ESTILOS DE VIDA SEDENTARIOS Grafico nº 3 pasatiempos

In this section I decompose the observed asset price into the first two of three components: fun- damental and bubble. Consider the standard asset pricing Euler equation for a dividend-producing asset:

Vt=EtQ[exp{−rt}(Vt+1+Dt+1)]

whereVtis the ex-dividend price,Dtis the dividend paid,rtis the log risk free rate of the market,

andE_tQ represents expectation taken with respect to the risk-neutral measure treating information up to time t as known. Under the assumptions of complete markets and no-arbitrage, the risk- neutral measure is unique. By denoting the Radon-Nikodym derivative asξt, the pricing equation

can be written in terms of the observed physical measure:

Vt=Et[exp{mt,t+1}(Vt+1+Dt+1)] (3.1)

where mt,t+1 = ln

ξt+1

paid for a dividend-producing asset must be equal to the expected discounted future revenue from holding the asset. The discounting is stochastic, because it depends on the state of the world in timet+ 1, which is unknown in timet.

Using forward recursion, equation (3.1) may be written:

Vt=Et " _∞ X n=1 exp ( _n X j=1 mt−1+j,t+j ) Dt+n # | {z } fundamental + lim n→∞Et " _n Y j=1 exp{mt−1+j,t+j}Vt+j # | {z } bubble (3.2)

The first term in equation (3.2) is the fundamental value of the asset, and it is equal to the expectation of all future discounted dividend streams to be paid by the asset. The second term is the limiting scrap value of the asset and equals zero when the asset is priced at its fundamental price. If the second term is positive, the asset is priced away from its fundamental, and I refer to this case as a bubble. The magnitude of the bubble is equal to this limiting scrap value in each time period and may or may not be time-varying: Bt = limn→∞Et

h Qn

j=1exp{mt−1+j,t+j}Vt+j

i

.11 Letting V_tf denote the fundamental value and Bt denote the bubble value, I refer to the asset

pricing equation asVt =Vtf +Bt.

Because of its relationship to the limiting value of the asset’s price, the bubble must grow according to a risk-adjusted martingale:

Bt =Et[exp{mt,t+1}Bt+1] (3.3)

which implies the first three necessary components for a statistical model of bubbles: (1)Bt≥0∀t, 11_{Often, tranversality conditions necessary for solving individual agent optimization problems are cited as reasons}

why bubbles cannot exist from a theoretical perspective. For an infinitely-lived representative agent under perfect information and rational expectations, that transversality condition islimn→∞Et

h_Qn

j=1exp{mt−1+j,t+j}Vt+j

i = Bt = 0∀t. There are a number of environments in which bubbles can still exist (Santos and Woodford 1997).

One direction of modern work in the rational bubble literature assumes models of the class considered in Kocher- lakota (1992a) where heterogenous agents and credit/collateral constraints support the existence of rational bub- bles, because individual stochastic discount factors are not necessarily equal to the market stochastic discount factor. Thus while limn→∞Et

h_Qn j=1exp mi t−1+j,t+j Vt+j i

= 0 ∀t, i for each agent i must hold, Bt = limn→∞Et

h_Qn

j=1exp{mt−1+j,t+j}Vt+j i

(2) ifBt = 0 thenBt+j = 0∀j > 0, and (3) bubbles must grow in expectation at the market rate

of discounting (Diba and Grossman 1988a). The proof is in Appendix C.1.1. Equation (3.3) is a condition which underpins the vast majority of rational bubble models and is derived directly from the asset pricing Euler equation. Thus the two key components for pinning down the dynamics of bubble growth are the stochastic discount factor process and its covariance with bubble growth.

For instance, with the assumption of a constant required rate of returnexp{−mt,t+1}=R >1

and a stationary dividends process, equation (3.3) implies that the bubble component of an asset’s price should come to dominate the the fundamental in finite time; an argument which was fre- quently used to rule out bubbles empirically. One approach to challenging this view was pioneered by Evans (1991) and Fukuta (1998), which formulated models of bubbles that grew at a constant market rate of return in expectation, but featured multiple regimes: explosive growth and partial collapse. This Markov-switching environment was subsequently heavily used in the empirical literature.12

However, the imposition of a finite number of regimes places an unnecessary, artificial limi- tation on the dynamics of the bubble component which can be relaxed. This paper proposes an alternative specification for the bubble growth process which does not rely on a regime-switching environment. By allowing for risk-averse agents and a theoretically-justified correlation between bubble growth and the aggregate states of the economy, this paper delivers a continuous process which exhibits stochastic growth and partial collapse in a flexible framework. The formulation for this process will be derived in Section 3.2.5.

Equation (3.2) decomposes the observed price of a productive asset (equity, housing, etc.) as a linear sum of the fundamental price and the bubble price. From equation (3.3), the dynamics of the bubble process resemble those of a separate, infinite-maturity, zero-coupon bond which gives inter-temporal returnBt+1/Bt. Thus we may consider the observed price of an asset to operate as

a portfolio of two infinitely-lived assets: a productive, dividend-producing asset and a zero-coupon

bond.13 _{The return to holding the fundamental portion of this portfolio is determined entirely by}

aggregate state dynamics. The return to holding the bubble portion of the portfolio is correlated with the returns to holding the fundamental portion of the portfolio only insofar as bubble growth is correlated with the aggregate state dynamics. As shown in Section 3.2.5, this correlation is key to matching observed price dynamics.

Estimation of the observed price of an asset depends on three elements: (1) a model of dividend growth, (2) a model of the stochastic discount factor, and (3) a model of the bubble which follows the dynamics given in equation (3.3). The following three subsections will address these elements in turn.