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AMBIGUITY AND INFORMATION PROCESSING IN A MODELOF INTERMEDIARY ASSET PRICING

LEYLA JIANYU HAN, KENNETH KASA, AND YULEI LUO

Abstract. This paper incorporates ambiguity and information processing constraintsinto a model of intermediary asset pricing. Financial intermediaries are assumed topossess greater information processing capacity. Households purchase this capacity, andthen delegate their investment decisions to intermediaries. As in He and Krishnamurthy(2012), the delegation contract is constrained by a moral hazard problem, which givesrise to a minimum capital requirement. Both agents have a preference for robustness,reflecting ambiguity about asset returns (Hansen and Sargent (2008)). We show thatambiguity aversion tightens the capital constraint, and amplifies its effects.

Indirect inference is used to calibrate the model’s parameters to the stochasticproperties of asset returns. Detection error probabilities are used to discipline the degreeof ambiguity aversion. The model can explain both the unconditional moments of assetreturns and their state dependence, even with DEPs in excess of 20%.

Keywords: Ambiguity, Information Processing, Asset Pricing, Financial Crisis.

JEL Classification Numbers: D81, G01, G12

1. Introduction

In a pair of influential papers, He and Krishnamurthy (2012, 2013) [henceforth HK12,HK13] argue that for many assets it is misleading to characterize prices using householdEuler equations. This is because many assets are not held by households. They areheld by financial intermediaries. Although these intermediaries may be investing onbehalf of households, the contractual relationships between them are plagued by frictions.In HK12, asymmetric information produces a moral hazard problem that leads to acapital constraint, requiring the intermediary to maintain a minimum amount of ‘skinin the game’. Due to these frictions, HK argue that it is better to relate asset pricesto the marginal value of intermediary wealth than to the marginal utility of householdconsumption. The resulting stochastic discount factor becomes more volatile, and the

Date: July, 2019.Leyla Jianyu Han: Faculty of Business and Economics, University of Hong Kong, [email protected];Kenneth Kasa: Department of Economics, Simon Fraser University, [email protected]; Yulei Luo: Faculty ofBusiness and Economics, University of Hong Kong, [email protected] would like to thank Hengjie Ai, Daniel Andrei, Martin Ellison, Paolo Fulghieri, Xiaowen Lei, Erica Li,Andrey Malenko, Jun Nie, Stavros Panageas, Yu Xu, Eric Young, Lei Zhang and seminar and conferenceparticipants in University of Hong Kong, New York University, 2019 American Economic AssociationAnnual Meeting, 2019 China International Conference in Macroeconomics, 2019 Asian Meeting of theEconometric Society, 2019 Summer Institute of Finance Conference, 31st Australasian Finance and BankingConference for helpful comments. Luo thanks the General Research Fund (GRF No. HKU791913) in HongKong for financial support.

1

2 LEYLA JIANYU HAN, KENNETH KASA, AND YULEI LUO

nonlinearity induced by the constraint can account for observed state dependence in riskpremia.1

Although the work of He and Krishnamurthy has been influential, it has not goneunquestioned. The key premise of HK12,13 is that some securities are too ‘complex’ forhouseholds to understand, so they delegate investment in these securities to specialists,whose actions cannot be precisely monitored. Cochrane (2017) questions how widespreadand insurmountable this complexity problem really is,

Furthermore, if there is such an extreme agency problem, that delegatedmanagers were selling during the buying opportunity of a generation, whydo fundamental investors put up with it? Why not invest directly, orfind a better contract?...So, in my view, institutional finance and smallarbitrages are surely important frosting on the macro-finance cake, neededto get a complete description of financial markets in times of crisis...Butare they also the cake?...Or can we understand the big picture of macro-finance without widespread frictions, and leave the frictions to understandthe smaller puzzles, much as we conventionally leave the last 10 basis pointsto market microstructure. (Cochrane (2017, p. 963-64))

Perhaps in anticipation of this critique, HK13 confine their empirical analysis to the marketfor mortgage-backed securities.

In this paper, we argue that intermediary asset pricing is indeed ‘the cake’. Werespond to Cochrane’s skepticism in two ways. First, we operationalize the vague notionof ‘complexity’ by assuming agents face limits on their ability to process information,giving rise to Rational Inattention (Sims (2003)). Mean dividend growth is assumed tobe stochastic and unobserved, so agents must solve a filtering problem. Intermediariesare assumed to have greater channel capacity, which enables them to solve this problemwith less estimation risk. By delegating their portfolios to intermediaries, households caneffectively purchase this additional capacity. Although households are free to invest forthemselves, most choose not to do so.2

Just as Merton’s (1973) ICAPM can co-exist with the Lucas (1978) model, our modeldoes not require that all agents delegate. What is important is that incentive constrainedintermediaries are the marginal investor, making their marginal value of wealth a validstochastic discount factor. It could well be that sophisticated private investors are alsomarginal. If so, their marginal utilities of consumption could also serve as stochasticdiscount factors. Unfortunately, we don’t observe their consumption. Hence, the case forIntermediary Asset Pricing is really a pragmatic and empirical one, based on the conjecturethat relatively well measured financial intermediary data is more closely related to assetprices than poorly measured consumption data.3

1Brunnermeier and Sannikov (2014), Adrian and Boyarchenko (2012), and Li (2013) develop related models.He and Krishnamurthy (2018) survey the subsequent literature.2Kacperczyk, Nosal, and Stevens (2018) argue that differences in information-processing capacitycontribute to wealth inequality, but do not allow agents to buy and sell this information-processing capacity.Pagel (2018) also bases portfolio delegation on inattention. However, in her model inattention is not basedon information processing limits, but rather on ‘information avoidance’, which arises from loss aversion.3Mankiw and Zeldes (1991) note that the consumption-CAPM works better using data on households thatown stocks. There are still significant shortcomings, however, which perhaps reflect the poor quality of

AMBIGUITY AND INFORMATION PROCESSING 3

Our second response to Cochrane is to suppose that investment is subject to KnightianUncertainty, or equivalently, ambiguity. Our approach is based on the work of Hansen andSargent (2008). Agents are assumed to have a correctly specified benchmark model of assetreturns, which they distrust in a way that cannot be captured by a conventional finite-dimensional Bayesian prior. Rather than commit to a single model/prior, agents entertaina set of unstructured alternative models, and then optimize against the worst-case model.Since the worst-case model depends on an agent’s own actions, agents view themselvesas being immersed in a dynamic zero-sum game. Solutions of this game produce robustportfolio policies. To prevent agents from being unduly pessimistic, in the sense that theyattempt to hedge against empirically implausible alternatives, the hypothetical ‘evil agent’who selects the worst-case model is required to pay a penalty that is proportional to therelative entropy between the benchmark model and the worst-case model.4

We show that ambiguity aversion tightens the capital constraint, and amplifies itseffects. With ambiguity, incentive constraints can bind in otherwise normal times. Afinancial crisis isn’t required. This makes intermediary asset pricing more than just‘frosting on the cake’.5 The reason ambiguity tightens the constraint is subtle. In contrastto Maenhout (2004), we do not scale the entropy penalty parameter by the value function.A constant entropy penalty produces horizon effects in portfolio choice. In particular,the effective degree of ambiguity aversion depends on an agent’s time preference. Agentswith a low rate of time preference are endogenously more ambiguity averse, since theycare more about the future. Following HK12, we assume specialists are more patientthan households, which makes them more ambiguity averse. As a result, their pessimisticdrift distortions are greater. This is important because it allows households to survivein the long-run, despite their greater impatience (Yan (2008)).6 Amplification occursbecause the relative pessimism of specialists increases once the constraint binds. As inHK12, when the constraint binds the relative risk exposure of specialists increases. Sincesubjective uncertainty ‘hides behind’ objective risk, the increasing leverage of specialistsmakes them endogenously more pessimistic. Since they are the marginal investor, theirrelative pessimism increases risk premia and the price of risk.7

PSID consumption data. Adrian, Etula, and Muir (2014) and He, Kelly, and Manela (2017) show thatleverage is a significantly priced factor in traditional cross-sectional regressions. Adrian, Moench, and Shin(2016) show that leverage also has time-series predictive ability.4There is a vast literature on asset pricing and Knightian Uncertainty. Epstein and Schneider (2010) surveythe early literature. Brenner and Izhakian (2018) provide a more recent example. Our contribution is tocombine Knightian Uncertainty with intermediation frictions and heterogeneous agents.5Caballero and Krishnamurthy (2008) also develop a model that combines scarce intermediary capital andKnightian Uncertainty. However, they focus on the role of central bank intervention rather than on assetpricing.6In contrast, the model in HK12 does not possess a nondegenerate stationary equilibrium, which makesit difficult to evaluate empirically. HK13 remedies this defect by introducing nontradeable labor income.However, to keep the analysis tractable, they assume households live for a single-period and have a ratherimplausible bequest motive.7The state dependence in the degree of robustness here bears some resemblance to the business cycle modelof Bidder and Smith (2012). However, the stochastic volatility in their model is exogenous. Here it arisesendogenously via equilibrium portfolio policies. It also bears some resemblance to the heterogeneous riskaversion models of Chan and Kogan (2002) and Garleanu and Panageas (2015). Once again, the differencehere is that heterogeneity is endogenous.


Although we are able to derive explicit expressions for asset prices and the distribution ofwealth, these processes are highly nonlinear, and feature an occasionally and endogenouslybinding constraint, which makes the model difficult to fit to the data using conventionalmethods. In response, we use the simulation-based methodology of ‘indirect inference’(Gourieroux, Monfort, and Renault (1993)). We assess fit using two types of auxiliaryfunctions. The first consists of simple unconditional means and variances of asset returns,price/dividend ratios, and risk premia. Since there are many frictionless models thatcan match unconditional moments, our second type of auxiliary functions are designedto assess the more challenging task of matching the persistence and cyclicality of assetreturns. In particular, we compare model-implied and empirically estimated autoregressiveprocesses for risk premia and price/dividend ratios. The free parameters consist ofdividend volatility, rates of time preference, and entropy penalty parameters designedto capture the degree of robustness in control and filtering.

With the exception of the price/dividend ratio, we find that the model does well atmatching unconditional moments. This is perhaps not too surprising given the results ofBarillas, Hansen, and Sargent (2009), who show that model uncertainty can explain theequity premium and risk-free rate puzzles, although it is noteworthy that we can do sowith higher detection error probabilities. The model does somewhat less well at capturingstate dependence. Although it can replicate the persistence of the equity premium andprice/dividend ratio, it generates only a 100-150 basis point increase in the equity premiumduring ‘crises’, and understates movements in the price/dividend ratio. We suspect thatpart of the problem here is that our model-implied relative wealth variable is a poor proxyfor financial sector capital. To check this we use data from He, Kelly, and Manela (2017).They construct market equity capital ratios for approximately two dozen New York Fedprimary dealers for the period 1970-2012. These are firms like JP Morgan, GoldmanSachs, and Citigroup. With this data, the model generates greater state dependence inrisk premia and the price/dividend ratio.

The remainder of the paper is organized as follows. The next section briefly discussesthe ongoing trend toward delegated wealth management, and compares our approach tothe already voluminous literature on this topic. Section 3 outlines the information andmarket structure, and the objective functions of households and specialists. Section 4imposes market-clearing, and solves for equilibrium asset prices. As in HK12/13, thekey endogenous state variable is the distribution of wealth. We show that ambiguityinduced heterogeneous beliefs are the crucial frosting on the Intermediary Asset Pricingcake. Section 5 outlines our indirect inference estimation strategy, and compares modelpredictions to US asset market data. Section 6 describes the simulation methodologywe use to compute detection error probabilities. We show that our agents’ doubts areempirically plausible. Finally, Section 7 provides a few concluding remarks and offersextensions for future research. A technical Appendix contains proofs and derivations.

2. Delegated Wealth Management

Between 1980 and 2006 the share of US equities held by institutional investors (e.g.,mutual funds and hedge funds) increased from 32% to 68% (Lewellen (2011)). Althoughprecise data do not exist, it is widely suspected that institutional investors hold even a


higher share of derivative securities. Growth in wealth management is even present invalue-added GDP statistics. During the same time period, the share of GDP accountedfor by financial services grew from 4.9% to 8.3% (Greenwood and Scharfstein (2013)). (Ithas since declined a bit, to 7.5% in 2018).

