Risks For The Long Run And The Real Exchange Rate: Technical Appendix Riccardo Colacito Mariano M. Croce∗

Abstract We propose a general equilibrium model that can explain a wide range of international finance puzzles. This includes the degree of volatility of the growth of the US dollar combined with the low correlation of consumption across countries and the high correlation of international stock markets despite the lack of fundamentals’ correlation. This is obtained by combining a model of highly correlated long run risks with Epstein and Zin (1989) preferences. We conduct an empirical analysis focusing on the US and the UK. The data confirm that these two countries have highly correlated long-run growth perspectives. This technical appendix complements the paper ‘Risks for the Long Run and the Real Exchange Rate’. This draft: November 20, 2007.

∗

Both authors are affiliated with the University of North Carolina at Chapel Hill, Kenan-Flagler Business School.

Appendix A. Derivation of moments We express the relevant moments of the variables of the model as a function of the set Υ of  hf deeper parameters defined as Υ = θ, ψ, δ, ρx , σc , σx , ρhf . We assume that: c , ρx εic,t ∼iid N 0, σc2

∆cit+1 = xit + εic,t+1 , xit+1 = ρix xit + εix,t+1 ,





εix,t ∼iid N 0, σx2 ,

∀i ∈ {h, f }

(A.1)

The Euler equation for the asset that pays one unit of the consumption bundle at each period is:  i  i Rc,t+1 1 = Et Mt+1 with log Mt+1 = θ log δ− ψθ log



i Ct+1 Cti



(A.2)

i +(θ−1) log Rc,t+1 . The return on the consumption asset

i = can be expressed in terms of the price-consumption ratio vc,t as Rc,t+1 {h, f }. This implies that (A.2) can be written as:

i θ (vc,t )

h

= Et δe

∆cit+1 (1−γ)

(1 +

i vc,t+1 )θ



δ = Et δ + 1−δ

exp ∆cit+1 , ∀i ∈

i

Linearizing (A.3) around the steady state of vc,t defined as vci,ss = i vc,t

i vc,t+1 +1 i vc,t

(A.3) δi , 1−δ i

we obtain:

   1 i i 1− ∆ct+1 + δvc,t+1 ψ

that can be solved forward delivering the price-consumption ratio as a function of the state variable xit nd of the deeper parameters of the model: i vc,t = αci + βci xit

αci

δi 1−δ i

βci

δi 1−δ i



1 ψi 1−ρix δ i

1−

(A.4)



with = and = , ∀i ∈ {h, f }. Log-returns on the assets that pay a stream of consumption follow immediately from (A.1 ) and (A.4): i rc,t+1

      1 i 1 1 = −log(δ) + xt + δ 1 − ix,t+1 + ic,t+1 ψ ψ 1 − ρx δ

1

(A.5)

Also the log-stochastic discount factor is: mit+1 = logδ −

δ (1 − γΨ) i 1 i xt − γic,t+1 +  ψ ψ(1 − ρx δ) x,t+1

(A.6)

The log-risk-free rate rfi is obtained as the solution of:    θ i θ i = −Et δ exp − ∆ct+1 + (θ − 1) rc,t+1 ψ "    2 # 1 1 1 1 = − log(δ) + [γ]2 σc2 + δ 1 − σx2 + xit 2 ψ 1 − ρx δ ψ

rfi

(A.7)

The exchange rate depreciation, by no-arbitrage, is: ∆et+1 ≡ πt+1 = mft+1 − mht+1 The following moments follow directly from the previous formulas and from the assumption that both countries share the same parametrization: V ar(π) = 2 corr(mht , mft ) = i V ar(rc,t+1 ) =





2 2 1 − ρhf Γ0 σx2 + 1 − ρhf γ σc x c

2 Γ0 ρhf x σx + Γ0 σx2 + Γ0 σx2 + σc2

2 γ 2 ρhf c σc γ 2 σc2

f h 2 hf 2 Cov(rc,t+1 , rc,t+1 ) = Γ0 ρhf x σx + ρc σc i Cov(∆cit+1 , rc,t+1 ) = Γ1 + σc2 f hf 2 Cov(∆cht+1 , rc,t+1 ) = Γ1 ρhf x + ρc σc

where Γ0 =

 2 1 ψ

1 1−ρ2x

+

h

δ(1−γψ) ψ(1−ρx δ)

i2

and Γ1 =

2

σx2 . ψ(1−ρ2x )

