Why are Countries' Asset Portfolios Exposed to Nominal Exchange Rates?
  • 1 0000000404811396https://isni.org/isni/0000000404811396International Monetary Fund

Contributor Notes

Author’s E-Mail Addresses: adamsjonathan@ufl.edu, pbarrett@imf.org

Most countries hold large gross asset positions, lending in domestic currency and borrowing in foreign. Thus, their balance sheets are exposed to nominal exchange rates. We argue that when asset markets are incomplete, nominal exchange rate exposure allows countries to partially insure against shocks that move real exchange rates. We demonstrate that asset market incompleteness can simultaneously generate realistic gross asset positions and resolve the Backus-Smith puzzle: that relative consumptions and real exchange rates correlate negatively. We also show that local perturbation methods that use stabilizing endogenous discount factors are inaccurate when average and steady state interest rates differ. To address this, we develop a novel global solution method to accurately solve the model.


Most countries hold large gross asset positions, lending in domestic currency and borrowing in foreign. Thus, their balance sheets are exposed to nominal exchange rates. We argue that when asset markets are incomplete, nominal exchange rate exposure allows countries to partially insure against shocks that move real exchange rates. We demonstrate that asset market incompleteness can simultaneously generate realistic gross asset positions and resolve the Backus-Smith puzzle: that relative consumptions and real exchange rates correlate negatively. We also show that local perturbation methods that use stabilizing endogenous discount factors are inaccurate when average and steady state interest rates differ. To address this, we develop a novel global solution method to accurately solve the model.

1 Introduction

Do gross international asset positions matter for macroeconomic outcomes? Most countries choose gross asset positions of bonds and debt that are long in their domestic currency and short in foreign currency: the median country’s nominal exchange rate exposure is 32% of GDP1. Until recently, tractability concerns led modern international macroeconomics to ignore the implications of gross asset positions - linearization is not well-suited for models with meaningful portfolio problems. But when balance sheet exposure is so large, macroeconomic shocks may have large wealth effects, so macroeconomic theories must account for gross asset positions to be quantitatively useful.2 In this paper, we make three contributions to the understanding of this issue:

  1. Both gross international asset positions and correlations between major macroeconomic variables are sensitive to asset market structure. So macroeconomists should consider asset markets that are consistent with data on both gross assets and key macroeconomic correlations. We demonstrate that imperfect risk-sharing with nominal non-contingent bonds is a simple asset market restriction which both generates gross asset positions with realistic exchange rate exposure and also resolves the Backus-Smith puzzle.3

  2. We show that the default solution approach, local perturbation with ad hoc stabilizing assumptions, is a poor approximation to the true model and results in inaccurate predictions for the Backus-Smith correlation and other moments.

  3. Because local perturbation methods fail to recover the true equilibrium dynamics, we develop a global solution algorithm generalizing Maliar and Maliar (2015)’s projections method, and demonstrate its accuracy. The accuracy gain of this method over a local one arises because it approximates near the average interest rate rather than the steady state.

The Backus-Smith puzzle is a well-known indicator that macroeconomic dynamics depend on asset markets, and we show that it is resolved when countries face a realistic and nontrivial portfolio choice. Backus and Smith (1993) show that when asset markets are complete, models of international business cycles generate strong positive correlations between a country’s consumption and their real exchange rate. In contrast, empirical estimates of this correlation vary across countries but are typically negative. The source of the correlation in a complete markets model is the equilibrium condition equating the real exchange rate to the ratio of marginal utilities. The most obvious way to resolve the consumption-real exchange rate puzzle is to break this link by introducing asset market incompleteness. However, early attempts to pursue this directly failed to yield much quantitative success (see, for example Chari, Kehoe, and McGrattan (2002)). And so the subsequent literature turned to resolving the puzzle with additional model ingredients or frictions. We demonstrate that the intuition of these early attempts to explain the puzzle was valid, but over-simplified the asset market, eliminating a crucial mechanism: portfolio choice.

In this paper, we revisit the issue and argue that asset market incompleteness can explain the puzzle, but the type of incompleteness matters. We show that when households are allowed to trade nominal bonds, asset market restrictions can produce negative correlations of consumption and the real exchange rate. The key mechanism that produces this result is households’ use of asymmetric holdings of foreign and domestic nominal bonds, which they use to hedge their income risk. Prior attempts to resolve the puzzle missed this mechanism by examining asset markets without all of these ingredients. Some papers considered only real bonds, to no avail; Lustig and Verdelhan (2016) prove that any portfolio decision over real non-contingent bonds yields a lower bound of zero for the Backus-Smith correlation. Other papers reduced the set of assets to a single nominal bond, so that standard perturbation methods could be employed without having to solve a portfolio decision over risky assets, as in Chari, Kehoe, and McGrattan (2002) or Evans and Hnatkovska (2014). This approach eliminates any role for gross asset positions. In contrast, we allow households to choose their holdings of multiple nominal bonds, allowing for large gross asset positions. As a result, the equilibrium conditions of the portfolio decision over nominal bonds in this model can resolve the Backus-Smith puzzle.

As motivation, Table 1 presents cross-country statistics for bond home bias and the Backus-Smith correlation. The Backus-Smith correlation is calculated using data from the Penn World Table (Feenstra, Inklaar, and Timmer, 2015). For each country, this is the correlation of real consumption growth relative to the United States, with the pairwise real exchange between that country and the United States. Consistent with previous findings, most countries have a negative correlation. Bond home bias is calculated from 1990-2012 using the Benetrix, Lane, and Shambaugh (2015) data set, defined as debt assets less debt liabilities, scaled by GDP.

