Econometric Tests of Asset Price Bubbles: Taking Stock

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Econometric Tests of Asset Price Bubbles: Taking Stock Refet S. Gurkaynak 2005-04 NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

Econometric Tests of Asset Price Bubbles: Taking Stock ∗ Refet S. Gürkaynak Division of Monetary Affairs Board of Governors of the Federal Reserve System Washington, DC 20551 rgurkaynak@frb.gov January 2005 Abstract Canassetpricebubblesbedetected? Thissurveyofeconometrictests of asset price bubbles shows that, despite recent advances, econometric detection of asset price bubbles cannot be achieved with a satisfactory degreeofcertainty. Foreachpaperthatfindsevidenceofbubbles,thereis anotherone that fits the data equally well without allowing for a bubble. We are still unable to distinguish bubbles from time-varying or regimeswitchingfundamentals,whilemanysmallsampleeconometricsproblems of bubble tests remain unresolved. Theopinionsexpressedarethoseoftheauthoranddonotnecessarilyreflecttheviewsof ∗ theBoard ofGovernorsorothermembersofitsstaff. Ithank Jim Clouse,BillNelson,Brian Sack and Jonathan Wrightforhelpfulsuggestions.

1 Introduction S&P Real Price Price 1800 1600 1400 1200 1000 800 600 400 200 0 1872 1880 1888 1896 1904 1912 1920 1928 1936 1944 1952 1960 1968 1976 1984 1992 2000 Figure 1: S&P Real Price, 1871-2003. Figure 1 shows the real S&P500 stock price index from 1871 to 2003, using annualdata.1 Therunupinequitypricesinthelate1990’sseemsextraordinary, especiallygiventheensuingdecline. Manycasualcommentatorsattributedthis steep rise in stock prices to the presence of a bubble. Can such a claim be substantiated using econometric methods? Alargeandgrowingnumberofpapersproposemethodstodetect“rational” bubbles. Equity prices contain a rational bubble if investors are willing to pay more for the stock than they know is justified by the value of the discounted dividendstreambecausetheyexpecttobeabletosellitatanevenhigherprice in the future, making the current high price an equilibrium price. Importantly, thepricingoftheequityisstillrational,andtherearenoarbitrageopportunities whentherearerationalbubbles. Section2belowdevelopsthebasicassetpricing 1The data is from Shiller(2003). 1

relationandrationalbubblefromautilitymaximizationproblemandpointsout the assumptions embedded in the ‘standard’ model. Section3isthemainbodyofthepaperandsurveystheliteratureontesting for rational bubbles in the context of the present value of dividends model. It begins with the variance bounds tests (section 3.1) of Shiller (1981) and LeRoy and Porter (1981), which were not designed as bubble tests but were later used in that fashion. West’s tests of bubbles (1987, 1988a) are taken up in section 3.2. Section 3.3 focuses on the integration/cointegration based tests (Diba and Grossman, 1988a, b) and Evans’ (1991) criticism of this approach. Tests of collapsing bubbles are also introduced in this section. Section 3.4 discusses intrinsic bubbles, their econometric detection, and related models of regimeswitching fundamentals. The bottom line is that available econometric tests are not that effective because they combine the null hypothesis of no bubbles with an overly simple model of fundamentals. Thus, rejections of the present value model that are interpreted by some as indicating the presence of bubbles can still be explained by alternative structures for the fundamentals. This is not only a theoretical possibility; for almost every paper in the literature that ‘finds’ a bubble, there is another one that relaxes some assumption on the fundamentals and fits the data equally well without resorting to a bubble. All of the papers surveyed in this paper are tests of rational bubbles, as explainedbelow. Amorerecent, alternativestrandofliteratureusesbehavioral models that allow for irrational pricing and associated “irrational bubbles.” These models, and their tests, are not covered in this paper; readers interested inthisstrandofliteraturearereferredtoVissing-Jorgensen(2004)forasurvey. Most of the tests surveyed below reject the standard model of stock pricing. Although they do not reject the null in a way that is consistent only with 2

a bubble, these tests do provide valuable information about the particular dimensions of the standard, present discounted value of dividends model that are inconsistent with the data. The best tests can show whether the data is inconsistent with the presence of a bubble, but there are no tests that would show the data is only consistent with a bubble and not with at least equally plausible alternatives. 2 Asset Prices and Bubbles Consumers’ optimization problem can be used to derive the basic asset pricing relationshipassumingno-arbitrageandrationalexpectations—standardassumptions in economics and finance. For simplicity let expected utility driven from consumption, u(c), be maximized in an endowment economy, Max E ∞ βiu(c ) t t+i { } i=o X s.t. c = y +(P +d )x P x , t+i t+i t+i t+i t+i t+i t+i+1 − where y is the endowment, β is the discount rate of future consumption, x t t is the storable asset, P is the after-dividend price of the asset, and d is the t t payoff received from the asset. In this paper the focus is on stock prices, thus P is a stock price, and d is dividend, however, in different contexts P may be t t t a house price and d rent, or P may be price of a mine and d the value of ore t t t unearthed every period. The optimization problem’s first order condition is E βu(c )[P +d ] =E u(c )P . (1) t 0 t+i t+i t+i t 0 t+i 1 t+i 1 { } { − − } Forassetpricingpurposes,itisoftenimplicitlyorexplicitlyassumedthatutility 3