(A.8)

Appendix B. Proof of Propositions Proof of Proposition 1. For any choice of the parameters that satisfy ρix 6= 1 and ρix δ i 6= 1 the following three partial derivatives ∂corr(mht , mft )

=

Γ0 σx2 >0 Γ0 σx2 + γ 2 σc2

∂ρhf x
2 2 hf 2Γ0 γ 2 ρhf σx σc ∂corr(mht , mft ) x − ρc = − ∂Ψ ψ (Γ0 σx2 + γ 2 σc2 )2 h i δ 3 (1−γψ)2 ρx 2 2 hf f + 2 γ 2 (ρhf h x − ρc )σx σc ∂corr(mt , mt ) (1−ρx δ)3 (1−ρ2x )2 = ∂ρx ψ 2 [Γ0 σx2 + γ 2 σc2 ]2

(B.1) (B.2)

(B.3)

exist and are well defined. (B.1) is positive for all the values of the parameters that respect the two conditions, implying that the correlation of the two stochastic discount factors is hf hf strictly increasing with respect to ρhf x . When ρx = ρc (B.2) is always zero, meaning that changes in Ψ, γ or δ do not affect the correlation of the two stochastic discount factors. If 1e hf e 1−2ρx δ+δ2 ρhf x 6= ρc , this derivative is zero only if ψ = γ δ, where δ = δ 2 (1−ρ2x ) . In particular, when 1e 1e hf ρhf x > ρc the sign of the derivative is positive for Ψ > γ δ and negative for ψ < γ δ. Notice that lim− γ1 δe ≥ γ1 and lim− lim− γ1 δe = γ1 ρx →1 δ→1

δ→1

So, for a high persistence of the long run component and an individual discount factor close to one, the minimum of the cross correlation of the two discount factors is achieved for ψ = γ1 , that is when the Epstein-Zin preferences collapse to the standard CES utility function. (B.3) hf hf hf hf hf is always positive when ρhf x > ρc , negative when ρx < ρc and equal to zero when ρx = ρc . hf hf hf Therefore if ρhf x > ρc , the minimum is achieved for ρx =0.  When ρx > ρc , (B.1),  (B.2) 1e and (B.3) imply the existence of one unique minimizer at ρx = 0, ρhf x = 0, ψ = γ δ .

Proof of Proposition 2. The partial derivative of the variance of the growth of the exchange rate with respect to ρhf x exists and is well defined provided that ρx 6= 1 and ρx δ 6= 1: ∂V ar(π) ∂ρhf x

"   2 # 2 1 1 δ (1 − γψ) = −2 + σx2 ≤ 0 ψ 1 − ρ2x ψ (1 − ρx δ)

(B.4)

In particular, this derivative is always negative, implying that the volatility of the log3

depreciation rate achieves its minimum when ρhf x = 1.

Proposition 3. Let σ mh and σ mf be the lower bounds on the volatilities of the stochastic discount factors in the home and in the foreign country, respectively. If markets are complete both domestically and internationally and the volatility of the depreciation of the home currency is strictly positive, then the lower bound on the correlation of stochastic discount factors is:

ρmh ,mf =

                    

σ2 σ mh σ2

mh

mh √

2 −σπ

2 σ mh −σπ

+σ 2

mf

2 −σπ

2σ mh σ mf σ2 σ mf

mf √

2 −σπ

2 σ mf −σπ

if

σ mf ≤ σ mh − σπ2

if

σ mf ∈ (σ mh − σπ2 , σ mh + σπ2 )

if

σ mf ≥ σ mh − σπ2

Proof of Proposition 3. Assuming complete markets, the problem to be solved takes the following form: choose

σmh , σmf

to

min ρmh ,mf

s.t.

