Stability of Equilibrium Asset Pricing Models: A Necessary and Sufficient Condition
SStability of Equilibrium Asset Pricing Models:A Necessary and Sufficient Condition
Jaroslav Boroviˇcka a and John Stachurski b a New York University, Federal Reserve Bank of Minneapolis and NBER b Research School of Economics, Australian National University
October 3, 2019 A BSTRACT . We obtain an exact necessary and sufficient condition for the existenceand uniqueness of equilibrium asset prices in infinite horizon, discrete-time, arbi-trage free environments. Through several applications we show how the conditionsharpens and improves on previous results. We connect the condition, and hencethe problem of existence and uniqueness of asset prices, with the recent literatureon stochastic discount factor decompositions. Finally, we discuss computation ofthe test value associated with our condition, providing a Monte Carlo method thatis naturally parallelizable.
JEL Classifications:
D81, G11
Keywords:
Asset pricing, equilibrium prices, spectral methods
1. I
NTRODUCTION
One fundamental problem in economics is the pricing of an asset paying a stochas-tic cash flow with no natural termination point, such as a sequence of dividends.
We thank Ippei Fujiwara, Sean Meyn and Thomas J. Sargent for valuable comments and sugges-tions, as well as participants at the plenary lecture of the SNDE 2018 Annual symposium at KeioUniversity. In addition, special thanks are due to Mirosława Zima for detailed discussion of localspectral radius conditions. Financial support from ARC grant FT160100423 and Alfred P. SloanFoundation grant G-2016-7052 is also gratefully acknowledged. a r X i v : . [ q -f i n . GN ] O c t In discrete-time no-arbitrage environments, the equilibrium price process { P t } t (cid:62) associated with a dividend process { D t } t (cid:62) obeys P t = E t M t + ( P t + + D t + ) for all t (cid:62)
0, (1)where { M t } t (cid:62) is the pricing kernel or one period stochastic discount factor pro-cess of a representative investor. Two questions immediately arise in connectionwith these dynamics:1. Given { D t , M t } t (cid:62) , does there exist a unique equilibrium price process?2. How can we characterize and evaluate such prices whenever they exist?Although these questions have been the subject of intensive analysis in the past,the number of settings where we lack a clear picture is rising rather than falling.The main reason is that models of dividend processes and state price deflators havebecome more sophisticated in recent years, in an ongoing effort to better match fi-nancial data and resolve outstanding puzzles in the literature (see, e.g., Campbelland Cochrane (1999), Barro (2006), or Bansal and Yaron (2004) and subsequent it-erations of these models). This complexity makes questions 1–2 increasingly chal-lenging to address, especially in quantitative applications with discount rates thatare close to the growth rates of underlying cash flows. In general there have beenfew sufficient conditions proposed that (a) imply existence and uniqueness of equi-librium prices, (b) are weak enough to be useful in modern quantitative analysis,and (c) are practical to implement in applied settings.To address this absence, we introduce a condition for existence and uniquenessof equilibria that is both weak enough to hold in realistic applications—in factnecessary as well as sufficient, and hence as weak as possible—and practical in thesense that testing the condition focuses on a single value. The value in question isthe stability exponent L M : = lim n → ∞ ln ψ n n , (2) See, for example, Rubinstein (1976), Ross (1978), Kreps (1981), Hansen and Richard (1987) orDuffie (2001). Here and below, prices are on ex-dividend contracts. Cum-dividend contracts are asimple extension to what follows. Throughout this paper, we consider only fundamental solutions to the asset pricing problem(1), setting aside rational bubbles (see, e.g., Santos and Woodford (1997)). where ψ n : = E ∏ nt = M t is the price of a default-free zero-coupon bond (ZCB)with maturity t , and the expectation averages over possible draws of the time zerostate. The value L M corresponds to the asymptotic growth rate of the average priceprocess { ψ t } , since existence of the limit in (2) implies thatln ψ t + ψ t ≈ L M for large t . (3)In a standard setting with uncertainty driven by an exogenous and time homo-geneous first order Markov state process, we show that, for the case where divi-dends are stationary and have finite first moment, existence and uniqueness of anequilibrium price process { P t } satisfying (1) is exactly equivalent to the statement L M <
0. When dividends are nonstationary we replace L M with an analogousquantity for a growth adjusted SDF process and obtain a parallel existence anduniqueness result for the price-dividend ratio.In addition to these existence and uniqueness results, we also study a method forcomputing equilibrium prices (or price-dividend ratios) for the dividend process { D t } using successive approximations via an equilibrium price operator. We showthat this algorithm is globally convergent if and only if L M <
0. In other words, thenegative growth condition necessary and sufficient for existence and uniqueness isalso necessary and sufficient for global convergence of successive approximations.One interesting implication is that convergence of the algorithm itself implies ex-istence and uniqueness of equilibrium asset prices.Regarding the intuition behind our result, recall that L M is the asymptotic growthrate for the average bond price ψ t . If we consider t (cid:55)→ ψ t as the average default-free yield curve, then the condition L M < exactly characterizes the set of models with well defined equilibrium prices. This result rests on irreducibility of the underlying state pro-cess that supplies persistent stochastic components to dividends and the stochasticdiscount factor—a condition that holds in all applications we consider. The neces-sity argument is built on a “local spectral radius” result for positive operators inBanach lattices due to Zabreiko et al. (1967) and Forster and Nagy (1991). Usingthis result, we show that, for any positive cash flow with finite first moment, theasymptotic mean growth rate of its discounted payoff stream is equal to the prin-cipal eigenvalue of an associated valuation operator. When a regularity conditionon this operator is satisfied, this in turn is equal to the exponential of L M . If theprincipal eigenvalue equals or exceeds unity and the state process is irreducible,then the sum of expected discounted payoffs grows without bound.As stated above, in the environment we consider, the condition L M < M t + in (1) as a random “contraction factor” around which a contractionmapping argument can be built, looking forwards in time. The operator in this ar-gument has, as its fixed point, an equilibrium price function, which maps statesinto equilibrium prices. If there exists a constant θ with M t (cid:54) θ < θ , yieldingexistence of a unique equilibrium. However, in most applications, M t + > L M <
0, which requires instead that M t < L M in those cases where no analytical solution exists. First,we show that, when the state space is finite, L M can be calculated using a famil-iar method for computing spectral radii via numerical linear algebra. Second, wepropose a Monte Carlo method that involves simulating independent paths forthe discount factor process and averaging to produce the expectation in (1). Thismethod is inherently parallelizable and suited to settings where the state space islarge.As one illustration of the method, we consider a model of asset prices with Epstein–Zin recursive utility, multivariate cash flows and time varying volatility studied in Schorfheide et al. (2018), which in turn builds on the long run risk framework de-veloped by Bansal and Yaron (2004). Hitherto no results have been available onexistence and uniqueness of equilibria in the underlying theoretical model, partlybecause Schorfheide et al. (2018) and other related studies have focused their atten-tion on approximations generated using perturbation methods. While these stud-ies have been insightful, Pohl et al. (2018) recently demonstrated the importanceof nonlinearities embedded in the original model for determining asset prices. Wefocus on the original model and show that L M < L M , so this standardresult is a special case of our main theorem. Since our results are necessary aswell as sufficient under irreducibility of the state process, they further extend ourunderstanding of the finite state case.We also encompass and extend the existence and uniqueness of Lucas (1978), whostudied a model with infinite state space and SDF of the form M t + = β u (cid:48) ( C t + ) u (cid:48) ( C t ) . (4)Here { C t } is a consumption process, β is a state independent discount componentand u is a period utility function. Using a change of variable, Lucas (1978) obtainsa modified pricing operator with contraction modulus equal to β , and hence, byBanach’s contraction mapping principle, a unique equilibrium price process. Weshow below that his theorem is a special case of our main theorem result. While Lucas (1978) frames his contraction based results in a space of boundedfunctions, our analysis admits unbounded solutions. This is achieved by embed-ding the equilibrium problem in a space of candidate solutions with finite firstmoments. Such a setting is arguably more natural for the study of forward look-ing stochastic sequences, since the forward looking restriction is itself stated interms of expectations. Adopting this setting allows us to generalize the existenceand uniqueness results for equilibrium prices obtained in Calin et al. (2005) andBrogueira and Schütze (2017), which extend Lucas (1978) by allowing for habitformation and unbounded utility. Some of the preceding results are analytical, based on exact expressions for theexponent L M , while others, such as the treatment of the asset pricing problem inSchorfheide et al. (2018), rely on numerical evaluation of L M due to complexityof the dividend and SDF processes. Of course one might object to a numericaltest for existence and uniqueness of equilibria, since such tests introduce round-ing or discretization errors, which, in the worst case, can qualitatively affect re-sults. However, we find that some modern quantitative asset pricing studies aretoo complex—and too close to the boundary between stability and instability—toallow for successful use of analytically tractable sufficient conditions. We showthat such restrictions are typically violated in practice, even at parameterizationswhere the asset pricing models in question do have unique and well defined equi-libria.Our work is also connected to the literature on stochastic discount factor decom-positions found in Alvarez and Jermann (2005); Hansen and Scheinkman (2009);Hansen (2012); Boroviˇcka et al. (2016); Christensen (2017); Qin and Linetsky (2017)and other recent studies. These decompositions are used to extract a permanentgrowth component and a martingale component from the stochastic discount pro-cess, with the rate in the permanent growth component being driven by the princi-pal eigenvalue of the valuation operator associated with stochastic discount factor. Not surprisingly, our results also generalize the simple risk neutral case M t ≡ β , which is linearand hence easily treated by standard methods (see, e.g., Blanchard and Kahn (1980)). The existenceof a unique equilibrium when β ∈ (
0, 1 ) is a special case of our results because the n period stateprice deflator is just β n , so, by the definition of the exponent L M in (2), we have L M = ln β . Thecondition β ∈ (
0, 1 ) therefore implies L M < We show that the log of this principal eigenvalue is equal to L M in our setting,using the local spectral radius result discussed above. Unlike to the literature onstochastic discount factor decompositions, which uses the permanent growth com-ponent to shed light on the structure of valuation for payoffs at alternative hori-zons, our concern is with existence and uniqueness of equilibria over infinite hori-zons. Thus, we extend the reach of the existing theory by mapping the permanentgrowth component to an exact necessary and sufficient condition for these prop-erties. Through this process we also offer new ways to compute the permanentgrowth component via its connection to the exponent L M .On a technical level, there is some overlap between this paper and the analysis ofexistence of Epstein–Zin utilities contained in Boroviˇcka and Stachurski (2017). Weuse some common results local spectral radius methods and Monte Carlo methods.However, the topic is different and so are the essential functional equations.Regarding the mathematical literature, the asymptotic growth rate L M of the zerocoupon bond price with maturity t can alternatively understood as a “forward Lya-punov exponent,” due to its connection with the integrated Lyapunov exponent(see, e.g., Furstenberg and Kesten (1960) or Knill (1992)), which, for the pricingkernel process { M t } , takes the form I M : = lim n → ∞ n n ∑ i = E ln M t . (5)When { M t } is stationary, this reduces to E ln M t , which is considerably simplerthan the forward Lyapunov exponent L M . However, it is immediate from Jensen’sinequality that I M (cid:54) L M , and, as L M < I M < I M < M t . This is not enough be-cause stability requires that we rule out long epochs during which the SDF exceedsunity. In other words, we must control persistence in the SDF process, which re-quires restrictions on the full joint distribution. More generally, the reason that theintegrated Lyapunov exponent is not suitable for studying the asset pricing equa-tion (1) is that traditional Lyapunov theory was developed for backward lookingequations, whereas (1) is forward looking. The rest of the paper is structured as follows: Our main theoretical result is pre-sented in section 2. Section 3 treats applications and section 4 extends the the-ory. Section 5 concludes. A discussion of numerical methods for implementingour test can be found in appendix A. All proofs are deferred until appendix B.Computer code that replicates our numerical results and figures can be found at https://github.com/jstac/asset_pricing_code .2. A N
ECESSARY AND S UFFICIENT C ONDITION
In this section we set up our framework in detail and state our main results.2.1.
