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A Bayesian DSGE Model of

Stock Market Bubbles and Business Cycles

∗

Jianjun Miao

†

, Pengfei Wang

‡

, and Zhiwei Xu

September 28, 2012

Abstract

We present an estimated DSGE model of stock market bubbles and business cycles using

Bayesian methods. Bubbles emerge through a positive feedback loop mechanism supported by

self-fulﬁlling beliefs. We identify a sentiment shock which drives the movements of bubbles and

is transmitted to the real economy through endogenous credit constraints. This shock explains

more than 96 percent of the stock market volatility and about 25 to 45 percent of the variations in

investment and output. It generates the comovements between stock prices and macroeconomic

quantities and is the dominant force in driving the internet bubbles and the Great Recession.

Keywords: Stock Market Bubbles, Bayesian Estimation, DSGE, Credit Constraints, Business

Cycle, Sentiment Shock

JEL codes: E22, E32, E44

∗

We thank Paul Beaudry, Francisco Buera, Christophe Chamley, Simon Gilchrist, Timothy Kehoe, Bob King,

Alb erto Martin, Rachel Ngai, Vincenzo Quadrini, Harald Uhlig, Jaume Ventura, and Tao Zha for helpful comments.

We are especially grateful to Zhongjun Qu for numerous conversations and to Zheng Liu and Tao Zha for kindly

providing us with the data. We have also beneﬁtted from comments from participants in the Boston University

Macro Workshop, 2012 AFR Summer Institute of Economics and Finance, and HKUST International Macroeconomics

Workshop. First version: April 2012.

†

Department of Economics, Boston University, 270 Bay State Road, Boston, MA 02215. Email: miaoj@bu.edu.

Tel.: 617-353-6675.

‡

Department of Economics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong. Tel:

(+852) 2358 7612. Email: pfwang@ust.hk

Department of Economics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong.

Email: zwxu@ust.hk. Tel.: (+852) 67334367.

1. Introduction

The U.S. stock market is volatile relative to fundamentals as is evident from Figure 1, which presents

the monthly data of the real Standard and Poor’s Composite Stock Price Index from January 1871

to January 2011, and the corresponding series of real earnings. Two recent boom-bust episodes are

remarkable. Starting from January 1995, the stock market rose persistently and reached the peak

in August 2000. Through this period, the stock market rose by about 1.8 times. This boom is often

attributed to the internet bubble. Following the peak in August 2000, the stock market crashed,

reaching the bottom in February 2003. The stock market lost about 47 percent. After then the

stock market went up and reaching the peak in October 2007. This stock market runup is often

attributed to the housing market bubble. Following the burst of the bubble, the U.S. economy

entered the Great Recession, with the stock market drop of 52 percent from October 2007 through

March 2009.

The U.S. stock market comoves with macroeconomic quantities. The boom phase is often

associated with strong output, consumption, investment, and hours, while the bust phase is often

associated with economic downturns. Stock prices, consumption, investment, and hours worked

are procyclical, i.e., they exhibit a positive contemporaneous correlation with output (see Table 3

presented later).

The preceding observations raise several questions. What are the key forces driving the boom-

bust episodes? Are they driven by economic fundamentals, or are they bubbles? What explains

the comovement between the stock market and the macroeconomic quantities? These questions

are challenging to macroeconomists. Standard macroeconomic models treat the stock market as

a sideshow. In particular, after solving for macroeconomic quantities in a social planner problem,

one can derive the stock price to support these quantities in a competitive equilibrium. Much

attention has been devoted to the equity premium puzzle (Hansen and Singleton (1983) and Mehra

and Prescott (1988)). However, the preceding questions have remained underexplored.

The goal of this paper is to provide an estimated dynamic stochastic general equilibrium (DSGE)

model to address these questions. To the best of our knowledge, this paper provides the ﬁrst

estimated DSGE model of stock market bubbles using Bayesian methods. Our model-based, full-

information econometric methodology has several advantages over the early literature using the

single-equation or the vector autoregression (VAR) approach to the identiﬁcation of bubbles. First,

because both bubbles and fundamentals are not observable, that literature fails to diﬀerentiate

between misspeciﬁed fundamentals and bubbles (see Gurkaynak (2008) for a recent survey). By

contrast, we treat bubbles as a latent variable in a DSGE model. The state space representation of

the DSGE model allows us to conduct Bayesian inference of the latent variables by knowledge of the

observable data. We can answer the question as to whether bubbles are important by comparing

the marginal likelihoods of a DSGE model with bubbles and an alternative DSGE without bubbles.