Not surprisingly, this growth has attracted the attention of financial economists. Muchof the early work was empirical. It asked whether professional money managers earnpositive (risk-adjusted) returns. There is by now a consensus that they do not. Forexample, a widely cited calculation by French (2008) suggests that the average investorcould increase his annual returns by 67 basis points by following a simple passive, market-portfolio, investment strategy.8

More recently, financial economists have turned their attention to the equilibriumpricing implications of delegated wealth management. The Intermediary Asset Pricingliterature is a direct descendant of this literature. Early work by Chevalier and Ellison(1997) showed that induced payoff convexity caused by the relationship between fundperformance and customer flows can create potential conflicts of interest between funds andtheir customers. Kaniel and Kondor (2012) introduce portfolio delegation and a convexflow/performance function into a Lucas-tree economy. They show that portfolio delegationcan generate increased Sharpe ratios, and can explain why on average funds outperformthe market during downturns, while underperforming during booms. Basak and Pavlova(2013) study the equilibrium pricing implications of another commonly observed featureof delegated wealth management, i.e., ‘benchmarking’, wherein funds are evaluated basedon their returns relative to a benchmark index (e.g., the S&P500). They show thatbenchmarking can generate countercyclical Sharpe ratios and increase market volatility.Compared to this literature, the HK12/13 models are relatively simple, in that customerflows do not depend on performance. Instead, the HK models focus on the nonlinearitythat is induced by an occasionally binding incentive constraint. Consequently, they aremore designed to address the asset pricing implications of financial crises.

As noted by Cochrane (2017), a potential criticism of the HK12/13 models is thatthey do not explain why households delegate. The absence of measurable excessreturns from delegation suggests that other motivations must be at work. Our modelmotivates delegation by introducing heterogeneity in information processing. Meanreturns are unobserved, and households delegate because financial institutions reduceestimation risk. Recent work by Gennaioli, Shleifer, and Vishny (2015) and Kacperczyk,Van Nieuwerburgh, and Veldkamp (2016) provides similar motivations. Gennaioli, Shleifer,and Vishny (2015) argue that delegation is based on ‘trust’, not performance, where trustis defined to be a reduction in the subjective perception of risk. Our model simply makesthis explicit by introducing filtering and estimation risk. Kacperczyk, Van Nieuwerburgh,and Veldkamp (2016) is even more closely related, as it is explicitly based on informationprocessing constraints and Rational Inattention. However, instead of focusing on assetprices, they focus on the cyclical allocation of attention. Their model predicts that fundmanagers focus on aggregate shocks during recessions, while focusing more on idiosyncraticshocks during booms.

8Berk and Green (2004) argue that in equilibrium the net-of-fees excess return should be zero.


3. Information, Market Structure, and Preferences

3.1. The HK12 Model. Our contribution is to introduce robust control and filteringinto the HK12 model. Before doing that, it is useful to begin by briefly summarizing theirmodel. Consider an infinite horizon continuous-time Lucas (1978) endowment economypopulated by two types of agents, specialists and households. There are two assets: onerisky and one risk-free. Only specialists can invest in the risky asset. However, in contrastto traditional segmented-markets models (e.g., Basak and Cuoco (1998)), households canindirectly invest in the risky asset by delegating some of their wealth to the specialist.At every t, households invest in intermediaries run by specialists. The relationship isshort term, i.e. only lasts from t to t + dt. Household’s cannot observe the portfoliochoices of the specialist, nor can they observe its ‘effort’ level. These unobserved choicesproduce a moral hazard problem. Figure 1 depicts the economy’s market structure. Theintermediary sector is indicated in the middle block.

!" #!" − %"%"

riskless

!", -%",!", − %", riskless

ExpertWealth

Intermediary Capital %".

HouseholdWealth

/". -0"/".(1 − 0")/".

≡ /"≡ /",

5"

1 − 5" 1 − /". riskless

Kt

Specialist

!"

Figure 1: Market Structure and Intermediation Relationship

Specialist wealth is Wt, and household wealth is W ht . Households allocate T ht to

purchase intermediary equities, and the remaining fraction is used to buy riskless bonds.Intermediaries absorb in total T It dollars, T ht from households and Tt from from their ownwealth. They then allocate a fraction αt to the risky asset and 1−αt to the riskless bond.Assuming there is no short-selling constraint for the intermediary, we expect αt to belarger than 1, i.e., specialists use leverage. In this case, specialists invest more than totalintermediary capital into risky equity and borrow (αt − 1)T It from the bond market. Thetotal risky asset exposure of the intermediary is εIt . Through an affine contract developedby HK12, βt ∈ [0, 1] is the share of returns going to specialist, while 1 − βt goes tohouseholds. Thus, at time t, the specialist bears a total risk exposure of εt = βtε

It , and

the household is offered an exposure of (1− βt)εIt .In practice, wealth management typically involves two fees - a one-time fixed cost,

and an ongoing variable cost, typically expressed as a percentage of profits. HK12 onlyconsiders the variable cost, denoted Kt. It is an endogenous variable. Later we model the


fixed cost K using filtering and information-processing constraints. It will be determinedby differences in channel capacity.

The population measures of households and specialists are normalized to one. Both areinfinitely lived and have log preferences over consumption. Denote households (specialists)consumption rate as Cht (Ct). The household’s objective is to:

maxCt,εht

E[∫ ∞

0e−ρ

ht lnCht dt

], (3.1)

while the specialist’s objective is to:

maxCt,εt,βt

E[∫ ∞

0e−ρt lnCtdt

], (3.2)

where ρh and ρ denote the time discount rates for households and specialists, respectively.The budget constraints are

dW ht = εht (dRt − rtdt)− ktεht dt+W h

t rtdt− Cht dt, (3.3)

and

dWt = εt(dRt − rtdt) + max

(1− βtβt

)ktε∗t

βt∈[ 11+m

,1]

+Wtrtdt− Ctdt, (3.4)

The endogenous risky asset return is defined as:

dRt =Dtdt+ dPt

Pt= µR,tdt+ σR,tdZt, (3.5)

where Pt is the risky asset price, µR,t is the expected return, and σR,t is the volatility of therisky asset. The riskless asset is in zero-net supply, and has return rt. The risk premiumis defined as: πR,t ≡ µR,t − rt. The equilibrium per-unit-of-exposure intermediation fee iskt, and

qt ≡Kt

Wt=

(1− βtβt

)πR,tσR,t

kt.

are total variable intermediation fees, expressed as a share of specialist wealth.Households obtain an exposure εht from the intermediary with an excess return indicated

by the first term in the budget constraint, i.e., εht (dRt − rtdt). Specialists bear a riskyexposure εt, by investing their own wealth into the intermediary. In order to use theintermediation service, households pay an intermediation fee ktε

ht ≡ Kt. The second

term in the household’s budget constraint represents this fee. The specialist chooses theoptimal contract share βt to maximize the intermediation fee. The third term is the risk-free interest earned by the household (specialist) on his own wealth. The last term is theconsumption expense. The optimal exposure supply schedule is β∗t = 1

1+m if kt > 0 and

β∗t ∈[

11+m , 1

]if kt = 0. The full-information rational expectations solutions for the above

two maximization problems are:

Ch∗t = ρhW ht and εh∗t =

πR,t − ktσ2R,t

W ht ,


andC∗t = ρWt and ε∗t =

πR,tσ2R,t

Wt.

The key endogenous state variable in HK12 is the scaled relative wealth of the specialist:xt ≡Wt/Dt. It is governed by the following stochastic process:

dxtxt

= µx,tdt+ σx,tdZt, (3.6)

where µx,t and σx,t are endogenously determined drift and diffusion coefficients. Theycapture the nonlinear dynamics of the model. (See Appendix 7.5 for the derivations ofµx,t and σx,t.)

3.2. Capacity Constrained Robust Filtering. As noted earlier, our model combinesrobust control and robust filtering. A key benefit of our log preference specification is thatwe can separate these two problems. Here we focus on the robust filtering problem, whichwe later use to motivate the portfolio delegation decision. Interestingly, we shall see thateven with log preferences, agents care about estimation risk when they distrust their priors.The analysis in this section can be interpreted as combining the backward-looking robustfiltering approach of Hansen and Sargent (2005) with the channel capacity-constrainedMerton model of Turmuhambetova (2005).9

Letting Dt be the exogenous dividend process, the capacity constrained robust filteringproblem can be written as follows:

dDt

Dt= gtdt+ σdZ0

t , (3.7)

dgt = ρg (g − gt) dt+ σgdZgt , (3.8)

dst = gtdt+ σsdZst , (3.9)

where Z0t , Zgt , and Zst are independent standard Brownian motions. In contrast to HK12,

we assume that mean dividend growth, gt, is both stochastic and unobserved. A preferencefor robustness arises because agents distrust the benchmark model for mean dividendgrowth (3.8). At the same time, the channel capacity constraint imposes an upper boundon the precision of the signal 1/σs in equation (3.9).

First consider the capacity constraint. Let κ denote the agent’s channel capacity:

limT→∞

sup1

TI(gT ; sT

)≤ κ, (3.10)

where I(gT ; sT

)is the Shannon mutual information between the true state process

gT = gt: 0 ≤ t ≤ T and the noisy signal process sT = st: 0 ≤ t ≤ T. Mutualinformation measures the average reduction in uncertainty per unit of time over an infinitehorizon. It follows from Duncan (1970) that in the stationary case the constraint becomes

limT→∞

sup1

TI(gT ; sT

)=

1

2

Q

σ2s

≤ κ,

9Using data on CDS credit spreads, Boyarchenko (2012) shows that state uncertainty and robust filteringwere important in the early phase of the financial crisis. She also provides evidence that the importanceof model uncertainty and robust control increased relative to state uncertainty as the crisis unfolded.


where gt = Et [gt|It] and Qt = Et[(gt − gt)2 |It

]are the conditional mean and variance of

gt, respectively, and Q ≡ E [Qt], is the steady state variance.Now let’s turn to the robustness part of the problem. Following Hansen and Sargent

(2005), the robust filter can be implemented by introducing a drift distortion into theKalman filtering equation for the conditional mean, and another distortion into the Ricattiequation for the conditional variance.10

The robust filtering equations for the conditional mean and variance can be writtenas:11

dgt = ρg (g − gt) dt+Qtσ

(ωtdt+ dZt

)+QtσsdZst , (3.11)

dQt =

[σ2g − 2ρgQt −Q2

t

(1

σ2+

1

σ2s

− θ2

)]dt, (3.12)

where dZt = dZt − ωtdt. dZt = 1σ

(dDtDt− gtdt

)and dZst = 1

σs(dst − gtdt) are innovations

corresponding to (3.7) and (3.9), respectively. The parameter θ2 penalizes the evil agent’sdistortions, ωt, to the conditional mean. Since the capacity constraint will always bind,we can finally write the capacity constrained robust filter as:

dgit =

[ρg(g − git

)+Qitσωit

]dt+

QitσdZit +

√2κiQitdZ

ist , (3.13)

dQit =

[σ2g − 2ρgQ

it −(

1

σ2− θi2

)Qi2t − 2κiQit

]dt, i = s, h . (3.14)

Note that a large channel capacity, κ, accelerates learning (i.e., causes Qt to decreasefaster). Although in general, households and specialists have their own filtering equations,in a delegated equilibrium, only specialists filter.

3.3. Robust Control. In addition to having doubts about the unobserved state, gt,agents also have doubts about the dividend process, (3.7), and equilibrium asset returns. Inresponse, they formulate robust portfolio policies (Anderson, Hansen, and Sargent (2003),Maenhout (2004)). Ambiguity about asset returns implies that the budget constraint in(3.3) is viewed as merely a useful approximating model, associated with a benchmarkprobability measure P. To ensure robustness, the agent surrounds the approximatingmodel by a convex set of unstructured alternatives, and then optimizes against the worst-case model within the set. The agent recognizes the worst-case model depends on hisown actions. If we let Q denote the probability measure of the worst-case model, thenGirsanov’s Theorem implies that the conditional relative entropy (or Kullback-Leiblerdistance) between the benchmark and worst-case models is given by a drift distortion,

10Note that we are assuming the evil agent can commit to previous distortions of the conditional mean.Hansen and Sargent (2007) develop a forward looking robust filter that does not permit the evil agent tocommit. In this case, the Riccati equation for the conditional variance is unaltered.11See Appendix 7.1 for the proof.


νit12 ∫

log

(dQt

dPt

)dQt =

1

2EQ∫ t

0

(νit)

2ds.