Environment.
We will work with the generic forward looking model Y t = E t [ Φ t + ( Y t + + G t + )] for all t (cid:62)
0. (6)Here { Φ t } and { G t } are given and { Y t } is endogenous. The equilibrium pricingequation (1) is obviously one special case of (6). While { Φ t } and { G t } will havedifferent interpretations in other applications, it is convenient to refer to { Φ t } asthe stochastic discount factor and { G t } as the cash flow . Example 2.1.
Aside from the pricing equation (1), one common version of (6) ariseswhen dividend growth is stationary, rather than dividends themselves. In this casewe divide (1) by D t , which yields Q t = E t (cid:20) M t + D t + D t ( Q t + + ) (cid:21) when Q t : = P t D t . (7)The price-dividend ratio Q t is the endogenous process to be obtained. This mapsto (6) when G t ≡ Φ t + = M t + D t + / D t .We say that a stochastic process { Y t } solves (6) if, with probability one, each Y t isfinite and (6) holds for all t (cid:62)
0. To obtain a solution we require some auxiliaryconditions on the state process, the cash flow and the stochastic discount process.The first of these is as follows:
Assumption 2.1. Φ t is a positive random variable and G t is nonnegative and non-trivial in the sense that G t > Neither of these assumptions cost any generality. Positivity of Φ t is equivalent toassuming no arbitrage in a complete market setting. The assumption that G t isnontrivial is also innocuous, since a trivial cash flow implies that Y t = t .To introduce the possibility of stationary Markov solutions, we assume that { Φ t } and { G t } admit the representations Φ t + = φ ( X t , X t + , η t + ) and G t + = g ( X t , X t + , η t + ) (8)where { X t } is an underlying X -valued state process, { η t } is a W -valued innovationsequence and φ and g are positive Borel measurable maps on X × X × W . The sets X and W may be finite, measurable subsets of R n , or infinite dimensional. Therepresentations in (8) replicate the general multiplicative functional specificationsconsidered in Hansen and Scheinkman (2009) and Hansen (2012) and are sufficientfor all problems we consider.The process { X t } is defined on some underlying probability space ( Ω , F , P ) , asis the innovation process { η t } . The innovation process is assumed to be IID andindependent of { X t } . Each η t has common distribution ν . The state process isassumed to be stationary and Markovian on X . The common marginal distributionof each X t is denoted by π . The conditional distribution of X t + given X t = x isdenoted by Π ( x , d y ) . Assumption 2.2.
The state process { X t } is irreducible.This means that, regardless of the initial condition, subsets of the state space withpositive probability under π are visited eventually with positive probability. Theassumption is satisfied in all applications we consider. Settings where assump-tion 2.2 fails can usually be rectified by appropriately minimal choice of the statespace. See, for example, Hansen and Richard (1987), lemma 2.3. We assume only that X and W are separable and completely metrizable topological spaces. Seesection B for details. More formally, the definition is that, for each Borel set B ⊂ X with π ( B ) > x ∈ X ,there exists an n ∈ N such that Π n ( x , B ) >
0. Here Π n represents n step transition probabilities andis defined recursively by Π = Π and Π n ( x , B ) = (cid:82) Π n − ( x , d z ) Π ( z , B ) . See, for example, Meynand Tweedie (2009), ch. 4. A measurable function h from X to R is called a Markov solution to the forwardlooking equation (6) if the process { Y t } defined by Y t = h ( X t ) for all t is a solutionto (6). Inserting Y i = h ( X i ) for all i into Y t = E t Φ t + ( Y t + + G t + ) and conditioningon X t = x , we see that h will be a Markov solution if h = Vh + ˆ g where ˆ g ( x ) : = (cid:90) (cid:90) φ ( x , y , η ) g ( x , y , η ) ν ( d η ) Π ( x , d y ) and V is the valuation operator defined by Vh ( x ) : = (cid:90) h ( y ) (cid:20) (cid:90) φ ( x , y , η ) ν ( d η ) (cid:21) Π ( x , d y ) . (9)The quantity Vh ( x ) is interpreted as the present discounted value of payoff h ( X t + ) conditional on X t = x . Letting T represent the equilibrium price operator defined atBorel measurable function h : X → R + by Th = Vh + ˆ g , (10)it is clear that h is a Markov solution if and only if h is a fixed point of T . We write T n for the n -th composition of T with itself.Before stating results, we need a candidate space for Markov solutions. To this end,for each p (cid:62)
0, we let L p ( X , R , π ) denote, as usual, the set of Borel measurable real-valued functions h defined on the state space X such that (cid:82) | h ( x ) | p π ( d x ) is finite.Let H p be all nonnegative functions in L p ( X , R , π ) . In other words, H p is the setof all nonnegative functions on X such that h ( X t ) has finite p -th moment. This willbe our candidate space, so if p =
2, say, then we seek solutions with finite secondmoment. Note that q (cid:54) p implies H p ⊂ H q , so existence of a solution in H p implies existence of a solution in H q whenever q (cid:54) p . Assumption 2.3.
There exists a p (cid:62) g ∈ H p and V is eventually com-pact as a linear operator from L p ( X , R , π ) to itself.The first part of assumption 2.3 is a finite moment restriction. The assumptionis weakest when p =
1, and in fact this minimal restriction cannot be omitted,since the forward looking restriction is stated in terms of expectations. Such arestriction is not well defined without finiteness of first moments. On the otherhand, we might wish to choose p to be larger when possible, in order to impose In what follows, all notions of convergence refer to standard norm convergence in L p . As usual,functions equal π -almost everywhere are identified. Appendix B gives more details. more structure on our solution (e.g., finiteness of second moments is necessary formany asymptotic results related to estimation).The “eventually compact” part of assumption 2.3 is a regularity condition, the de-tails of which are given in appendix B. Analogous conditions can be found in theliterature on eigenfunction decompositions of valuation operators (see, e.g., as-sumption 2.1 in Christensen (2017)). In section 4 we show that the sufficiency com-ponent of theorem 2.1 below continues to be valid when this regularity conditionis dropped.2.2. Existence and Uniqueness.
In addition, we introduce the p-th order stabilityexponent of the SDF process { Φ t } as L p Φ : = lim n → ∞ np ln E (cid:40) E x n ∏ t = Φ t (cid:41) p . (11)This is a generalization of the (first order) stability exponent L Φ introduced in (12).The simplest cases arise when p =
1, since by the Law of Iterated Expectations, wehave
E E x = E and hence L Φ = L Φ : = lim n → ∞ n ln (cid:40) E n ∏ t = Φ t (cid:41) . (12)As discussed in (2)–(3), when { Φ t } is the discount factor process, the exponent L Φ can be interpreted as the growth rate in t of the expected price of a zero-couponbond with maturity t . We show below that, in many important applications, wecan concentrate all our attention on L Φ , since L p Φ = L Φ for all p . Theorem 2.1.
