JOURNAL OF APPLIED ECONOMETRICS J. Appl. Econ. 14: 143±154 (1999)

DETECTING PERIODICALLY COLLAPSING BUBBLES: A MARKOV-SWITCHING UNIT ROOT TEST STEPHEN G. HALL,a,b ZACHARIAS PSARADAKISc* AND MARTIN SOLAc,d aManagement

School, Imperial College of Science, Technology and Medicine, 53 Prince's Gate, London SW7 2PG, UK for Economic Forecasting, London Business School, Sussex Place, London NW1 4SA, UK cDepartment of Economics, Birkbeck College, 7±15 Gresse Street, London W1P 2LL, UK d Departamento de EconomõÂa, Universidad Torcuato di Tella, Minores 2159/77, Buenos Aires, Argentina bCentre

SUMMARY This paper addresses the problem of testing for the presence of a stochastic bubble in a time series in the case that the bubble is periodically collapsing so that the asset price keeps returning to the level implied by the market fundamentals. As this is essentially a problem of identifying the collapsing periods from the expanding ones, we propose using a generalization of the Dickey±Fuller test procedure which makes use of the class of Markov regime-switching models. The potential of the new methodology is illustrated via simulation, and an empirical example is given. Copyright # 1999 John Wiley & Sons, Ltd.

1. INTRODUCTION This paper addresses the problem of testing for the presence of periodically collapsing rational bubbles in economic time series, a problem which has attracted considerable attention in recent years. A popular testing strategy is based on investigating the integration and cointegration properties of (say) asset prices and observable underlying fundamentals, as the existence of an explosive bubble would imply that prices are more explosive than the fundamentals. However, unit root and cointegration tests of this type are known to have little power to detect bubbles that collapse periodically, even in cases where the bubble is substantial both in magnitude and in volatility (cf. Evans, 1991). It is argued here that, since the problem of testing for collapsing bubbles is essentially one of identifying the expanding periods from the collapsing ones, a test for bubbles should allow for the possibility of changes in the dynamic behaviour of asset prices during the sample. In particular, we propose an approach based on a generalization of the so-called Augmented Dickey± Fuller (ADF) unit root test which makes use of the class of dynamic Markov-switching models explored in Hamilton (1989, 1990). By allowing the ADF regression parameters to switch values between di€erent regimes, the ADF formulation can cope e€ectively with the dynamics generated by a collapsing bubble process. Within this framework, the existence of a rational bubble is consistent with one of the regimes that control the process of interest being characterized by the presence of an explosive autoregressive root. Although such a test constitutes only an indirect test of the presence of a bubble, we will argue that it overcomes some of the drawbacks of conventional unit root tests previously considered in the literature. * Correspondence to: Zacharias Psaradakis, Department of Economics, Birkbeck College, 7±15 Gresse Street, London W1P 2LL, UK. E-mail: [email protected]; fax: ‡44 171 6316416. Contract grant sponsor: Economic and Social Research Council.

CCC 0883±7252/99/020143±12$17.50 Copyright # 1999 John Wiley & Sons, Ltd.

Received 2 August 1996 Revised 15 January 1998

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The plan of the paper is as follows. In Section 2 we show how the standard ADF testing strategy may be generalized to allow for the possibility of Markov-switching in the structure of the series. Section 3 discusses the problem of detecting stochastic bubbles and draws attention to the key paper by Evans (1991). The data-generating processes (DGPs) of Evans (1991) are then used to illustrate, by means of simulations, the potential of the switching ADF test. Section 4 presents an application of the methodology to the case of Argentinian hyperin¯ation. Some conclusions are o€ered in Section 5. 2. A MARKOV-SWITCHING UNIT ROOT TEST A popular approach to constructing tests of the hypothesis that the stochastic component of the time series fyt gntˆ1 is characterized by the presence of a unit autoregressive root is based on regression models of the form Dyt ˆ m ‡ fytÿ1 ‡

