics 2 (2011)

Table 3. Critical values of max DF tr obtained in simulations.

Sample Size Test Statistic 10% 5% 1%

500 max DF tr 1.1800 1.4603 2.0043

100 max DF tr 1.1914 1.5073 2.1899

and worldwide. Recognition of the global recessionary effects of the financial crisis pre-
cipitated a collapse in the commodity price and the bond market bubbles.

Hypothesis C. Bubbles in commodity prices and the bond market collapsed as the
global economic implications of the crisis became apparent.

These hypotheses provide a timeline of market bubble phenomena that can be sub-
jected to empirical evaluation. To do so, we need to identify a bubble in each of the rel-
evant time series within the sample period, fit the dating of these bubbles to the given
timeline, and, if possible, test the migration of the bubbles from the housing market to
commodity prices and the bond market.
We now report and discuss our empirical findings in relation to these hypotheses.
First, we check for statistical evidence of the presence of bubble activity in each time
series using the recursively calculated sup statistic max DF tr .7 Table 3 reports critical val-
ues for the statistic obtained by simulation for the two sample sizes, 100 and 500. The
critical values for max DF tr are essentially identical to those reported in PWY. Note that
these critical values are substantially larger than those of the DF tr statistic.
The first row in Table 4 reports the max DF tr statistic for the three time series using
endogenized initializations. All cases show overwhelming evidence for the presence of
bubbles. The p values are substantially below 1% for each of these time series, suggest-
ing that the bubble characteristics are strong in all three cases, in sharp contrast to the
result from the full sample analysis.
Time series plots of the recursively calculated DF tr statistic are shown in Figures 3–5.
Superposed on these plots are the critical value paths, −008 + ln(n)/5 (where the con-
df
stant C in cvβn = −008 + ln(
nr)/C is chosen as C = 5 to give a conservative test), the
estimated dates τ̂e (τ̂0 ) and τ̂f (τ̂0 ), and the onset date for the subprime crisis. Recall that
Figure 1 plots τ̂e (τ̂0 ) and τ̂f (τ̂0 ), together with the time series data. In all these cases, we
clearly identify an explosive subperiod in the data.

Table 4. Testing the presence of bubbles and date stamping.a

Home Price Oil Baa/Aaa

max DF tr 14.867 3.0849 8.7515

τ̂e (τ̂0 ) February/02 March/08 September/22/08

τ̂f (τ̂0 ) December/07 July/08 April/20/09

a The reported estimates of τ and τ are based on DF t .

e f t

7 Similar results were obtained with the max DF δr statistic and, for brevity, they are not reported.
Quantitative Economics 2 (2011) Timeline of financial bubbles 479

Figure 3. Recursive values of the t statistic for the rental-adjusted home price Composite-10
index from January 1990 to January 2009 obtained from forward recursive regressions.

Some general conclusions can be drawn from the estimates and Figures 1 and 3–5.
First, the estimated origination and collapse dates seem to cover a subperiod of signifi-
cant 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 process. The present tests have substantial discrimi-
natory power for mildly explosive roots beyond those of O(n−1 ) neighborhoods and are
consistent against such alternatives.
The following conclusions can be drawn for the individual time series and Hypothe-
ses A–C.

Figure 4. Recursive values of the t statistic for the supply-adjusted crude oil price from January
1999 to January 2009 obtained from forward recursive regressions.
480 Phillips and Yu Quantitative Economics 2 (2011)

Figure 5. Recursive calculation of the t statistic for the spread between Baa and Aaa bond rates
from January 3, 2006 to July 2, 2009, obtained from forward recursive regressions.

• For the rental-adjusted house price series, a significant bubble is found by the DF tr
statistic during the early part of 2000. Our estimate of the bubble origination date
in May 2002 strongly supports the argument by Baker (2002), who claimed that
there was a housing bubble at that time. In addition, according to DF tr , the bub-
ble collapsed in December 2007, soon after the subprime crisis erupted, which is
consistent with Hypothesis A.