Table 1:

Cross-Country Statistics on bond home bias and the Backus-Smith correlation

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Almost all countries have a home biased portfolio, which is consistent with earlier estimates of exchange rate exposure by Eichengreen, Hausmann, and Panizza (2003), Goldstein and Turner (2004), and Lane and Shambaugh (2010a).4 Developed economies5 show slightly less home bias. Figure 1 demonstrates that this pattern holds over the sample period. Although the US features less home bias than the international median, the two converge after the financial crisis.

Figure 1:
Figure 1:

Bond Home Bias 1990-2012

Citation: IMF Working Papers 2017, 291; 10.5089/9781484335468.001.A001

We introduce a two-country model driven by two types of shocks - real shocks to productivity and nominal shocks to prices of country-specific tradable intermediates. We consider a simple model with only the ingredients necessary to both produce home bias in bonds6 and resolve Backus-Smith. The crucial asset market restriction we impose is that households may only hold non-contingent nominal bonds. Our structure implies that real shocks generate a positive correlation between labor income and foreign asset returns; higher domestic productivity makes the foreign intermediate good relatively scarcer, raising both the real and nominal exchange rates. Households use this correlation to hedge labor income risk, taking an asymmetric asset position similar to that seen in the data - going relatively long domestic bonds. However, because this hedge comes at the cost of increasing households’ exposure to nominal shocks, they do not go long enough to fully insure the real shocks. As a result, this portfolio loses value when the nominal exchange rate rises (i.e. the currency depreciates). If the portfolio is sufficiently home biased, this effect can reduce consumption. So if nominal and real exchange rates covary, the Backus-Smith correlation can be negative.

The Backus-Smith correlation is sensitive to parameterization of endogenous discount factors (EDFs). In open economy models, approximations around a deterministic steady state are typically nonstationary, so economists assume ad hoc features to induce stationarity, most commonly an EDF which is decreasing in the consumption level (Schmitt-Grohe and Uribe, 2003). But in general equilibrium, standard international macroeconomic models are typically stationary7, so the ad hoc endogenous discount factor assumption is unnecessary when using global solution methods. Recently, Rabitsch, Stepanchuk, and Tsyrennikov (2015) have shown that policy functions can be inaccurate when an EDF is assumed. We show that this matters for a key macroeconomic indicator: the Backus-Smith correlation is particularly sensitive to the parameterization of the EDF, even when the EDF’s elasticity is small. This is troubling for macroeconomic modelers, because the EDF is a commonly used ad hoc assumption which is not well disciplined by the data.

To achieve accurate global solutions, we generalize the simulation and projection algorithm of Maliar and Maliar (2015) to a model where equilibrium conditions cannot easily be expressed purely in terms of the states.8 As such, our paper is proof-of-concept that highly accurate global solution techniques can be applied to model equation listings that include prices. We use global methods because our model contains no ad hoc endogenous discount factor chosen to impose stationarity in the linear model. We show that our global method is substantially more accurate than the standard perturbation method for international models with portfolio problems, the Devereux-Sutherland algorithm.9 Section 4 documents that the Devereux-Sutherland algorithm accurately calculates average portfolio holdings, but inaccurately predicts macroeconomic dynamics. The reason our global method succeeds when local ones fail is that it allows us to solve near the true average interest rate, which is less than in the deterministic steady state.

2 Model

We consider a two-country RBC model, similar to the workhorse model of Backus, Kehoe, and Kydland (1992) but with two goods and without capital. There are two identical countries, indexed by i = H and i = F, which we refer to as Home and Foreign. They are each populated by mass one of identical, infinitely lived households. Each country produces a tradable intermediate good and a nontradable consumption good. Consumption in each country aggregates the nontradable consumption good with a tradable consumption bundle, which itself is an aggregate of domestic and foreign intermediates.

2.1 Households

The representative household in country i maximizes


where Ci,t is the domestic consumption good for country i in period t.

The household earns wage income Wi,t per unit of labor, denominated in domestic currency10. The household inelastically supplies 1 unit of labor every period. The price level of the consumption good is Pi,t. The household has access to two asset markets. It can hold non-contingent domestic bonds Bi,t+1i at price 1Rt+1i, which pays one unit of domestic currency in period t+1, and it can hold non-contingent foreign bonds Bi,t+1j at price 1Rt+1j, which pays one unit of foreign currency. Both of these bonds are in zero net supply. εti,j is the nominal exchange rate: the relative price of the country j consumption good in country j’s currency, to the price of the country i consumption good in country i’s currency.

The household’s period budget constraint (denominated in domestic currency) is


for ji.

2.2 Firms

There are four types of firms in each country: intermediate goods producers, nontradable goods producers, tradable consumption goods producers, and final goods producers. All firms are perfectly competitive.

The intermediate goods producers hire domestic labor Ni,tT to produce a country specific tradable intermediate good Xi,t. Their production function is linear in labor, with productivity

Ai,tT, which is exogenous and stochastic


The nontradable goods producers hire domestic labor Ni,tN to produce a country specific nontradable good Ci,tN. Their production function is linear in labor, with productivity Ai,tN:


For simplicity, we assume that Ai,tN is constant, so we normalize it to 1.