k X j ˆ1

cj Dytÿj ‡ t

…1†

where D is the ®rst-di€erence operator (Dyt ˆ yt ÿ ytÿ1 †, ft g is a zero-mean white noise, and k is a suitably chosen integer (see Dickey and Fuller, 1981; SaõÈ d and Dickey, 1984).1 The coecient on ytÿ1 provides the basis for testing the null hypothesis of an autoregressive unit root in fyt g, and the ADF statistic is the ordinary least-squares t-statistic associated with f ˆ 0. In this section we show how the ADF test procedure can be generalized to allow for the possibility that the dynamic behaviour of fyt g might be di€erent for di€erent parts of the sample. In particular, we consider the case where the parameters of the autoregression in equation (1) can be in¯uenced by the state (or regime) that the process is in at any given date. One can think of such a formulation in the speci®c context of periodically collapsing rational bubbles, where the motivation for two separate regimes, re¯ecting the expanding and collapsing phases of the bubble, is quite obvious. More generally, it can be thought of as a generalization of the speci®cations analysed in Perron (1990) and Perron and Vogelsang (1992) where the structure of the series was assumed to be subject to a one-time exogenous break in mean. Here it is assumed that the parameters governing equation (1) are time-varying, changing with an unobserved indicator st 2 f0; 1g so that Dyt ˆ m0 …1 ÿ st † ‡ m1 st ‡ ‰f0 …1 ÿ st † ‡ f1 st Šytÿ1 ‡

k X ‰c0 j …1 ÿ st † ‡ c1 j st ŠDytÿj ‡ se et

…2†

j ˆ1

where fet g is a sequence of independent and identically distributed (i.i.d.) random variables with zero mean and unit variance.2 Following Goldfeld and Quandt (1973) and Hamilton (1989, 1990), nature is assumed to select regime at date t with a probability that depends on what regime

1

When the DGP for fyt g is a ®nite-order autoregressive process, k ‡ 1 must be set at least equal to the true order of the process. If, however, a moving average component is present in the DGP, k needs to increase with the sample size at a controlled rate (n1=3 ). 2 Theoretically, the analysis can be easily extended to any ®nite number of regimes. Copyright # 1999 John Wiley & Sons, Ltd.

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the process was in at date t ÿ 1. The random sequence {st} is thus speci®ed to be a homogeneous Markov chain on the state space {0,1} with transition probabilities: Pr…st ˆ 1 j stÿ1 ˆ 1† ˆ p Pr…st ˆ 0 j stÿ1 ˆ 1† ˆ 1 ÿ p Pr…st ˆ 0 j stÿ1 ˆ 0† ˆ q

…3†

Pr…st ˆ 1 j stÿ1 ˆ 0† ˆ 1 ÿ q We further require that the innovations fet g be independent of the state variables fsm g for all t and m. The speci®cation in equations (2) and (3) generalizes the linear model (1) by allowing the model's parameters to be functions of the stochastically chosen regime that controls the process at date t. Moreover, the probability law that governs these regime changes is speci®ed to be ¯exible enough to encompass a wide range of empirically plausible outcomes, allowing the data to determine the speci®c form of parameter non-constancy that is consistent with the sample information. Test criteria for the null hypothesis of a unit root in either regime (that is, f0 ˆ 0 and/or f1 ˆ 0) may be constructed using likelihood-based procedures. In particular, assuming that fet g are Gaussian random variables, inference in equation (2) can proceed by making use of a recursive, non-linear ®ltering algorithm similar to that described in Hamilton (1994, pp. 692± 694). This is designed to produce probabilistic inferences of the form Pr(st ˆ 1 j I t , W ), where I t ˆ fy1 ; . . . ; yt g and W ˆ …m0 , m1 , f0 , f1 , se , c01 , . . ., c0k , c11 , . . ., c1k)0 , but gives, as a byproduct, the likelihood for the observed data. The value W^ of W that maximizes this likelihood can then be found by means of a numerical optimization algorithm. By analogy to the standard ADF statistic, a test of the unit root null hypothesis may be based on the (asymptotic) t-ratios associated with the maximum likelihood estimates of f0 and f1 . Moreover, once W^ is obtained, probabilistic inferences about the unobserved regimes fst g may be made on the basis of the `®lter' probabilities Pr(st ˆ 1 j I t , W^ ) or the `smoothed' probabilities Pr(st ˆ 1 j I n , W^ ) (see Hamilton, 1989, 1994 for details). 3. EXPLOSIVE RATIONAL BUBBLES 3.1.