• For the supply-adjusted crude oil price, DF tr does not identify a bubble before the
subprime crisis broke. However, a significant bubble is found by DF tr from March
τ−τ
to July 2008. In the left panel of Figure 6, we plot (θ̂X (τ) − 1) mpX , where θ̂X (τ) is
obtained from the property market, and θ̂Y (τ) − 1, where θ̂Y (τ) is obtained from
the crude oil market. To test bubble migration from house prices to oil prices, we

(a) Migration from Housing to Oil (b) Migration from Housing to Bonds
τ−τ
Figure 6. Bubble migration between markets. Panel (a) plots (θ̂X (τ) − 1) mpX for the housing
market and (θ̂Y (τ) − 1) from the oil market between November 2005 and June 2008. Panel (b)
τ−τ
plots (θ̂X (τ) − 1) mpX for the house market and (θ̂Y (τ) − 1) from the bond market between May
2006 and October 2008.
Quantitative Economics 2 (2011) Timeline of financial bubbles 481

Table 5. Values of Zβ to test migration from the

housing market to the oil market.

a = 1/3 a=1 a=3

m = 32 −20.85 −6.95 −2.33

ran the empirical regression (19) with rp being selected by the t statistic. The point
estimate of β1 is −10.46. Table 5 shows the values of Zβ for L(m) = a log(m) with
a = 1/3 1, and 3. These values are all significant at the 1% level. The results suggest
very strong evidence of bubble migration from the housing market to the oil price
market, consistent with Hypothesis B(i).

• For the time series spread between the Baa and Aaa bond rates, the DF tr statistic
suggests random wandering behavior in the series for much of the period but re-
veals a significant bubble from September 22, 2008 to April 20, 2009. This period
corresponds to the rapid acceleration of financial distress in the weeks following
the Lehman Brothers bankruptcy on September 15, 2008. To test bubble migration
from house prices to bond prices, we ran the empirical regression (19) with rp be-
ing selected by the t statistic. In the right panel of Figure 6, we plot (θ̂X (τ) − 1) nτ ,
where θ̂X (τ) relates to the house market, and θ̂Y (τ) − 1, where θ̂Y (τ) relates to the
bond market. The point estimate of β1 is −12.00. Table 6 shows the values of Zβ
for L(m) = a log(m) with a = 1/3 1, and 3. These values are all significant at the
1% level. The results suggest very strong evidence of bubble migration from the
housing market to the bond market, consistent with Hypothesis B(ii).

• The bubble in the oil price market collapsed in July 2008 and the bubble in the
bond market collapsed on April 20, 2009. Both findings are consistent with Hy-
pothesis C.

In sum, the tests reveal bubble characteristics in the data that are consistent with
Hypotheses A–C. The empirical estimates of the crisis timeline broadly support the pre-
dictions made in the CFG (2008a) model. Figure 7 shows the complete timeline of the
bubble phenomena in a profile using the recursive tests. The timeline shows how bub-
bles migrated from the property market following the subprime crisis to certain goods
in the commodity markets and then to the bond market.
To assess whether these bubble characteristics were generic or specific features of
commodity and financial markets during the subprime crisis and its aftermath, we ap-
plied the methods more broadly to many series in a secondary data set. To preserve

Table 6. Values of Zβ to test migration from the

housing market to the bond market.

a = 1/3 a=1 a=3

m = 30 −24.37 −8.12 −2.71

482 Phillips and Yu Quantitative Economics 2 (2011)

Figure 7. Timeline of financial bubbles in the real estate, commodity, and bond markets. The
panels show recursive calculations of the DF t statistic and critical values highlighting the suc-
cessive bubble episodes.

space, we present only summary empirical results of these findings in Table 7 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 flight-to-quality and flight-to-liquidity rea-
sons, the results in Table 7 suggest that investors were selective in transferring assets.
For example, in the commodity market, we identify a bubble in heating oil prices, with
similar origination and collapsing dates as those for crude oil prices. However, we find
no evidence of bubbles in coffee, cotton, sugar, and feeder cattle prices.

