A More Detailed and Complete Appendix for Macroeconomic Volatilities and Long-run Risks of Asset Prices

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1 A More Detailed and Complete Appendix for Macroeconomic Volatilities and Long-run Risks of Asset Prices This is an on-line appendix with more details and analysis for the readers of the paper. B.1 Derivation for the A i s, risk-free rate and market price of risk First, we re-write the normalized aggregator f defined in Equation (5) as f(c, J) = β 1 1 (1 γ)j[g 1], ψ where G ( C ((1 γ)j) 1 1 γ ) 1 1 ψ. (B1) Then, taking partial derivatives of f(c, J) with respect to J and C, we have f J = (θ 1)βG βθ (B2) and f C = β G (1 γ)j. C (B3) where we use the notation θ = 1 γ. Theoretically, the aggregator f(c, J) should be an 1 1 ψ increasing function of the value function J (see, e.g., Skiadas, 29, Chapter 6.3). Otherwise, the monotonicity axiom of preferences will be violated. This places joint restrictions on γ and ψ such that θ 1 or θ <. This is because f J > implies that If θ > 1 : G > θ θ 1 If θ < 1 : G < θ θ 1. If θ > 1, the first inequality is possible to have solutions. However, if < θ < 1, the second inequality is impossible as G > always. Hence, the necessary restriction on γ and ψ is either θ > 1 or θ <. If θ = 1, as shown by Duffie and Epstein (1992), we obtain the standard additive expected utility of constant relative risk aversion (CRRA). So θ > 1 can be extended to θ 1. 1

2 Conjecturing a solution for J of the following form, J(W t, X t, V 1t, V 2t ) = exp(a + A 1 X t + A 2 V 1t + A 3 V 2t ) W 1 γ t 1 γ, (B4) and using the standard envelope condition f C = J W, we have C = J ψ W [(1 γ)j]1 γψ β ψ. (B5) Substituting (B3) and (B4) into (B5), we obtain C W = βψ exp and hence J can be re-written as [ (A + A 1 X t + A 2 V 1t + A 3 V 2t ) 1 ψ 1 γ ]. (B6) J(C t, X t, V 1t, V 2t ) = β ψ(1 γ) exp[ψ(a + A 1 X t + A 2 V 1t + A 3 V 2t )] C1 γ t 1 γ. (B7) Further substituting (B6) and (B4) into (B1), we get βg = C t W t. Applying the log-linear approximation, we obtain βg = C t W t g 1 g 1 log g 1 + g 1 log(βg). (B8) This implies that f = θj(βg β) θj [ ] 1 ψ g 1 1 γ (A + A 1 X t + A 2 V 1t + A 3 V 2t ) + ξ, (B9) where θ = 1 γ 1 1 ψ and ξ = g 1 g 1 log g 1 + g 1 ψ log β β. (B1) Substituting (B9) into the HJB Equation (6), f(c, J) + C (µ + X)J C [δ cv 1 + (1 δ c )V 2 ]C 2 J CC + J X ( αx) φ2 x[δ x V 1 + (1 δ x )V 2 ]J XX +J V1 κ 1 ( V 1 V 1 ) σ2 1v 1 J V1 V 1 + J V2 κ 2 ( V 2 V 2 ) σ2 2V 2 J V2 V 2 =, (B11) where {C t } is the optimal consumption process, and we have used the definition of A c J i b(z) J(z) z + i,j (σσ T ) i,j (z) 2 J z i z j, 2

