Exchange rate bubbles: Fundamental value estimation and rational expectations test

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  • Exchange rate bubbles:rational expectations tes

    Wilfredo L. Maldonado a,1, OctaUniversidade Catlica de Braslia, SGAN 916, MdulbUniversidade do Estado do Rio de Janeiro, Rua SocBanco Central do Brasil, SBS, Quadra 3 Bloco B, Bra

    JEL classication:E32

    2011 Elsevier Ltd. All rights reserved.

    * Corresponding author. Tel.: 55 21 2334 0172.E-mail addresses: wilfredo@pos.ucb.br (W.L. Maldonado), octavio.tourinho@terra.com.br (O.A.F. Tourinho), marcos.valli@

    bcb.gov.br (M. Valli).1 Tel.: 55 61 3448 7135.2 Tel.: 55 61 3414 2351.

    Contents lists available at SciVerse ScienceDirect

    Journal of International Moneyand Finance

    journal homepage: www.elsevier .com/locate/ j imf

    Journal of International Money and Finance 31 (2012) 10331059three different structural models are considered for the determina-tion of the fundamental value. The rst two imply that the exchangerate satises either purchasing power parity (PPP), or a modiedversion of it. The third structural model is a version of the monetarymodel of exchange rate determination, tted to the period underconsideration. We obtain the maximum likelihood estimate of theparameters of themodels, explore theproperties of theerrors, test itsrestricted versions, and compare the three specications for thefundamental. We nd that the models we propose t well the data,and are useful in the heuristic interpretation of the exchange ratemovementsof theperiod. Finallyweselect the structuralmodels thatdisplay the best performance, according to several criteria.C22

    Keywords:Exchange rate bubblesRegime-switching regression0261-5606/$ see front matter 2011 Elsevier Ltdoi:10.1016/j.jimonn.2011.12.009Fundamental value estimation andt

    vio A.F. Tourinho b,*, Marcos Valli c,2

    o B, Braslia, DF 70790-160, BrazilFrancisco Xavier 524, Bloco F, Sala 8039, Rio de Janeiro, RJ 20550-013, Brazilslia, DF 70074-900, Brazil

    a b s t r a c t

    We propose a model of periodically collapsing bubbles whichextends the Van Norden (1996) model, and nests it, by consideringa non-linear specication for the bubble size in the survival regime,and the endogenous determination of the level of the fundamentalvalue of the stochastic process. They allow us to test for rationality inthe formation of expectations, and remove the arbitrariness ofexogenously setting the level of the fundamental value. This generalmodel is applied to the exchange rate of the Brazilian real to the USdollar from March 1999 to February 2011. The futures marketexchange rate is used as a proxy of its expected future value, andd. All rights reserved.

  • 1. Introduction

    Brazil adopted a oating exchange rate regime in February of 1999, and the Brazilian real to US

    W.L. Maldonado et al. / Journal of International Money and Finance 31 (2012) 1033105910343 Evans (1991) points out that, when applied to this type of bubble, the Diba and Grossman (1988b) unit root based testwould lead to incorrect conclusions regarding the presence of a bubble. However, Charemza and Deadman (1995) later rejoindollar exchange rate has experienced wide uctuations since then, in spite of the fact that the Brazilianeconomy exhibited a relatively stable behavior throughout that period. This suggests that to charac-terize the trajectory of that exchange rate one should use a formulation that is able to account for theoccurrence of large deviations, as well as for some mean-reversal behavior. Of course, it must also takeinto account the asset nature of foreign exchange stocks, as well as its role in pricing tradable goodsrelative to non-tradable ones. This can be done with a model that allows for the occurrence of bubbles.

    It is assumed that in equilibrium the exchange rate is the solution of a structural system of equationsthat has two components: the fundamental value and the bubble. Tirole (1982, 1985) and Blanchardand Fisher (1989) present several simple macroeconomic models that produce bubbles. Some alter-natives can be considered regarding the stochastic specication of the bubble. In Obstfeld and Rogoff(1983) and Engsted (1993) the bubble only disappears when the system itself ceases to exist. Blanchard(1979) and Blanchard and Watson (1982) consider a more exible specication with an exogenousprobability of the bubble collapsing. Several tests for his type of bubble have been proposed in theliterature, and have been applied to study foreign exchange and stockmarkets, as indicated below. Dibaand Grossman (1988b) discuss the inception and survival of bubbles.

    Evans (1986) proposes a nonparametric procedure that takes into account the possibility ofoccurrence of more than one bubble, each covering part of the sample period, and nds evidence ofa negative bubble in the excess return to holding sterling rather than dollar assets during 19811984.Meese (1986) considers the occurrence of bubbles in a monetarymodel of the exchange rate. He rejectsthe hypothesis of inexistence of bubbles in the dollar to deutsche mark and in the dollar to poundexchange rates over the period 19731982. West (1987) proposes a parametric approach based on theHausman specication test, and rejects the null hypothesis of no bubbles in the Standard and Poors500 index (18711980) and the Dow Jones index (19281978). Diba and Grossman (1988a) proposetesting the stock price and the dividend series for the presence unit roots, and for cointegrationbetween them, to test for bubbles. They conclude that the empirical evidence is inconsistent with theexistence of an explosive rational bubble in prices.

    Some of these models have a restrictive feature which may be undesirable in some contexts: theyassume that the bubble increases indenitely until some exogenous structural change occurs. Thislimitation can be removed by specifying a regime-switching model for the bubble, which can beappropriate if agents contemplate the existence of two alternative dynamics of the bubble size, andincorporate the possibility of a regime change in their expectations. In this formulation the bubble iseither: (i) collapsing, and its expected size is decreasing, or (ii) it survives, and its expected size isincreasing. Some examples of this approach, which is also followed in the model proposed here, arediscussed below.

    Evans (1991) is the standard reference for a bubble that collapses periodically and switches itsgrowth regime, depending on its size. Whenever it exceeds a certain value, it enters a regime wherea stochastic variable with a Bernoulli distribution shifts its local behavior between two trajectories: itcan either grow faster, or can it collapse, reverting to a mean value.3 Van Norden and Schaller (1993)provide an alternative specication of the periodically collapsing bubble, where the bubble regime isa non-observable stochastic binary variable whose probability of occurrence is a function of the bubblesize. In addition to the different specication of the bubble, these models display another crucialdifference: in Evans (1991) the regime is observable, while in Van Norden and Schaller (1993) it is not.Since the relation between the innovation and the bubble size is determined by the regime, the latermodel allows for a more exible interaction between them.

    Van Norden and Schaller (1993) apply their model to the Toronto stock market index from January1956 to November 1989, and nd evidence of regime switches in stock market returns that areinuenced by apparent deviations from fundamentals. Van Norden (1996) used this model to modelthis discussion, and propose a unit root based test of explosive regime-switching bubbles.

  • the dynamics of speculative bubbles in the exchange rate between the US dollar and three other majorcurrencies, and nds mixed evidence regarding the occurrence of regime-switching bubbles of thattype. Several other studies test for the presence of regime-switching bubbles in a variety of data sets.For example, Funke et al. (1994) examine the ination rate in Poland, Roche (2001) analyzes the price ofland in Iowa, and Brooks and Katsaris (2005) study the S&P 500, with this type of model.

    A related body of literature employs switching models with Markovian state dependence(Hamilton, 1994). This type of regime switching is not considered in the models mentioned above, andsubstantially complicates the estimation because it requires the use of more than two states, asindicated by Evans (1996).

    The model proposed here generalizes the Van Norden (1996) formulation of two regimes for thebubble component by considering the possibility of a non-linear relation between the innovation andthe bubble size in the survival regime. This allows us to test for rational expectations, using theprocedures proposed by White (1982, 1987) and Hamilton (1996). We also include the endogenousestimation of the reference level of the fundamental value, which is exogenous in his model.

    We apply our model to study the occurrence of regime-switching bubbles in the Brazilian real to USdollar exchange rate on monthly data extending fromMarch 1999 to February 2011. The chosen periodstarts with the adoption of the oating exchange rate regime, and ends at the last month for which thedata was available when the empirical analysis reported here was performed. It is sufciently long topermit the identication of bubbles, but is not long enough to raise questions regarding the possibilityof signicant structural shifts within the sample. It is reasonable to specify a regime-switching bubblefor the stochastic process of the Brazilian exchange rate in that period because it was determined ina context where there existed a governmental authority (the Central Bank) which frequently inter-vened in the market.

    To dene the fundamental value and the bubble component of the exchange rate we consider threestructural models. Models I and II are based on the application of the inter-temporal non-arbitrageprinciple to the investment decision of rms engaged in international trade. They imply that thefundamental exchange rate satises one of two dynamic pricing rules: it either maintains purchasingpower parity (PPP) between the Brazilian real and the US dollar (Model I), or it satises a modiedversion of that principle, which accounts for the interest rate differential of Brazilian government bondstraded domestically and abroad (Model II). Model II can be interpreted as adding the uncovered interestrate parity hypothesis to the PPP hypothesis of Model I. In Model III the dynamic pricing rule of thefundamental exchange rate is obtained from a suitably modied version of the monetary model ofexchange rate determination proposed in Meese (1986). It essentially adds a monetary equation toModel II.

    The evidence regarding the presence of regime-switching bubbles of the type we postulate isdifferent in each of the models indicated above. This is to be expected since, as widely pointed out inthe literature, there is generally a way in which bubbles can be reinterpreted in terms of the processdriving fundamentals.

