Bad Advice, Herding, and Bubbles

1

Bad Advice, Herding, and Bubbles

by

Mark Thoma*
University of Oregon

March 14, 2012
Revised September 22, 2012

Abstract: Prior to the crash of the housing market, many experts told the public that the upward trend in housing prices was not a bubble, it could be explained by fundamentals. This paper shows that an increase in the propensity of individuals to herd toward trend-chasing behavior caused, for example, by bad advice from experts, increases the likelihood that a destructive bubble will occur.

JEL Classification: G01, E32, B40,
Keywords: Herding, bubbles, economists

* Mark A. Thoma, Department of Economics, University of Oregon, Eugene, OR 97403-1285, (541) 346-4673, fax (541) 346-1243, . Thanks to the participants at the Workshop on Methodology, Systemic Risk, and the Economics Professionheld at Duke University in December 2011, and to the participants at the Duke INET Workshop at the 2012 American Economic Association Meetings in Chicago for helpful comments.

Introduction

The belief that housing prices would continue to rise into the foreseeable future was an important factor in creating the housing price bubble. But why did people believe this? Why did they become convinced, as they always do prior to a bubble, that this time was different? One reason is bad advice from academic and industry experts. Many people turned to these experts when housing prices were inflating and asked if we were in a bubble. The answer in far too many cases – almost all when they had an opinion at all – was that no, this wasn’t a bubble. Potential homebuyers were told there were real factors such as increased immigration, zoning laws, resource constraints in an increasingly globalized economy, and so on that would continue to drive up housing prices.

When the few economists who did understand that housing prices were far above their historical trends pointed out that a typical bubble pattern had emerged – both Robert Shiller and Dean Baker come to mind –they were mostly ignored. Thus, both academic and industry economists helped to convince people that the increase in prices was permanent, and that they ought to get in on the housing boom as soon as possible.

But why did so few economists warn about the bubble? And more importantly for the model presented in this paper, why did so many economists validate what turned out to be destructive trend-chasing behavior among investors?

One reason is that economists have become far too disconnected from the lessons of history. As courses in economic history have faded from graduate programs in recent decades, economists have become much less aware of the long history of bubbles. This has caused a diminished ability to recognize the housing bubble as it was inflating. And worse, the small amount of recent experience we have with bubbles has led to complacency. We were able to escape, for example, the stock bubble crash of 2001 without too much trouble. And other problems such as the Asian financial crisis did not cause anything close to the troubles we had after the housing bubble collapsed, or the troubles other bubbles have caused throughout history.

Economists did not have the historical perspective they needed, and there was confidence that even if a bubble did appear policymakers would be able to clean it up without too much damage. As Robert Lucas said in his 2003 presidential address to the American Economic Association, the “central problem of depression-prevention has been solved.” We no longer needed to worry about big financial meltdowns of the type that caused so many problems in the 1800s and early 1900s. But in reality economists hardly knew what to look for, did not fully understand the dangers, and were hence unconcerned even if they did suspect that housing prices were out of line with the underlying fundamentals.

A second factor is the lack of deep institutional knowledge of the markets academic economists study. Theoretical models are idealized, pared down versions of reality intended to capture the fundamental issues relative to the question at hand. Because of their mathematical complexity, macro models in particular are highly idealized and only capture a few real world features such as sticky prices and wages. Economists who were intimately familiar with these highly stylized models assumed they were just as familiar with the markets the models were intended to represent. But the models were not up to the task at hand,[1] and when the models failedto signal that a bubble was coming there was no deep institutional knowledge to rely upon. There was nothing to give the people using these models a hint that they were not capturing important features of real world markets.

These two disconnects – from history and from the finer details of markets – made it much more likely that economists would certify that this time was different, that fundamentals such as population growth, immigration, financial innovation, could explain the run-up in housing prices.

The model in this paper examines the implications of these two disconnects and shows that whenexperts endorse the idea that this time is different and cause herding toward incorrect beliefs about the future, it increases the likelihood that a large, devastating bubble will occur.