The νit process conveniently parameterizes the alternative models. A hypothetical evilagent chooses νit subject to a relative entropy cost. Using this distortion, we can define achange of measure dZt = dZ0

t − νitdt, where dZt is a Brownian motion under Q.Under the distorted probability measure, the household’s problem becomes

V(ght , Q

ht ,W

ht ;Y h

t

)= supCht ,εht

infνht ,ωht

E[∫ ∞

0e−ρ

ht

(lnCht +

1

2θh1

(νht

)2+

1

2θh2

(ωht

)2)dt

]subject to (3.13), (3.14), and

dW ht =

[εht (πR,t − kt) + rtW

ht − Cht

]dt+ σhW,t

(νht dt+ dZt

), (3.15)

where σhW,t ≡ σR,tεht . Note that the household’s dynamic programming problem features

four state variables: (ght , Qht ) summarize his current beliefs, W h

t is his current wealth,and Y h

t captures the endogenous dynamics of equilibrium asset prices. As in HK12, Y ht

is determined by a second order ODE in xt. Following Hansen and Sargent (2007), weassume that state and model uncertainty are constrained by separate relative entropypenalties, θh2 and θh1 . 13

Using Ito’s Lemma, the HJB equation is:

ρhV = supCht ,εht

infνht ,ωht

[lnCht +DV + νht σ

hW,tVw + ωht

QtσVg +

1

2θh1

(νht

)2+

1

2θh2

(ωht

)2]

where D[·] is the Dynkin operator,

DV = Vw

[εht (πR,t − kt) + rtW

ht − Cht

]+

1

2Vww(εht )2σ2

R,t

+ VwgσR,tσεhtQ

ht + Vgρg

(g − ght

)+

1

2Vgg

[(Qht)2

σ2+ 2κhQht

]

+ VQ

[σ2g − 2ρgQ

ht −

(1

σ2− θh2

)Qh2t − 2κhQht

]+ µhY,t,

and where µhY,t is the endogenously determined drift of Y ht .

Solving first the infimization part and substituting back into the HJB equation gives:

ρhV = supCht ,εht

[lnCht +DV − θh1

2(σhW,tVw)2 − θh2

2

(QhtσVg

)2]. (3.16)

Note that ambiguity makes the agent dislike variance of continuation utility. Thefollowing proposition summarizes the solution:

12Absolute continuity between P and Q requires their diffusion coefficients to be the same.13Note that in doing this, we are combining the T1/T2 operator approach of Hansen and Sargent (2007)with the commitment filter of Hansen and Sargent (2005).


Proposition 3.1. The consumption policy is:

Ch∗t = ρhW ht , (3.17)

and the optimal risk exposure is:

εh∗t =πR,t − ktγhσ2

R,t

W ht , (3.18)

and the optimal entropy constrained drift distortions of the control and filtering problemsare:

νh∗t = −θh1

ρhεht σR,t

W ht

, (3.19)

ωh∗t = − θh2ρh (ρh + ρg)

Qhtσ

(3.20)

where γh = 1 + θh1/ρh is the household’s effective degree of ambiguity aversion and θh1 , θh2

summarize the degree of model and state uncertainty.Given log preferences, the household’s value function takes the additively separable form

V(ght , Q

ht ,W

ht ;Y h

t

)=

1

ρhlnW h

t + F h(ght , Q

ht

)+ Y h

t ,

where F h(ght , Q

ht

)is determined by the following PDE

ρiF i =git − gρi

+ F ig

[ρg(g − git

)− θi2

2

Qi2tσ2

F i2g

]+

1

2F igg

(Qi2tσ2

+ 2κiQit

)+F iQ

[σ2g − 2ρgQ

it −(

1

σ2− θi2

)Qi2t − 2κiQit

], (3.21)

and Y ht (a function of aggregate state xt) solves the following ODE

ln ρh − 1 +(Y ht

)′µx,txt +

1

2

(Y ht

)′′σ2x,tx

2t = ρhY h

t −(πR,t − kt)2

2ρhγhσ2R,t

− ritρh. (3.22)

where rit = rt −(git − g

), i = h.

Proof. See Appendix 7.2 for the derivations.

We now turn to the specialist’s problem. The specialist’s problem can be written as:

J (gt, Qt,Wt;Yt) = supCt,εt

infνt,ωt

E[∫ ∞

0e−ρt

(lnCt +

1

2θ1ν2t +

1

2θ2ω2t

)dt

]subject to (3.13), (3.14), and

dWt = [εtπR,t + (qt + rt)Wt − Ct] dt+ σW,t (νtdt+ dZt) (3.23)

where σW,t = σR,tεt. The HJB equation is:

ρJ = supCt,εt

infνt,ωt

[lnCt +DJ + νtσW,tJw + ωt

QstσJg +

1

2θ1ν2t +

1

2θ2ωs2t

],


where

DJ = Jw [εtπR,t + (qt + rt)Wt − Ct] +1

2Jwwεt

2σ2R,t + Jwg

σR,tσεtQ

st + Jgρg (g − gst )

+1

2Jgg

(Qs2tσ2

+ 2κQst

)+ JQ

[σ2g − 2ρgQ

st −

(1

σ2− θs2

)Qs2t − 2κiQst

]+ µY,t.

Solving first the infimization part and substituting back into the HJB equation gives:

ρJ = supCt,εt

[lnCt +DJ − θ1

2(σW,tJw)2 − θ2

2

(QstσJg

)2]. (3.24)

The following proposition summarizes the solution:

Proposition 3.2. The specialist’s consumption policy is:

C∗t = ρWt, (3.25)

and the optimal risk exposure is:

ε∗t =πR,tγσ2

R,t

Wt, (3.26)

and the optimal entropy constrained drift distortions of the control and filtering problemsare:

ν∗t = −θ1

ρ

εtσR,tWt

, (3.27)

ω∗t = − θ2

ρ (ρ+ ρg)

Qstσ, (3.28)

where γ = 1 + θ1/ρ is the specialist’s effective degree of ambiguity aversion and θ1, θ2

summarize the degree of model and state uncertainty.The specialist’s value function takes the additively separable form

J (gst , Qst ,Wt;Yt) =

1

ρlnWt + F (gst , Q

st ) + Yt,

where F (gst , Qst ) is determined by the PDE (3.21) (where i = s) and Yt solves the following

ODE

ln ρ− 1 + Y ′t µx,txt +1

2Y ′′t σ

2x,tx

2t = ρYt −

qt + ritρ−

π2R,t

2ργσ2R,t

. (3.29)

Proof. See Appendix 7.2 for the derivations.

3.4. Portfolio Delegation. Portfolio delegation is based on the fact that if κ > κh,then J > V . This value function difference motivates trade in channel capacity.Households are willing to pay to use the specialists’ channel.14 We assume the decisionis made once at time-0, and so is based on an ex ante expected value calculation.Once the household delegates, it no longer needs to filter.15 The probability measure

14Note that in contrast to the investment decision, we assume that the filtering efforts of the specialist arenot subject to any incentive constraints.15Yin (2019) provides evidence in support of this. Using data from the Survey of Consumer Finances, hefinds that households which delegate pay less attention to their portfolios.


used to compute this expectation is the probability measure associated with a delegatedcompetitive equilibrium, reflecting our assumption that households make unilateraldelegation decisions. As a result, only the F

(ght , Q

ht

)component of V influences the

delegation decision. In the steady state, dQit = 0, which from equation (3.14) implies:

(1− θi2σ2

)Qi2 + 2σ2

(κi + ρg

)Qi − (σgσ)2 = 0.

Solving this quadratic,

Qi =−(κi + ρg

)+√

(κi + ρg)2 +

(1− θi2σ2

)(σg/σ)2

1/σ2 − θi2. (3.30)

It is straightforward to then show

dQi

dκi=

σ2

1− θi2σ2

[−1 +

κi + ρg

κi + ρg + (Qi/σ2)(1− θi2σ2

)] .It is clear that dQi

dκi< 0 when θi2 is sufficiently small. That is, a higher channel capacity

reduces the steady state conditional variance. In the steady state, (3.21) reduces to asecond order ODE:

ρiF i =g − gρi

+ ρg (g − g)− 1

2θi2Qi2

σ2F i2g +

1

2F iggσ

i2,

where σi2 = σ2g − 2ρgQ

i. The solution to this ODE can be written as follows:

F i (g) = Ai (g − g) +Bi,

where Ai = 1ρi(ρi+ρg)

and Bi = − 1ρi

[θi22

(Qitσ

)21

ρi2(ρi+ρg)2

]. Evaluating at g = g, this

implies the household’s expected value function difference arising from delegation is givenby

F h (κ)− F h(κh)

=θh2(Qh2 −Qs2

)2σ2ρh3 (ρh + ρg)

2 . (3.31)

Note that what matters is the interaction between state uncertainty θi2 and channelcapacity κi. Capacity differences become more important when state uncertainty isgreater. In fact, without state uncertainty, θi2 = 0, estimation variance does not matter,as you would expect given log preferences. Clearly, F h (κ) > F h

(κh)

as long as θi2 > 0,

and κ > κh. One can view equation (3.31) as determining a maximum delegation fee,K(κ, κh, θ2, θ

h2

). Using our benchmark parameter values we find that this fee turns out

to be 2.41% of wealth.16

16Define ∆F = Fh (κ) − Fh(κh

)as the welfare loss of limited capacity. To calculate the corresponding

money loss, we follow Cochrane (1989) and use ∆F/V ′(Cht ) to measure the money loss due to the limitedcapacity.


4. Market Equilibrium

We now combine the policy functions derived in the previous section with marketclearing conditions and derive equilibrium asset prices. We shall see that ambiguityaversion tightens the capital constraint and amplifies its effects.

4.1. Capital Constraint and Exposure Fee. The key assumption in HK12 is thatspecialists can shirk. Assume the specialist’s flow benefit from doing so is B. Assume theflow cost to the intermediary from the specialist’s shirking is X. The cost incurred by thespecialist depends on his exposure to the risky asset. If we define βt as the intermediary’sexposure, then to prevent shirking, βt > B/X. Alternatively, if we define B/X ≡ 1

1+m ,then we can write the incentive constraint as follows,

εht ≤ mεt (4.32)

where εt, εht are the specialist’s and household’s exposures given in Propositions (3.1) and

(3.2). The constraint in (4.32) is the key ingredient in the HK12 model. It implies thatthe household is less willing to supply capital to the intermediary when the specialist isless exposed. m can be interpreted as an inverse measure of the severity of the agencyproblem. The lower is m, the more severe the agency problem. If we substitute the optimalexposure policies given in Propositions (3.1) and (3.2) into the incentive constraint, wecan write the constraint in terms of wealth17:

W ht ≤ mWt,

where m ≡ γh

γ m is the effective capital constraint. From this, we have the following result:

Proposition 4.1. If θ1/ρ > θh1/ρh, then ambiguity aversion tightens the capital constraint.

The intuition is as follows. If θ1/ρ > θh1/ρh, then the specialist is effectively more

ambiguity averse than the household. Agents worry more about model uncertainty whenthey are more patient.18 Since the constraint binds when the household wants to investwhile the specialist does not, ambiguity tightens the constraint since this makes thespecialist less willing to invest compared to the household.

The exposure fee kt is determined by supply and demand. The specialist’s exposuresupply is a step function:

1−β∗tβ∗t

ε∗t ∈ [0,mε∗t ], for any β∗t ∈ [ 11+m , 1] if kt = 0,

mε∗t with β∗t = 11+m if kt > 0.

In contrast, the household’s exposure demand depends negatively on kt, and is εht =πR,t−ktγhσ2

R,tW ht . Notice that both the exposure supply and demand curves are influenced by

ambiguity aversion.

17As in HK12, the incentive constrained financial relationship between households and specialists can besupported via equity trade. Households purchase a share of the intermediary’s outstanding equity subjectto a constraint Tht ≤ mWt.18Note that if we had scaled the entropy penalty by the value function as in Maenhout (2004), this linkagebetween time preference and ambiguity aversion would disappear (See Maenhout (2004) Appendix B fordetails).


Figure 2 depicts the equilibrium intermediary fee. There are two cases. The left paneldepicts the case when the capital constraint binds, and kt > 0, reflecting the scarcityof intermediary capital. The right panel depicts the case when the constraint is slack,intermediary capital is plentiful, so that kt = 0.

Exposure

Price !"

Constrained Region 0"( ≥ 230"

34"∗

Exposure Demand1

1 + ⁄&'( )(*+," − !".+,"/

0"(

Exposure Supply

0, 31 + ⁄&' )

*+,".+,"/

0" 78 !" = 0,

31 + ⁄&' )

*+,".+,"/

0" 78!" > 0.

Exposure

Price !"

Exposure Demand1

1 + ⁄&'( )(*+," − !".+,"/

0"(

Unconstrained Region 0"( < 230"

34"∗

Exposure Supply

0, 31 + ⁄&' )

*+,".+,"/

0" 78 !" = 0,

31 + ⁄&' )

*+,".+,"/

0" 78!" > 0.

Figure 2: Constrained and Unconstrained Region in Equilibrium

4.2. Equilibrium Definition. Here we provide a detailed definition of marketequilibrium in our model economy:

Definition 4.2. An equilibrium for the economy is a set of progressively measurable priceprocesses Pt, rt, Rt, kt, and households’ decisions Ch∗t , εh∗t , and specialists’ decisionsC∗t , ε∗t , β∗t such that:

(1) Given the price processes, agents consumption and portfolio decisions solve theobjective functions (3.1) and (3.2).