Model and Tests

The standard approach to testing for the presence of rational bubbles has been based on the formulation and estimation of a complete parametric speci®cation of the bubble process (see, for instance, Flood and Garber, 1980; Flood et al., 1984). However, such direct tests are susceptible to the criticism that they are unable to detect bubbles other than those that belong to the speci®c parametric class under consideration, so failure to reject the no-bubble hypothesis does not necessarily imply the absence of other unspeci®ed types of bubbles. In this paper, we are interested in a class of indirect tests which does not make use of an explicit parametric bubble speci®cation. These procedures, put forward by Diba and Grossman (1984, 1988) and Hamilton and Whiteman (1985), are based on checking the order of integration of a given pair of variables.3 If prices (say stock prices or in¯ation) are not more explosive than the 3

Other notably indirect procedures include those based on variance-bounds tests (e.g. Shiller, 1981; LeRoy and Porter, 1981), speci®cation tests (e.g. West, 1987; Durlauf and Hooker, 1994; Hooker, 1996), and regime-switching regressions (van Norden, 1996).

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S. G. HALL ET AL.

relevant driving fundamental variable (say dividends or money expansion), then it can be concluded that rational bubbles are not present since, if they were, they would generate an explosive component in the respective prices. Such hypotheses have been typically tested using autocorrelation patterns, unit root tests, and cointegration tests.4 However, this indirect procedure is not always reliable. In the majority of applications, part of the diculty is that the stationarity properties of the relevant time series are analysed by testing the null hypothesis of a unit root in the levels and di€erences of the series against one-sided stationary alternatives rather than the more relevant explosive ones. Although such tests should, in theory, be capable of revealing the existence of a rational bubble (since bubbles imply that di€erencing of prices will not be sucient to induce stationarity), this is not an easy task in samples of the size that is typical for many applications. In small samples, series with explosive bubble components could look very much like stationary processes when di€erenced a sucient number of times. In addition to these diculties, Evans (1991) has demonstrated that the use of standard unit root and cointegration tests for prices and underlying fundamentals can erroneously lead to acceptance of the no-bubble hypothesis for an important class of rational bubbles that collapse periodically. In such cases, even a direct test for explosive behaviour in the levels of the relevant series may fail to detect the bubble since such a test tends to have low power. We will discuss Evans' analysis at some length here as we will use this later as the basis for our own numerical experiments to illustrate the potential of the test procedure outlined in Section 2. Evans' theoretical model consists of the following equations: ÿ1

Pt ˆ …1 ‡ r† Et …Pt‡1 ‡ dt‡1 †

…4†

1 X ÿj …1 ‡ r† Et dt‡j

…5†

Ft ˆ

j ˆ1

ÿ1

Bt ˆ …1 ‡ r† Et Bt‡1

…6†

Pt ˆ Ft ‡ Bt

…7†

where Pt is the real stock price at date t, dt‡1 is the real dividend paid to the owner of the stock between dates t and t ‡ 1, r is a constant real interest rate appropriate for discounting expected capital gains, fBt g is any sequence of random variables that satis®es the homogeneous expectational di€erence equation (6), and Et denotes conditional expectation with respect to the information set of market participants at date t. Equation (4) describes a standard ecientmarkets model for stock prices, while (5) de®nes the forward-looking solution to (4); equation (7) gives the general solution to (4) as the sum of a market-fundamentals component (Ft ) and a rational-bubble component (Bt ) .

4

Of course, a potential problem with both direct and indirect tests is the observational equivalence between bubbles and expected future changes in the driving fundamentals; a testing procedure that overcomes this diculty is explored in Blackburn and Sola (1996). Copyright # 1999 John Wiley & Sons, Ltd.