If assumptions 2.1–2.3 hold, then the limit in (11) exists and all of thefollowing statements are equivalent:(a) L p Φ < .(b) A Markov solution h ∗ exists in H p .(c) A unique Markov solution h ∗ exists in H p .(d) There exists an h in H p such that T n h converges to some limit in H p as n → ∞ .(e) A unique Markov solution h ∗ exists in H p and T n h converges to h ∗ for everyh ∈ H p . Moreover, if one and hence all of (a)–(e) are true, then the unique Markov solution h ∗ satisfies h ∗ ( x ) = ∞ ∑ n = E x n ∏ i = Φ i G n . for all x in X . (13)Parts (a)–(c) of theorem 2.1 tell us that L p Φ < p -th moment to exist, and, moreover,that existence itself fails (rather than uniqueness) when L p Φ (cid:62)
0. Condition (d) isvaluable from an applied perspective because it shows that if iteration with T doesconverge from some starting point, then the limit is necessarily a Markov solution,and also the only Markov solution in H p . Part (e) provides a globally convergentalgorithm for computing the unique equilibrium whenever L p Φ < Connection to Spectral Radius Arguments.
In appendix B we show that,when assumptions 2.1–2.3 hold, the value L p Φ is always finite and L p Φ = ln r ( V ) (14)where r ( V ) is the spectral radius of the valuation operator when regarded as alinear self-map on L p ( X , R , π ) . This result is central to our necessity result in par-ticular and useful for computation. The spectral radius r ( V ) also appears in the literature on stochastic discount factor decompositions discussed in the introduc-tion, since it equals the log of the principal eigenvalue of the valuation operator V ,which in turn the determines the permanent growth component of the stochasticdiscount factor. Theorem 2.1 and (14) show that this permanent growth componentexactly determines the boundary between existence and nonexistence of equilibria.The ideas behind (14) can be understood by considering the following line of ar-gument. Suppose, for simplicity, that the state space X is finite. Let Π ( x , y ) repre-senting the probability of transitioning from x to y in one step. Note that, for all x in X and all n in N , we have V n ( x ) = E x n ∏ t = Φ t , (15)where V is the valuation matrix V ( x , y ) = (cid:20) (cid:90) φ ( x , y , η ) ν ( d η ) (cid:21) Π ( x , y ) (16)corresponding to the valuation operator defined in (9). In particular, V n ( x ) iselement x of the column vector V n , where V n is the n -th power of V . The expec-tation E x conditions on X = x . The identity in (15) can be confirmed by induction(consider, for example, the case n =
1) and the intuition is straightforward: apply-ing V to a payoff vector yields a present discounted value per unit of dividend.Thus, V n ( x ) is the present discounted value of a zero coupon default-free bondthat matures in n periods, contingent on current state n . Since Φ t is the (growthadjusted) stochastic discount factor, E x ∏ nt = Φ t gives the same value. By (15) and the law of iterated expectations, we have (cid:107) V n (cid:107) : = ∑ x ∈ X V n ( x ) π ( x ) = E n ∏ t = Φ t , (17)where (cid:107) · (cid:107) is the L vector norm defined by (cid:107) h (cid:107) = ∑ x ∈ X | h ( x ) | π ( x ) . Gelfand’s for-mula tells us that (cid:107) V n (cid:107) n → r ( V ) as n → ∞ whenever (cid:107) · (cid:107) is a matrix norm, andthis result can be modified to show that (cid:107) V n (cid:107) n → r ( V ) also holds. Connecting Lemma B.2 in the appendix provides a proof of (15) in a general (i.e., finite or infinite) statesetting. This last convergence claim is a so-called “local spectral radius” result. The local spectral radiusargument yielding (cid:107) V n (cid:107) n → r ( V ) in the general case uses a theorem for positive operators inBanach lattices due to Zabreiko et al. (1967) and Forster and Nagy (1991), which is notable for the the last result with (17) gives L Φ = lim n n ln (cid:40) E n ∏ t = Φ t (cid:41) = lim n ln (cid:110) (cid:107) V n (cid:107) n (cid:111) = ln r ( V ) ,which confirms the claim in (14).2.4. The Finite State Case.
The problem treated in this paper simplifies when thestate space is finite, which is important partly because some theoretical specifica-tions are finite and partly because numerical implementations inevitably reducecomputations onto a finite set of floating point numbers. The following result iskey:
Proposition 2.2.
If assumptions 2.1–2.2 hold and, in addition, the state space X is finite,then assumption 2.3 also holds and L p Φ = L Φ for all p (cid:62) . In particular (b)–(e) oftheorem 2.1 all hold at every p (cid:62) if and only if L Φ < . One implication is that, in the finite state case, one can always work with the sim-pler exponent L Φ = L Φ , which is easier to calculate than L p Φ at p >
1. The proof ofproposition 2.2 rests on the fact that all norms are equivalent in finite dimensionalnormed linear space. 3. A
PPLICATIONS
We now turn to applications of theorem 2.1, showing how its results can be ap-plied by testing the condition L p Φ < fact that it can handle Banach lattices where the positive cone has empty interior. The fact thatlim can be used instead of lim sup in the definition of L p Φ is based on a theorem of Daneš (1987).Appendix B provides a detailed treatment. The technical results listed in this footnote are essentialto our proofs. Constant Volatility and Relative Risk Aversion.
We first treat a simple set-ting where the stability exponent can be computed analytically. This provides in-tuition and a benchmark for testing numerical calculations (see appendix A for thelatter). As in Lucas (1978), we set the SDF to the standard time separable form M t + = β u (cid:48) ( C t + ) u (cid:48) ( C t ) , (18)where β ∈ (
0, 1 ) is a state independent discount factor. Agents have CRRA utility u ( c ) = c − γ − γ where γ (cid:62) γ (cid:54) =
1. (19)Dividends and consumption growth obey the constant volatility specification fromsection I.A of Bansal and Yaron (2004), which isln ( D t + / D t ) = µ d + ϕ X t + σ d ξ t + (20a)ln ( C t + / C t ) = µ c + X t + σ c (cid:101) t + (20b) X t + = ρ X t + σ η t + (20c)Here − < ρ < { ( ξ t , (cid:101) t , η t ) } is IID and standard normal in R . We solve forthe price dividend ratio using (7), which means, by the discussion following thatequation, that G t = Φ t + = M t + D t + D t = β exp { ( µ d + ϕ X t + σ d ξ t + ) − γ ( µ c + X t + σ c (cid:101) t + ) } . (21)It follows that n ∏ i = Φ i = β n exp (cid:40) n ( µ d − γµ c ) + ( ϕ − γ ) n ∑ i = X i + σ d n ∑ i = ξ i − γσ c n ∑ i = (cid:101) i (cid:41) .Using (20c), we then have (cid:32) E x n ∏ i = Φ i (cid:33) p = β np exp ( pa n x + pb n ) , (22)where a n : = ( ϕ − γ ) ρ ( − ρ n ) / ( − ρ ) and b n : = n ( µ d − γµ c ) + ( ϕ − γ ) s n + n σ d + n ( γσ c ) s n is the variance of ∑ ni = X i . The next step in calculating L p Φ is to take theunconditional expectation of (22), which amounts to integrating with respect to the stationary distribution π = N ( σ / ( − ρ )) . This yields E (cid:32) E x n ∏ i = Φ i (cid:33) p = β np exp (cid:18) ( pa n σ ) ( − ρ ) + pb n (cid:19) ,and hence L p Φ = lim n → ∞ (cid:26) ln β + pn ( a n σ ) ( − ρ ) + b n n (cid:27) = ln β + lim n → ∞ b n n , (23)where the second equality uses the fact that a n converges to a finite constant. Somealgebra yields s n n = σ − ρ (cid:26) + ( n − ) n ρ − ρ − ρ n · − ρ n − ( − ρ ) (cid:27) . (24)Combining this with (23), we find that L p Φ = ln β + µ d − γµ c + σ ( ϕ − γ ) ( − ρ ) + σ d + ( γσ c ) L p Φ represents the long-run growth rate of the discounted dividend Φ t . In expression (25), the term µ d + σ d /2 + [ ϕσ / ( − ρ )] /2 corresponds to thelong-run dividend growth rate, ln β − γµ c + ( γσ c ) /2 + [ γσ / ( − ρ )] /2 to the(negative of) the long-run discount rate, and ϕγσ / ( − ρ ) is the long-run covari-ance between the two. When do the conditions of theorem 2.1 hold? Since G t = Φ t is given by(21), assumption 2.1 is clearly valid. The state process (20c) is certainly irreducible,so assumption 2.2 holds. Moreover, in view of (27), the valuation operator V hasthe form Vh ( x ) = β exp { ax + b } (cid:82) h ( y ) q ( x , y ) d y for suitably chosen constants a and b , where q is the Gaussian transition density associated with (20c). From thisexpression it can be verified that assumption 2.3 holds at p = Notice that L p Φ in (25) does not depend on p . In particular, we have L p Φ = L Φ : = L Φ for all p . This matches the finding that L p Φ = L Φ for all p in the finite dimensional case, as shown inproposition 2.2. In other words, this Gaussian constant volatility model is simple enough to retainkey features of the finite dimensional setting. A Finite State Application.
