Using Long-Run Consumption-Return Correlations to Test Asset Pricing Models : Internet Appendix

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1 Using Long-Run Consumption-Return Correlations to Test Asset Pricing Models : Internet Appendix Jianfeng Yu University of Minnesota April 0, 0 Proof. [For Equation 6]: To derive the log-linear approximation to the price-dividend ratio and asset returns, let m t = log M t be the log intertemporal marginal rate of substitution IMRS. Plugging in the log-linear approximation to returns r t+ k 0 +g d,t+ + κ z t+ z t and the linear approximation to the log price-dividend ratio z t a 0 + a s t + a δ t in the Euler equation, I obtain = E t exp m t+ + r t+ γ φ s s + log β + k 0 + µ c γ +a 0 κ a 0 + a κ φ s s a κ σδ = exp + γ φ s a + a κ φ s s t + a κ ρ δ a + ρ δ δ t a κ λ s t γ + λ s t + + a κ λ s t γ + λ s t + a κ σ cδ. Replacing the sensitivity function λ s with its linear approximation λ s a λ s s max in the above equation, and setting the coefficients in front of the state variables to zero, it follows that a = ρ δ κ ρ δ, The parameters κ and κ 0 are determined endogenously as a function of mean price-dividend ratio by κ = expe[zt] +expe[z, and κ t] 0 = log κ κ log κ.

2 and a can be determined as the unique positive root of the following quadratic equation: 0 = a λ κ κ a κ φ s + a λσ c,δ κ [ + a κ ] a λ κ + γ + 4γκ a + a 0.5 λ [ + a κ ] σ c,δ γ + a λ γ γ φ s γ. 0.5 Finally, a 0 can be determined by the equation a 0 = κ σ c a κ γ + s + a κ a κ γ + a λ s max + a κ + γ + a κ γ a λ s max + a κ σ c,δ 0.5σ c +γ φ s s + log β + k 0 + µ c γ +a κ φ s s a κ σ δ 3 Plug this linear approximation on the price-dividend ratio back into Campbell-Shiller loglinear approximation to returns to obtain r t+ κ 0 + µ c a 0 + a 0 κ + a κ φ s s + ρ δ a + a κ ρ δ δ t + a κ φ s a s t + + a κ λ s t ɛ c,t+ + + a κ ɛ δ,t+ St κ 0 + µ c a 0 + a 0 κ + a κ φ s s + a κ φ s a + log + ρ δ a + a κ ρ δ δ t + + a κ λ s t ɛ c,t+ + + a κ ɛ δ,t+ κ 0 + µ c a 0 + a 0 κ + a κ φ s s + a κ φ s a log + a κ φ s a S t + + a κ λ s t ɛ c,t+ + + a κ ɛ δ,t+. Notice that λ s ; hence, it is natural to choose 0 < a λ < to approximate the sensitivity function. Hence, the coefficient a λ κ κ term in the quadratic equation satisfies 0.5 a λ [ + a κ ] σ c,δ γ + a λ γ. in the quadratic equation is negative. Further, the constant γ φ 0.5σ c γ [ ] = κ,δ κ ρ δ + a λ γ. If we assume σ c,δ σ, then the constant term in the quadratic equation is positive. Hence, there is a c unique positive root for equation. Notice that ɛ c,t is the innovation in c t, and ɛ δ,t is the innovation in d t minus the innovation in c t. As long as the innovations in c t and the innovations d t are positively correlated i.e., consumption growth and dividend growth are positively correlated, σ c,δ σ > holds. c

3 Letting ˆα = κ 0 + µ c a 0 + a 0 κ + a κ φ s s + a κ φ s a log 4 β S = a κ φ s, 5 β c = + a κ 6 β δ = + a κ = + ρ δ κ = κ > 0 7 κ ρ δ κ ρ δ and further approximating λ s t with λ s = obtained from the defintion of λ s t in CC, it follows that r t+ ˆα β S S t + β c ɛ c,t+ + β δ ɛ δ,t+. 8 Moreover, the habit level X t can be further approximated as an exponentially weighted average of past consumption, X t φ s φ s φ k sc t k, 9 k= where φ s is the measure of habit persistence. Since φs φ s k= φk s =. Substitute equation 9 back into the definition of the surplus ratio, approximate to the first order, and simplify to obtain S t S t j= φ j s g c,t+ j. 0 Plugging equation 0 into equation 8, I can write down excess asset returns as follows: r ex t+ α β S j= φ j s g c,t+ j + β c ɛ c,t+ + β δ ɛ δ,t+, where the constant α = ˆα r f. Thus, the derivation for equation 6 is complete. Proof. [For Propositions &]: First, replacing each term in equations 6 and by its spectral representation, and noting that the resulting equations are valid for all t, I obtain the following equations for the orthogonal increment processes Z gc, Z r, Z ɛc, and Z ɛδ in the spectral representations of {g t }, {rt ex }, {ɛ c,t }, and {ɛ δ,t }: dz gc λ = dz ɛc λ dz r λ = β S φ j s e ijλ dz gc λ + β c dz ɛc λ + β δ dz ɛδ λ. j= 3