5. Conclusions
This paper provides an empirical study of the bubble characteristics in several key fi-
nancial variables over an historical time period that includes the subprime crisis and its

Table 7. Test results for the presence of bubbles and date stamps.

max DF tr τ̂e τ̂f

Heating oil 22416 March/08 August/08

Coffee −07002 NA NA
Cotton −00866 NA NA
Cocoa 09872 NA NA
Sugar −02220 NA NA
Feeder cattle 04327 NA NA
Quantitative Economics 2 (2011) Timeline of financial bubbles 483

sequel, including global effects. The econometric methods employed are based on re-
cursive regression, right-sided unit root tests, and a newly developed dating technology
and associated limit theory from Phillips and Yu (2009) and Phillips, Shi, and Yu (2011).
The methods are complemented by a mechanism for testing potential migration across
markets. The dating techniques enable us to track the timeline of the crisis in terms of
the individual series by empirically determining 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 specific sequential hypothesis concerning bubble migrations that are predicted
in the theoretical model proposed by CFG (2008a). Our estimates suggest that bubbles
emerged in the housing market before the subprime crisis and collapsed with the sub-
prime crisis. The bubble then migrated from the housing market to selected commodity
markets and the bond market after the crisis erupted into the public arena. All these
bubbles collapsed as the financial crisis impacted real economic activity. The estimated
sequence of the bubble migration phenomenon is broadly consistent with the predic-
tions of CFG (2008a).
The methods used here can be used to provide early warning diagnostics for market
exuberance as they provide consistent tests for mildly explosive behavior. Such diag-
nostics may assist policy makers in framing early monetary policy responses or other
regulatory actions or interventions to combat speculative bubbles in financial markets.

Appendix: Limit theory for the bubble migration test

Consistent with the notion of bubble migration and to fix ideas, it is subsequently as-
sumed that the period of bubble collapse in Xt is relatively short prior to the emer-
gence of exuberance in Yt . We therefore set τpX = τeY + o(n). Then τpX =
nrpX  =

nreY {1 + o(1)} and we define m =

nrpY  −
nrpX .
The empirical regression version of (19) has the form

τ − τpX
θ̂Y (τ) − 1 = β̂0n + β̂1n (θ̂X (τ) − 1) + error (21)
m

over the interval τ ∈ (τpX  τpY ]. Our first interest is in the null behavior of the fitted slope
coefficient

nrpY 

Z̃Y (τ)Z̃X (τ)
τ=
nrpX 
β̂1n = 

nrpY 

Z̃X (τ)2
τ=
nrpX 

where
τ − τpX
ZY (τ) = θ̂Y (τ) − 1 ZX (τ) = (θ̂X (τ) − 1)
m
484 Phillips and Yu Quantitative Economics 2 (2011)

and

nrpY 

Z̃a (τ) = Za (τ) − m−1 Za (s) for a = X Y
s=
nrpX 

Under the null hypothesis θY (τ) = 1 + cnYα , the estimation error θ̂Y (τ) − θY (τ) has the
asymptotic behavior (as shown in Phillips, Shi, and Yu (2011))
 reY
1
τ−τ BY (s) ds
nα θn eY r 0 1
(θ̂Y (τ) − θY (τ)) ⇒ =: C say. (22)
2cY BY (re ) r
r
Here C = 0 eY BY (s) ds/BY (re ) is a Cauchy-like random variable, being a ratio of corre-
lated normal variables, and BY (s) is the Brownian motion (or more general Gaussian
process) corresponding to the limit under weak convergence of the standardized series
n−1/2 Y
ns prior to exuberance in Yt (i.e., for s < reY ). Then

nZY (
nr) = n(θ̂Y (τ) − 1) = n(θY (τ) − 1) + n(θ̂Y (τ) − θY (τ))
(23)
1−α n1−α 2cY
=n cY + τ−τeY C{1 + op (1)} for r > reY 
rθY

It follows that

nrpY 

ZY (k)
k=
nrpX 
nZ̃Y (τ) = nZY (τ) − n

nrpY  −
nrpX 
⎧ τpY
⎫
⎪
⎪  1 ⎪
⎪
⎪
⎪ ⎪
⎪
⎪
⎪ ⎪
⎪
⎨ 1 k=τ θ
k−τeY
(k/n) ⎬
1−α eY Y
= n 2cY C − {1 + op (1)}
⎪
⎪
τ−τ
rθY eY
nrpY  −
nrpX  ⎪⎪
⎪
⎪ ⎪
⎪
⎪
⎪ ⎪
⎪
⎩ ⎭
 
e−cY (s−reY )n
rpY 1−α
1−α 1 1
=n 2cY C − (24)
τ−τeY
rθY (rpY − reY ) reY s
× {1 + op (1)}

