Macroeconomic Models with IncompleteInformation and Endogenous Signals ∗

Jonathan J Adams†

January 6, 2020

Link to Most Current Version

Abstract

This paper characterizes a general class of macroeconomic models with in-complete information, which feature both endogenous signal processes and en-dogenous state variables. I derive conditions for existence and uniqueness ofequilibrium, and I introduce an algorithm to solve the general model. As anapplication I consider a business cycle model with capital where firms mustmake inferences about aggregate shocks through the movements of endogenousprices. In this model, the central bank’s policy rule determines the real effectsof nominal shocks, by controlling how informative prices are about the aggre-gate state. The optimal policy targets acyclical inflation, which makes moneyneutral. Finally, I demonstrate an advantage of models with endogenous infor-mation: the noisy signals are driven by fundamental shocks, rather than ad hocnoise, so data can discipline the information structure. Accordingly, I calibratethe model using US industry-level panel data.

JEL-Codes: D84, E32, C62, C63Keywords: Endogenous Signals, Incomplete Information, Heterogeneous Be-liefs, Business Cycles, Real Effects of Monetary Policy

∗I am grateful for helpful comments from Chunrong Ai, Ryan Chahrour, Thorsten Drautzburg,

Igor Livshits, Kristoffer Nimark, James Pascoe, Richard Romano, David Sappington, Todd Walker,

seminar participants at the University of Florida, Federal Reserve Bank of Philadelphia, and the

Summer Meetings of the European Econometric Society. This paper previously circulated under

the title “Rational Expectations with Endogenous Information.” Replication code and resources for

applying the solution algorithm to other models are available on my website.†University of Florida. Website: www.jonathanjadams.com Email: adamsjonathan@ufl.edu

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1 Introduction

I derive a method to solve macroeconomic models with dispersed information and twochallenging features: endogenous signals and endogenous state variables. Endogenoussignals1 improve models of incomplete information by allowing for agents to learnfrom realistic sources, such as quantities and prices. But solving dynamic modelswith endogenous signals has proven difficult, and lacked characterizations of generalproperties such as equilibrium existence, uniqueness, or stability. When endogenousstate variables such as capital are included, the challenges in finding equilibriumbecome even more difficult, and few such models have been solved without shortcutsthat recover full information after some number of periods or for some subset ofagents. This paper resolves these difficulties, and gives conditions for existence anduniqueness of equilibrium in a general macroeconomic model with endogenous signalsand endogenous state variables.

The paper’s foremost contribution is a contraction theorem. Endogenous signalsmake models difficult to solve because the usual methods for solving models with ex-ogenous signals cannot necessarily be applied. When signals are endogenous, agents’choices depend on their information processes, which depend on agents’ choices, whichdepend on their information processes, and so forth. Does this sequence converge toan equilibrium process for information and actions? In general, this is an open ques-tion.2 I derive sufficient conditions on a general macroeconomic model that ensurethe information process does converge, that the equilibrium is unique, and can besolved for with an algorithm: Signal Operator Iteration.

As an application, I study a macroeconomic model of local information featur-ing the previously challenging combination: both endogenous signals and capital.3

Agents can observe prices and quantities in their local market, but not the aggre-gate state of the economy. Rather, when making investment decisions, they musttry to infer the magnitude of aggregate shocks by observing the realizations of local

1“Endogenous” signals or information has different meanings in different literatures. In thiscontext it is Huo and Takayama (2015)’s definition: the endogeneity refers agents’ observation ofheterogeneous noisy signals containing endogenous variables. This contrasts with the large liter-ature of endogenous information acquisition, where agents choose to utilize a subset of availableinformation, such as in the rational inattention literature following Sims (2003).

2In dynamic models without endogenous state variables, uniqueness or multiplicity can some-times be characterized. For example, some asset pricing models with endogenous information havedemonstrable multiplicity, as in Angeletos and Werning (2006), Hellwig, Mukherji, and Tsyvinski(2006), or Angeletos, Hellwig, and Pavan (2007). However, other dynamic asset pricing models withendogenous information such as Nimark (2017) demonstrate a unique equilibrium. In static modelssuch as Grossman (1976), uniqueness is typically straightforward to characterize.

3To my knowledge, the only paper studying such a model without additional assumptions thatreveal the true state is Graham and Wright (2010), who apply the Nimark (2017) asset pricingalgorithm to a version of the Neoclassical growth model with dispersed information. Their modelfeatures two signals and two shocks and does not meet the Nimark (2017) conditions for a uniqueequilibrium, so Graham and Wright (2010) study a useful “market-consistent” subset of equilibria.

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variables that are affected by the shocks. Signals such as prices and quantities areendogenous, so Signal Operator Iteration must be used to solve the model. In equi-librium, money is non-neutral, and the size of its effects on real variables depends onthe strength of the feedback from endogenous variables to signals. I demonstrate apractical conclusion: the monetary authority can control the magnitude of the realeffects of monetary shocks by choosing the parameters of its policy rule. With theright choice, it can eliminate the non-neutrality entirely and minimize the aggregatecosts of the information friction.

Why is endogenous information an appealing model feature? When agents tradein markets, they see endogenous variables – prices, quantities, counterparties, etc. –and these variables contain information about the state of the economy. Individualslearn from these variables; for example, when homeowners see the price of housingin their area rise, they infer that demand for housing might be increasing, and maychoose to invest in renovations. In a model without endogenous information, theopportunity for this learning is assumed away. Many models with only exogenousinformation assume that agents cannot learn from the prices or quantities that theyare trading, and instead assume that agents receive informative signals with ad hocexogenous noise. Moreover, an endogenous information process can be disciplined bythe data. When agents learn from exogenous noisy signals, in many cases the signalshave no empirical analog, so their properties are typically estimated using the impliedaggregate dynamics.4 With endogenous information, an econometrician can directlyuse the time series properties of the very same observables that agents learn from. Forexample, in Section 4’s local information model, idiosyncratic sectoral demand shocksobfuscate inference of aggregate demand, and these demand shocks are measurablewith sectoral price and quantity data from national accounts.

The literature has employed several methods to solve dynamic models with en-dogenous information, but these methods are limited to special cases, and have notbeen successfully applied to models that also feature endogenous state variables.5 Forexample, Kasa (2000), Acharya (2013), and Rondina and Walker (2015) use Blaschkeroot flipping to find the Wold representation of the information process. This methodrequires that there are the same number of shocks as signals; otherwise finding theWold representation is more difficult. An alternative approach is to assume thatshocks become common knowledge after some fixed number of periods, as in themonetary model of Hellwig and Venkateswaran (2009), the model with housing capi-tal of Chahrour and Gaballo (2019), or originally in Lucas (1972). Huo and Takayama(2016) use a finite ARMA to approximate the solution to a version of Angeletos andLa’O (2009b) with endogenous information and without endogenous states. They

4For example, this approach is used in Angeletos and La’O (2013), Makowiak and Wiederholt(2015), among many others.

5A large literature has developed the solution methods for dispersed information models moregenerally, following the seminal work of Townsend (1983). Huo and Takayama (2016) review thisliterature.

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show there is no finite ARMA representation to a true solution, and go on to provesuch a solution exists, but do not demonstrate uniqueness.6

In closely related work, Nimark (2017) uses a similar iterative algorithm to cal-culate higher order expectations in a general asset pricing model with endogenousinformation. My results are complementary to Nimark’s in two primary respects.First, the Nimark algorithm is useful for solving asset pricing models characterizedby a forward-looking Euler equation, but cannot be directly applied to macroeconomicmodels with backward-looking state variables.7 My approach allows for such statevariables, but requires some further conditions to guarantee uniqueness, so it cannotbe directly applied to all asset pricing models. Second, the Nimark algorithm is ableto characterize the dynamics of higher order expectations concurrent to solving themodel, which are inherently valuable; my algorithm simply solves for the equilibriumexpectations.

This paper contributes to a large literature studying the relationship between in-formation frictions and monetary non-neutrality.8 Lucas (1972) is the seminal paperand featured endogenous information in a 2-period overlapping generations model.Research followed to translate the intuition to a dynamic setting which might allowfor capital, but Sargent (1991) suggests that the profession abandoned this approachdue to the difficulty of solving these sorts of models. The literature returned tothese questions with papers such as Woodford (2003), Angeletos and La’O (2009a),Makowiak and Wiederholt (2009), and Lorenzoni (2010), which added dynamic learn-ing from dispersed information, but restricted analysis to exogenous signal processes.More recently, Melosi (2016) allowed firms to learn from an endogenous signal – thenominal interest rate – in a New Keynesian model without capital, which Melosisolved by applying Nimark’s asset pricing algorithm.

The strategy for the remainder of the paper is outlined as follows. In Section 2I define a general linear rational expectations model where agents respond to shocksand signals they perceive as exogenous. Then in Section 3.1 I allow signal processes tobe equilibrium objects that depend on aggregate behavior. In Section 3.2 I define thealgorithm to solve for the equilibrium, and in Section 3.4 I prove that the algorithm isa contraction under some conditions, so that there must exist a unique equilibrium tothe model. Section 4 contains the application and examines optimal monetary policy.

6Other simplifying assumptions that are used to make asset pricing models with endogenousinformation tractable include: having uninformed traders (e.g. Townsend (1983) or Wang (1993))or by assuming a ”day of reckoning” structure with a constant payoff (e.g. Back, Cao, and Willard(2000) or Amador and Weill (2012)).

7Nimark (2008) and Melosi (2016) use this algorithm to solve New Keynesian models with en-dogenous information but without endogenous state variables. It is applied to empirical asset pricingmodels with endogenous information in Barillas and Nimark (2017), Barillas and Nimark (2018),and Struby (2018).

8This paper also joins the larger literature on information frictions in macroeconomics, whichAngeletos and Lian (2016) survey.

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2 Exogenous Information

In this section I describe a general method for solving macroeconomic models withexogenous information. The purpose of this section is to introduce notation, thegeneral model, and the solution when information is exogenous, which is necessary asan intermediate step once information is endogenized (Section 3).

Consider a stationary linear macroeconomic model of the following form.9 Theequilibrium conditions for agent i at time t are:

0 = Ei,t[BS0XS,i,t +BS1XS,i,t+1 +BC0XC,i,t +BC1XC,i,t+1 +BZ0Zi,t +BZ1Zi,t+1] (1)

XS,i,t is an nS × 1 vector containing the endogenous state variables and XC,i,t isan nC × 1 vector containing the control variables at time t. The mZ × 1 vectorZi,t contains exogenous variables such as economic shocks and signals. Additionally,Zi,t may include variables that agent i takes as exogenous, but are endogenous inequilibrium, such as an economy-wide interest rate or price level.

The matrices BS0, BS1, BC0, BC1, BZ0, BZ1 contain coefficients corresponding tothe n ≡ nC + nS equilibrium conditions of the model. The expectations operatorEi,t applies to the information set of agent i at time t. Information is exogenous, soindividual choices do not affect the information available to other agents.

A linear solution to the model is policy that expresses XS,i,t and XC,i,t as a functionof exogenous variables Zk for k ≤ t such that (1) holds with equality for all t. Thisis not necessarily a recursive policy function; it may depend on the entire historyof Zk. Specifically, define policy functions to be linear in the history of white noiseinnovations Wi,t implied by the Wold decomposition of Zi,t. Let Z(L) be the Woldrepresentation expressed as a polynomial in the lag operator L. Then Zi,t is given by

Zi,t = Z(L)Wi,t ≡j=∞∑j=0

ZjLjWi,t (2)

Stack the endogenous variables into a single n × 1 vector Xi,t ≡(XC,i,t

XS,i,t

). The

policy function can be expressed as a polynomial in the lag operator:

Xi,t = X(L)Wi,t ≡j=∞∑j=0

XjLjWi,t (3)

Expressing policy functions in terms of information is convenient because forecast-ing is straightforward: Ei,t[Wi,t+k] = 0 for all k > 0.10 Frequently the policy function

9This general encompasses a broad class of DSGE models, and is solved without informationfrictions in Uhlig (1995), among many others.

10This is the Wiener-Kolmogorov prediction formula. See Hansen and Sargent (1981) for a de-

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is expressed in term of the history of signals, and this form is easily recovered becausethe Wold decomposition is invertible:

Xi,t = X(L)Wi,t = X(L)Z(L)−1Zi,t (4)

where Xj∞j=0 are n × mZ matrices. When expressed in terms of the innovationsWi,t, the equilibrium condition (1) becomes

0 = [BX0X(L)Wi,t +BX1L−1X(L)Wi,t +BZ0Z(L)Wi,t +BZ1L

−1Z(L)Wi,t]+ (5)

where [·]+ is the annihilation operator, which annihilates negative powers of L. Thecoefficient matrices are stacked so thatBX0 ≡

(BC0 BS0

)andBX1 ≡

(BC1 BS1

).

