This is the fourth post in a series on bubbles in the U.S. Equity Market. The first part can be found here.

# Introduction

Although a lot of work has been done on bubbles since this paper was written (including the Phillips et al. paper discussed in Part 2), I still think it is a good survey of econometric methods for detecting “bubbles”. The conclusion is mostly negative.
The problem is, these tests are unable to distinguish between bubbles and a variety of alternatives including time-varying discount factors and regime-switching fundamentals. For those who haven’t seen regime switching before, imagine dividends follow $$d_t=d_{t-1} + \mu_0 (1-s_t) + \mu_1 s_t + [\sigma_0(1-s_t) + \sigma_1 s_t]\epsilon_t$$ where $$s$$t is a state variable that denotes the different regimes.
This is a problem inherent in any purely statistical test - there’s no way to distinguish between bubbles and bubble-like behavior (it also happens that the definition of a bubble changes in every paper, further complicating the issue).
My first thought reading the paper was kind of impossibility result - is there a theoretical reason bubble tests should not work? Although the author does not say so explicitly, he seems to suggest this. In any proclaimed bubble someone can just tweak the process for fundamentals and fit the data equally well without a bubble. The Nasdaq paper discussed yesterday does exactly that.
Another theoretical concern was raised by Tirole (1982). In any infinitely lived rational expectations model there can be no bubbles. Suppose not: if the agents sell the asset, the money received today is higher than the discounted present value of future income (as the non-fundamental term is non-zero). So all agents want to sell and it’s not an equilibrium price. This raises a bigger issue of whether or not we should even try to model bubbles, but that will be the topic of a future post.

# The Paper

The paper starts by simplifying the standard asset pricing model discussed in previous posts. Assume linear utility, along with the existence of a risk-free bond with interest rate $$r$$ (where $$r =\frac{1}{\beta}-1)$$ and we get: $$P_t=\left[ \sum \limits_{i=1}^{\infty} \left(\frac{1}{1+r} \right)^i E_t(d_{t+i}) \right ]+B_t$$ where $$E_t(B_{t+1})=(1+r)B_t$$. If dividends grow slower than $$r$$, the term in brackets converges, but the bubble term does not. The problem here is that fundamentals alone do not pin down the asset price. There will be a different law of motion for $$P_t$$ depending on the initial value of $$B_t$$. If $$B_t$$ is zero for all $$t$$, however, the solution is uniquely pinned down by the fundamentals. All of the tests discussed in the paper build on this model.
Although there are 5 tests discussed in the paper, I focus on three: the Variance Bounds, the West Test and the Integration/Cointegration test.

# Variance Bounds Tests from Shiller (1981)

Consider the no bubbles solution to the asset pricing problem above: $$P_t=\sum \limits_{i=1}^{\infty} \left(\frac{1}{1+r} \right)^i E_t(d_{t+i})$$ Now consider the ex-post rational price, $$\tilde{P}$$ - the price if there was no uncertainty about future dividends: $$\tilde{P_t}=\sum \limits_{i=1}^{\infty} \left(\frac{1}{1+r} \right)^i d_{t+i}$$ Under rational expectations, the difference between actual and expected dividends should be unforecastable and mean zero, call this $$\epsilon_t$$ (in case you are not convinced, suppose not - then agents would trade today based on expected dividends and the price would adjust until this is true).
Using this we get: $$\tilde{P_t}=P_t+\sum \limits_{i=1}^{\infty} \left(\frac{1}{1+r} \right)^i \epsilon_{t+i}$$ If $$\epsilon_t$$ is uncorrelated with all information at time $$t$$, then we can easily bound the variance Letting $$V(X)$$ denote the variance of a random variable $$X$$: $$V(\tilde{P_t})=V(P_t)+ K V(\epsilon_t) \geq V(P_t)$$ where $$K$$ is a constant that depends on $$r$$. This gives us an upper bound on the variance of the price series. If variance is higher than $$V(P_t)+ K V(\epsilon_t)$$ the prices do not follow Equation 3.
The theory is nice, but there are some piratical problem in implementation:
1) We don’t actually observe $$\tilde{P}$$ so we need to set a terminal value (for example, use todays price and construct $$\tilde{P}$$ recursively using observed dividends). This introduces measurement error that can bias the test results.
2) If you believe the the variance of $$\epsilon_t$$ or the discount factor is changing over time, the test will not work, as the variance bound changes over time as well (note $$K$$ depends on $$r$$).