Second, the single-equation or the VAR approach does not produce time series of the bubble

component and the shock behind the variation in bubbles. Thus, it is diﬃcult to evaluate whether

the properties of bubbles are in line with our daily-life experience. By contrast, we can simulate our

model based on the estimated parameters and shocks to generate a time series of bubbles. Third,

because our model is structural, we can do counterfactual analysis to examine the role of bubbles

in generating ﬂuctuations in macroeconomic quantities.

We set up a real business cycle model with three standard elements: habit formation, invest-

ment adjustment costs, and variable capacity utilization. The novel element of our model is the

assumption that ﬁrms are subject to idiosyncratic investment eﬃciency shocks and face endogenous

credit constraints as in Miao and Wang (2011a,b, 2012a,b), and Miao, Wang, and Xu (2012). Under

this assumption, a stock market bubble can exist through a positive feedback loop mechanism sup-

ported by self-fulﬁlling beliefs. The intuition is as follows. Suppose that households have optimistic

beliefs about the stock market value of the ﬁrm. The ﬁrm uses its assets as collateral to borrow

from the lender. If both the lender and the ﬁrm believe that ﬁrm assets have high value, then the

ﬁrm can borrow more and make more investment. This makes ﬁrm value indeed high, supporting

people’s initial optimistic beliefs. Bubbles can burst if people believe so. By no arbitrage, a rational

bubble on the same asset cannot re-emerge after a previous bubble bursts. To introduce recurrent

bubbles in the model, we introduce exogenous entry and exit. New entrants bring new bubbles in

the economy, making the total bubble in the economy stationary.

We introduce a sentiment shock which drives the ﬂuctuations in the bubble and hence the

stock price. This shock reﬂects households’ beliefs about the relative size of the old bubble to

the new bubble. This shock is transmitted to the real economy through the credit constraints. Its

movements aﬀect the tightness of the credit constraints and hence a ﬁrm’s borrowing capacity. This

aﬀects a ﬁrm’s investment decisions and hence output.

In addition to this shock, we incorporate six

other shocks often studied in the literature: persistent and transitory labor-augmenting technology

(or TFP) shocks, persistent and transitory investment-speciﬁc technology (IST) shocks, the labor

supply shock, and the credit shock. We estimate our model using Bayesian methods to ﬁt the U.S.

data of consumption, investment, hours, the relative price of investment goods, and stock prices.

Stanley and Merton (1984), Barro (1990), Chirinko and Schaller (2001), Baker, Stein, and Wurgler (2003), Goyal

and Yamada (2004), Gilchrist, Himmelberg and Huberman (2005) ﬁnd empirical evidence that investment responds

to the stock market value beyond the fundamentals. See Gan (2007) and Chaney, Sraer, and Thesmar (2009) for

empirical evidence on the relation between collateral constraints and investment.

Our full-information, model-based, empirical strategy for identifying the sentiment shock exploits

the fact that in the theoretical model the observable variables react diﬀerently to diﬀerent types of

shocks. We then use our estimated model to address the questions raised earlier. We also use our

model to shed light on two major bubble and crash episodes: (i) the internet bubble during the

late 1990s and its subsequent crash, and (ii) the recent stock market bubble caused by the housing

bubble and the subsequent Great Recession.

Our estimation results show that the sentiment shock explains more than 96 percent of the

ﬂuctuations in the stock price over various forecasting horizons. It is also the dominant force

in driving the ﬂuctuations in investment in the medium run, explaining about 40 percent of its

variations. Overall, it explains about 25 to 45 percent of the variations in investment and output

over various horizons. Historical decomposition of shocks shows that the sentiment shock explains

almost all of the stock market booms and busts. In addition, it is the dominant driving force

behind the movements in investment during the internet bubble and crash and the recent stock

market bubble and the subsequent Great Recession. The sentiment shock accounts for a large share

of the consumption fall during the Great Recession. But it is not a dominant driver behind the

consumption movements during the internet bubble and crash. For both boom-bust episodes, the

labor supply shock, instead of the sentiment shock, is the major driving force behind the movements

in labor hours.

