TESTING FOR SPECULATIVE BUBBLES USING SPOT AND FORWARD PRICES

• Author: Efthymios G. Pavlidis,David A. Peel,Ivan Paya
• DOI: http://doi.org/10.1111/iere.12249
• Date: 01 November 2017
• Published date: 01 November 2017

INTERNATIONAL ECONOMIC REVIEW

Vol. 58, No. 4, November 2017

TESTING FOR SPECULATIVE BUBBLES USING SPOT AND FORWARD PRICES∗

BYEFTHYMIOS G. PAVLIDIS,IVAN PAYA,AND DAVID A. PEEL1

Lancaster University Management School, U.K.

The probabilistic structure of periodically collapsing bubbles creates a gap between future spot and forward

(futures) asset prices in small samples. By exploiting this fact, we use a recently developed recursive unit root

test and rolling Fama regressions for detecting bubbles. Both methods do not rely on a particular model of asset

price determination, are robust to explosive fundamentals, and allow date stamping. An application to U.S.

dollar exchange rates provides evidence of bubbles during the interwar German hyperinflation, but not during

the recent floating-rate period. A further application to S&P 500 supports the existence of bubbles in the U.S.

equity market.

1. INTRODUCTION

“A rational bubble reflects a self-confirming belief that an asset’s price depends on a variable

(or a combination of variables) that is intrinsically irrelevant—that is, not part of market

fundamentals” (Diba and Grossman, 1988, p. 520). Since the widespread adoption of rational

expectations into macroeconomics, a large number of studies have attempted to empirically

separate the contribution of rational bubbles and market fundamentals to asset price movements

(e.g., Balke and Wohar, 2009; Phillips et al., 2011).2A variety of econometric tests have been

developed in the literature that can be used for this purpose. The most popular include volatility,

Hausman-type, and (co)integration tests (LeRoy and Porter, 1981; West, 1987, 1988; Diba and

Grossman, 1988; Cochrane, 1992; Taylor and Peel, 1998; Engsted and Nielsen, 2012; Phillips

et al., 2015).

A common feature of all tests for bubbles is that they actually examine a composite hypothesis

of no bubbles and a model for market fundamentals. Because rejection of the null may be due

to the presence of bubbles or model misspecification, or to both of these reasons, results are

deemed to be inconclusive (Hamilton and Whiteman, 1985; Flood et al., 1994; G ¨

urkaynak, 2008).

To make matters worse, even if the specification of the true model for market fundamentals is

known, econometric tests may still fail to detect bubbles if these processes periodically crash

(Blanchard, 1979; Blanchard and Watson, 1983; Evans, 1991; van Norden and Schaller, 1993;

van Norden, 1996). Following Evans (1991), several studies have shown that, in finite samples,

∗Manuscript received May 2014; revised May 2016.

1For comments and suggestions, we are grateful to the editor, two anonymous referees, Karim Abadir, Mehmet

Caner, Tom Engsted, Atshushi Inoue, Bruce Mizrach, Bent Nielsen, Garry Phillips, Peter C. B. Phillips, Phil Rothman,

Michalis Stamatogiannis, Mike Tsionas, and participants of the 5th CSDA International Conference on Computational

and Financial Econometrics, the 19th Annual Symposium of the Society for Nonlinear Dynamics and Econometrics, the

2015 Annual Conference of the Royal Economic Society, the North Carolina State University Econometrics Workshop,

the Workshop on Recent Developments in Econometric Analysis, the University of Liverpool, the Federal Reserve

Bank of Dallas, and the Bank of England. Documentation about the data used in this article is available from the

Lancaster University data archive at https://doi.org/10.17635/lancaster/researchdata/81. Please address correspondence

to: Efthymios G. Pavlidis, Department of Economics, Lancaster University Management School, Lancaster, LA1 4YX,

U.K. Phone: +44(0)1524594849. Fax: +44(0)1524594244. E-mail: e.pavlidis@lancaster.ac.uk.