Foreign and domestic tradable goods aggregate into a tradable consumption good Ci,tT by


The final goods producers in each country produce the consumption good from the tradable consumption good Ci,tT and the nontradable good Ci,tN. Their production function is CES with elasticity of substitution ξ and tradable expenditure share μ


The price of intermediate good Xi,tj in units of the country i consumption good is Pi,tj, so intermediate goods firms solve


The price of the nontradable good Ci,tN in units of the country i consumption good is Pi,tN, so nontradable goods firms solve


Tradable goods firms aggregate the domestic and foreign intermediate goods. Where Pi,tT is the price of the tradable consumption good, the tradable consumption goods firms solve


Final goods firms in country i face prices Pi,tT for the tradable consumption good and Pi,tN for the nontradable good. They solve


2.3 Goods Prices

The first order conditions for both the intermediate goods firms and the nontradable goods firms imply that wages are proportional to their respective value marginal products of labor:


The first order conditions for the tradable consumption goods firms imply


and the first order conditions for the final goods firms imply


The real exchange rate is the price of foreign relative to domestic consumption. The law of one price holds for both intermediate goods, so the nominal exchange rate εti,j is


and the real exchange rate Qti,j as


The terms of trade in this economy is the relative price of the foreign intermediate in terms of the domestic intermediate:


2.4 The Price Level

Without further discipline, the price level is undetermined. So for both countries, we assume that the domestic intermediate price Pi,ti is exogenous and stochastic. When calibrating the model in Section 3, we estimate a joint process for price and productivity shocks to insure that their correlation is realistic.

We assume that the price level is stochastic so that perfect risk-sharing is impossible. If the only shocks were the two productivity shocks, households could choose portfolios of the two assets to recover perfect risk-sharing, as in Devereux and Sutherland (2007) and Engel and Matsumoto (2009). With four shocks, asset markets are incomplete, so relative consumption and the real exchange rate will not be perfectly correlated. Because two of the shocks are nominal, it is crucial that bonds are also nominal; if households were trading real bonds instead, perfect risk-sharing would be attainable.

One interpretation of this assumption is that the domestic monetary authorities in both countries commit to stabilizing domestic tradable producer prices, but do so with error. Alternatively, this assumption could be thought of as implying that domestic tradable producer prices are very sticky, but that this price stickiness does not have real effects on the production side of the economy as it does in Chari, Kehoe, and McGrattan (2002). Rather, it determines how inflation behaves in response to real shocks11. And inflation dynamics will have real effects through the household balance sheet.

The price of the aggregate consumption good is the CES price index


and the price of the tradable consumption good Ci,tT is the CES price index


When Pi,ti is fixed at 1, the terms of trade Sti,j moves one for one with the nominal exchange rate εi,j. An increase in domestic productivity Ai,t relative to foreign productivity will increase the terms of trade, increasing the nominal exchange rate and the price level, which is why productivity growth is positively correlated with inflation in equilibrium.

2.5 Equilibrium Asset Prices

The household’s optimal behavior includes an Euler Equation for each asset. The domestic bond’s Euler equation is


while the Euler equation for the foreign bond ji is


Both bonds suffer inflation risk due to variation in the price of consumption, while the foreign bond also suffers exchange rate risk because εt+1i,jεti,j varies stochastically in equilibrium.

Given the same interest rate, a household has incentive to decrease its holdings of foreign bonds if exchange rate growth (currency depreciation) logεt+1i,jεti,jet+1i,j covaries sufficiently with consumption growth logCi,t+1Ci,tCi,t+1. This is the central force that leads to home bias in assets; households choose to be long domestic bonds and short foreign bonds to minimize the correlation of their wage income with the returns on foreign bonds.

A second order approximation of the Euler equations provides a condition for home bias in assets. The second order log approximations12 around the period t+1 average for the domestic Euler equation13 (with lower case variables denoting growth rates) is


where inflation is defined logPi,t+1Pi,tπi,t+1. The Euler equation for the foreign bond ji is


Differencing the two Euler equations yields an approximate indifference condition for holding the two bonds:


The left hand side of this approximation is the premium of foreign bonds over domestic bonds, which is zero when uncovered interest rate parity holds. We use this equation to explain the intuition behind our main results.

2.6 Home Bias and the Backus-Smith Puzzle

How do households choose their portfolio allocations? When optimizing, they must satisfy approximation (24). The symmetry of the model implies that uncovered interest rate parity holds on average. Taking unconditional expectations of this approximation therefore yields:


The most obvious implication of this approximation is that when risk aversion γ is large, the covariance of consumption and the nominal exchange rate must be small in absolute value. The incomplete markets framework can therefore produce a weak correlation between at least the nominal exchange rate and consumption.

In addition to simply being small, depending on the other terms in this approximation, the consumption-nominal exchange rate correlation may also be negative. In particular, if Cov(et+1i,j,πi,t+1)Var(et+1i,j)2>0 (a result which holds in some calibrations of the model) then this approximation dictates that γCov(et+1i,j,ci,t+1)<0.

The mechanism whereby this occurs is through households’ adjustment of their international bond portfolio. If households were to hold foreign and domestic bonds equally, we would expect Cov(et+1i,j,ci,t+1)>0, because their total income would covary positively with the nominal exchange rate. In order to deliver γCov(et+1i,j,ci,t+1)<0, they go long domestic bonds and short foreign bonds in order to reduce their exposure to exchange rate risk. Therefore, home bias in bond holdings is crucial part of explaining the consumption-real exchange rate anomaly; households achieve a low (or even negative) correlation of the consumption with the real exchange rate by shorting foreign assets.

The Backus-Smith puzzle concerns consumption and real exchange rates, but Equation (25) only characterizes the relationship between consumption and nominal exchange rates. To understand the covariance of relative consumption with the real exchange rate, express approximation (24) in real terms, by defining real exchange rate growth as logQt+1i,jQti,jqt+1i,j.