Equation (5) follows from (1) because forecasting is linear, which I assume for tractabil-ity and is consistent with how the DGSE literature handles expectations in linearizedmodels.

The equilibrium policy function can be expressed as a linear function of the Wolddecomposition. Before deriving the formula, some notation must be defined. LetΛC denote a diagonal matrix of the eigenvalues of −B−1

X1BX0 that are outside thecomplex unit circle, and let ΛS denote a diagonal matrix of eigenvalues that areinside the complex unit circle. Let Q denote a matrix of eigenvectors of −B−1

X1BX0

ordered so that

−B−1X1BX0 = Q

(ΛC 00 ΛS

)Q−1 (6)

and partition the Q−1 matrix into blocks:

Q−1 =

(RCC RCS

RSC RSS

)(7)

where RCC is nC × nC , RSC is nS × nC and so forth.Define the matrices Zk, Ξk, and Θk by

Zk ≡

0 k < 0

−B−1X1(BZ1Zk+1 +BZ0Zk) k ≥ 0

(8)

Ξk ≡

Q

(−Λk−1

C 0

0 0

)Q−1 k ≤ 0

Q

(0 0

−Λk−1S RSCR

−1CC Λk−1

S

)Q−1 k > 0

(9)

scription in the context of rational expectations models. The Wiener filter in this case is used forcharacterizing expectations in lieu of the Kalman filter is which more common in the literature;the Kalman filter is less convenient in this situation because time is infinite, and when endogenousinformation is introduced, the state space becomes infinite as well.

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Θk ≡

0 k < 0

Q

(I 0

RSCR−1CC I

)Q−1 k = 0

Q

(0 0(

ΛkSRSCR

−1CC − Λk−1

S RSCR−1CCΛC

)0

)Q−1 k > 0

(10)

and lastly define the polynomials Z(L), Ξ(L) and Θ(L) by

Z(L) ≡∞∑

k=−∞

ZkLk Ξ(L) ≡

∞∑k=−∞

ΞkLk Θ(L) ≡

∞∑k=−∞

ΘkLk (11)

Theorem 1 If BX1 is invertible and if −B−1X1BX0 has nC eigenvalues outside the unit

circle, and nS nonzero eigenvalues inside the unit circle, then the model has a uniquesolution and the policy function is given by

X(L) = Θ(L)[Ξ(L)Z(L)

]+

Proof: See Appendix A.1.The purpose of Theorem 1 is to express the policy function in a way that can be

easily applied to the endogenous information case in Section 3. The requirement that−B−1

X1BX0 has nC eigenvalues outside the unit circle is not novel; it is the Blanchardand Kahn (1980) condition that there must be as many unstable eigenvalues as thereare contemporaneous jump variables for the equilibrium to be uniquely determined.This condition prevents application of the Nimark (2017) solution algorithm whenthe model includes endogenous state variables. The Nimark condition requires thatthe matrix norm of (−B−1

X1BX0)−1 is less than one.11 (−B−1X1BX0)−1 has nS eigen-

values outside the unit circle, and the norm of a matrix is weakly greater than thelargest absolute eigenvalue, so if there are endogenous state variables then the Nimarkalgorithm cannot be directly applied.

This method of finding the policy function is linear, which is useful in order toprove existence and uniqueness when information is endogenous in Section 3. Addingand multiplying lag operator polynomials are linear operations, as is applying theannihilation operator (although it is not generally commutative with multiplication,which creates challenges). Other approaches to solving linear models, such as themethod of undetermined coefficients, require inverting a matrix, which makes demon-strating a contraction more challenging. In addition, this method has computationaladvantages. When information is endogenous, the relevant state space becomes largeand the policy function must be calculated for many different information processes,so it is preferable to avoid inverting a large matrix to find the policy function at everyiteration.

11(−B−1X1BX0)−1 is equivalent to Λ in Nimark (2017) Section 6.

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3 Endogenous Information

This section shows how to solve macroeconomic models with endogenous signal pro-cesses. I describe a general model of dispersed information with endogenous signals,present a generally applicable solution algorithm, and characterize conditions for ex-istence of a unique equilibrium.

In this section, I assume that the conditions for Theorem 1 are satisfied, so thatgiven an information process, there is a unique equilibrium. This requires that BX1

is invertible, and that −B−1X1BX0 has nC eigenvalues outside the unit circle, and nS

nonzero eigenvalues inside the unit circle (the Blanchard-Kahn condition).

3.1 Endogenous Information Formation

When information is endogenous, the signals Zi,t are jointly determined in equilibriumwith the rest of the model. For Zi,t to be endogenous variables, the model requiresadditional equilibrium conditions. These conditions are informational consistency :that the signal dynamics are consistent with the dynamics of the other endogenousvariables. I proceed by outlining a general framework for how endogenous informationis formed, characterize it in terms of lag operator polynomials, and then define theinformational consistency condition.

Suppose the signals Zi,t observed by agent i are a sum of exogenous signals SZ,i,tand endogenous signals SX,i,t:

Zi,t = SZ,i,t + SX,i,t (12)

where all of these signals are mZ × 1 vectors. These signals can be expressed aslag polynomials times the white noise process of fundamental exogenous shocks, εi,t,which has dimensionality mε ≥ mZ with positive definite variance matrix Σε:

SZ,i,t = SZ(L)εi,t SX,i,t = SX(L)εi,t (13)

The causal square-summable polynomial SZ(L) is a primitive of the model. But thepolynomial SX(L) depends on equilibrium behavior and aggregation. Define the sumof the two polynomials as

Zi,t = S(L)εi,t ≡ SZ(L)εi,t + SX(L)εi,t (14)

Endogenous signals are determined by macroeconomic aggregates. This assumesthat the actions of atomistic agent i do not affect the information of any agent be-yond their effect on the aggregate economy. The square-summable polynomial A(L)codifies exactly how aggregate variables affect the endogenous signal. For example,it may include aggregate resource constraints or adding up constraints, asset pricingequations, sectoral demand, or other conditions relating aggregate allocations to id-iosyncratic prices observed by the decision makers. A(L) is a primitive of the model,and generates signals by

SX,i,t = [A(L)Xt]+ (15)

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The right hand side of (15) includes no idiosyncratic terms, so SX,i,t is the same forall agents; it is determined only by macroeconomic aggregates Xt. The annihilationoperator is included so that A(L) might have terms associated with negative powersof L. This allows A(L) to include expectational equations if needed. By assumption,both the causal and non-causal terms of A(L) are square-summable.

3.1.1 The Wold Representation

Before describing aggregation, it is necessary to characterize how white noise innova-tions Wi,t are determined by the fundamental shocks εi,t.

The signal Zi,t is equivalent to two polynomials: S(L)εi,t is a lag polynomial offundamental shocks, while Z(L)Wi,t is a lag polynomial of white noise innovations.The Wold representation Z(L)Wi,t is invertible, which implies that the white noiseinnovations can be written as

Wi,t = Z(L)−1S(L)εi,t

How can Z(L)−1 be calculated? The innovation polynomial Z(L) and the signalpolynomial S(L) both produce the same series of Zi,t, so they must have the sameautocovariance function. Let the mZ ×mZ matrix Γj denote the jth autocovarianceof the signal Zi,t. The fundamental shock εi,t is a white noise process with varianceΣε, so Γj satisfies

Γj =∞∑k=0

SkΣεS′k−j (16)

The innovation polynomial Z(L) is the Wold decomposition of the signal polynomialS(L), so its inverse Z(L)−1 solves the Yule-Walker Equations:

Γ0 Γ1 Γ2 ...Γ1 Γ0 Γ1 ...Γ2 Γ1 Γ0 ......

......

. . .

−(Z−1)′1−(Z−1)′2−(Z−1)′3

...

=

Γ1

Γ2

Γ3

...

(17)

where the polynomial Z(L) is normalized so that Z0 = I.

3.1.2 Aggregation

Aggregate variables affect the endogenous signal, so I must characterize how shocksaggregate, and how aggregated shocks determine aggregate allocations.

The shock εi,t contains both aggregate and idiosyncratic dimensions. Supposethere is a unit measure λ of agents i in the set I. Assume the idiosyncratic dimensionsare mean zero in the population. The aggregate shock εt is defined

εt ≡∫Iεi,tdλ(i)

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Then the average signal Zt ≡∫I Zi,tdλ(i) satisfies

Zt =

∫IS(L)εi,tdλ(i) = S(L)εt

because S(L)εi,t is linear in the sequence of shocks. Similarly, the aggregate endoge-nous vector Xt ≡

∫I Xi,tdλ(i) satisfies

Xt =

∫IX(L)Wi,tdλ(i) =

∫IX(L)Z(L)−1S(L)εi,tdλ(i) = X(L)Z(L)−1S(L)εt (18)

Finally, let the projection matrix Pε denote the diagonal matrix with ones in dimen-sions corresponding to aggregate shocks and zeros elsewhere, so that

εt = Pεεi,t ∀i ∈ I (19)

3.1.3 Informational Consistency

The lag polynomial SX(L) is determined by combining equations (15), (18), and (19):

SX(L)εi,t = [A(L)X(L)Z(L)−1S(L)Pεεi,t]+ (20)

combining equations (14) and (20) provides the informational consistency condition:

S(L)εi,t = SZ(L)εi,t + [A(L)X(L)Z(L)−1S(L)Pεεi,t]+ (21)

Equation (21) states that the dynamics of the signal process S(L) must be consistentwith the dynamics it implies for the endogenous variables X(L)Z(L)−1S(L).

The informational consistency condition is related to the higher order expectationsstudied in e.g. Morris and Shin (2002), Woodford (2003), Allen, Morris, and Shin(2006), Makarov and Rytchkov (2012), and Nimark (2017). In such models, agentsmust forecast the forecasts of others, which themselves depend on the forecasts of oth-ers, and so forth. Characterizing these orders of thinking pose a challenging problemin many models; where do those challenges appear here? The information consis-tency condition implicitly imposes that the equilibrium is consistent with the limitof this thinking. There is no advantage that agents can gain by trying to forecastothers’ expectations because they are already making the best linear forecast. Allagents’ actions are consistent with their best possible forecasts, and the informationprocess is consistent with their actions. Thus agents need not spend time trying toforecast the forecasts of others, or working through some math of how such higherorder expectations affect them; their linear forecast is already their best possible fore-cast conditioned on their entire information set. Huo and Pedroni (2020) solve forexpectations in a general class of beauty contest models with this same insight.

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3.2 Solution Algorithm

This section describes the algorithm to solve the endogenous information model, whileAppendix C details how to compute this algorithm in practice.

The algorithm is straightforward to describe informally. Begin by guessing a signalprocess Sn(L). Then, find the policy function Xn(L) implied by the signal process bycalculating the Wold decomposition and applying the solution method from Section2. Next, use the assumed relationship between endogenous variables and endogenousinformation that is encoded in A(L) to calculate the implied signal process Sn+1(L).Repeat until the signal process converges. Formally, the algorithm is:

Algorithm 2 (Signal Operator Iteration) Conjecture a square-summable causallag polynomial S0(L). Then proceed with iteration n = 0 as follows:

1. Find the autocovariance function Γn(L) implied by Sn(L) using equation (16).

2. Use Γn(L) to solve the Yule-Walker equations (17) for Zn(L).

3. Given the Wold representation Zn(L), generate the polynomial Zn(L) in equa-tion (8).

4. Calculate the policy function Xn(L) from Zn(L) by Theorem 1.

5. Calculate the next signal polynomial Sn+1(L) by combining equations (14) and(20):

Sn+1(L) = SZ(L) + [A(L)Xn(L)(Zn)−1(L)Sn(L)Pε]+ (22)

6. If ||Sn+1 − Sn|| is sufficiently close to zero, conclude that S(L) = Sn+1(L).Otherwise return to Step 1 with guess Sn+1.

Theorem 6 (below) specifies conditions under which the Signal Operator Iterationis a contraction mapping, and has a unique fixed point.