    The paper is divided into ve sections. Section 2 describes the structural equation for the stochasticprocess that has solutions with bubbles. It also presents the regime-switching model that will describethe dynamics of the innovations to the process, and its relation to the bubble size. In Section 3, wedetail the empirical model, specifying the functions chosen to characterize the behavior of the bubblecomponent of the exchange rate, describe the data, discuss the estimation and testing procedures, andpresent the results. In Section 4 we make a heuristic evaluation of the results, contrasting the severalmodels we test with the actual dynamics of the Brazilian exchange rate. In Section 5, we summarize themain conclusions. Appendix 1 shows the derivation of the three structural equations used to calculatethe fundamental values for the exchange rate and Appendix 2 describes the formulae used forstatistical inference.

    2. The stochastic process with bubbles

    In macro dynamic models it is common to describe the trajectory of a variable as depending on theexpectations of its future value, and on the values of other exogenous variables. The simplest linear

    W.L. Maldonado et al. / Journal of International Money and Finance 31 (2012) 10331059 1035model with such a structure is described below, and is the basis of our model.

  • Let Yt be the stochastic process for the asset value, and assume the system that governs its priceevolves according to:

    Yt aYet1 f Xt (1)

    W.L. Maldonado et al. / Journal of International Money and Finance 31 (2012) 103310591036where t denotes the time period, Yt is the value of the endogenous variable, and Xt is a vector ofexogenous variables, Yet1 is the expected value of the endogenous variable for period t 1, a (0,1) isa constant, and f is a well behaved function of the exogenous variables that reects the equilibriumconditions of our system, the absence of riskless arbitrage opportunities, and any rationality conditionsthat are imposed on the behavior of the agents.

    The rational expectations hypothesis requires that Yet1 EYt1jJt , where Jt is the s-algebragenerated by {Xt,Yt,Xt1,Yt1,.}. Henceforth, we will denote E$jJt as Et $. Therefore, the rationalexpectations solution to (1) is a stochastic process (Yt)t0 that satises

    Yt aEt Yt1 f Xt (2)Any solution of (2) may be written as a sum of two components: the fundamental value, which only

    depends on the exogenous variables and their expected values, and the bubble component, which isequal to the present value of the expected long-run level of the endogenous variable. To prove this,solve equation (2) forward in time to obtain the following decomposition of the endogenous variable:

    Yt XNi0

    aiEt f Xti limT/N

    aTEt YtT (3)

    where the rst term in the right hand side (r.h.s.) is the fundamental value, denoted Yt and dened by:

    Yt XNi0

    aiEt f Xti (4)

    Noting that the value of the exogenous variables at time t is known, and using the law of iteratedexpectations, the fundamental value can be written as:

    Yt XNi1

    aiEt f Xti f Xt aEtYt1

    f Xt (5)Letting Jt be the s-algebra generated by fXt ;Yt ;Xt1;Yt1;.g note that Jt4Jt implies

    EE$jJt jJt E$jJt . Therefore, taking the expectation of (5) conditional on Jt shows that funda-mental value Yt is also a solution of equation (2), since Y

    t aEYt1jJt f Xt. This is denoted by

    calling it the fundamental solution associated with (Yt)t0.The difference between the solution and the fundamental value is the bubble component. It is equal

    to the last term in the r.h.s. of (3), assuming that the limit exists4:

    bt Yt Yt limT/N aTEt YtT (6)

    The dynamic equation for the bubble size can be obtained by substituting (2) and (5) in (6):

    bt aEt bt1 (7)Note from (7) that the bubble size is governed by the same type of expectations mechanism present

    in the asset price, but it is not affected by the exogenous variables. Therefore, it isolates the speculativemechanism, which functions autonomously: its current size depends only on its expected size nextperiod. This dependence, however, is not unrestricted, since (7) is a linear function, and the parametera is exogenous and constant. Since it is assumed that a (0,1), the bubble size will be expected toincrease in the next period if the current bubble size is not zero.

    4 Regarding a as a discount factor, the hypothesis that the limit exists is the conventional specication of the transversalitycondition for the dynamic optimization of the agents behavior, and can be interpreted as stating that the present value of the

    terminal asset stock is nite, when the horizon is indenitely extended.

  • W.L. Maldonado et al. / Journal of International Money and Finance 31 (2012) 10331059 1037Yt1 Yt1 bt1 in equation (11) and using equation (7).

    Rt1 Rt1 bt1 bta

    (12)

    where Rt1 is the innovation in the fundamental solution, i.e. Rt1 Yt1 Et Yt1.

    The expected value of the innovation, conditioned on the bubble collapse and survival, is calculatedas:

    Et Rt1jrt1 0 EtRt1

    rt1 0 ubt bta (13a)Et Rt1jrt1 1 Et

    Rt1

    rt1 1 1qbt 1

    bta ubt

    (13b)

    Assumption 2. The current forecast of the next period fundamental value is not affected by theknowledge of regime of the bubble in the next period:

    EtYt1

    rt1 EtYt1; for all t (14)Noting that Assumption 2 implies that, conditional to either regime, the expected value of theof the bubble dynamics, and the second hypothesis makes precise the sense in which the fundamentalvalue is unaffected by the bubbles regime.

    Assumption 1. There exist two functions u:R/ R and q:R/ [0,1] that satisfy equations (8) and (9):

    Et bt1jrt1 0 ubt (8)

    qt1 qbt (9)

    Equation (8) asserts that if, in addition to the information in Jt , it is known that collapse will occurin period t 1, then the expected bubble size only depends on the current bubble size. Equation (9)implies that probability of survival of the bubble in period t 1 is independent of all the variablesin Jt , and depends only on bt.

    The expected bubble size in the next period if it survives (equation (10)) can be calculated byusing (7)(9) to substitute for the expectations terms in the following identity:Et bt1 qt1Et bt1jrt1 1 1 qt1Et bt1jrt1 0.

    Et bt1jrt1 1 bt

    aqbt ubtqbt ubt (10)

    As can be seen in equation (6), the bubble is not observable because the fundamental value isunobservable. However it is feasible to infer from the data the size of the innovations to the asset value,Rt dened by:

    Rt1 Yt1 Et Yt1 (11)The innovation is closely related to the bubble, as shown by equation (12), obtained by letting

    Following Blanchard (1979) and Blanchard and Watson (1982), assume that in each period one oftwo events occurs in the bubble dynamics: either collapse (C) or survival (S). These events willdetermine the current regime of the bubble. However, since by hypothesis the regime is not observable,only a probabilistic statement can be made about it. This can be formalized by letting (rt)t0 be thestochastic process generated by the occurrence of those events, dened as rt 1 if S occurs, and rt 0 ifC occurs, in period t. Let qt1 be the probability of that the bubble survives in period t 1, given theinformation available in period t: qt1 Prt1 1jJt.

    The two assumptions below characterize the dynamics of the bubble and of the fundamental value,specifying how they depend on the bubbles regime. Loosely speaking, the rst hypothesis says that thesize of the bubble itself summarizes all the information required to determine the endogenous aspectsinnovation in the fundamental solution is null, (13a) and (13b) can be written as follows:

  • Et Rt1jrt1 0 ubt bta

    (15a)

    Et Rt1jrt1 1

    1qbt 1

    bta ubt

    (15b)

    3. The empirical model

    W.L. Maldonado et al. / Journal of International Money and Finance 31 (2012) 103310591038as a form of limited rationality, because agents are only unable to fully recognize the nature of the dependence of the size of theinnovation if the bubble survives on probability of survival.

    7 To preserve identication of the model and the ability to distinguish between bS0 and bS00, we always include thehypothesis bS00 0 when the test considers the possibility that bp2 0.

    8 The hypothesis of homoscedasticity and lack of serial autocorrelation of the errors are tested when the model is estimated.The normality of the errors cannot be tested because the regime-dependent errors are not observable. Nevertheless, we test fornormality of the observed average error using the JB test, and note that it is rejected in all models, due observations that appeardynamics, it is necessary to further restrict the model, and specify functional forms for the functions qand u.

    Following Van Norden (1996), we use a logit function for the probability of survival:

    qb h1 exp

    bq0 bq2b2

    i1; bq2 > 0 (16)

    Note, however, that unlike his formulation, the exponent of the exponential function in (16) doesnot include a linear term.5 The fact that the probability of survival is decreasing in the bubble size i.e.dq=djbj < 0, controls the explosive behavior of the bubble.

    Noting that in (15a) and (15b) the expected value of the innovation is linear with respect to u, weadopt a linear specication for the function u(bt). When used in (15a), it yields (17a). Expanding ther.h.s. of (15b), and using (16) and (17a), yields the following: Et Rt1jrt1 1 expbq0expbq2b2t bC0 bC1bt. Equation (17b) is a generalization of this expression, since it can beobtained by setting bS0 bC0expbq0; bS1 bC1expbq0; bS00 0 and bp2 bq2 in (17b). Themain advantage of this more general formulation is allowing us to test for rational expectations in thefutures market for foreign exchange.6

    Et Rt1jrt1 0 bC0 bC1bt (17a)

    Et Rt1jrt1 1 bS00 bS0 bS1btexpbp2b

    2t

    (17b)

    This derivation is exact and, unlike Van Nordens (1996) formulation, does not require the log-linearization of the system resulting from the combination of (15) and (16). Our empirical imple-mentation nests his, and his model can be tested against ours by testing for bp2 0. If bp2s 0, it is alsopossible to test the rational expectations hypothesis, a verication that is impossible in his model.7

    System (17) is written in terms of the expected innovation. To obtain an empirical model, weassume that, conditional on the regime that will prevail in t 1, the deviations of the innovation fromits expected value are independent, homoscedastic, serially uncorrelated, normally distributedstochastic errors8:

    3Ct1 Rt1 Et Rt1jrt1 0; 3CtwN

    0; s2C

    (18a)

    3St1 Rt1 Et Rt1jrt1 1; 3StwN

    0; s2S

    (18b)

    5 The linear term in the exponential for the logit distribution of the probability of survival appears in Van Norden (1996)when he log-linearizes the bubble. In our approach we do not approximate the model so that term does not appear.