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DETECTING PERIODICALLY COLLAPSING BUBBLES

147

Since a rational bubble is de®ned only via equation (6), there are, in principle, in®nite functional forms which satisfy this di€erence equation. Evans describes an empirically plausible class of bubbles which are rational, positive, and periodically collapsing. They take the form: 8 …1 ‡ r†Bt ut‡1 if Bt 4 a <    …8† Bt‡1 ˆ 1‡r d : d‡ Bt ÿ xt‡1 ut‡1 if Bt 4 a p 1‡r where d and a are real scalars such that 0 5 d 5 …1 ‡ r†a, fut g is a sequence of non-negative exogenous i.i.d. random variables with Et ut‡1 ˆ 1, and fxt g is an exogenous i.i.d. Bernoulli process, independent of fut g, such that Pr(xt ˆ 0† ˆ 1 ÿ p and Pr(x ˆ 1† ˆ p …0 5 p 4 1). Hence, the bubble has two di€erent rates of growth: if Bt 4 a, it grows at mean rate 1 ‡ r; when eventually Bt 4 a, the bubble grows at the faster mean rate (1 ‡ r†=p, but collapses with probability 1 ÿ p per period. Evans' basic argument is that integration and cointegration tests are likely to have some power to detect a rational bubble only when the latter lasts for most of the period under investigation. In theory, a deterministic bubble will continue to in®nity, and integration tests should clearly provide a good way of detecting such an event. Nevertheless, the real world cannot be characterized by deterministic bubbles; if bubbles exist, they must be stochastic, or periodic, in some way, so that periods of expansion will eventually be followed by a collapse or contraction (cf. equation (8)). Evans' point is that, despite having explosive conditional means, such collapsing bubbles will appear to standard unit root tests as stationary processes, and tests (with either stationary or explosive alternatives) will have little or no power to detect the bubbles. He demonstrates this by constructing a DGP with an explosive stochastic bubble which expands and contracts, and by showing that standard unit root tests are heavily biased towards stationarity of fDPt g.5 The basic problem encountered in these situations is that rational bubbles of the type described by equation (8) only exhibit characteristic bubble behaviour during their expansion phase. As a consequence, a test is more likely to ®nd evidence of systematic divergence between asset prices and fundamentals if it is based only on data points that are associated with the expansion phase of the bubble. We propose identifying such periods by making use of the methodology discussed in Section 2. Within the Markov-switching ADF framework, the existence of an explosive rational bubble in prices is consistent with f0 4 0 or f1 4 0, indicating that one of the regimes governing the process of interest is characterized by the presence of an explosive autoregressive root. Evidence, on the other hand, that f0 ˆ f1 ˆ 0 for fPt g and fdt g is inconsistent with the existence of rational bubbles for the reasons explained in Diba and Grossman (1988). 3.2.

A Simulation Study

To investigate the ability of the proposed procedure to detect the presence of periodically collapsing rational bubbles, we carry out some Monte Carlo experiments. All computations are based on 500 independent realizations of fPt g from DGPs identical to those considered by Evans (1991). These are de®ned by equations (5), (7) and (8). The equation describing the evolution of

5

Charemza and Deadman (1995) show that similar problems arise for the class of multiplicative rational bubbles with stochastic explosive roots.

Copyright # 1999 John Wiley & Sons, Ltd.

J. Appl. Econ. 14: 143±154 (1999)

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S. G. HALL ET AL.

Table I. Data-generating processes DGP1

Equation (9) a d s2z s2e c r d0 B0 p k n

DGP2

Dividends

1 0.5 0.0025 0.1574 0.0373 0.05 1.3 0.5 0.85 20 100

Equation (10) 1 0.5 0.0025 0.016 0.013 0.05 0.26 0.5 0.85 250 100

the fundamentals is derived by solving (5) under two alternative assumptions about the generating mechanism of real dividends, namely dt ˆ c ‡ dtÿ1 ‡ et