islinear,whichimpliesconstantmarginalutilityandriskneutrality.Inthiscase, equation (1) simplifies to βE (P +d )=E (P ). t t+i t+i t t+i 1 − Assuming further the existence of a riskless bond available in zero net supply with one period net interest rate, r, no-arbitrage implies 1 E (P )= E (P +d ). (2) t t+i − 1 1+r t t+i t+i Equation(2) isthestartingpointofmostempiricalassetpricingtests. This first-degree difference equation can be iterated forward to reveal the solution ∞ 1 i P = E (d )+B (3) t 1+r t t+i t i=1µ ¶ X such that E (B ) = (1+r)B (4) t t+1 t The asset price has two components, a “market fundamental” part, which is the discounted value of expected future dividends, the first term in the lefthand-side of equation (3), and a “bubble” part, the second term. In this setup, the rational bubble is not a mispricing effect but a basic component of the asset price. Despite the potential presence of a bubble, there are no arbitrage opportunities—equation (4) rules these out. Under the assumption that dividends grow slower than r, the market fundamental part of the asset price converges. The bubble part, in contrast, is non-stationary.2 The price of the asset may exceed its fundamental value as long as agents expect that they can sell the asset at an even higher price in a future date. Notice that the expectation of making high capital gains from the 2This fact is exploited by some of the econometric tests of bubbles that are considered in this survey. 4

saleoftheassetinthefutureisconsistentwithno-arbitragepricingasthevalue of the right to sell the asset is priced in. Importantly, the path of the bubble (and consequently the asset price) is not unique. Equation (4) only restricts the law of motion of the non-fundamental part of the asset price, but it implies a different path for each possible value of the initial level of the bubble. An additional assumption about B is required to determine the asset price. t AspecialcaseofthesolutionthatpinsdowntheassetpriceisB =0,which t impliesthatthevalueofthebubbleiszeroatalltimes. Thisisthefundamental solutionthatformsthebasisofpresentvaluepricingapproachestoequityprices. In theremainder of the paper this solution is alternativelycalled “the standard model,” “the present value model,” and “the market fundamentals model.” It is useful to explicitly spell out the assumptions other than the absence of bubbles that are embedded in this formulation of the present value pricing model: 1. There are no informational asymmetries. Price movements are not amplified (or driven) by uninformed (e.g. momentum) traders who try to extract information from prices. 2. Therepresentativeconsumerisriskneutral. Acorollaryofthisassumption is that there are no risk premia. This, obviously, rules out time-varying riskpremiaduetovariationinthepriceoramountofriskasanexplanation of volatility of stock prices. 3. The discount rate is constant. Note that this is a restriction on r, rather thanonβ,althoughtheyarenotreallydifferentiatedinthismodel. Ifthe discount rate is constant at r, and dividends grow at the constant rate g, r must be greater than g for sum of the discounted dividend stream to be finite. 5

4. Theprocessthatgeneratesdividendsisnotexpectedtochange. Although this is not an assumption about the model per se, it is an assumption commonlymadeintheeconometrictestsofthismodel. Manyeconometric testsneedtogenerateanestimateofexpecteddividendsbasedonhistory. This exercise is meaningful only if the dividend generating process is not expected to change in the future. As stated above, the market fundamentals model is a special case of a more general model that allows for bubbles. The no bubbles special case is justified by a transversality condition in infinite horizon models. The price of the asset todayisthesumofthenetpresentvalueofexpecteddividendsandtheexpected resale value: ∞ 1 i 1 i P = E (d )+ lim P . t 1+r t t+i i 1+r t+i i=1µ ¶ →∞µ ¶ X Thetransversalityconditionassertsthatthesecondtermontherighthandside iszero. Thisisjustifiedbythefollowingargument: Ifthereisapositivebubble, and this term is not zero, the infinitely lived agent could sell the asset and the lost utility, which is the discounted value of the dividend stream, will be lower than the sale value. This cannot be an equilibrium price as all agents will want to sell the asset and the price will fall to the fundamental level. Tirole (1982) argues that bubbles can be ruled out in infinitely lived rational expectations models, but the same author (1985) shows that bubble paths for asset prices are possible in overlapping generations models. Thecurrentliteratureusuallytakesitasgiventhatnon-fundamentalsbased assetpricesarepossible,skippingthetheoreticalexistenceproblem,andtreating bubbles as an empirical issue. The empirical tests usually start from equations (3) and (4), without delving into general equilibrium arguments. 6

3 Econometric Tests of Rational Bubbles 3.1 Variance Bounds Tests Variance bounds tests for equity prices were initiated by Shiller (1981) and LeRoy and Porter (1981). Shiller’s test only generates point estimates of variances so statistical significance cannot be tested, whereas LeRoy and Porter treat equity prices and dividends as a bivariate process, constructing estimates ofvarianceswithstandarderrors.3 HerewefollowShillerforeaseofexposition.4 The null hypothesis is that the ‘market fundamental’ solution to equation (3) forms the basis of asset prices, so that ∞ 1 i P = E (d ). (5) t 1+r t t+i i=1µ ¶ X ThenP ,theex-postrationalprice,canbedefinedasthepresentvalueofactual ∗ (as opposed to expected) dividends: ∞ 1 i P = d . t∗ 1+r t+i i=1µ ¶ X Under rational expectations the difference between actual and expected dividends is an unforecastable, mean zero variable. Denoting this difference by ε , t ∞ 1 i ∞ 1 i P = [E (d )+ε ]=P + ε (6) t∗ 1+r t t+i i t 1+r t+i i=1µ ¶ i=1µ ¶ X X Thevarianceboundstestsrestontheobservationthat,asε isuncorrelated t with all information at time t, including P , the variance of P can be written t t∗ 3LeRoyandPorter’stestisessentiallyaVARbasedtestofthemarketfundamentalprices, and in this sense is close to the work ofCampbelland Shiller(1988,1989). 4Gilles and LeRoy(1991)provideacomprehensivesurvey ofvarianceboundstests. Their discussion does notincludethese tests’applications forbubble detection. 7