1−α 1 1
∼ n 2cY C τ−τeY − 1−α 
rθY n (rpY − reY )reY cY

since as n → ∞,
τpY  rpY −cY (s−reY )n1−α
1  1 e
k−τeY
= ds{1 + o(1)}
n s
k=τeY θY (k/n) reY
Quantitative Economics 2 (2011) Timeline of financial bubbles 485
 rpY −reY e−cY pn
1−α

= dp{1 + o(1)}
0 p + reY
 (25)
1 (rpY −reY )n1−α e−cY q
= dq{1 + o(1)}
n1−α 0 reY + q/n1−α
1
= {1 + o(1)}
n1−α r eY cY

A more general version of this type of asymptotic approximation is developed in (28)

below.
To obtain the corresponding limit theory for θ̂X (τ), we assume that from the bubble
peak τ =
nrpX , the autoregressive coefficient θX (τ) falls rapidly during the collapse
period to the immediate vicinity of unity. We use data from [τpX  τpY ] to compute θ̂X (τ)
recursively for τ =
nr by least squares regression, giving data for the regressor in (21),
namely
τ − τpX
nZX (
nr) = n(θ̂X (τ) − 1)
m
τ − τpX τ − τpX
= n(θ̂X (τ) − θX (τ)) + n(θX (τ) − 1)
m m
 r (26)
r − rpX
W̃X (s)dWX (s)
rpY − rpX rpX r − rpX
∼  r + cX
2 rpY − rpX
W̃X (s)
rpX

r − rpX r − rpX
=: hX (r) + cX =: gX (r)
rpY − rpX rpY − rpX
r
where W̃X (s) = WX (s) − 1r 0 WX (p) dp for s ∈ [0 r] is a Gaussian process (a demeaned
Brownian motion if θX (τ) = 1 and a demeaned diffusion if θX = 1 + cX /n for some con-
τ−τ
stant localizing coefficient cX ). Since the regressor in (21) is ZX (τ) = (θ̂X (τ) − 1) mpX ,
we define the demeaned and scaled variable for τ =
nr over the sample period
(τpX  τpY ] as

nrpY 

ZX (k)
k=
nrpX 
nZ̃X (τ) = nZX (τ) − n

nrpY  −
nrpX 
 rpY
1
∼ gX (r) − gX (s) ds =: g̃X (r)
rpY − rpX rpX

An important aspect of the migration test is that it does not require explicit characteriza-
tion of the limit process W̃X (s) or the demeaned process g̃X (r), thereby accommodating
any value of θX that is local to unity and various forms of unconditional and conditional
heterogeneity in the Xt error process. The null hypothesis is therefore highly composite
in terms of potential generating mechanisms leading to the limit process W̃X .
486 Phillips and Yu Quantitative Economics 2 (2011)

Under the null, we have

nrpY 
nrpY 
 1 
Z̃Y (τ)Z̃X (τ) {nZ̃Y (τ)}{nZ̃X (τ)}
n
τ=
nrpX  τ=
nrpX 
β̂1n = =

nrpY 
nrpY 
 1 
2
Z̃X (τ) {nZ̃X (τ)}2
n
τ=
nrpX  τ=
nrpX 

nrpY 
 
1−α 1 1 1
n 2cY C {nZ̃X (τ)} τ−τe −
n
τ=
nrpX 
θY τ/n n1−α (r pY − reY )reY cY
∼  rpY
g̃X (s)2 ds
rpX
 rpY
g̃X (s)s−1 e−cY n
1−α (s−r
n1−α 2cY C eY ) ds
reY
=  rpY (27)
2
g̃X (s) ds
rpX

n1−α 2cY C g̃X (reY )

∼ rpY
n1−α cY reY
g̃X (s)2 ds
rpX

Cg̃X (reY )
∼2  rpY 
2
reY g̃X (s) ds
rpX

Here we use the fact that

 τ−τe
! cY
∼ e−cY n
τ−τe 1−α (r−r
1/θY =1 1+ α eY )
n

and employ the asymptotic approximation

 b  b−a
f (s)e−cn f (p + a)e−cn
γ (s−a) γp
ds = dp
a 0
 (b−a)nγ  
q
= f a + γ e−cq dq/nγ (28)
0 n
 ∞
f (a) f (a)
∼ e−cq dq = 
nγ 0 nγ c

to the integral in the numerator of (27), of which (25) above is a special case and for
which there are other uses. It follows that

