شناسایی حباب های سوداگرانه با استفاده از مدلهای فضای حالت به همراه سوئیچینگ مارکوف

In this paper we use a state-space model with Markov-switching to detect speculative bubbles in stock-price data. To this end we express a present-value stock-price model in state-space form which we estimate using the Kalman filter. This procedure enables us to estimate a two-regime Markov-switching specification of the unobservable bubble process. The respective Markov-regimes represent two distinct phases in the bubble process, namely one in which the bubble survives and one in which it collapses. We ultimately identify bursting stock-price bubbles by statistically separating both Markov-regimes from each other. In an empirical analysis we apply our methodology to a plethora of artificial and real-world data sets. Our study has two major findings. First, we find significant Markov-switching structures in real-world stock-price bubbles. Second, in the stock markets considered our identification procedure correctly detects most speculative periods which have been classified as such by economic historians.

A classical belief is that under rational expectations and rational behaviour of economic agents any asset price should be in line with its market fundamental value. According to this view, economists often regard a persistent and substantial divergence between an asset price and its fundamental value as market irrationality. However, recent work elaborates that the dynamics of an asset price may well contain a self-fulfilling bubble component and that the explosive asset-price behaviour caused by the bubble may be consistent with rational behaviour among market participants. Up to date, a multitude of theoretical studies examine the emergence of (stock-market) bubbles and their structural properties under rational expectations (e.g. Allen and Gale, 2000 and Abreu and Brunnermeier, 2003). A closely related strand of literature is concerned with the econometric detection of speculative bubbles. We can roughly divide these papers into two groups. The first group of studies is based on so-called indirect bubble tests. Here, the authors apply sophisticated cointegration and unit-root tests to a dividend-price relationship and try to overcome the well-known econometric weaknesses of the standard tests. Most influential articles belonging to this category include Diba and Grossman, 1988 and Evans, 1991 and several other contributions cited in McMillan, 2007 and Chen et al., 2009. The second group of studies, which are the more relevant to our paper, implement direct tests for speculative bubbles by explicitly formulating the existence of a bubble in the alternative hypothesis. Examples of such direct test procedures are West, 1987 and Wu, 1997. The key idea of West’s (1987) direct bubble test is to compare two alternative estimators (i.e. an indirect and a direct estimator) for one particular parameter. More concretely, West constructs the indirect estimator from two different estimations, namely (1) from the estimation of the observable no-bubble Euler equation, and (2) from the estimation of a stationary autoregressive process which he assumes to govern dividends. West combines both estimations to obtain an indirect estimate of the linear relationship between dividends and stock prices. Alternatively, one can directly estimate the linear relationship between dividends and stock prices by performing a straightforward linear regression of stock prices on dividends. Under the null hypothesis of ‘no bubble’, the direct and the indirect estimates of the linear relationship should be equal (within the limits of statistical accuracy) while under the alternative of ‘a rational bubble’ both estimates should differ significantly from each other. Hence, the basic idea of West’s (1987) test is to interpret a statistically significant difference between the direct and the indirect estimates as an indication of a speculative bubble. We may strengthen this interpretation further by additionally applying specification tests to the Euler equation and the autoregressive representation of dividends in order to rule out all model misspecification and leaving bubbles as the only possible source of the discrepancy between the two estimates. Essentially, West’s procedure tests the standard present-value model against an unspecified alternative which he interprets as an emergence from a speculative bubble. However, the test does not generate a time series of the bubble component. By contrast, Wu (1997), who also considers the deviation of stock prices from the present-value model, uses these discrepancies to construct a bubble time series. As in West (1987) he assumes that dividends follow an autoregressive process and treats the bubble as an unobservable variable which he estimates using the Kalman filter. In his empirical analysis Wu ascribes large portions of stock-price movements within the S&P 500 to speculative bubbles. Another class of econometric models intensively used for the detection of bubble components are so-called Markov-switching (or regime-switching) models. These models try to capture discrete shifts in the generating process of time series data and were introduced by Hamilton (1989). Hall et al. (1999) add an important application of Markov-switching models to the bubble literature by treating each component of a simulated bubble process as a separate Markov-regime with constant transition probabilities between the regimes. Within a Monte Carlo experiment they analyze the power of Augmented-Dickey-Fuller unit-root tests with Markov-switching (Markov-switching ADF tests) and use these test procedures to detect bubble episodes. Although Vigfusson and Van Norden (1998) criticize this methodology on econometric grounds, Markov-switching approaches constitute a useful tool for modeling stock-market fluctuations and bubbles that switch between two or more states (e.g. Driffill and Sola, 1998, Brooks and Katsaris, 2005 and Chen, 2009). In this paper we treat the bubble as an unobservable variable as in Wu (1997) but extend his framework by allowing the bubble to switch between alternative regimes. Through this we aim at separating two distinct periods in the bubble process from each other, namely one in which the bubble survives and one in which it collapses. Technically speaking, we implement Markov-switching in our unobserved-components framework by adopting the methodology from Kim and Nelson (1999) who show how to use state-space models that are subject to regime-switching. Hitherto, this econometric technique has mainly been used for the detection of turning points in business-cycle research (see Chauvet, 1998inter alia) and its application to the bubble literature constitutes the innovation of this paper. In line with several previous studies from the bubble literature, we analyze both artificially generated bubble data as well as real-world data sets. The inclusion of artificial bubble processes has the advantage of knowing exactly when a bubble starts to evolve over time. Thus, we obtain precise information on the quality of our bubble-detection method. By contrast, the identification of bubble periods in real-world data sets turns out to be a more complicated matter. For this data type we are reliant on what economic historians classify ex-post as bubble periods. In our empirical analysis below we rely on the work of Kindleberger and Aliber (2005) who classify bubble periods in real-world stock-market data. Our study has two major findings. First, we show that Markov-switching in the data-generating process of real-world stock-price bubbles appears to be a statistically significant phenomenon. Second, we obtain the encouraging overall result that our econometric framework is able to detect many bubble periods in our artificial data sets and is even more successful in tracking down real-world stock-price bubbles as classified by Kindleberger and Aliber (2005). This paper contains six sections. Section 2 briefly reviews the basic present-value model. Section 3 transforms the present-value model into a state-space representation. We demonstrate how to estimate the state-space model including the unobserved asset-price bubble via the Kalman filter. In Section 4 we incorporate Markov-switching elements into the state-space model. Section 5 describes our artificial bubble processes, motivates the selection of our real-world data sets on the basis of historical bubble periods and presents the estimation results. Section 6 offers some concluding remarks.