    6 Note that what we are considering here is a rather specic failure of rational expectations that could possibly be describedTo estimate equations (15a) and (15b), and recover the parameters that characterize the bubbleto be outliers.

  • These errors can be interpreted as noise, admitting that all the relevant information present in theinnovation is contained in its expected value. Further, in each regime, they are equal to the innovationto Yt1, computed under the assumption that, in addition to Jt , the information that the regime will

    W.L. Maldonado et al. / Journal of International Money and Finance 31 (2012) 10331059 1039prevail in t 1 is also available. To show this, use the denition Rt1 Yt1 Et[Yt1] in (18), and applythe law of iterated expectations to obtain:

    3Ct1 Yt1 Et Yt1jrt1 0 (19a)

    3St1 Yt1 Et Yt1jrt1 1 (19b)

    The average error 3t1 is a convex combination of 3Ct1 and 3St1, and is equal to the innovation to Yt1,

    as follows:

    3t1h1qbt 3Ct1qbt 3St1 Yt1f1qbtEt Yt1jrt1 0qbtEt Yt1jrt1 1g Yt1Et Yt1

    (20)

    This implies that Y is in fact an AR(1) process, where the average error is the innovation. This can beshown writing the structural model of equation (2) as Et[Yt1] a1Yt a1f(Xt), adding 3t1 to bothsides of this equation, and using (20):

    Yt1 a1Yt a1f Xt 3t1 (21)The regime-switching equations are obtained by substituting for the expectation terms in (17) from

    (18):

    Rt1 (bC0 bC1bt 3Ct1; Prob 1 qbt abS00 bS0 bS1btexp

    bp2b2t

    3St1; Prob qbt b

    (22)

    We obtain maximum likelihood estimates for the parameters in (22), and use the Likelihood Ratiotest to verify the following hypotheses.9

    C Rational expectations hypothesis

    To test if expectations regarding the innovations are rational we verify if the following conditionsare satised by the estimated parameters.10

    bS00 0; bp2 bq2; bS0 bC0 expbq0

    0; bS1 bC1expbq0 0 (23)

    C Two linear regimes

    The empirical implementation in Van Norden (1996) assumes that the innovation is a linearfunction of the bubble size in both regimes. We test if the more general model proposed here bettersupports our data by testing the conditions:

    bp2 0; bS00 0 (24)We also reproduce the following tests proposed by Van Norden (1996):

    C Two linear regimes with the same slope

    9 Goldfeld and Quant (1973) and Kiefer (1978) proved that maximum likelihood estimators for this type of model areconsistent and efcient. See Appendix 2 for the details on how the tests of the different hypothesis were implemented.10 To test the rational expectations hypothesis we use (22a) and (22b), in the identity

    Et Rt1 qbtEt Rt1jrt1 1 1 qbtEt Rt1jrt1 0, and test Et Rt1 0; cbt .

  • Test if the impact in the innovation of changes in the bubble size is the same in both regimes.

    bp2 0; bS00 0; bC1 bS1 (25)

    C Linear regression model

    Test if the single regime stochastic process ts the data better.

    bp2 0; bS00 0; bC1 bS1; bC0 bS0 (26)

    C Normal mixture model

    Test if distribution of the innovations is a mixture of normal random variables that occur withprobabilities q and 1 q, as indicated in (22), with regime probabilities that are independent of thebubble size.

    bp2 0; bS00 0; bC1 bS1 0; bq2 0 (27)

    C Restricted normal mixture model

    Test a special case of the previous hypothesis: having just one regime that is independent of thebubble.

    bp2 0; bS00 0; bC1 bS1 0; bq2 0; bC0 bS0 (28)

    3.1. Data

    In this sectionwe describe the data used to estimate de model described above for the Brazilian realto US dollar exchange rate, report the values of the estimated parameters, and discuss the results of thestatistical inference analysis. We denote this particular time series St, to emphasize that this is anapplication of the general model described in the previous section, where the time series wasdenoted Yt.

    For all variables we use monthly data from March 1999 to February 2011. The data on prices,production and money stocks has been seasonally adjusted, but the exchange and interest rates havebeen used in their original form. The sample period was constrained by the availability of the proxyvariable for the innovations, as discussed below.

    For the spot exchange rate (St) we use sell price of the US dollar on the rst day of the month, asreported by the Central Bank of Brazil. We obtain the one month forward exchange rate (Ft) from dataof the Brazilian Mercantile Exchange database (BM&F of Brazil), and use it as a proxy for the expectedvalue of St, i.e. Et[St1] Ft. Therefore, the innovation is: Rt1 St1 Ft.

    To calculate the bubble it is necessary to specify the fundamental value St . We consider threestructural models to determine it. Model I assumes only the PPP (Purchasing Power Parity) hypothesis.Model II is an extension of Model I that takes into account the interest rate differential for the Braziliangovernment debt in the domestic and international markets. Model III is an extension of the Meese(1986) monetary model for the exchange rate. It essentially adds a monetary equation to Model II.Appendix 1 justies the use of Models I and II as the rational pricing relation if rms arbitrage theirinvestment opportunities in the country and abroad, and develops Model III in detail.

    For Model I, St is determined by (29), where PDt and PFt are the domestic and foreign price levels.

    St;I PDtPFt

    (29)

    In this case the bubble size is given by (30), where PDI and PFI are domestic and foreign price

    W.L. Maldonado et al. / Journal of International Money and Finance 31 (2012) 103310591040indexes, and k is a constant that must be estimated in the maximization of the likelihood function.

  • bt;I St kPDItPFIt

    (30)

    For the Model II, St is given by:

    St;II 1 IF 0t

    1 IDtEt

    PDt1PFt1

    (31)

    W.L. Maldonado et al. / Journal of International Money and Finance 31 (2012) 10331059 1041where ID and IF0 are the nominal interest rates observed in the domestic and foreign markets forBrazilian government bonds, respectively. The expected value of the next periods price ratio is taken tobe the realized price ratio, assuming that expectations are conrmed. In this case the bubble size isgiven by (32), where PDI and PFI are as before.

    bt;II St k1 IF 0t

    1 IDt

    PDIt1PFIt1

    (32)

    In Models I and II the domestic price index (PDI) is the Producer Price Index (IPA-OG), as suggestedby the development in Appendix 1.11 We consider two versions (A and B) of the above models, eachcorresponding to one of two alternatives for the foreign price index (PFI). For models of type A, it is theProducer Price Index for the USA (US-PPI).12 For models type B, we take a trade-weighted index of theproducer price index of the more signicant partners of Brazil in international trade in the sampleperiod.13 In Model III we used the same price indexes are as in Models IA and IIA, for comparability.

    For ID we consider the domestic xed interest rate for 30 days implied by traded contracts forswapping xed nominal interest rate bonds for indexed bonds whose return is post-xed on the basisof the realized very short term interest rate. For IF0 we use a proxy: the interest rate paid by the Bra-zilian Government on its foreign debt, calculated as the sum of the interest rate of the U.S. Treasury Bills(IF) and the average spread on Brazilian foreign debt, denoted IS. For the latter we use the monthlyequivalent rate of the interest rate spread of Brazilian debt over US treasuries. It is measured by theEmerging Markets Bond Index (EMBI) calculated by JP Morgan as the average interest rate differentialbetween Brazilian Bonds and US treasuries of same duration, weighted by their debt volume.14 Allinterest rates are specied in decimals, not in basis points, i.e. 1% per month is represented as 0.01. Thisfact is important when comparing our estimates for the model coefcients with those in otherempirical studies.

    Model III is presented in detail in Appendix 1, but its implications for the fundamental rate can besummarized by pointing out that, given an initial value for the exchange rate (S0), the fundamentalexchange rate of the monetary model can be calculated from the following equations:

    DxthDmt a1Dyt a2DISt (33)

    Dxt cDxt1 dt (34)

    Dst Dxt gc

    1 gc Dxt Dxt1 (35)

    St expst

    (36)

    where st is the log of the spot exchange rate (St), ft is the log of the forward exchange rate (Ft),mt and ytare the logs of the ratio of the domestic aggregate to the foreign aggregates for the nominal moneystock m lnMD=MF and for industrial production y lnYD=YF. The parameters a1 > 0 anda2 > 0 are, respectively, the income elasticity and the interest semi-elasticity of real money balances,

    11 The variable IPA-OG is the ndice de Preos no Atacado Oferta Global calculated by Instituto Brasileiro de Geograa eEstatstica (IBGE), obtained from IPEAdata: http://www.ipea.gov.br.12 As reported by the Federal Reserve Bank of St. Louis.13 Specically, we choose twenty countries among those with greater trade volume (imports plus exports) with Brazil. Theyare responsible for 76% of the total trade with Brazil. It is available from the authors, on request.14 This proxy variable for IF has a weakness: the duration of the long term instruments it considers is much longer than that of

    short term government debt used in measuring ID. However, no other better measure was readily available.

