Here I describe the approximation to the equilibrium conditions and state variable dy-namics used to solve the model with Evans and Hnatkovska’s (2006) method. I draw extensively on this paper and Evans and Hnatkovska (2005).
The equilibrium in this model is defined by the Euler equations for assets and con-sumption, the two budget constraints, and the market clearing conditions for both assets and goods. In summary, we have the following conditions:
P1tC1tU S = α1(1−δ)WtU S P1tC1tEU = ˆα1(1−δ)WtEU StP2tC2tU S = α2(1−δ)WtU S
StP2tC2tEU = (1−αˆ1−αˆ2)(1−δ)WtEU PU StCN tU S = (1−α1−α2)(1−δ)WtU S StPEU tCN tEU = (1−αˆ1−αˆ2)(1−δ)WtEU
1 = Et Rθt+1Mt+1U S 1 = Et
Rt+1ψ Mt+1U S 1 = Et Rγt+1Mt+1U S 1 = Et Rθt+1Mt+1EU 1 = Et
Rt+1ψ Mt+1EU 1 = Et Rηt+1Mt+1EU 1 = Et RtMt+1U S 1 = Et RtMt+1EU
Wt+1U S = Rt+1U S(WtU S−P1tC1tU S−StP2tC2tU S−PU StCN tU S) Wt+1EU = Rt+1EU(WtEU−P1tC1tEU −StP2tC2tEU−StPEU tCN tEU) BRW = BtU S+BtEU
1 = θtU S+θtEU 1 = ψtU S+ψtEU 1 = γtU S+γtEU 1 = ηU St +ηEUt
where wealth and its return are defined in (21) - (22) and their equivalent expressions for EU.
To economize space, I present the approximation only to the case where γtU S = 1 and ηU S = 0. To generalize it to the case where there are cross border equity holdings in the nontradable goods sectors is relatively straightforward. The approximate solution comes from linearizing the equilibrium conditions around a non-stochastic steady-state and some initial wealth distribution. Evans and Hnatkovska’s (2006) approach consists in log-linearizing up to the first order the equations and variables of the real side of the model, while the financial side is approximated up to the second order. From this point on, all small case letters mean log-deviations from either a non-stochastic steady state or the initial wealth distribution.
Log-returns are given by Campbell, Chan, and Viceira’s (2003) approximation:
rtU S =rt+aU Sθt (rθt−rt) +aU Sψt (rψt−rt) +aU Sγt (rγt−rt) +1
2(diag(ΣU St )−ΣU St aU St ) rEUt =rt+aEUθt (rθt−rt) +aEUψt (rψt−rt) +aEUηt (rηt−rt) + 1
2(diag(ΣEUt )−ΣEUt aEUt ),
where Σct, c ∈ {US, EU} is the conditional covariance matrix for the relevant excess returns for each country. Each individual return is then approximated as a linear function of prices and shocks (in log-deviations), like in Campbell and Shiller (1989):
rθt+1 ≃ δqθt+1+ (1−δ)d1t+1+p1t+1−qθt−p1t
rψt+1 ≃ δqψt+1+ (1−δ)d2t+1+st+1+p2t+1−qψt−st−p2t
rγt+1 ≃ δqγt+1+ (1−δ)yU St+1+pU St+1−qγt−pU St
rηt+1 ≃ δqηt+1+ (1−δ)yEU t+1+st+1+pEU t+1−qηt−st−pEU t.
The Euler equations for consumption of nontradables are easily log-linearized and after imposing market clearing for these goods’ markets can be written as follows:
pU St = wtU S−ytU S
pEU t = wtEU−yEUt −st. (B-1) The six approximate Euler equations for returns can be written in vector form as:
Eterct+1 = Σctact− 1
2diag(Σct). (B-2)
for c∈ {US, EU}. These Euler equations are combined with the approximations to the returns and iterated forward, giving solutions to current equity prices as functions of future dividends (as in Lucas (1978)), goods prices, and excess equity returns.
Because of logarithmic utility, the consumption to wealth ratios are constant which makes it possible to solve for the stochastic discount factor as a function of wealth only.
Using log-normality of returns, the budget constraints for each country c ∈ {US, EU}, combined with the Euler equations for the bond, are approximated by
Etwt+1c =wtc+rt+1
2act′Σctact. (B-3) It is worthwhile inspecting the Euler equations for European equities. For instance, the Euler equation of the US investor for equity on EU tradable sector is
qψt = ((πs+πp2+ (1−δ)πd2)A−πr−πs−πp2−πerψ) (I−δA)−1Xt
= −πs−πp2+ ((1−δ)(πs+πp2+πd2)A−πr−πerψ)(I−δA)−1 Xt
where assumptions (25) and (26) are used, and at=πaXt for any variablea. This shows that the price of traded equity responds to exchange rate changes. In fact, the impact of the currency depreciation in the dollar value of the European (tradable) equity market should be at least partially neutralized by a fall in the price of European equity in euro terms. This possibility was raised in Obstfeld and Rogoff (2005a) as a reason to look with caution at the valuation effects of currency depreciations as potential facilitators of external adjustment.