(2) The intermediation market reaches equilibrium with risk exposure clearingcondition:

εh∗t =1− β∗tβ∗t

ε∗t . (4.33)

(3) The stock market clears:

ε∗t + εh∗t = Pt. (4.34)

(4) The goods market clears:

C∗t + Ch∗t = Dt. (4.35)

(5) Transversality conditions satisfy: limt→∞

E[exp

(−ρht

)V (W h

t , t)]

= 0 and

limt→∞

E [exp (−ρt) J(Wt, t)] = 0.

4.3. Equilibrium Asset Prices. The following propositions summarize the influence ofambiguity and the capital constraint on the dynamics of asset prices.

(a) The price/dividend ratio. Since bonds are in zero net supply, the asset market clearswhen aggregate wealth equals the market value of the risky asset,

W ht +Wt = Pt. (4.36)


In equilibrium, from the goods market clearing condition (4.35) and the optimalconsumption rules of households and specialists, we have

ρWt + ρhW ht = Dt. (4.37)

Proposition 4.3. The equilibrium price/dividend ratio is given by:

PtDt

=1

ρh+

(1− ρ

ρh

)xt =

1 +∆ρxtρh

, (4.38)

where ∆ρ ≡ ρh − ρ.

Notice that ambiguity aversion only affects the price/dividend ratio through its influenceon xt. Since specialists are relatively patient, the price/dividend ratio falls as their relativewealth decreases. Therefore, since crises are characterized by declines of specialist’s wealth,the model generates procyclical movements in the price/dividend ratio.

When the risk sharing constraint just starts to bind, the threshold level of the state xc

can be written as:εht = mεt,

which means that Pt−Wt

γh= mWt

γ . Together with the equilibrium price/dividend ratio, this

allows us to restate the capital constraint in terms of the specialist’s scaled wealth:

xc =1

mρh + ρ. (4.39)

When xt ≤ xc, the economy is within the constrained region; otherwise, when xt > xc,the economy is unconstrained. Notice that the ambiguity aversion of both households andspecialists influence the critical level of xc. As the relative ambiguity of the specialistincreases, the threshold wealth level increases.

(b) Specialist’s portfolio share. The specialist invests all his wealth into the intermediary.The specialist then makes a portfolio choice to invest a share αt of the total firm wealthT It = Wt + T ht into the risky asset, and the rest into the riskless bond. Thus, theintermediary’s total exposure is:

εIt = αtTIt ,

or in other words,ε∗t + εh∗t = αt(Wt + T ht ). (4.40)

Given the specialist’s optimal ε∗t choice, this can be interpreted as a constraint on theintermediary’s portfolio. It requires the specialist to choose αt in order to reach the optimalrisk exposure ε∗t . Similarly, households obtain their desired exposure εh∗t by choosing howmuch of their wealth T ht to contribute to the intermediary. Notice that when αt > 1, theintermediary is leveraged.

Later we shall see that the specialist’s share of the intermediary’s return plays animportant role in the dynamics of asset prices. The following proposition characterizesthis share:

Proposition 4.4. In the unconstrained region, the specialist’s share of the intermediary’sreturn is

βUt =

[1 +

γ

γh

(1

ρhxt− ρ

ρh

)]−1

. (4.41)


In contrast, in the constrained region,

βt =1

1 +m. (4.42)

Proof. See Appendix 7.3 for the proof.

In other words, within the unconstrained region, the specialist’s share of theintermediary’s return declines as his wealth share declines. However, once the constraintbinds, his share remains fixed at β = 1

1+m .Above it was noted that the households’ and specialists’ desired exposures can be

restated in terms of the intermediary’s portfolio choice. The following propositionsummarizes this relationship.

Proposition 4.5. In the unconstrained region, the specialist’s desired risk exposure andthe implied optimal portfolio choice of the intermediary are

εU∗t =

[1 +

γ

γh

(1

ρhxt− ρ

ρh

)]−1

Pt and αU∗t = 1,

respectively. In the constrained region,

ε∗t =1

1 +mPt and α∗t =

1xt

+ ρh − ρ(1 + m)ρh

. (4.43)


Notice that when the constraint binds, the intermediary becomes leveraged (αt > 1),and its exposure increases as the specialist’s scaled wealth declines. Also notice thatwhen specialists are effectively more ambiguity averse than households, so that m < m,ambiguity aversion magnifies leverage.

(c) Return volatility of the risky asset.Return volatility is a central feature of financial crises. In general, the effects of the

specialist’s scaled wealth on return volatility are subtle. However, the following propositionshows that return volatility unambiguously increases once the constraint binds.

Proposition 4.6. In the unconstrained region return volatility is,

σUR,t =1

Pt/Dt

σ

ρh −∆ρβUt. (4.44)

In the constrained region it is,

σR,t =1

Pt/Dt

σ

ρh −∆ρβt. (4.45)


When the constraint is not binding, declines in xt have offsetting effects on σR,t. Theinduced decline in Pt/Dt increases σR,t. However, the decline in the specialist’s returnshare offsets this. Once the constraint binds and the return share remains fixed, only thePt/Dt effect remains, and σR,t unambiguously increases as xt falls further.

(d) The risk premium and financial constraint.


In addition to increases in volatility, financial crises are also accompanied by largeincreases in risk premia. The following proposition shows that risk premia increase duringcrises, and that ambiguity amplifies this increase.

Proposition 4.7. In the unconstrained region the risk premium is given by,

πUR,t =σ2γγh

(1 +∆ρxt)

[(ρhγh − ργ

)xt + γ

][ρ (γh − γ)xt + γ]

2 . (4.46)

In the constrained region,

πR,t =σ2ρhγ

xt (1 +∆ρxt)

1 +m

(mρh + ρ)2 . (4.47)


Notice, the risk premium is highly state dependent. In the Online Appendix A wederive the following comparative static results, which show that ambiguity amplifies therisk premium.

Corollary 4.8.dπUR,tdθ1

> 0,dπUR,t

dθh1≥ 0,

dπR,tdθ1

> 0,dπR,t

dθh1= 0.

Proof. See Online Appendix A for the proof.

The intuition is straightforward, as agents become more ambiguity averse, they mustbe compensated with a higher equilibrium risk premium. It is interesting to notice thatθ1 positively changes the risk premium both in the unconstrained and constrained region,while θh1 also has a positive impact, but only in the unconstrained region. This reflectsthe fact that when the constraint binds, the specialist is the only marginal investor.

(e) The market price of risk and uncertainty.The market price of risk is defined as the Sharpe ratio. In our model, the conventional

Sharpe ratio measures a combination of risk aversion and ambiguity aversion. Ambiguityincreases it unambiguously. Barillas, Hansen, and Sargent (2009) call this induced increase‘the price of model uncertainty’. Propositions 4.6 and 4.7 summarize the results:

Proposition 4.9. In the unconstrained region, the market price of risk is

πUR,t

σUR,t=

σγγh

ρ (γh − γ)xt + γ. (4.48)

In the constrained region,πR,tσR,t

= σγ

mρh + ρ

1

xt. (4.49)

Proof. The proof is straightforward by combining Equations (7.81), (7.82), (4.46), and(4.47).

The following corollary summarizes the comparative static effects of ambiguity on theSharpe ratio.


Corollary 4.10.

d(πUR,t/σ

UR,t

)dθ1

> 0,d(πUR,t/σ

UR,t

)dθh1

≥ 0;

d (πR,t/σR,t)

dθ1> 0,

d (πR,t/σR,t)

dθh1= 0.


As with the risk premium, in the constrained region only the specialist’s ambiguityhas a direct effect on the Sharpe ratio. Again, this reflects the central argument in theintermediary asset pricing literature that specialists are the marginal investors.

(g) The interest rate.The real challenge of asset pricing is to explain both a high equity premium and a low

average riskless interest rate. Recursive preference can do this. However, implausibly highrisk version coefficients are still needed. Barillas, Hansen, and Sargent (2009) show thatan empirically plausible ‘model uncertainty premium’ can substitute for high risk aversion.Our model shares this advantage, and at the same time captures observed state dependencein these returns. The following proposition characterizes the equilibrium interest rate.

Proposition 4.11. In the unconstrained region, the interest rate is

rUt = ρh + gt − ρ∆ρxt + σ2γγh −

(γh + γ

) [ρxt(γh − γ

)+ γ]

[ρ (γh − γ)xt + γ]2 . (4.50)

In the constrained region,

rt = ρh + gt − ρ∆ρxt − σ2 (1− ρxt)[ρ (1 + γm) + ρhm2γh

]+ ρhm2

(ρhxt − γh

)(1− ρxt) (ρ+mρh)

2xt

. (4.51)


This yields the following comparative statics.

Corollary 4.12. When γh ≤ γ and θ1 = θh1 = θ1, then

drUtdθ1≤ 0,

drtdθ1

< 0.


The intuition is that, as ambiguity increases, the specialists want to borrow less andhouseholds want to save more, which drives down the interest rate.

(f) The intermediation fee.The information processing superiority of the specialist allows him to earn two kinds of

fees. The one time participation fee characterized in section 3.4, and an on-going portfoliomanagement fee kt, which is state-dependent and varies with the specialist’s scaled wealth.The following proposition characterizes the equilibrium depicted in Figure 2.


Proposition 4.13. In the unconstrained region, the per-unit exposure price kUt = 0. Inthe constrained region,

kt =σ2(1 +m)

(mρh + ρ)2

(γ − ρhγhmxt

1− ρxt

)ρh

(1 +∆ρxt)xt. (4.52)


This leads immediately to the following comparative static results.

Corollary 4.14. The intermediation fee increases when the specialist’s ambiguityincreases but decreases when the household’s ambiguity decreases:

dktdθ1

> 0,dkt

dθh1< 0.


This reflects simple supply and demand. The fee increases when the household wantsto invest but the specialist does not.

Since a picture is worth a thousand words (or equations), the following figure depictsthe intermediary’s optimal portfolio policy along with equilibrium asset prices. Each panelplots the results as a function of the specialist’s scaled wealth xt. We use the benchmarkparameter values estimated in the following section, which are contained in Table 1. In allcases, we assume that θ1 = θh1 . Since ρ < ρh, this implies that specialists are effectivelymore ambiguity averse in all plots. To illustrate the effects of ambiguity, each panelcontains four lines, pertaining to four alternative values of θ1 = θh1 . For comparison, theblue solid line in each plot pertains to the no ambiguity case of HK12.

The top left plot shows that leverage increases when the constraint binds and thatambiguity both amplifies the effect and causes it to occur at higher levels of specialistwealth. The top right panel shows that when the constraint binds, return volatilityincreases. Interestingly, it also shows that ambiguity depresses return volatility awayfrom the constraint. This comes from a wealth reallocation to the more ambiguity aversespecialist, who would bear less risk exposure and this reduces the equilibrium returnvolatility. The middle plots show the risk premium and Sharpe ratio. Evidently, theeffects of xt and θ1, θ

h1 on them are very similar. Both the risk premium and Sharpe ratio

increase as xt decreases within the constrained region, as in HK12. More importantly,ambiguity increases the effect. For example, at the HK12 constraint, the equity premiumrises from only 2% in the HK12 model to more than 10% with our benchmark parametersθ1 = θh1 = 0.03. Also notice that its mild increase in the unconstrained region reflectsthe fact that wealth is being redistributed toward the relatively ambiguity averse agent.A similar mechanism is at work in Garleanu and Panageas (2015). The lower left plotdepicts the specialist’s state-dependent intermediation fee. At the HK12 constraint, thevariable intermediation fee is approximately 3%, which is close to estimates provided byGreenwood and Scharfstein (2013). Finally, the lower right plot depicts the equilibriuminterest rate. Notice that the interest rate is insensitive to the specialist’s scaled wealthwhen the constraint does not bind, but it drops sharply when the constraint binds.


15 20 25 30 35 40 45 50 55 60

Scaled Specialist Wealth x

0.8

1

1.2

1.4

1.6

1.8

2

t

Specialist Portfolio Share

= h=0

= h=0.01

= h=0.03

= h=0.05

15 20 25 30 35 40 45 50 55 60


0.12

0.125

0.13

0.135

0.14

0.145

0.15

R,t

Risky Asset Volatility

= h=0

= h=0.01

= h=0.03

= h=0.05

15 20 25 30 35 40 45 50 55 60


0

0.05

0.1

0.15

0.2

R,t

Risk Premium

= h=0

= h=0.01

= h=0.03

= h=0.05

15 20 25 30 35 40 45 50 55 60


0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

R,t /

R,t

Sharpe Ratio

= h=0

= h=0.01

= h=0.03

= h=0.05

15 20 25 30 35 40 45 50 55 60


0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

kt

Intermediation Exposure Fee

= h=0

= h=0.01

= h=0.03

= h=0.05

15 20 25 30 35 40 45 50 55 60


-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0.025

r t

Interest Rate

= h=0

= h=0.01

= h=0.03

= h=0.05

Figure 3: Policy Functions and Asset Prices

This graph plots the policy functions of the specialist’s portfolio share for risky asset αt, risky assetvolatility σR,t, risk premium πR,t, intermediation exposure fee kt and risk free rate rt for different

ambiguity parameters (θ1 = θh1 ). The threshold value xc (vertical line) separates the constrained (left)and unconstrained (right) regions. The solid blue line shuts down ambiguity. All other parameter values

are from Table 1.