In many asset pricing problems, the exogenousstate evolves as a finite Markov chain. For concreteness, we maintain the settingof section 3.1, apart from switching the state process { X t } from the AR(1) dynamicsin (20c) to a Markov chain taking values in finite set X and obeying stochastictransition matrix Π . The pricing problem in (7) then reduces to solving for a q of the form Q t = q ( X t ) where q satisfies q ( x ) = E [ Φ t + ( Q t + + ) | X t = x ] for each x ∈ X . Using (20a)–(20b) and the formula for the lognormal expectation,this can be written as q ( x ) = β exp (cid:34) µ d − γµ c + ( − γ ) x + σ d + ( γσ c ) (cid:35) ∑ y ∈ X ( q ( y ) + ) Π ( x , y ) ,or, stated as a vector equation, q = V ( q + ) . Here is a column vector of ones ofsize | X | and V is the valuation matrix, with ( x , y ) -th element V ( x , y ) : = β exp (cid:34) µ d − γµ c + ( − γ ) x + σ d + ( γσ c ) (cid:35) Π ( x , y ) . (26)As is well understood, the Neumann Series Theorem implies that a solution to q = V ( q + ) exists whenever r ( V ) <
1, where r ( V ) is the spectral radius of V .In view of (14) and proposition 2.2, r ( V ) < L Φ < X is finite. Thus, in the finite state setting, the condition L Φ < r ( V ) <
1. This means that thestandard condition r ( V ) < { X t } is irreducible. Habit Persistence.
There is a large literature on asset prices in the presence ofconsumption externalities and habit formation (see, e.g., Abel (1990) and Campbell See, for example, Mehra and Prescott (1985), Rietz (1988), Weil (1989), Kocherlakota (1990),Alvarez and Jermann (2001), Cogley and Sargent (2008), or Collin-Dufresne et al. (2016). The logic is as follows: theorem 2.1 states that L Φ < X is finite, assumption 2.3 holds automatically, by proposition 2.2. Assump-tion 2.1 is also true in the current setting. Hence necessity requires only assumption 2.1, whichis irreducibility. and Cochrane (1999)). In the “external” habit formation setting of Abel (1990) andCalin et al. (2005), the growth adjusted SDF takes the form M t + D t + D t = k exp (( − γ )( ρ − α ) X t ) (27)where k : = β exp ( b ( − γ ) + σ ( γ − ) /2 ) and α is a preference parameter. Thestate sequence { X t } obeys X t + = ρ X t + b + σ η t + with − < ρ < { η t } IID ∼ N (
0, 1 ) . (28)The parameter b is equal to x + σ ( − γ ) where x represents mean constantgrowth rate of the dividend of the asset.The price-dividend ratio associated with this stochastic discount factor satisfies theforward recursion (7) from example 2.1 and, by theorem 2.1, there exists a uniqueprice-dividend with finite second moment (we set p = L Φ < p = L Φ < L Φ can be obtained using similar techniques to thoseemployed in section 3.1. Stepping through the algebra shows that L Φ = ln k + ( − γ )( ρ − α ) b − ρ + ( − γ ) ( ρ − α ) σ ( − ρ ) . (29)A unique equilibrium price-dividend ratio exists in H if and only this term is neg-ative. The intuition behind the expression (29) is analogous to (25) in section 3.1.To give some basis for comparison, let us contrast the condition L Φ < τ <
1, where τ depends on the parameters ofthe model (see equation (7) of Calin et al. (2005) for details). Since the condition L Φ < τ at a range of pa-rameterizations. The right sub-figure shows L Φ at the same parameters, evaluatedusing (29). The horizontal and vertical axes show grid points for the parameters β . ln . F IGURE
1. Alternative tests of stability for the habit formation modeland σ respectively. For both sub-figures, ( β , σ ) pairs with test values strictly lessthan zero (points to the south west of the 0.0 contour line) are where the respectivecondition holds. Points to the north west of this contour line are where it fails. Inspection of the figure shows that the sufficient condition in Calin et al. (2005)fails for many parameterizations that do in fact have unique stationary Markovequilibria. That is, there are many empirically relevant ( β , σ ) pairs such that L Φ <
0, indicating existence and uniqueness of a solution with finite second moment,and yet ln τ >
0. Note also that, because L Φ < Long-Run Risk With Stochastic Volatility.
Next we turn to an asset pricingmodel with Epstein–Zin utility and stochastic volatility in cash flow and consump-tion estimated by Bansal and Yaron (2004). Preferences are represented by the con-tinuation value recursion V t = (cid:104) ( − β ) C − ψ t + β {R t ( V t + ) } − ψ (cid:105) ( − ψ ) , (30)where { C t } is the consumption path and R t is the certainty equivalent operator R t ( Y ) : = ( E t Y − γ ) ( − γ ) . (31) The parameters held fixed in figure 1 are ρ = − γ = x = α = The parameter β ∈ (
0, 1 ) is a time discount factor, γ governs risk aversion and ψ is the elasticity of intertemporal substitution. Dividends and consumption growaccording to ln ( C t + / C t ) = µ c + z t + σ t η c , t + , (32a)ln ( D t + / D t ) = µ d + α z t + ϕ d σ t η d , t + , (32b) z t + = ρ z t + ϕ z σ t η z , t + , (32c) σ t + = max (cid:110) v σ t + d + ϕ σ η σ , t + , 0 (cid:111) . (32d)Here { η i , t } are IID and standard normal for i ∈ { d , c , z , σ } . The state X t can berepresented as X t = ( z t , σ t ) . The (growth adjusted) SDF process associated withthis model is Φ t + : = M t + D t + D t = β θ D t + D t (cid:18) C t + C t (cid:19) − γ (cid:18) W t + W t − (cid:19) θ − , (33)where W t is the aggregate wealth-consumption ratio and θ : = ( − γ ) / ( − ψ ) . To obtain the aggregate wealth-consumption ratio { W t } we exploit the fact that W t = w ( X t ) where w solves the Euler equation β θ E t (cid:34)(cid:18) C t + C t (cid:19) − γ (cid:18) w ( X t + ) w ( X t ) − (cid:19) θ (cid:35) = w ( z , σ ) = + [ Kw θ ( z , σ )] θ ,where K is the operator Kg ( z , σ ) = β θ exp (cid:26) ( − γ )( µ c + z ) + ( − γ ) σ (cid:27) Π g ( z , σ ) (34)In this expression, Π g ( z , σ ) is the expectation of g ( z t + , σ t + ) given the state’s lawof motion, conditional on ( z t , σ t ) = ( z , σ ) .The existence of a unique solution w = w ∗ to (3.4) in H under the parameteri-zation used in Bansal and Yaron (2004) is established in Boroviˇcka and Stachurski(2017) when the innovation terms { η i , t } are truncated, so that the state space is a For a derivation see, for example, Bansal and Yaron (2004), p. 1503. compact subset of R . In what follows, we compute w ∗ using the iterative methoddescribed in Boroviˇcka and Stachurski (2017) and recover W t as w ∗ ( X t ) for each t .As discussed in detail in appendix A, to approximate the stability exponent L Φ ,we can use Monte Carlo, generating independent paths for the SDF process { Φ t } and averaging over them to estimate the expectation on the right hand side of (12).In computing the product ∏ nt = Φ t we used (32) and (33) to express it as n ∏ t = Φ t = ( β θ exp ( µ d − γµ c )) n × exp (cid:32) ( α − γ ) n ∑ t = z t − γ n ∑ t = σ t η c , t + + ϕ d n ∑ t = σ t η d , t + + ( θ − ) n ∑ t = ˆ w t (cid:33) , (35)where ˆ w t + = ln [ W t + / ( W t − )] .At the parameter values using in Bansal and Yaron (2004) and based on the MonteCarlo method discussed above, we estimate that L Φ = − H . While thisvalue is close to zero, we find that significant shifts in parameters are required tocross the contour L Φ = L Φ calculated at a range of parameter values in theneighborhood of the Bansal and Yaron (2004) specification via a contour map. Theparameter α is varied on the horizontal axis, while µ d is on the vertical axis. Otherparameters are held fixed at the Bansal and Yaron (2004) values. The black contourline shows the boundary between stability and instability. Not surprisingly, the testvalue increases with the cash flow growth rate µ d . In this region of the parameterspace, it also declines with α , because an increase in α with γ > α reduces thecovariance between cash flow growth and discounting captured by the term ( α − γ ) ∑ nt = z t in (35). However, we can see that L Φ < α and µ d from their estimated values. The value shown is the mean of 1,000 Monte Carlo draws L Φ ( n , m ) , where the latter is definedin (39) of appendix A. For each draw, n and m in in this calculation were set to 1,000 and 10,000respectively. The standard deviation was less than 0.001. Following Bansal and Yaron (2004), theparameters used were γ = β = ψ = µ c = ρ = ϕ z = v = d = ϕ σ = µ d = α = ϕ d = d Bansal-Yaron . -0.010000-0.008000-0.006000-0.004000-0.0020000.0000000.0020000.004000 F IGURE
2. The exponent L Φ for the Bansal–Yaron model3.5. Long-Run Risk Part II.
Now we repeat the analysis in section 3.4 but usinginstead the dynamics for consumption and dividends in Schorfheide et al. (2018),which are given byln ( C t + / C t ) = µ c + z t + σ c , t η c , t + ,ln ( D t + / D t ) = µ d + α z t + δσ c , t η c , t + + σ d , t η d , t + , z t + = ρ z t + ( − ρ ) σ z , t υ t + , σ i , t = ϕ i ¯ σ exp ( h i , t ) , h i , t + = ρ h i h i + σ h i ξ i , t + , i ∈ { z , c , d } .The innovation vectors η t = ( η c , t , η d , t ) and ξ t : = ( υ t , ξ z , t , ξ c , t , ξ d , t ) are IID over time,mutually independent and standard normal in R and R respectively. The statecan be represented as the four dimensional vector X t : = ( z t , h z , t , h c , t , h d , t ) . Other-wise the analysis and methodology radius is similar to section 3.4. The product of the growth adjusted stochastic discount factors over n period from t = n ∏ t = Φ t = ( β θ exp ( µ d − γµ c )) n exp (cid:32) ( α − γ ) n ∑ t = z t + ( δ − γ ) n ∑ t = σ c , t η c , t + + n ∑ t = σ d , t η d , t + + ( θ − ) n ∑ t = ˆ w t (cid:33) As in section 3.4, we generate this product many times and then average to obtainan approximation of L Φ . At the parameterization used in Schorfheide et al. (2018),this evaluates to − Figure 3 shows the stability exponent L Φ calculated at a range of parameter valuesin the neighborhood of the Schorfheide et al. (2018) specification. The parameter φ d is varied on the horizontal axis, while µ d is on the vertical axis. Other param-eters are held fixed at the Schorfheide et al. (2018) values. The interpretation isanalogous to that of figure 2 from section 3.4, as is the method of computation,with the dark contour line shows the exact boundary between stability and insta-bility. Increases in both µ d and ϕ d increase the long-run growth rate of the levelof the discounted cash flow, and hence increase L Φ . As with figure 2, significantdeviations in estimated parameter values are required to change the sign of L Φ .3.6. Unbounded Utility and Stationary Dividends.
