Modeling Uncertainty and Disagreement

We consider a simple specification for the uncertainty and disagreement in our economy. Economic agents update their beliefs based on the available information using Bayes’ rule. The difference in their posteriors can arise either from a difference in agents’ priors or a difference in some subjective parameter value of the dynamics of cash flows or signals. In the second case, a parsimonious model can be based on the realistic assumption that the uncertainty parameterσµz of the market-wide expected component

µz(t) is agent dependent. We follow this assumption below when we derive a set of testable empirical predictions of our model.

Given that the model–implied state dynamics is conditionally Gaussian, the Bayesian prior updating rules of each agent follow with standard arguments, and the heterogeneity in beliefs can be completely summarized by the differences in meansmi(t) and covariance matrices γi_{(t) across agents.}

Let mi(t) := (mi_A(t), mi_z(t))0 _{:= E}i _(µ

A(t), µz(t))0|FtY

, where FY

t := FtA,z is the in- formation generated by A(t) and z(t) up to time t, and Ei_{(·) denotes expectation rel-} ative to the subjective probability of investor i= 1,2. Let also Y(t) = (logA(t), z(t)),

b1 _{= diag(σ}1

µA, σ

1

µz), a0 = (a0A, a0z)0, a1 = diag(a1A, a1z), B = diag(σA, σZ) and

A= 1 0

α β

!

. The (posterior) belief dynamics of agent one then follows by a standard application of the Kalman-Bucy filter:

dm1(t) = (a0+a1m1(t))dt+γ1(t)A0B−1dWY1(t), (2.1)

dγ1(t)/dt = a1γ1(t) +γ1(t)a01+b1b10−γ 1

(t)A0_(BB0₎−1_Aγ1_{(t), (2.2)} with initial conditionsm1_{(0) =m}1

0andγ1(0) =γ01, wheredWY1(t) :=B−1 dY(t)−Am1(t)dt

is the innovation process induced by the first investor’s belief and filtration.7 To com- pletely specify the disagreement structure in the economy, we finally specify the dis- agreement process implied by the learning dynamics of agent two. This process is the

7_{A formal proof of this result can be found in Liptser and Shiryaev (2000); see also the technical}

key state variable driving all equilibrium quantities in our economy. It is defined by the two dimensional process:

Ψ(t) := ΨA(t) Ψz(t) ! = (m 1 A(t)−m2A(t))/σA (m1 z(t)−m2z(t))/σz ! .

The first component of Ψ(t) measures disagreement about the expected growth rate of future firm cash flows, while the second component measures the disagreement about the market-wide signalz(t). Since the market wide uncertainty parameterσµz influences

the subjective dynamics of each individual belief mi(t), it also has implications for the stochastic properties of the disagreement process itself. The dynamics for Ψ(t) follows directly:

dΨ(t) =B−1 _a

1B+γ2(t)A0B−1Ψ(t)dt+B−1(γ1(t)−γ2(t))A0B−1dWY1(t), (2.3) with initial conditions Ψ(0) = (m1

A(0)−m2A(0))/σA,(m1z(0)−m2z(0))/σz, where γ2(t) satisfies the same differential equation as γ1_{(t), but with agent specific parameter b}1 replaced byb2. The dynamics of m1(t) and Ψ(t) completely characterize the beliefs in- duced by investors’ priors and filtrations. Heterogeneity in prior beliefs alone is sufficient to let investors disagree aboutµA(t) andµz(t) at all times, even when they agree on the dynamics of cash flows and signals (i.e. b1 _=b2_{), but in this case heterogeneity in beliefs} vanishes asymptotically.8 If prior variances are also identical (γ1(0) =γ2(0)) then the disagreement process is deterministic, as in the model solved by Buraschi and Jiltsov (2006). We consider a model with truly stochastic disagreement dynamics, in which heterogeneity in beliefs does not vanish asymptotically, by assuming that b1 ₆=b2, i.e., we allow for the presence of heterogeneous subjective uncertainty across agents. Note that since both µA(t) and µz(t) are unobservable, the parameter bi cannot be uniquely inferred by investors from the quadratic variation of these processes. Moreover, we emphasize that the average level and the heterogeneity of the subjective uncertainty parameters across agents both impact directly on the steady-state distribution of the disagreement process implied by dynamics (3.1). This feature offers a natural interesting link between economic uncertainty and the stochastic properties of the heterogeneity in beliefs, which motivates the use of belief disagreement proxies as indicators of economic uncertainty in our tests of the main empirical predictions of the model.

8_{Acemoglu, Chernozhukov, and Yildiz (2008) show that when agents are uncertain about a random}

variable and about the informativeness of a signal even an infinite sequence of signal observations does not lead agents’ heterogeneous prior beliefs to converge.

Remark: Our way of modeling disagreement and the economy with heterogeneous beliefs is related to, but distinct from models that study the impact of behavioral biases on asset prices. In Scheinkman and Xiong (2003) and Hong, Scheinkman, and Xiong (2006), risk neutral investors subject to short-selling constraints update their beliefs based on their personal interpretations of incoming news. It follows that trading occurs whenever investors valuations “cross”, i.e., whenever the more optimistic investor switches to being the more pessimistic. In a world in which investors interpret news differently, the greater news stimulus associated with an asset leads to a higher time-series variability of investors’ relative valuations and larger trading volumes via the “crossing”. At the same time, the short-sales constraint makes some assets overpriced relative to others, so that their valuation reflects more the opinion of the optimist. In contrast, in an economy without constraints the stock price simply reflects the value-weighted average opinion.9 In our model, agents are rational, but risk averse, and short-selling is allowed. The local volatility of the growth rate of the underlying state variables is unknown and cannot be estimated from the quadratic variations of the drift, because the drift itself is unobservable. Thus, agents can disagree without imposing an axiomatic behavioral bias assumption. Even without short-selling constraints, uncertainty affects asset prices in our economy. This is consistent with the less important role of short-selling restrictions in bond markets; see, e.g., Longstaff, Mithal, and Neis (2005).