3 with z = (C, X, V 1, V 2 ) and b(z) and σ(z) the drift and diffusive terms for z defined in Equation (2). Collecting the terms containing constant, X t, V 1t and V 2t, resp, we have X : V 1 : V 2 : θg 1 1 ψ 1 γ A + θξ + (1 γ)µ + κ 1 V1 ψa 2 + κ 2 V2 ψa 3 = θg 1 1 ψ 1 γ A 1 + (1 γ) αψa 1 = θg 1 1 ψ 1 γ A γ(1 γ)δ c φ2 xδ x ψ 2 A 2 1 κ 1 ψa σ2 1ψ 2 A 2 2 = θg 1 1 ψ 1 γ A γ(1 γ)(1 δ c) φ2 x(1 δ x )ψ 2 A 2 1 κ 2 ψa σ2 2ψ 2 A 2 3 =. Solving the above algebraic equations, we obtain A = 1 g 1 ψ [θξ + (1 γ)µ + κ V 1 1 ψa 2 + κ 2 V2 ψa 3 ], A 1 = 1 γ (g 1 + α)ψ, A 2 = b 1 b 2 1 4a 1 c 1 2a 1, (B12) with A 3 = b 2 b 2 2 4a 2 c 2 2a 2, a 1 = 1 2 σ2 1ψ 2, b 1 = (g 1 + κ 1 )ψ, c 1 = 1 2 γ(1 γ)δ c φ2 xδ x (1 γ) 2 (g 1 +α) 2, a 2 = 1 2 σ2 2ψ 2, b 2 = (g 1 + κ 2 )ψ, c 2 = 1 2 γ(1 γ)δ c(1 δ c ) φ2 x(1 δ x ) (1 γ)2 (g 1 +α) 2. We then derive the risk-free rate and market prices of risks. Recall that the pricing kernel is given by Equation (A6). Based on the definition for f, we have where f J = ξ 1 g 1 (A 1 X t + A 2 V 1t + A 3 V 2t ) 1 γψ 1 γ, [ f C = β ψγ exp (B + A 1 X t + A 2 V 1t + A 3 V 2t ) 1 γψ ] C γ t, 1 γ Applying Ito s Lemma to π t in Equation (A6), we have ξ 1 = (θ 1)ξ β g 1 1 γψ 1 γ A. (B13) dπ t π t = (r f dt + λ 1 dz 1t + λ 2 dz 2t + λ 3 dw 1t + λ 4 dw 2t ), (B14) where the risk-free rate r f and the market prices of risks, λ i, i = 1, 2, 3, 4, are given below. 3

4 First, the risk-free rate is r f = r + r 1 X t + r 2 V 1t + r 3 V 2t, (B15) where r = (ξ 1 + (κ 1 A 2 V1 + κ 2 A 3 V2 ) 1 γψ 1 γ γµ), r 1 = 1 ψ, r 2 = (g 1 + κ 1 )A 2 1 γψ 1 γ 1 2 r 3 = (g 1 + κ 2 )A 3 1 γψ 1 γ 1 2 Second, the market prices of risks are ( ) 2 1 γψ (A 2 1 γ 1φ 2 xδ x + A 2 2σ1) γ(γ + 1)δ c, ( 1 γψ 1 γ ) 2 [A 2 1φ 2 x(1 δ x ) + A 2 3σ 2 2] 1 2 γ(γ + 1)(1 δ c). (B16) Q.E.D. λ 1 = γ V 1t δ c + V 2t (1 δ c ), λ 2 = 1 γψ 1 γ A 1φ x V1t δ x + V 2t (1 δ x ), λ 3 = 1 γψ 1 γ A 2σ 1 V1t, λ 4 = 1 γψ 1 γ A 3σ 2 V2t. (B17) B.2 Derivation for the A im s Let D t P t = exp{(a m + A 1m X t + A 2m V 1t + A 3m V 2t )}. (B18) A key step in the derivation is to use the following pricing relation given in ( ) dpt E t + D [ ] t dπt dp t dt = r f dt E t. (B19) P t P t π t P t With similar loglinear approximation as Equation (B8), we can approximate the ratio as where D t P t g m + g 1m log D t P t = g m + g 1m ((A m + A 1m X t + A 2m V 1t + A 3m V 2t )), (B2) g m = g 1m g 1m log g 1m. 4