We can use theorem 2.1 togeneralize a recent result of Brogueira and Schütze (2017). In their setting, { X t } is a stationary Markov process with stochastic density kernel q ( x , y ) , consumptionsatisfies C t = c ( X t ) for some measurable and positive function c and, in the for-ward looking equation (6), Φ t = β and G t = u (cid:48) ( c ( X t )) c ( X t ) , where β ∈ (
0, 1 ) and u is a concave and strictly increasing (but not necessarily bounded) utility functionon R + . We used the posterior mean values from Schorfheide et al. (2018), setting β = γ = ψ = µ c = ρ = φ z = σ = φ c = ρ hz = σ hz = √ ρ hc = σ hc = √ µ d = α = δ = φ d = ρ hd = σ hd = √ n =
1, 000 and m =
10, 000, and then drew 1,000 observations of the statistic L Φ ( n , m ) , as defined in (39) of appendix A. The mean of these 1,000 draws was − d d Schorfheide-Song-Yaron . -0.003200-0.002400-0.001600-0.0008000.0000000.0008000.0016000.0024000.003200 F IGURE
3. The exponent L Φ for the Schorfheide–Song–Yaron modelConsider the conditions of theorem 2.1 at p =
2. Since L Φ = ln β <
0, a uniqueequilibrium price process with finite second moment exists whenever assump-tions 2.1–2.3 hold. Assumptions 2.1 and 2.2 are obviously true and assumption 2.3holds whenever M : = E [ u (cid:48) ( c ( X t )) c ( X t )] is finite and Vh ( x ) = β (cid:82) h ( y ) q ( x , y ) d y is an eventually compact linear operator on L ( X , R , π ) . These conditions will holdif, as in Brogueira and Schütze (2017), we take X = R , utility is CRRA as in (19), c ( x ) is the exponential function a exp ( x ) for some a > q ( x , y ) = N ( ρ x , σ ) for some σ > | ρ | <
1. Indeed, under this specification, we have M = E exp ( ( − γ ) X t ) < ∞ , since X t is Gaussian. Eventual compactness of V is trueby proposition B.1 in the appendix.4. E XTENSIONS
The result on necessity and sufficiency of L p Φ < L Φ : = L Φ < An Existence Result.
One way to obtain a solution to the forward lookingmodel (6) is to iterate forward in time, which leads to, in the limit¯ Y t : = E t ∞ ∑ n = (cid:32) n ∏ i = Φ t + i (cid:33) G t + n . (36)In asset pricing terms, when prices are well defined, current price equals currentexpectation of the total sum of lifetime cash flow, appropriately discounted. Wenow seek conditions required for the right hand side of (36) to be finite, and for theresulting stochastic process { ¯ Y t } to solve (6). Assumption 4.1.
There exists a constant m such that P { G t (cid:54) m } = t (cid:62) Proposition 4.1.
Let assumption 4.1 hold. If, for some t (cid:62) , we have lim sup n → ∞ n ln (cid:40) E n ∏ i = Φ t + i (cid:41) <
0, (37) then ¯ Y t defined in (36) is finite with probability one. If, in addition, (37) holds for all t (cid:62) ,then the stochastic process { ¯ Y t } solves (6) . Proposition 4.1 only provides a sufficient condition, rather than a necessary andsufficient one. But the conditions imposed on the primitives are far weaker thantheorem 2.1, which required a stationary Markov structure and assumptions 2.1–2.3. Below is a simple but useful corollary to proposition 4.1, focused on the sta-tionary case. It relates to the stability exponent of the SDF process, as defined in(12).
Corollary 4.2.
Let assumption 4.1 hold. If { Φ t } is stationary and L Φ < , then thestochastic process { ¯ Y t } is finite with probability one and solves (6) . Corollary 4.2 is immediate from stationarity of { Φ t } and proposition 4.1. Application: A Theorem of Lucas.
Corollary 4.2 can be used to obtain a clas-sic result of Lucas (1978) on existence of equilibrium asset prices under far weakerconditions. In the problem of Lucas (1978), the price process obeys (1), where P t isthe price of a claim to the aggregate endowment stream { D t } , and the stochasticdiscount factor is as given in (18). In equilibrium, C t is equal to an endowment D t ,which is itself a continuous function of a stationary Markov process. FollowingLucas (1978) we take Y t : = P t u (cid:48) ( C t ) to be the endogenous object rather than P t .After dividing the fundamental asset pricing equation (1) through by u (cid:48) ( C t ) andsetting D t = C t for all t , we obtain Y t = β E t [ Y t + + u (cid:48) ( C t + ) C t + ] . (38)This is a version of (6) with Φ t = β and G t = u (cid:48) ( C t ) C t . Assumption 4.1 holdsbecause Lucas (1978) assumes that u is nonnegative, concave and bounded, whichin turn gives G t = u (cid:48) ( C t ) C t (cid:54) m for some finite constant m . Finally, since Φ t = β ,the left hand side of (37) is just ln β , which is strictly negative because β <
1. Hencecorollary 4.2 applies and a P -almost surely finite solution to (38) exists. Note thatthis argument does not use any part of the assumption that consumption is a fixedand continuous function of a stationary Markov process.5. C ONCLUSION
We developed a practical test for existence and uniqueness of equilibrium assetprices in infinite horizon arbitrage free settings. By eschewing the one-step con-traction methods of earlier work and seeking instead restrictions that ensure con-traction occurs “on average, eventually,” we find a test that is necessary as wellas sufficient, and hence can provide exact delineation between stable and unstablemodels in realistic applications. Computational techniques are provided to ensurethat the test can be implemented in complex quantitative applications.In our applications, we focused on consumption-based asset pricing models. How-ever, the theoretical results apply in the same way to other no-arbitrage settingswhere asset prices can be represented using recursion (1) with a positive marginalrate of substitution. Embedding this analysis into frameworks with endogenouslydetermined consumption is left to future research. A PPENDIX
A. C
OMPUTING THE S TABILITY E XPONENT
The stability exponent L p Φ plays a key role our results. In some cases it can becalculated analytically, as in (25) or (29). In others it needs to be computed. If thestate space is small and finite, then, as shown in section 3.2, the exponent L p Φ = L Φ is equal to the log of the spectral radius of a valuation matrix, as shown in(14), and can therefore be obtained by numerical linear algebra: First compute theeigenvalues of V and then take the maximum in terms of modulus to obtain r ( V ) .Now set L Φ = ln r ( V ) .For cases where the state space is infinite and no analytical expression for L Φ ex-ists, we consider two methods, both of which have advantages in certain settings.The first is based on discretization and the second is a Monte Carlo method. Thediscretization method works well for low dimensional state processes but is sus-ceptible to the curse of dimensionality. The Monte Carlo method is slower whenthe state space is finite with a small number of elements but less susceptible to thecurse of dimensionality—and also highly parallelizable.A.1. Discretization. If Φ t is a function of a stationary Markov process but the statespace for that Markov process is not finite, we can discretize it (see, e.g., Tauchenand Hussey (1991), Rouwenhorst (1995), or Farmer and Toda (2017)). Once thatprocedure has been carried out, the remaining steps are the same as for the finitestate Markov case described above. In what follows we investigate this procedure,finding support for its efficiency when the state space is relatively small.Our experiment is based on the constant volatility model from section 3.1, wherethe analytical expression for L p Φ = L Φ exists. We discretize the Gaussian AR(1)state process (20c) using Rouwenhorst’s method, compute the valuation matrix V in (26) corresponding to this discretized state process, calculate the spectral radius r ( V ) using linear algebra routines and, from there, compute the associated valuefor the stability exponent via (14). Finally, we compare the result with the truevalue of L Φ obtained from the analytical expression (25).Figure 4 shows this comparison when the utility parameter γ is set to 2.5 and theconsumption and dividend parameters are set to the values in table I of Bansal andYaron (2004). The vertical axis shows the value of L Φ . The horizontal axis shows In particular, µ c = µ d = ρ = σ = σ c = σ d = ϕ = F IGURE
4. Accuracy of discrete approximation of L Φ the level of discretization, indexed by the number of states for { X t } generated atthe Rouwenhorst step. The true value of L Φ at these parameters, as calculate from(25), is − L Φ is accurate up to six deci-mal places whenever the state space has more than 6 elements. Thus, the discreteapproximation is sufficiently accurate to implement the test L Φ < L Φ converges to the true value as the number of states increases. We experi-mented with other parameter values and found similar results.A.2. A Monte Carlo Method.
