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LECTURE 11 : RATIONAL BUBBLES

LECTURE 11 : RATIONAL BUBBLES. Contents. M arket price of stocks may deviate, possibly substantially , from their fundamental value even when agents are homogeneous and rational and the market is informationally efficient

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LECTURE 11 : RATIONAL BUBBLES

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  1. LECTURE 11 :RATIONAL BUBBLES

  2. Contents • Market price of stocks may deviate, possibly substantially, from their fundamental value even when agents are homogeneous and rational and the market isinformationallyefficient • Insurgence of rational bubbles and periodically collapsing bubbles as solutions to the Euler equation • Tests for the presence of ‘exogenous’ bubbles. • Bubbles may also depend on fundamentals • Test for the presence of these ‘intrinsic bubbles’.

  3. Introduction • Debate on idea of self-fulfilling ‘bubbles’ or ‘sunspots’ in asset prices has been discussed almost since organised markets began • South sea bubble • Toulipmania…but also • Actual Estate evalutation bubbles • US/USA stock markets 1994/2000 • $ spot FX-rate 1982/1985….

  4. Introduction Andamento NASDAQ 1994/2012

  5. Mathematics • Indeterminate aspect of solutions to rational expectations models (Euler equation) • The price for a stock also depends on the price you think you can obtain at some point in the future. • The Euler equation determines a sequence of prices but does not ‘pin down’ a unique price level unless somewhat arbitrarily, we impose a terminal condition (i.e. the transversality condition) to obtain the unique solution. • In general, the Euler equation does not rule out the possibility that the price may contain an explosive bubble.

  6. Mathematics • Recent work emphasises that such sharp movements or ‘bubbles’ may be consistent with the assumption of rational behaviour. • Even if traders are perfectly rational, the actual stock price may contain a ‘bubble element’, and, therefore, there can be a divergence between the stock price and its fundamental value.

  7. Mathematics • Pt = δ(EtPt+1 + EtDt+1) * • δ = 1/(1 + k); k = constant (real) rate of return • * under RE by repeated forward substitution • Pt = Pft = ∑i;(i=1, ∞)δiEtDt+i ** • The TVC ensures a unique price which we denote as the fundamental value Pft

  8. Mathematics • The basic idea behind a rational bubble is that there is another mathematical expression for Pt that satisfies the Euler equation, namely • Pt = ∑i;(i=1, ∞)δiEtDt+i+ Bt = Pft+ Bt*** • Btis the ‘rational bubble’ term.

  9. Mathematics • Properties of Bt?: clearly, if Btis large relative to fundamental value, then actual prices can deviate substantially from their fundamental value. • restrictions on the dynamic behaviour of Btto avoid contradiction. We do so by assuming *** is a valid solution to * and this then restricts the dynamics of Bt. • (simple) procedure for consistency in the text (p. 399) • δ[EtDt+1 + EtPt+1] = Pft+ δEtBt+1 • (differentthan *** unless EtBt+1= Bt/δ = (1 + k)Bt • (consistency formula)

  10. Mathematics • Notice: EtBt+1 = Bt/δ = (1 + k)Bt • the bubble is a valid solution if it is expected to grow at the rate of return required to hold the stock • We have E(Bt+1/Bt) − 1 = k. • Investors do not care if they are paying for the bubble (rather than fundamental value) because the bubble element of the actual market price pays the required rate of return, k • The bubble is a self-fulfillingexpectation.

  11. Extensions of the model • Extension (Blanchard 1979) to include the case where the bubble collapses with probability (1 − π) and continues with probability π • Bt+1 = Bt(δπ)−1 with probability π • Bt+1 = 0 with probability 1 − π • These models tell us nothing about how bubbles start or end.The bubble is ‘exogenous’ to the ‘fundamentals model’ and the usual RVF for prices.

  12. Empirics • Some tests for the presence of rational bubbles are based on investigating the stationarity properties of Ptand Dt. • An exogenous bubble introduces an explosive element into prices, which is not (necessarily) present in the fundamentals (i.e. dividends or discount rates). • Hence, if the stock price Pt ‘grows’ faster than Dt , • this could be due to the presence of a bubble term Bt . • These intuitive notions can be expressed in terms of the literature on unit roots and cointegration. • Under the assumption of a constant discount rate, if Pt and Dt are unit root processes, then they should be cointegrated.