as V(P t∗ )=V(P t )+ϕV(ε t )>V(P t ). (7) where ϕ is [1/(1+r)]2/[1 (1/(1+r))2]. Equation (7) places an upper bound − on the variance of the observed price series, under the assumption that prices are formed according to (5). The ex-post rational price should be at least as variable as the observed prices because observed prices are based on expected dividends and do not have the variation introduced by future forecast errors, which the ex-post price includes. If the variance bound is violated in data, this will be evidence that equity prices do not follow equation (5). The implementation of the variance bound test is more complicated than its theory because P is never observed as the values of d out to infinity are t∗ t unrealized. Forempiricalapplicationsitisapproximatedbyassumingaterminal valueofP ,whereT istoday,thelastdatapoint,andconstructingtheP series T∗ ∗ recursively using observed values of dividends. For the terminal price Shiller (1981) uses the sample average of detrended real price. Shiller’s test shows that actual price volatility exceeds the bound imposed by the variance of ex-post rational price by an order of magnitude.5 Although Shiller (1981) and Grossman and Shiller (1981) used this evidence as a critique of the present value model in general, without attributing the high volatility of equity prices to bubbles, other authors, including Tirole (1985) and Blanchard andWatson(1982)havesuggestedthatthevarianceboundmaybeviolateddue the presence of bubbles. Although a violation of the variance bound constructed as above, might be due to the presence of bubbles, these test have problems with implementation that makes them unsuitable for bubble detection. Some of these are broad 5HissampleisrealS&P500pricesanddividendsgoingbackto1871,atanannualfrequancy, observedatthebeginningoftheyear. AlmostallstudiesofstockpricebubblesintheUSuse this data set. 8

problems that are present when variance bounds tests are used to evaluate the present value model, and are not specific to testing for bubbles. Flavin (1983) hasshownthatusingthemeanpriceastheterminalex-postrationalpricebiases the test towards rejection in small samples. Kleidon (1986) argues a subtler point: The variances in question, theoretically, are cross-section variances at a point in time, but in estimation time-series variances are used. He shows that data constructed from the net present value model violates the variance bound when non-stationary time series variances are used. Marsh and Merton (1983) also provide a striking example of variance bounds test failing when dividends and stock prices are non-stationary. These criticisms apply to the use of variance bounds tests to refute the present value model. To get around the Flavin criticism, the test has been modifiedtousethelastobservedpriceastheterminalprice,whichindeedmakes theactualpriceexpectedvalueof theex-postrationalprice. This approachhas a problem that is specific to bubble detection. Authors as early as Mankiw, Romer, and Shapiro (1985), who employed this approach, have noted that in this case variance bounds tests are not well suited for bubble detection, as explained below.6 This terminal value assumption defines the observable counterpart of the ex-post rational price as P˜ = T 1 i − t d + 1 T − t P . (8) t 1+r i 1+r T i=t+1µ ¶ µ ¶ X Under the null hypothesis that there are no bubbles present, this adds some noise to the ex-post rational price, but does not reverse the variance bound inequality. The important point is that, under the assumption that there is a 6Its inapplicability to bubble detection notwithstanding, this paper is a noteworthy attempttousevarianceboundstotestthepresentvaluemodelwithoutbeingsubjecttoearlier criticisms. 9

rational bubble in the data, the variance bound still stands, i.e. this is not a test of bubbles (Flood, Hodrick, and Kaplan, 1994)). To see this, assume that ∞ 1 i P = E (d )+B t 1+r t t+i t i=1µ ¶ X as in equation (3), and B =0. Then, with some algebra, P˜ can be written as t t 6 P˜ = P + T 1 i − t ε t t 1+r i i=t+1µ ¶ X ∞ 1 i − t 1 T − t + [E (d ) E (d )]+ B B . i=T+1µ 1+r ¶ T i − t i "µ 1+r ¶ T − t # X The last three terms on the right-hand-side are forecast errors or forecast updates,andarealluncorrelatedwithP ,thereforecollectivelyaddanonnegative t amount to the variance of P˜. Thus, the variance bound is once again t P˜ t >P t . (9) Rememberingthatinequality(9)wasderivedundertheassumptionofarational bubble, it is clear that if the variance bound is violated in data, this cannot be attributed to the presence of a rational bubble. In general, variance bounds tests are tests of the present value model and rejection (even when there are no econometric problems) may be due to any assumption of the model failing. In a later strand of the variance bounds literature, Campbell and Shiller (1988, 1989) provide a log linear approximation to thedividend/priceratioandestimateaVARsystemallowingfortime-variation in the discount rates. In the absence of a bubble, the dividend/price ratio will be stationary even if dividends and prices have unit roots (more on this in subsection3.3). Theyfindthatevenwhentheconstantdiscountfactorassumption 10