σmh ≥ σ mh σmf ≥ σ mf

where σ mh and σ mf are the Hansen and Jagannathan (1991) bounds. Attaching Lagrange multipliers λh and λf to the two constraints, the first order necessary conditions are: ∂ρmh ,mf − λh = 0 ∂σmh ∂ρmh ,mf − λf = 0 ∂σmf along with the two constraints. Four cases must be taken into account. Case 1: λh = λf = 0. This implies σπ2 = 0, a contradiction.

4

Case 2: λh = 0, λf > 0. The system of first order conditions implies:   σmf = σ mf    2 σmh = σ 2mf − σπ2 ≥ σ 2mh ⇒ σ 2mf ≥ σπ2 + σ 2mh 2 +σ 2  σ 2 f −(σ 2 f −σπ ) π  f f m m  q  2σ2 σ2 −σ2 − λ = 0 ⇒ λ > 0 mf

σ 2mf h

σπ2 f

π

mf

σ 2mh ,

≥ + the minimum is achieved for σmf = σ mf and σmh that is, if Case 3: λ > 0, λ = 0. This case is symmetric to the previous one. Case 4: λh > 0, λf > 0. The system of first order conditions implies:

q = σ 2mf − σπ2 .

 σmf = σ mf       σmh = σ mh

2 +σπ mh 2 2σ f σ mh m 2 σ 2 h −σ 2 f +σπ m m 2 2σ h σ mf m

σ2

mf

     

−σ 2

− λf = 0 − λh = 0

that is, if σ mf ∈ (σ mh − σπ2 , σ mh + σπ2 ) the minimizer is the Hansen and Jagannathan (1991) bound itself. Combining the four cases, we obtain:

ρmh ,mf =

                    

σ2 σ mh σ2

mh

mh √

2 −σπ

2 σ mh −σπ

+σ 2

mf

2 −σπ

2σ mh σ mf σ2 σ mf

mf √

2 −σπ

2 σ mf −σπ

if

σ mf ≤ σ mh − σπ2

if

σ mf ∈ (σ mh − σπ2 , σ mh + σπ2 )

if

σ mf ≥ σ mh − σπ2

That concludes the proof.

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Appendix C. Numerical algorithm for the approximation of the price dividend ratios We describe the procedure to numerically approximate the price-consumption and pricedividend ratios for the most general case in which stochastic volatility is in the model, too. We discretize the support of x and σ into Ix and Iσ points respectively, to get I = Ix · Iσ nodes (x, σ)i , ∀i = 1, ..., I. In what follows, we will refer to xi and σi as the first and the second entry of (α, s)i respectively. Specify J known linearly independent basis functions φj ((x, σ)i ), j ∈ {1, ..., J}. In our solution, we employ a third order polynomial in x combined with a first order polynomial in σ, implying that J = 6. The goal is to find basis coefficients cj , j = 1, ..., J that best approximate the Euler equation vci

= V ((x, σ)i ) ≈

J X

cj φj ((x, σ)i ) =

j=1

J X

cj φj,i

(C.1)

j=1

∀i = 1, ..., I or, in the equivalent matrix notation: vc ≈ Φc where vc is the I × 1 vector of approximated value functions at each node, Φ is the I × J collocation matrix and c = [c1 , ..., cJ ]0 is the vector of approximation coefficients. We also discretize the support of the three shocks in K1 , K2 and K3 points and denote w1,k , w2,k and w3,k the approximated probability masses associated to each of the nodes. The shocks are assumed to independent. Under these assumptions, we get for each node i ∈ {1, ..., I}: "

vc,i

    1 0 ∆ci,εc = δ w1,εc w2,εx w3,εσ exp θ 1 − ψ εc ∈K1 εx ∈K2 εσ ∈K3 !θ  θ1 J X
0  1+ cj φj x0i,εx , σi,ε σ X X X

j=1

6

(C.2)

where 0 σi,ε = σ + ν1 (σi − σ) + σw εσ σ 0 ∆c0i,εc = xi + σi,ε ε σ c 0 x0i,εx = ρx xi + ϕe σi,ε ε σ x