In this paper we propose a new methodology for detecting speculative bubbles in stock-price data. The methodology constitutes a state-space approach that we enrich by Markov-switching elements. Hitherto, this procedure has mainly been used in the field of business-cycle research and our innovation is to adapt it to Campbell and Shiller’s (1988) present-value stock-price model. In the state-space representation of this model, the unobservable state-vector constitutes the bubble component. By allowing the bubble process to switch between two regimes according to a first-order Markov chain, we are able to statistically discriminate surviving from collapsing phases in the bubble process. In order to check the validity of our econometric procedure, we apply it to a number artificial bubble processes which we generate according to the algorithm suggested by Evans (1991). Furthermore, we analyze real-world data using the well-known stock-price and dividend data set by Robert Shiller as well as stock-price data for various other countries which are known to have experienced severe bubble periods. The results of our investigation are twofold. First, we find statistically significant regime-switching structures in the stock-price bubble processes of our real-world data sets. Consequently, any theoretical bubble model and, in particular, any econometric (bubble) specification should take regime-switching structures into account. Second, our Markov-switching approach is able to detect the majority of bubbles in our artificial Evans-processes as well as in our real-world data sets. For the countries considered, our (smoothed) regime-probability technique identifies most speculative periods as classified by Kindleberger and Aliber (2005). We analyze a set of heterogeneous countries by exploring stock-price data for three emerging economies (Brazil, Indonesia, Malaysia) and two industrialized countries (Japan and the USA). Interestingly, our methodology identifies the relevant bubbles in both groups of countries although the respective bubble periods as well as the subsequent crash phases may feature qualitative differences. Analyzing policy-side reactions to asset-price booms, we find that for Indonesia and Malaysia the IMF acted as a lender of last resort in the Asian Crisis of 1997 (see Kindleberger and Aliber, 2005, pp. 180–181) and insisted that (1) both countries’ governments balanced their budgets and that (2) the respective central banks increased interest rates. In contrast to this, the authorities in the US and in Japan responded to the recent bubble periods by pursuing expansive fiscal and monetary policies which, compared to the emerging countries, helped both economies to better recover from financial distress. However, our present econometric framework is not designed to capture such qualitative differences. A line for future research in this direction may lie in the development of alternative econometric specifications of the bubble term Bt which are more complex than our two-regime autoregressive Markov-switching process. However, the econometric procedure applied in this paper illustrates that Markov-switching specifications may make a substantial contribution to the dynamics of stock-price bubbles. We therefore propose integrating this empirical finding into conventional methods designed to discover real-time stock-price bubbles in order to considerably improve the chances of detecting future bubbles at an early stage.