Using (B-2) and (B-3), the Euler equation for the nontraded equity of the European investor can be simplified to
qηt = (πyf + (12ΛEUt −πerη)(I−δA)−1)Xt
where ΛEUt Xt = aft′ΣEUt aft and Etrηt+1−rt = covt(wEUt+1, rηt+1)− 12vart(rηt+1) = πerηXt, as in Evans and Hnatkovska (2006). So, the price of nontraded equity only changes in response to the exchange rate through its effect in the equity risk premium. In fact, any shock to the economy that changes the equity risk premia has an impact on asset prices and equilibrium portfolio shares, as pointed out in Evans and Hnatkovska (2005).
Specifically, shocks that change the covariances between consumption expenditures and dollar returns or the relative prices of consumption goods have this effect.
The market clearing in the bond market uses the definitions of asset shares, combined with the clearing conditions of the each equity. The approximate version is:
P1Qθ(p1t+qθt)+SP2Qψ(st+p2t+qψt)+PhQγ(pU St+qγt)+SPfQη(st+pEU t+qηt)−BRW/Rrt
=δ(WU SwU St +WEUwtEU). (B-4) The market clearing conditions for equity are approximated, with minor modifica-tions, as in the appendix of Evans and Hnatkovska (2005). Conditions (B-1)-(B-4) and the approximated market clearing conditions for equity are finally written in the same form as (24) as prescribed in Evans and Hnatkovska (2006). The resulting system of equations is solved for the policy functions (Π), the law of motion of the state vector (A), the variances of wealth, and covariances of wealths with one another and the forcing variables. When solving for the variance matrix Ω0 we have to make sure that the result satisfies positive definiteness. This is done by solving for the coefficients of a Cholesky decomposition of that matrix and not for the matrix itself.
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Table 1: Correlation of EUR-USD exchange rate and contemporaneous and one-month ahead net debt security and net stock purchases by US residents from Eurozone residents.
rt stands for contemporaneous correlation and rt+1 for one month ahead correlation.
Net Purchase of rt p-value rt+1 p-value US debt securities 0.0633 0.1502 -0.0931 0.0299 Foreign debt securities -0.0525 0.2500 0.0652 0.1799
US stocks 0.0701 0.1006 0.0531 0.2880
Foreign stocks -0.1133 0.0117 0.0085 0.8760
Table 2: Calibration parameters for static model
Parameter Value Definition
α1 0.35 Share of tradable good 1 in US consumption α2 0.15 Share of tradable good 2 in US consumption ˆ
α1 0.15 Share of tradable good 1 in EU consumption.
ˆ
α2 0.35 Share of tradable good 2 in EU consumption.
β 0.335 Share of US debt owed to EU.
b -0.09 Total US foreign debt as share of household net worth.
r 0.01 Interest rate.
D1 10 Endowment of tradable good 1.
D2 7 Endowment of tradable good 2.
Yh 10 Endowment of US nontradable good.
Yf 7 Endowment of EU nontradable good.
P1 1 Price of tradable good 1 in dollars.
P2 1 Price of tradable good 2 in euros.
θ 0.806 US’s share of equity on tradable good 1.
ψ 0.232 US’s share of equity on tradable good 2.
γ 0.95 US’s share of equity on US’s nontradable good.
η 0.05 US’s share of equity on Europe’s nontradable good.
Table 3: Percent changes in consumption, welfare, and distribution of wealth of a 10%
dollar depreciation. No asset choice.
Baseline values
Impact on: With Home Bias in Equity No Home Bias in Equity
C1U S 1.996% 3.929%
C2U S -6.167% -2.497%
C1EU 6.464% 3.924%
C2EU -1.967% -2.505%
UU S -0.114% 0.520%
UEU 0.172% -0.167%
Real agg. consumption in US -0.343% 0.623%
Real agg. consumption in EU 0.107% -1.163%
Real wealth in US -0.327% 0.675%
Real wealth in EU 0.090% -0.843%
WU S
WU S+WEU -1.932% 0.000%
Table 4: Percent changes in consumption, welfare, and distribution of wealth after a permanent 10% dollar depreciation. No portfolio choice.
Model without Model with
nontradables nontradables
No home bias Home bias No home bias Home bias Impact in consumption in consumption in consumption in consumption
on: of tradables of tradables of tradables of tradables
C1U S 1.645% 1.645% 1.647% 1.647%
C2U S -7.283% -7.283% -7.281% -7.281%
C1EU 6.750% 6.750% 6.748% 6.748%
C1EU -2.178% -2.178% -2.181% -2.181%
UU S -1.831% -0.628% -0.733% -0.261%
UEU 1.725% 0.361% 0.698% 0.149%
Real agg. cons. in US -2.820% -1.035% -1.410% -0.517%
Real agg. cons. in EU 2.287% 0.501% 1.142% 0.250%
Real wealth in US -2.518% -0.967% -1.351% -0.507%
Real wealth in EU 2.042% -0.427% 0.948% 0.202%
WU S
WU S+WEU -2.281% -2.281% -2.281% -2.281%
Table 5: Percent changes in discounted values of consumption, and welfare after a tem-porary 10% dollar depreciation (half-life of about one year). Optimal asset choice.
Optimal Portfolio Choice Static Model
C1U S -0.383% 0.001%
C2U S -0.737% -0.003%
C1EU 0.854% 0.003%
C1EU 0.496% -0.001%
UU S -0.130% -0.010%
UEU 0.176% 0.007%
Real agg. consumption in US -0.246% -0.020%
Real agg. consumption in EU 0.300% 0.011%
Real wealth in US -0.197% -0.020%
Real wealth in EU 0.316% 0.011%