We can now use the following algorithm to solve the Euler equation recursively: 1. guess an initial vector of basis coefficients c1 2. for each node (s, α)i compute the right hand side of equation (C.2) using c1 and call v(c1 ) the outcome 3. solve for c2 = (Φ0 Φ)−1 Φ0 v(c1 ) 4. replace c1 with c2 and iterate until convergence. Having solved for the price-consumption ratio vc , we can solve the Euler equation for the price-dividend ratio in a similar way: vd,i = δ θ

X X X X

 w1,εc w2,εx w3,εσ w4,εd exp m0i,(εc ,εx ,εσ )

εc ∈K1 εx ∈K2 εσ ∈K3 εd ∈K1

1+

J X

0 dj φj x0i,εx , σi,ε σ

!


(C.3)

j=1

where m0i,(εc ,εx ,εσ )

  θ = θ−1− ∆c0i,εc (θ − 1) log ψ

7

0 1 + vi,(ε c ,εx ,εσ )

vi

!

Appendix D. Spectral analysis i h Denoting as Yt = ∆cht ∆cft the vector of consumption growth in the US and the UK at time t, the population spectrum at frequency ω is SY (ω) =

k=+∞ 1 X Γk e−ikω 2π k=−∞

(D.1)

  0 where Γj = E Yt − Y Yt−j − Y and Y = E[Yt ]. We follow Hamilton (1994) in estimating Γj , ∀j with their sample counterparts: PT bj = Γ

t=j+1

Yt − Y T




Yt − Y


0

and smoothing (D.1) with a 10 period Bartlett (1964) window SbY (ω) =

 k=10  1 X |k| b −iωk Γk e 1− 2π k=−10 10

(D.2)

As a measure of the covariance explained at different frequencies, we consider the coherence, defined as: 2 b S∆ch ,∆cf (ω) 2 = K∆c h ,∆cf (ω) Sb∆ch (ω) Sb∆cf (ω)

Appendix E. Details of the estimation Appendix E.1. Log-linearization procedure To better account for the non-linearities in the dynamics of asset returns, we approximate the model around a stochastic steady state. The formulas derived in this appendix are then used to construct the relevant moments that are used in the GMM estimation.

8

Let εt ≡

h

εc,t εd,t εx,t

i

∼ N (0, S)

The Campbell and Shiller (1988) log linearization of the returns implies: vc,t = v c +

∞ X

κc Et [∆ct+1+i ] −

i=0

κc ≡

∞ X

κic Et [rc,t+1+i ]

(E.1)

i=0

exp(vc )

(E.2)

1 + exp(vc )

When Epstein and Zin (1989) preferences are adopted, a log-linearization of the first order conditions of the representative agent implies: 1 γ − 1/ψ xc,t − κc x,t+1 − γc,t+1 ψ 1 − ρκc 1 1 − 1/ψ x,t+1 + c,t+1 = rc + xt + κc ψ 1 − ρκc 1 = r f + xt ψ 1 − 1/ψ = vc + xt 1 − κc ρ

mt+1 = m − rc,t+1 rf,t vc,t

(E.3)

Given the results above, the Euler Equation for the asset that pays consumption (evaluated at xt =0) provides the following non linear equation in κc :  κc = δ exp

1 1− ψ



−1


µ − .5(γ − 1)V ar[c,t+1 + κc (1 − ρκc ) x,t+1 ]



Rewriting the stochastic discount factor mt+1 and the return in vector form, we obtain: 1 mt+1 = m − xc,t + Γm t+1 Ψ h i γ−1/Ψ Γm ≡ −γ 0 −κc 1−ρκc 1 rc,t+1 = rc + xt + Γc t+1 Ψ h i Γc ≡ 1 0 κc 1−1/Ψ 1−ρκc