The Model

The model is based upon the Brock and Hommes (1998)generalization of the Lucas (1978) asset pricing model.In the version of the model used here, there are two different beliefs about the evolution of prices, fundamentalists and trend-chasers. Agents switch between these two beliefs endogenously based upon which of the two was the most profitable in the previous time period.[2]The endogenous evolution of the share of agents pursuing each strategy is a key component of the dynamics of the model

Brock and Hommes show that a model of this type can produce chaotic prices, and that the chaotic fluctuations have features that resemble asset price bubbles. In this paper, the conditions under which chaotic fluctuations arise in these models is not the main concern, so a slightly simpler version of the Brock and Hommesmodel is adopted. The model still produces bubbles, but does so without the need to resort to chaos.[3]

The choice to focus on just two belief types, fundamentalists who have rational expectations, and trend-chasers who do not, is based upon two considerations. First, Anufriev and Hommes (2007) present evidence that participants in laboratory experimentstend to adopt a narrow set of distinct belief types, adaptive expectations, weak trend chasing, strong trend chasing, and anchoring adjustment. Thus, trend-chasing in one form or another appears often in laboratory experiments. The second reason comes from the Brock and Hommes paper. Their results show that, among the belief types they consider, trend-chasing is most strongly associated with chaotic fluctuations and bubbles.[4] Since the focus of this paper is the frequency that bubbles appear as herding behavior changes, and since trend-chasing is a common belief type in any case, trend-chasing is one of the belief types adopted here.

The other belief type used in the model, fundamentalist, represents the baseline for the model. This type of agent expects prices to return to their fundamental values in future time periods, and expectations are fully rational. If all agents in the model have fundamentalist beliefs, bubbles do not appear and expectations are fully rational.When beliefs differ, prices can depart from these baseline, fundamental values and produce the bubble dynamics in the model.

The Baseline Model

There is one risky asset and one risk free asset, and the risk-free asset is supplied perfectly elastically at gross return.[5] The price of the risky asset at timeis, and isa stochastic dividend process. Lettingbe the number of shares, wealth in this model evolves according to:

(1)

That is, wealth at time is plus the interest rate times the risk-free wealth carried forward from the last period, plus the excess return on the investment in the risky asset.[6]

The conditional variance of is times the variance of excess returns per share Following Brock and Hommes, let and represent the beliefs of agents of type about the conditional mean and conditional variance, assume that beliefs about the conditional variance are the same for all agents, and also assume thatbased upon time information.

Each investor is assumed to be a myopic mean-variance maximizer. Thus, agents with belief type have a demand for shares, , equal to the solution to the maximization problem:

(2)

The solution is:

(3)

The parameter represents risk aversion which, for simplicity, is assumed to be the same for all agents in the model.

If we let be the supply of shares per investor and be the fraction of investors of type at time , setting demand equal to supply gives:

(4)

If there is only one type , then market equilibrium can be expressed through the pricing equation:

(5)

Now, as in Brock and Hommes assume the special case where the number of outside shares is zero, i.e. assume that. If we let the information set be all variables dated time or earlier, and let denote the fundamental solution for prices, then if dividends follow an i.i.d. process the fundamental solution solves:

(6)

As usual for this type of model, there is an infinite number of solutions, but only one solutionsatisfies the transversality condition:

(7)

Brock and Hommes find it convenient to express the model in terms of deviations from this “benchmark fundamental,” where the deviation is defined as:

(8)
Heterogeneous Beliefs

In this sectionheterogeneous beliefs are introduced into the model. This is accomplished by generalizing the solution given in (7) to allow for different belief types. This gives:

(9)

The set of allowable beliefs are assumed to be of the form

(10)

where a * indicates the fundamental solution. This restricts the deviation of beliefs from the underlying fundamental values to be a deterministic function of past deviations from fundamentals.

The next step is to define how the fraction of traders following various beliefs changes over time. The assumption is that traders look to the last period or a weighted average of previous periods and adopt the strategy that was the most profitable in the past.

Profit in the model for a trader with beliefs of type is defined as:

(11)

Where the term is the i.i.d. shock to the dividend process, e.g. in the specific case discussed in Brock and Hommes, and more generally represents the usual efficient market theory term in these models.

Brock and Hommes also assume a more general function to measure past performance of beliefs of type :

(12)

That is, the measure is a weighted average of past profit. However, for most of what they do, and for the results below, it is assumed that .

The evolution of the shares of agents following each strategy can now be specified. Brock and Hommes assume the following structure:

, where (13)

Notice that the evaluation function is indexed with . That ensures that the fraction of agents pursuing each strategy evolves based upon observable quantities. The parameter represents the “intensity of choice.” When this parameter is zero, the distribution is uniform. When it is infinite, all agents pursue the strategy with the highest profit in the past.

However, the Brock and Hommes model of how the fraction of agents pursuing each strategy evolves over time is not well suited for the application considered here. In particular, the model in this paper begins with all agents pursuing a rational, fundamentalist strategy and then allows some or all of the agents switch to trend-chasing when it looks more profitable based upon past performance. Because all agents begin as fundamentalist with rational expectations, and it’s possible for all agents to switch to trend-chasing as the other polar extreme, the uniform distribution doesnot work as a limiting case.