5 Applying Ito s lemma to (B18), we have dp t P t = dd t D t (A 1m dx t + A 2m dv 1t + A 3m dv 2t ) A2 1m(dX t ) A2 2m(dV 1t ) A2 3m(dV 2t ) 2. Hence, E t ( dp t P t )/dt = µ d + ϕx t + αa 1m X t κ 1 A 2m ( V 1 V 1t ) κ 2 A 3m ( V 2 V 2t ) A2 1mφ 2 x[v 1t δ x + V 2t (1 δ x )] A2 2mσ 2 1V 1t A2 3mσ 2 2V 2t. (B21) The risk premium term in Equation (B19) can thus be written as [ ] dπt dp t E t /dt = σ dc λ 1 V1t δ c + V 2t (1 δ c ) (A 1m φ x σ dx )λ 2 V1t δ x + V 2t (1 δ x ) π t P t (A 2m σ 1 σ dv )λ 3 V1t (A 3m σ 2 σ dv2 )λ 4 V2t, where λ 1, λ 2, λ 3 and λ 4 are market prices of risks defined in Equation (B17). (B22) Now, substituting (B2), (B21), (B22), and risk-free rate (B15) into Equation (B19), and collecting terms containing X t, we obtain A 1m = ϕ 1 ψ g 1m + α. (B23) Collecting terms containing V 1t and V 2t, resp, we obtain an equation for A 2m, with a 2m A 2 2m + b 2m A 2m + c 2m = a 2m = 1 2 σ2 1, b 2m = g 1m + κ 1 1 γψ 1 γ A 2σ 2 1, c 2m = ( 1 2 A2 1m 1 γψ 1 γ A 1A 1m )φ 2 xδ x + r 2. Solving it, we have A 2m = b 2m ± b 2 2m 4a 2m c 2m 2a 2m. (B24) We choose the root that goes to zero when σ 1 goes to zero. This is because when σ 1, or a 2m goes to zero, the price sensitivity to V 1 should be zero. Similarly, we obtain an equation for A 3m, a 3m A 2 3m + b 3m A 3m + c 3m = 5

6 with a 3m = 1 2 σ2 2, b 2m = g 1m + κ 2 1 γψ 1 γ A 3σ 2 2, c 3m = ( 1 2 A2 1m 1 γψ 1 γ A 1A 1m )φ 2 x(1 δ x ) + r 3. The solution is A 3m = b 3m ± b 2 3m 4a 3m c 3m 2a 3m, (B25) where we choose the root in a similar fashion as for A 2m above. Finally, collecting the constant terms in Equation (B19), we obtain µ d κ 1 A 2m V1 κ 2 A 3m V2 + g m + g 1m A m + r =, and re-arrange terms to get A m = 1 g 1m [ µd κ 1 A 2m V1 κ 2 A 3m V2 + g 1m g 1m log g 1m + r ]. So far, we obtain all the A im coefficients. obtain To obtain the market return volatility, we apply Ito s Lemma to Equation (B18) and dp t P t = [µ d (A 2m κ 1 V1 + A 3m κ 2 V2 ) + (ϕ + αa 1m )X t +( 1 2 A2 1mφ 2 xδ x A 2mσ A 2m κ 1 A 1m σ dx φ x δ x A 2m σ 1 σ dv )V 1t +( 1 2 A2 1mφ 2 x(1 δ x ) A 3mσ A 3m κ 2 A 1m σ dx φ x (1 δ x ) A 3m σ 2 σ dv2 )V 2t ]dt +φ d V1t δ d + V 2t (1 δ d )db t + σ dc V1t δ c + V 2t (1 δ c )dz 1t +(σ dx A 1m φ x ) V 1t δ x + V 2t (1 δ x )dz 2t +(σ dv A 2m σ 1 ) V 1t dw 1t + (σ dv2 A 3m σ 2 ) V 2t dw 2t = [c 3 + c 4 X t + c 5 V 1t + c 6 V 2t ]dt + c 1 V 1t + c 2 V 2t dz t, where c i (i = 1 to 6) are constants, dz t is a new Brownian motion defined accordingly, and hence the variance of the price process is V t = c 1 V 1t + c 2 V 2t, with c 1 = φ 2 dδ d + σ 2 dcδ c + (σ dx A 1m φ x ) 2 δ x + (σ dv A 2m σ 1 ) 2, c 2 = φ 2 d(1 δ d ) + σ 2 dc(1 δ c ) + (σ dx A 1m φ x ) 2 (1 δ x ) + (σ dv2 A 3m σ 2 ) 2, (B26) 6