Then differencing this condition for country H and country F implies


The implication of this approximation is that deviations from the complete markets consumption ratio are uncorrelated with exchange rate growth, which is the risky premium of foreign over domestic bonds14. Then use the relationship et+1i,j=qt+1i,j+πi,t+1πj,t+1 to rearrange this approximation in terms of the covariance between relative consumption and the real exchange rate:


This approximation reveals why the portfolio decision can resolve the real exchange rate consumption puzzle. Covt(qt+1i,j,et+1i,j) is positive, so if the intertemporal elasticity of substitution

1γ is small, then the covariance of the real exchange rate and relative consumption will be negative if consumption covaries more with domestic inflation than foreign inflation. This holds in our model because productivity shocks make households richer, and so consume more, but at the same time cause a depreciation of the currency and so increase imported inflation.

2.7 Equilibrium

A competitive equilibrium in this economy consists of sequences for t ≥ 0 of prices, PH,t, PF,t, PH,tF, PF,tF, WHt, WF,t, εtH,F, RtH, RtF; allocations, CH,t, CF,t, NH,t, NF,t, XH,tH, XF,tH, XH,tF, XF,tF and assets BH,tH, BF,tH, BH,tF, BF,tF; given initial assets BH,0H, BF,0H, BH,0F, BF,0F and realizations of the exogenous stochastic productivities AH,t, AF,t, PH,tH, PF,tH; such that:

  1. Households maximize their intertemporal utility (1).

  2. Firms maximize profits, satisfying factor demands (11), (12), (13) and (14).

  3. Markets clear, satisfying the household budget constraint (2), the labor market constraint Ni,tT+Ni,tN=1 and production functions (3), (4), (5) and (6).

  4. Household assets are in net zero supply: BH,tH+BH,tF=0 and BF,tH+BF,tF=0

3 Results from a Calibrated Economy

In this section we discuss two main issues. First, we show that a calibration of the model including within-country correlated shocks can produce a small, negative correlation between the real exchange rate and relative consumption. The same calibration, though, fails to reproduce this fact if asset market restrictions are either tighter (the one bond case) or looser (complete markets).

The second issue we discuss is solution technique. The model is challenging to solve because a) the linearized model is non-stationary near the steady state, and b) the steady state is far from the solution manifold. We show that standard modifications that solve the first problem fail to produce accurate perturbation solutions simply because the second problem remains15. The results of Schmitt-Grohe and Uribe (2003), who argue that this approach works well in a small open economy model, do therefore not carry over to the two-country case.

3.1 Calibration

To illustrate how the model works to deliver a negative correlation between the real exchange rate and relative consumption, we calibrate at a quarterly frequency using standard parameters from the literature. The main restriction we impose at this point is that the shock processes in the two countries are symmetric and independent16. We assume that the stochastic variables, productivity and prices, follow a joint autoregressive process:


To calibrate this process, we estimate a VAR for detrended log tradable productivity log logAi,tT and detrended log tradable prices log logPi,ti on US KLEMS data17. The US KLEMS data on productivity and tradable prices are annual, but we are interested in macroeconomic statistics typically measured at a quarterly frequency, such as the Backus-Smith correlation. To generate a quarterly shock process, we estimate the VAR using the annual data, and then impute a quarterly VAR consistent with the annual process18. The estimated persistence and innovation covariance matrices are reported in Table 2 with standard errors from Maximum Likelihood Estimation.

Table 2:

VAR estimates of the shock processes for output and prices

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Although cross-persistence is minimal (the off-diagonal elements of B are small) the cross-correlation of the innovations is large, at −0.6. The key model mechanism is that households use their portfolio choice to insure against competing nominal and real shocks. So including correlation of these shocks is an important test of the quantitative validity of our results. The remaining parameter values are summarized in Table 3:

Table 3:

Calibrated Parameters

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While there is not strong agreement in the literature over the true value of some of these parameters (particularly γ and η), we endeavor to choose reasonable values. We consider the sensitivity of our results to these assumptions in Section 4.

3.2 The Backus-Smith Correlation in the Calibrated Model

The main object we wish to measure is the correlation of relative consumption growth with the real exchange rate. That is, given by:


Our main contention is that risky gross asset positions matter for this correlation. So we also compare our results those of two other models, representing alternative views. The first, complete markets, represents the notion that asset-based risk is not a meaningful driver of this correlation. The second, a symmetric one-bond model, captures the idea that only risks to the net asset position matter. In this case, we permit only an average bond, which costs one unit of each currency in period t and returns Rt+1 units of each currency in period t + 1. The corresponding Euler equations are then:


Table 4 compares the prediction of these different models. The model with two bonds delivers a value of the Backus-Smith correation of −0.23. This is shown in the third line of Table 4. Benigno and Thoenissen (2008) report a median estimate of −0.16 for advanced economies. And in Table 5 above, we find a mean correlation between a sample of countries and the USA of −0.05. While the model-based and empirical moments differ slightly, the model with a meaningful portfolio choice is able to match this data moment substantially better than models with less rich portfolio choices. The complete markets model, which delivers a correlation of 1: the Backus-Smith puzzle. And the symmetric one-bond example produces a correlation of −119.