4.4. The Stationary Wealth Distribution. To map the previous plots into empiricalpredictions, we need the stationary distribution of xt, which is endogenous. In fact, withoutambiguity, a stationary distribution does not exist, because when ρh > ρ, households areeventually driven out of the economy, the specialists accumulate all the wealth, and theconstraint never binds. However, when the specialist is effectively more ambiguity averse,the greater impatience of the households can be offset by the greater ambiguity of thespecialists (Yan (2008)). The following result characterizes this distribution.

Proposition 4.15. The specialist’s scaled wealth process is given by the followingdiffusion:

dxtxt

= µx,tdt+ σx,tdZt. (4.53)

In the unconstrained region the drift and volatility are given by

µUx,t = ∆ρ (1− ρxt) + σ2 + σ2 AU0 x2t +AU1 xt +AU2

(1− ρxt) [ρ (γh − γ)xt + γ]2 , (4.54)

σUx,t = σ

[γh

ρ (γh − γ)xt + γ− 1

], (4.55)

where AU0 = ρ2(γh − γ

) (2γh + γ

), AU1 = ρ

(2γ2 − 3

(γh)2

+ 2γγh)

, and AU2 =(γh)2 −

γ2 − γγh.In the constrained region they are given by

µx,t = ∆ρ (1− ρxt) + σ2 + σ2 A0x2t +A1xt +A2

(mρh + ρ)2

(1− ρxt)x2t

, (4.56)

σx,t = σ

[1

(mρh + ρ)xt− 1

], (4.57)

where A0 = ρh(ργh − ρh

)m2 + ρ

(ργ + ρh

)m+ 2ρ2, A1 = −ρhγhm2−

(ρh + 2ργ

)m− 3ρ,

and A2 = γm+ 1.


Whether this diffusion process yields a stationary distribution depends on the boundaryproperties of µx,t and σx,t. Intuitively, µx,t should be negative for sufficiently large xt andpositive for sufficiently small xt. Assuming it exists, let f(x) be the stationary densityassociated with the diffusion in equation (4.53). Since xt is univariate, we can obtain thefollowing expression for the solution to the steady state Kolmogorov Fokker-Planck (KFP)equation, which characterizes this density.

Proposition 4.16. The stationary density of specialist scaled wealth satisfies

f(x) =C1

σ2x(x)

exp

(∫ xc

0

2µx(s)

σ2x(s)

ds

)+

C2

(σUx (x))2 exp

(∫ 1ρ

xc

2µUx (s)

(σUx (s))2ds

), (4.58)

where C1 and C2 satisfy:

(1)∫f(x)dx = 1;

(2) continuous at xc.


Proof. See Online Appendix B for the proof.

Existence rests on the convergence of these integrals.19 A formal proof and explicitsolution would be challenging, since the drift and diffusion coefficients are non-linear andchange at the constraint. Instead, we simply use Monte-Carlo simulations to informallycheck whether the stationary distribution exits. Figure 4 depicts the empirical distributionof xt for the benchmark parameter values in Table 1. It is computed from 5000 samplepaths, each consisting of 200 years.20

Figure 4: Stationary Distribution

This figure plots a histogram of the specialist’s scaled wealth using the benchmark parameters. Thevertical red line separates the constrained (left) and unconstrained (right) regions.

5. Quantitative Implications

5.1. Indirect Inference. Due to the endogenously binding capital constraint, our modelwould be difficult to estimate using traditional methods. However, it is relatively easy togenerate sample paths from the model. Hence, it is a good candidate for the simulation-based estimation methodology of indirect inference (Gourieroux, Monfort, and Renault(1993), Smith Jr (1993)). The basic idea behind indirect inferences is to specify a setof (easily estimated) reduced form ‘auxiliary functions’, which are designed to capturefeatures of the data of interest, i.e., comovement, volatility and persistence. The reduced-form parameters are estimated twice - once using the actual data, and again using repeatedsamples generated from the structural model. The model is then evaluated based on howclose (averaged) estimates from the simulated data are to those from the actual data. If

19See Karlin and Taylor (1981).20The figure is not changed with longer simulations. We also simulated 5000 sample paths of 5000 years.The distribution is roughly the same.


there are more auxiliary function parameters than model parameters, then some sort ofminimum distance/weighting matrix must be specified. The minimum distance can beused to test the null that the structural model is correctly specified. Note, the auxiliaryfunctions can be misspecified.

We use two sets of auxiliary functions. The first captures the model’s ability toexplain unconditional moments. Because there are many models capable of explainingunconditional moments, we also want to assess our model’s ability to explain statedependence in risk prices and return premia. This is more challenging. Our secondset of auxiliary functions are therefore based on our model’s ability to match simpleautoregressions of equity premia and price/dividend ratios.

Several of our model’s parameters can be calibrated to outside data. For example,m relates to the specialist’s return share. Following HK12, we set m = 4, reflectingthe assumption that specialists maintain a 20% return share. HK12 cite evidence insupport of this. Mean dividend growth is calibrated to match US dividend data. Thisimplies g = 0.02, σg = 0.03, and ρg = 0.055. Channel capacities are based on twoconsiderations. First, notice from eq. (3.28) that the magnitude of the specialist’sfiltering distortion depends positively on estimation variance, Q, which in turn dependsnegatively on capacity, κ (eq. 3.30). Hence, we calibrate κ to keep g close to g. Thisprevents detection error probabilities from becoming implausibly low. Second, we thencalibrate the household’s capacity, κh, so that the implied delegation fee is close to thedata, which suggest that upfront management fees are about 1.5-2.5% of invested assets(Greenwood and Scharfstein (2013))). This leaves (σ, ρ, ρh, θ1, θ

h1 ) as the free parameters.

The first set of auxiliary functions consists of the following five unconditional moments:the mean equity premium, the mean risk-free rate, the mean price/dividend ratio, andthe standard deviations of the risky return and price/dividend ratio. The second setof of auxiliary functions consists of the intercept and autoregressive coefficient in simpleunivariate AR(1) processes for the equity premium and price/dividend ratio. The thirdauxiliary function consists of the correlation between the data and model implied timeseries of equity premium. For the equity premium data we use estimates from Gagliardini,Ossola, and Scaillet (2016). They fit a 4-factor conditional linear factor model to USmonthly stock returns for the period 1964-2009. We use the risk premium on the marketfactor for the period 1970-2009. The price/dividend data are from Shiller’s webpage, againfor the period 1970-2009. The weighting matrix used to match the 5 parameters to the 10targets is described in Appendix 7.6.

5.2. Estimates. Table 1 contains calibrated and estimated benchmark parameter values.The grid search constrained θ1 = θh1 , θ2 = θh2 , and ρh > ρ. This ensures a stationary wealthdistribution and imposes the restriction that, absent discounting effects, the ambiguityaversion of households and specialists is the same. For each candidate parameter value wesimulate 5000 observations, and use the final 40 years to compute moments and regressions.We then average across repetitions to compute distances.

The data moments and model estimates are summarized in Table 2. Estimates are basedon averages across 5000 simulations using the benchmark parameter values. To assess theimportance of model uncertainty, we also compute moments in the HK12 model. The


Table 1: Benchmark Parameters

Panel A. Preferences and Intermediationρ Specialist Time Discount Rate 0.007

ρh Household Time Discount Rate 0.014θ1 Specialist Model Uncertainty Preference 0.03

θh1 Household Model Uncertainty Preference 0.03θ2 Specialist State Uncertainty Preference 1

θh2 Household State Uncertainty Preference 1κ Specialist Information Channel Capacity 0.01

κh Household Information Channel Capacity 0.00999m Intermediation Multiplier 4Panel B. Equity Marketg Mean Dividend Growth Rate 0.02σ Dividend Growth Volatility 0.1389σg Unobservable Dividend Growth Volatility 0.03ρg Mean Reversion Rate of Unobservable Dividend Growth 0.055Q0 Initial Value of Posterior Variance 0.005%

HK12 model has the ambiguity aversion parameter values close to zero.21 The results arein the column labeled ‘Non-Robust’. The column labeled ‘Model 1’ contains results whenthe two agents are ambiguity averse (θ1 = θh1 = 0.03).

There are several points to notice. First, without ambiguity, the HK12 model cannotexplain the Equity Premium Puzzle. It only produces a 1.94% equity premium and a14.02% Sharpe ratio. Both are significantly lower than in the data. However, whenagents are concerned about model misspecification, the model’s performance improvessignificantly. It now produces a 6.27% equity premium and 48.12% Sharpe ratio. Theseare both close to the data, although the Sharpe ratio is a little high. Second, our modelproduces a mean interest rate of 1.24% compared to 1.29% in the data. Hence, we avoid therisk-free rate puzzle. Third, and perhaps most important, the Non-Robust HK12 modeldoes not produce crises. The constraint never binds because the relative impatience ofhouseholds causes the specialist to accumulate most of the economy’s wealth. In contrast,in our model the constraint binds more than 10% of the time, and when it does, it producessignificant increases in the price of risk. For example, 2.86% of the time the Sharperatio exceeds its long-run mean by more than 105 basis points. The only significantdiscrepancy is the model’s price/dividend ratio, which is more than double the observedvalue. This reflects the model’s relatively low rates of time preference. For comparison,the column labeled ‘Model 2’ estimates the model with a higher weight on the meanprice/dividend ratio. Notice this produces higher estimates of time preference, and alower price/dividend ratio. However, it also drives the estimated equity premium belowthe data. More importantly, as we shall see, it also erodes the model’s ability to generatestate-dependence. For that reason, we prefer the benchmark parameterization in Model1.

21In order to ensure non-degenerate distribution, we set ambiguity aversion as 0.0001 instead of 0.


Table 2: Measurements and Estimates

Data Non-Robust Model 1 Model 2

θ2 0.0001 1 1

θh2 0.0001 1 1θ1 0.0001 0.03 0.155

θh1 0.0001 0.03 0.02ρ 0.007 0.007 0.016

ρh 0.014 0.014 0.0387γ 1.014 5.29 10.69

γh 1.007 3.14 1.52Risk Premium (%) 5.91 1.94 6.27 3.83Sharpe Ratio (%) 31.91 14.02 48.12 38.78Interest Rate (%) 1.29 1.26 1.24 1.25

Interest Rate Volatility (%) 2.25 2.34 2.33 1.28Return Volatility (%) 17.30 13.87 13.04 9.89

P/D mean 41.31 93.00 87.09 42.69k(%) 2 0 0.23 0.79

DEP Specialist (%) 49.25 24.98 22.44DEP Household (%) 49.36 25.06 23.05

Prob. of Sharpe Ratio Exceed 105bps (%) 0 2.86 26.30Prob. of Constraint Binds (%) 0 11.57 43.52

This table reports unconditional moments. We simulate 200 years and 5000 sample paths using our

benchmark parameters. To correspond to our empirical sample, 1970-2009, we report means from the

final 40 periods of each sample paths.

Table 3 shows that our model also captures the persistence of the equity premium andprice/dividend ratio.

Table 3: Persistence

Data Non-Robust Model 1 Model 2

θ1 0.0001 0.03 0.155

θh1 0.0001 0.03 0.02Risk Premium: AR(1) Coefficient 0.9556 1.0007 0.9978 0.9980

Risk Premium: Intercept 0.2741 -0.0014 0.0136 0.0078PD: AR(1) Coefficient 0.9964 1.0004 0.9995 0.9997

PD: Intercept 0.1838 -0.0247 0.0435 0.0127

This table reports AR(1) coefficients and constants estimated from risk premium and price/dividend

ratio monthly data (1970-2010) as well as the model implied coefficients and constants. We simulate 40

years and average across 5000 sample paths with implied monthly frequency.

The autoregressive coefficients are quite close to those in the data, and the impliedlong-run means are also close.