One issue with the discretization based method justdiscussed is that the algorithm is computationally inefficient when the state spaceis large. For this reason we also propose a Monte Carlo method that requires onlythe ability to simulate the SDF process { Φ t } . This method is less susceptible to thecurse of dimensionality and has the advantage that simulation of the SDF processcan be targeted for parallelization across CPUs or GPUs.The idea behind the Monte Carlo method is to approximate L Φ via L Φ ( n , m ) : = n ln (cid:40) m m ∑ j = n ∏ i = Φ ( j ) i (cid:41) , (39) T ABLE
1. Monte Carlo spectral radius estimates when L Φ = − m = 1000 m = 2000 m = 3000 m = 4000 m = 5000n = 250 -0.0033183 -0.0032524 -0.0032434 -0.0032533 -0.0032353(0.000099) (0.000065) (0.000056) (0.000047) (0.000042)n = 500 -0.0032045 -0.0032149 -0.0031948 -0.0031907 -0.0031922(0.000080) (0.000058) (0.000045) (0.000040) (0.000036)n = 750 -0.0031985 -0.0031841 -0.0031748 -0.0031784 -0.0031890(0.000080) (0.000054) (0.000044) (0.000041) (0.000038) where each Φ ( j ) , . . . , Φ ( j ) n is an independently simulated path of { Φ t } , and n and m are suitably chosen integers. The idea relies on the strong law of large numbers,which yields m ∑ mj = ∏ ni = Φ ( j ) i → E ∏ ni = Φ i with probability one, combined withthe fact that Z n → Z almost surely implies g ( Z n ) → g ( Z ) almost surely whenever g : R → R is continuous. However, these are only asymptotic results and ourconcern here is sufficiently good performance in finite samples.Table 1 analyzes this issue. Here we again use the constant volatility model fromsection 3.1, comparing Monte Carlo approximations of L Φ with the true value ob-tained via the analytical expression given in (25). The consumption and dividendgrowth parameters are again chosen to match table I of Bansal and Yaron (2004), asin footnote 17. The true value of L Φ calculated from the analytical expression (25)is − n and m in the table is consistent with the left hand side of (39). For each n , m pair, we com-pute L Φ ( n , m ) n = L Φ . A PPENDIX
B. P
ROOFS If E is a Banach lattice, then an ideal in E is, as usual, a vector subspace L of E with x ∈ L whenever | x | (cid:54) | y | and y ∈ L . The spectral radius of a bounded linear operator M from E to itself is the supremum of | λ | for all λ in the spectrum of A . The operator M is called compact if the image under M of the unit ball in E has compact closer. M is called eventually compact if there exists an i ∈ N suchthat M i is compact. M is called positive if it maps the positive cone of E into itself.A positive linear operator M is called irreducible if the only closed ideals J ⊂ E satisfying M ( J ) ⊂ J are { } and E . See Abramovich et al. (2002) or Meyer-Nieberg(2012) for more details.If X is an Polish space, π is a finite Borel measure on X and p (cid:62)
1, then L p ( π ) : = L ( X , R , π ) denotes is the set of all Borel measurable functions f from X to R sat-isfying (cid:82) | f | p d π < ∞ . The norm on L p ( π ) is (cid:107) f (cid:107) : = ( (cid:82) | f | p d π ) p . Functionsequal π -almost everywhere are identified. Convergence on L p ( π ) is with respectto the norm topology generated by (cid:107) · (cid:107) . We write f (cid:54) g for f , g in L p ( π ) if f (cid:54) g holds pointwise π -almost everywhere. We write f (cid:28) g if f < g holds pointwise π -almost everywhere. The positive cone of L p ( π ) is all f ∈ L p ( π ) with f (cid:62) H p , which conforms with our previous definition (see, inparticular, theorem 2.1).B.1. Operator Compactness in Spaces of Summable Functions.
Assumption 2.3requires that, for some p (cid:62)
1, the operator V is eventually compact as a linear mapfrom L p ( X , R , π ) to itself. In this section we discuss some sufficient conditions andstate a result that was used in section 3. One sufficient condition is as follows: V will be eventually compact if there is abounded linear operator M such that V (cid:54) M pointwise on L p ( π ) and M is eventu-ally compact. Indeed, in that case there exists a k ∈ N such that M k is compact and,since each L p space has order continuous norm (Meyer-Nieberg (2012), §2.4), it fol-lows from corollary 2.37 in Abramovich et al. (2002) that V k is compact. Hence V is eventually compact.Next we give a sufficient condition focused on the applications in section 3. Take X = R and p =
2. In the proposition below, q is a stochastic density kernel on R with stationary density π and two step density kernel q . The statement that q is time-reversible means that q ( x , y ) π ( x ) = q ( y , x ) π ( y ) for all x , y ∈ R . (A number of It is worth nothing that, since V is a positive operator and obviously linear, V is a bounded lin-ear operator on L p ( π ) whenever it maps L p ( π ) to itself (see Abramovich et al. (2002), theorem 1.31).In particular, boundedness of V need not be separately checked. our results use the fact that q ( x , · ) = N ( ρ x , σ ) for some σ > | ρ | < q is time-reversible. See, e.g., O’Donnell (2014).) Proposition B.1.
Let M be an operator that maps f in L ( π ) intoM f ( x ) = g ( x ) (cid:90) f ( y ) q ( x , y ) d y ( x ∈ R ) , (40) where g is a measurable function from R to R + . If q is time-reversible and (cid:90) g ( x ) q ( x , x ) d x < ∞ , (41) then M is a compact linear operator on L ( π ) .Proof. We can express the operator M as M f ( x ) = (cid:90) f ( y ) k ( x , y ) π ( y ) d y where k ( x , y ) : = g ( x ) q ( x , y ) π ( y ) .By theorem 6.11 of Weidmann (2012), the operator M will be Hilbert–Schmidt in L ( π ) , and hence compact, if the kernel k satisfies (cid:90) (cid:90) k ( x , y ) π ( x ) π ( y ) d x d y < ∞ .Using the definition of k and the time-reversibility of q , this translates to (cid:90) g ( x ) (cid:90) q ( x , y ) q ( y , x ) d y d x < ∞ .This completes the proof because, by definition, q ( x , x ) = (cid:82) q ( x , y ) q ( y , x ) d y . (cid:3) B.2.
Remaining Proofs.
Let X be the state space, as in section 2.1. Throughout allof the following we take assumptions 2.1, 2.2 and 2.3 to be in force. The symbol p represents the constant in assumption 2.3. The positive cone of L p ( π ) is all f ∈ L p ( π ) with f (cid:62)
0. We denote this set by H p , which conforms with our previousdefinition (see, in particular, theorem 2.1).As before, Π is a stochastic kernel on X and { X t } is a stationary Markov processon X with stochastic kernel Π and common marginal distribution π . Let Π n de-note the n -step stochastic kernel corresponding to Π . The symbol E x will indicate In other words, Π is a function from ( X , B ) to [
0, 1 ] such that B (cid:55)→ Π ( x , B ) is a probabilitymeasure on ( X , B ) for each x ∈ X , and x (cid:55)→ Π ( x , B ) is B -measurable for each B ∈ B . The process { X t } satisfies P { X t + ∈ B | X t = x } = Π ( x , B ) for all x in X and B ∈ B . conditioning on the event X = x , so that, for any h ∈ L ( π ) and any n ∈ N , wehave E x h ( X n ) = (cid:90) h ( x ) Π n ( x , d y ) . (42) Lemma B.2.
For any h ∈ H p and all x ∈ X we haveV n h ( x ) = E x n ∏ i = Φ i h ( X n ) . (43) Proof.
Equation (43) holds when n = Vh ( x ) = (cid:90) (cid:90) φ ( x , y , η ) ν ( d η ) h ( y ) Π ( x , d y ) = E x Φ h ( X ) .Now suppose (43) holds at arbitrary n ∈ N . We claim it also holds at n +
1. Indeed, V n + h ( x ) = E x Φ V n h ( X ) = E x Φ E X n + ∏ i = Φ i h ( X n + ) = E x E X n + ∏ i = Φ i h ( X n + ) .An application of the law of iterated expectations completes the proof. (cid:3) Lemma B.3.
For each h ∈ H p , x ∈ X and n ∈ N we haveV n h ( x ) = = ⇒ (cid:90) h ( y ) Π n ( x , d y ) = Proof.
Fix h ∈ H p , x ∈ X and n ∈ N , and suppose that V n h ( x ) =
0. It follows fromlemma B.2 that E x ∏ ni = Φ i h ( X n ) =
0, which in turn implies that ∏ ni = Φ i h ( X n ) = P x -a.s. But then h ( X n ) = P x -a.s., and hence E x h ( X n ) =
0. By (42),this is equivalent to (cid:82) h ( y ) Π n ( x , d y ) = (cid:3) Lemma B.4.
For given h ∈ H p , the following statements are true:(a) If h (cid:29) , then V n h (cid:29) for all n ∈ N .(b) If h (cid:54) = , then there exists an n in N such that V n h (cid:29) .Proof. Regarding part (a), it suffices to show this is true when n =
1, after whichwe can iterate. To this end, fix h ∈ H p with h > B ∈ B with π ( B ) = Vh ( x ) = (cid:90) h ( y ) (cid:20) (cid:90) φ ( x , y , η ) ν ( d η ) (cid:21) Π ( x , d y ) = φ is positive, we must then have Π ( x , B ) =
0. But π is invariant, so π ( B ) = (cid:82) Π ( x , B ) π ( d x ) =
0. Contradiction. (cid:3) Lemma B.5.
The valuation operator V is irreducible on L p ( π ) .Proof. Suppose to the contrary that there exists a closed ideal J in L p ( π ) such that V is invariant on J and J is neither ∅ nor L p ( π ) itself. Since J is a closed ideal in L p ( π ) , there exists a set B ∈ B such that J = { f ∈ L p ( π ) : f = π -a.e. on B } . Moreover, since J is neither empty nor the whole space, it must be that, for this set B that defines J , we have 0 < π ( B ) < V is invariant on J , we have V n h ∈ J for all h ∈ J and n ∈ N . In particular, V n B c is in J for all n ∈ N . This means that V n B c ( x ) = π -almost all x ∈ B andall n in N . Fixing an x ∈ B and applying lemma B.3, we then have Π n ( x , B c ) = n ∈ N . But π ( B ) <
1, so π ( B c ) >
0. This contradicts irreducibility of thestochastic kernel Π (see footnote 6 for the definition of the latter), which in turnviolates assumption 2.2. (cid:3) The following is a local spectral radius result suitable for L p ( π ) that draws onresults from Zabreiko et al. (1967) and Krasnosel’skii et al. (2012). (Suitability for L p ( π ) is due to the fact that the interior of the positive cone can be empty.) Theproof provided here is due to Mirosława Zima (private communication). In thestatement of the theorem, a quasi-interior element of the positive cone of a Banachlattice E is a nonnegative element h satisfying (cid:104) h , g (cid:105) > E ∗ . (See Krasnosel’skii et al. (2012) for moredetails.) Theorem B.6.