To overcome this problem, it is assumed that the proportion of agents following the fundamental strategy at time t is , where evolves according to:

if (14)

Thus, all agents begin as fundamentalists, i.e. is set to 1.0 initially, but when profit is higher in the previous period for trend-chasing, of the agents switch to the trend-chasing belief. The parameter , which is assumed to be between zero and one so that the evolution of shares is stationary, captures the speed at which agents switch from the fundamental strategy to trend chasing. In particular, measures the “stickiness” to the fundamentalist strategy, while can be interpreted as the propensity to herd. The shock is white noise.[7]

As agents jump on the trend-chasing bandwagon, the beliefs put upward pressure on prices so that it’s possible for herding to drive prices away from fundamentals for several time periods. However, for reasons explained in detail below eventually the bubble pops causing agents to herd back to the fundamental beliefs. In particular, herding back to the fundamental belief evolves according to:

if (15)

It is assumed that , i.e. that agents herd back to fundamental beliefs faster than they abandon them.[8]

Finally, to capture the idea that poor advice might have increased the propensity for agents to adopt trend-chasing behavior, it is assumed that

(16)

The parameter represents the proportion of experts giving bad advice, and <0, i.e. the propensity to herd toward trend-chasing goes up when bad expert advice is more frequent. The simulations conducted below will show how the probability of bubbles appearing in the model varies with changes in .

Belief Types

The beliefs of both types of agents in the model are restrictedto the linear model adopted by Brock and Hommes:

(17)

The left-hand side term is the belief or expectation of an agent of belief type , and the coefficient is the trend. The constant termrepresents any bias in the beliefs.

Brock and Hommes adopt the following terminology. When and the agent is categorized as a pure trend chaser. If and the agent is a contrarian, and if , the agent is a strong trend chaser or strong contrarian.If , then the agent is said to be purely biased. The agent is upward biasedif , and downward biased if .

When if , the type agents are fundamentalists. This type of agent believes that prices will return to their fundamental values.[9]

Fundamentalists versus Trend Chasers

In the specific application of the model used in this paper, it is assumed that fundamentalists have the beliefs , and that trend chasers, who believe prices follow the process ,, i.e. that prices are trending upward, have the corresponding belief that.[10]

Finally, it is assumed that dividends follow the first-order autoregressive process

(18)

Before proceeding, it may be helpful to summarize the main components of the model:

Baseline asset pricing

Dividends

Asset pricing w/diff beliefs

if , Share of agents of each type

if Share of agents of each type

Herding propensity

Fundamentalist beliefs

Trend-chasing beliefs

This model has the fundamental solution:

(19)

When agents are allowed to switch between fundamental and trend-chasing beliefs, the solution is:

(20)

Equations (19) and (20), along with equation (18) describing the evolution of dividends, are the key equations in the simulations described below.

Simulation of the Model

This section simulates the model, and then examines how the probability of bubbles varies with changes in the parameters of the model.

The simulations are based upon the following baseline values for the parameters:

= 1.0/1.02

= .95

= .90

= 1.05

= 1.0

= .70

= .02

= .075

= .10

The model is parameterized as quarterly data, the simulations run over 1,000 observations or 250 years,[11] and the frequency of bubblesover the 250 year time period is tabulated.[12]

The first graph shown below shows the last 100 quarters of a 1,000 quarter simulation in which there were no large deviations from fundamental values. Thus, the model can deliver periods of calmness.

Figure 1 - Normal Time Period

Figure 2 shows a time period in which a moderate size bubble appears, and Figure 3 shows two bubbles in succession, a relatively large bubble and a smaller bubble following on its heels.

In the cases where bubbles appear, there is generally a time period in which the fundamental prices increase for a period of time. This validates the trend chasing strategy, and the fact that trend-chasing itself puts upward pressure on prices combined with a series of shocks that reinforce the trend-chasing expectation causes a bubble to emerge.

Figure 2 - Small Bubble

However, eventually the bubble pops for one of two reasons. First, the trend-chasers base their expectation of on the lagged value of the price level, i.e. on . In particular, the one-period ahead expectation is . However, since the price level is a function of the lagged value of dividends, trend-chasers are in essence basing their expectation of on. Information on is available, but trend-chasers choose to base their expectations on rather than . Fundamentalists, however, do take advantage of in setting their expectations.[13] This means that sufficiently large differences in dividends in successive time periods can flip which strategy is the most profitable and end a bubble.