5.3. Sample Paths. Although it is encouraging that the model matches theunconditional moments of asset returns, many other models that can do this as well. Thevalue-added of the HK12/13 models is to explain the nonlinear, state-dependent dynamicsof asset returns. For example, a significant fraction of the average equity premium isgenerated during infrequent crisis episodes when it rises dramatically. Generating strongcounter-cyclicality in risk prices and risk premia is more challenging. Perhaps the bestway to assess this ability is to simply look at sample paths produced by the model. Figure5 displays a representative sample path using our benchmark parameters. The time unitis a quarter. The red rectangular areas indicate times when the constraint binds. The topleft panel plots the specialist’s scaled wealth, which is the model’s key endogenous statevariable. The top right panel plots the drift distortions of the specialist and household,while the middle left panel plots their difference. The middle right and bottom left panelsplot the risk premium and Sharpe ratio, respectively. The blue horizontal lines depictsample means. Finally, the bottom right panel plots the interest rate.

These plots nicely reveal the model’s key forces. As in HK12/13, the top left plotshows that crises occur when the specialist’s wealth declines. The top right and middleleft plots depict the new mechanism produced by model uncertainty. Note that asthe specialist’s wealth declines, he becomes relatively pessimistic, i.e., his relative driftdistortion increases. This occurs because he endogenously becomes more exposed to therisky asset. This endogenous belief heterogeneity amplifies the rise in risk premia and riskprices, as depicted in the middle right and bottom left plots. The risk premium rises byabout 100 basis points, from about 6.2% to 7.2%. It is interesting to observe that theconstraint can easily bind for a decade or more, although the peak itself tends to be rathershort-lived. In contrast, HK13 emphasize that most crisis models have difficulty capturingthe duration of crises. Finally, note that these crisis episodes contribute significantly toaverage equity premia and Sharpe ratios. As you would expect from Figure 3, when theconstraint is slack, the equity premium and Sharpe ratio are nearly constant.

5.4. Implications From Observed Capital Ratios. A 100-150 basis point increase inthe risk premium is significant, but still less than the increases observed during actualcrises, where spreads often increase by several hundred basis points. We suspect that thefailure of our model to fully match the magnitude of the increase arises from the fact thatour model-generated wealth process produces less variation than observed in the data. Wecan check this by using data from He, Kelly, and Manela (2017). They collect data onmarket-value capital ratios for the New York Fed’s primary dealers. These institutionsactively trade in most, if not all, asset markets. Although there is heterogeneity acrossdealers, they compute a simple value-weighted average to correspond with our model’sassumed ‘representative’ specialist.22 We can then use this observed capital ratio in placeof xt in the equilibrium pricing equations of our model. Figure 6 depicts the results. Thetop two figures pertain to the benchmark parameterization in Model 1, while the bottomtwo pertain to that of Model 2.

22There is also considerable heterogeneity across assets. However, their empirical results suggest that themodel’s assumption of a single risky asset might not be a bad approximation, since the estimated price ofrisk is quite similar across asset classes.


20 40 60 80 100 120 140 160

21

22

23

24

25

26

27

Specialist Scaled Wealth

20 40 60 80 100 120 140 160

0.042

0.044

0.046

0.048

0.05

0.052

0.054

0.056

0.058

0.06

0.062

Two Agents Dividend Distortion (Pessimism)

SpecialistDistortion

HHDistortion

20 40 60 80 100 120 140 160

0.008

0.009

0.01

0.011

0.012

0.013

0.014

0.015

0.016

0.017

0.018

Two Agents Pessimism Difference

20 40 60 80 100 120 140 160

0.062

0.064

0.066

0.068

0.07

0.072

Risk Premium

20 40 60 80 100 120 140 160

0.47

0.48

0.49

0.5

0.51

0.52

0.53

0.54

Sharpe Ratio

20 40 60 80 100 120 140 160

0.01

0.012

0.014

0.016

0.018

0.02

0.022

0.024

0.026

Risk Free Rate

Figure 5:

This figure plots a representative sample path using the benchmark parameters. Red rectangular regionsindicate the constrained region. The light blue horizontal lines in the first two figures are unconditional

means for risk premium and Sharpe Ratio, respectively.


Figure 6: Model Implied Risk Premium and Price/Dividend Ratio

This figure plots the model implied risk premium and price/dividend ratio by imputing He, Kelly, andManela (2017) data against the risk premium from Gagliardini, Ossola, and Scaillet (2016) and real

price/dividend ratio from Robert Shiller’s website.

The left panel displays the risk premium and the right panel displays the price/dividendratio. For comparison, the blue lines plot the data. As before, for the equity premium weuse the market factor from Gagliardini, Ossola, and Scaillet (2016), and the price/dividendratio from Robert Shiller’s website. Perhaps not surprisingly given Figure 3, during normaltimes the model generates little variation in the equity premium and price/dividend ratio.Notice, however, that during the recessions the model generates significant increases. Forexample, during the financial crisis of 2008-09, the model generates an increase in theequity premium of about 10 percentage points. Still less than in the data, but muchcloser.


6. Detection-Error Probabilities

We have seen that the ambiguity parameters (θ1, θh1 , θ2, θ

h2 ) play an important role in

our model’s ability to fit the unconditional moments and state dependent dynamics ofasset returns. It is possible to view these parameters as solely a reflection of preferencesand allow them to be unrestricted. However, following Hansen and Sargent (2008), weprefer to interpret them as reflecting both the presence of model uncertainty and theagent’s preference for robustness.23 Under this interpretation, it is important to disciplinethe magnitude of (θ1, θ

h1 , θ2, θ

h2 ). In particular, we do not want to allow agents to hedge

against empirically implausible alternative models.Plausibility is quantified using detection error probabilities (DEPs). Agents are viewed

as statisticians, who attempt to discriminate among models using likelihood ratio statistics.When likelihood ratio statistics are large, DEPs are small, and models are easy todistinguish. DEPs will be small when models are very different, or when there is a lot ofdata. DEPs are based on an equally-weighted average of Type I and Type II errors:

DEP =1

2Prob (HA|H0) +

1

2Prob (H0|HA) .

Hence, a DEP is analogous to a p-value. We compute DEPs using the Monte-Carlosimulation strategy outlined in Chapter 9 of Hansen and Sargent (2008).

Since ex post only specialists are filtering, the null/approximating model is:

dDt

Dt= gtdt+ σdZt, (6.59)

dgt = ρg (g − gt) dt+QtσdZt +

√2κQtdZ

st , (6.60)

while the distorted/alternative model is:

dDit

Dit

=(gt + σνit

)dt+ σdZit , (6.61)

dgt =

[ρg (g − gt) +

Qtσωt

]dt+

QtσdZt +

√2κsQtdZ

st . (6.62)

Let LP and LQ be the likelihood of null model and alternative model, the log-likelihood

ratios are l|P = log(LPLQ

)and l|Q = log

(LQLP

), respectively. When the null model generates

the data,

l|P =1

T

T∑t=1

(1

2wn′t w

nt − wn

′t ε

nt

), (6.63)

(l|P)h =1

T

T∑t=1

(1

2wnh

′t wnht − wnh

′t εnht

)(6.64)

23In discrete time models, it is possible to distinguish the presence of ambiguity from the agent’s aversionto ambiguity (see e.g., Klibanoff, Marinacci, and Mukerji (2005) and Ju and Miao (2012)). However, in thecontinuous time limit, one must rescale θ1 to maintain ambiguity aversion (see Hansen and Miao (2018) fordetails). Given the hidden state in our model, we could in principle distinguish ambiguity from ambiguityaversion. We leave this for future work.


for specialists and households, respectively, where wnt =

[νntωt

], wnht =

[νnhtωt

],

εnt =

[ZtZt

], εnht =

[ZtZht

]and Zt, Zt, and Zht are i.i.d. If the alternative model is

the data-generating process,

l|Q = − 1

T

T∑t=1

(1

2wa′t w

at + wa

′t ε

at

), (6.65)

(l|Q)h =1

T

T∑t=1

(1

2wah

′t waht − wah

′t εaht

). (6.66)

where εat = εaht =

[ZtZt

]. The relative entropy computed from the household’s and

specialist’s robust control problems can be written as:24

νht = σγ(

1− γh)[ mρh

γ (mρh + ρ) (1− ρxt)1xt∈(xmin,xc] +

1

ρ (γh − γ)xt + γ1xt∈

(xc, 1

ρ

]] ,(6.67)

νt = σγh (1− γ)

[1

γh (mρh + ρ)xt1xt∈(xmin,xc] +

1


(xc, 1

ρ

]] , (6.68)

where xmin = 1/(mρh + ρ

)and 1 denotes the indicator function. The negative drift

distortion ωt comes from the robust filtering solution, (3.28).Figure 7 depicts the results using our benchmark parameters.

Figure 7: Detection Error Probability

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.080.05

0.1

0.15

0.2

0.25

0.3

DEP Specialist

DEP Household

0 0.5 1 1.5 2 2.5 30

0.05

0.1

0.15

0.2

0.25

0.3

DEP Specialist

DEP Household

This figure plots detection error probabilities for the specialist and household. The left panel fixes

θ2 = θh2 = 1 and then plots the DEP as a function of θ1(= θh1 ). The right panel fixes θ1 = θh1 = 0.03 and

then plots the DEP against θ2(= θh2 ). Other parameters are from Table 2.

24See Appendix 7.7 for the proof.


As in Hansen and Sargent (2007), each agent has two robustness parameters, onepertaining to the control problem, θi1, and one pertaining to the filtering problem, θi2.The left panel of Figure 7 plots DEPs as a function of the robust control parameter,θi1, holding constant the filtering parameter at its benchmark value. The right panel ofFigure 7 plots DEPs as a function of the filtering problem, θi2, holding constant the controlparameter. From the left panel, we can see that DEPs remain above 10% for values of θi1as large as 0.07. However, such a large value of θi1 produces counterfactually large equitypremia and Sharpe ratios.25 The right panel shows that DEPs actually increase with θi2.This reflects our use of the commitment filter of Hansen and Sargent (2005). In this case,as θi2 increases, the agent’s state estimation variance Q increases (see equation (3.30)).Notice that as θi2 → 0, DEPs become quite small. Hence, state uncertainty is important.In section 3 we saw that when θi2 = 0, there would be no portfolio delegation. Here we seethat θi2 6= 0 is also important in generating plausible DEPs.

7. Conclusion

This paper has argued that Intermediary Asset Pricing offers more than 2nd-ordermarket micro-structure corrections to existing asset pricing models. We’ve done this byintroducing two new elements into the influential model of He and Krishnamurthy (2012,2013). First, we formalize the notion of ‘complexity’ and motivate portfolio delegationby assuming agents face information processing constraints. Portfolio delegation allowshouseholds to purchase the higher channel capacity of financial intermediaries. Second,we assume that households and specialists are ambiguity averse. We show that ambiguityaversion tightens incentive constraints and amplifies their effects.

Although Lucas-style models are convenient for studying asset pricing, they have thedrawback of eliminating feedbacks from asset prices to the real economy. It is widelybelieved that the financial crisis of 2007-08 featured such feedbacks. Therefore, a usefulextension would be to introduce ambiguity and information processing constraints intothe production-based asset pricing model of He and Krishnamurthy (2019). In addition tohelping us understand asset prices during the crisis, perhaps the combination of ambiguityand financial constraints can help explain the depth and duration of the ensuing recession.

Another extension would be to combine our analysis with the Complex Asset Marketsmodel of Eisfeldt, Lustig, and Zhang (2017). They also study the pricing implications of amodel that combines ‘experts’ and ‘nonexperts’. In contrast to our model, where channelcapacity differences are exogenous, agents in their model can choose whether to becomeexperts. Becoming an expert is beneficial because it reduces idiosyncratic investmentrisk. The key mechanism in their model is endogenous entry and exit, and their inducedselection effects. However, in their model funds cannot be reallocated across investors,and there is no trade in expertise. As a result, there is no moral hazard problem orcapital constraint. Combining endogenous expertise, equilibrium entry and exit, portfoliodelegation, and moral hazard would be challenging, but also potentially quite fruitful.

25It is worth noting that Barillas, Hansen, and Sargent (2009)’s frictionless model could only attain theHansen-Jaganathan bound with DEPs below 0.05.


Finally, it would be useful to further examine the robustness of our quantitative results.Our model is relatively complex, with many free parameters. We have attempted tocontrol over-fitting by only estimating a small subset of the parameters, while calibratingthe rest to outside values. Still, Chen, Dou, and Kogan (2017) have recently proposedmore formal methods of assessing model ‘fragility’, which could perhaps be profitablyapplied in our setting. They note that Rational Expectations models tend to be fragile,since changes in parameters, which are presumed to be known by agents but unknown tothe outside econometrician, often elicit strong reactions by the agents, which makes themodel’s predictions fragile. They go on to observe that models which attribute ambiguityaversion to agents tend to be less fragile, since agents’ responses to parameter changes aremore muted. This suggests that our model’s empirical results might be relatively robust.