7 and the parameters for the drift term are c 4 = ϕ + αa 1m, c 5 = ( 1 2 A2 1mφ 2 xδ x A 2mσ A 2m κ 1 A 1m σ dx φ x δ x A 2m σ 1 σ dv ), (B27) Q.E.D. c 6 = ( 1 2 A2 1mφ 2 x(1 δ x ) A 3mσ A 3m κ 2 A 1m σ dx φ x (1 δ x ) A 3m σ 2 σ dv2 ). B.3 Solutions to g 1 and g 1m Note that the derived solutions depend on the approximation constant g 1, which can be solved endogenously. Given the model parameters, we can compute the unconditional mean of consumption-wealth ratio as a function of the parameters, ( ) { C 1 g 1 = E = β ψ exp {A a } exp W 4 A2 1aφ 2 ( V 1 δ x + V } 2 (1 δ x )) x α { exp 2κ V 1 1 log(1 A } { 2a ) exp 2κ V 2 2 log(1 A 3a σ1 2 2κ 1 /σ1 2 σ2 2 2κ 2 /σ2 2 Note that the A ia s on the right hand side are also functions of g 1. } ). (B28) Substituting A ia as function of g 1 into Equation (B28), we obtain a nonlinear function in terms of g 1 only, and hence g 1 can be solved in terms of the fundamental parameters of the model, and can be computed numerically with many available algorithms. Similarly, we can solve g 1m endogenously based on dividend-price ratio given as ( ) { D 1 g 1m = E = exp {A m } exp P 4 A2 1mφ 2 ( V 1 δ x + V } 2 (1 δ x )) x α { exp 2κ V 1 1 log(1 A } { 2m ) exp 2κ V 2 2 log(1 A } 3m ). (B29) σ1 2 2κ 1 /σ1 2 σ2 2 2κ 2 /σ2 2 This can be solved numerically as above. Q.E.D. B.4 Predictability of variables The regressors of the three regressions given in Equation (14)-(16) all have the generic functional form of dy t = [a + a 1 X t + a 2 V 1t + a 3 V 2t ]dt + b 1 V 1t + b 2 V 2t dz t, 7