Let h be an element of a Banach lattice E and let M be a positive andcompact linear operator. If h is quasi-interior, then (cid:107) M n h (cid:107) n → r ( M ) as n → ∞ .Proof. Let h and M be as in the statement of the theorem and let E + be the positivecone of E . Let r ( h , M ) : = lim sup n → ∞ (cid:107) M n h (cid:107) n . From the definition of r ( M ) itis clear that r ( h , M ) (cid:54) r ( M ) . To see that the reverse inequality holds, let λ be aconstant satisfying λ > r ( h , M ) and let h λ : = ∞ ∑ n = M n h λ n + . (44)The point h λ is a well-defined element of E + by lim sup n → ∞ (cid:107) M n h (cid:107) n < λ andCauchy’s root test for convergence. It is also quasi-interior, since the sum in (44) See, for example, Gerlach and Nittka (2012), p. 765. includes the quasi-interior element h , and since M maps E + into itself. Moreover,by standard Neumann series theory (e.g., Krasnosel’skii et al. (2012), theorem 5.1),the point h λ also has the representation h λ = ( λ I − M ) − h , from which we obtain λ h λ − Mh λ = h . Because h ∈ E + , this implies that Mh λ (cid:54) λ h λ . Applying thislast inequality, compactness of M , quasi-interiority of h λ and theorem 5.5 (a) ofKrasnosel’skii et al. (2012), we must have r ( M ) (cid:54) λ . Since this inequality wasestablished for an arbitrary λ satisfying λ > r ( h , M ) , we conclude that r ( h , M ) (cid:62) r ( M ) .We have now established that r ( h , M ) = r ( M ) . Since M is compact, corollary 1 ofDaneš (1987) gives lim sup n → ∞ (cid:107) M n h (cid:107) n = lim n → ∞ (cid:107) M n h (cid:107) n .The proof is now complete. (cid:3) Theorem B.7.
The growth exponent L Φ satisfies exp ( L Φ ) = r ( V ) .Proof. Let = X be the function equal to unity everywhere on X . In view oflemma B.2, to prove theorem B.7 it suffices to show thatlim n → ∞ (cid:107) V n (cid:107) n = r ( V ) . (45)By assumption 2.3 we can choose an i ∈ N such that V i is a compact linear operatoron L p ( π ) . Fix j ∈ N with 0 (cid:54) j (cid:54) i −
1. By lemma B.4 we know that V j ispositive π -almost everywhere on X , and is therefore quasi-interior. As a result,theorem B.6 applied to V i with initial condition h : = V j yields (cid:107) V in V j (cid:107) n = (cid:107) V in + j (cid:107) n → r ( V i ) ( n → ∞ ) .But r ( V i ) = r ( V ) i , so (cid:107) V in + j (cid:107) ( in ) → r ( V ) as n → ∞ . It follows that (cid:107) V in + j (cid:107) ( in + j ) → r ( V ) .As this is shown to be true for any integer j satisfying 0 (cid:54) j (cid:54) i −
1, we canconclude that (45) is valid. (cid:3) By the Riesz Representation Theorem, the dual space of L p ( π ) is isometrically isomorphic to L q ( π ) where 1/ p + q =
1. If g is a nonnegative and nonzero element of L q ( π ) then it is positiveon a set of positive π measure. Since f (cid:29) X , the produce f g must be positive on a set ofpositive π measure. Hence (cid:82) f g d π >
0, so f is quasi-interior. To prove theorem 2.1, we will also need the following two lemmas:
Lemma B.8.
T has a fixed point in H p if and only if there exist elements g , h in H p suchthat T n g → h as n → ∞ .Proof. Suppose first that there exist g , h in H p such that T n g → h as n → ∞ . Since T f = V f + ˆ g and V is a bounded linear operator on L p ( π ) , we know that T iscontinuous as a self-map on L p ( π ) . Letting g n = T n g , we have g n → h and hence,by continuity, Tg n → Th . But, by the definition of the sequence { g n } , we must alsohave Tg n → h . Hence Th = h .Conversely, if T has a fixed point f ∈ H p , then the condition in the statement oflemma B.8 is satisfied with g = h = f . (cid:3) Proposition B.9.
If T has a fixed point in H p , then L p Φ < .Proof. Let V ∗ be the adjoint operator associated with V . Since V is irreducible (seelemma B.5) and V i is compact for some i , the version of the Krein–Rutman the-orem presented in lemma 4.2.11 of Meyer-Nieberg (2012) together with the RieszRepresentation Theorem imply existence of an e ∗ in the dual space L q ( π ) such that e ∗ (cid:29) V ∗ e ∗ = r ( V ) e ∗ . (46)Let h be a fixed point of T in H p . Clearly h is nonzero, since T = V + ˆ g = ˆ g and ˆ g is not the zero function (see assumption 2.1). Moreover, since h is a fixed point, wehave h = Vh + ˆ g and hence, with the inner production notation (cid:104) φ , f (cid:105) : = (cid:82) φ f d π , (cid:104) e ∗ , h (cid:105) = (cid:104) e ∗ , Vh (cid:105) + (cid:104) e ∗ , ˆ g (cid:105) = (cid:104) V ∗ e ∗ , h (cid:105) + (cid:104) e ∗ , ˆ g (cid:105) = r ( V ) (cid:104) e ∗ , h (cid:105) + (cid:104) e ∗ , ˆ g (cid:105) .In other words, ( − r ( V )) (cid:104) e ∗ , h (cid:105) = (cid:104) e ∗ , ˆ g (cid:105) .Both h and ˆ g are nonzero in L p ( π ) and e ∗ is positive π -a.e., so (cid:104) e ∗ , h (cid:105) > (cid:104) e ∗ , ˆ g (cid:105) >
0. It follows that r ( V ) <
1. By theorem B.7, we have L p Φ = ln r ( V ) , whichproves the claim in the lemma. (cid:3) Proof of theorem 2.1.
By lemma B.8, (b) and (d) of theorem 2.1 are equivalent, so itsuffices to show that (e) = ⇒ (c) = ⇒ (b) = ⇒ (a) = ⇒ (e). Of these, theimplications (e) = ⇒ (c) = ⇒ (b) are trivial, and the fact that (b) = ⇒ (a) wasestablished in proposition B.9. Hence we need only show that (a) = ⇒ (e). To see that (a) implies (e), suppose that L p Φ <
0. Then, by theorem B.7, we have r ( V ) <
1. Using Gelfand’s formula for the spectral radius, which states that r ( V ) = lim n → ∞ (cid:107) V n (cid:107) n with (cid:107) · (cid:107) as the operator norm, we can choose n ∈ N such that (cid:107) V n (cid:107) <
1. Then, for any h , h (cid:48) ∈ H p we have (cid:107) T n h − T n h (cid:48) (cid:107) = (cid:107) V n h − V n h (cid:48) (cid:107) = (cid:107) V n ( h − h (cid:48) ) (cid:107) (cid:54) (cid:107) V n (cid:107) · (cid:107) h − h (cid:48) (cid:107) .Observe that H p is closed in L p ( π ) , since L p ( π ) is a Banach lattice. Hence H p iscomplete in the norm topology. Existence, uniqueness and global stability nowfollow from a well-known extension to the Banach contraction mapping theorem(see, e.g., p. 272 of Wagner (1982)).Lastly, to see that (13) holds, suppose that (a)–(e) are true. Then r ( V ) <
1, whichimplies that ( I − V ) − is well-defined on H p and equals ∑ ∞ i = V i (see, e.g., theo-rem 2.3.1 and corollary 2.3.3 of Atkinson and Han (2009)). In particular, the fixedpoint of T is given by h ∗ = ∑ ∞ n = V n ˆ g . Applying (43) to this sum verifies the claimin (13). (cid:3) Proof of proposition 2.2.
Fix p (cid:62)
1. If assumptions 2.1–2.2 hold and X is a finite setendowed with the discrete topology, then all functions from X to R are measur-able and have finite p -th moment, so L p ( X , R , π ) = R X and H p = R X + . It followsthat ˆ g ∈ H p and V is a bounded linear operator from L p ( X , R , π ) to itself (sinceevery linear operator mapping a finite dimensional normed vector space to itselfis bounded). By the Heine–Borel theorem, bounded subsets in finite dimensionalspace have compact closure, so V is also (eventually) compact. Thus, assump-tion 2.3 holds. Finally, L p Φ = L Φ by the identity in (14), since, in a finite dimen-sion normed linear space, the spectral radius is independent of the choice of norm(due to equivalence of norms combined with Gelfand’s formula for the spectralradius). (cid:3) Proof of proposition 4.1.