References

Adrian, T., and N. Boyarchenko (2012): “Intermediary Leverage Cycles and FinancialStability,” Unpublished Working Paper.

Adrian, T., E. Etula, and T. Muir (2014): “Financial Intermediaries and the Cross-Sectionof Asset Returns,” Journal of Finance, 69(6), 2557–2596.

Adrian, T., E. Moench, and H. S. Shin (2016): “Dynamic Leverage Asset Pricing,” CEPRDiscussion Paper No. DP11466.

Anderson, E. W., L. P. Hansen, and T. J. Sargent (2003): “A Quartet of Semigroups forModel Specification, Robustness, Prices of Risk, and Model Detection,” Journal of the EuropeanEconomic Association, 1(1), 68–123.

Barillas, F., L. P. Hansen, and T. J. Sargent (2009): “Doubts or Variability?,” Journal ofEconomic Theory, 144(6), 2388–2418.

Basak, S., and D. Cuoco (1998): “An Equilibrium Model with Restricted Stock MarketParticipation,” The Review of Financial Studies, 11(2), 309–341.

Basak, S., and A. Pavlova (2013): “Asset Prices and Institutional Investors,” AmericanEconomic Review, 103(5), 1728–58.

Berk, J. B., and R. C. Green (2004): “Mutual Fund Flows and Performance in RationalMarkets,” Journal of Political Economy, 112(6), 1269–1295.

Bidder, R. M., and M. E. Smith (2012): “Robust Animal Spirits,” Journal of MonetaryEconomics, 59(8), 738–750.

Boyarchenko, N. (2012): “Ambiguity Shifts and the 2007–2008 Financial Crisis,” Journal ofMonetary Economics, 59(5), 493–507.

Brenner, M., and Y. Izhakian (2018): “Asset Pricing and Ambiguity: Empirical Evidence,”Journal of Financial Economics.

Brunnermeier, M. K., and Y. Sannikov (2014): “A Macroeconomic Model with a FinancialSector,” American Economic Review, 104(2), 379–421.

Caballero, R. J., and A. Krishnamurthy (2008): “Collective Risk Management in a Flightto Quality Episode,” Journal of Finance, 63(5), 2195–2230.

Chan, Y. L., and L. Kogan (2002): “Catching up with the Joneses: Heterogeneous preferencesand the dynamics of asset prices,” Journal of Political Economy, 110(6), 1255–1285.

Chen, H., W. W. Dou, and L. Kogan (2017): “Measuring the ‘Dark Matter’ in Asset PricingModels,” Unpublished Working Paper.

Chevalier, J., and G. Ellison (1997): “Risk Taking by Mutual Funds as a Response toIncentives,” Journal of Political Economy, 105(6), 1167–1200.

Cochrane, J. (1989): “The sensitivity of tests of the intertemporal allocation of consumption tonear-rational alternatives,” The American Economic Review, 79(3), 319–337.

Cochrane, J. H. (2017): “Macro-Finance*,” Review of Finance, 21(3), 945–985.Dai Pra, P., L. Meneghini, and W. J. Runggaldier (1996): “Connections between Stochastic

Control and Dynamic Games,” Mathematics of Control, Signals and Systems, 9(4), 303–326.Duncan, T. E. (1970): “On the Calculation of Mutual Information,” SIAM Journal on Applied

Mathematics, 19(1), 215–220.Eisfeldt, A. L., H. Lustig, and L. Zhang (2017): “Complex Asset Markets,” NBER Working

Paper 23476.Epstein, L. G., and M. Schneider (2010): “Ambiguity and Asset Markets,” Annual Review of

Financial Economics, 2, 315–346.French, K. R. (2008): “Presidential Address: The Cost of Active Investing,” Journal of Finance,

63(4), 1537–1573.


Gagliardini, P., E. Ossola, and O. Scaillet (2016): “Time-Varying Risk Premium in LargeCross-Sectional Equity Data Sets,” Econometrica, 84(3), 985–1046.

Gennaioli, N., A. Shleifer, and R. Vishny (2015): “Money Doctors,” Journal of Finance,70(1), 91–114.

Gourieroux, C., A. Monfort, and E. Renault (1993): “Indirect inference,” Journal ofApplied Econometrics, 8(S1), S85–S118.

Greenwood, R., and D. Scharfstein (2013): “The Growth of Finance,” Journal of EconomicPerspectives, 27(2), 3–28.

Garleanu, N., and S. Panageas (2015): “Young, Old, Conservative, and Bold: TheImplications of Heterogeneity and Finite Lives for Asset Pricing,” Journal of Political Economy,123(3), 670–685.

Hansen, L. P., and J. Miao (2018): “Aversion to Ambiguity and Model Misspecification inDynamic Stochastic Environments,” Proceedings of the National Academy of Sciences, 115(37),9163–9168.

Hansen, L. P., and T. J. Sargent (2005): “Robust Estimation and Control underCommitment,” Journal of Economic Theory, 124(2), 258–301.

Hansen, L. P., and T. J. Sargent (2007): “Recursive Robust Estimation and Control withoutCommitment,” Journal of Economic Theory, 136(1), 1–27.

Hansen, L. P., and T. J. Sargent (2008): Robustness. Princeton University Press.He, Z., B. Kelly, and A. Manela (2017): “Intermediary Asset Pricing: New Evidence from

Many Asset Classes,” Journal of Financial Economics, 126(1), 1–35.He, Z., and A. Krishnamurthy (2012): “A Model of Capital and Crises,” The Review of

Economic Studies, 79(2), 735–777.(2013): “Intermediary Asset Pricing,” American Economic Review, 103(2), 732–770.(2018): “Intermediary asset pricing and the financial crisis,” Annual Review of Financial

Economics, 10, 173–197.(2019): “A Macroeconomic Framework for Quantifying Systemic Risk,” American

Economic Journal: Macroeconomics, forthcoming.Ju, N., and J. Miao (2012): “Ambiguity, learning, and asset returns,” Econometrica, 80(2),

559–591.Kacperczyk, M., J. Nosal, and L. Stevens (2018): “Investor Sophistication and Capital

Income Inequality,” Journal of Monetary Economics.Kacperczyk, M., S. Van Nieuwerburgh, and L. Veldkamp (2016): “A Rational Theory of

Mutual Funds’ Attention Allocation,” Econometrica, 84(2), 571–626.Kaniel, R., and P. Kondor (2012): “The Delegated Lucas Tree,” The Review of Financial

Studies, 26(4), 929–984.Karlin, S., and H. M. Taylor (1981): A Second Course in Stochastic Processes, vol. 25.

Academic Press, Philadelphia.Kasa, K. (2006): “Robustness and Information Processing,” Review of Economic Dynamics, 9(1),

1–33.Klibanoff, P., M. Marinacci, and S. Mukerji (2005): “A smooth model of decision making

under ambiguity,” Econometrica, 73(6), 1849–1892.Lewellen, J. (2011): “Institutional Investors and the Limits of Arbitrage,” Journal of Financial

Economics, 102(1), 62–80.Li, K. (2013): “Asset Pricing with a Financial Sector,” Unpublished Working Paper.Maenhout, P. J. (2004): “Robust Portfolio Rules and Asset Pricing,” The Review of Financial

Studies, 17(4), 951–983.


Mankiw, N. G., and S. P. Zeldes (1991): “The Consumption of Stockholders andNonstockholders,” Journal of Financial Economics, 29(1), 97–112.

Pagel, M. (2018): “A News-Utility Theory for Inattention and Delegation in Portfolio Choice,”Econometrica, 86(2), 491–522.

Pan, Z., and T. Basar (1996): “Model Simplification and Optimal Control of StochasticSingularly Perturbed Systems under Exponentiated Quadratic Cost,” SIAM Journal on Controland Optimization, 34(5), 1734–1766.

Sims, C. (2003): “Implications of Rational Inattention,” Journal of Monetary Economics, 50(3),665–690.

Smith Jr, A. A. (1993): “Estimating Nonlinear Time-series Models using Simulated VectorAutoregressions,” Journal of Applied Econometrics, 8(S1), S63–S84.

Turmuhambetova, G. (2005): “Decision Making in an Economy with Endogenous Information.,”Unpublished PhD dissertation University of Chicago.

Ugrinovskii, V. A., and I. R. Petersen (2002): “Robust Filtering of Stochastic UncertainSystems on an Infinite Time Horizon,” International Journal of Control, 75(8), 614–626.

Yan, H. (2008): “Natural Selection in Financial Markets: Does It Work?,” Management Science,54(11), 1935–1950.

Yin, P. (2019): “The Optimal Amount of Attention to Capital Income Risk and HeterogeneousPrecautionary Saving Behavior,” Unpublished Working Paper.


Appendix

7.1. Proof for Robust Filtering Problem. The basic equations in the main text are(3.7) (3.8) and (3.9). Following the robust filtering literature, applying the change ofmeasure from the approximating model P to the distorted model Q to the state transitionand observation equations yields the following distorted filtering model:

dgt = [ρg (g − gt) + σgv1,t] dt+ σgdZgt , (7.69)

dDt

Dt= (gt + σv2,t) dt+ σdZ0

t , (7.70)

dst = (gt + σsv3,t) dt+ σsdZst , (7.71)

where Zgt , Z0t , and Zst are Wiener processes that are related to the corresponding

approximating processes:

Zgt = Zgt −∫ t

0v1,sds, Z

0t = Z0

t −∫ t

0v2,sds, and Zst = Zst −

∫ t

0v3,sds

and ν1,t, ν2,t, and v3,s are distortions to the conditional means of the three shocks, Zgt ,

Z0t , and Zst , respectively. As shown in Pan and Basar (1996), Ugrinovskii and Petersen

(2002), and Kasa (2006), a robust filter can be characterized by the following dynamiczero-sum game:

Lt = infmj

supQ

lim supT→∞

EQ [F ]− θ−12 H∞ (Q|P)

, (7.72)

subject to (7.69), (7.70), and (7.71), where F ≡ 1T

∫ T0 (gj − gj)2 dj is the loss function and

H∞ is the relative entropy and is bounded from above:

H∞ (Q|P) = lim supT→∞

EQ[

1

2T

∫ T

0

(v2

1,t + v22,t + v2

3,t

)dj

]≤ η0, (7.73)

where η0 defines the set of models that the consumer is considering, and θ−12 is the Lagrange

multiplier on the relative entropy constraint, (7.73). As shown in Dai Pra, Meneghini, andRunggaldier (1996), the entropy constrained robust filtering problem, (7.72), is equivalentwith the following risk-sensitive filtering problem:

1

θ2log

(∫exp (θ2F (g, g)) dP

)= sup

Q

∫F (g, g) dQ− θ−1

2 H∞ (Q|P)

. (7.74)

The following proposition summarizes the results for this robust filtering problem:

Proposition 7.1. When θ2 < 1/σ2 + 1/σ2s , there is a unique solution for the robust

filtering problem, (7.74):

dgt = ρg (g − gt) dt+Qtσ

(ωtdt+ dZt

)+QtσsdZst , (7.75)

dQtdt

= σ2g − 2ρgQt −Q2

t

(1

σ2+

2κ

Qt− θ2

), (7.76)


where Qt = Et[(gt − gt)2

]is the conditional variance of gt, dZt = dZt − ωtdt, dZt =

1σ

(dDtDt− gtdt

)and dZst = 1

σs(dst − gtdt) are innovations. In the steady state, the

conditional variance converges to

Q =− (ρg + κ)σ2 + σ

√(ρg + κ)2 σ2 + (1− θ2σ2)σ2

g

1− θ2σ2> 0. (7.77)

Proof. The right-hand side of (7.74) is an entropy constrained filtering problem in termsof the scaled quadratic objective function F , while the left-hand side is a risk-sensitivefiltering problem in terms of the same function F , with risk-sensitivity parameter, θ2.The solution of this risk-sensitive filtering problem is just a special case of Theorem 3 ofPan and Basar (1996). Here we also use the result that Qt/σ

2s= 2κ to replace 1/σ2

s with2κ/Qt.

0 0.05 0.1 0.15

2

2.5

3

3.5

4

4.5

10-4

=0.01

=1

=10

Figure 8: Steady state posterior variance and channel capacity

7.2. Solving the Agents’ Optimization Problems. Proof of Proposition 3.1. Solving

first the infimization part yields νh∗t = −θh1σhW,tVw and ωht = −θh2Qhtσ Vg.

Optimal household consumption and portfolio rules under robustness are

Cht =1

Vw(7.78)

and

εht =−Vw

Vww − θh1V 2w

(πR,t − kt)σ2R,t

, (7.79)


respectively. Guess the value function takes the following form:

V(ght , Q

ht ,W

ht ;Y h

t

)=

1

ρhlnW h

t + F h(ght , Q

ht

)+ Y h

t ,

where Y ht is a function of aggregate state variable xt. Now define Y h

t and Yt as a functionof xt:

dY h(xt) = µhY,tdt+ σhY,tdZt.

