Learning about Consumption Dynamics

• Date: 01 April 2016
• DOI: http://doi.org/10.1111/jofi.12246
• Published date: 01 April 2016
• Author: YIQUN MOU,LARS A. LOCHSTOER,MICHAEL JOHANNES

THE JOURNAL OF FINANCE •VOL. LXXI, NO. 2 •APRIL 2016

Learning about Consumption Dynamics

MICHAEL JOHANNES, LARS A. LOCHSTOER, and YIQUN MOU∗

ABSTRACT

This paper characterizes U.S. consumption dynamics from the perspective of a

Bayesian agent who does not know the underlying model structure but learns over

time from macroeconomic data. Realistic, high-dimensional macroeconomic learning

problems, which entail parameter, model, and state learning, generate substantially

different subjective beliefs about consumption dynamics compared to the standard,

full-information rational expectations benchmark. Beliefs about long-run dynamics

are volatile, with counter-cyclical conditional volatility, and drift over time. Embed-

ding these beliefs in a standard asset pricing model significantly improves the model’s

ability to match the stylized facts, as well as the sample path of the market price-

dividend ratio.

AT THEIR CORE,CONSUMPTION-BASED ASSET pricing theories link beliefs about

macroeconomic outcomes and aggregate asset prices (Lucas (1978)). Funda-

mentally, how do these beliefs arise? Traditional asset pricing models presume

that agents know the “true” structure of the economy, that is, the model specifi-

cation and the parameters. As Hansen (2007) argues, however, this assumption

is extreme.1Instead of being endowed with full structural knowledge of the

economy, agents in reality must form their beliefs about states, parameters,

and models via difficult, high-dimensional learning problems, similar to those

confronting econometricians. This paper studies these learning problems from

the perspective of a Bayesian agent learning about aggregate consumption

dynamics from observed post–World War II macroeconomic data.

∗Michael Johannes and Lars A. Lochstoer are at Columbia Business School, Department of Fi-

nance and Economics. Yiqun Mou is at Clarke Capital Management. Wewould like to thank Fran-

cisco Barillas (CEPR Barcelona discussant), Pierre Collin-Dufresne, Kent Daniel, Lars Hansen

(AFA discussant), ˇ

Luboˇ

sP

´

astor (NBER discussant), Tano Santos, two anonymous referees and the

Editor, as well as seminar participants at AFADenver 2011, CEPR conference in Barcelona (May

2011), Columbia University, NBER Asset Pricing meeting in Chicago 2011, Norwegian School of

Economics, Stanford, University of Lausanne, University of Texas at Austin, and the University

of Wisconsin at Madison for helpful comments. The authors do not have any potential conflicts of

interest, as identified in the Journal of Finance’s Disclosure Policy.

1Hansen (2007, p. 2) states: “In actual decision making, we may be required to learn about

moving targets, to make parametric inferences, to compare model performance, or to gauge the

importance of long-run components of uncertainty. As the statistical problem that agents confront

in our model is made complex, rational expectations’ presumed confidence in their knowledge of

the probability specification becomes more tenuous. This leads me to ask: (a) how can we burden

the investors with some of the specification problems that challenge the econometrician, and (b)

when would doing so have important quantitative implications.”

DOI: 10.1111/jofi.12246

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552 The Journal of Finance R

Our main contribution is empirical. We document that these realistic and dif-

ficult learning problems generate subjective beliefs about consumption dynam-

ics that differ substantially from those generated by standard implementations

of the same models, which fix parameters at the most likely full-sample values

and assume the model is known. In particular, over the postwar sample, we em-

pirically find that uncertainty about the underlying economic structure is high

and resolves slowly, leading to beliefs that drift considerably over time. Shocks

to beliefs about long-run properties of consumption dynamics are highly volatile

and strongly countercyclical, due to parameter and model uncertainty. Since

the market portfolio is a long-duration asset, its price is particularly sensitive

to these long-run shocks (see, for example, Barsky and De Long (1993)and

Timmermann (1993)). Embedded in an equilibrium asset pricing model, the

specific time series of beliefs generated over the postwar period helps account

for the sample path of the market price-dividend ratio as well as the standard

asset pricing moments observed over the same sample.