What explains these different values? When markets are complete, a correlation of 1 is an equilibrium condition, because marginal utilities must be proportional to the real exchange rate

Table 4:

Model values for bond home bias-output ratio and Backus-Smith correlation

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When there is one bond and nontradables, positive productivity shocks to the tradable sector lead to consumption growth, because the risk is uninsured and makes countries richer But tradables and nontradables are complements so the real exchange rate falls, and the Backus-Smith correlation is negative 20

Yet when there are two bonds, countries are partially insured. After a domestic productivity increase, the nominal exchange rate rises because the foreign tradable becomes dear relative to the domestic tradable, and their nominal prices are fixed without a contemporaneous nominal shock. Households choose a portfolio exposed to this exchange rate risk, so that their portfolio loses value after a productivity shock, partly offsetting their gains from higher future productivity. To do this, they chose a home-biased portfolio. As Table 4 makes clear, of the three model types considered here, only the one with two bonds can produce this phenomenon.

As a result of this portfolio choice, the correlation between consumption growth and income is therefore less than one, indicating partial insurance. Table 5 verifies this, reporting this and other correlations for the model, as well as comparable empirical values.21 The model’s most prominent deviation from the data is the low correlation between real and nominal exchange rates. This is common in international macro models, and is referred to as the purchasing power parity puzzle (Rogoff, 1996).

Table 5:

Other model correlations

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Home bias in the calibrated model is illustrated futher in Figure 2, which shows the asset distributions in a ten-million-period simulation. Saving in the domestic currency is typically funded by borrowing in the foreign currency. Mean holdings of domestic assets equal 0.38, or 38% of average output22. The slight asymmetry in the distribution results from the fact that even though assets are not a random walk, they are very persistent, so even a very long simulation results in a slightly asymmetric asset distribution. As discussed in the next section, the persistence of assets is determined by the size of the average interest rate relative to 1/β. Accurately calculating the interest rate is therefore critical for describing the dynamics of the model, including the Backus-Smith correlation.

Figure 2:
Figure 2:

Distribution of bond holdings

Citation: IMF Working Papers 2017, 291; 10.5089/9781484335468.001.A001

This introduces the key issue of solution technique. Although the model itself is stationary, linearizing around the steady state interest rate will produce an average interest rate of 1/β, and so deliver non-stationary debt dynamics. One approach to dealing with this issue is to modify the model to stationarize it near the steady state, as in (Schmitt-Grohe and Uribe, 2003). In following section we show that such modifications have non-trivial effects in our model. So instead, we solve the model using a nonlinear global method, based on that of Maliar and Maliar (2015). Details of the solution method are presented in Appendix B.

3.3 Perturbation Solution near the Steady State

In our model, the deterministic steady state interest rate is RtH=RtF=1/β. Approximating the model near this point results in explosive asset dynamics, as the average rate of interest equals the rate of time preference, and so induces a unit root in asset dynamics23. This is a common problem in small open economy models where the world interest rate is fixed, and is typically set to 1/β. Schmitt-Grohe and Uribe (2003) show that a number of simple model modifications which stationarize the canonical small open economy model have no impact on a large range of economically relevant model moments. This is a key contribution to the solution of open economy models, as it allows them to be modified slightly and then solved by standard perturbation methods.

The modification that is most commonly used is to alter the consumer’s preferences so that the discount rate is a function of current consumption24. This is often termed an “endogenous discount factor”. This introduces a new parameter θ in the Euler equations, which become:


The idea behind this approach is to pick a θ large enough that a local approximation is stationary, but small enough that the underlying economic properties of the model are unaffected. Figure 3 shows that this is a hopeless task, at least in the current model. This chart presents the Backus-Smith coefficient as a function of θ25. Small changes in θ have a very large effect on this coefficient. As θ varies from 0 to 0.1, the Backus-Smith statistic ranges from −0.9 to nearly 0.1. In other words, the parameterization of the ad hoc problem modification is economically meaningful.

Figure 3:
Figure 3:

Consumption-real exchange rate correlation

Citation: IMF Working Papers 2017, 291; 10.5089/9781484335468.001.A001

Common practice is to try to calibrate θ by targeting some other moment of the data, typically some of the properties of net foreign assets26. If this can inform the choice of θ and select a stable value for the Backus-Smith correlations, then there may still yet be hope for the EDF-perturbation approach. So in Figure 3 we also show the results from calibrating θ to three different moments of the Benetrix, Lane, and Shambaugh (2015) cross-country data on net foreign debt assets (NFDA) - the pooled autocorrelation, and the mean and median within-country standard deviations - as well as a fourth alternative - matching the average bond home bias. Alas, these cannot pin down a single value for the Backus-Smith correlation, and produces numbers varying from −1 to −0.1. Not only does this approach fail to select a unique value for the Backus-Smith correlation, none of them appear to be very close to the correct value (computed via the global solution), at around −0.23.

Nor should Figure 3 be read as saying that θ ≃ 0.01 is a “good” choice (the intersection of the curve and dashed “Global quadratic” line in Figure 3), simply because the local and global Backus-Smith correlations are similar. Not only does the “best” choice of θ vary with the particular calibration, but without having an accurate (e.g. global) solution, there is no way to know in advance what value of θ will deliver the correct Backus-Smith correlation. In the next section, we discuss how to accurately solve the model, before preceding to compare the two solution approaches.

3.4 A Global Solution

The true model is stationary because the average interest rate in the model is less than 1/β. This is related to home bias in bond holdings: households save in domestic bonds and lend in foreign bonds, as already shown in Figure 2. Because domestic production is a majority of domestic consumption, the domestic price level is more correlated with domestic than foreign shocks. As a result, the real return on foreign bonds (which is susceptible to exchange rate risk) is riskier than that on home bonds. So the borrowers of each bond bear more risk than savers. The average interest rate is therefore less than the steady state level to compensate borrowers for this extra risk. By ignoring risk, the steady state would eliminate this effect, producing average interest rates that are a too high and leading to nonstationary debt dynamics.