is relaxed, there is still substantial unexplained variance in the divided/price ratio. They do not, however, make an argument about bubbles. Cochrane (1992) explicitly tests for a bubble using the variance of the dividend/price ratio. His test, essentially, asks whether there exists a discount rate process that ‘explains’ the dividend/price volatility. If no discount rate process can generate the observed dividend/price behavior one can conclude that must be a bubble that drives prices. Note that the ‘standard’ model imposes some conditions on the discount rates so that the discounted sum of dividends converges and the discount rate cannot be negative, so finding a process that justifies the dividend/price ratio process is not trivial. Cochrane finds that there exists a time-varying discount rate process that fits the data (without requiring a bubble), and that this process satisfies the model restrictions and is “reasonable” compared to the Hansen-Jagannathan (1991) bound. 3.2 West’s two-step tests It is clear from the discussion of the variance bounds tests that testing for the validity of the standard model and bubbles are related but different endeavors. Fora‘testofbubbles,’abubbleshouldatleastbeinthesetofalternativeswhen the test rejects the standard model. A milestone test of equity price bubbles that explicitly put a bubble in the alternative hypothesis was West’s (1987) test. This cleverly designed test also tries to tackle the “simultaneous test of model specification and bubbles” problem by testing the model and no-bubbles hypotheses sequentially. West’sinsightwastoobservethat,intheabsenceofbubbles,theEulerequation that forms the basis of no-arbitrage asset pricing can be estimated alone, which provides information about the discount rate. Then, if dividends can be representedasanautoregressiveprocess, knowingthediscountrateandthepa- 11

rametersoftheARprocessthatgovernsdividendsprovidesenoughinformation to pin down the relationship between dividends and the market fundamental stock price. The actual relationship between stock prices and dividends can be directly estimated by regressing the stock prices on dividends. Under the null hypothesis that there are no bubbles, the ‘actual’ relationship should not differ from the ‘constructed’ one. The beauty of this method is that if the two estimates of the impact of dividends on equity prices differ it is possible to trace the discrepancy to model misspecification or bubbles. The econometrician can apply specification tests totheEulerequationandtheARrepresentationofdividends, rulingoutmodel misspecificationandleavingbubblesastheonlypossiblereasonforthedifference between the two estimates. Thus, this bubble detection method is conceptually very appealing, but it does have problems in implementation. These are discussed within the context of a simple example that West (1987) works out. The Euler equation derived from the consumer’s optimization problem, under the assumptions discussed in Section 2, implies 1 P = E (P +d Ω ), (10) t 1+r t t+1 t+1 | t µ ¶ which is the same as equation (2) but makes the dependence of the pricing equation on the consumer’s information set, Ω , explicit. Equation (10) can be t cast in a regression form using observable variables: 1 P = (P +d )+u , (11) t 1+r t+1 t+1 t µ ¶ where u is 1 [E (P +d Ω ) P +d ). The correlation of the t 1+r t t+1 t+1 | t − t+1 t+1 ³ ´ error term with the regressors is bad news for OLS but in this context the past history of dividends are natural candidates for instruments, which West uses. 12

The IV estimation of (11) provides an estimate of the discount rate. Notice that this intertemporal relationship between P and P is independent of the t t+1 presence of a bubble. It only asserts that there are no arbitrage opportunities, with or without a bubble. The next step is characterizing the dividend process. Assume for the sake of this example that dividends are exogenous and follow a stationary AR(1) process of the form d =φd +ud. (12) t t 1 t − The autoregressive parameter is easily recovered by an OLS regression. Given this setup, the market fundamental stock price should be Pf = ∞ 1 i E (d Ω )=βd , (13) t 1+r t t+i | t t i=1µ ¶ X φ (1+r) where β = . Ã1 φ ! − (1+r) The actual stock price, on the other hand, may contain a bubble. P is the t sum of the market fundamental price and possibly a bubble component, which the null hypothesis sets to zero. If the null hypothesis is true, estimating the stock price equation P =βd +B (14) t t t without taking into consideration a bubble (regressing P on d ) will provide t t the ‘correct’ estimate of β. If, however, there does exist a bubble in data, and if the bubble is correlated with dividends, the estimate of β in equation (14), β, will be biased. Note that in this set up β will only be biased if the bubble is correlatedwithdividendsandthusthetestwill‘detect’onlythiskindofbubble. b b West’s test exploits being able to estimate β in two ways. If the estimated Euler equation in (11) correctly characterizes intertemporal asset pricing, and 13

an autoregressive dividend process can be estimated, one estimate of the relationship between dividends and market fundamental stock prices is given by β. The second estimate, β, is expected to be the same as this in the absence of bubbles, but will differ from β if bubbles are present in the data. Comparing b these two estimates is the essence of West’s test of speculative bubbles.7 UsingaHausmancoefficientrestrictiontestWest stronglyrejectstheequality of β and β coefficients, indicating the presence of a bubble. There are numerous practical issues that arise in performing this test. The first issue is b nonstationarity; West points out that if data are nonstationary, the test can beappliedtoappropriatelydifferenceddata. Becausedetectingnonstationarity with a reasonable degree of certainty is difficult, he runs his tests in levels and indifferences. ThesecondissueisdeterminingtheorderoftheARprocessthat governsdividends,whichwetooktobe1inequation(12)forsimplicity. Related to this is the issue of information available to agents but not to the econometrician: Investors form their expectations about futures dividends taking into account more information than just the history of the dividend process. West’s test, in its general form, is designed to handle ∞ 1 i P = E (d )+B +εw, t 1+r t t+i |F t t t i=1µ ¶ X ∞ 1 i where εw = [E (d Ω ) E (d )]. t 1+r t t+i | t − t t+i |F t i=1µ ¶ X The information set is a subset of Ω and includes the past history of t t F dividends. In this case εw is uncorrelated with past dividends, but it will be t autocorrelated. West derives coefficient restrictions for this case, where the restrictions are more involved but have the same underlying idea as the AR(1) 7West (1988a) presents a variance bounds version of this test. The underlying idea is similar, but rather than testing parameter restrictions, the variance bounds version tests a restriction on the variances calculated in two differentways. 14

