RATIONAL BUBBLES AND LEARNING

RATIONAL BUBBLES AND LEARNING Rational bubbles arise because of the indeterminate aspect of solutions to rational expectations models, where the process governing stock prices is encapsulated in the Euler equation. The price you are prepared to pay today for a stock depends on the price you think you can obtain at some point in the future. But the latter depends on the expected price even further 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. However, in general, the Euler equation does not rule out the possibility that the price may contain an explosive bubble. 1

While one can certainly try and explain prolonged rises or falls in stock prices as being due to some kind of irrational behaviour such as herding or market psychology, nevertheless, 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. Euler Equation and the Rational Valuation Formula We investigate how the market price of stocks may deviate, possibly substantially, from their fundamental value even when agents are homogeneous and rational and the market is informationally efficient. To do so, we show that the market price may equal its fundamental value plus a bubble term and yet the stock is still willingly held by rational agents and no excess profits can be made. 2

To simplify the exposition, assume (i) agents are risk-neutral and have rational expectations and (ii) investors require a constant (real) rate of return on the asset E t R t+i = k. The Euler equation is P t = δ(e t P t+1 + E t D t+1 ) (1) where δ = 1/(1 + k). This may be solved under RE by repeated forward substitution (2) where we assume the transversality condition holds (i.e. lim(δ n E t D t+n ) = 0, as n ). The transversality condition ensures a unique price given by (2) which we denote as the fundamental value P f t. 3

The basic idea behind a rational bubble is that there is another mathematical expression for P t that satisfies the Euler equation, namely (3) and B t is the rational bubble. Thus, the actual market price P t deviates from its fundamental value P f t by the amount of the rational bubble B t. In order that (3) should satisfy (1), we have to place some restrictions on the dynamic behaviour of B t. Start by leading (3) by one period and taking expectations at time t E t P t+1 = E t [δe t+1 D t+2 + δ 2 E t+1 D t+3 + +B t+1 ] = [δe t D t+2 + δ 2 E t D t+3 + +E t B t+1 ] (4) where we have used the law of iterated expectations E t (E t+1 D t+j ) = E t D t+j. 4

after manipulating δ[e t D t+1 + E t P t+1 ] = δe t D t+1 + [δ 2 E t D t+2 + δ 3 E t D t+3 + +δe t B t+1 ] (5) Substituting the definition of P f t from (2) in the RHS of (5), we have δ[e t D t+1 + E t P t+1 ] = P f t+ δe t B t+1 (6) Substituting from (6) into (1), P t = P f t+ δe t B t+1 (7) We can make these two solutions (3) and (7) equivalent if E t B t+1 = B t /δ = (1 + k)b t (8) Then, (3) and (7) collapse to the same expression and satisfy (1). 5

An alternative approach to showing the bubble must be a martingale is to note that from (1), we can write P t + B t = δ(e t P t+1 + E t D t+1 + E t B t+1 ), providing B t = E t B t+1. More generally, (8) implies E t B t+m = B t /δ m (9) Hence (apart from the known discount factor), B t must behave as a martingale: the best forecast of all future values of the bubble depends only on its current value. While the bubble solution satisfies the Euler equation, it violates the transversality condition (for B t 0) and because B t is arbitrary, the stock price in (3) is non-unique. Note that the bubble is a valid solution, providing it is expected to grow at the rate of return required for investors to willingly hold the stock, from (8), we have E(B t+1 /B t ) 1 = k. 6

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-fulfilling expectation. Consider a simple case where expected dividends are constant and the value of the bubble at time t, B t = b (> 0) a constant. The bubble is deterministic and grows at the rate k, so that E t B t+m = (1 + k) m b. Thus, once the bubble exists, the actual stock price at t + m, even if dividends are constant, is from (3) P t+m = [δd / (1 δ)] + b(1 + k) m (10) Even though fundamentals (i.e. dividends) indicate that the actual price should be constant, the presence of the bubble means that the actual price can rise continuously, since (1 + k) > 1. 7

In the above example, the bubble becomes an increasing proportion of the actual price since the bubble grows but the fundamental value is constant. In fact, even when dividends are not constant, the stock price always grows at a rate that is less than the rate of growth of the bubble (= k) because of the payment of dividends (E t P t+1 /P t ) 1 = k E t D t+1 /P t (11) In the presence of a bubble, the investor still uses all available information to forecast prices and rates of return. Hence, forecast errors are independent of information at time t and excess returns are unforecastable. Tests of informational efficiency are, therefore, useless in detecting bubbles. However, the bubble does not allow (supernormal) profits, since all information on the future course of dividends and the bubble is incorporated in the current price: the bubble satisfies the fair game property. 8

Our bubbles model can be extended (Blanchard 1979) to include the case where the bubble collapses with probability (1 π) and continues with probability π B t+1 = B t (δ) 1 with probability π (12a) = 0 with probability 1 π (12b) This structure also satisfies the martingale property. These models of rational bubbles, tell us nothing about how bubbles start or end; they merely tell us about the time-series properties of the bubble once it is under way. The bubble is exogenous to the fundamentals model and the usual RVF for prices. As noted above, investors cannot distinguish between a price rise that is due solely to fundamentals or because of the bubble. 9

Individuals do not mind paying a price over the fundamental s price as long as the bubble element yields them the required rate of return next period and is expected to persist. One implication of rational bubbles is that they cannot be negative (i.e. B t < 0). This is because the bubble element falls at a faster rate than the stock price. Hence, a negative rational bubble ultimately ends in a zero price. Rational agents realise this and they, therefore, know that the bubble will eventually burst. But by backward induction, the bubble must burst immediately, since no one will pay the bubble premium in the earlier periods. Thus, if the actual price P t is below fundamental value P f t, it cannot be because of a rational bubble. 10