To prove the theorem, it is useful to treat lag operator polynomials as linearoperators on a Hilbert space. Specifically, an arbitrary n ×m square summable lagpolynomial Y (L) =

∑∞j=−∞ YjL

j is a bounded linear operator on a bi-infinite sequence

of shocks. Each random shock εt is represented as an object ~ε in `2(Zm), the Hilbertspace of square-summable bi-infinite sequences of m × 1 vectors.12 The entries inthe vector ~ε determine how the shock is generated by some underlying fundamentalunit variance process. The time series of shocks is the bi-infinite matrix ε, which hascolumns ~ε for all t ∈ (−∞,∞). ε is also a bounded linear operator on `2, mappingfundamental processes into shocks. Because the shock process is i.i.d., ε and Y (L)

12Economists are typically familiar with representing random variables in the function space L2,but this paper employs machinery from the signal processing field, which frequently represents dis-crete time series in the vector space `2. The two spaces are both Hilbert spaces and thus isomorphicif they have the same cardinality; Appendix D makes this mapping explicit.

11

are both Laurent operators, which implies they commute with the lag operator L.For notation, let Y denote the Laurent operator of the polynomial Y (L), which maps`2(Zm) to `2(Zn).13 Thus Y ε is the time series of n× 1 vectors generated by the timeseries of m× 1 shocks.

Laurent operators on `2(Zm) can be represented as bi-infinite dimensional blockToeplitz matrices, also known as Laurent matrices.14 For the arbitrary operator Y ,the matrix form is:

. . ....

......

... . ..

... Y0 Y−1 Y−2 Y−3 ...

... Y1 Y0 Y−1 Y−2 ...

... Y2 Y1 Y0 Y−1 ...

... Y3 Y2 Y1 Y0 ...

. .. ...

......

.... . .

The product of two operators is the product of the Laurent matrices, the inverse Y −1

of the operator is the inverse of the Laurent matrix, and so forth. The lag operatorL is the Laurent matrix with identity matrices along the first block diagonal belowthe main block diagonal.

When Y (L) is a constant matrix so that Yj = 0 for all j 6= 0, then the operatorY is block diagonal with Y0 along the main block diagonal. To ease notation, I letthe matrix Y0 also denote its corresponding block diagonal operator, so that I do nothave to define a new operator for every matrix that is added to or multiplied by a lagpolynomial. An important case is the shock variance Σε. The i.i.d. shock process εsatisfies εε∗ = Σε, which is the Laurent matrix with the variance Σε along the mainblock diagonal.

3.3 The Sandpaper Lemma

It is useful to prove an intermediate lemma before introducing the Signal OperatorIteration theorem. This lemma makes up the most novel component in the proofof the contraction theorem. I begin by defining some important properties of signalprocesses.

The first property is whether the signal is causal. A causal signal process is onlyaffected by current and past fundamental shocks.

13All separable Hilbert spaces are isomorphic to `2(Z), but in order to keep track of the sizesof blocks, I find it useful to denote the spaces as sequences of vectors, e.g. `2(Zm) is the space ofbi-infinite sequences of m×1 vectors. In this case, m is sometimes referred to as the “multiplicity” of`2 (Frazho and Bhosri, 2010). ε can also be interpreted as the operator that converts an underlyingdata generating process in `2(Z) into the bi-infinite sequence of shocks in `2(ZmZ ).

14The Toeplitz terminology is usually reserved for finite or singly infinite matrices. For this andother useful properties of operators on `2, see Conway (2007), or Frazho and Bhosri (2010) forLaurent operators in particular.

12

Definition 3 A signal process S(L) is causal if Sj = 0 for all j < 0.

Equivalently, a causal operator S is lower block triangular.The second property is whether the signal is “white noise”. As a time series

process, this has the usual meaning: being identically distributed over time anduncorrelated across periods. I use the notation W to denote white noise operators,which generates a white noise process Wε for any time series ε of i.i.d. shocks withvariance Σε. Specifically, ε is a Laurent operator on `2(Z) such that εε∗ = Σε.

Definition 4 A signal operator W is a white noise operator on ε if Wεε∗W ∗ isblock diagonal with positive semi-definite blocks.

When Σε = εε∗ is the variance of the fundamental shock process, then WΣεW∗ = ΣW

is the variance of the white noise process.15

The main challenge with proving existence and uniqueness of equilibrium is thatthe Signal Operator Iteration algorithm is not a linear operator. This is becausecreating a forecast of future signals is nonlinear in the signal process; in particular,the annihilation operator [·]+ is nonlinear. This nonlinearity creates challenges for themain proof, because although addition commutes through the annihilation operator,multiplication does not. Lemma 5 resolves these challenges.

Lemma 5 (Sandpaper Lemma) If W 1 and W 2 are both causal operators, whitenoise on ε, and if the nullity of their initial matrix blocks W j

0 satisfy

mZ − rank(W j0 ) ≤ mε −mZ

then for any Laurent operators Y 1 and Y 2 mapping `2(ZmZ )→ `2(ZmZ )

||[Y 1]+W1 − [Y 2]+W

2|| ≤ ||Y 1W 1 − Y 2W 2||

|| · || denotes the operator norm on the Hilbert space of random shocks E . The set Eis the set of columns of the bi-infinite Laurent matrix ε representing the entire timeseries of shocks. E is a subset of the Hilbert space `2(Zm).

Proof: See Appendix A.3.

3.4 Equilibrium Existence and Uniqueness

Define the operator B as applying steps 1-5 of Algorithm 2, so that guesses of thesignal operator Sn and Sn+1 are related by

BSn = Sn+1

15To relate white noise to a more familiar operator property, left multiply a white noise operatorW by the inverse of the Cholesky decomposition CW of variance matrix ΣW and right multiply bythe Cholesky decomposition Cε of Σε: C

−1W WCε is a co-isometry. This operator cannot be unitary

because its blocks are not square.

13

= SZ + [AXn(Zn)−1SnPε]+

Let S denote the set of causal Laurent operators that map the space of random shocksE ⊂ `2(Zmε) to `2(ZmZ ). S is a Banach space, and the distance metric on this spaceis the operator norm || · ||. Lastly, let mV denote the number of linearly independentcontemporaneous private shocks in the exogenous signal, i.e. mV = rank(SZ,0(I−Pε)).

Theorem 6 (Signal Operator Contraction) Let

β ≡ ||AΘ||||ΞB−1X1(BZ1L

−1 +BZ0)||

If β < 1 andmZ −mV ≤ mε −mZ (23)

then B is a contraction on S with modulus β, and Signal Operator Iteration has aunique fixed point S∗ ∈ S satisfying S∗ = BS∗.

Proof. First I prove that B is an operator mapping S to S, then I prove the con-tractive property.

For B to be S → S, SZ + [AXn(Zn)−1SnPε]+ must be causal and map `2(Zmε)→`2(ZmZ ) when Sn ∈ S. Starting with the right-most term:

• Pε is block mε × mε and has non-zero values on the main diagonal only, soSnPε ∈ S

• The Wold representation Zn is causal and block mZ×mZ ; the inverse must existand also be causal, square-summable (Strohmer, 2002), and block mZ ×mZ , so(Zn)−1SnPε ∈ S

• By assumption the conditions for Theorem 1 are satisfied, so Xj is square-summable, causal, and block n×mε. A is square-summable and block mZ ×n,but may include square-summable non-causal terms. The annihilation operator[·]+ removes the non-causal terms, so [AXn(Zn)−1SnPε]+ ∈ S.

• SZ ∈ S by assumption, so SZ + [AXn(Zn)−1SnPε]+ ∈ S

Therefore B maps S to S.Next, consider any two operators S1 and S2 in S. The difference BS1 − BS2 is

given byBS1 − BS2 = [AX1(Z1)−1S1Pε]+ − [AX2(Z2)−1S2Pε]+

per Equations (14) and (20). (Zj) transforms the signal Sj into a white noise operatorW j, so the equation becomes

BS1 − BS2 = [AX1W 1Pε]+ − [AX2W 2Pε]+

= [A(X1W 1 −X2W 2)Pε]+

14

and the expression for the policy function from Theorem 1 gives

BS1 − BS2 = [AΘ([ΞZ1]+W1 − [ΞZ2]+W

2)Pε]+ (24)

Take norms of both sides of (24):

||BS1 − BS2|| = ||[AΘ([ΞZ1]+W1 − [ΞZ2]+W

2)Pε]+||The annihilation operator has operator norm 1, and the operator norm is sub-multiplicative,implying the inequality

||BS1 − BS2|| ≤ ||AΘ([ΞZ1]+W1 − [ΞZ2]+W

2)Pε||≤ ||AΘ||||[ΞZ1]+W

1 − [ΞZ2]+W2||

Pε disappears from this last step because it is a projection operator, so ||Pε|| = 1.W 1 and W 2 are white noise, and have the same main diagonal blocks as their

signal processes: Sj0 = W j0 . The rank condition (23) ensures that Sj0 has nullity

no larger than mε − mZ : the matrix Sj0 must have mV linearly independent rows.Therefore W 1 and W 2 satisfy the conditions to apply Lemma 5. This implies:

||BS1 − BS2|| ≤ ||AΘ||||ΞZ1W 1 − ΞZ2W 2|| (25)

Equation (8) defines Zj, which in terms of operators is

Zj = −B−1X1(BZ1[L−1Zj]+ +BZ0Z

j) (26)

Substituting (26) into the inequality in (25) gives

|BS1 − BS2|| ≤ ||AΘ||||ΞB−1X1(BZ1[L−1Z1]+ +BZ0Z

1)W 1 − ΞB−1X1(BZ1[L−1Z2]+ +BZ0Z

2)W 2||

Again, Lemma 5 implies

||BS1 − BS2|| ≤ ||AΘ||||ΞB−1X1(BZ1L

−1Z1 +BZ0Z1)W 1 − ΞB−1

X1(BZ1L−1Z2 +BZ0Z

2)W 2||

Next, the Wold representation Zj converts white noise back to signals by ZjW j =Sj:

||BS1 − BS2|| ≤ ||AΘ||||ΞB−1X1(BZ1L

−1S1 +BZ0S1)− ΞB−1

X1(BZ1L−1s2 +BZ0S

2)||and rearranging and applying sub-multiplicativity yields

||BS1 − BS2|| ≤ ||AΘ||||ΞB−1X1(BZ1L

−1 +BZ0)||||S1 − S2||Therefore, B is a contraction on S with modulus β. By the Banach fixed point

theorem, there exists a unique fixed point S∗ ∈ S such that

S∗ = BS∗

Theorem 6 proves that the Signal Operator Iteration algorithm is a contractionwhen β < 1. In addition to guaranteeing existence and uniqueness of equilibrium, itensures the solution algorithm converges to the equilibrium, when starting with anycausal signal process as an initial guess. The modulus β is the smallest Lipschitz con-stant that I have found, although it is possible that the algorithm is still a contractionif β ≥ 1. Appendix C describes how to compute β in practice.

15

4 A Local Information Model

The solution method is applied to a model of local information. The model’s struc-ture is similar to Lucas (1972) albeit with the additional challenges of an endogenouscapital stock and dynamic learning; or Lorenzoni (2010) with the addition of capitaland allowing firms to observe and learn from quantities and prices. The economyis made up of many “islands” which represent local markets. Firms and householdsobserve prices and quantities on each island, but not the aggregate state of the econ-omy. They get some information by observing local market conditions, but thereare more shocks than informative signals, so the aggregate state is not revealed. Theshocks that confound information map directly into observable quantities in the data:idiosyncratic productivity shocks, sectoral demand shocks, and monetary shocks. Inequilibrium, money is non-neutral because firms cannot perfectly distinguish betweenmonetary shocks and real aggregate shocks.

4.1 Households

There is a continuum of islands I indexed by i. On each island, there is a unitmeasure λ(i) = 1 identical and infinitely lived households.

The island i representative household’s preferences over current and future con-sumption are represented by the utility function

Ei,t

[∞∑s=0

βsC1−γi,t+s − 1

1− γ

](27)

where Ci,t is the household’s consumption in period t, β is their discount factor,and γ is their coefficient of relative risk aversion. The expectation operator Ei,t isconditional on the representative household i’s information set Ωi,t.

Households earn two types of income. They inelastically supply one unit of laboron their island, for which they are paid nominal wage Wi,t. They also own the capitalon their island, Ki,t, which they rent to firms at nominal rental rate RK,i,t.