  13. Empirics • Diba and Grossman (1988) find that Pt and Dt are non-stationary and Pt and Dt are cointegrated, thus rejecting the presence of explosive bubbles • The interpretation of the above tests has been shown to be misleading in the presence of what Evans (1991) calls periodically collapsing bubbles.

  14. Empirics

  15. Empirics • Clearly, there is no strong upward trend in Figure • and although the variance alters over time, this may be difficult to detect particularly if the bubbles have a high probability of collapsing (within any given time period). • If the bubbles have a very low probability of collapsing, we are close to the case of ‘explosive bubbles’ (i.e. EtBt+1 = Bt/δ) examined by Diba and Grossman, and here one might expect standard tests for stationarity to be more conclusive.

  16. PHILLIPS, WU AND YU • During the 1990s, led by DotCom stocks and the internet sector, the U.S. stock market experienced a spectacular rise in all major indices, especially the Nasdaq index • At the same time there was much popular talk among economists about the effects of the internet and computing technology on productivity and the emergence of a “new economy” • What caused the unusual surge and fall in prices? • Rational bubbles or pricing of new investment opportunities?

  17. PHILLIPS, WU AND YU Recall Pt = ∑i;(i=1, ∞)δiEtDt+i + Bt = Pft + Bt *** Bt is the ‘rational bubble’ term.

  18. PHILLIPS, WU AND YU

  19. PHILLIPS, WU AND YU • To achieve this goal, we first define financial exuberance in the time series context in terms of explosive autoregressive behavior and then introduce some new econometric methodology based on forward recursive regression tests and mildly explosive regression asymptotics to assess the empirical evidence of exuberant behavior in the Nasdaq stock market index. In this context, the approach is compatible with several different explanations of this period of market activity, including the rational bubble literature, herd behavior, and exuberant and rational responses to economic fundamentals.

  20. PHILLIPS, WU AND YU • All these propagating mechanisms can lead to explosive characteristics in the data. Hence, the empirical issue becomes one of identifying the origination, termination, and extent of the explosive behavior. • Although with traditional test procedures “there is little evidence of explosive behavior” (Campbell et al., 1997, p. 260),with the recursive procedure, the authors successfully document explosive periods of price exuberance in the Nasdaq

  21. PHILLIPS, WU AND YU • Method: forward recursive regression technique • Nasdaqindex over the full sample period from 1973 to 2005 and some subperiods. • Date stamp the origin and conclusion of the explosive behavior. The statistical evidence from these methods • indicates that explosiveness started in 1995, thereby predating and providing empirical content to the Greenspan remark in December 1996. The empirical evidence indicates that the explosive environment continued until sometime between September 2000 and March 2001.

  22. PHILLIPS, WU AND YU

  23. PHILLIPS, WU AND YU • The relationship between Pt and Dt suggest that a direct way to test for bubbles is to examine evidence for explosive behavior in Ptand Dtwhen the discount rate is time invariant. Of course, explosive characteristics in Ptcould in principle arise from Dtand the two processes would then be explosively cointegrated. • However, if Dtis demonstrated to be nonexplosive, then the explosive behavior in Ptwill provide sufficient evidence for the presence of bubbles because the observed behavior may only arise through the presence of Bt.

  24. PHILLIPS, WU AND YU • It is possible to look directly for explosive behavior in ptand nonexplosive behavior in dtvia right-tailed unit root tests • However, as Evans (1991) noted, explosive behavior is only temporary when economic bubbles periodically collapse, and in such cases the observed trajectories may appear more like an I(1) or even stationary series than an explosive series, thereby confounding empirical evidence. He demonstrated by simulation that standard unit root tests had difficulties in detecting such periodically collapsing bubbles.

  25. PHILLIPS, WU AND YU • Recursive regression techniques • The tests are implemented as follows. For each time series xt(log stock price or log dividend), the ADF test for a unit root against the alternative of an explosive root (the right-tailed). • H0 : δ = 1 and the right-tailed H1 is H1 : δ > 1.

  26. PHILLIPS, WU AND YU • In forward recursive regressions, the model is estimated repeatedly, using subsets of the sample data incremented by one observation at each pass. • Denote the r-th pass t-statistic by ADFr • Criterion:

  27. PHILLIPS, WU AND YU RECURSIVE CASE

  28. PHILLIPS, WU AND YU ROLLING CASE

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