  • and g a2/(1 a2) is an auxiliary parameter. Themarket fundamentals process, denoted Dxt, is denedby (33), and its autocorrelation is jcj< 1. Sample determination of the trajectory for the fundamental

    W.L. Maldonado et al. / Journal of International Money and Finance 31 (2012) 103310591042unable to reject the presence of a unit root with zero drift inm p, y, IS and s p, and barely rejects itin ID IF. However, the presence of a unit root in ID IF is not rejected at the 10% signicance level, andinspection of the graph of that series suggests the presence of a unit root if the possibility of a break inthe time series is taken into account. This is conrmed by applying the Perron (1997) tests, which areunable to reject the presence of a unit root at the 5% condence level.16 Finally, the innovation in the logof the exchange rate s f is classied as stationary by using the methodology above. Nevertheless,examination of the graph of the series indicates the presence of signicant noise superimposed ona stochastic trend. To handle this, the series was subjected to the Hodrick and Prescott (1980, 1997)lter, and it was veried that the presence of unit root cannot be rejected in the ltered series.Therefore, all the variables listed in the beginning of this paragraph are classied as I(1).

    Second, the parameters of the realmoney balances demand equation are obtained by estimating thecointegrating vector of m p, y, and ID IF, by regressing the rst variable on the other two, via theStock and Watson (1993) dynamic OLS approach, using 2 lags. The estimated values are signicantlydifferent from zero (standard errors in parenthesis): a1 0.95895 (0.47277) and a2 2.70749(1.07285).17 These are, approximately, double and half (respectively) the values found in Ball (2001).However, considering the variability of estimates documented in that survey and in Stock and Watson(1993), they appear to be within the range of reasonable values for the US in the period considered inour sample. The estimates for Brazil, obtained using the methodology in Tourinho (1995) are similar tothe values above. The auxiliary parameter g 0.73028 is calculated using its denition.

    Third, the two other structural hypotheses of the monetary model are veried. The UIP hypothesis,extended to include a country risk spread variable, is tested by estimating the cointegrating vectorbetween ID IF, s f, and IS. It is obtained by regressing the rst variable on the other two, via the StockandWatson (1993) dynamic OLS approach, using 1 lag. The estimated values are, respectively: 1.323089(0.917568) and 0.933017 (0.212349) (standard errors in parenthesis). The estimated coefcients areclose to 1, and the expandedUIP hypothesis is not rejected. The importance of including the interest ratespread,which reects the country risk and the foreigncapital supplyconditions, in theUIP is evident inits relatively small standard error in the estimated equation. The PPP hypothesis, that s p is a randomwalk, is conrmed by the presence of a unit root in that series, already discussed above.

    Fourth, the fundamental market process Dxt is calculated from (33), using the sample values ofDmt ; Dyt and DISt . Theparameter c0.46646 is obtainedbyestimating (34) inrst differences byOLS.

    Fifth, using the values of the parameters discussed above and the time series of the marketfundamentals process, equation (35) can be used to calculate Dst . Using an initial value s0 we can useequation (37) to obtain the following fundamental exchange rate:

    St exps0 exp Xt

    i1Dsi

    ! k exp

    Xti1

    Dsi

    !(37)

    So for Model III the bubble is:

    bt;III St k exp Xt

    i1Dsi

    !(38)

    In all three models the estimation of k jointly with the parameters of the bubble process (equations(30), (32) and (38), respectively), allows the endogenous estimation of the fundamental value of theexchange rate. In Van Norden (1996) the value of k is calibrated in such a way that the average value of

    15 Unit root testing was done using the URAUTO procedure of RATS 7.0 (Estima, 2007).16 The results of the PERRON97 procedure or RATS 7.0 (Estima, 2007) are in Appendix B.exchange rate in this model is done as follows.First, the variables of the model, m p, y, ID IF, IS, s p, and s f are tested for the presence of

    a unit root, using the procedure suggested by Doldado et al. (1990) to address the issue of inclusion ofa constant and a time trend in the test equations, as described in Enders (1995). The number of lagsused in the tests is chosen by the Akaike information criterion.15 This test, at the 5% signicance level, is17 Estimation was performed using the SWDOLS standard procedure of RATS 7.0 (Estima, 2007).

  • the bubble over the sample period is null.18 Here we estimate it simultaneously with all the otherparameters within the likelihood maximization procedure.

    In the next sections we describe the estimation and the tests of the ve models we consider: IA, IB,IIA, IIB, and III.

    function for the survival regime (bp2 and bS00) are displayed in the last two columns of the table.

    W.L. Maldonado et al. / Journal of International Money and Finance 31 (2012) 10331059 104318 More precisely, the parameter which is calibrated in Van Norden (1996) is the analog of k in the log-linearization of themodel for equation (25).19 The normalization factor of the likelihood function was omitted in the calculations; therefore the log-likelihood is notnecessarily negative. Since the sample is the same for all models, that statistic is comparable across them.3.2. Estimation

    Table 1 shows the maximum likelihood estimates of the parameters of the regime-switchingempirical equation (22) that relates the innovation to the bubble size, for each model, for theirunrestricted and restricted versions. The rst two columns indicate the model and the version.

    The third column contains the maximum value of the log-likelihood in each case.19 These are usedto test the hypothesis previously described with the likelihood ratio test. Looking at the unrestrictedversion of the models, in the rst line in each section of Table 1, it can be seen that for this metric theranking of the models is as follows: IA, IIA, IB, IIB, III. Therefore, model I, where the fundamentalexchange rate is specied by the pure PPP relation, dominates model II which uses an interest rateadjusted PPP rule for that purpose. The worse performance is that of model III, where the fundamentalrate is determined by the monetary model.

    This ranking is strengthened by noting that the best models are also the more parsimonious. Thevariant Aofmodels I and II dominates the variant B, i.e. themodels that adopt as a foreignprice index theProducer Price Index for the USA (US-PPI) dominates those that use trade-weighted index of theproducerprice indexof themoresignicantpartnersof Brazil in international trade in the sampleperiod.

    The other columns of Table 1 show the values of the estimated parameters. Column 4 refers toparameter k, which scales the reference trajectory of the fundamental exchange rate, which arecomparable between models.20 It varies from 0.51 to 1.44 across models and versions. The smallestvalues are produced by model IB, and the largest values are produced by the restricted versions ofmodel III. Examining the unrestricted versions of themodels, it can be seen that the variant B of modelsI and II produces smaller estimated values for k. Generally speaking, the base trajectory of thefundamental value has to be scaled down by a larger proportion whenever the weighted foreign priceindex is used. However, for the same variant of the two models, the estimated k is similar, as can beseen, for example, by comparing its value for models IA and IIA, equal to 0.87 and 0.9 respectively.

    It is also interesting to note that the estimate of k in the models restricted to satisfying the rationalexpectations hypothesis is signicantly smaller than those obtained in the unrestricted models (anaverage decrease of 30.8%). This means that the fundamental rate is smaller under rational expecta-tions, and that the likelihood of occurrence of negative bubbles is much smaller in those restrictedmodels than in the unrestricted model.

    Columns 5 and 6 show the estimated values of the parameters of the logit function q(b) for theprobability of the bubble collapsing in the next period (equation (16)). The plot of that function for theunrestricted version of the models IIB and III is displayed in Fig. 1. For model III it shows that theprobability of survival is very close to one whenever jbj 1:2, i.e. until a large bubble develops. Afterthat occurs, andwhile the bubble does not burst, that probability drops steeply and becomes null for jbj 3, signaling an imminent regime change. This situation was observed in 2002. For Model IIB theprobability of survival is at most 0.57, and declines steadily as the bubble size increases, becoming nullfor jbj 2.

    The columns 711 of Table 1 show the estimated values of the parameters for linear part of thefunctions that relate the innovation to the bubble size in the two regimes (see equations (22)). Theparameters with the C subscript (bC0 and bC1) refer to the collapse regime, while those with the Ssubscript (bS0 and bS1) refer to the survival regime. The parameters of the non-linear part of the20 The base trajectory of the exchange rate starts at approximately R$1.2/US$ in march 1999.

  • Table 1Maximum likelihood estimates of the parameters.

    Restrictions Scaledlog-likelihood

    k Transition Regime S Regime C Rational expectation Constant

    bq0 bq2 bS0 bS1 sS bC0 bC1 sC bp2 bS00

    Model IA Unrestricted model 148.66 0.87 (0.77) 1.43 0.06 (0.02) 0.050 0.01 (0.06) 0.106 1.64 (0.12)Rational expectations model 143.77 0.60 (0.60) 0.89 (0.04) 0.04 0.038 0.07 (0.07) 0.111 0.89 Two linear regimes (Van Norden) 141.45 0.55 (11.56) 3.10 0.01 (0.04) 0.077 0.90 (0.37) 0.320 Two linear regimes w/ same slope 140.95 0.77 (10.23) 4.81 (0.01) (0.02) 0.078 0.11 (0.02) 0.339 Linear regression model 140.30 0.85 (9.62) 5.64 (0.02) (0.02) 0.078 (0.02) (0.02) 0.354 Normal mixture model 138.01 0.85 2.52 0.14 0.358 (0.03) 0.072 Restricted normal mixture model 136.98 0.85 (2.53) (0.03) 0.072 (0.03) 0.398

    Model IB Unrestricted model 146.79 0.76 0.07 1.40 0.25 (0.11) 0.034 0.01 (0.06) 0.097 1.15 (0.30)Rational expectations model 144.73 0.51 0.84 0.56 (0.09) 0.09 0.008 0.04 (0.04) 0.096 0.56 Two linear regimes (Van Norden) 139.65 0.92 (6.04) 5.64 (0.02) (0.02) 0.076 0.24 (0.15) 0.321 Two linear regimes w/ same slope 139.45 0.90 (6.16) 5.29 (0.02) (0.02) 0.077 0.09 (0.02) 0.327 Linear regression model 139.12 0.59 (9.51) 3.14 0.01 (0.04) 0.077 0.01 (0.04) 0.357 Normal mixture model 138.01 0.59 2.52 0.14 0.358 (0.03) 0.072 Restricted normal mixture model 136.98 0.59 (2.53) (0.03) 0.072 (0.03) 0.398