case discussed above. Thenextissueisthechoiceofeconometricmethodtotestmodelspecification and coefficient restrictions. West uses a number of specification tests for the Euler equation and thedividend equation, including structural break tests. His coefficient restriction test, as mentioned above is a Hausman test that leads to a rejection of the equality of coefficients null hypothesis. Dezbakhsh and Demirguc-Kunt(1990)criticizeWest’seconometricmethodologyonthegrounds that his tests have size distortion in small samples (reject the null too often), and are inconsistent under the bubble alternative.8 They propose tests with better small sample properties to check whether β is indeed different from β, and find no evidence of bubbles. b Thequestionabouttheinterpretationofrejectingtheno-bubbleshypothesis is still valid. As West points out, a rejection may be due to the presence of a bubble, but it may also bedueto failure of the model in some otherdimension. Indeed, when he allows for time varying discount rates, he finds no evidence of bubbles under the difference stationarity assumption. Although his approach allowsforseparatetestingofmodelmisspecificationandbubbles,itisdifficultto testforeverycontingencyintermsofmodelmisspecification. Forexample,West tests for a structural break in mid-sample in his model equations and does not findone,howeverifdiscountratesaretimevaryingbutstillmeanreverting,his testwouldnotdetectthis,whichmayexplainwhyhisEulerequationpassedthe specification testbut allowingfortimevaryingdiscount ratesmadeadifference in rejecting the no bubbles hypothesis. Flood, Hodrick, and Kaplan (1994) point out a related issue. The Euler equation in (11) is derived and tested for two consecutive periods but it should hold in its more general form to price long lived assets. The general form is a 8Westalso points outthe inconsistency ofthetestwhen a bubble is present. 15

relationship between any two periods in the future: 1 k k 1 i P = P + d +uk, (15) t 1+r t+k 1+r t+i t µ ¶ i=1µ ¶ X whereuk isonceagainacompositeerrorterm, reflectingthedifferencebetween t expectedandactualoutcomes.9 Themarketfundamentalsprice,equation(13), reliesuponthisrelationshipholdingnotjustforconsecutiveperiods,butforperiodsinfinitelyapart. Flood,Hodrick,andKaplanarguethatalthoughequation (15) holding for consecutive periods exactly implies that it should hold for any two periods, the statistical error in its estimation may be small for consecutive periods(notleadingtoarejection),butaccumulateandbeverylargeforperiods furtherapart. Theytestequation(15)fork equalingoneandtwoandfindthat while they replicate West’s results for k = 1, for k = 2 the specification tests rejectequation(15). Noticethatthisrejectiondoesnotpointtowardsarbitrage opportunities or irrationality; it suggests that the risk-neutral agent-constant discount rate Euler equation is not a good approximation to reality. Flood, Hodrick, and Kaplan also point out that even if the model did not have any problems detectable with specification tests, a rejection of the coefficient restrictions may still be due to factors other than a bubble. Their alternative is one that is also suggested by Hamilton and Whiteman (1985) and Flood and Hodrick (1986): agents might attribute a small probability to an event that will have a large impact on the asset price (the so called peso problem). The standard example of this is a tax law change that agents put a positive probability on, and therefore incorporate into stock prices, but the change does not happen in sample. If there are such large impact events that happen very seldom, these events may not be captured even in samples of one k k k i k i 9uk t =Et " ³ 1+ 1 r ´ P t+k + X i=1³ 1+ 1 r ´ dt+i | Ωt #− ³ 1+ 1 r ´ P t+k + X i=1³ 1+ 1 r ´ dt+i. 16

hundred years of annual data. Expected regime switches, especially those that fail to materialize, pose a major problem for bubble detection because their observed impact on stock prices is similar to bubbles. 3.3 Integration/cointegration based tests The tests so far have imposed very little structure on bubbles. Both variance bounds tests and West’s two-step tests try to detect ‘something other than fundamentals.’ West’s test would ‘find’ a bubble by eliminating all other alternatives by appropriate specification tests. Bubbles, however, have certain theoretical properties that may be exploited for their detection. Diba and Grossman (1987, 1988a) observe that a rational bubble cannot start, thus if it exists now, it must always have existed. The reasoning depends on lack of arbitrage opportunities and impossibility of negative prices. Lack of arbitrage opportunities imply that there are no excess returns from holding an asset with a bubble component, i.e., E (B )=(1+r)B , t t+1 t as in equation (4). In this case, the actual bubble process (assuming it is a stochastic bubble) follows a stochastic difference equation: B (1+r)B = z , (16) t+1 t t+1 − with E t (z t+i ) = 0 i>1. (17) ∀ If B is zero, the bubble will start with the next nonzero realization of z. If t this realization is a negative number, the bubble will be negative and progressively larger in absolute value in expectation, according to its law of motion. This implies that thestock price will be negativeinfinite time, which is impos- 17

sible given free disposal.10 If the expected realization of z cannot be negative when the bubble component is zero, it cannot be positive either, because it has to be zero in expectation to rule out arbitrage opportunities. Thus, when B t is zero, all future realizations of z must be zero with probability one, and the bubble cannot (re)start. Given this argument, Diba and Grossman conclude that, if there is a bubble it must have existed from the first day of trading. They see this as an argument to rule out rational bubbles, and propose a way to empirically test the absence of bubbles. Their test for bubbles (1988b) allows for unobserved fundamentals, and imposes some structure on which deviations from fundamentals in data may be blamed on the presence of bubbles. Diba and Grossman specify the market fundamental price to be Pf = ∞ 1 i E (d +o ), (18) t 1+r t t+i t i=1µ ¶ X o denoting the fundamentals unobservable to the econometrician.11 Under the t assumptionthato isnotmorenonstationarythand (ifdividendsarestationary t t when twice differenced, o is assumed to be stationary when at most twice t differenced, for example), the market fundamentals price will be as stationary asthedividends. Intheabsenceofbubbles,ifdividendsarestationaryinlevels, stockpriceswillbeequaltomarketfundamentalsandshouldalsobestationary in levels; if dividends are stationary in nth differences, stock prices should be stationary in nth differences. Thisrelationshipbreaksdowninthepresenceof bubbles, whichprovidesan intuitive bubbles test. The nth difference of the bubble process, from equation 10Tirole (1982)also notesthatbubbles mustbe positive. 11DibaandGrossmanalsoallowfordifferentvaluationoffuturedividendsandcapitalgains, butthispointis notcentraltothebubble analysis. 18