Households spend their income on two types of goods. They purchase genericoutput goods from an economy-wide market at aggregate price level Pt. This genericgood can be used for either consumption Ci,t or investment Ii,t. Therefore their budgetconstraint is

Wi,t +RK,i,tKi,t = PtCi,t + PtIi,t (28)

Investment faces an adjustment cost ϕ(Ii,tKi,t

) which affects the productivity of

investment goods at producing new capital. A household owning Ki,t capital andinvesting Ii,t faces the law of motion:

Ki,t+1 = Ii,t(1− ϕ(Ii,tKi,t

)Ki,t

Ii,t) + (1− δ)Ki,t (29)

16

where the adjustment cost function satisfies ϕ(δ) = 0, ϕ′(δ) = 0 and ϕ′′(δ) > 0.The household’s problem is to choose sequences of Ci,t, Ii,t and Ki,t+1 to maximize

(27) subject to the budget constraint (28) and law of motion (29). The solution tothis problem is characterized by an Euler equation (30):

1

1− ϕ′( Ii,tKi,t

)=

Ei,t

[β

(Ci,tCi,t+1

)γ (RK,i,t+1

Pt+1

+1

1− ϕ′( Ii,t+1

Ki,t+1)

(ϕ′(

Ii,t+1

Ki,t+1

)Ii,t+1

Ki,t+1

− ϕ(Ii,t+1

Ki,t+1

) + 1− δ))]

(30)

where expectations Ei,t are conditional on representative household i’s informationset Wi,t. On the left-hand side of the Euler equation (30), 1

1−ϕ′(Ii,tKi,t

)is Tobin’s Q, the

marginal cost of an additional unit of capital for firms in market i at time t. On theright-hand side of (30), households discount the real return on their capital, plus themarginal units of capital they carry over.

4.2 Firms

There are two types of firms in the economy. There are intermediate goods firms thateach operate on an island indexed by i, and there are final goods firms that aggregatethe intermediate goods into a final output good in an economy-wide market.

Final goods firms aggregate specialized goods Yi of type i ∈ I with a CES pro-duction function,

Yt =

(∫i∈I

G1η

i,tYη−1η

i,t dλ(i)

) ηη−1

(31)

with η 6= 1. Gi,t is a good-specific stochastic shock and is i.i.d across types.The final goods sector is perfectly competitive, so the price of output Pt is given

by the CES price aggregator,

Pt =

(∫i∈I

Gi,tP1−ηi,t dλ(i)

) 11−η

(32)

and final goods firms’ demand for intermediates is given by the CES demand function

Pi,t = Pt(Gi,tYtYi,t

)1η (33)

Intermediate goods firms are perfectly competitive and have constant returns.The representative firm on island i in period t uses specialized capital Ki,t and laborLi,t with stochastic productivity Ai,t to produce output Yi,t by

Yi,t = Ai,tKαi,tL

1−αi,t (34)

17

Firms rent capital at nominal rental rate RK,i,t and hire labor at nominal wage Wi,t

from the households on island i in period t. They sell their output at price Pi,t. Therepresentative firm chooses inputs to maximize their profits, which implies that laborand capital demands for island i are given by

Pi,tαYi,tKi,t

= RK,i,t Pi,t(1− α)Yi,tLi,t

= Wi,t (35)

4.3 Money and Information

Money’s only role is to determine the price level. Specifically, money Mt is a stochasticprocess for nominal aggregate output:

Mt = PtYt (36)

Households observe the aggregate price level because they buy goods for consumptionand investment at price Pt, but they cannot observe Mt directly, so they cannotdirectly infer Yt. This is the information friction that allows monetary shocks toaffect the real economy.

The money supply is determined by a stochastic money supply rule, in whichmoney growth depends on aggregate output and a stochastic term µt.

lnMt = φY lnYt + µt (37)

The rule might be set by a monetary authority that observes the aggregate stateof the economy, and for which the parameter φY determines the dependence of themoney supply on real aggregate output. A value of φY = 1 implies that money supplywill move one-for-one with aggregate output, as if the monetary authority is targetinga particular price level (or a particular inflation rate if µt has a trend component.)The stochastic shock µt might represent error with which the monetary authoritymeasures Yt, or it could represent other stochastic factors it considers when settingmoney supply.

The price level is one of three endogenously-noisy signals that inform islands aboutthe state of the aggregate economy. The other signals are demand for specializedgoods and island-specific productivity.

Firms can see demand for their goods, which is a noisy signal of the aggregateoutput level Yt. They observe local prices and quantities, and the aggregate pricelevel Pt, but cannot directly observe whether changes in real demand for their spe-cialized goods Yi,t is driven by aggregate output Yt or by the sector-specific shock Gi,t.Through the sectoral demand equation (33), they observe the value of the demandsignal

Hi,t ≡ (Gi,tYt)1η (38)

but not the individual components.

18

In logs, productivity lnAi,t is the sum of an aggregate component lnAt and a

mean zero idiosyncratic component ln Ai,t satisfying

lnAi,t = lnAt + ln Ai,t (39)

Firms cannot observe aggregate productivity directly, but must make inference basedon their idiosyncratic productivity.

The firm’s information set Wi,t includes all of the local endogenous variables onisland i, plus the aggregate price level Pt, and demand signal Hi,t, and the informationset evolves by

Ωi,t = Ωi,t−1, Pi,t, Yi,t, Ai,t, Ki,t+1, Ii,t,Wi,t, RK,i,t, Li,t, Hi,t, Pt (40)

Many of these quantities are redundant in equilibrium. Together, agents observe threenoisy signals of the aggregate state: productivity Ai,t, the demand signal Hi,t, andthe price level Pt. Because each island faces four shocks, the aggregate state of theeconomy cannot be revealed if none of the shocks are perfectly collinear.

The novel characteristic of the information structure is that signals are endogenousand their noise is measurable in the data. Households see prices and quantities, whichinform their forecasts. This differs from papers such as Melosi (2016) and Woodford(2003), where agents observe exogenous noisy signals of aggregate shocks.

4.4 Equilibrium Definition

Given infinite sequences of exogenous variables Gi,t, Ai,t, At, µt for all i ∈ I, a com-petitive equilibrium in this economy consists of infinite sequences of prices, Pi,t, Pt,Wi,t, RK,i,tfor all i ∈ I; allocations Ci,t, Ii,t, Ki,t, Li,t, Yi,t, Yt for all i ∈ I; and information setsΩi,t for all i ∈ I such that:

1. Households maximize utility (27), subject to the constraints (28) and (29)

2. Intermediate firms choose allocations to maximize profits, satisfying the pro-duction function (34) and factor demands (35).

3. Final goods firms choose allocations to maximize profits, satisfying the produc-tion function (31) and input demands (33).

4. Money determines the aggregate price level by (36)

5. Firm productivity is given by (39)

6. Information sets evolve by (40)

7. The labor market clears: Li,t = 1 for all i ∈ I

19

4.5 Linearization

The model must be put in a linear form that can be solved by Signal OperatorIteration.

4.5.1 Linear Equilibrium Conditions

First, the equilibrium conditions must be linearized. Let lower case variables denotelog deviations from the deterministic steady state. By combining equations, house-hold i’s choice variables can be reduced to one control ii,t and one state ki,t. Thesequantities are determined by two linear equilibrium conditions expressed in terms ofthe log-linearized observable signals: productivity ai,t, demand hi,t, and inflation πt.

Log linearizing the household’s Euler equation (30) and substituting in with thecapital demand equation, sectoral demand equation, and budget constraint yields

ϕ(ii,t − ki,t) = Ei,t

[γ(YC

(η−1η

(ai,t + αki,t − (ai,t+1 + αki,t+1)) + hi,t − hi,t+1)− IC

(ii,t − ii,t+1))]

+Ei,t

[βα Y

K(hi,t+1 + η−1

ηai,t+1 + (αη−1

η− 1)ki,t+1) + β(1− δ)(1 + δ)ϕ(ii,t+1 − ki,t+1)

](41)

where β ≡ 11+ρ

, ϕ ≡ ϕ′′(δ)δ, and variables with overlines, e.g. Y , denote steady statelevels. This linearized Euler equation is derived explicitly in Appendix B.

The second equilibrium condition for island i is the linearized law of motion forcapital:

ki,t+1 ≈I

Kii,t − (1− δ)ki,t (42)

With these two linear equations, the model can be expressed in matrix form,corresponding to equation (1). The endogenous vector Xi,t and the exogenous vector(from the perspective of island i) Zi,t are given by

Xi,t =

(ii,tki,t

)Zi,t =

ai,thi,tπt

(43)

while the coefficient matrices encoding equations (41) and (42) are given by

BX0 =

(−ϕ− γ I

Cϕ+ γα Y

Cη−1η

IK

−(1− δ)

)(44)

BX1 =

(β(1− δ)(1 + δ)ϕ+ γ I

C−β(1− δ)(1 + δ)ϕ− γα Y

Cη−1η

+ βα YK

(αη−1η− 1)

0 −1

)(45)

20

BZ0 =

(γ YCη−1η

γ YC

0

0 0 0

)BZ1 =

(−γ Y

Cη−1η

βα YK− γ Y

C0

0 0 0

)(46)

Inflation πt does not show up in the linearized equilibrium conditions, which areexpressed in real terms. Inflation serves only as a noisy signal of the aggregate state.

4.5.2 Linear Signal Formation

Next, the three observable signals of the aggregate economy (ai,t, hi,t, πt) must belinearly expressed in terms of the exogenous variables and endogenous aggregates.

By assumption, log-linearized productivity is given by

ai,t = ai,t + at (47)

The linearized demand signal (38) is given by

hi,t ≡1

ηgi,t +

1

η(at + αkt) (48)

where log-linearized aggregate output is replaced by yt = at + αkt.Inflation is used instead of the price level, so that the money supply can have a

unit root while ensuring that the linearized system remains stationary. Accordingly,inflation is given by

πt = (φY − 1)(at − at−1 + α(kt − kt−1)) + µt (49)

where the exogenous term µt is given by µt ≡ µt − µt−1, the first difference of themoney supply shock in equation (37).

These three linear equations determining the signals are encoded into the lagpolynomials given by the following matrix equation:

Zi,t = SZ(L)εt + A(L)Xt (50)

where εi,t =(ai,t at µt gi,t

)′is the 4 × 1 vector of fundamental innovations

ot the exogenous variables and SZ(L) is a 3 × 4 lag polynomial encoding the time

series properties of the exogenous shocks. To produce Zi,t =(ai,t hi,t πt

)′, these

polynomials must satisfy Equations (47), (48), and (49):

SZ(L)εt =

ai,t + at1ηat + 1

ηgi,t

(φY − 1)(at − at−1) + µt

A(L)Xt =

01ηαkt

(φY − 1)α(kt − kt−1)

(51)

This implies that the lag polynomial A(L) =∑∞

j=0AjLj is given by

A0 =

0 00 1

ηα

0 (φY − 1)α

A1 =

0 00 00 −(φY − 1)α

(52)

21

with Aj = 0 otherwise.I model the exogenous variables as AR(1) processes:

ai,t = ρaai,t−1 + εa,i,t εa,i,t ∼ N(0, σa) (53)

at = ρaat−1 + εa,t εa,t ∼ N(0, σa) (54)

µt = ρmµt−1 + εm,t εm,t ∼ N(0, σm) (55)

gi,t = ρggi,t−1 + εg,i,t εg,i,t ∼ N(0, σg) (56)

For this autoregressive form, the lag polynomial for the exogenous informationcomponent SZ(L) has coefficients given by

SZ,j =

ρja ρja 0 00 1

ηρja 0 1

ηρjg

0 (φY − 1)(ρja − ρj+1a ) 0 ρjm

j ≥ 0 (57)

4.6 Calibration

The local information model is calibrated to resemble the US economy. Values forpreference and production parameters are chosen, while the time series processes forfundamental shocks are estimated from national accounting and industry-level data.

Most of the preference and production parameters are chosen in a standard way.The model period is one year, in order to match the frequency of the sectoral-leveldata. The discount factor is set to β = 0.95 while the coefficient of relative riskaversion is set to γ = 3. For levels of risk aversion above 4, the modulus rises above1 for the baseline calibration and uniqueness is no longer guaranteed. The annualdepreciation rate is set to δ = 0.1, a typical value used by King and Rebelo (1999),although they suggest that it should be lower. The adjustment cost parameter ϕ isusually estimated from aggregate dynamics, and depends on the underlying model.To be agnostic I choose ϕ = 1, which is higher than the values typically estimated bymacro models (e.g. Levin, Onatski, Williams, and Williams (2006)) but lower thatthe values estimated with micro data (Groth and Khan, 2010). The capital share isset to the usual α = .33.