Our solution method is and extension of Maliar and Maliar (2015) and is described in detail in Appendix B27. The basic principles are quite straightforward, though. We guess a policy function for each variable in terms of the states28, simulate the model, and update the policy functions based on the equation errors. Because our solution is global, we do not have to find a steady state first and then approximate the solution nearby. Instead, the algorithm fits an approximation to the entire solution manifold simultaneously. And so the average gross quarterly interest rate in the solution is less than 1/β. Asset dynamics are therefore stable. A similar effect occurs in Huggett (1993) and Aiyagari (1994) where idiosyncratic risk leads the equilibrium interest rate to be less than the discount rate.

Figure 4 displays the equilibrium relationship between interest rates and asset holdings in the calibrated solution, when all other variables are at their long-run averages29. There are two main points to note. First, that the interest rate is always less than 1/β, stabilizing the model. This holds elsewhere in the state space too; the quarterly average log interest rate is 0.099, close to the lines shown in Figure 4. And while the difference between the true average interest rate and the steady state is small - only around one basis point per quarter - the steady state is the knife-edge separating stable from explosive dynamics, so even small differences can have large effects.

Figure 4:
Figure 4:

Equilibrium Asset Returns

Citation: IMF Working Papers 2017, 291; 10.5089/9781484335468.001.A001

Second, as home’s domestic saving increases above its average value (moving to the right in the left-hand panel), the return on home-currency bonds, RH, falls relative to the return on foreign-currency bonds, RF. This occurs because more domestic saving by country H requires more foreign borrowing by country F. Because international borrowers are more exposed to risk than savers, households in country F must be compensated through a lower cost of borrowing relative to the return on their assets. So in equilibrium, RH falls relative to RF.

3.5 Comparing solution methods

To check that the global method we develop is indeed more accurate than competing alternatives, we compute the errors on the four Euler equations and compare across solution methods.

Figure 5 shows the distribution of the log errors on the four Euler equations30. The blue and red lines show the errors from the linear and quadratic global solutions respectively. The black line shows the errors using the Devereux and Sutherland (2011) method, using an EDF of θ = 0.01 to stationarize the model. Figure 6 summarizes these densities, displaying the mean and mean absolute error, reported in log10 units.

Figure 5:
Figure 5:

Distribution of log errors on Euler equations

Citation: IMF Working Papers 2017, 291; 10.5089/9781484335468.001.A001

Figure 6:
Figure 6:

Log errors on Euler equations (expressed in log10 units)

Citation: IMF Working Papers 2017, 291; 10.5089/9781484335468.001.A001

As these figures makes clear, the local method is biased, as it approximates around a point where the interest rate is 1/β, which is not the true long-run average. This leads to systematic errors on the Euler equations, equivalent to nearly two basis points per quarter. This means that in the long run, households are incurring substantial losses relative to their optimal portfolio by following the policy rules computed in the local approximations. Not only is the local solution highly sensitive to changes in the stabilizing assumptions, it is also systematically wrong.

In the global quadratic approach, however, there is almost no bias at all; the average error on the Euler equations is in the order of one hundredth of a basis point. The global solution does not suffer from the systematic errors displayed by the local one.

The quadratic global solution method also generates very small absolute equation errors. The results from our method produce absolute log errors on the Euler equations that average to less than 10−5, or below one tenth of one basis point per quarter. At least on some equations, this is an order of magnitude smaller than the alternative solution methods.

4 Parameter Sensitivity

In this section we examine how the equilibrium of the model varies with the chosen calibration. We find that the equilibrium correlations, especially the Backus-Smith correlation, are highly sensitive to parameter values. This includes parameter values that are not well identified in the literature, such as the risk aversion coefficient γ, the elasticity of substitution between foreign and domestic tradables η, and the elasticity of substitution between tradables and nontradables ξ. Figure 7 plots the equilibrium Backus-Smith correlation versus values for these three parameters.

Figure 7:
Figure 7:

Parameter Sensitivity: Backus-Smith Correlation

Citation: IMF Working Papers 2017, 291; 10.5089/9781484335468.001.A001

The Backus-Smith correlation is especially sensitive to η, the elasticity of substitution between foreign and domestic tradables. Values of η much larger than 2, as is commonly estimated in the trade literature, imply that consumption and real exchange rates are almost perfectly correlated, contrary to the data. However, when the elasticity of substitution is much less than 2, as commonly assumed in the international macro literature31, the correlation is negative, approaching —1.

Why is it so sensitive to η? When foreign and domestic tradables are more substitutable, changes in relative quantities have smaller effects on relative prices. So as η gets larger, domestic productivity shocks are associated with smaller real exchange rate depreciations. Accordingly, a home biased bond portfolio becomes less effective at insuring productivity shocks. Choosing more home bias to achieve the same insurance from real shocks would expose country portfolios to greater nominal risk, so households choose less home bias (Figure 9). With less home bias, countervailing valuation effects from productivity shocks are reduced, so productivity shocks and real exchange rate depreciations are associated with higher consumption. Thus the Backus-Smith correlation increases with η.