(16) is (1 L)n[1 (1+r)L]B =(1 L)nz . t t − − − DibaandGrossmannotethatforstandardsimpleprocessesforz (suchaswhite noise) the first difference of the bubble is generated by a nonstationary and noninvertible process. Indeed, the bubble process is nonstationary regardless of how many differences are taken and this is a property that can be tested econometrically. Note that it is the argument that the bubble does not pop and restart that makes this assertion correct in realized values, and not only in expectation. A process that is unit root in expectation but falls to zero and restarts periodically in realization may have different econometric implications, as will be discussed below. A natural way to test for the existence of a bubble in the data, then, is to seewhetherstockpricesarestationarywhentheyaredifferencedthenumberof times required to make dividends stationary. They also observe that although both dividends and stock prices are integrated of order one, equation (18) imposes an equilibrium relationship between these two series. Under the null hypothesis of no bubbles in stock prices, and assuming that o is stationary, t dividends and stock prices should be cointegrated.12 Note that the assumption madeabout theunobserved fundamentals is morestringent this time; they should be stationary in levels although dividends only need to be stationary in differences for the test to work. Using Dickey-Fuller tests, Diba and Grossman find that both dividends and stock prices are integrated in levels, but stationary in differences. Thus, their first test indicates that there are no bubbles. When they test for cointegration using Bharghava (1986) ratios, they also find strong evidence for cointegration of stock prices and dividends. They interpret these findings as indicating that 12Pt − 1 r dt willbe stationary if thereare no bubbles and the assumption about ot holds. 19

a stock price bubble is not present in the data. BeforemovingontoEvans’(1991)criticismofthesetests,itisusefultothink about the interpretation of the results had they indicated that stock prices are morenonstationarythandividends, orthatdividendsandstockpricesareboth I(1)butarenotcointegrated. Oneproblemwithintegration/cointegrationbased tests is the econometric problems of detecting nonstationarity and estimating cointegrating relationships. This is a problem regardless of the outcome of the bubble tests; there are many competing tests with different size/power propertiesandtheseneednotagreeontheresult. Inthecasethetestsdoindicatethe presence of a bubble, the correct interpretation is that they suggest the presenceof‘somethingnonstationary’inthe(appropriatelydifferenced)stockprice. Thiscouldofcoursebebecauseofabubble,butitcanalsobethattheassumption made on the unobserved fundamentals does not hold, and the o series is, t say, integrated of order two while dividends are I(1). It would of course then be an open question whether one can come up with a reasonable unobserved fundamental that would be I(2). Diba and Grossman also allude to this point andarguethatalthougharejectionofthestationarity/cointegrationconditions would not be proof of a bubble, failing to reject is proof of nonexistence of bubbles. Evans (1991) disagrees. Evans points out that although Diba and Grossman’s argument about bubbles only starting on the initial date of trading implies a bubble cannot pop and restart, it is possible that the bubble will collapse to a small nonzero value and then continue increasing, and still follow equation (4). His example of a periodically collapsing bubble is B = (1+r)B v if B α (19) t+1 t t+1 t ≤ B = δ+π 1(1+r)θ [B (1+r) 1δ] v if B >α (20) t+1 − t+1 t − t+1 t { − } 20

where E v =1, and θ takes the value of 1 with probability π and 0 with t t+1 t+1 probability (1 π). This formulation of the bubble satisfies equation (4), the − expected gross return from the bubble is always (1+r). For small values of B the bubble increases slowly, once it is larger than a threshold value, α, it t expands faster but may collapse each period with probability (1 π). In case − of a collapse, the bubble’s value does not shrink to zero; rather, it becomes a smallpositivequantity,δ. InthiscasethebubbleisnotsubjecttotheDibaand Grossman criticism of restarting because it never ‘pops,’ it only gets discretely smaller periodically. This example of bubbles exploits the fact the bubble only has to increase at rate r in expectation, but it may collapse in realization. Evans generates data from a model with bubbles and does Monte Carlo experiments of the Diba and Grossman bubble detection test, using their specification of a bubble (approximated by setting π close to unity). He finds that in this case the test works well, as Diba and Grossman claim. He then uses lowervaluesofπ sothatthebubbleperiodicallycollapses. Inthiscase,evenfor values of π as high as 0.95, the tests perform much worse, failing to reject the no-bubbles hypothesis more often than not. For π smaller than 0.75, the tests almost never detect bubbles. Theunitrootbasedtestshavedifficultydetectingcollapsingbubblesbecause these behave more like stationary processes than like explosive processes as a result of the periodic collapses involved. This, of course, does not bode well for the Diba and Grossman testing strategy. As noted above, rejecting the nobubbles hypothesis with these tests may be due to time variation in someother componentofthepresentvaluemodel, impartingnonstationaritytodifferenced stockprices. FromEvans’study, itappearsthatfailingtorejecttheno-bubbles hypothesiswiththesetestsmaynotbeconclusiveproofthatbubblesareindeed absent from data, either. 21