If negative bubbles are not possible, then if the bubble is ever zero, it cannot restart. This arises because the innovation (B t+1 E t B t+1 ) in a rational bubble must have a zero mean. If the bubble started again, the innovation could not be mean zero since the bubble would have to go in one direction only, that is, increase, in order to start up again. In principle, a positive bubble is possible since there is no upper limit on stock prices. However, in this case, we have the rather implausible state of affairs where the bubble element B t becomes an increasing proportion of the actual price and the fundamental part of the price becomes relatively small. One might conjecture that this implies that individuals will feel that at some time in the future, the bubble must burst. 11

Again, if investors think that the bubble must burst at some time in the future (for whatever reason), then it will burst now. To see this, suppose individuals think the bubble will burst in the year 2030. They must realise that the market price in the year 2029 will reflect only the fundamental value because the bubble is expected to burst over the coming year. But if the price in 2029 reflects only the fundamental value, then by backward induction, this must be true of the price in all earlier years. Therefore, the price now will reflect only fundamentals. Thus, it seems that in the real world, rational bubbles can really only exist if each individual s horizon is shorter than the time period when the bubble is expected to burst. 12

The idea here is that one would pay a price above the fundamental value because one believes that someone else will pay an even greater price in the future. Here, investors are myopic, and the price at some future time t + N depends on what they think, and other investors think, the price will be. Tests of Rational Bubbles An ingenious test for bubbles is provided by West (1987a). The test involves calculating a particular parameter by two alternative methods. Under the assumption of no bubbles, the two parameter estimates should be equal within the limits of statistical accuracy, while in the presence of rational bubbles, the two estimates should differ. Any bubble that is correlated with dividends can in principle be detected. 13

To illustrate the approach, note first that δ can be estimated using estimation of the observable Euler equation P t = δ(p t+1 + D t+1 ) + u t+1 (16) where u t+1 = δ[(p t+1 + D t+1 ) E t (P t+1 + D t+1 )]. Now assume an AR(1) process for dividends D t = αd t 1 + v t α < 1 (17) Under the no-bubbles hypothesis, the RVF and (17) give P t = ψd t + ε t (18) where ψ = δα/(1 δα) and ε t arises because the econometrician has a subset of the true information set. An indirect estimate of ψ, denoted ψ, can be obtained from the regression estimates of δ from (16) and α from (17). 14

However, a direct estimate of ψ, denoted ψ can be obtained from the regression of P t on D t in (18). Under the null of no bubbles, the indirect and direct estimates of ψ should be equal. Consider the case where bubbles are present and hence P t = P f t+ B t = ψ D t + B t. The regression of P t on D t now contains an omitted variable, namely, the bubble and the estimate of ψ, denoted ψ, will be inconsistent: plimψ (19) If the bubble B t is correlated with dividends, then ψ will be biased (upwards if cov(d t,b t ) > 0) and inconsistent. 15

But the Euler equation and the dividend forecasting equations still provide consistent estimators of the parameters and hence of ψ. The above test procedure is used by West (1987a) whose data consists of the Shiller (1981) S&P index 1871 1980 (and the Dow Jones index 1928 78). West finds a substantive difference between the two sets of estimates, thus rejecting the null of no bubbles. Some tests for the presence of rational bubbles are based on investigating the stationarity properties of price and dividend data. 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 P t grows faster than D t, this could be due to the presence of a bubble term B t. These intuitive notions can be expressed in terms of the literature on unit roots and cointegration. 16

Using the RVF (under the assumption of a constant discount rate), if P t and D t are unit root processes, then they should be cointegrated. If dividends (or log dividends) are integrated of order one I(1) and P t = [δ/(1 δ)]d t, then P t must be I(1) and cointegrated with D t and z t = P t δ/(1 δ)d t is stationary I(0). Using aggregate stock price and dividend indexes, Diba and Grossman (1988) find that ΔP t and ΔD t are stationary and P t and D t are cointegrated, thus rejecting the presence of explosive bubbles of the type represented by equation (8). Unfortunately, the interpretation of the above tests has been shown to be potentially misleading in the presence of what Evans (1991) calls periodically collapsing bubbles. The type of rational bubble that Evans examines is one that is always positive but can erupt and grow at a fast rate before collapsing to a positive mean value, when the process begins again. 17

The path of the periodically collapsing bubble (see Figure 1) can be seen to be different from a bubble that grows continuously. Intuitively, one can see why testing to see if P t is non-stationarity I(1) might not detect a bubble component like that in Figure 1. The (Dickey Fuller) test for stationarity essentially tries to measure whether a series has a strong trend or an unconditional variance that is non-constant. 18

Clearly, there is no strong upward trend in Figure 1, 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. E t B t+1 = B t /δ) examined by Diba and Grossman, and here one might expect standard tests for stationarity to be more conclusive. Evans proceeds by using MCS to generate a series for a periodically collapsing bubble. Adding the bubble to the fundamentals P f t (e.g. under the assumption that D t is a random walk with drift) gives the generated stock price series, which is then subject to standard tests for the presence of unit roots. He finds that the results of his unit root tests depend crucially on π, the probability (per period) that the bubble does not collapse. For values of π < 0.75, more than 90% of the simulations erroneously indicate that ΔP t is stationary and P t and D t are cointegrated. 19