…9†

ln dt ˆ c ‡ ln dtÿ1 ‡ et

…10†

and

where fet g are i.i.d. N …0; s2e † random variables. Moreover, fut g in equation (8) are generated according to ut ˆ exp…zt ÿ s2z =2), where fzt g are i.i.d. N …0; s2z † random variables independent of fet g. The parameter values used for the experiments are listed in Table I, where k is scaling factor for the arti®cial bubbles, chosen as in Evans (1991). For each realization from the DGP, we ®t model (2) with k ˆ 4 to fPt g, where we normalize by setting f1 4 f0 .6 The unit root tests are then based on the t-statistics associated with f0 ˆ 0 and f1 ˆ 0, with covariance matrix estimates obtained from the negative of the Hessian of the loglikelihood function evaluated at the optimum. Since the null distribution of such statistics is unknown, we resort to using simulated critical values for the tests. These are calculated by parametrically bootstrapping the null model (corresponding to f0 ˆ f1 ˆ 0) using the estimates of (m0 , m1 , se , c01 , . . ., c0k , c11 , . . ., c1k) obtained for the particular realization of fPt g. The number of bootstrap replications is 199; this makes the simulations extremely computer-intensive for they entail a total of 99,500 replications per experiment. The results are summarized in Table II which shows the proportion of replications in which the null hypotheses f0 ˆ 0 and f1 ˆ 0 are rejected, at the 5% level, against the one-sided alternative f0 5 0 and f1 4 0, respectively.7 For comparison, we also show the percentage rejection frequencies of conventional ADF tests based on equation (1) with k ˆ 4; the critical values for these tests are taken from Hylleberg and Mizon (1989).8 It is clear that the Markov-switching 6

The value of k was chosen to re¯ect the situation encountered in the empirical application discussed in Section 4. Notwithstanding that the alternative f0 4 0 is more relevant than f0 5 0, we have chosen to consider the latter since the estimated f0 is negative in the majority of Monte Carlo replications. 8 Very similar results are obtained using standard Dickey±Fuller critical values from Table B.6 in Hamilton (1994). 7

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DETECTING PERIODICALLY COLLAPSING BUBBLES

Table II. Percentage rejections of ADF and switching ADF tests t-test for

DGP1 DGP2

f0 ˆ 0 versus f0 5 0

f1 ˆ 0 versus f1 4 0

fˆ0 versus f50

fˆ0 versus f40

32.8 33.2

76.8 74.2

4.40 82.2

7.80 2.20

ADF procedure has considerable power to detect the presence of bubbles in fPt g. For either DGP, the tests correctly indicate the existence of an explosive regime (associated with st ˆ 1) and a homogeneously nonstationary regime (associated with st ˆ 0) in more than 65% of the cases. In sharp contrast, conventional ADF tests fail to reveal any explosiveness in the simulated price series.9 These results do not, of course, imply that switching ADF tests would successfully detect all types of periodically collapsing bubble components. For example, if the contribution of the bubble to the volatility of prices is not substantial or the probability of the bubble collapsing (1 ÿ p) is relatively large, it would be dicult, for any test, to con®rm the presence of the bubble. However, if the excess volatility that is observed in real-world price series were attributable to rational bubbles, these would typically be longer lasting and more pronounced (in magnitude and volatility) than the bubbles considered in our experiments. In such cases, switching ADF tests for prices and observable fundamentals will be powerful enough to reveal the existence of bubbles. 4. AN APPLICATION To illustrate the ideas presented in this paper, we analyse the integration properties of the monetary base, consumer prices, and exchange rate (in terms of the US dollar) in Argentina.10 The data set consists of 82 monthly observations from 1983:1 to 1989:11. During this period, the Argentinian economy experienced several episodes of hyperin¯ation (in 1985, 1988 and 1989),11 and the ruling Radical Party implemented at least three stabilization plans (Austral I and II, and Primavera). Our objective here is to investigate whether the non-stationarity of consumer prices may be attributable to explosive rational bubbles. Since the time-series properties of the underlying economic fundamentals (i.e. money supply) are unknown, it is essential that these be considered along with the properties of prices.12 The reason for including a third series, namely the exchange rate, in the analysis is that it is often dicult to distinguish between a rational bubble in prices and fundamentally determined bubble-like behaviour. One way of overcoming this diculty is to 9