2In line with most of the empirical literature, we focus on rational extrinsic bubbles. Froot and Obstfeld (1992) and

Driffill and Sola (1998) examine intrinsic bubbles, i.e., bubbles that depend on market fundamentals. There are also

studies that relax the rational expectations assumption and incorporate behavioral aspects into the analysis, such as

overconfidence and differences in priors and beliefs (Abreu and Brunnermeier, 2003; Scheinkman and Xiong, 2003;

Brunnermeier, 2008; Lansing, 2010). For a survey on speculative bubbles, see LeRoy (2004).

1191

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(2017) by the Economics Department of the University of Pennsylvania and the Osaka University Institute of Social

and Economic Research Association

1192 PAVLIDIS,PAYA,AND PEEL

periodically collapsing bubbles can make asset prices look nonexplosive and cointegrated with

market fundamentals, thus leading to a severe power loss of standard statistical tests.

In this article, we propose two novel methods for detecting bubbles that periodically collapse.

The methods do not rely on a particular asset price determination model, impose minimal

assumptions on the process for market fundamentals, and, thus, alleviate the misspecification

problem. The methods have two further appealing features. First, because they are recursive,

they allow the identification of the starting and ending points of bubble episodes. Second,

they only require data on spot and forward (or futures) prices, which, contrary to market

fundamentals, are readily available at high frequencies and over long time periods. This latter

feature is especially important in light of the low power of statistical tests (Evans, 1991; West,

1994).

Although we focus on the foreign exchange market, the methods are of general interest since

they are applicable to any market for which spot and forward (or futures) prices are available.

The underlying idea is simple: When a bubble is occurring, both the forward exchange rate

and the future spot rate incorporate it but with different weights. This is due to the fact that

rational agents, when forming expectations, correctly attach a nonzero probability to the bubble

bursting. As a consequence, the forward rate becomes a downward biased predictor of the future

spot rate, with the difference between the two rates (i.e., the degree of the bias) depending on

the bubble process and, as such, being explosive. As long as the forecast errors for fundamentals

are not explosive, the presence of explosive dynamics in the difference between the two rates

can be attributed to the existence of bubbles.

On this basis, instead of applying (co)integration tests to the spot rate and a set of observed

market fundamentals, which is the common practice in the literature, we advocate the applica-

tion of right-tailed unit root tests to the difference between future spot and forward rates. Of

the various unit root tests proposed in the literature, we employ the Generalized Supremum

Augmented Dickey–Fuller (GSADF) recently developed by Phillips et al. (2015). This choice

is based on Monte Carlo simulation results that indicate that the GSADF exhibits good power

properties in our setting.

The second method we propose is based on the failure of the unbiasedness hypothesis in small

samples and consists of estimating rolling Fama regressions. The fact that the variables that enter

the Fama regression are highly persistent (potentially explosive) implies that conventional tests

may be invalid (Cavanagh et al., 1995; Jansson and Moreira, 2006; Phillips and Magdalinos,

2008; Nielsen, 2010; Engsted and Nielsen, 2012). To circumvent this obstacle, we adopt the

IVX instrumentation procedure of Phillips and Magdalinos (2009), Phillips and Lee (2013), and

Kostakis et al. (2015).

We provide two empirical applications of the proposed bubble detection methods to U.S.

dollar exchange rates. In the first application, we examine one of the most studied events in

monetary history: the interwar German hyperinflation. The existence of speculative bubbles

in hyperinflation Germany has been the subject of a considerable amount of research (see,

e.g., Sargent, 1977; Burmeister and Wall, 1982, 1987; Christiano, 1987; Casella, 1989; Taylor,

1991; Engsted, 1993; Durlauf and Hooker, 1994; Engsted, 1994; Hooker, 2000). However, no

general consensus has been reached in the literature. Previous studies, which focus on prices

and money, find mixed results that depend crucially on the validity of a variety of different

assumptions. For instance, Burmeister and Wall (1982) find that bubbles cannot be rejected

if the money supply process depends on expected inflation. Casella (1989) shows that the

presence of bubbles in prices can be rejected if the money supply process is assumed exogenous

to current inflation but not when there is a feedback rule. Hooker (2000) shows through Monte

Carlo simulations that the small sample properties of tests for bubbles may differ substantially

from their asymptotic counterparts. By using finite-sample critical values, he concludes that

there is evidence of misspecification in the Cagan money demand model but no evidence

of bubbles. Engsted (2003) points out that Hooker’s methodology neglects the magnitude of

misspecification noise in Cagan’s model. Taking the magnitude of noise into account, he also

does not find any evidence of bubbles.