To see what drives the comovement between the stock market and the macroeconomic quanti-

ties, we compute counterfactual simulations of history from the model based on the estimated time

series of sentiment shocks. We then compute the impulse responses of the stock price, consumption,

investment, and hours following a shock to the stock price from a four-variable Bayesian vector au-

toregression (BVAR) based on the simulated data. We compare these responses to those estimated

from the actual data. We ﬁnd that the impulse responses from the simulated data conditional on

the sentiment shock alone mimic those from the actual data, suggesting that the sentiment shock

is the major driver of the comovement.

The intuition behind the comovement is as follows. In response to a positive sentiment shock,

the bubble and stock price rise. This relaxes credit constraints and hence raises investment. But

Tobin’s marginal Q falls, causing the capacity utilization rate to rise. This induces the labor

demand to rise. The wealth eﬀect due to the rise in stock prices causes consumption to rise and the

labor supply to fall. It turns out that the rise in the labor demand dominates the fall in the labor

supply, and hence labor hours rise. The increased hours and capacity utilization raises output.

In our model, the sentiment shock is an unobserved variable. We infer its properties from our

ﬁve time series of the U.S. data using an estimated model. Given its importance for the stock

market and business cycles, one may wonder whether there is a direct measure of this shock. We

ﬁnd that the consumer sentiment index published monthly by the University of Michigan and

Thomson Reuters is highly correlated with our sentiment shock (the correlation is 0.61).

Thus,

this index can provide an observable measure of the sentiment shock in our model and should be

useful for understanding the stock market and business cycles.

It is challenging for standard DSGE models to explain the stock market booms and busts.

One often needs a large investment adjustment cost parameter to make Tobin’s marginal Q highly

volatile. In addition, one also has to introduce other sources of shocks to drive the movements

of the marginal Q b ecause many shocks often studied in the literature cannot generate either the

right comovements or the right relative volatility. For example, the TFP shock cannot generate

large volatility of the stock price, while the IST shock generates counterfactual comovements of

the marginal Q (hence stock prices) and the relative price of investment goods if both series are

used as observable data. The credit shock typically makes investment and consumption move in

an opposite direction and makes the marginal Q move countercyclically.

Recently, two types of shocks have drawn wide attention: the news shock and the risk (or

uncertainty) shock. The idea of the news shock dates back to Pigou (1926). It turns out that

the news shock cannot generate the comovement in a standard real business cycle model (Barro

and King (1984) and Wang (2012)). To generate the comovement, Beaudry and Portier (2004)

incorporate multisectoral adjustment costs, Christiano et al. (2008) introduce nominal rigidities

and inﬂation-targeting monetary policy, and Jaimovich and Rebelo (2009) consider preferences that

exhibit a weak short-run wealth eﬀect on the labor supply. These three papers study calibrated

DSGE models and do not examine the empirical importance of the news shock.

Fujiwara, Hirose,

and Shintani (2011) and Schmitt-Grohe and Uribe (2012) study this issue using the Bayesian

DSGE approach. Most Bayesian DSGE models do not incorporate stock prices as observable data

for estimation. As Schmitt-Grohe and Uribe (2012) point out, “as is well known, the neoclassical

model does not provide a fully adequate explanation of asset price movements.”

By incorporating the stock price data, Christiano, Motto, Rostagno (2010, 2012) argue that the

risk shock, related to that in Bloom (2009), displaces the marginal eﬃciency of investment shock

and is the most important shock driving business cycles.

They also introduce a news shock to the

The consumer conﬁdence index issued monthly by the Conference Board is also highly correlated with our

smo othed sentiment shock. The correlation is 0.5.

Beaudry and Portier (2006) study the empirical implications of the news shock using the VAR approach.

In Section 6.8 of their paper, Schmitt-Grohe and Uribe (2012) discuss brieﬂy how the share of unconditional

variance explained by anticipated shocks will change when stock prices are included as observable data. But they do

not include stock prices in their baseline estimation.

It is diﬃcult for shocks to the TFP shock’s variance (uncertainty shocks) to generate comovements among

investment, consumption, hours, and stock prices in standard DSGE models (see, e.g., Basu and Bundick (2011)).

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