4 Notice that j= φj s e ijλ = e i λ φ se i λ. Solve for dz gc λ and dz r λ to obtain dz gc λ = dz ɛc λ βs e i λ dz r λ = φ s e + β i λ c dz ɛc λ + β δ dz ɛδ λ. It follows that the multivariate spectrum is given by πf λ = πf λ = β S e i λ φ s e + β i λ c + βδ σδ e +Re κ i λ φ s e + β i λ c β δ σ cδ βs e i λ πf λ = φ s e + β i λ c + β δ σ cδ βs φs e i λ = + φ s φ s cos λ + β c + β δ σ cδ. For the cospectrum C sp λ, the real part of the cross spectrum f λ, βs φ s cos λ πc sp λ = + φ s φ s cos λ + β c + β δ σ cδ. Therefore, the derivative of the cospectrum is C sp λ = β S sin λ + φ s φ s cos λ + φ s sin λ cos λ φ s + φ s φ s cos λ = β S σ c sin λ + φ s φ s cos λ φ s 0, and the portion of covariance contributed by components at frequency λ is increasing in the frequency λ. By definition, the coherence and the phase are, respectively, h λ = tan φ λ = f f f β S sinλ +φ s φ s cosλ σ ɛ βs φ s cosλ +φ s φ s cosλ + β c σ c + β δ σ cd. 4

5 At the frequency λ = 0, the cospectrum is φ s C sp 0 = β S + β + φ c + β δ σ cδ s φ s = a κ a κ + β δ σ cδ. φ s Therefore, the low-frequency correlation between consumption growth and asset returns is negative if and only if a κ a κ σ φ s + β cδ δ < 0. By differentiating equation, the sign of the slope of the phase spectrum can be examined. To see this, start with φ λ [ β S φ s cos λ σ c + β c σ c + β δ σ cδ + φ s φ s cos λ ] β S cos λ +β S sin λ [β S sin λ σ c + φ s βc σ c + β δ σ cδ sin λ ], where denotes that both sides of have the same sign. Rearrange and simplify to obtain φ λ a κ φ s { + a κ + φ s + a κ σ cδ + φ s β δ + β δ σ cδ σ c [ a κ a κ φ s + β δ + φ s + a κ φ s ] σ cδ φ s. + a φ s + β δφ s } σ cδ cos λ The inequality above requires the assumption a κ + φ s + a κ + a φ s + β δ + φ s σ cδ σ c > 0, 3 which is true if the correlation between the innovations of return and consumption is positive. Thus, the phase spectrum is increasing as long as a κ a κ σ φ s + β cδ δ < 0 and the correlation between the innovations of return and consumption is positive. Proof. [For Proposition 3]: Replacing each term in equations and 3 by its spectral representation, and noting that the resulting equations are valid for all t, I obtain the following equations for the orthogonal increment processes Z gc, Z r, Z x, Z e, Z u, and Z η in 5

6 the spectral representations of {g c,t }, {r ex t }, {x t }, {e t }, {u t }, and {η t }: dz gc = e iλ dz x + σdz η dz x = ρe iλ dz x + ϕ e σdz e dz r = φ ψ κ m ρ κ mϕ e σdz e + ϕ d σdz u. Define A m = φ ψ κ m ρ, rearrange, and solve for dz g c and dz gc to obtain Thus, the multivariate spectrum is given by dz gc = ϕ eσe iλ ρe dz iλ e + σdz η dz r = κ m A m ϕ e σdz e + ϕ d σdz u. f rr = σξ = κ m A m ϕ e σ + ϕ d σ f gg = ϕ e σ e iλ ρe iλ + σ e iλ f gr = κ ρe iλ m A m ϕ eσ + ϕ d σ ρ ηu = e iλ ρ + ρ ρ cos λ κ ma m ϕ eσ + ϕ d σ ρ ηu Solving for the cospectrum C sp λ, the real part of the cross spectrum f λ, yields C sp λ = cos λ ρ + ρ ρ cos λ κ ma m ϕ eσ + ϕ d σ ρ ηu. Taking the derivative and rearranging the equation yield C sp λ = sin λ ρ + ρ ρ cos λ φ ψ κ m ρ κ mϕ eσ. From the expression of f gr, under the assumption of ρ ηu = 0, I can solve for the phase spectrum φ λ : tan φ λ = sin λ ρ cos λ ρ. 6