Cg̃X (reY )
β̂1n ⇒ 2  rpY  (29)
reY rpX g̃X (s)2 ds
Quantitative Economics 2 (2011) Timeline of financial bubbles 487

which gives the limit distribution of β̂1n under the null hypothesis. The form of this limit
distribution depends on the components (C g̃X ), which in turn depend on the precise
form of the generating mechanism for Xt , including the localizing coefficient cX , and
the generating mechanism for Yt . Thus, as it stands β̂1n is not well suited for testing
because of the difficulty in constructing critical values for the nonpivotal limit theory.
However, it is possible to construct a conservative test, as explained in the text using
the statistic Zβ = β̂1n /L(m), where L(m) → ∞ is slowly varying at infinity and satisfies
L(m) = o(nε ) for all ε > 0 as n → ∞.
To develop a limit theory under the alternative, we consider the specific form
⎧
⎨ 1 τ < τeY ,
 2
θY (τ) = cY cX τ − τpX (30)
⎩1 + +d α  τ ≥ τeY ,
n n m
cX τ−τpX
where θX (τ) = 1 + n m , possibly with cX < 0, so that for τ ≥ τeY we have
 2
cY cX τ − τpX τ − τpX
θY (τ) − 1 = α + d α = β0n + β1n (θX (τ) − 1)  (31)
n n m m

with β1n = dn1−α = 0 and β0n = cnYα . For Yt to be mildly explosive from τ ≥ τeY  we require
that
   
τ − τpX 2
min cY + dcX >0
τ∈[τe τp ] m

and, in general, we expect dcX > 0 or d < 0 if cX < 0, so that under the alternative hy-
pothesis, the collapse in Xt has a migratory effect that raises exuberance in Yt . The
model (31) allows for explicit evolution in the autoregressive coefficient θY (τ) that de-
pends on the presence of the localizing autoregressive coefficient cX and the trend factor
τ−τpX
m .
The specific alternative (31) involves a linear dependence of θY (τ) on cX but more
general dependencies might be formulated as indicated above in (18). In (18), we func-
τ−τ
tionalize the dependence as cX ( mpX ) to allow for changes in the autoregressive struc-
ture of Xt during the collapse period in which Xt returns to a unit root process, with
an implied impact on the generating mechanism of θY (τ). In such a case, the localizing
τ−τ
coefficient cX = cX ( mpX ) captures changes in the autoregressive process for Xt as the
bubble in Xt collapses, and this evolution is mapped into the generating mechanism for
Yt by means of a localizing coefficient of the form
 
τ − τpX
cX
cY m τ − τpX
θY (τ) = 1 + α + d α

n n m
Development of the limit theory under this more general functional alternative is, of
course, more complicated than under (31) and is not pursued here.
The model (31) does not hold exactly for the empirical regression which uses esti-
mated versions of θY (τ) − 1 and θX (τ) − 1. However, it can be verified that since the
488 Phillips and Yu Quantitative Economics 2 (2011)

formulation given in (31) still produces a mildly explosive process in Yt , we have the
relationship, (just as in the basic case (22) where β1n = 0 and θY (τ) = 1 + cnYα )

θ̂Y (τ) − 1 = θY (τ) − 1 + (θ̂Y (τ) − θY (τ))

 
1
= θY (τ) − 1 + Op τ−τeY 
nα θnmin

with
  2
τ − τpX
min cY + dcX
τ∈[τe τp ] m
θnmin = 1 + 
nα

which involves a negligible estimation error. The estimation error in θ̂X (τ) is, as before,
 