The multidimensional nature of our learning is a key differentiator from

extant work that focuses on learning about latent states or a single parameter

(see P´

astor and Veronesi (2009) for a review). A crucial aspect of our learning

problem is confounding, which occurs when uncertainty about one quantity

makes learning about another quantity more difficult. For example, it is more

difficult to learn about persistent latent state variables when the parameters

governing state dynamics are unknown, and vice-versa. Joint learning about

parameters, states, and models magnifies total uncertainty, slows down the

learning process, and is crucial for empirically relating updates in estimated

beliefs and asset prices.

Given this overview, we now describe the main features of our analysis. We

study learning using popular Markov switching models of consumption growth:

unrestricted two- and three-state models and a restricted two-state model gen-

erating i.i.d. consumption growth. The hidden states capture business cycle

fluctuations and can be labeled “expansion” and “recession” in two-state mod-

els, with an additional “depression” state in three-state models. Our agent does

not know the parameters, states, or specific model, and uses Bayes’s rule to up-

date beliefs from realized consumption data, as well as additional data such as

GDP growth, over the postwar sample. We calibrate the priors using historical

macroeconomic data.

Empirically, we find that parameter estimates and model probabilities drift

over time. In particular, the agent perceives a strong secular decline in con-

sumption growth volatility, longer expected duration of expansions, shorter

expected duration of recessions, and a decline in the probability of large drops

in consumption growth. The nonstationarity of beliefs is not surprising. In fact,

it is a signature of parameter and model uncertainty as beliefs are martingales,

which implies that shocks to beliefs are permanent.2

2This is easy to see by iterated expectations: for a given parameter θ,Eθ|yt=

EEθ|yt+1|yt,where ytdenotes data up to time t.

Learning about Consumption Dynamics 553

Despite a rather long sample, there is still substantial uncertainty about

various model parameters and states at the end of the sample. Thus, learning

is difficult. We also find that the speed of learning (the rate at which the

posterior standard deviation declines) varies substantially across parameters.

This is due to three factors: (i) it is more difficult to learn about, for example,

the persistence of a state than the volatility of shocks within the state, (ii) some

states, such as recessions or depressions, are not commonly observed, and (iii)

confounding. As an example of the latter, it is harder to learn the parameters

governing the state dynamics when states are unobserved.

We quantify confounding using simulations by comparing two cases: learn-

ing about parameters with known states, and “confounded” learning about

both parameters and states. For samples of the sizes we observe, there are

very strong confounding effects as learning about parameters, especially those

governing rare states, is much slower when states are unknown. Agents find

it difficult to determine from noisy data whether shocks are transient (tradi-

tional i.i.d. errors), reflect a state transition, or reflect a change in beliefs about

parameters.

One of our main contributions is to empirically identify subjective long-run

risks, which are of first-order importance for asset pricing (see, for example,

Barsky and De Long (1993), Bansal and Yaron (2004), and Collin-Dufresne,

Johannes, and Lochstoer (2016)), from macroeconomic data alone. We show

that learning induces significant time-variation in long-run beliefs. Defining

long-run shocks as changes in beliefs about expected discounted long-run con-

sumption growth, we find that the volatility of long-run shocks from a two-state

model with parameter uncertainty is more than three times the volatility of

the fixed-parameters counterpart over the postwar sample. Thus, these long-

run risks are driven mainly by parameter and not state uncertainty. Long-run

shocks are largest during recessions, as there is more uncertainty about the

parameters governing infrequent bad states (see also Chen, Joslin, and Tran

(2012)), and therefore there is more updating when these states are visited.

This contributes to the high volatility of returns in recessions.

The timing of estimated belief revisions is important for the empirical rel-

evance of the learning channel: if the agent’s subjective beliefs change, then

asset prices should change at the same time. To test this conjecture, we regress

quarterly excess stock market returns on quarterly revisions in beliefs about

expected consumption growth and find a positive and highly significant rela-

tionship. These regressions control for contemporaneous realized consumption

growth, as well as updates in beliefs about expected consumption growth from

the fixed-parameter models. Realized returns are also significantly negatively

related to shocks to predictive consumption growth volatility. This is a par-

ticularly stringent test of a macroeconomic learning story since beliefs are

estimated sequentially using only available macroeconomic information.

We embed these subjective beliefs into an equilibrium asset pricing model

assuming Epstein-Zin (1989) preferences. At each time t, we price a levered

claim to future consumption given beliefs over parameters, models, and states,

computing quantities such as ex ante expected returns and dividend-price