8 given in Equation (17) where dy t corresponds to excess return d ln P t + Dt P t r f dt, consumption growth d ln C t and dividend growth d ln D t, respectively. For stock market excess return, we have a 1 = c 4 + r 1 + g 1m A 1m, a 2 = c 5 c r 2 + g 2m A 2m, a 3 = c 6 c r 3 + g 3m A 3m, where c 1, c 2, c 4, c 5 and c 6 are defined in Equations (B26) and (B27). For consumption growth, we have For dividend growth, we have a 1 = φ, a 2 = φ2 d δ d + σ 2 dc δ c + σ 2 dv + σ2 dx δ x 2 (B3) a 1 = 1, a 2 = δ c 2, a 3 = 1 δ c. (B31) 2, a 3 = φ2 d (1 δ d) + σdc 2 (1 δ c) + σdv2 2 + σ2 dx (1 δ x). 2 (B32) We want to show Equations (A13). Given Equations (B3) and (B18), and denoting Cov(x, y) < x, y >, and pd t p t d t, we have where = = < = t+τ t t+τ t t+τ t dy s, p t d t > ds < a + a 1 X s + a 2 V 1s + a 3 V 2s, pd t > ds[a 1 < x s, pd t > +a 2 < V 1s, pd t > +a 3 < V 2s, pd t >] ds [ a 1 A 1m σ 2 x 2α e αs + a 2 A 2m σ 2 1 V 1 2κ 1 e κ 1s + a 3 A 3m σ 2 2 V 2 2κ 2 e κ 2s ] σ 2 x = φ 2 x[ V 1 δ x + V 2 (1 δ x )] Integrating the above equation, we obtain Equation (A13), where [ σx 2 Cov( τ y, p d) = a 1 A 1m 2α (1 σ 2 2 e ατ ) + a 2 A V 1 1 2m (1 e κ1τ ), 2κ 2 1 σ 2 +a 3 A V ] 2 2 3m (1 e κ2τ ) 2κ 2 2 Var(p d) = A 2 σx 2 1m 2α + σ 2 V A m + A 2 σ 2 V 2 2 3m, 2κ 1 2κ 2 (B33) (B34) 8

9 with σ 2 x = φ 2 x[ V 1 δ x + V 2 (1 δ x )] which are the same given in the text. We have used the unconditional covariance: for i = 1, 2. < X t, X s > = σ2 x 2α e α t s < V it, V is > = σ2 i V i 2κ i e κ i t s Similar computation applies to obtain Equation (A17). Q.E.D. B.5 Predictability of volatilities First, we prove Equation (A16). To do so, we apply the following approximation: 1 τ exp(x s )ds exp( 1 τ x s ds) (B35) for any process x s. This is equivalent to approximating the arithmetic mean by the geometric mean. The approximation is good when the variation of x t is small, which is true for our variance processes because the magnitude is generally in the order of , and the variation of log V t is within 1. Applying the approximation to log V t, we have Hence, 1 τ ln 1 τ which is Equation (A16). Vt dt = 1 τ Vt dt 1 2τ = 1 2τ exp( 1 2 ln V t)dt exp( 1 2τ ln V t dt [ ln V ln V + 1 2τ = Const + 1 2τ V ln(1 + V t V V V t V dt V V t dt, ln V t dt). ] )dt (B36) (B37) Because of the approximation above, we can express the volatilities as an integral of b 1 V 1s + b 2 V 2s over (t, t + τ). Plugging these terms into the definition of the covariance, we then obtain Equation (A17). 9

10 Then we provide the derivation of the AR(1) coefficient. Consider a stochastic process of the form b 1 V 1t + b 2 V 2t, where b 1 and b 2 are constants. Due to independence between V 1 and V 2, the unconditional auto-covariance can be evaluated as b 2 σ 2 V exp( κ 1 τ) + b 2 σ 2 V exp( κ 2 τ) 2κ 1 2κ 2 and the unconditional variance can be evaluated as b 2 σ 2 V b 2 σ 2 V κ 1 2κ 2 Hence, the AR(1) coefficient can be computed easily based on above. Q.E.D. B.6 Derivation of VRP We derive the time t expected future realized variance over time period τ under the risk-neutral probability. The market prices of risk for V 1t and V 2t are λ 3 and λ 4 of Equation (B17), hence the risk premia associated with V 1t and V 2t are where λ 3 σ 1 V1t = ν 1 V 1t, and λ 4 σ 2 V2t = ν 2 V 2t, (B38) ν 1 = 1 γψ 1 γ A 2σ 2 1, and ν 2 = 1 γψ 1 γ A 3σ 2 2. (B39) Hence, the risk-neutral processes for V 1t and V 2t are dv 1t = κ Q 1 ( κ 1 κ Q 1 dv 2t = κ Q 2 ( κ 2 κ Q 2 V 1 V 1t )dt + σ 1 V1t dw Q 1t, V 2 V 2t )dt + σ 2 V2t dw Q 2t, (B4) where the risk-neutral mean-reversion coefficients for V it are defined as κ Q i = κ i ν i (B41) for i = 1, 2. In order for well-defined risk-neutral processes in Equation (B4), we need to have κ Q i s to be positive such that ν 1 < κ 1 and ν 2 < κ 2. (B42) 1