It isimportanttonotethat Evansdoesnotshowtheexistenceofbubblesin stock prices, he only shows that unit root tests are not adequate to reject this hypothesis.13 However, we do learn from Diba and Grossman’s unit root tests that monotonically increasing bubbles are indeed not in stock prices. We can at least rule out a certain class of bubbles. Evans’ criticism of unit root tests of rational bubbles led to a number of papers trying to overcome the difficulty of detecting collapsing bubbles. The favorite method of attack was to think of expanding and collapsing periods of the bubble as different regimes. This way of modeling the bubble leads to unit root tests where regime shifts in the mean that follow a Markov process are allowed for under the null.14 Hall,Psaradakis,andSola(1999)treateachcomponentoftheEvanscollapsing bubble (equations 19 and 20) as a separate regime with constant switching probabilities. TheirMonteCarloexperimentshowsthatMarkovswitchingADF testsperformwellin detectingbubbleepisodes, buttheydonot haveanempirical application to stock price bubbles. Van Norden and Vigfusson (1998) study the regime switching bubbles tests of Hall and Sola (1993) and van Norden (1996) and conclude that “...even with severalhundredobservations,thetestsshowsometimesconsiderablesizedistortion.”In theirapplication, theHallandSolatest, whichhasconstantswitching 13Charemza and Deadman (1995) conduct a similar study of bubbles that are stochastic explosive root processes. In this case there are no probabilistic collapses but the AR(1) coefficientinthebubbleprocess(thereturnonthebubble)isstochastic. Theyfindthatunit roottestsareunable to detectbubbles in this setup as well. 14An exception is the work of Taylor and Peel (1998). They propose a cointegration test thatisrobusttosknewnessandkutosisintheerrorterm,whichwillbethecaseforacollapsing bubble. In Monte Carlo simulations their test is superiorto Dickey-Fuller test in detecting a periodically collapsing bubble. They do notfind evidence ofa stock pricebubble in the data (1871 to 1987)when they apply theirrobusttest. InunpublishedworkWuandXiao(2002)proposeatestofcollapsingbubblesbasedonthe sizeoftheresidualsofthecointegratingrelationship. Intuitively,evenifperiodicallycollapsing bubbles do not generate unit root residuals, they will still generate large residuals. Wu and Xiaoquantify‘large’andbasetheirtestontheorderofmagnitudeoftheresiduals. Theirtest alsodoes notsuggestthe presenceofbubbles in US stock marketdata. 22

probabilities,suggeststheexistenceofbubblesintheS&P500,butthevanNordentest,whichmodelstheswitchingprobabilitiesasfunctionsofthesizeofthe bubble, does not indicate the presence of a bubble in the same data set. There are many ways to model collapsing bubbles, and Van Norden and Vigfusson’s comparisonoftwooftheseseemstosuggestthattheexactchoiceoftheprocess to be tested does matter. Markov switching tests of collapsing bubbles allow the bubble to switch between two states, but the fundamentals do not change. Driffill and Sola (1998) provide a striking example of switching fundamentals that match the dataequallywellinthecontextofapossibleintrinsicbubble,onceagaindemonstratingthelackofidentificationinbubbletesting. Theirapproachisdescribed at the end of the next section. 3.4 Intrinsic bubbles Bubbles may or may not be correlated with fundamentals. If they are uncorrelated with fundamentals, they must grow exogenously at an expected rate of (1+r) per period to be arbitrage free. In this case the bubble and the fundamentals diverge at an explosive rate. Froot and Obstfeld (1991) suggest a different formulation of bubbles, one in which the bubble is tied to the level of dividends. The stock price, fundamental price, and bubble processes are once 23

again given by15 1 P = E (D +P ), (21) t 1+r t t t+1 Pf = ∞ 1 i E (D ), (22) t 1+r t t+i i=1µ ¶ X 1 B = E (B ). (23) t 1+r t t+1 To tie the bubble to fundamentals, dividends should be explicitly modeled. Froot and Obstfeld assume that log dividends, denoted by d , follow a random t walk with drift: d =µ+d +ξ (24) t t 1 t − where ξ t vN(0,σ2). It is easy to verify that a bubble process of the form B(D )=cDλ, (25) t t whereλisthepositiveroot of λ2σ2/2+λµ ln(1+r)=0andcis anarbitrary − positiveconstant,satisfiesequation(23).16 Thisbubbleprocessdependsentirely on the level of dividends, and does not take off on its own.17 If such a bubble is present, stock prices will be more sensitive to dividend innovations than is justified by the linear pricing equation in (22). Given the law of motion of dividends, and assuming that D is known at the beginning of the period, the t 15Froot and Obstfeld define the stock price inclusive of the dividend and r as the instantaneous interest rate which leads to more elegant algebra. The model is recast into an ex-dividends price and r as the period interest rate format to make it comparable to earlier examples ofbubble tests. 16Dt has an error term that is log-normal, and the expected value of the log normal is a function ofits variance,hence the varianceterms in evaluating expectations. 17Testing for a bubble that is correlated with dividends is also the centerpiece of West’s test. Frootand Obstfeld impose more structure on the bubble process. 24

