# Bubbles

Following {cite:t}`blanchardwatson1982bubbles`, consider the stochastic expectational difference equation

```{math}
:label: eq-203
y_t = \lambda P_t y_{t+1} + x_t,\quad |\lambda|<1
```

where $x_t$ is a stationary autoregressive process

```{math}
:label: eq-204
x_t = \rho x_{t-1} + \epsilon_t,\quad |\rho|<1
```

where $\epsilon_t$ is a white noise that is fundamental for $x_t$. In {eq}`eq-203`, $P_t$ is the linear least squares projection operator, conditioned on information known at $t$. An application of formula {eq}`eq-90` shows that the *stationary* solution of {eq}`eq-203` given {eq}`eq-204` is

```{math}
:label: eq-205
y_t = \frac{1}{1-\lambda\rho}x_t
```

Blanchard and Watson noted that in addition to the stationary solution {eq}`eq-205` there are many nonstationary solutions. These nonstationary solutions can be characterized as follows. Let $c_t$ be any martingale, that is, let $c_t$ be any stochastic process that satisfies $P_t c_{t+1} = c_t$. Then a solution of {eq}`eq-203` is

```{math}
:label: eq-206
y_t = \frac{1}{1-\lambda\rho}x_t + \left(\frac{1}{\lambda}\right)^t c_t
```

That {eq}`eq-206` is a solution of {eq}`eq-203` can be verified directly.

Three examples of martingales $c_t$ can usefully be given. A first is the constant $c_t = c$ for all $t$. This is the sort of deterministic bubble encountered in {doc}`Chapter IX <ch09_difference_equations>`. A second is the one proposed by Blanchard and Watson, namely, the process

$$
c_{t+1} = \begin{cases}
  c_t/\pi & \text{with prob } \pi, \quad 0<\pi<1 \\
  0 & \text{with prob } 1-\pi
\end{cases}
$$

The process $c_{t+1}$ is readily verified to be a martingale. A third example is generated from the $x_t$ process itself. Simply set $c_t = \rho^{-t}x_t$, which is a martingale in light of {eq}`eq-204`: $P_t c_{t+1} = \rho^{-(t+1)}P_t x_{t+1} = \rho^{-(t+1)}\rho x_t = c_t$.

In Chapter XIV, we study a model in which a transversality condition serves to make setting $c_t= 0$ the only admissible solution for a version of equation {eq}`eq-203`. There has recently been work designed to estimate and test models in which there are insufficient boundary conditions to justify setting $c_t = 0$. (See {cite:t}`blanchardwatson1982bubbles`, {cite:t}`meese1986testing`, Sargent and Wallace (1985), and {cite:t}`hamiltonwhiteman1985observable`.) Such models have been proposed as candidates for understanding the stock market, foreign exchange rates, and hyperinflations.


```{seealso}
{doc}`36d_explosive_decomposition` applies the same forward solution to a scalar explosive
autoregression and reads the two resulting solutions as two ways of representing a nonstationary
process.
```