7 Thus, taking the derivative, it follows that φ λ d sinλ ρ cosλ ρ dλ cos λ + ρ cos λ sin λ ρ sin λ = ρ cos λ sin λ < 0, where denotes that both sides of have the same sign. Thus, the phase spectrum is always decreasing. Assume that the consumption and return dynamics are given by rt+ ex α β C x t + ɛ r,t+. x t = ρ x x t + ɛ x,t c t = T t + x t T t = T t + ξ t, 4 where β C > 0 captures the countercyclical expected return behavior in the model, ɛ r,t, ɛ x,t, and ξ t are corresponding i.i.d. shocks with corr ξ t, ɛ x,t < 0. This trend-cycle representation of consumption is equivalent to the one-channel long-run risk model. We only need to define ρ = ρ x, µ = E ξ t+, x t ρ x x t, η t+ ξ t+ µ+ɛ x,t+, and σ e t+ ρ x ɛ x,t+ ϕ eσ. Then, consumption growth can be represented as: g c,t+ = µ + x t + ση t+, and x t+ = ρx t + ϕ e σe t+. Hence, when the cyclical component is low i.e., x t < 0, the expected consumption growth rate in the BY model x t is high. Moreover, corr η t, e t = 0 implies that corr ξ t+, ɛ x,t+, = σɛ x σ ξ < 0. Proposition 4: It can be shown that consumption and excess returns has the following property: dc sp λ < 0 for λ 0, π dλ if and only if Proof. below: β C σ + ρ x + σ ɛ x ξ,ɛ x + σ ɛr,ɛ x + ρ x < 0. 5 ρ x [For Proposition 4]: First, rewrite the consumption and return dynamics as g c,t+ = ξ t+ + x t+ x t x t+ = ρ x x t + ɛ x,t+ rt+ ex = α β C x t + ɛ r,t+, where β C > 0. 6 Then, replacing each term in equation 6 by its spectral representation, and noting that the resulting equations are valid for all t, I obtain the following equations for the orthogonal increment processes Z gc, Z r, Z x, Z ɛ x, and Z ɛr in the spectral representations of {g c,t }, {rt ex }, 7

8 { x t }, {ɛ x,t }, and {ɛ r,t }: Rearranging the above equations yields dz gc = dz ξ + e iλ dz x dz x = ρ x e iλ dz x + dz ɛ x dz r = β C e iλ dz x λ + dz ɛr λ. dz gc = dz ξ + e iλ ρ x e iλ dz ɛ x dz r = β Ce iλ ρ x e iλ dz ɛ x + dz ɛr. Thus, the cross spectrum between consumption and excess returns is given by β C e iλ ρ x β C e iλ + β C πf λ = σ ρ x cos λ + ρ x ξ,ɛ x + σ ρ x cos λ + ρ x ɛ x ρ x e iλ e iλ + ρ x + ρ x cos λ + ρ x σ ɛr,ɛ x + σ ξ,ɛr. Solving for the cospectrum C sp λ, the real part of the cross spectrum f λ, it follows that πc sp λ = β C cos λ ρ x σ ξ,ɛ x β C cos λ σ + + ɛ x ρ x cos λ σ ɛr,ɛ x +σ ρ x cos λ + ρ x ξ,ɛr. Taking the derivative of the above equation and rearranging the resulting equation yield C sp λ β C sin λ σ ξ,ɛ x + β C sin λ σ ɛ x + ρ x sin λ + sin λ σ ɛr,ɛ x ρ x cos λ + ρ x + ρ x sin λ β C cos λ ρ x σ ξ,ɛ x + β C cos λ σ ɛ x + ρ x cos λ σ ɛr,ɛ x = ρ x sin λ β C σ + β ɛ x C sin λ σ ξ,ɛ x ρ x + σɛr,ɛ x sin λ ρ x + ρ x = [ ρ x sin λ ] ] [β C σ + ρ x + σ ɛ x ξ,ɛ x + σ ρ x ɛr,ɛ x + ρ x. Thus, dc sp λ dλ < 0 for λ 0, π if and only β C σ ɛ x + σ ξ,ɛ x +ρ x ρ x + σ ɛr,ɛ x + ρ x < 0. Moreover, if the correlation between innovation in the trend and the cycle is positive, the derivative of the cospectrum can be positive, and hence the cospectrum can be increasing from low to high frequencies. 8