    Model IIA Unrestricted model 148.65 0.90 (1.15) 1.92 0.02 (0.01) 0.050 0.02 (0.07) 0.115 2.56 (0.07)Rational expectations model 144.75 0.60 (0.67) 0.94 (0.04) 0.03 0.036 0.07 (0.07) 0.110 0.94 Two linear regimes (Van Norden) 141.50 0.69 (11.71) 4.45 (0.00) (0.03) 0.078 0.82 (0.38) 0.323 Two linear regimes w/ same slope 140.97 0.82 (10.30) 5.73 (0.01) (0.02) 0.078 0.10 (0.02) 0.337 Linear regression model 140.33 0.87 (9.89) 6.49 (0.02) (0.02) 0.078 (0.02) (0.02) 0.352 Normal mixture model 138.01 0.87 2.52 0.14 0.358 (0.03) 0.072 Restricted normal mixture model 136.98 0.87 (2.53) (0.03) 0.072 (0.03) 0.398

    Model IIB Unrestricted model 145.08 0.72 (0.30) 1.37 0.01 (0.00) 0.037 0.02 (0.06) 0.106 2.49 (0.04)Rational expectations model 143.62 0.55 (0.43) 0.91 (0.04) 0.04 0.033 0.07 (0.07) 0.106 0.91 Two linear regimes (Van Norden) 138.62 0.93 (7.43) 7.75 (0.02) (0.03) 0.077 0.34 (0.24) 0.322 Two linear regimes w/ same slope 138.31 0.97 (5.99) 7.13 (0.02) (0.02) 0.076 0.09 (0.02) 0.321 Linear regression model 137.76 0.63 (8.73) 3.11 0.01 (0.04) 0.077 0.01 (0.04) 0.358 Normal mixture model 136.38 0.63 (2.52) (0.03) 0.072 0.14 0.359 Restricted normal mixture model 135.35 0.63 (2.53) (0.03) 0.072 (0.03) 0.399

    Model III Unrestricted model 143.10 0.84 (5.07) 1.16 0.08 (0.02) 0.077 2.36 (1.38) 0.224 (0.09) (0.09)Rational expectations model 129.96 0.56 1.92 0.72 0.26 (0.11) 0.189 (0.04) 0.02 0.079 0.72 Two linear regimes (Van Norden) 143.08 0.83 (5.08) 1.15 (0.01) (0.02) 0.077 2.37 (1.38) 0.225 Two linear regimes w/ same slope 138.85 1.44 2.51 0.15 0.15 (0.01) 0.364 (0.03) (0.01) 0.072 Linear regression model 137.78 1.39 2.59 0.02 (0.03) (0.01) 0.403 (0.03) (0.01) 0.072 Normal mixture model 138.01 1.32 2.52 0.14 0.358 (0.03) 0.072 Restricted normal mixture model 136.98 1.32 2.53 (0.03) 0.398 (0.03) 0.072

    W.L.M

    aldonadoet

    al./Journal

    ofInternational

    Money

    andFinance

    31(2012)

    10331059

    1044

  • 0.4

    0.6

    0.8

    1.0q

    PROBABILITY OF SURVIVAL OF THE BUBBLE

    Model IB

    I

    I

    Model II

    W.L. Maldonado et al. / Journal of International Money and Finance 31 (2012) 10331059 1045Examining the coefcients for the linear part of the unrestricted models, it can be seen that theirmagnitude is generally small, indicating aweak linear dependence of the innovation on the bubble size,for both regimes. Model III an exception in this respect, because for it bC0 and bC1 are one order omagnitude larger than in the other models. Broadly speaking, this occurs because in models I and II thenon-linear partof the function for the survival regimeplays themost important part, as can be seen in thelarge relative values for bp2 in the next-to-last column,while inmodel III that parameter is small. In otherwords, the function for the survival regime in model III is approximately linear. This is conrmed bycomparing the lines for the Unrestricted and the Two linear regimes formodel III inTable1, andnotingthat the estimated parameters are very similar and that their log-likelihood is also almost identical.

    The estimated standard deviation of the errors in the two regimes (sS and sC), displayed in columns9 and 12, shows some interesting patterns, which can be seen by looking at their values for theunrestrictedmodels. They are summarized in Table 2, where themodels are listed in increasing order ofthe standard deviation of the errors.

    This ranking is different from the one obtained earlier by comparing the log-likelihood value, sincethis metric indicates that the variant B of models I and II ts the data better, offering evidence in favorof the superiority of the trade-weighted foreign price index. The dominance of model I over II ismaintained and, as previously found, the t of model III to the data on the innovations on the exchangerate is inferior to that of the other models. The information on the standard deviation of the errors,however, provides some quantitative indication on its performance relative to model IB, since theestimated standard deviation of its errors is about 2.2 times larger, in both regimes.

    3.3. Tests

    Table 3 shows the results of the likelihood ratio test of the several hypotheses discussed earlier, forall ve models. For Models I and II, the last column shows that all of them can be rejected at the usual5% of condence level, except the rational expectations hypothesis. Due to the results of these tests, inwhat follows we test misspecication andmake inferences on the parameters only for the unrestricted

    3 2 1 1 2 3b

    0.2

    Fig. 1. Probability of survival of the bubble.version of these 4 models.For Model III, all hypotheses, except Van Nordens (1996) specication of two linear regimes, can be

    rejected at the 10% condence level. The rejection of rational expectations in Model III may be relatedto the fact that its fundamental exchange rate is more informative than in the other models, a fact thatmay have elicited the following important feature of this application on our bubble model: since we

    Table 2Scaled log-likelihood of the models.

    Model Scaled log-likelihood sS sC

    IB 146.79 0.034 0.097IIB 145.08 0.037 0.106IA 148.66 0.050 0.106IIA 148.65 0.050 0.115III 143.10 0.077 0.224

  • Table 3

    W.L. Maldonado et al. / Journal of International Money and Finance 31 (2012) 103310591046Likelihood ratio tests.

    Null hypothesis Restrictionrank (n)

    UnrestrictedLog-L

    RestrictedLog-L

    LR statisticsChi2(n)

    p-Value (%)

    Model IA Rational expectations model 5.00 148.66 143.77 9.78 8.17%Two linear regimes (Van Norden) 2.00 148.66 141.45 14.43 0.07%Two linear regimes w/ same slope 3.00 148.66 140.95 15.42 0.15%Linear regression model 5.00 148.66 140.30 16.73 0.51%Normal mixture model 5.00 148.66 138.01 21.30 0.07%Restricted normal mixture model 7.00 148.66 136.98 23.37 0.15%

    Model IB Rational expectations model 5.00 146.79 144.73 4.12 53.18%Two linear regimes (Van Norden) 2.00 146.79 139.65 14.29 0.08%Two linear regimes w/ same slope 3.00 146.79 139.45 14.68 0.21%Linear regression model 5.00 146.79 139.12 15.34 0.90%Normal mixture model 5.00 146.79 138.01 17.56 0.36%Restricted normal mixture model 7.00 146.79 136.98 19.63 0.64%

    Model IIA Rational expectations model 5.00 148.65 144.75 7.79 16.80%Two linear regimes (Van Norden) 2.00 148.65 141.50 14.30 0.08%Two linear regimes w/ same slope 3.00 148.65 140.97 15.35 0.15%Linear regression model 5.00 148.65 140.33 16.63 0.53%Normal mixture model 5.00 148.65 138.01 21.27 0.07%Restricted normal mixture model 7.00 148.65 136.98 23.34 0.15%

    Model IIB Rational expectations model 5.00 145.08 143.62 2.93 71.11%Two linear regimes (Van Norden) 2.00 145.08 138.62 12.93 0.16%Two linear regimes w/ same slope 3.00 145.08 138.31 13.55 0.36%Linear regression model 5.00 145.08 137.76 14.64 1.20%measure the innovations by a proxy variable produced in a derivatives market, we in fact test the jointhypothesis of rational expectations in the spot and in the derivative markets for foreign exchange.Therefore, the rejection of RE could be due to imperfections in it, or in the modeling of the relationbetween the spot and futures market.

    Table 4 displays the results of the signicance test for the coefcients of equation (22) for theunrestricted models.21 Greater signicance (smaller p-value) of the coefcients indicates larger like-lihood that the bubble dynamics is relevant in explaining the innovations to the exchange rate. ModelsIA and IIB have sharply dened bubble dynamics, since four of the ve parameters are signicant.Models IB, IIA and III have only one signicant parameter, at the conventional 5% level. Model III hastwo signicant parameters at the less stringent 10% signicance level.

    The relevance of one of our extensions to Van Nordens model, represented by including a non-linear term in the bubble dynamics in the survival regime (represented by nding bp2 signicantlydifferent from zero), is conrmed in Models IA and IIB. In the others we cannot exclude the possibilitythat bp2 0.