9

Since ex Et [rc,t+1 ] = −cov (mt+1 − Et [mt+1 ], rc,t+1 − Et [rc,t+1 ]) − .5V (rc,t+1 − Et [rc,t+1 ])

then the following holds: ex Et [rc,t+1 ] = −Γm CΓ0c − .5Γc CΓ0c

By rearranging the definition of the stochastic discount factor, the following holds: 1 (1 − θ) 1 ex E(∆c) + Et [rc,t+1 ] − Vt [mt+1 ] Ψ θ 2θ 1 (1 − θ) 1 = −log(δ) + µ + (−Γm CΓ0c − .5Γc CΓ0c ) − Γm CΓ0m Ψ θ 2θ

rf = E[rf ] = −log(δ) +

We are now able to find the intercept of the log stochastic discount factor: m = θ log δ −

θ µ + (θ − 1) (E[rcex ] + E[rf ]) Ψ

We follow the same strategy in order to find the log-return of the asset that entitles to the stream of dividends: λ − 1/ψ xt 1 − κd ρ 1 λ − 1/ψ x,t+1 + d,t+1 = rd + xt + κd ψ 1 − ρκd

vd,t = v d + rd,t+1

An equivalent vector form representation is: 1 rd,t+1 = rd + xt + Γd t+1 ψ h i Γd ≡ 0 1 κd λ−1/ψ 1−ρκd

10

(E.4) (E.5)

Given the results above, the Euler Equation for the asset that pays dividend (evaluated at xt = 0) provides the following non linear equation in κd : κd = exp {m + µ + .5V ar[(Γm + Γd )vt+1 ]} The equity premium in this case is: ex 0 0 rex d = Et [rd,t+1 ] = −Γm CΓd − .5Γd CΓd

(E.6)

and note that rd = rex d + rf . h Now, define hf t ≡ [t

ft ] and S hf = V (hf ). The exchange rate growth can be written as:

∆et+1 ≡ πt+1

f

h

(m − m ) +

=

xft xht − Ψf Ψh

! + Γπ hf t+1

(E.7)

where Γπ

[Γfm

=

− Γhm ]

The unconditional volatility of the exchange rate growth is:

σπ2

= Γπ S

hf

Γ0π

 + V ar

xh xf − Ψh Ψf

 (E.8)

The correlation of the excess returns expressed in local units can be computed as follows:

→ − ρhf Γfd ]S hf [Γhd rex = [ 0

→ − 0 0]

(E.9)

while the correlation of the excess returns expressed in domestic units is:

h ρhf rex |domestic = [−Γm

Γfd + Γfm ]S hf [Γhd

11

→ − 0 0]

(E.10)

All other moments can be easily computed in a similar manner.

Appendix E.2. Unconditional GMM The GMM in section 3.5 is based on the following 46 unconditional moments:
• cov ∆cit , ∆cit−j − (ρi )j σxi2 /(1 − ρi2 ), ∀i ∈ {h, f } ∧ j ∈ {0, 1, ..., 4}   • cov ∆cht , ∆cft−j − (ρh )j σxh σxf /(1 − ρh ρf ), ∀j ∈ {0, 1, ..., 4}   • cov ∆cft , ∆cht−j − (ρf )j σxh σxf /(1 − ρh ρf ), ∀j ∈ {1, ..., 4}
• cov ∆dit , ∆dit−j − (ρi )j (λi σxi )2 /(1 − ρi2 ), ∀i ∈ {h, f } ∧ j ∈ {0, 1, ..., 4}   • cov ∆dht , ∆dft−j − (ρh )j (λh σxh )(λf σxf )/(1 − ρh ρf ), ∀j ∈ {0, 1, 2}   • cov ∆dft , ∆dht−j − (ρf )j (λh σxh )(λf σxf )/(1 − ρh ρf ), ∀j ∈ {0, 1, 2} • V ar (∆et+1 ) − σπ2 = 0  i   i  • E rf,t − rif , V ar rf,t −