TESTING FOR SPECULATIVE BUBBLES USING SPOT AND FORWARD PRICES 1193

Unlike existing studies on the German hyperinflation, we examine the presence of speculative

bubbles in the foreign exchange market. The lack of research in this area is surprising given the

wide use of the German mark as a vehicle for speculation at that time. Keynes, for instance, notes

that foreign exchange speculation during the German hyperinflation was “on a tremendous scale

and was, in fact, the greatest ever known.”3Our results reveal two bubble episodes, one in July

to August 1922 and another a year later in the summer of 1923, almost a month before forward

trading was suspended by the German authorities. These two bubble periods coincide with

large and persistent speculative movements in the foreign exchange market (Bresciani-Turroni,

1937).

Our second empirical application focuses on the recent floating-rate period. This period is

particularly interesting because, following the collapse of the Bretton Woods system, exchange

rates among major currencies became volatile and highly persistent, moving in one direction

over long time spans (see, e.g., Engel and Hamilton, 1990). A notable example of such move-

ments is the large appreciation of the U.S. dollar against the British pound in the early 1980s. The

magnitude of this appreciation motivated several researchers to investigate whether exchange

rate changes only reflect changes in fundamentals or if speculative bubbles are also present

in the foreign exchange market (Sarno and Taylor, 2002; De Grauwe and Rovira-Kaltwasser,

2012). For instance, Evans (1986) develops a nonparametric bubble-detection test, where the

null hypothesis is that the median of excess foreign exchange returns is zero. By applying the

test to data on the British pound–U.S. dollar rate for the period 1981 to 1984, he rejects the null

hypothesis and concludes that speculative bubbles were present over the sample period exam-

ined. In another study, Meese (1986) employs an econometric methodology based on Hausman’s

specification test. The joint null hypothesis of the test is that there are no bubbles and that the

exchange rate is determined in a hybrid monetary model. By using data for the period 1973

to 1982, Meese (1986) also rejects the null hypothesis. Wu (1995), on the other hand, uses

the Kalman filter to decompose the British pound–U.S. dollar exchange rate into two parts:

a part that depends on monetary fundamentals and another that depends on the unobserved

bubble process. Contrary to the previous studies, Wu (1995) finds that the unobserved bubble

component is not statistically different from zero throughout the period 1974 to 1988. In line

with the findings of the latter study, our empirical analysis of the British pound–U.S. dollar spot

and forward rates provides no evidence in favor of speculative bubbles during the recent float.

To demonstrate the applicability of our methods to markets other than the foreign exchange,

as a final application, we investigate the behavior of the Standard & Poor’s 500 (S&P 500)

index. The long history of the S&P 500 is characterized by several episodes of large price

run-ups as well as severe market crashes, such as the boom and bust of the 1920s, Black

Monday in October 1987, and the millennium boom. Due to its rich history, the index is one

of the most commonly examined series for speculative bubbles. Early studies related to this

topic show that the variance of the actual price index exceeds by a substantial margin the

bound imposed by the variance of the estimated ex post rational price (see, e.g., LeRoy and

Porter, 1981; Shiller, 1981). As noted by Blanchard and Watson (1983) and Tirole (1985),

a potential cause of such excess asset price volatility is the presence of speculative bubbles.

Cochrane (1992), however, using a decomposition of the variance of the S&P 500 price-to-

dividend ratio, provides evidence against this hypothesis. West (1987) also examines the S&P

500 index. Contrary to Cochrane (1992), he rejects the null of no bubbles by employing a novel

Hausman-type misspecification test. Following Diba and Grossman (1988), a large literature

has developed that explores the presence of explosive bubbles in the U.S. equity market by

adopting a (co)integration framework. The results of this literature are mixed. In general,

standard (co)integration tests applied to long spans of data fail to provide evidence in favor

of bubbles (Diba and Grossman, 1988; Taylor and Peel, 1998). On the other hand, recursive

unit root tests and tests that allow for either discrete changes in the integration properties

of the time series under investigation or for Markov switching regimes produce much more

3This quote is taken from Ferguson (1995).