To estimate the remaining parameters, I run several time series regressions. Theestimated parameters are reported in Table 1. In all cases, the series are logged anddetrended using the Christiano-Fitzgerald filter (Christiano and Fitzgerald, 2003),removing only the highest and lowest frequencies: below 2 years and above 50 years.The nominal shock process is estimated from the US National Accounts. Whenexpressed in terms of output, the linearized inflation equation (49) is given by

πt = (φy − 1)(yt − yt−1) + µt

I estimate this equation by OLS, with yt measured as log detrended real GDP percapita, and with πt measured as the inflation rate in the GDP deflator. I find φy =

22

1.17, so inflation is slightly pro-cyclical. Then I use the residuals µt to estimate theAR(1) process (55) to find ρm and σm.

The real shock processes are estimated from the TFP series in the US KLEMSindustry-level data (Jorgenson, Ho, and Samuels, 2012). I interpret an island i to bea KLEMS industry top-level industry, of which there are 29. I use the TFP seriesto estimate Equations (53) and (54). In the model, the idiosyncratic demand shockGi,t is identified by the CES demand function (33). When linearized, this demandfunction is

pi,t − pt =1

η(yt − yi,t) +

1

ηgi,t (58)

When mapping this equation to the data, yt − yi,t uses the log total output quantityrelative to the quantity index of industry i, and the log relative price level pi,t − ptuses the price index of industry i relative to the output price index. Estimating (58)implies an elasticity of substitution of η = 8.2 across industries.16 Then I use theresiduals gt to estimate equation (56).

The two parameters φy and η are particularly important for guaranteeing unique-ness of equilibrium. To understand how these parameters affect the algorithm, con-sider the matrices in the lag polynomial A(L), which controls how endogenous vari-ables feed back into the signal process (equation (52)). η could be set arbitrarilylarge, and φY could be set arbitrarily close to one, to ensure that the terms in thesematrices becomes arbitrarily close to zero. Accordingly, choosing parameter valuesof η and φY can insure that the operator norm ||A|| is arbitrarily close to zero, and||A|| is a coefficient in the definition of the Signal Operator Iteration modulus, soconvergence can be guaranteed.

Intuitively, the parameters φY and η control the information content in agents’endogenous signals of the aggregate state. For example, φY close to one makes theinflation rate an extremely noisy signal of aggregate output growth. The effect ofη is less straightforward, because larger η simultaneously increases ||Ξ||. The base-line estimate of φY = 1.17 implies that inflation is only modestly procyclical, but inSection 4.8 I consider alternative values to illustrate the impact of endogenous infor-mation content on the economy’s dynamics. The implied modulus of the algorithmfor the baseline calibration is 0.67, so convergence and uniqueness are guaranteed byTheorem 6.

Crucially, all four shock processes are estimated from industry-level or aggregatedata. This is unusual for macroeconomic models with incomplete information, whichusually feature exogenous noisy signals, whose noise does not map to any quantitymeasured in the data. In such cases, the time series process for the noise must bepinned down by the equilibrium aggregate dynamics, and it is difficult to tell whetherthe signal process is realistic or not. However with the endogenous information struc-

16This estimation should be viewed as motivating an illustrative calibration, and as measuringµt, rather than as a precise estimate of η. A direct regression of price on quantity is almost surelybiased.

23

Parameter Interpretation Value Justificationβ Discount factor 0.95 5% Annual real returnδ Depreciation factor 0.1 Standard valueα Capital share 0.33 Standard valueγ Risk aversion 3 Standard valueϕ Investment adjustment cost 1 Standard valueη Elasticity of substitution 8.2 US KLEMSφy Output coefficient in money rule 1.17 US National Accountsρa Persistence of idiosyncratic technology shock 0.80 US KLEMSρa Persistence of aggregate technology shock 0.78 US KLEMSρg Persistence of idiosyncratic demand shock 0.88 US KLEMSρm Persistence of nominal shock 0.60 US National Accountsσa Standard deviation of idiosyncratic technology shock 0.08 US KLEMSσa Standard deviation of aggregate technology shock 0.01 US KLEMSσg Standard deviation of idiosyncratic demand shock 0.04 US KLEMSσm Standard deviation of nominal shock 0.02 US National Accounts

Table 1: Illustrative calibration

ture in this model, all of the noisy processes are quantities that are measurable ex-post,so the time series properties of information are disciplined by the data.

4.7 Equilibrium Dynamics

I solve the model with Algorithm 2, using the computational methods detailed inAppendix C. For numerical approximation length N = 400, the model solves inunder 30 seconds on a standard desktop computer.

Agents observe three noisy signals: their productivity, real demand for their goods,and the inflation rate. But agents face four independent fundamental shocks, so thesesignals cannot be perfectly revealing. Figure 1 plots islands’ impulse responses tounit innovations in each signal. These graphs demonstrate the responses to agents’prediction errors of each signal, each of which is a linear combination of current andpast fundamental shocks.

First, a productivity innovation has classical effects (Figure 1, panel (a)); islandsinvest more, increasing their capital and output in future periods. But they areuncertain about whether their productivity shock is aggregate or idiosyncratic.

Second, a positive demand innovation can be driven either by an idiosyncraticincrease in demand for a firms’ goods, or a positive aggregate productivity shock thatraises aggregate output. Both possibilities encourage investment (Figure 1, panel(b)). Expecting higher future prices for their output, agents invest and accumulatecapital.

Third, an inflation innovation affects agents by informing their expectations; it hasno direct real effect on an agents’ equilibrium conditions. Inflation is a noisy signal ofaggregate output growth (equation (49)). When prices rises, agents on island i inferthat aggregate output is likely to have increased. Without a corresponding increase

24

in idiosyncratic productivity, goods on island i would be relatively scarce in thefuture after aggregate capital accumulates and aggregate output rises (equation (33)).Expecting higher prices for their output in the future, island i increases investmentimmediately (Figure 1, panel (c)). However, islands quickly learn if inflation is notassociated with increases in demand or productivity, and investment falls. In thebaseline calibration, φY is close to one, so inflation is a very uninformative signalabout the aggregate state, and the investment response to an inflation innovation issmall.

(a) Productivity Innovation (b) Demand Innovation (c) Inflation Innovation

Figure 1: Innovation Impulse Responses: Baseline

In the aggregate, idiosyncratic demand and productivity shocks sum to zero, soonly aggregate productivity and monetary shocks have aggregate effects. Figure 2plots the impulse responses of consumption and capital to aggregate productivityand monetary shocks. A productivity shock resembles the standard response in anRBC model, although an agent’s response is different than if they could observe theshock directly.

In contrast, the monetary shock is non-classical. It has a real effect because agentscannot perfectly distinguish its effect on inflation from a possible productivity shock.In response to a positive monetary shock, aggregate investment increases as all firmsincorrectly believe that aggregate output might have expanded. Then, firms rapidlylearn that a monetary shock was likely the cause when they don’t see increased returnsto their investment. Still, investment remain because capital increases the persistenceof the initial boom; the monetary shock increased aggregate output for many periods,which increases demand for each islands’ good for many periods. This explains whythe aggregate monetary shock (2, panel (b)) does not resemble the response to aninflation innovation: the monetary shock also produces positive demand innovations.In the baseline, these dynamics are small because inflation is a relatively noisy signalof aggregate real output, so firms place little weight on this confounding signal whenmaking forecasts.

However, when the inflationary signal contains more information, the nominalshock is more distortionary. I demonstrate this effect in the next section.

25

(a) Productivity Shock (b) Monetary Shock

Figure 2: Aggregate Impulse Responses: Baseline

4.8 Effects of Increasing Information in the Endogenous Sig-nal

In the baseline calibration, inflation is relatively uninformative about the aggregatestate of the economy. In this section, I increase the precision of this endogenous signalby varying the monetary policy parameter, and show that it increases the real effectof nominal shocks.

I modify the baseline calibration by setting φY = 2, so that the money supply ismore elastic with respect to output. This increases the weight placed on aggregateoutput growth in the determination of the inflation rate (equation (49)). The resultingimpulse responses to inflation innovations and aggregate monetary shocks are plottedagainst the baseline in Figure 3.

(a) Inflation Innovation (b) Monetary Shock

Figure 3: Impulse Responses: High Money Supply Elasticity (φY = 2) vs. Baseline

When agents see a positive inflation innovation with φY = 2, they can infer that apositive aggregate productivity shock was likely. They increase investment, becausethey expect demand for their island’s output to rise as the economy booms in response

26

to the aggregate productivity improvement. This increase is much larger than in thebaseline calibration of φy = 1.17, when inflation told them relatively little aboutaggregate output. Agents quickly increase their certainty about whether the inflationinnovation was driven by real or nominal shocks, and an inflation innovation withoutcorresponding demand innovations would result in a deep bust. But capital generatespersistence from the monetary shock; aggregate output is increased for an extendedperiod, raising demand for each islands’ goods. With φY = 2, the resulting boom ismore dramatic than in the baseline.

When the inflation signal contains more information about the aggregate stateof the economy, then nominal shocks have stronger real effects. Figure 4 (left axis)plots the cumulative effect of an aggregate monetary shock on investment for differentvalues of φY . When inflation is acyclical (φY = 1), inflation reveals no informationabout the level of real aggregate output, and nominal shocks have no real effects.In this case, islands have full information and the Pareto optimal equilibrium is re-covered. When φY deviates from 1 in either direction, the model exhibits monetarynon-neutrality. As the inflationary signal becomes more informative about the aggre-gate state of the economy, nominal shocks have larger real effects in absolute value.This result corresponds to the finding of Morris and Shin (2002), who show that in astatic model, increasing the precision of public information reduces social welfare.17

The convergence properties of the algorithm are also affected by the informationcontent of inflation. Figure 4 (right axis) also plots the modulus of the algorithm,which increasing in the distance between φY and 1, as is the absolute cumulativeeffect of a monetary shock. When inflation becomes more informative, the feedbackfrom the real economy to the endogenous signal increases. As a result, the speedof convergence of the Signal Operator Iteration slows. Convergence still occurs forall of these calibrations despite moduli above one, suggesting that there might beless restrictive sufficient criteria to insure uniqueness and convergence than those ofTheorem 6.

5 Conclusion

This paper proved a condition for existence and uniqueness of equilibrium in a gen-eral class of macroeconomic models with endogenous information. In macroeconomicsettings, demonstrating uniqueness has been a challenge, and even showing existencehas been difficult when endogenous state variables are included. The existence and

17The result differs from that of Angeletos and La’O (2019), who find that monetary policy shouldbe countercyclical in a model with dispersed information. The difference is due to source of monetarynon-neutrality. Angeletos and La’O study a model with exogenous information and sticky prices;in this case monetary policy must be countercyclical so that the most optimistic firms will chargelower prices and produce more after a positive shock. In contrast, I study a model with endogenousinformation and flexible prices; monetary policy needs to be acyclical so that agents ignore the pricelevel when forecasting output.

27

Figure 4: Effects of Different Money Supply Elasticities

uniqueness result is general enough to apply to models with endogenous state vari-ables, and I present an algorithm – Signal Operator Iteration – to solve such models.To demonstrate the method, I considered a local information model where monetaryshocks had real effects and agents learned from endogenous quantities and prices theyobserved. The model has a role for policy because the monetary authority can affectthe time series properties of information, and the optimal policy is to choose monetarypolicy such that inflation is acyclical.

Endogenous information may prove valuable for many applications. Macroeco-nomic models with information frictions that previously relied on exogenous noise, orthat made approximations to the information structure, can now be solved with fullyendogenous signals. Such models can be used to answer questions that were impossi-ble when information was exogenous. How can a policymaker influence expectationsby affecting endogenous variables? What is the optimal monetary policy in such anenvironment when additional frictions and complexities are introduced? What aboutfiscal stabilization or financial regulation? A wide range of policies that affects assetprices or other endogenous quantities from which agents might draw information cannow be addressed.

28

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32

A Proofs

This appendix contains the proofs of Theorem 1 and Lemma 5, as well as some otherlemmas that used as intermediate steps in the proof of Lemma 5.