The correlation between nominal and real exchange rates (Figure 8) is non-monotonic because the correlation between productivity and the relative price of foreign to home consumption PF,tPH,t depends on two opposing effects. First, a positive productivity shock in country H causes both countries to increase consumption of country H‘s tradable good, but due to home bias, the price of country H‘s tradable bundle decreases relative to country F‘s tradable bundle. The second effect is that the price of the nontradable good relative to the tradable bundle increases in both countries, and increases by more in country H, where the tradable bundle increase is larger due to home bias. For low substitutability, the real-nominal exchange rate correlation is decreasing in η because the first effect dominates, so movements in tradable prices affect the price level more than the countervailing movements in nontradable prices. For high substitutability, the correlation is increasing in η because the second effect dominates, so tradable prices are less responsive to quantities.

Figure 8:
Figure 8:

Parameter Sensitivity: Real-Nominal Exchange Rate Correlation

Citation: IMF Working Papers 2017, 291; 10.5089/9781484335468.001.A001

The elasticity of substitution between tradables and nontradables, ξ, is also poorly identified in the literature and has opposite effects as η. When tradables are complements, as commonly assumed, real exchange rates and consumption growth are loosely negatively correlated. But as ξ rises, the correlation approaches −1. A higher ξ increases the response of the real exchange rate to a change in the terms of trade. When domestic productivity increases, the real exchange rate depreciates by more when ξ is larger. This increases the valuation effect of a productivity shock. The home biased portfolio loses more money when the exchange rate depreciates, so consumption decreases by more. The reduces the Backus-Smith correlation (an exchange rate depreciation is an increase in q).

Bond home bias is increasing in risk aversion, γ (Figure 9). As risk aversion rises, households are less willing to consume out of an income shock. Productivity shocks generate smaller changes in relative consumptions across countries, so the real exchange rate is less responsive to real shocks. This makes a home biased bond portfolio less effective at insuring against real shocks. Households reduce their home bias, as a given portfolio position conveys less real income insurance for the same exposure to nominal shocks. A higher γ also decreases the Backus-Smith correlation (Figure 7), because as consumption becomes less responsive to productivity shocks, valuation effects begin to dominate.

Figure 9:
Figure 9:

Parameter Sensitivity: Bond Home Bias

Citation: IMF Working Papers 2017, 291; 10.5089/9781484335468.001.A001

Uncovered interest rate parity (UIP) does not hold exactly in our model. Figure 10 plots the UIP coefficient for different parameter values. When the UIP coefficient is one, interest rate differentials are perfectly offset by expected exchange rate growth, and uncovered interest rate parity holds. When the coefficient is zero, interest rate differentials do not predict exchange rate growth at all. The empirical UIP puzzle is that the coefficient is much less than one32, which is difficult to reconcile with standard international macro models when solved with linearization. In our model, the relative demands for domestic and foreign bonds move in response to shocks, which requires small differences in expected returns to clear the bond market. Regular deviations from UIP suggest that global solution methods with endogenous portfolios might be a promising set of tools for researchers working to understand the uncovered interest rate parity puzzle.

Figure 10:
Figure 10:

Parameter Sensitivity: Uncovered Interest Rate Parity

Citation: IMF Working Papers 2017, 291; 10.5089/9781484335468.001.A001

5 Concluding Remarks

We have shown that a standard international macroeconomic model can jointly explain home bias in bonds and the Backus-Smith puzzle by incorporating incomplete asset markets and nominal non-contingent bonds. Home bias in bonds exposes a country’s balance sheet to nominal exchange rate movements. The portfolio position occurs because household use home bias in bonds to partially insure against real income shocks, which are negatively correlated with exchange rates. This exchange rate exposure is large in the data, so macroeconomic models should incorporate endogenous portfolio choice to capture this channel. The asset market incompleteness must also be realistic, to yield accurate dynamics - in this case, the consumption-real exchange rate correlation.

We also show that solution methods are crucial to drawing accurate conclusions from a model with endogenous portfolios. Standard linear methods fail in this situation. The Devereux-Sutherland algorithm can correctly recover average portfolio holdings, but using it in conjunction with linearization and endogenous discount factors yield inaccurate predictions for macroeconomic dynamics.

Instead, we employed a global solutions method which generalizes the projections approach of Maliar and Maliar (2015). We show that our method is highly accurate and may be useful for research a variety of international macroeconomic puzzles. There are many cases in which linear methods are insufficient to capture crucial economic forces - particularly when risk and portfolio choice are involved. Our method is general, efficient, and easily implemented. We provide the code and documentation on our websites.


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A Approximations

Any asset with return Xt+1 satisfies the pricing equation for the household in country i:


Define ci,t+1 = log (Ci,t+1/Ci,t), xt+1 = log Xi,t+1, x¯t=Etxt+1, and c¯t=Etct+1. Then the second-order expansion of the integrand in the pricing equation is:


Substituting into (29) gives


For the foreign bond,


Let rt+1j=logRt+1j,et+1ij=log(εt+1ij/εtij), and πi,t+1=log(Pt+1i/Pti) Then:


Substituting in for this then gives equation (23)


Likewise for the domestic Euler equation, we recover equation (22):


B Computation

B.1 Model Specification

The method we develop can be used to solve a general model with exogenous state Xt, endogenous state Yt and controls Ct=(Ct1,Ct2), where the equilibrium conditions can be written as:


Here, the partition of the control space allows for more model descriptions with more forward-looking equations than endogenous states, as is the case ni our model when B21,t and B12,t are written out via the market clearing conditions. A solution to this model is a pair of functions Yt = y(Xt, Yt-1) and Ct = c(Xt, Yt-1) satisfying:


Where C(·) = (c1(·), c2(·)). We can write the model in this paper in this form by letting:


Where lower case variables are the logarithms of their upper case counterparts (this is simply a change of variables that improves the accuracy of the resulting solution). The forward-looking equations for the endogenous states are defined by:


So these reduce to B1,t+11=B1,t+11 and B2,t+12=B2,t+12 if and only if the two international Euler equations hold. For the forward-looking controls:


And the function h(·) is given by the remaining contemporaneous identities.