    Following Van Norden (1996), we used the White (1987) method to explore the possibility ofmisspecication of the error terms, testing for autocorrelation, heteroscedasticity, and for Markovianstate dependence in the two regimes ( 3Ct and 3

    St ). We also test the joint hypothesis that the errors satisfy

    all these hypotheses, and refer to it as a test that the errors are well behaved.To test for presence of an AR(1) error process, we have to detect serial correlation in the derivatives

    of the likelihood functionwith respect to bS00 and bC0 in the survival and collapse regimes. Analogously,

    Normal mixture model 5.00 145.08 136.38 17.41 0.38%Restricted normal mixture model 7.00 145.08 135.35 19.47 0.68%

    Model III Rational expectations model 5.00 143.10 129.96 26.27 0.01%Two linear regimes (Van Norden) 2.00 143.10 143.08 0.03 98.62%Two linear regimes w/ same slope 3.00 143.10 138.85 8.49 3.70%Linear regression model 5.00 143.10 137.78 10.63 5.93%Normal mixture model 5.00 143.10 138.01 10.17 7.06%Restricted normal mixture model 7.00 143.10 136.98 12.24 9.30%

    21 We use the Quasi-Maximum Likelihood method (Hamilton, 1994) to estimate the standard errors.

  • Table 4Unrestricted models estimates and tests of the coefcients.

    W.L. Maldonado et al. / Journal of International Money and Finance 31 (2012) 10331059 1047to test for ARCH(1) error process in each regime, we test for serial correlation between the derivativesof that functionwith respect to sS and sC. To test if a state-dependentMarkov-switching regime ismoreappropriate to the data, we investigate if there exists rst order correlation in the series dened by thederivatives with respect to bq0.22 In these tests we adopt a condence level of 1%, because of the sizedistortion of nite samples.

    The results of the specication tests are shown in Table 5. The p-values in the last column indicatethat: (i) there is evidence of AR(1) errors in the collapse regime for all models except IB, and in thesurvival regime only for models IB and IIB, (ii) in both regimes there is no evidence of heteroscedasticerrors in anymodel, (iii) theMarkov property can be rejected formodel IB, (iv) all models, except IA, failthe joint specication test.

    The results of this section can be summarized as follows. Model IA is the best model because itachieves the largest likelihood in the unrestricted model, rejects all the restricted models and acceptsrational expectation, displays sharply dened bubble dynamics, and passes the joint specication testson the errors. Model IIB comes in second in the overall ranking because it displays the second-loweststandard deviation of the errors, in spite of placing third in terms of the value of the log-likelihood ofthe unrestricted model, also rejects all the restrictedmodels, accepts rational expectation, and displaysNull hypothesis bq2 0 bS1 0 bC1 0 bp2 0 bS00 0Model IA Estimate 1.43 (0.02) (0.06) 1.64 (0.12)

    Std. dev. 0.56 0.03 0.02 0.47 0.04p-Value (%) 1.03% 47.55% 0.16% 0.05% 0.65%

    Model IB Estimate 1.40 (0.11) (0.06) 1.15 (0.30)Std. dev. 4.54 0.26 0.18 10.24 0.01p-Value (%) 75.87% 66.93% 75.36% 91.09% 0.00%

    Model IIA Estimate 1.92 (0.01) (0.07) 2.56 (0.07)Std. dev. 3.44 3.07 0.03 2.02 0.28p-Value (%) 57.66% 99.75% 2.36% 20.51% 80.94%

    Model IIB Estimate 1.37 (0.00) (0.06) 2.49 (0.04)Std. dev. 0.49 0.00 0.02 0.78 0.01p-Value (%) 0.55% 55.90% 0.98% 0.14% 0.25%

    Model III Estimate 1.16 (0.02) (1.38) (0.09) (0.09)Std. dev. 0.63 0.04 0.33 0.92 0.34p-Value (%) 6.53% 70.33% 0.00% 92.41% 78.98%sharply dened bubble dynamics.

    4. Heuristic evaluation of the results

    The t of models has been compared in the last section in terms of the estimated equation (22),which relates the innovation to the bubble in the two regimes. However, there is another dimensionover which the models be contrasted: the extent to which they track the level of the exchange rate. Todiscuss this, Figs. 26 display the sample values of the fundamental value of the exchange rate St , thebubble (given by St St ), and the probability of collapse (1 q(bt)) for each model.

    Since the fundamental rate is determined endogenously in our approach, it is instructive to compareits value to spot exchange rate at the end of the sample, which is equal to 1.66 in February of 2011. Theevidence regarding the presence or absence of a speculative bubble in recent data is important becauseit may have policy implications.

    The variant A of models I and II, that use the US-PPI as the foreign price, display a trajectory for thefundamental value which implies a negative bubble of size approximately equal to 0.6 at that date. The

    22 Recall that the model here displays is a particular regime-switching behavior, where the transition probability does notdepend on the state (equation (9), of hypothesis 1). We actually test for indications of Markovian dependence in the dynamicsof the states.

  • W.L. Maldonado et al. / Journal of International Money and Finance 31 (2012) 103310591048Table 5Specication tests.

    Specication Degrees of freedom (n) Value of statistic chi2(n) p-Value

    Model IA AR(1) in the error for the survival regime (S) 1 0.81 36.68%AR(1) in the error for the collapse regime (C) 1 8.44 0.37%ARCH(1) in the error for the survival regime (S) 1 0.05 81.87%ARCH(1) in the error for the collapse regime (C) 1 2.69 10.12%Markovian effect 1 4.25 3.94%Joint test 5 9.87 7.91%

    Model IB AR(1) in the error for the survival regime (S) 1 10.06 0.15%AR(1) in the error for the collapse regime (C) 1 5.11 2.38%ARCH(1) in the error for the survival regime (S) 1 0.33 56.79%ARCH(1) in the error for the collapse regime (C) 1 2.52 11.27%Markovian effect 1 6.74 0.94%Joint test 5 19.57 0.15%

    Model IIA AR(1) in the error for the survival regime (S) 1 1.16 28.16%AR(1) in the error for the collapse regime (C) 1 10.44 0.12%ARCH(1) in the error for the survival regime (S) 1 1.04 30.79%ARCH(1) in the error for the collapse regime (C) 1 2.27 13.16%Markovian effect 1 0.93 33.44%Joint test 5 14.50 1.27%

    Model IIB AR(1) in the error for the survival regime (S) 1 11.44 0.07%AR(1) in the error for the collapse regime (C) 1 7.17 0.74%ARCH(1) in the error for the survival regime (S) 1 0.96 32.65%ARCH(1) in the error for the collapse regime (C) 1 2.68 10.19%variant B of those models, that use a trade-weighted foreign price index, yields very small values forthe end-of-sample bubble. The bubble in Model III is virtually also null at that date. Therefore, themodels with the richer description of the fundamental exchange rate suggest that no signicantchange of exchange rate policy is needed, since nomajor distortions (produced by rational bubbles) aredetected recently.

    In a general sense, as can be seen in Fig. 6, the evolution of the fundamental exchange rate in modelIII more closely follows that of the spot exchange rate, i.e., there exists a higher correlation between thefundaments and the observations. This implies a linear adjustment between the innovation and thebubble. This has been observed earlier, in the discussion of Table 1, when it was noted that the esti-mated value of bp2 is not signicant, and in that of Table 3, when it was pointed out that the two linearregimes hypothesis cannot be rejected.

    Comparison of the models with respect to the general behavior of the bubble indicates that all themodels are very similar in the rst part of the sample. The bubble is positive and increasing until theend of 2002, and then decreases until it becomes null, several years later. For models IA, IIA and III thisoccurs in the third trimester of 2007, and one year later for models IB and IIB.

    The speculative bubble centered in the end of 2002 is easily correlated with the political andeconomic events of that period, and can be associated with the uncertainty with respect to the result ofthe presidential elections in that year and the policies that would be implemented in case the leftistleaning candidatewon the election. Lula da Silva, of the Laborers Party was elected, but the worst fearsof the agents, which had driven the spot exchange rate to almost 4.00 at the end of 2002, did notmaterialize. The policies he implemented were essentially orthodox, especially the monetary policyand, the speculative bubble shrank steadily until it disappeared six years later.

    Markovian effect 1 4.55 3.29%Joint test 5 20.11 0.12%

    Model IIB AR(1) in the error for the survival regime (S) 1 0.75 38.67%AR(1) in the error for the collapse regime (C) 1 26.13 0.00%ARCH(1) in the error for the survival regime (S) 1 2.64 10.40%ARCH(1) in the error for the collapse regime (C) 1 0.23 63.05%Markovian effect 1 0.81 36.77%Joint test 5 31.12 0.00%

    Note: Bold values indicate rejection of the tested hypothesis at the 1% signicance level.

  • Fig. 2. Model IA: spot and fundamental rates versus collapse probability.

    W.L. Maldonado et al. / Journal of International Money and Finance 31 (2012) 10331059 1049After the bubble size becomes null toward the later part of the sample, however, its behavior differsacross models. In models IA and IIA two negative bubbles develop. One starts in third trimester of 2007and lasts for one year, and the other starts in the second trimester of 2009, and lasts to the end of the

    sample. In models IB and IIB only one positive bubble develops after the third trimester of 2008. Inmodel III one negative bubble starts growing in the third trimester of 2007 (like in models IA and IIA)

    Fig. 3. Model IB: spot and fundamental rates versus collapse probability.

  • Fig. 4. Model IIA: spot and fundamental rates versus collapse probability.

    W.L. Maldonado et al. / Journal of International Money and Finance 31 (2012) 103310591050followed by a positive bubble after the third trimester of 2008 (as in models IIA and IIB) that lasts untilthe end of the sample.

    The collapse probability is also displayed in Figs. 26. The general shape of its trajectory is similar inboth variants of models I and II, and is consistent with the description above. At the peak of the bigspeculative bubble (at the end of 2002), the collapse probability reaches 100%, and then decreases toabout 40%, toward the end of the sample. In model III, however, the collapse probability has a very

    different behavior. It is null almost always, except for a period that extends from the rst trimester of

    Fig. 5. Model IIB: spot and fundamental rates and collapse probability.