11 Now we compute the squared VIX, or more generally, variance swap rate V S t with maturity τ, defined as the risk neutral expectation of the variance. Because the risk-neutral process and the physical process of Equation are both Heston (1993) processes, we obtain Equation VS t = 2 c i (A Q i i=1 + B Q i V it), where the constants A Q i and B Q i (i = 1, 2) are given by A Q i = κ V [ ] i i 1 1 Q e κ i τ κ Q i κ Q i τ, B Q i = 1 Q e κ i τ κ Q i τ. (B43) Q.E.D. B.7 The GMM test First, it will be useful to see why we can assume V 1 = V 2. In our model, the combination, V 1t δ c +V 2t (1 δ c ), is the variance of the consumption growth. The relative importance of the two volatility factors in driving consumption variance is characterized by δ c, hence without loss of generality, we can assume V 1 = V 2. This is because, if we have V 1 V 2, we can redefine another latent variable V 2t bv 2t, where b = V 1 V 2, such that dv 2t = κ 2 ( V 1 V 2t)dt + σ 2 V 2tdw 2t (B44) with σ 2 = bσ 2. By adjusting δ c accordingly, the new process match exactly the same variance of the consumption growth. Denote h(θ) as the vector of target moments implied by the model given parameter set θ. We choose 23 target moments as described in the text. Let h T be sample vector from data with size T corresponding to the target moments, and expressed as h T = ϕ(g T ) (B45) with g T 1 T T t=1 x t (B46) where x t is a vector representing market data, the details are given below. The GMM estimator {θ T : T 1} is defined as min θ T [h(θ T ) h T ] W [h(θ T ) h T ] (B47) 11

12 for some positive definite weighting matrix W. If the model is true and data is stationary, then the GMM estimator must be consistent (Hansen 1982). By optimizing the quadratic form of Equation (B47), and substituting Equation (B45) into the first order condition, we obtain A T [h(θ T ) h T ] = A T [ϕ(g(θ T )) ϕ(g T )] =, (B48) with and A T = h (θ T ) θ T W, (B49) D T = h(θ T ) θ T (B5) For a consistent estimator θ T, asymptotically we have Taylor expansion Let A plima T target moments is plim[ϕ(g(θ T )) ϕ(g T )] = dϕ(θ ) dµ plim[g(θ T ) g T ] (B51) and D plimd T, following Zhou (1994), the covariance matrix for the Λ T = 1 T (I D(AD) 1 A) where S is the spectral matrix defined as S [ ] dϕ S dµ j= [ ] dϕ (I D(AD) 1 A) dµ Ex t x t j. Denote J = (h(θ T ) h T )Λ T (h(θ T h T )) which measures the sum of squared errors of target moments, J χ 2 (# ofmoments # of parameters), In addition, if J r is J-statistics with the same covariance matrix for a restricted version of the model, then J r J χ 2 (# of restrictions) where the number of restrictions is the number of parameters that is restricted in one-factor model. Q.E.D. 12