A Monte Carlo analysis of the performance of the switching ADF test is also presented in van Norden and Vigfusson (1996). Their results are not, however, directly comparable to ours since they apply the tests to arti®cial data for fBt g rather than fPt g. More importantly, their tests are carried out using Gaussian critical values the validity of which is questionable. 10 An application of the methodology to the case of the Polish hyperin¯ation can be found in Funke et al. (1994). 11 We use th term `hyperin¯ation' loosely here since the episodes in questions are not always consistent with de®nitions of hyperin¯ation such as Cagan's (1956). 12 This approach was not taken up in the simulations of Section 3.2 since the underlying fundamentals (dividends) were known to evolve according to a random walk. Copyright # 1999 John Wiley & Sons, Ltd.

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consider the behaviour of a pair of series that are driven by common fundamentals. The natural partner for prices is the foreign-exchange rate since, both series are primarily driven by monetary expansion during a hyperin¯ation. Then, evidence of a simultaneous change in the behaviour of both prices and exchange rates would suggest that the nonstationarity in prices is attributable to their market-fundamentals component. If, on the other hand, the changes in behaviour are not synchronous across the two series, the likely explanation must be the presence of a rational bubble. We begin by considering conventional unit root tests for the natural logarithms of the three series. These include the ADF test as well as the Z…^a† and Z…ta^ † † tests of Phillips and Perron (1988). The last two tests are used in conjunction with a lag-window spectral estimate at zero based on the quadratic-spectral window, with the bandwidth chosen using the data-driven procedure discussed in Andrews and Monahan (1992). For the ADF test, the autoregressive truncation lag k in equation (1) is selected from the range 0 4 k 4 5 using the sequential (10%-level) t-tests for the signi®cance of the coecient on the longest lag recommended in Ng and Perron (1995).13 The results of these tests are summarized in Table III. It is clear that, when the alternative hypothesis is that of stationarity, the tests provide no evidence against the unit root hypothesis for any of the three series. Using, on the other hand, right-tailed tests, the unit root null can be rejected (at the 5% level) in favour of an explosive alternative for all series. This result, however, does not necessarily imply the existence of rational bubbles in consumer prices or the exchange rate since the monetary base also exhibits explosive behaviour over the whole sample. To investigate further, we ®t the Markov-switching model (2) with k ˆ 4 to the logarithmically transformed Argentinian time series.14 The results are shown in Table IV, where the ®gures in square brackets are the p-values of t-type tests of the null hypotheses f0 ˆ 0 and f1 ˆ 0 against the one-sided alternative f0 5 0 and f1 4 0, respectively. These were obtained by bootstrapping the null model as in Section 3.2, using 499 replications. For all three series a two-regime speci®cation appears to be consistent with the data. In the regime represented by st ˆ 0, the estimate of f0 is negative, and the unit root hypothesis cannot be rejected. The regime represented by st ˆ 1 is, on the other hand, characterized by positive values of f1 , and the unit root hypothesis can be rejected in favour of an explosive alternative for all three series. Such explosive episodes will typically persist for around 5 months for prices, as Table III. Unit root tests ADF [k] Raw data Money Exchange rate Prices First-di€erenced data Money Exchange rate Prices

1.058 [1] 0.749 [4] 0.339 [2] ÿ5.017 [0] ÿ4.739 [3] ÿ4.115 [1]

Z(^a) 1.201 0.392 ÿ0.026 ÿ34.58 ÿ32.96 ÿ29.73

Z(ta^ ) 1.326 0.342 ÿ0.019 ÿ4.887 ÿ4.593 ÿ4.044

13 A linear time trend is not included in any of the test regressions since its coecient is not signi®cant on the basis of either Gaussian critical values or the critical values given in Dickey and Fuller (1981, Table III). 14 The value of k is chosen using a data-dependent method analogous to that employed in the case of the ADF test.