  • W.L. Maldonado et al. / Journal of International Money and Finance 31 (2012) 10331059 10512001 to the rst trimester of 2005, during the 2002 speculative bubble. It briey reaches 50%, butgenerally remains below 20%. When compared to the bubble of models I and II, the bubble in model IIIis muchmore persistent. This was shown earlier in the discussion of Fig.1, when it was pointed out thatthe probability of survival of the bubble in model III is close to one for jbjh1:2.

    Fig. 6. Model III: spot and fundamental rates and collapse probability.As can be seen in (11), the innovation is the gain of an investor that detains a long position inforward contracts for the exchange rate. The expectation of a positive innovation associated to theoccurrence of a positive bubble in period t 1 implies a positive gain for the agent. Therefore, rational

    Fig. 7. Average error relative to fundamental (absolute value).

  • expectations implies Et[Rt1] 0. The expected innovation in the exchange rate can be calculated bythe following expression, and tested:

    Et Rt1 qbtEt Rt1jS 1 qbtEt Rt1jC (39)As seen in the previous section, rational expectations hypothesis is only rejected in model III.The average error over the sample can be calculated by the identity in equation (20), with the

    sample errors given by equations (19a) and (19b). In Figs. 7 and 8 they are compared, in absolute value,to S* (the fundamental exchange rate) and to 1 jbj for models IIB and III, respectively.

    Fig. 8. Average error relative to 1 bubble (absolute value).W.L. Maldonado et al. / Journal of International Money and Finance 31 (2012) 103310591052Fig. 7 shows that the error is generally less than 10% of the fundamental, except during the 20022003 bubble. Relative to 1 jbj, the average error is also around 10%most of the time, except during the2002 and 2008 bubbles. These two graphs suggest that the t of the model is relatively good.

    5. Conclusions

    In this paper we propose a model of periodically collapsing bubbles which extends the Van Norden(1996) model, and nests it, by considering a non-linear specication for the bubble size in the survivalregime, and the endogenous determination of the level of the fundamental exchange rate. Thesegeneralizations allow us to test for rationality in the formation expectations regarding the level of thestochastic process, which was impossible in the original model, and remove the arbitrariness ofexogenously setting a reference level of the fundamental value, which is another feature of it. We feelthat these are important improvements.

    This general model is applied to the exchange rate of the Brazilian real to the US dollar fromMarch1999 to February 2011, a period characterized by very large variations in its level and its rate of growth.The possibility that these may have been caused by speculative bubbles that may have occurred in thatperiod is considered. Accordingly, the spot exchange rate is modeled as the sum of a fundamental valueand a bubble component.

    The futures exchange rate is used as a proxy of its expected future value, and three differentstructural models, of increasing generality, are considered for the determination of the fundamentalvalue. The rst two (models I and II) are based on the application of the inter-temporal non-arbitrageprinciple to the investment decision of rms engaged in international trade, and imply that theexchange rate satises either purchasing power parity (PPP), or a modied version of it that accounts

  • graduate studies. The reasoning, interpretation and conclusions presented here do not necessarily

    PDt St 1 PDt1 Set1 (A1.1)PFt 1 r PFt1where St is the exchange rate, quoted as the price of foreign exchange in domestic currency, PDt and PFtare the domestic and the foreign prices of the representative tradable good, and r is the riskless interestrate. Note that PDt/PFt is the revenue in domestic currency per unit of foreign exchange invested in theimport operation, while St is the cost.We assume for simplicity there are no import tariffs or other costsof importing.

    Assuming rational expectations, equation (A1.1) can be written as:

    St aEt St1 PDtPFt

    aEt

    PDt1PFt1

    (A1.2)

    where a (1 r)1 is the discount factor. Noting that equation (A1.2) is analogous to equation (2) in themain text, it follows that f is equal to the last 2 terms in the r.h.s.

    Recalling the denition of the fundamental value of the process from equation (4), the fundamentalreect that of the institutions to which the authors are afliated.

    Appendix 1. The determination of the fundamental exchange rateIn this appendix we present three models that suggest the formulation to be used to calculate the

    fundamental solution of the stochastic process proposed in the text for the exchange rate. They arebased on asset approach to the determination of the exchange rate, the principle of absence of risklessarbitrage opportunities in the international markets for goods and capital, uncovered interest rateparity, and the determination of the demand for monetary aggregates.

    Model I: The PPP pricing relation

    Suppose that the balance of trade is in equilibrium in (at least) two periods, and that it involvesa strictly positive ow of tradable goods. Then an agent which is a net importer must be indifferentbetween importing a given good in period t or in period t 1. If he is neutral to risk, then the prot ofdoing it immediately must be equal to the discounted expected future prot of doing it next period: e for the difference in the interest rate of Brazilian government bonds traded domestically and abroad.Two variants of models I and II are tested, varying the foreign price index. Models IA and IIA use the PPI,and models IB and IIB use the trade-weighted foreign producers prices. The third structural model(model III) is a suitably modied version of the monetary model of exchange rate determination. TheMeese (1986) formulation is extended to take into account the interest rate spread that characterizesthe country risk, as measured by the EMBI rate. The model is calibrated to the period of analysis.

    The empirical implementation adequately models the dynamics of the Brazilian foreign exchange inthe period we consider. All ve models are tested along several dimensions, and the overall bestmodels are deemed to be IA and IIB in their unrestricted forms. For all models, except the monetaryexchange rate model (model III), we reject Van Nordens empirical implementation, but are unable toreject rational expectations, at the 5% signicance level.

    In summary,wewereable toconstructamodel for theBrazilian foreignexchangerate for theperiodafterit was allowed to oat that contemplates bubble dynamics and describes well the dynamics of that rate.

    Acknowledgments

    The authors thank Sren Johansen, Simon Van Norden, and an anonymous referee of this journal,for extensive comments on earlier versions, but retain full responsibility of any remaining errors.Maldonados work was supported, in part, by CNPq (Brazil) though grants 304844/2009-8 and 401461/2009-2, and acknowledges that the nal revision was made during his academic visit to the AustralianNational University. Vallis work was supported, in part, by a scholarship granted by CAPES (Brazil) for

    W.L. Maldonado et al. / Journal of International Money and Finance 31 (2012) 10331059 1053exchange rate process is given by:

  • same as the one employed to derive the expression in the l.h.s of (A1.1), adjusted for the fact that when

    W.L. Maldonado et al. / Journal of International Money and Finance 31 (2012) 103310591054the import operation is conducted in period t 1, when the capital of the importer would have beenincreased by the nancial gain of purchasing a foreign bond with a maturity of 1 period yielding1 IF 0t in period t 1.

    Analogously, if the importer delays the operation for one period, the present value of his expectedprot per unit of foreign exchange will be given by:

    1 IF 0t11 IDt1

    PDt2PFt2

    eSet1 (A1.7)

    Equilibrium in international trade with strictly positive quantities of imports in t 1 and t 2 willimply that the expected prot in both strategies be the same. Therefore the expressions in (A1.6) and(A1.7) must be equal under rational expectations:

    1 IF 0t1 IDt

    Et

    PDt1PFt1

    St aEt

    1 IF 0t11 IDt1

    PDt2PFt2

    St1

    (A1.8)

    Rearranging terms,

    St aEt St1 1 IF 0t1 IDt

    Et

    PDt1PFt1

    aEt

    "1 IF 0t11 IDt1

    PDt2PFt2

    #(A1.9)St PNi0

    aiEt f Xti limPTi0

    aiEt

    PDtiPFti

    aEti

    PDti1PFti1

    St PDtPFt

    limT/N

    aT1Et

    PDTt1PFTt1

    (A1.3)which is just the Purchasing Power Parity (PPP) value for the exchange rate minus the present value ofthe expected long-run PPP value. We can discard this last term, assuming that the present value of thedomestic currency in terms of the foreign currency tends to zero:

    limT/N

    aTEt

    PDTt1PFTt1

    0 (A1.4)

    This is a familiar transversality condition in dynamic models: in essence it implies that the relativeprice will not increase faster than the discount factor, as the terminal time increases without bound.

    Using (A1.4) in (A1.3), we obtain the PPP pricing equation.

    St PDtPFt

    (A1.5)

    Model II: Modied PPP pricing relation

    In this model we will consider as exogenous variables not only the price levels in the country andabroad, but also the respective nominal interest rates. The arbitrage argument is similar to the oneemployed in the previous model.

    Suppose that in a period t an importer is evaluating an operation to be undertaken either in periodt 1 or t 2. He must be indifferent between these two dates.

    If he decides to do it in period t 1, then the present value of the expected prot for each unit offoreign currency is given by:

    1 IF 0t1 IDt

    PDt1PFt1

    eSt (A1.6)

    where IDt and IF 0t are the nominal interest rates observed in the domestic and foreign markets forBrazilian government bonds, respectively, and the other variables are as before. The reasoning is the

  • The domestic and foreign interest rates (ID and IF) are the rates paid by the country and abroad onIDt IFt Et st1 st ISt (A1.16)The interest rate spread (IS) is a measure of the risk premium assigned by the market to foreign

    investments in the country, and is often denominated country risk. It reects not only risks associatedwith the performance of the domestic economy, but also the supply of capital in the internationalcapital markets. This variable is very important in explaining the interest rate differential between theBrazil and the U.S. in the sample considered in the empirical analysis of this paper. The interest rate ongovernment debt.The model has three equations which are specied below. The deterministic terms (a constant and

    seasonal dummies) are suppressed in the theoretical model specication, to simplify the analysis, butare included in the empirical model.

