2. Literature review
2.2 Methodologies for testing bubble
The conception of bubbles is the part of observed real stock prices above the equilibrium level. According to the dynamic optimization problem of a representative consumer, the equilibrium level of real stock prices can be calculated from present value model. If the components of real stock prices only reflect market fundamental, the present value model can be a pricing model. However, the observations of real stock price may not always exhibit what pricing model predicts. The difference between stock price and its equilibrium level will trigger the proceeding of arbitrage.
By adopting the theoretical view, due to the existence of arbitrage, real stock prices eventually converges to equilibrium level. For this reason, according to the present value model, after excluding possible non-rational trade in the stock market;
meanwhile, real stock prices and dividends are not cointegrated, the stock market may have shown the evidence of rational bubbles. Therefore, the common approaches to test the null hypothesis of rational bubbles are unit-root tests to the price-dividend ratio or cointegration test between these two variables with long span data.
In that the arduously to obtain the real pricing model, the unobservable variables are not easily distinguished into fundamentals and bubbles components.
Different assumptions on the Euler equation, potential switching in market beliefs and
types of bubbles have made a long debate on testing technique. This section will introduce the reasons. Final part of this section will briefly introduce such debate that could be solved by the conception of mixture contingent state.
The price-dividend cointegration approach was applied by Campbell and Shiller (1987); and Diba and Grossman (1988a), following early empirical work includes Timmermann (1995)
. However, as discussed by
Diba and Grossman (1988a), the rejection of cointergation between real stock prices and real dividends would not necessarily conclude the evidence of existing rational bubbles. The present value model which they considered contained the unobservable variables that could have an effect on market fundamentals. From their present value model, if the first differences of the real dividends and unobservable variables are stationary, therejection of
the unit-root on first differences of real stock prices may be resulted from non-existence of rational bubbles. As the first differences on stock prices is stationary implies the transversality condition holds, Diba and Grossman (1988a), further derived a cointegrating vector between real stock prices and real dividends. But the cointegration test they provided requested unobservable variables in market fundamentals must stationary in levels. This is a more restrictive condition than unit-root test on the first differences of real stock prices.These methods had opened up new avenues for empirical strategy to overcome the identification problem pointed out by Flood and Garber (1980). The problem was difficulty in distinguishing from the market fundamentals components that researcher cannot observe when using the specification tests introduced by Singleton (1979);
Hansen and Sargent (1979). For instance, emphasized by Hamilton (1986), the changes in the expectations of market participants, such as future tax treatment of
dividend income, a researcher may be unable to observe or to infer. Other misspecification includes adding a constant discount rate restriction on the present value model or neglecting the inter-temporal marginal rate of substitution due to the assumption of risk neutral. All of these misspecifications should not be worried as long as they are stationary in levels when adopting the method of unit-root test on first differences of the real stock price and cointegration test between real stock prices and real dividends.
Nevertheless, different restrictions on the unobservable variables in market fundamentals of these tests had generated mixed results shown by Diba and Grossman (1988a). This suggests the fact that real stock prices are stationary in first differences results from the unobservable market fundamentals following the unit-root process.
Timmermann (1995) illustrated that real stock prices and real dividends did not cointegrated was caused by the assumption of constant discount rate. By allowing time-varying discount rate in the present value model, Timmermann (1995) showed that regression residuals contained linear component of real dividend. This implied that the unobservable market fundamentals are non-stationary in levels but stationary in first differences. For this reason, price-dividend ratio test provided by Craine (1989), Cochrane (1992) and log price-dividend ratio test provided by Campbell and Shiller (1988) were suitable methodologies for testing bubbles if the persistence of the discount rate was not very high. Therefore, Timmermann (1995) also investigated the impact of persistence in discount rate in present value model on cointegration test in levels and in logarithms by Monte Carlo experiment. But samples were generated by nonlinear model which followed the approach of Poterba and Summers (1986).
However, that the argument of the linear present value model is suitable for the assumption of constant discount rate which had been discussed by Campbell et al.,
1997, (CLM) had enlightened nonlinear econometric techniques to test present value model. Manzan, 2004, also followed the approach of Poterba and Summers (1986) to derived the nonlinear present value model by allowing time-varying discount rate which is known in the literature as the dynamic Gordon model. To capture such nonlinear phenomenon, Manzan, 2004, studied the demeaned log price-dividend ratio, that is, log deviations of the price from static Gordon valuation, by using smooth transition regression (STR) approach.
Although log dividend-price ratio test implies releasing the assumption of constant discount rate, some researches had provided other reasons of nonlinear adjustment towards equilibrium. Gallagher and Taylor (2001), in their investigation of risky arbitrage and the limits of arbitrage hypothesis also utilized STR approach and the variable was a linear combination of log dividend-price ratio and discount rate.
The linear combination was a cointegration vector between log dividend-price ratios and discount rates. Not only does the existence of risky arbitrage and the limits of arbitrage suggests that log dividend-price ratio displays nonlinear trajectory, irrational fads or investor sentiment and misspecification of market fundamentals had also been argued repeatedly. Coakley and Fuertes (2005) argued nonlinear effects may occasionally dominate the linear components because the log dividend-price ratio test conflates bull and bear market phases may suffer the problem of potential switching in government policy and other mispricing part which belongs to fads or waves of optimism. In their investigation of asymmetrical adjustments during bull and bear market phases, the nonlinear model they applied was a parsimonious two-regime model that belongs to the momentum threshold autoregressive (MTAR) class. The MTAR method was proposed by Enders & Granger (1998); Enders and Siklos (2001).
Except nonlinearities behavior, the more serious problem of linear model argued by Evans (1991) was incorrect conclusions with respect to the presence of bubbles if periodically collapsing bubbles appears in stock market. Relying on Evans’
Monte Carlo simulations with periodically collapsing bubbles presence, these simulations results incorrectly showed the absence of bubbles when using standard unit root and cointegration tests. Base on these simulations results, Evans (1991), further argued that periodically collapsing bubbles would appear to be a stable linear autoregressive process unless the probability of bubbles collapsing was not very high.
Afterward, a nonlinear model literature to overcome the Evans critique which was proposed by Hall, Psaradakis, and Sola (1999), used Markov switching ADF test in which time series switching between explosive and stable process. Since such bubbles exhibit the characteristic of sudden collapse, standard tests may incorrect conclusion with respect to mean reversion. Hall, Psaradakis, and Sola (1999), argued if data only contain the expansion phase of bubbles, a test would more likely to find the evidence of divergence process. Basing on this criticism, Markov switching ADF test may provide more testing power of null hypothesis. Relying on Monte Carlo experiments shown by Hall, Psaradakis, and Sola (1999), the Markov switching ADF test had considerable power to detect the presence of periodically collapsing bubbles.
Bohl (2002) applied momentum threshold autoregressive (MTAR) model to capture the characteristics of periodically collapsing bubbles. The results of the Bohl’s Monte Carlo studies displayed that the MTAR approach provided a sufficiently powerful test to detect the presence of periodically collapsing bubbles.
It appears nonlinear techniques is adequate to test present value model. On the contrary, a noticeable dilemma of these nonlinear models is difficulty on specifying a
specific form for the sake of the testing price-dividend relation or bubbles. In that these nonlinear components are difficulty to detect whether they originations from the fundamentals or bubbles, a researcher would have perplexity on specifying. However, such indeterminate originates could be regarded as a mixture contingent state. By manipulating inter-temporal marginal rate of substitution, a partial of unobservable elements can be derived which belongs to fundamentals to solve some of identification problem. Detail econometric methodology will discuss in next section.