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DETECTING PERIODICALLY COLLAPSING BUBBLES

Table IV. Maximum likelihood estimates of parameters for Markov-switching ADF regressionsa Money f0

ÿ0.006

m0 c01 c02 c03 c04 f1

ÿ0.151 0.142 0.171 0.086 0.083 0.038

m1 c11 c12 c13 c14 se p q

0.127 0.244 ÿ0.606 2.584 ÿ1.101 0.071 0.640 0.972

aFigures

Exchange rate (0.004) [0.14] (0.289) (0.082) (0.076) (0.071) (0.064) (0.018) [0.03] (0.060) (0.136) (0.170) (0.397) (0.402) (0.006) (0.206) (0.020)

ÿ0.007 0.040 0.380 0.122 0.073 ÿ0.264 0.130 0.105 ÿ0.393 0.051 ÿ0.320 ÿ1.747 0.049 0.740 0.935

(0.003) [0.09] (0.092) (0.054) (0.047) (0.066) (0.064) (0.009) [0.00] (0.019) (0.063) (0.587) (0.436) (0.103) (0.004) (0.130) (0.033)

Prices ÿ0.013 ÿ0.321 0.381 ÿ0.037 0.012 0.069 0.069 0.082 1.009 0.316 ÿ0.722 0.296 0.043 0.786 0.968

(0.012) [0.22] (0.106) (0.056) (0.064) (0.065) (0.051) (0.039) [0.04] (0.009) (0.250) (0.406) (0.554) (0.518) (0.004) (0.129) (0.023)

in parentheses are asymptotic standard errors and those in square brackets are p-values of unit root tests.

opposed to 4 months and 3 months for exchange rates and money supply, respectively. The average duration of the non-explosive regime (st ˆ 0) is approximately 31, 15 and 36 months for prices, exchange rates and money supply, respectively. The inferred ®lter probabilities of being in the explosive regime (st ˆ 1) at each point of the sample are shown in Figure 1, together with plots of the ®rst-di€erenced series. In all cases there is a clear switch to the explosive regime in 1989:4. Since, however, this episode is common to all three series under study, the likely explanation for the 1989 hyperin¯ation is the rapid growth in the money supply. The ®lter probabilities also identify the period 1988:6±1988:8 as associated with explosive prices (Pr(st ˆ 1 j I t , W^ † 4 0
5). Such explosive behaviour is not present in either the money supply or the exchange rate series, a ®nding which points towards the existence of a rational bubble in consumer prices during the period in question. Finally, a bubble appears to be present in the exchange rate series in 1984±1985. The collapse of this explosive episode coincides with the implementation of Austral I which introduced price ceilings, a massive one-o€ increase in the money supply and a ®xed parity of the currency relative to the US dollar. CONCLUSIONS This paper has examined the possibility of detecting the presence of rational bubbles through an analysis of the integration properties of the relevant observable time series. We have o€ered a new approach based on a generalization of the ADF test procedure which makes use of the idea of Markov-switching regressions. Unlike standard unit root tests, which have little power to detect rational bubbles that collapse periodically, the switching ADF tests have been found to be capable of giving sensible inferences for the DGPs analysed by Evans (1991). The methodology proposed here clearly has applications beyond the context of testing for periodically collapsing bubbles. In particular, it o€ers a convenient way of constructing tests of the hypothesis of a unit root against alternatives characterized by Markov-switching parameter Copyright # 1999 John Wiley & Sons, Ltd.

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Figure 1. Monthly rate of growth of prices, money supply and exchange rates (solid lines) and inferred Pr(st ˆ 1 j I t ; W^ † (dashed lines)

variation, when the number and location of change points are unknown. As structural changes and regime shifts are endemic, both in the sense of changing policy regimes and in the sense of changes in the general economic structure, such tests can be very useful for the development of good descriptions of the generating mechanisms of economic and ®nancial time series. ACKNOWLEDGEMENTS

The authors are indebted to Steven Durlauf, Brendan McCabe, Garry Phillips, Stephen Satchell, Ron Smith, Elias Tzavalis, two anonymous referees, and seminar participants at the University of Southampton and the European University Institute, Florence, for helpful comments and/or discussions. Thanks also go to Ricardo Lopez Murphy (FIEL) for kindly providing the data used Copyright # 1999 John Wiley & Sons, Ltd.

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in the empirical application of the article. The research of Stephen Hall was supported, in part, by the UK Economic and Social Research Council.

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