13 B.8 Moment conditions for GMM test In this section, we present the moment conditions for GMM estimation. The 23-dimensional vector h T (θ) in the quadratic form h T (θ)w T (θ)h T (θ) that we choose to minimize are the differences between the model functions and their sample values. The first 15 moment functions are E( c) σ( c) AC1( c) E( d) σ( d) AC1( d) E(r e ) σ(r e ) AC1(r e ) E(r f ) σ(r f ) AC1(r f ) E(p d) σ(p d) AC1(p d) Denote r x as consumption growth c, dividend growth d, excess return r e, risk free rate r f, and price-dividend ratio p d, the above moments are given as σ(r x ) = E[(r x ) 2 ] E(r x ) 2 AC1(r x ) = E[r x,t+1r x,t ] E[r x ] 2 E[r 2 x] E[r x ] 2 where E(r x ) are easy to compute analytically given the processes in the paper. The 16th and 17th moments are E[VRP], σ(vrp), the expectation and standard deviation of variance risk premium (VRP) defined in the text. The 18th to 2th moments are the regression coefficients β s. All the 3 regression coefficients are of the form of β = Cov(r x,t+1, (p t d t )) Var(p t d t ) = E[r x,t+1, (p t d t )] E[r x,t+1 ] E[p t d t ] E[(p t d t ) 2 ] E[p t d t ] 2 where r x are consumption growth, dividend growth, excess return, resp. (B52) The 2th to 23rd moments are the three regression β s of volatility regressions for t = 1 year. Specifically, they are β vol = Cov(ln Vol t,t+τ, (p t d t )) Var(p t d t ) = E[ln Vol t,t+τ (p t d t )] E[ln Vol t,t+τ ] E[p t d t ] E[(p t d t ) 2 ] E[p t d t ] 2 where Vol t,t+τ is given in Equation (A15) and stands for volatility of consumption, dividend, and excess return, resp. With specification of all the moment conditions, and the analyt- 13

14 ical formula of the moments implied by the model that is solved in the paper, the GMM estimation and tests can be carried out as usual (see, e.g., Singleton, 26). We show the 26 elements of the moments in g T as follows. The first 15 moments are: E[r x ], E[rx], 2 E[r x,t+1, r xt ] where r x stands for consumption growth, dividend growth, excess return, risk free rate, and price-dividend ratio. Moments 16 and 17 are: E[VRP t ], E[VRP 2 t] where VRP is the variance risk premium. Moments 18 to 2 are: E[r x,t+1, (p t d t )] where r x stands for consumption growth, dividend growth, and excess return. Moments 21 to 26 are: E[log Vol xt ], E[(log Vol xt ), (p t d t )] where x stands for consumption, dividend, and excess return. The data can be obtained through quarterly data regression r x,t+1 = α + βr x,t + ϵ x,t and annual expected volatility Vol xt are obtained from Vol xt = 4 ϵ t+k (B53) k=1 Finally, the form of function h T = ϕ(g T ) that links the target functions h T and the moments g T, as well as its first-order derivative matrix are elementary, and can be obtained from authors upon request. Q.E.D. B.9 Accuracy of the Log-linear Approximation To show that the log-linear approximation (which is accurate when ψ = 1) is accurate enough for the parameter values of interest, we take a three step approach. First, we show 14

15 the standard deviation of the log consumption-wealth ratio is small. Second, we show that the second factor of the two-factor volatility model contributes less than 2% to it. Third, we show that the exact solution of a one-factor with the same magnitude of the standard deviation is very close to the log-linear approximation. First, extending the approximation in discrete-time models by many, Chacko and Viceira (25) show that the approximation works for continuous models too as long as the standard deviation of the log consumption-wealth ratio does not vary too much around its unconditional mean. Specifically, the approximation is a Taylor expansion of the consumption-wealth ratio around its unconditional mean level, denoted as g 1, C t W t = e c t ω t e E[c t ω t ] + e E[c t ω t] [(c t ω t ) E(c t ω t )], g 1 g 1 log g 1 + g 1 log(c t /W t ). (B54) This implies that a small enough standard deviation of log(c t /W t ) yields a good approximation. In Panel A of Table 1, we, like Chacko and Viceira (25), show that the standard deviation of this ratio is indeed small at less than 1.8% for a range of preference parameters, and is much smaller for ψ closer to 1 (this is not surprising as the approximation is accurate when ψ = 1). In particular, for our model parameterization, the standard deviation is 1.6%. Second, the relative contribution of the second volatility factor to the total standard deviation of the log consumption wealth ratio is small, as shown by the results in Panel B of Table 1. This means that the approximation error for our two factor model is almost the same as a one-factor model. Finally, we have to show that the approximation error of a one-factor model is indeed small for the parameter values of interest. To do so, we design a one-factor version of our model with the one factor calibrated to the first volatility in our model and provide the exact solution. (Ideally, we want to compare the exact solution of our two-factor model to the linear approximation. But that is too complex to solve.) The one-factor model is a non-trivial version of the two-factor one, dc t C t = µdt + V t dz t, dv t = κ( V V t ) + σ V t dw t. We calibrate the parameters to match the first two moments of consumption growth. The 15