A.1 Proof of Theorem 1

Proof. The equilibrium conditions (5) must hold for all realizations of the shocks,so it’s possible to collect terms, restricting the values of the matrices Xj∞j=0. Thisimplies a recursive equation for j ≥ 1:

0 = BX0Xj +BX1Xj+1 +BZ0Zj +BZ1Zj+1 (59)

Left multiply by BX1, substitute with Zj, and rearrange to get

Xj+1 = −B−1X1BX0Xj + Zj (60)

for j ≥ 0. Eigendecompose the matrix −B−1X1BX0, ordering the eigenvalues as in (6),

and left multiply by Q−1:

Q−1Xj+1 =

(ΛC 00 ΛS

)Q−1Xj +Q−1Zj

The recursive relationship can now be separated into a stable recursive equation andan unstable recursive equation. Let (Q−1X)C,j and (Q−1Z)C,j denote the first nCrows of Q−1Xj and Q−1Zj respectively. Then the unstable recursive equation is

(Q−1X)C,j+1 = ΛC(Q−1X)C,j + (Q−1Z)C,j (61)

And where (Q−1X)S,j and (Q−1Z)S,j denote the corresponding last nS rows, the stablerecursive equation is

(Q−1X)S,j+1 = ΛS(Q−1X)S,j + (Q−1Z)S,j (62)

Because ΛC is diagonal with all values outside the unit circle, the unstable recur-sive equation (61) allows (Q−1X)C,j to be expressed as the infinite sum

(Q−1X)C,j = −∞∑k=0

Λ−(k+1)C (Q−1Z)C,j+k ∀j ≥ 0 (63)

Similarly, the stable recursive equation (62) implies the sum

(Q−1X)S,j =

j∑k=1

Λk−1S (Q−1Z)S,j−k + Λj

S(Q−1X)S,0 ∀j > 0 (64)

33

Equation (63) determines (Q−1X)C,j for all j ≥ 0, but equation (63) only deter-mines (Q−1X)S,j for j > 0. Instead, (Q−1X)S,0 is determined by the restriction thatnS state variables are predetermined: XS,0 = 0. To calculate the initial matrix X0,relate it to the transformed Q−1X0 by(

(Q−1X)C,0(Q−1X)S,0

)=

(RCC RCS

RSC RSS

)(XC,0

XS,0

)The restriction XS,0 = 0 implies

(Q−1X)C,0 = RCCXC,0

Q−1 is full rank by assumption, so RCC must be full rank, and (Q−1X)S,0 can befound by

(Q−1X)S,0 = RSCR−1CC(Q−1X)C,0 (65)

Equations (63), (64), and (65) are enough to calculate X(L) in terms of Z(L) andXC,0, so it only remains to derive XC,0 in terms of Z(L). The recursive equation (61)implies

(Q−1X)C,j − ΛC(Q−1X)C,j−1 =

(Q−1Z)C,j−1 j > 0

(Q−1X)C,0 j = 0

Expressing (Q−1X)C(L) and (Q−1Z)C(L) as the corresponding lag operator polyno-mials, this relationship is written as

(Q−1X)C,0 = (I − ΛCL)(Q−1X)C(L)− L(Q−1Z)C(L) (66)

Stack equations (63) and (64) and redefine the indices to yield((Q−1X)C,j(Q−1X)S,j

)=

∞∑k=−∞

(−Λk−1

C 1k ≤ 0 00 Λk−1

S 1k > 0

)((Q−1Z)C,j−k(Q−1Z)S,j−k

)+

(0

ΛjS(Q−1X)S,0

)which holds for all j ≥ 0. Equation (65) allows the final term to be expanded as(

(Q−1X)C,j(Q−1X)S,j

)=

∞∑k=−∞

(−Λk−1

C 1k ≤ 0 00 Λk−1

S 1k > 0

)((Q−1Z)C,j−k(Q−1Z)S,j−k

)+

(0 0

ΛjSRSCR

−1CC 0

)((Q−1X)C,0

0

)(67)

At this point it is useful to switch to lag operator notation. This uses the prop-erty that the product of lag operators YA(L) = YB(L)YC(L) is equivalent to the

34

convolution of the matrix coefficients i.e. YA,j =∑∞

k=−∞ YB,kYC,j−k . It also usesthe property that for any matrix Y with eigenvalues inside the unit circle, the lagoperator

∑∞k=0 Y

kLk is equivalent to (I − Y L)−1. Accordingly, equation (67) can bewritten with lag polynomials as:

Q−1X(L) =

[(−Λ−1

C (I − Λ−1C L−1)−1 0

0 L(I − ΛSL)−1

)Q−1Z(L)

]+

+

(0 0

(I − ΛSL)−1RSCR−1CC 0

)((Q−1X)C,0

0

)The annihilator operator [·]+ appears because equation (67) only holds for j ≥ 0 sothat X(L) is causal. Substituting in for (Q−1X)C,0 with equation (66) yields

Q−1X(L) =

[(−Λ−1

C (I − Λ−1C L−1)−1 0

0 L(I − ΛSL)−1

)Q−1Z(L)

+

(0 0

(I − ΛSL)−1RSCR−1CC 0

)((I − ΛCL)(Q−1X)C(L)− L(Q−1Z)C(L)

0

)]+

which simplifies to

Q−1X(L) =

[(−Λ−1

C (I − Λ−1C L−1)−1 0

−L(I − ΛSL)−1RSCR−1CC L(I − ΛSL)−1

)Q−1Z(L)

]+

+

(0 0

(I − ΛSL)−1RSCR−1CC(I − ΛCL) 0

)Q−1X(L)

Collecting terms on Q−1X(L) gives(I 0

−(I − ΛSL)−1RSCR−1CC(I − ΛCL) I

)Q−1X(L) =[(

−Λ−1C (I − Λ−1

C L−1)−1 0−L(I − ΛSL)−1RSCR

−1CC L(I − ΛSL)−1

)Q−1Z(L)

]+

Inverting

(I 0

−(I − ΛSL)−1RSCR−1CC(I − ΛCL) I

)and left multiplying by Q yields

X(L) = Q

(I 0

(I − ΛSL)−1RSCR−1CC(I − ΛCL) I

)Q−1[

Q

(−Λ−1

C (I − Λ−1C L−1)−1 0

−L(I − ΛSL)−1RSCR−1CC L(I − ΛSL)−1

)Q−1Z(L)

]+

and substituting in Θ(L) and Ξ(L) completes the proof:

X(L) = Θ(L)[Ξ(L)Z(L)]+

35

A.2 Signal Properties and Intermediate Lemmas

Before proving the general result of Lemma 5, it is first necessary to prove the resultfor a specific subset of signal processes (Lemma 10), which satisfy an importantproperty. This property is whether the signal is “contemporaneous”, i.e. whether thesignal contains enough information about the contemporaneous shocks εi,t:

Definition 7 A signal process S is contemporaneous if S0 has full row rank.

The contemporaneous property is important because it ensures that two signalprocesses W 1 and W 2 mapping `2(Zmε)→ `2(ZmZ ) can be related by Lemma 8:

Lemma 8 If the operators W 1 and W 2 are causal, contemporaneous, invertible, andwhite noise on ε, then they satisfy

W 1 = Φ12W2 W 2 = Φ21W

1

where Φ12 and Φ21 are block diagonal operators, with full rank blocks on the maindiagonal.

Proof of Lemma 8. W 1 and W 2 are invertible, so the operators Φ12 and Φ21 arethe change of basis operators given by

Φ12 = W 1(W 2)−1 Φ21 = W 2(W 1)−1

Φ12 and Φ21 must be lower block triangular with full rank blocks on the main diag-onal (i.e. causal and contemporaneous) because W 1 and W 2 are both lower blocktriangular with full row rank blocks on the main diagonal. To complete the proof, itremains to be shown that operators Φ12 and Φ21 are block diagonal.

Write the variance ΣW 1 = W 1ε(W 1ε)∗ as

ΣW 1 = Φ12W2ε(Φ12W

2ε)∗

ΣW 1 = Φ12ΣW 2Φ∗12 (68)

The variance matrices ΣW 1 and ΣW 2 are both positive semi-definite and can beCholesky decomposed by CW 1 and CW 2 respectively. Equation (68) becomes

I = C−1

W 1Φ12CW 2(C−1

W 1Φ12CW 2)∗

Therefore (C−1

W 1Φ12CW 2) is unitary with inverse (C−1

W 1Φ12CW 2)∗. Φ12 is lower block

triangular and contemporaneous, while C−1

W 1 and CW 2 are block diagonal along the

main diagonal and contemporaneous, so C−1

W 1Φ12CW 2 is lower block triangular andcontemporaneous. However, the inverse of any contemporaneous lower block trian-gular operator must also be contemporaneous and lower block triangular. Therefore

36

(C−1

W 1Φ12CW 2)∗ is both upper and lower block triangular: it must be block diagonal,so Φ12 must be block diagonal.

Finally, W 1 and W 2 are contemporaneous, so the main diagonal blocks of Φ12 arefull rank. By the same logic, Φ21 is block diagonal with full rank blocks on the maindiagonal.

Lemma 8 states that any two causal, contemporaneous, and invertible white noiseprocesses of the same fundamental shocks can be written as a linear transformation ofone another. Invertibility ensures that W 1ε can be written in terms of W 2ε withouta residual, while the causal, white noise, and contemporaneous properties ensure thatΦ12 and Φ21 are block diagonal, so that any realization W 1εt can be written in termsof only the contemporaneous realization W 2εt, without needing any lagged terms. Ofcourse, the white noise processes in most interesting models are not invertible, butthey are a sub-block of an invertible process, which is enough for the proof of thenext result, Lemma 9.

Lemma 9 characterizes the norm of the sum of operators mapping white noiseprocesses to orthogonal processes. This result is related to the usual variance decom-position of orthogonal random variables: the variance of the sum of two orthogonalrandom variables is equal to the sum of their variances. Let [·]− denote the idempotentoperator that annihilates the causal terms of an operator, i.e.

[Y ]− ≡ Y − [Y ]+

For a Laurent operator Y , [Y ]+ is the lower block triangle, and [Y ]− is the upperblock triangle beginning on the block diagonal above the main block diagonal.

Lemma 9 If W mapping `2(Zmε)→ `2(ZmZ ) is causal and white noise on ε, then forany Laurent operator Y mapping `2(ZmZ )→ `2(ZmZ ) and any random shock ~ε ∈ E

||YW || ≥ ||[Y ]+W ||

Proof. Decompose Y to get

||YW~ε|| = ||[Y ]+W~ε+ [Y ]−W~ε||

Consider any random shock ~ε ∈ E ⊂ `2(Zmε), the set of bi-infinite column vectors inthe time series matrix ε. The Euclidean vector norm ||YW~ε||2 is given by definition

||YW~ε||2 =√

(YW~ε)∗YW~ε

=√

(W~ε)∗([Y ]∗+[Y ]+ + [Y ]∗−[Y ]− + [Y ]∗+[Y ]− + [Y ]∗−[Y ]+)W~ε (69)

W~ε is white noise, uncorrelated with lags of itself. Thus for any operator Ψ ≡∑∞j=−∞ΨjL

j,(W~ε)∗ΨW~ε = (W~ε)∗Ψ0W~ε

37

That is, all terms associated with non-zero powers of the lag operator are zero, i.e.(W~ε)∗ΨjL

jW~ε = 0 for j 6= 0.Crucially, the operator [Y ]∗+[Y ]− + [Y ]∗−[Y ]+ has only coefficients associated with

non-zero powers of L: [Y ]∗+[Y ]− has only terms above the main block diagonal, while[Y ]∗−[Y ]+ has only terms below the main block diagonal. Thus this operator dropsout of equation (69), which becomes

||YW~ε||2 =√

(W~ε)∗([Y ]∗+[Y ]+ + [Y ]∗−[Y ]−)W~ε

||YW~ε||22 = ([Y ]+W~ε)∗[Y ]+W~ε+ ([Y ]−W~ε)∗[Y ]−W~ε

||YW~ε||22 = ||[Y ]+W~ε||22 + ||[Y ]−W~ε||22||YW~ε||22 ≥ ||[Y ]+W~ε||22||YW~ε||2 ≥ ||[Y ]+W~ε||2 (70)

because all norms are positive.Lastly, this relationship between vector norms imply a relationship between oper-

ator norms. In this case the operator norm evaluated on the set E , so by definition,the operator norm ||YW || is the scalar satisfying

||YW || = infc : ||YW~ε||2 ≤ c||~ε||2 ∀~ε ∈ E

Inequality (70) holds for all ~ε ∈ E , so it follows that

||[Y ]+W~ε||2 ≤ ||YW ||||~ε||2

Therefore the smallest scalar infc : ||[Y ]+W~ε||2 ≤ c||~ε||2 ∀~ε ∈ E must be less thanor equal to ||YW ||. By definition of the operator norm:

||YW || ≥ ||[Y ]+W ||

Lemma 9 relates the norm of an operator on white noise to the norm when thenon-causal portion is annihilated. If the conditions for Lemma 8 are satisfied, thenthis will also be true of the difference between operators on two different white noiseprocesses, which allows for the proof of Lemma 10.