Using Ito’s formula, we have

µhY,t =(Y h(xt)

)′µx,txt +

1

2

(Y h(xt)

)′′σ2x,tx

2t ,

σhY,t =(Y h(xt)

)′σx,txt.

Under this conjecture, Vw = 1ρhWh

tand Vww = − 1

ρh(Wht )2

. Substituting these expressions

into the FOCs, (7.78) and (7.79), gives the consumption and portfolio rules in the maintext. Further substituting back into the HJB equation, we could obtain the PDE forF h(ght , Q

ht

)and ODE system for Y h

t (xt).Similarly, the specialist’s problem can be solved in the same procedure.

7.3. Proofs of Propositions 4.4 and 4.5. Proof of Proposition 4.4. In theunconstrained case, per-unit exposure price is zero. Recall that the share of return contractβt ≡ ε∗t /εIt . Since the robust concern distorts the specialist’s desired risk exposure ε∗t , thechoice of share contract turns into

βUt =Wt

Wt + γγhW ht

and kt = 0.

Now the specialist and household no longer hold the equity claims according to theirwealth contributions as the benchmark case, but with a distortion term γ

γhwhich equals

the inverse of distortion on the capital constraint. Note that although agency friction mdoesn’t enter βUt in unconstrained region, both robustness parameters distort the contractshare alternatively. Replacing W h

t with asset market clearing condition (4.36) yields:

βUt =1

1 + γγh

(1

ρhxt− ρ

ρh

) .

By the imposed assumption that 0 < βUt ≤ 1, xt should be limited within (0, 1/ρ].In the constrained region, the share of return is determined by the incentive constraint

of specialist. In order to prevent the specialist from shirking, households need to pay apositive intermediation fee and exposure price to the intermediary, thus

βt =1

1 +mand kt > 0.

Proof of Proposition 4.5. In the unconstrained region, T ht = W ht , both households and

specialists put all their wealth into the intermediation, such that the total risk exposureequals εIt = αt(Wt + W h

t ). The equilibrium conditions (4.34) and (4.36) yield αU∗t = 1.


Given that ε∗t + εh∗t = Wt + W ht , the risk exposure for the specialist can be derived as

follows:

ε∗t +1− βUtβUt

ε∗t = Pt,

which means that εU∗t = 1

1+ γ

γh

(1

ρhxt− ρ

ρh

)Pt. From Figure 4.3, in the unconstrained region,

αt = 1 such that the specialist invests all of the intermediary’s equity capital into the riskyasset. Once the constraint is binding. αt > 1 means the specialist holds above 100% ofthe total equity and borrows (αt − 1)(Wt + T ht ) riskless bonds.

In the constrained region, the specialist holds βt = 11+m share of risk. Hence, the

specialist’s risk exposure is:

ε∗t = βtεIt =

1

1 +mPt.

Furthermore, the specialist’s portfolio share is:

α∗t =εIt

Wt + T ht=

Pt(1 + m)Wt

=1xt

+ ρh − ρ(1 + m)ρh

.

7.4. Solving for the Key Moments of Asset Prices. The return volatility. The returnvolatility can be derived from matching the diffusion terms of equation (3.7), (3.5), and(4.38) that

σR,t =σDt

ρhPt − (ρh − ρ)ε∗t=

1

Pt/Dt

σ

ρh − (ρh − ρ)βt. (7.80)

Using Proposition 4.5 and 4.4, we have

σUR,t =σ

ρh1

Pt/Dt

ρ(γh − γ

)xt + γ

(ρhγh − ργ)xt + γand σR,t =

1

Pt/Dt

σ

ρh − ρh−ρ1+m

.

Thus,

σUR,t =σ

1 +∆ρxt

(ρhγh − ργ

)xt + γ

ρ (γh − γ)xt + γ, (7.81)

σR,t =σρh

1 +∆ρxt

1 +m

mρh + ρ. (7.82)

The risk premium. Given that

πR,t =γσ2

R,tε∗t

Wt=γσ2

R,tβtPt

Wt=γσ2

R,tβt(Pt/Dt)

xt,

we have

πUR,t =γσU2

R,tβUt (Pt/Dt)

xt=

σ2γγh

(1 +∆ρxt)

[(ρhγh − ργ

)xt + γ

][ρ (γh − γ)xt + γ]

2

in the unconstrained region. By contrast, in the constrained region,

πR,t =γσ2

R,tβt(Pt/Dt)

xt=

σ2γρh

xt (1 +∆ρxt)

1 +m

(mρh + ρ)2 .


Solving the exposure price and intermediation fee. In the constrained region, kt ≥ 0.When household desired exposure demand (3.18) equals specialist exposure supply (3.26),i.e., εh∗t (kt) = mε∗t , we have

πR,t − ktγhσ2

R,t

W ht = m

πR,tγσ2

R,t

Wt,

which gives

kt =

(1− mρhxt

1− ρxt

)πR,t =

σ2(1 +m)

(mρh + ρ)2

(γ − ρhγhmxt

1− ρxt

)ρh

(1 +∆ρxt)xt,

where we use the fact that m =(γh/γ

)m.

Solving for the risk free rate. Using the household’s consumption function, we have

dCh∗tCh∗t

=d(ρhW h

t

)ρhW h

t

=d (Pt −Wt)

Pt −Wt. (7.83)

Combining it together with the distorted model (3.24) yields:

dWt

Wt= (φW,t + rt) dt+

πR,tγσR,t

dZt, (7.84)

where φW,t =(πR,tγσR,t

)2+ qt − ρ.

From Equations (4.36) and (4.37), we have:

d (Pt −Wt)

Pt −Wt=

(gtdt+ σdZt)Dt − ρdWt

Dt − ρWt=

(gtdt+ σdZt)− ρdWtWt

xt

1− ρxt(7.85)

where we use the facts that Pt − Wt = Dtρh− ρ

ρhWt and Pt − Wt = (gtdt+σdZt)Dt−ρdWt

ρh.

Taking conditional expectations and variances on both sides of the above equation gives:

Et[d(Pt −Wt)

Pt −Wt

]=gt − ρxt (φW,t + rt)

1− ρxtdt and vart

[d(Pt −Wt)

Pt −Wt

]=

(σ − ρxt

γπR,tσR,t

1− ρxt

)2

dt.

(7.86)Substituting Expressions (7.83) and (7.86) into the household’s Euler equation under

the distorted model,

rtdt = ρhdt+ Et[dCh∗tCh∗t

]− vart

[dCh∗tCh∗t

],

yields the expression for the risk-free rate:

rt = ρh +gt − ρxt (φW,t + rt)

1− ρxt−

(σ − ρxt

γπR,tσR,t

1− ρxt

)2

= rt = ρh + gt − ρ(ρh − ρ

)xt − ρqtxt −

ρxt

([ πR,tγσR,t

]2 − 2σπR,tγσR,t

)+ σ2

1− ρxt.


Using the expressions for πR,t/σR,t and qt in the constrained and unconstrained regionsby Propositions 4.9 and 4.13, we have

rUt = ρh + gt − ρ∆ρxt + σ2γγh −

(γh + γ

) (ρxt(γh − γ

)+ γ)

[ρ (γh − γ)xt + γ]2 ,

rt = ρh + gt − ρ∆ρxt − σ2 (1− ρxt)[ρ (1 + γm) + ρhm2γh

]+ ρhm2

(ρhxt − γh

)(1− ρxt) (ρ+mρh)

2xt

.

From Equation (4.36),

dW ht

W ht

=d(Pt −Wt)

Pt −Wt=

(gtdt+ σdZt)− ρdWtWt

xt

1− ρxt=

(1− 1

1− ρxt

)dWt

Wt+

1

1− ρxtdDt

Dt,

which gives:

dW ht

W ht

− dWt

Wt= − 1

1− ρxt

(dxtxt

+σ

γ

πR,tσR,t

dt− σ2dt

).

7.5. Solving the Stochastic Process of Aggregate State. In order to derive theunconditional mean and variance of risk premium and interest rate, we need to know thedistribution of the state variable, xt = Wt/Dt. Using Ito’s formula, we have

dxtxt

=dWt

Wt− dDt

Dt− dDt

Dt

dWt

Wt+

(dDt

Dt

)2

.

Substituting the dividend process (3.7) and derived specialist wealth process (7.84) intothe above equation yields:

dxtxt

=

(σ2 − gt − ρ+ qt + rt +

1

γ2

π2R,t

σ2R,t

− σ

γ

πR,tσR,t

)dt+

(πR,tγσR,t

− σ)dZt.

The drift and diffusion coefficients of the aggregate state process, µx,t and σx,t, can bewritten as:

µx,t = σ2 − gt − ρ+ qt + rt +

(πR,tγσR,t

)2

− σ(πR,tγσR,t

)

= σ2 + ρh − ρ− ρ(ρh − ρ

)xt + (1− ρxt) qt +

(πR,tγσR,t

)2(1− 2ρxt) + (3ρxt − 1)σ

(πR,tγσR,t

)− σ2

1− ρxt,

σx,t =

(πR,tγσR,t

)− σ.

Note that

qt ≡Kt

Wt=mktε

∗t

Wt=mktγ

Ptβt

=m

1 +m

Pt/Dt

xtkt =

σ2m

(mρh + ρ)2

(γ − ρhγhmxt

1− ρxt

)(1

xt

)2

.

Putting the expressions of qt, πt/σR,t and rt from Propositions 4.9 and 4.11 back, wecould obtain the scaled wealth process in the main text.


7.6. Indirect Inference. The distance is measured by the quadratic form

GT (Ψ) = GT (Ψ)′ΩTGT (Ψ), γ > γh and ρ < ρh,

where Ψ =[ρ ρh θ1 θ

h1 σ]′

is a 5 × 1 vector.26 GT (Ψ) is the moment condition involvingthe data and parameters. We use T = 5000 to calculate the sample averages. ΩT is an10× 10 symmetric and positive definite weighting matrix,

ΩT =

1λ21

0 0 0

0 1λ22

0 0

0 0. . . 0

0 0 0 1λ210

.

Sample moment conditions are as follows: GiT (Ψ) =(λi − λi

), i = 1, . . . , 9, G10T (Ψ) =

[1− corr (πτ , πτ )], where λ1 = AR(1) coefficient of equity premium, λ2 = AR(1) interceptof equity premium, λ3 = AR(1) coefficient of price/dividend ratio, λ4 = AR(1) interceptof risk premium, λ5 = mean equity premium, λ6 = mean risk free rate, λ7 = meanprice/dividend ratio, λ8 = price/dividend ratio volatility, λ9 = mean return volatility,λ10 = 1/

√2. πτ is the time series of equity premium with τ = 480 (40 years). denotes

the model estimated corresponding values.We set the priors of the parameters as follows: ρ ∈ [.005, .03], ρh ∈ [.005, .03],

θ1 ∈ [.01, .1], θh1 ∈ [.01, .1], σ ∈ [.1, .15]. We take 10 grids for each parameter.

Finally, we use indirect inference to obtain estimated parameters ΨGMM =arg min

ΨGT (Ψ)′ΩTGT (Ψ).

7.7. Detection Error Probabilities. The relative entropy computed from thehousehold’s robust control problem can be written as:

νht = −θh1σR,tε

ht

ρhW ht

= − θh1ρhγh

πR,t − ktσR,t

=1− γh

γh

(πR,tσR,t

− ktσR,t

). (7.87)

In the constraint case, we have:

πR,tσR,t

− ktσR,t

=mρhσγ

(1− ρxt) (mρh + ρ).

In the unconstrained case in which kt = 0, we have

πR,tσR,t

− ktσR,t

=σγγh

ρ (γh − γ)xt + γ.

26At the beginning, we put another restriction θ1 ≤ θh1 . However, the optimal parameter choice is alwaysθ1 = θh1 . Thus, we impose θ1 = θh1 to simplify the calibration and reduce the grid points matrix.


Substituting these expressions into (7.87) yields:

νht =1− γh

γh

[γhmρhσ

(1− ρxt) (mρh + ρ)1xt∈(xmin,xc] +

σγγh


(xc, 1

ρ

]] .

= σγ(

1− γh)[ mρh

γ (mρh + ρ) (1− ρxt)1xt∈(xmin,xc] +

1


(xc, 1

ρ

]] ,where 1 denotes the indicator function.

Similarly, for the specialist’s problem, we have

νt = −θ1σR,tεtρWt

= − θ1

ργ

πR,tσR,t

=1− γγ

πR,tσR,t

,

which implies that

νt = σγh (1− γ)

[1

γh (mρh + ρ)xt1xt∈(xmin,xc] +

1


(xc, 1

ρ

]] .

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