1 τ
θ̂X (τ) − 1 = (θX (τ) − 1) + (θ̂X (τ) − θX (τ)) ∼ (θX (τ) − 1) + hX 
n n

since from (26) we have n(θ̂X (τ) − θX (τ)) ⇒ hX (r) for τ =

nr. So the fitted model can
be written in the form
 
cY n(θX (τ) − 1) τ − τpX 1
θ̂Y (τ) − 1 = + d + Op τ−τeY
nα nα m nα θnmin

cY n(θ̂X (τ) − 1) τ − τpX (θX (τ) − θ̂X (τ))n τ − τpX

= +d +d
nα nα m nα m
 
1
+ Op τ−τeY
α
n θnmin
 
cY n(θ̂X (τ) − 1) τ − τpX d τ τ − τpX
∼ α +d α
− α
h X
n n m n n m
  
τ − τpX 1 τ τ − τpX
= β0n + β1n (θ̂X (τ) − 1) − hX 
m n n m

If d = 0, then β1n = 0 and null behavior is the same as that described above. If d = 0, then
the limit behavior under the alternative is

nrpY 

Z̃Y (τ)Z̃X (τ)
τ=
nrpX 
β̂1n =

nrpY 

Z̃X (τ)2
τ=
nrpX 
Quantitative Economics 2 (2011) Timeline of financial bubbles 489
⎧
nr 
nrpY    ⎫
⎪
⎪  pY
1  τ τ − τpX ⎪
⎪
⎪
⎪ Z̃X (τ)2 Z̃X (τ)hX ⎪
⎪
⎪
⎪ ⎪
⎪
⎨ τ=
nrpX  n
τ=
nrpX 
n m ⎬
= β1n −
⎪
⎪
nrpY 

nrpY 
 ⎪
⎪
⎪
⎪ ⎪
⎪
⎪
⎪ Z̃X (τ) 2
Z̃X (τ)2 ⎪
⎪
⎩ ⎭
τ=
nrpX  τ=
nrpX 
⎧
nrpY    ⎫
⎪
⎪ 1  τ τ − τpX ⎪
⎪
⎪
⎪ (nZ̃X (τ))hX ⎪
⎪
⎪
⎪ ⎪
⎪
⎨ n
τ=
nrpX 
n m ⎬
= β1n 1 −
⎪
⎪
nrpY 
 ⎪
⎪
⎪
⎪ 1 ⎪
⎪
⎪
⎪ (nZ̃X (τ))2 ⎪
⎪
⎩ n ⎭
τ=
nrpX 
⎧   ⎫
⎪
rpY s − rpX
⎪
⎪
⎨ g̃X (s) g̃x (s) − hX (s) ds ⎪
⎪
⎪
⎬
rpX rpY − rpX
∼ dn1−α  rpY 
⎪
⎪ ⎪
⎪
⎪
⎩ g̃X (s)2 ds ⎪
⎭
rpX

s−r s−r
From (26), we have hX (s) rpY −rpXpX + cX rpY −rpXpX = gX (s). Then
    rpY
rpY s − rpX s − rpX
g̃X (s) g̃X (s) − hX (s) ds = cX g̃X (s) ds = 0
rpX rpY − rpX rpX rpY − rpX
 reY s−r
provided that rpX g̃X (s) rpY −rpXpX ds = 0 and cX = 0. It follows that the statistic β̂1n =
Op (n1−α ) and is divergent under the alternative hypothesis H1 : β1n = 0. We deduce that
the test statistic Zβ = β̂1n /L(m) leads to an asymptotically conservative test under the
null when a fixed critical value from the standard normal distribution is used because
Zβ →p 0 and the test is consistent under the alternative H1 : β1n = 0, provided that
cX = 0.
Figures 8 and 9 show densities of Zβ = β̂1n / ln(m) under H0 and H1 for various
sample sizes for a model with localizing coefficients cX = −2 (for the period of col-
lapse for the Xt series), cY = 1, α = 085, and d = −3 for the mildly explosive au-

Figure 8. Densities of Zβ under the null.

490 Phillips and Yu Quantitative Economics 2 (2011)

Figure 9. Densities of Zβ under the alternative.

toregressive parameter settings for Yt (i.e., θY = 1 + cY /nα under the null and θY =
τ−τ
1 + cnY + d cnXα mpX under the alternative in (30)) and date settings reX = 025, rpX = 050,
reY = 0625, and rpY = 0875. The densities are computed from 50,000 replications with
m = 25 50 75 100.

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Submitted May, 2010. Final version accepted July, 2011.