Fix t (cid:62)
0. To prove that P { ¯ Y t < ∞ } =
1, it suffices toshow that E ¯ Y t < ∞ , which, by the definition of ¯ Y t and the law of iterated ex-pectations, will hold whenever the infinite sum ∑ ∞ n = E ∏ ni = Φ t + i G t + n converges.(The expectation is passed through the sum by nonnegativity of the sum compo-nents combined with the Monotone Convergence Theorem, which is valid regard-less of whether or not the sum is finite. See, e.g., Dudley (2002), theorem 4.3.2.) By Cauchy’s root criterion for convergence of sums, the sum converges whenever A <
1, where A : = lim sup n → ∞ a nn with a n : = E n ∏ i = Φ t + i G t + n .We have ln A (cid:54) lim sup n → ∞ n ln E n ∏ i = Φ t + i m = lim sup n → ∞ n ln E n ∏ i = Φ t + i < m is the constant in assumption 4.1 and the final inequality is due to (37).Hence A < E ¯ Y t < ∞ , as was to be shown.Now suppose that (37) holds for all t (cid:62)
0. Then ¯ Y t is finite with probability one forall t . Substituting the definition of ¯ Y t + into the right hand side of (6) and using thelaw of iterated expectations, it is straightforward to show that { ¯ Y t } is a solution to(6). Details are omitted. (cid:3) R EFERENCES A BEL , A. B. (1990): “Asset prices under habit formation and catching up with theJoneses,”
The American Economic Review , 38–42.A
BRAMOVICH , Y. A., Y. A. A
BRAMOVICH , AND
C. D. A
LIPRANTIS (2002):
Aninvitation to operator theory , vol. 1, American Mathematical Soc.A
LVAREZ , F.
AND
U. J. J
ERMANN (2001): “Quantitative Asset Pricing Implicationsof Endogenous Solvency Constraints,”
Review of Financial Studies , 14, 1117–1151.——— (2005): “Using Asset Prices to Measure the Persistence of the Marginal Util-ity of Wealth,”
Econometrica , 73, 1977–2016.A
TKINSON , K.
AND
W. H AN (2009): Theoretical Numerical Analysis: A FunctionalAnalysis Framework , vol. 39, Springer Science & Business Media.B
ANSAL , R.
AND
A. Y
ARON (2004): “Risks for the long run: A potential resolutionof asset pricing puzzles,”
The Journal of Finance , 59, 1481–1509.B
ARRO , R. J. (2006): “Rare Disasters and Asset Markets in the Twentieth Century,”
Quarterly Journal of Economics , 121, 823–866.B
LANCHARD , O. J.
AND
C. M. K
AHN (1980): “The solution of linear differencemodels under rational expectations,”
Econometrica , 1305–1311.B
OROVI ˇCKA , J., L. P. H
ANSEN , AND
J. A. S
CHEINKMAN (2016): “Misspecifiedrecovery,”
The Journal of Finance , 71, 2493–2544. B OROVI ˇCKA , J.
AND
J. S
TACHURSKI (2017): “Necessary and sufficient conditionsfor existence and uniqueness of recursive utilities,” Tech. rep., NBER.B
ROGUEIRA , J.
AND
F. S
CHÜTZE (2017): “Existence and uniqueness of equilibriumin Lucasâ ˘A ´Z asset pricing model when utility is unbounded,”
Economic TheoryBulletin , 5, 179–190.C
ALIN , O. L., Y. C
HEN , T. F. C
OSIMANO , AND
A. A. H
IMONAS (2005): “Solvingasset pricing models when the price–dividend function is analytic,”
Economet-rica , 73, 961–982.C
AMPBELL , J. Y.
AND
J. H. C
OCHRANE (1999): “By Force of Habit: AConsumption-Based Explanation of Aggregate Stock Market Behavior,”
TheJournal of Political Economy , 107, 205–251.C
HRISTENSEN , T. M. (2017): “Nonparametric stochastic discount factor decompo-sition,”
Econometrica , 85, 1501–1536.C
OGLEY , T.
AND
T. J. S
ARGENT (2008): “The Market Price of Risk and the EquityPremium: A Legacy of the Great Depression?”
Journal of Monetary Economics , 55,454–476.C
OLLIN -D UFRESNE , P., M. J
OHANNES , AND
L. A. L
OCHSTOER (2016): “Parame-ter Learning in General Equilibrium: The Asset Pricing Implications,”
AmericanEconomic Review , 106, 664–698.D
ANEŠ , J. (1987): “On local spectral radius,” ˇCasopis pro pˇestování matematiky , 112,177–187.D
UDLEY , R. M. (2002):
Real Analysis and Probability , vol. 74, Cambridge UniversityPress.D
UFFIE , D. (2001):
Dynamic Asset Pricing Theory , Princeton University Press.F
ARMER , L. E.
AND
A. A. T
ODA (2017): “Discretizing nonlinear, non-GaussianMarkov processes with exact conditional moments,”
Quantitative Economics , 8,651–683.F
ORSTER , K.-H.
AND
B. N
AGY (1991): “On the local spectral radius of a nonneg-ative element with respect to an irreducible operator,”
Acta Universitatis Szegedi-ensis , 55, 155–166.F
URSTENBERG , H.
AND
H. K
ESTEN (1960): “Products of random matrices,”
TheAnnals of Mathematical Statistics , 31, 457–469.G
ERLACH , M.
AND
R. N
ITTKA (2012): “A new proof of Doob’s theorem,”
Journalof Mathematical Analysis and Applications , 388, 763–774. H ANSEN , L. P. (2012): “Dynamic valuation decomposition within stochasticeconomies,”
Econometrica , 80, 911–967.H
ANSEN , L. P.
AND
S. F. R
ICHARD (1987): “The Role of Conditioning Informationin Deducing Testable Restrictions Implied by Dynamic Asset Pricing Models,”
Econometrica , 55, 587–613.H
ANSEN , L. P.
AND
J. A. S
CHEINKMAN (2009): “Long-Term Risk: An OperatorApproach,”
Econometrica , 77, 177–234.K
NILL , O. (1992): “Positive Lyapunov exponents for a dense set of bounded mea-surable SL (2, R )-cocycles,” Ergodic Theory and Dynamical Systems , 12, 319–331.K
OCHERLAKOTA , N. (1990): “On Tests of Representative Consumer Asset PricingModels,”
Journal of Monetary Economics , 26, 285–304.K
RASNOSEL ’ SKII , M., G. V
AINIKKO , R. Z
ABREYKO , Y. R
UTICKI , AND
V. S
TET ’ SENKO (2012):
Approximate Solution of Operator Equations , SpringerNetherlands.K
REPS , D. M. (1981): “Arbitrage and equilibrium in economies with infinitelymany commodities,”
Journal of Mathematical Economics , 8, 15–35.L
UCAS , R. E. (1978): “Asset prices in an exchange economy,”
Econometrica , 1429–1445.M
EHRA , R.
AND
E. C. P
RESCOTT (1985): “The equity premium: A puzzle,”
Journalof Monetary Economics , 15, 145–161.M
EYER -N IEBERG , P. (2012):
Banach lattices , Springer Science & Business Media.M
EYN , S.
AND
R. L. T
WEEDIE (2009):
Markov chains and stochastic stability , Cam-bridge University Press.O’D
ONNELL , R. (2014):
Analysis of boolean functions , Cambridge University Press.P
OHL , W., K. S
CHMEDDERS , AND
O. W
ILMS (2018): “Higher Order Effects in As-set Pricing Models with Long-Run Risks,”
The Journal of Finance , 73, 1061–1111.Q IN , L. AND
V. L
INETSKY (2017): “Long-Term Risk: A Martingale Approach,”
Econometrica , 85, 299–312.R
IETZ , T. A. (1988): “The Equity Risk Premium: A Solution,”
Journal of MonetaryEconomics , 22, 117–131.R
OSS , S. A. (1978): “A Simple Approach to the Valuation of Risky Streams,”
Journalof Business , 51, 453–475.R
OUWENHORST , K. G. (1995): “Asset pricing implications of equilibrium businesscycle models,” in
Frontiers of Business Cycle Research , Princeton University Press,294–330. R UBINSTEIN , M. (1976): “The Valuation of Uncertain Income Streams and the Pric-ing of Options,”
Bell Journal of Economics , 7, 407–425.S
ANTOS , M. S.
AND
M. W
OODFORD (1997): “Rational Asset Pricing Bubbles,”
Econometrica , 65, 19–57.S
CHORFHEIDE , F., D. S
ONG , AND
A. Y
ARON (2018): “Identifying long-run risks:A Bayesian mixed-frequency approach,”
Econometrica , 86, 617–654.T
AUCHEN , G.
AND
R. H
USSEY (1991): “Quadrature-based methods for obtainingapproximate solutions to nonlinear asset pricing models,”
Econometrica , 371–396.W
AGNER , C. H. (1982): “A Generic Approach to Iterative Methods,”
MathematicsMagazine , 55, 259–273.W
EIDMANN , J. (2012):
Linear operators in Hilbert spaces , vol. 68, Springer Science &Business Media.W
EIL , P. (1989): “The Equity Premium Puzzle and the Risk-Free Rate Puzzle,”
Journal of Monetary Economics , 24, 401–421.Z
ABREIKO , P., M. K
RASNOSEL ’ SKII , AND
V. Y. S
TETSENKO (1967): “Bounds for thespectral radius of positive operators,”