    Assume a log-linear demand for real money balances which accounts for transactions and specu-lative motives for holding money:

    mt pt a1yt a2IDt IFt (A1.15)where a1 > 0 is the income elasticity and a2 > 0 is the interest semi-elasticity of real money balances.Following customary practice in specifying the monetary model of exchange rates, a disturbance termis omitted from the money equation.

    An extended uncovered interest parity relation is assumed, where the difference between domestic(ID) and foreign (IF) interest rates is equal to the expected devaluation plus the interest rate spreadbetween foreign-denominated domestic bonds and foreign bonds of similar maturity, denoted IS.Equation (A1.9) has the same form as equation (2) in the main text, and the function f is equal to thelast two terms in the r.h.s. Again the fundamental solution can be found using equation (4):

    St X

    aiEt f Xti (A1.10)

    St 1 IF 0t1 IDt

    Et

    PDt1PFt1

    lim

    T/NaT1Et

    1 IF 0t1 IDt

    PDTt1PFTt1

    (A1.11)

    Therefore, the fundamental value for the exchange rate is the expected future value of the PPPadjusted by the spread of the nominal interest rate, minus a residual termwhich represents the presentrevenue value of investing one unit of foreign currency in period t during T t periods. As in theprevious model, we will assume that this residual vanishes as T goes to innity:

    limT/N

    aTEt

    1 IF 0Tt1 IDTt

    PDTt1PFTt1

    0 (A1.12)

    Using (A1.11) and (A1.12) we conclude that the fundamental value for the exchange rate is given bythe PPP adjusted for the interest rate differential:

    St 1 IF 0t1 IDt

    Et

    PDt1PFt1

    (A1.13)

    Model III: The monetary model of exchange rate determination

    In this model we consider a modied version of the Meese (1986) monetary model of exchange ratedetermination. As in his model, the variables are specied in logs of the ratio of the domestic aggregateto the foreign aggregate, for the money stock (MD andMF), prices (PD and PF), and income (YD and YF).The exchange rate (S) is also specied in logs.

    mt lnMD=MF; pt lnPD=PF; yt lnYD=YF; st lnSt (A1.14)

    W.L. Maldonado et al. / Journal of International Money and Finance 31 (2012) 10331059 1055Brazilian government bonds denominated in US dollar used in Model II is, therefore, IF 0t IFt ISt .

  • Lastly, assume that the deviations from PPP are a random walk.

    st pt ut ; ut ut1 3t (A1.17)where 3t is white noise, and ut is the real exchange rate.

    Substituting the interest rate differential in (A1.15) for its value in (A1.16) yields:

    mt pt a1yt a2Et st1 st IStUsing (A1.17) to substitute for pt in the above equation,

    mt st a1yt a2Et st1 st a2ISt utCollecting terms,

    st gEt st1 1 gmt a1yt a2ISt 1 gut (A1.18)where g a2=1 a20 1 g 1=1 a2. This is the structural equation for the dynamics of thelog of the exchange rate.

    Following Meese (1986), assuming that st follows a borderline stationary process, we rely on rstdifference of (A1.18) for empirical applications:

    Dst 1 gDmt a1Dyt a2DISt gEt st1 Et1st 1 g 3t (A1.19)For notational simplicity, dene the market fundamental process Dxt variables:

    DxthDmt a1Dyt a2DISt cDxt1 dt ; jcj < 1 (A1.20)Equation (A1.19) can be solved forward to yield:

    Dst 1 gXT1s0

    gsEt xts Et1xt1s gT Et xtT Et1xtT1 1 gXT1s0

    gs 3ts

    If the transversality condition holds

    limT/N

    gT Et xtT Et1xtT1 0 (A1.21)

    then the unique, no bubbles solution to (A1.19) is:

    Dst 1 gXNs0

    gsEt xts Et1xt1s 3t (A1.22)

    From (A1.20) the optimal prediction formula for xts is:

    Et xts xt Xsk1

    ckDxk (A1.23)

    Then, the rational expectations solution to (A1.19) can be shown to be:

    Dst Dxt gc1 gc Dxt Dxt1 3t (A1.24)

    Recalling that the error 3t is white noise, the changes to the fundamental value of the log of theexchange rate can be calculated as:

    Dst Dxt gc

    1 gc Dxt Dxt1 (A1.25)

    Appendix 2. The statistics for tests of the modelThis section describes formally the statistics used to test the several models presented in the text.

    C Likelihood ratio test

    The Likelihood Ratio test (LR test) is one of the most popular approaches for testing hypotheses on

    W.L. Maldonado et al. / Journal of International Money and Finance 31 (2012) 103310591056parameter restrictions, when they are estimated by the maximum likelihood method. Typically, the

  • W.L. Maldonado et al. / Journal of International Money and Finance 31 (2012) 10331059 1057hypotheses are restrictions on the parameters of the model, which are represented by a linear systemof equations with these parameters. The LR statistics is dened by the expression in (A2.1):

    2hLq^ L

    ~qi

    (A2.1)

    where q^~q is the unrestricted (restricted) maximum likelihood estimator of the parameters and L is thelog-likelihood function. It is proved that this statistics has a c2m probability distribution, wherem isthe rank of the linear system, which denes the restrictions. See Hamilton (1994) for details on the LRtest.

    C Tests on the standard errors of the parameters

    We used the tests based on the quasi-maximum likelihood estimation proposed by White (1982),which uses the matrix T1^I2D I^

    1OP I^2D as an approximation of the variancecovariance of maximum

    likelihood estimates.In this approximation the second-derivative estimate of the information matrix is given by (A2.2):

    I^2D T1v2Lvq vq0

    qqq^

    (A2.2)

    and I^OP is the outer-product estimate of the same information matrix.

    I^OP T1XTt1

    grad Lt!q^05grad Lt!q^ (A2.3)

    where Lt represents the tth term of the log-likelihood function Lq PT

    t1 Ltq, and where thegradient line vector grad Lt

    !q^ is the vector of scores of the tth observation.C Specication tests

    The specication tests of regime-switching time series models allow us to test for omitted auto-correlations, omitted ARCH and Markovian effects on errors. These are described in detail in Hamilton(1996), that presents specication tests for Markov-switching models based on special properties ofthe vector of scores (dened above). The main property is that the vector series of scores have zerotime-correlation when evaluated at the true parameters, namely:

    Egrad Lt!q05grad Lt1!q 0 for t 2;3;.; T (A2.4)

    As proposed by White (1987), if Ct(q) is any k-dimensional line vector whose coordinates areelements of the corresponding outer-product matrix of scores (inside the expectation above) and if themodel is correctly specied, then:"

    T1=2XTt1

    Ctq^#B^hT1=2

    XCtq^

    i0zc2k (A2.5)

    where B^ is the (2,2) sub-block of the inverse of the following partitioned matrix:

    bA T126664PTt1

    grad Lt!bq05grad Lt!bq PT

    t1

    grad Lt!bq05Ctbq

    PTt1

    Ctbq05grad Lt!bq PCtbq05Ctbq

    37775 (A2.6)In our analysis we use six specications for the vector Ct(q): two for AR(1)s (one for each regime)

    which correspond to the derivatives of the log-likelihood with respect to bS00 and bC0, two forARCH(1)s (one for each regime) which correspond to the derivatives with respect to s and s , one forS C

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    Kiefer, N.M., 1978. Discrete parameter variation: efcient estimation of a switching regression model. Econometrica 46, 427434.

    Meese, R., 1986. Testing for bubbles in exchange markets: a case of sparkling rates? Journal of Political Economy 94 (2), 345373.

    Obstfeld, M., Rogoff, K., 1983. Speculative hyperinations in maximizing models: can we rule them out? Journal of PoliticalEconomy 91 (4), 675687.

    Perron, P., 1997. Further evidence on breaking trend functions in macroeconomic variables. Journal of Econometrics 80 (2), 355385.

    Tirole, J., 1982. On the possibility of speculation under rational expectations. Econometrica 50 (5), 11631180.

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    Appendix 3The results of the PERRON97 procedure or RATS 7.0 (Estima, 2007) are presented in Table B1.

    References

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    Economic and Financial Structure. D.C. Heath and Company, Lexington, MA, pp. 295316.Brooks, C., Katsaris, A., 2005. A three-regime model of speculative behaviour: modelling the evolution of bubbles in the S&P 500

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    Table B1The results of the PERRON97 procedure or RATS 7.0 (Estima, 2007).

    Break date TB 2005:05 statistic t(alpha 1) 3.79603Critical values at 1% 5% 10% 50% 90% 95% 99%

    For 100 obs. 6.21 5.55 5.25 4.22 3.35 3.13 2.63Innite sample 5.57 5.08 4.82 3.98 3.25 3.06 2.72

    Number of lag retained: 12.

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    World Congress of the Econometric Society, vol. II. Cambridge University Press, Cambridge.

    W.L. Maldonado et al. / Journal of International Money and Finance 31 (2012) 10331059 1059

    Exchange rate bubbles: Fundamental value estimation and rational expectations test1. Introduction2. The stochastic process with bubbles3. The empirical model3.1. Data3.2. Estimation3.3. Tests

    4. Heuristic evaluation of the results5. ConclusionsAcknowledgmentsAppendix 1. The determination of the fundamental exchange rateModel I: The PPP pricing relationModel II: Modified PPP pricing relationModel III: The monetary model of exchange rate determination

    Appendix 2. The statistics for tests of the modelAppendix 3References

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