σxi2 Ψi2 (1−ρi2 )

 i  σxi2 i and cov rf,t , rf,t−1 − ρi Ψi2 (1−ρ i2 ) , ∀i ∈ {h, f }

 i

 i i i and V ar r − r − Γd CΓ0d , ∀i ∈ {h, f } rd,t − rf,t − rex d d,t f,t h i
 f f h h • cov rd,t − rf,t , rd,t − rf,t − ρhf rex

• E



We implement a standard two-steps GMM procedure. In the first iteration we use a diagonal weighting matrix. Each element of the diagonal is equal to the inverse of the variance of the sample moment we are interested in. In this way our estimate is not unit sensitive. In the second step we minimize the quadratic criterion using the optimal weighting matrix computed at the point estimate of the first step. The statistics we report are obtained using the optimal weighting matrix computed at the final point estimate of the second step. We calibrate µ and µd using the sample mean of the consumption and dividends growth both for US and UK.

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Appendix E.3. GMM on Euler Equations restirctions The predictable components of consumption growths are constructed according to either equation (4.6) or (4.7) reported in the paper. Innovations to consumption growth and its low frequency components are computed as follows: ic,t+1 = ∆cit+1 − xit ix,t = xit − ρi xit−1 We then use a log-linearized version of the model in order to recover the time series of stochastic discount factors and exchange rates. We calibrate the mean of consumption growth to µ = .02 and the subjective discount factor to δ = .99 for both US and UK. The continuous updating GMM procedure of Hansen, Heaton, and Yaron (1996) is applied to the sample counterparts of the following set of Euler equations   i i Rj,t+1 − 1 = 0, ∀i = {U S, U K} and ∀j = {m, f } • E Mt+1   US UK Rj,t+1 exp∆et+1 − 1 = 0, ∀j = {m, f } • E Mt+1 h
−1 i UK US • E Mt+1 Rj,t+1 exp∆et+1 − 1 = 0, ∀j = {m, f } and of the following exchange rate moment conditions  US  UK • E Mt+1 /Mt+1 − exp∆et+1 = 0 h

2 i UK 2 US • E Mt+1 /Mt+1 − exp∆et+1 =0 When preference parameters and the coefficients governing the dynamics of the predictable components of consumption growth are jointly estimated, we enrich the above system of moment conditions with the appropriate set of orthogonality conditions.

Appendix E.4. GMM on the reduced form of the model The innovations to consumption growth and its predictable component can be computed as in the previous section. The innovation to cash flows is εid,t+1 = ∆dit+1 − λi xit , ∀i ∈ {U S, U K} 13

The log-linearized version of the model discussed in the Technical Appendix is then used to construct the sample counterparts of the following moments conditions: • E





 i i i i rem,t+1 − ref,t − rm,t+1 − rf,t+1 = 0, ∀i ∈ {U S, U K}

  i i = 0, ∀i ∈ {U S, U K} − rf,t+1 • E ref,t   US UK − exp∆et+1 = 0 /Mt+1 • E Mt+1 h

2 i UK 2 US =0 /Mt+1 − exp∆et+1 • E Mt+1 i where rem and refi are the predictions of the model and are computed as shown in Appendix E.1. Also in this case, when preference parameters and the coefficients governing the dynamics of the predictable components of consumption growth are jointly estimated, we enrich the above system of moment conditions with the appropriate set of orthogonality conditions.

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References Bartlett, M. (1964). The spectral analysis of two-dimensional point processes. Biometrika 51, 299–311. Campbell, J. Y. and R. J. Shiller (1988). Stock prices, earnings, and expected dividends. The Journal of Finance XLIII (3), 661–676. Epstein, L. G. and S. E. Zin (1989). Substitution, risk aversion, and the temporal behavior of consumption and asset returns: A theoretical framework. Econometrica 57 (4), 937–69. Hamilton, J. (1994). Time Series Analysis. Princeton University Press. Hansen, L. P., J. Heaton, and A. Yaron (1996). Finite-sample properties of some alternative gmm estimators. Journal of Business and Economic Statistics 14(3), 262–280. Hansen, L. P. and R. Jagannathan (1991). Implications of security market data for models of dynamic economies. Journal of Political Economy 99, 225–262.

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