Lemma 10 If W 1 and W 2 mapping `2(Zmε)→ `2(ZmZ ) are both causal, white noiseon ε, and contemporaneous, then for any Laurent operators Y 1 and Y 2 mapping`2(ZmZ )→ `2(ZmZ )

||[Y 1]+W1 − [Y 2]+W

2|| ≤ ||Y 1W 1 − Y 2W 2||

38

Proof. Let Cε denote the Cholesky decomposition of the shock variance Σε, andlet CW j denote the Cholesky decomposition of the white noise variance ΣW j . Rightmultiply W j by Cε, left multiply by C−1

W j , and take the singular value decomposition:

C−1W jW

jCε = UW j

(I 0

)V ∗W j (71)

where UW j and VW j are unitary operators. W j is contemporaneous and causal, soits rows span the space `2(ZmZ ), i.e. W j is surjective. W j is also white noise soC−1W jW

jCε is a co-isometry, and thus all of its singular values are 1.The final mε −mZ rows in the blocks of V ∗W j form an orthonormal basis for the

kernel of W j. Let kj denote these rows, with V ∗W j chosen so that kj is causal andcontemporaneous. V ∗W j can be chosen in such a way because it is only determined upto a unitary transformation, so its rows can be arbitrarily shifted forwards and back-wards. kj is also white noise because V ∗W j is unitary so kj is orthonormal: kj(kj)∗ = I.

Construct the operator W j by

W j ≡(

W j

kjC−1ε

)In matrix form, this construction adds rows within each block, increasing the size ofeach block of rows from mZ to mε.

W j is invertible because

(C−1W j 00 I

)W jCε is unitary:

(C−1W j 00 I

)W jCε =

(UW j

(I 0

)V ∗W j

kj

)

=

(U j 00 I

)(V j)∗

with

(U j 00 I

)and V j both unitary operators. By construction, W j is also causal,

contemporaneous, and white noise on ε.Construct the block mz ×mε operator Y j by

Y j ≡(Y j 0

)where Y j is block mZ ×mZ and 0 is block mZ × (mε −mZ).

By construction Y j and Y j are related by the equations:

Y jW j = Y jW j (72)

[Y j]+Wj = [Y j]+W

j (73)

Let [·]− denote the idempotent operator that annihilates the causal terms of anoperator, i.e.

[Y j]− = Y j − [Y j]+

39

For a Laurent operator Y j, [Y j]+ is the lower block triangle, and [Y j]− is the upperblock triangle beginning on the block diagonal above the main block diagonal.

W jε is orthogonal to LkW jε for all k 6= 0, because W jε is white noise on ε.Therefore

[Y j]−Wjε ⊥ [Y j]+W

jε

Moreover, Lemma 8 implies W 1ε is orthogonal to LkW 2ε for all k 6= 0, giving

[Y i]−Wiε ⊥ [Y j]+W

jε i, j ∈ 1, 2

Next, consider the norm of the object in question, Y 1W 1−Y 2W 2. Apply equation(72):

||Y 1W 1 − Y 2W 2|| = ||Y 1W 1 − Y 2W 2|| (74)

Decompose Y 1 and Y 2 to write

||Y 1W 1 − Y 2W 2|| = ||[Y 1]+W1 − [Y 2]+W

2 + [Y 1]−W1 − [Y 2]−W

2|| (75)

Apply Lemma 8 to get

||Y 1W 1 − Y 2W 2|| = ||[Y 1]+W1 − [Y 2]+Φ21W

1 + [Y 1]−W1 − [Y 2]−Φ21W

1||

and because Φ21 is block diagonal it commutes through the annihilation operator:

= ||[Y 1 − Y 2Φ21]+W1 + [Y 1 − Y 2Φ21]−W

1||

Lemma 9 implies the inequality:

||Y 1W 1 − Y 2W 2|| ≥ ||[Y 1 − Y 2Φ21]+W1||

Again, Lemma 8 allows W 2 to return to the inequality:

||Y 1W 1 − Y 2W 2|| ≥ ||[Y 1]+W1 − [Y 2]+W

2||

and finally apply equation (73) to complete the proof:

||Y 1W 1 − Y 2W 2|| ≥ ||[Y 1]+W1 − [Y 2]+W

2||

Lemma 10 is straightforward to prove for signals that are contemporaneous, butnot all admissible signals satisfy this property. However, all causal white noise signalprocesses with sufficient rank of their diagonal blocks will be arbitrarily close to acontemporaneous white noise signal, so that a modification to the lemma can beapplied in order to prove the general result of Lemma 5. Intuitively, it is a knife-edgecase for a causal process to be non-contemporaneous, because that implies that someset of rows are not full rank. The strategy to prove Lemma 5 is to show that it ispossible to add an arbitrarily small operator to a causal white noise operator suchthat the sum remains causal and white noise, but is also contemporaneous so thatLemma 10 can be applied.

40

A.3 Proof of Lemma 5

This proof uses the definitions and lemmas from Section A.2.Proof. The proof consists of two steps. The first step creates a white noise operatorW j that is necessarily contemporaneous and arbitrarily close to W j. The second stepuses W j to prove the main result of the lemma.

Step 1 If W j is contemporaneous, then let W j ≡ W j.If W j is not contemporaneous, then at most mZ − rank(W j

0 ) rows of W j0 are

linearly dependent on the other rows. If mZ − rank(W j0 ) ≤ mε −mZ , an arbitrarily

small operator can be added to create a contemporaneous operator. Take the singularvalue decomposition of W j:

W j = UW j

(DW j 0

)V ∗W j (76)

where UW j and VW j are unitary operators,(DW j 0

)is an mZ×mε diagonal matrix

(i.e. a block diagonal operator), and DW j is the mZ ×mZ matrix of singular valuesof W j. The block of zeros 0 is mZ × (mε −mZ).

Divide the operator V ∗W j into blocks:

V ∗W j =

(V ∗W j ,1

V ∗W j ,2

)V ∗W j ,1 denotes the first mZ rows of each block of V ∗W j , while V ∗W j ,2 denotes the last

mε −mZ rows. The null space of W j is given by

null(W j) ⊇ span(V ∗W j ,2)

Let vj ∈ span(V ∗W j ,2) denote an arbitrary causal, white noise operator on ε in

the null space of W j, with vj0 full rank and linearly independent of W j0 . vj maps

`(Zmε)→ `(Zmε−mZ ).Order the rows in each block row of W j so that all mZ − rank(W j

0 ) linearlydependent rows appear last. Stack vj with 2mz −mε zero rows into the operator dj:

dj = α

(0vj

)dj is an operator mapping `(Zmε) → `(ZmZ ). α is an arbitrary scalar so that theoperator satisfies

||dj|| ≤ δ

Define the signal operatorW j ≡ W j + dj (77)

The operator W j is contemporaneous by construction if mZ − rank(W j0 ) ≤ mε−mZ ,

i.e. if there are enough rows in the null space to add a contemporaneous operator

41

to each of the public rows. Additionally, the operator W j is white noise on ε, i.e.W jΣε(W

j)∗ is block diagonal:

W jΣε(Wj)∗ = (W j + dj)Σε(W

j + dj)∗

dj is in the null space of W j, so djε(W jε)∗ = 0 implies

= W jΣε(Wj)∗ + djΣε(d

j)∗

and dj is also white noise on ε, so W jΣε(Wj)∗ and djε(djε)∗ = Σdj are both block

diagonal, thus W j is white noise on ε.Step 2 W 1 and W 2 are causal, white noise, and contemporaneous. Thus Lemma

10 implies||Y 1W 1 − Y 2W 2|| ≥ ||[Y 1]+W

1 − [Y 2]+W2|| (78)

If both signals W 1 and W 2 are contemporaneous, then W j = W j. If (78) holds withequality in this case, then the proof is complete. Otherwise, the inequality is strict,and substituting in Equation (77) to the left side implies

||Y 1W 1 − Y 2W 2 + Y 1d1 − Y 2d2|| > ||[Y 1]+W1 − [Y 2]+W

2||

and the triangle inequality gives

||Y 1W 1 − Y 2W 2|| + ||Y 1d1 − Y 2d2|| > ||[Y 1]+W1 − [Y 2]+W

2|| (79)

Similarly, the triangle inequality and Equation (77) imply that the right hand side of(79) satisfies

||[Y 1]+W1 − [Y 2]+W

2|| ≥ ||[Y 1]+W1 − [Y 2]+W

2|| − ||[Y 1]+d1 − [Y 2]+d

2||

substituting this into (79) and rearranging gives

||[Y 1]+W1 − [Y 2]+W

2|| < ||Y 1W 1 − Y 2W 2||+ ||Y 1d1 − Y 2d2|| + ||[Y 1]+d

1 − [Y 2]+d2||

By construction, the norm of the added terms satisfy ||dj|| ≤ δ. Again applying thetriangle inequality and substituting in this value yields:

||[Y 1]+W1− [Y 2]+W

2|| < ||Y 1W 1−Y 2W 2||+δ(||Y 1|| + ||Y 2|| + ||[Y 1]+|| + ||[Y 2]+||

)Therefore there exists some sufficiently small δ > 0 such that

||[Y 1]+W1 − [Y 2]+W

2|| ≤ ||Y 1W 1 − Y 2W 2||

42

B Linearization Details

The Euler equation is

1

1− ϕ′( Ii,tKi,t

)=

Ei,t

[β

(Ci,tCi,t+1

)γ (RK,i,t+1

Pt+1

+1

1− ϕ′( Ii,t+1

Ki,t+1)

(ϕ′(

Ii,t+1

Ki,t+1

)Ii,t+1

Ki,t+1

− ϕ(Ii,t+1

Ki,t+1

) + 1− δ))]

(80)

Substituting for the rental rate RK,i,t+1 with capital demand (35) gives

1

1− ϕ′( Ii,tKi,t

)=

Ei,t

[β

(Ci,tCi,t+1

)γ (Pi,t+1αYi,t+1

Pt+1Ki,t+1

+1

1− ϕ′( Ii,t+1

Ki,t+1)

(ϕ′(

Ii,t+1

Ki,t+1

)Ii,t+1

Ki,t+1

− ϕ(Ii,t+1

Ki,t+1

) + 1− δ))]

(81)

and substituting for the price Pi,t+1 with sectoral demand (33) gives

1

1− ϕ′( Ii,tKi,t

)=

Ei,t

[β

(Ci,tCi,t+1

)γ (Hi,t+1

αY1−1/ηi,t+1

Ki,t+1

+1

1− ϕ′( Ii,t+1

Ki,t+1)

(ϕ′(

Ii,t+1

Ki,t+1

)Ii,t+1

Ki,t+1

− ϕ(Ii,t+1

Ki,t+1

) + 1− δ))]

(82)

Combining labor market clearing with the production function gives Yi,t = Ai,tKαi,t,

which implies

1

1− ϕ′( Ii,tKi,t

)=

Ei,t

[β(

Ci,tCi,t+1

)γ (Hi,t+1αA

1−1/ηi,t+1 K

α(1−1/η)−1i,t+1 + 1

1−ϕ′(Ii,t+1Ki,t+1

)

(ϕ′(

Ii,t+1

Ki,t+1)Ii,t+1

Ki,t+1− ϕ(

Ii,t+1

Ki,t+1) + 1− δ

))](83)

43

Log-linearize this equation to get

1

(1− ϕ′( IK

))2ϕ′′(

I

K)I

K(ii,t − ki,t) =

Ei,t

[γ (ci,t − ci,t+1) + βα

Y

K(hi,t+1 +

η − 1

ηai,t+1 + (α

η − 1

η− 1)ki,t+1)

]+

Ei,t

[βϕ′( I

K) IK− ϕ( I

K) + 1− δ

1− ϕ′( IK

)

(ϕ′′( I

K)

1− ϕ′( IK

)

I

K+ ϕ′′(

I

K)(I

K)2

)(ii,t+1 − ki,t+1)

](84)

The law of motion for capital implies I/K = δ, and by assumption ϕ′(δ) = ϕ(δ) = 0,so the linearized Euler equation simplifies to

ϕ(ii,t − ki,t) =

Ei,t

[γ (ci,t − ci,t+1) + βα

Y

K(hi,t+1 +

η − 1

ηai,t+1 + (α

η − 1

η− 1)ki,t+1)

]+

Ei,t [β(1− δ)(1 + δ)ϕ(ii,t+1 − ki,t+1)] (85)

where I have defined the term ϕ ≡ ϕ′′(δ)δ.To get the Euler equation in terms of ii,t and ki,t, consumption must be substituted

out using the budget constraint (28), which expressed in terms of Hi,t becomes

Hi,tY1− 1

η

i,t = Ci,t + Ii,t

so the linearized budget constraint is

Y (hi,t +η − 1

ηyi,t) = Cci,t + Iii,t (86)

The demand shock is normalized so that G = 1, implying HY1− 1

η = Y per equation(38). The linearized production function is

AKα(ai,t + αki,t) = Y yi,t (87)

which implies that consumption is given by ci,t = YC

(η−1η

(ai,t + αki,t) + hi,t) − ICii,t.