16 value function in steady state can be written as It can be shown that the solution for G(V ) follows an ODE as [ 1 ϵ (β βeϵg(v ) ) + (1 γ)(µ γ ] 2 V ) +κ( V V ) dg(v ) dv +1 2 σ2 V G(Vt) C1 γ t J(V t, C t ) = e 1 γ. (B55) ( dg(v ) dv ) σ2 V d2 G(V ) dv 2 =, (B56) where ϵ = 1/ψ 1. We solve this ODE numerically and report the results in Figure 1 with 1 γ both the function values and their differences (errors). It is seen that the solution is almost the same as the linear approximation for the parameter range we consider. B.1 Monotonicity of the Aggregator Theoretically, it is very important to note that the log-linear approximation of the aggregator f should be an increasing function of J. Otherwise, it will be in violation of the monotonicity axiom of preferences (see, e.g., Skiadas, 29, Chapter 6.3). But this is not always the case for all possible parameter values, which is a drawback of certain approximations. However, it should and must be so in the domain of interest of the state variables. Indeed, based on the partial derivative f J of (B2), we know that the variation of f J f J = (θ 1)βG βθ, is driven only by βg, which is the consumption wealth ratio based on (B8). In order to check whether f J > for the relevant state variables, we need to verify whether f J is positive for the reasonable range of consumption wealth ratio, βg, which has mean value equal to g 1. Based on (B8), the standard deviation of βg can be computed as σ cw = A 2 1σ 2 x + A 2 2σ 2 v1 + A 2 3σ 2 v2 g 1 1 ψ 1 γ where σ x, σ v1, σ v2 are unconditional standard deviations of state variables X, V 1, V 2. As a result, we only need to check the positivity of f J in the range of βg (g 1 2σ cw, g 1 + 2σ cw ) of interest. Figure 1 shows the numerical values f J in terms of the number of standard deviation from the mean of βg. Within the range of our interest, f J is indeed positive as it should. 16

17 Table 1: Unconditional Standard Deviation of the Log Consumption-Wealth Ratio This table shows the consumption-wealth ratio variability around its long term mean level as well as the percentage contribution of the components to the total variation. It shows that the standard deviation of log consumption wealth ratio is less than 2%, hence the approximation of log-linearization is a good one. In addition, the new factor contribution to this variability is small, with less than 1%, due to its short-run nature. EIS ψ γ Standard Deviation of log C-W Ratio (%) γ Contribution by the New Factor (%)

18 Figure 1: Log-linear Approximation vs Numerical Solution The figure plots the log-linear approximated vs. exact numerical solution for G(V ) of Equation (B56). The parameters are: ψ = 1.5, γ = 1, β =.1, µ =.2, κ =.35, V =.4, σ =.26, which are designed to match the unconditional moments of consumption growth and its volatility. Loglinear Approximation vs Numerical Solution 1 Error: Loglinear Numerical Log linear Numerical V x V 6 8 x

19 Figure 2: Monotonicity of the Aggregator The figure plots the partial derivative of aggregator f with respect to J, f J, vs. the number of standard deviation from the mean value of f J. The parameters are taken from Table 3 of the paper..4 f J Standard Deviation 19