sum in that equation converges to Pf = κD , (26) t t e(µ+σ2/2 ln(1+r)) where κ = − . (1+r) e(µ+σ2/2) − Under the null hypothesis of no (intrinsic) bubbles, prices are a linear function of dividends and the price dividend ratio is a constant, κ, as suggested by equation (26). Intrinsic bubbles impart nonlinearity into the relationship between stock prices and dividends. In this case, the price/dividend ratio is P t =κ+cDλ 1+ι , (27) D t− t t whereι isawellbehavederrorterm.18 Thedifferentbehavioroftheprice/dividend t ratio in the absence and presence of bubbles can be exploited to form a bubble test. FrootandObstfeldtestforbubblesbyrunningregressionsofprice/dividend ratios on a constant and dividends. Not finding any significant coefficients exceptfortheconstantintheseregressionswillindicatelackofbubbles,whilefinding a nonlinear relationship between prices and dividends will be interpreted as signalling the presence of an intrinsic bubble. In the event, Froot and Obstfeld find strong evidence for positive values of c; however, they point out that the resultsmay“...merelyshowthatthereisacoherentcasetobemadeforbubbles alongside ... alternative possibilities. If that is so, then we should not feel too comfortable about how well we really understand stock prices.” Indeed, their tests show that there exists a nonlinear relationship between stock prices and dividends, but this is interpreted as asign of bubbles onlybecausethe model is assumed to be linear. What if the ‘true’ model is nonlinear? Driffill and Sola (1998) formalize this argument about the underlying stock 18Theexistenceofthiserrorterm isnotwellmotivated. FrootanfObstfeldsuggestitmay arise because ofwithin-period predictable excess returns. 25

pricing model being nonlinear. They note that the time invariance of Froot and Obstfeld’s random walk characterization of the log dividends is central to the analysis and results, and show that this assumption can be rejected when specification tests (in particular, an ARCH specification test) are applied to data. They propose a regime switching model of dividends: d =d +µ (1 s )+µ s +[σ (1 s )+σ s ](cid:18) t t − 1 0 − t 1 t 0 − t 1 t t where s is a state variable that follows a Markov process with constant trant sition probabilities. In this case, growth rates of dividends, ∆d , is distributed t N(µ ,σ2) in the s = 0 state, and N(µ ,σ2) in the s = 1 state. Driffill and 0 0 t 1 1 t Sola verify that this formulation of the dividend process fits that data better, and then test the model with regime switching fundamentals. When they include both regime switching fundamentals and intrinsic bubbles, they find that the explanatory contribution of bubbles is low. Their more striking finding is that the fit of a model with regime switches but no bubbles and that of a model with intrinsic bubbles but no regime switches is about the same. There is a certain nonlinearity in the data that will be attributed to whatever is nonlinear in the model. 3.5 Bubble as an unobserved variable Theeconometricbubbledetectiontestsdiscussedaboveimposeverylittlestructure on the bubble process. Indeed, many of these are tests of the standard model against an unspecified alternative, which is interpreted to be a bubble. Thesetestsdonotproduceatimesseriesofthebubblecomponent,soitisdifficult to evaluate whether the implied properties of the bubble are reasonable or not. Wu (1997) takes the ‘bubble as a deviation from the present value model’ detection scheme seriously and presents estimated values of the bubble under 26

this interpretation. His paper specifies the present value model as in section 2, assuming that differenceddividendsfollowanARprocess(likeWest),andestimatesthebubble asanunobservedvariablesubjecttotheno-arbitrageconditionusingaKalman filter. He finds that the bubble explains a large proportion of the movement in stock prices; however, his estimated bubble is often negative. One of the few strong theoretical predictions about bubbles is that, if one exists, it can never be negative.19 Thus, Wu’s bubble process clearly proxies for the failure of the model in other dimensions. 4 Conclusion Whathavewelearnedfrombubbletests? Thissurveyshowedthatbubbletests do not do a good job of differentiating between misspecified fundamentals and bubbles. This is not only a theoretical concern: For every test that ‘finds’ a bubble, there is another paper that disputes it. The finding of a bubble, at best, suggests that the data is either consistent with a bubble or a myriad of other extensions of the standard model. The Driffill and Sola paper highlights this central difficulty of bubble detection perfectly. It is a matter of taste and personal preference that makes the econometrician choose between bubble and fundamentals-based explanations of stock price behavior. The tests fare somewhat better in detecting a lack of bubbles, but there still are issues about the specification of the bubble that is shown not to exist, as pointed out by Evans. The tests are powerful against only certain types of bubbles. The bubble tests teach us little about whether bubbles really exist or not. However we do learn valuable stylized facts about the dimensions in which the presentvaluemodelofstockpricesfails. Thevarianceboundstests,forexample, 19Seethediscussion in integration/cointegration based tests of the bubble. 27

haveshownthatsomethingismorevolatilethanassumedinthemodel. Intrinsic andcollapsingbubbleargumentshaveledtomodelingandestimationofregime switching fundamentals. In the end, the underlying model remains a matter of belief. One can as well argue that regime switching fundamentals are a misspecified model that captures the effects of bubbles. Giventheseshortcomingsofthestandardpresentvaluemodelofstockprices, while a strand of the literature is still focusing on theoretically justifying and detectingbubbles,anotherstrandislookingfornon-bubbleexplanationsforthe apparent anomalies in asset prices, often involving time varying discount rates. For example, Campbell and Cochrane (1999) and the following literature on habit formation essentially make risk aversion a function of consumption and thus cause the discount factor to vary with the business cycle. In general, having a less restrictive fundamentals model—for example by allowingfortime-varyingdiscountrates,risk-aversion,orstructuralbreaks—allows the fundamentals part of the model fit the data better, leaving less room for a bubble. In this sense, the bubble is a catch-all for stock price movements not explained by the model. We have learned a lot about asset pricing models from bubble detection tests, but we have not learned definitively whether bubbles exist or not. 28

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