This implies that the final linear Euler equation in terms of capital, investment, andsignals is:

ϕ(ii,t−ki,t) = Ei,t

[γ

(Y

C(η − 1

η(ai,t + αki,t − (ai,t+1 + αki,t+1)) + hi,t − hi,t+1)− I

C(ii,t − ii,t+1)

)]+Ei,t

[βα

Y

K(hi,t+1 +

η − 1

ηai,t+1 + (α

η − 1

η− 1)ki,t+1) + β(1− δ)(1 + δ)ϕ(ii,t+1 − ki,t+1)

](88)

44

To find YK

, consider the Euler equation in the deterministic steady state:

1

1− ϕ′( IK

)= β

(αY

K+

1

1− ϕ′( IK

)

(ϕ′(

I

K)I

K− ϕ(

I

K) + 1− δ

))

Then, substituting with the law of motion’s steady state I = δK and the assumptionϕ′(δ) = ϕ(δ) = 0 gives

1 = β

(αY

K+ 1− δ

)which implies a steady state output-to-capital ratio of

ρ+ δ

α=Y

K

where ρ is defined by β ≡ 1/(1 + ρ).

C Computation

This appendix describes a method for computing the Signal Operator Iteration algo-rithm.

The theoretical algorithm defined in Section 3.2 finds the equilibrium signal opera-tor S∗, which is a bi-infinite Toeplitz (i.e. Laurent) matrix. However, when computinga solution, some approximation to a finite object is required. My approach is to choosea truncation length N , and set all matrix coefficients Sj = 0 for j > N .

C.1 Properties of the Approximation

Approximating operators in this way allows for arbitrarily precise approximation ofthe solution S∗. The signal operators S map `2(Zmε) → `2(ZmZ ), implying thecorresponding lag operator polynomials have square summable coefficients, and the bi-infinite matrix features exponential decay off the main diagonal. When the operator Sis approximated by an operator SN which has truncation length N , Strohmer (2002)proves that the error to linear operations and inversion can be made arbitrarily smallby choosing a large enough value of N .

It is practical to select a large value of N , given that the solution algorithm isnot computationally intensive. Because the solution S∗ should be square summable,a strategy for checking whether N is large enough is to select a small bound b > 0below which terms are considered sufficiently close to zero, and then check that allterms s in the Nth matrix S∗N of the computed solution are within the bounds, sothat |s| < b. If not, increase the truncation length N .

This approximation method is well-suited for this problem specifically becausethe algorithm uses causal operators. Usually, approximating operators on bi-infinite

45

Toeplitz matrices also requires embedding into a circulant matrix, which introducesperturbation error in addition to the truncation error. This is because even thoughS is approximated by SN , SN is still a bi-infinite matrix. However, causal operatorshave lower block triangular Toeplitz matrices, so it’s possible to calculate the trun-cated product of two truncated Toeplitz matrices without any additional operators.Theorem 11 formalizes this property.

Suppose A,B,C are operators mapping `2(Zm)→ `2(Zn) for some m and n. LetTN(A), TN(B), TN(C) denote the Nn × Nm matrices with the same main diagonalblocks as the bi-infinite matrices.

Theorem 11 If operators A,B,C mapping `2(Zm)→ `2(Zn) are causal and satisfy

A = BC

Then the finite approximations TN(A), TN(B), TN(C) satisfy

TN(A) = TN(B)TN(C)

Proof. Partition the bi-infinite matrices A,B,C into blocks of arbitrary but equalsize. The equation A = BC becomes

. . ....

......

... . ..

... A0 0 0 0 ...

... A1 A0 0 0 ...

... A2 A1 A0 0 ...

... A3 A2 A1 A0 ...

. .. ...

......

.... . .

=

. . ....

......

... . ..

... B0 0 0 0 ...

... B1 B0 0 0 ...

... B2 B1 B0 0 ...

... B3 B2 B1 B0 ...

. .. ...

......

.... . .

. . ....

......

... . ..

... C0 0 0 0 ...

... C1 C0 0 0 ...

... C2 C1 C0 0 ...

... C3 C2 C1 C0 ...

. .. ...

......

.... . .

These matrices are block lower triangular, so the blocks A0, B0, C0 satisfy

A0 = B0C0

If the operatorsA,B,C are partitioned intoNn×Nm blocks, then TN(A), TN(B), TN(C)appear on the main block diagonals. Therefore, they must satisfy

TN(A) = TN(B)TN(C)

C.2 Computing the Algorithm

To compute the Signal Operator Iteration Algorithm with finite Toeplitz approxima-tions, I use the following steps. Begin then by conjecturing a causal square-summablesignal process S0 which is approximated by the finite block Toeplitz matrix TN(S0).Then proceed by:

46

1. Find the autocovariance’s finite block Toeplitz approximation implied by signalprocess Sn using using equation (16). For j ∈ [−N,N ], the blocks in the TN(Γn)block Toeplitz matrix are given by

Γj =N∑k=0

SnkΣεSnk+j

′

2. Use TN(Γn) to find the Wold representation TN(Zn) that solves the Yule-Walkerequations (17). This is done efficiently by calculating the Cholesky decompo-sition of TN(Γn), taking the final row as TN(ZnCε), and right-multiplying byC−1ε .

3. Given the Wold representation TN(Zn), generate the matrix TN(Zn) by equa-tion (8). If TN(L−1) is the finite approximation to the inverse lag operator (i.e.a block matrix with identity matrices along the first block above the main diag-onal and zeros elsewhere) and if T (B−1

X1) is the block matrix with B−1X1 along the

main diagonal (and similarly for matrices BZ1 and BZ0) then TN(Zn) is givenby

TN(Zn) =[−TN(B−1

X1)(TN(BZ1)TN(L−1) + TN(BZ0)

)TN(Zn)

]LT

where the operator [·]LT is the finite matrix equivalent of the annihilation op-erator [·]+ setting all blocks above the main diagonal to zero.

4. Calculate the block Toeplitz approximation of the policy function TN(Xn) byapplying Theorem 1:

TN(Xn) = TN(Θ)[TN(Ξ)TN(Zn)

]LT

5. Calculate the implied approximation of the signal TN(Sn+1) using equation(22):

TN(Sn+1) = TN(SZ) + [TN(A)TN(Xn)TN(Zn)−1(L)TN(Sn)T (Pε)]LT

6. If the Euclidean matrix norm of ||TN(Sn+1)− TN(Sn)||2 is sufficiently close tozero, conclude that the equilibrium signal process is S(L) = Sn+1(L). Otherwisereturn to Step 1 with guess Sn+1.

The finite matrix approximation introduces some error into this algorithm, al-though this error can be reduced by choosing an arbitrarily large approximation lengthN . This is practical even for large values of N because only the matrix inversion inStep (5.) is computationally intensive; the other steps are linear matrix operations.Concatenation error occurs in Steps (1.) and (2.), but Caines and Gerencsr (1991)

47

prove that TN(Z) calculated by Cholesky decomposition converges to the true Woldrepresentation as N becomes large. Theorem 11 ensures that the remaining stepsintroduce no additional error.

Lastly, the algorithm modulus β is computed by the approximation

β ≈ ||TN(A)TN(Θ)||2||TN(Ξ)TN(B−1X1)(TN(BZ1)TN(L−1) + TN(BZ0)

)||2

D Time Series in `2

This Appendix describes how to represent an identically distributed time series in theHilbert space `2. The primary reference for this section is Conway (2007).

D.1 Mapping L2 to `2

The usual space for random variables is the function space L2(Ω,F ,P), with samplespace Ω, σ-algebra F , and probability measure P . When Ω is countable, L2 is aseparable Hilbert space, and all separable Hilbert spaces are isomorphic to the Hilbertspace `2(Z), the space of square-summable bi-infinite dimensional sequences.18

To construct the discrete time series process, suppose Ω is countably infinite. L2

can be translated to `2 in the following way. L2 is a Hilbert space, so it has anorthonormal basis. This basis is a set of independent random variables with unitvariance; let time t represent the index of these random variables and denote therandom variables Wt∞t=−∞. Any random variable X in L2 can be written as a linearcombination of the basis:

X =∞∑

t=−∞

αXt Wt

with coefficients αXt ∞t=−∞.The textbook way to construct `2 from L2 is to map X ∈ L2 to x ∈ `2 by

x =∞∑

t=−∞

αXt et

where et∞t=−∞ are the standard basis vectors of `2, i.e. the vectors with a one in thetth position, and zeros elsewhere.

D.2 Time Series in `2

It is possible to construct a time series by stacking the vectors associated with randomvariables into matrices. For example, if x1 and x2 are random variables, then the time

18More generally, all Hilbert spaces of a particular cardinality are isomorphic to the vector space`2 of the same cardinality.

48

series of xtt=2t=1 is the matrix | |

x1 x2

| |

Similarly, we can construct infinite time series by creating infinite matrices. When

time is bi-infinite, the vector space is `2(Z), the space of bi-infinite square-summablesequences, indexed by t ∈ (−∞,∞). The bi-finite time series xtt=∞t=−∞ is the bi-infinite matrix

X ≡

| | |... x1 x0 x−1 ...| | |

(89)

An important example is the time series of i.i.d. unit variance random variables,which in `2(Z) is the standard basis et∞t=−∞. Ordered by t, this time series is thebi-infinite identity matrix I.

The vectors xtt=∞t=−∞ map to scalar random variables, but it is straightforwardto construct random vectors. For example, let yτ be the `2 representation of a k × 1random vector Yτ . Let each dimension of Yτ correspond to a random variable xt.Then yτ can be represented as the matrix | ... |

xτk+k ... xτk+1

| ... |

A new time series can be made with time index τ by

Y ≡

| | |... y1 y0 y−1 ...| | |

which is equivalent to the matrix X in (89).

D.3 Time Series as Laurent Matrices

When a time series is identically distributed over time, its `2 representation xtt=∞t=−∞is the same for every t, except shifted up or down by t places. This gives the matrixrepresentation X a Toeplitz structure: all of its diagonals are constant. When X isbi-infinite and Toeplitz, it is referred to as a Laurent matrix.

When the random variable is a k× 1 vector, its `2 representation yττ=∞τ=−∞ is the

same for every τ , but each time period corresponds to k realizations of the underlyingrandom variables. Accordingly, when Y is identically distributed over time, its `2

representation is the same for every τ , except shifted up or down by τk places. Thisgives the matrix representation Y a block Toeplitz structure, with k × k blocks. k issometimes referred to as the “multiplicity” of the `2 space.

49

D.4 The Autocovariance Function as a Laurent Matrix

The identically distributed time series xtt=∞t=−∞ has an autocovariance function γjwhere j indicates the order of the autocovariance (i.e. γ0 is the variance, γ1 is the firstautocovariance, and so forth.) In L2, the jth autocovariance is the inner product:

γj = E[XtXt+j] = 〈Xt, Xt+j〉

and the same is true in `2, where the inner product is also the dot product:

= 〈xt, xt+j〉 = x∗txt+j

Here, x∗t refers to the conjugate transpose, although in most cases the vector xt isreal.

Because of the Toeplitz structure in X, the tth row of X is x∗t but with entries inreverse order. Similarly, the tth column of X∗ is xt but with entries in reverse order.Thus, the autocovariance function is given by the diagonals of the Laurent matrix

Γ ≡ XX∗

because the terms on the jth diagonal are all x∗txt+j.This relationship holds as well when the random variable Y is vector valued.

For the block Laurent matrix Y, the autocovariance function is given by the blockdiagonals of YY∗.

50