# West Test

As can be seen in the variance bound discussed above, a bubble test is really two tests at the same time:
1) Is our model (price equals discounted present value of future dividends) correct?
2) Is there a bubble in the data?
The West is designed to test both sequentially. First, we use the Euler equation to get the discount rate, $$r$$ (this does not depend on presence of a bubble). Given $$r$$, we model dividends as an AR(p) process. Using these together, we can recover a theoretical relationship between the price and dividends, which we test against the relationship in the data. Differences between theory and the data might be evidence of a bubble.
Let’s formalize the model. Write the one period Euler Equation: $$P_t=\left( \frac{1}{1+r} \right) E_t (P_{t+1} + d_{t+1} | \Omega_t)$$ where $$\Omega_t$$ is the information set at $$t$$.
Now consider running the regression: $$P_t=\left( \frac{1}{1+r} \right) (P_{t+1} + d_{t+1}) + u_t$$ The first obvious problem is that $$u_t$$ is correlated with the regressors, but West fixes this by using an IV regression instead of OLS (past dividends are used as an instrument for dividends today). Note that by no arbitrage, the Euler Equation must hold both in the the presence and the absence of a bubble, so the $$r$$ we get from this works in both cases.
Now we want to model the dividend process as an AR(p). For simplicity let’s work with an AR(1): $$d_t = \phi d_{t-1} + u_t^d$$ In the absence of bubbles and $$u_t^d$$ uncorrelated with $$u_t$$ at all leads and lags, we can use this to substitute into Equation 3, which gives us our theoretical relationship: $$P_t= \left (\frac{ \frac{\phi}{1+r} }{ 1-\frac{\phi}{1+r} } \right)d_t = \bar{\beta} d_t$$ To test for bubbles we run the OLS regression: $$P_t=\hat{\beta} d_t + v_t$$ With no bubble, $$\bar{\beta}$$ should equal $$\hat{\beta}$$, so we test s for the equality of these two parameters.
There are several issues with this test:
1) The correct $$p$$ for the dividends process is not obvious. We cannot just go to the data, as information criteria will be biased if we have a bubble.
2) Dividends might not be stationary, in which case Equation 10 is not valid (no longer get convergence of fundamental part).
3) Investors may be using information not observable (read: not modeled) to the econometrician. This is important because we only get $$\bar{\beta} \neq \hat{\beta}$$ if the bubble is correlated with dividends. The test cannot detect bubbles related to unmodeled components.
5) The Euler equation may hold approximately for one period, but theoretically should also hold for long lived assets. Problem is that small errors accumulate will accumulate over time. For any $$k$$ we have: $$P_t=\left( \frac{1}{1+r} \right)^k P_{t+k} + \sum \limits_{i=1}^{k} \left(\frac{1}{1+r} \right)^i d_{t+i} + u_t^k$$ Flood, Hodrick and Kaplan (1994) find that although this holds in the data for $$k=1$$ (specification used by West), this does not hold for $$k=2$$.

# Integration/Cointegration

As mentioned above, the no arbitrage condition implies no excess returns from holding the bubble asset so $$E_t(B_{t+1})=(1+r)B_t$$. In this context, we can think about the bubble process as a difference equation: $$B_{t+1}-(1+r)B_t=z_{t+1}$$

$$E_t(z_{t+i}) =0 \quad \forall i\geq 1$$ From here we start running into problems very quickly. Suppose at some $$t$$ there is no bubble. Then the bubble starts with the first positive realization of $$z$$. Suppose the first non-zero realization of $$z$$ is negative - then we get a “negative bubble” where (in expectation) the price eventually goes negative, which contradicts free disposal. Further, if the expected realization of $$z$$ cannot be negative, then it cannot be positive either, as this violates $$E_t(z_{t+i}) =0$$. This together implies if $$B_t$$ is zero then must be zero forever. So if there is a bubble, it would need bubble to exist from first day of trading.
To get around this they start making some assumptions: Suppose there is no bubble. Further, suppose unobserved fundamentals are not more non-stationary than dividends (i.e. not integrated of a higher order). Then, if dividends are stationary in $$n$$ differences, stock prices should be as well.
This will not hold if we have bubbles. To see this, suppose $$z$$ is white noise - then the first difference of the bubble is non-stationary and non-invertible. To see this, consider an AR(1) with a unit root: $$x_t=x_{t-1} + \epsilon_t$$. Taking a first difference makes the right hand side (RHS) stationary. Now think of the bubble equation $$B_{t+1}=(1+r)B_t + z_t$$. First differencing gives $$(1-L)B_{t+1}=r B_t + z_t$$ which is not stationary. In fact, any number of differences will not make the RHS stationary. Note the similar flavor to the Phillips et al. paper where we have have a right tailed alternative to a unit root (i.e. AR coefficient$$>1$$).
So now to the actual test: First, find the correct number of differences to make dividends stationary. Now, apply the same number to stock prices and see if that makes them stationary as well. Finally, test for cointegration between stock prices and dividends. This last part is important, as it comes from the theoretical relationship between prices and dividends in our asset pricing equation (both being integrated of the same order is not enough). If all of this holds, we conclude there were no bubbles.
The author presents two main criticisms of this test:
1) If some of the above conditions fail to hold, all we can really say is that there is some nonstationarity in the data - there’s no way to be sure if it’s caused by a bubble or something else. In particular, this would happen if fundamentals are integrated of a higher order than dividends and stock prices.
2) Periodically collapsing bubbles “act” like stationary process. Although this test can rule out monotonic bubbles, it has no power against bubbles that periodically collapse and re-inflate. Although I did not mention this in the previous post, this issue is discussed in the Phillips et al. paper and is one of the motivation for their multiple bubbles tests.

# Connecting the Dots

Before reading the paper I was not familiar with the variance bound. Despite all the issues mentioned above, some part of it should apply to the discussion yesterday. My guess is that the observed volatility in the Nasdaq “bubble” was higher than the theoretical limit (there’s no way annualized volatility of over 100% respects any bound!)

# Summary

Bubble tests have problems both theoretically, and empirically: There are many alternatives against which the tests will have no power. Although not discussed above, the section on intrinsic bubbles shows that a regime switching model with no bubble, and a bubble model with no regime switching fit the data equally well.
That being said, I think the paper uses these problematic tests to make a strong conclusion: The “classical” asset pricing model (Equation 3) does not fit the data. To match fundamentals with observed prices, we need to add more factors, such as time varying discount rates, changing risk aversions and regime switching. This uncovers the definition implicit in many papers on bubbles - a residual unexplained by simple models.