B.2 Functional forms

We approximate the functional solutions y(X, Y) and c(X, Y) with polynomials. The notation d = ϕ(δ) means that the function d(X, Y) is defined by the sum over the polynomials of (X, Y) with weights given by the elements of the vector δ. That is:


Where qn(z) is the vector of all possible nth-order combinations of the elements of z. For example, if z = (x,y) ∈ ℝ2, then:


So the approximate solutions for y(X, Y) and c(X, Y) can be described by vectors δy and δc33.

In our calculations we use standard polynomials to approximate the functional forms of the solution, but in general one can use Chebychev or other polynomials as instead34. We also scale the basis polynomials depending on the expected variance of the states.

B.3 The algorithm

Given guesses δjy and δjc the instructions for updating them to coefficient guesses δj+1y and δj+1c are:

  1. Simulate (Xt-1, Yt-1, Xt, Yt) for a very large number of points. In the paper we use a simulation of 1,000,000 points, discarding the first 10,000 periods.

  2. Use the EDS algorithm of Maliar and Maliar (2015), reduce this to an almost-ergodic set of many fewer points, denoted ξj={Xj,Yj,Xj,Yj}=1m. In the paper we use a target of m = 1000 points.

  3. Solve for the set {C˜j,2}=1m satisfying C˜j,2=h(Yj,Xj,Yj,C˜j,)

  4. Regress the {C˜j,2}=1m on q2(Yj,Xj) to get updated coefficients δj+1c,2

  5. Using monomial integration to evaluate the integrals, solve for {Y˜j,C˜j,1}=1m solving:
  6. Regress {Y˜j,C˜j,1}=1m on q2(Yj,Xj) to get updated coefficients δj+1y, δj+1c,1

The preceding description outlines the simplest way to update the vectors of coefficients that describe the model solution. However, in practice, a number of additional complications arise, omitted from the algorithm description for clarity. The first of these is the stopping rule. In the implementation used in the paper we terminate the algorithm when successive updates of the solution coefficients are sufficiently close. However, the ultimate test of convergence is the error on the model equations. With a novel method, such as this, evaluating the solution errors is of primary importance. Hence further discussion of them is promoted to the main body of the text.

Also omitted from this description is the role of damping. The method is, at heart, a policy function iteration algorithm. As is well know, policy function iteration often requires very heavy damping in order to converge, as discussed in Judd (1998). When solving the model in this paper, the initial gain is set to between 0.02 and 0.5 depending on the particular problem. Finally, to partially offset the slow convergence induces by damped policy function iteration we also allow for an adaptive gain. By making the gain a decreasing function of the distance between successive coefficient updates, we reduce the extent of the damping as the model converges.

Because solution can be expressed as a coefficient vector, this method also enables the enforcement symmetry of symmetry. We simply identify the symmetry restrictions that we wish to enforce and average out any deviations in the coefficients from symmetry. This also diminishes error propagation - small solution asymmetries can quickly lead to divergence.

To aid performance, we also write most of the solution code in C++. This leads to fairly swift solutions. A modern laptop takes approximately 2 minutes to produce a second-order solution to the six-state model in this paper. Linear solutions are much faster, but (as discussed in the body of the paper) are also less accurate, producing larger average Euler equation errors.


We are grateful for helpful comments from Adrian Peralta Alva, Thorsten Drautzburg, Burcu Eyigungor, Lars Hansen, Daniel Garcia Macia, Divya Kirti, Adrien Verdelhan, Mark Wright, and participants of the Chicago Economic Dynamics working group, the Midwest Macro Conference, the University of Florida economics seminar, the Royal Economic Society Conference, the Federal Reserve Bank of Philadelphia, and the Southern Economic Association Conference.


i.e. when a country’s currency depreciates by 1%, the value of its balance sheet loses 0.32% of GDP, ceteris paribus.


In examining home bias in bonds, we join a large recent literature researching exchange rate exposure and valuation effects on international portfolios, including Cavallo and Tille (2006), Tille (2008), Benigno (2009), Lee, Ghironi, and Rebucci (2009), Matsumoto and Engel (2009), Mendoza, Quadrini, and RiosRull (2009), Lane and Shambaugh (2010b), Corsetti, Dedola, and Leduc (2014), and Maggiori, Neiman, and Schreger (2017) among many others. Gourinchas and Rey (2014) provide a summary of the literature on valuation effects.


To the best of our knowledge, ours is the first paper which features endogenous portfolio decisions and achieves a realistic consumption-real exchange rate correlation. Coeurdacier and Rey (2013) note that most models with endogenous portfolios recover perfect risk-sharing, and that in papers where asset markets are incomplete, consumption and real exchange rates typically remain correlated, as in Benigno and Kucuk-Tuger (2008) and Coeurdacier, Kollmann, and Martin (2010).


After updating the Lane and Shambaugh (2010a) data, Benetrix, Lane, and Shambaugh (2015) show that aggregate exchange rate exposures reversed after 2004 when also accounting for changes to the composition of equity and FDI. However, the debt exposure remains negative – or equivalently, bond home bias remains positive (Figure 1). We consider the debt exposure to be the relevant measure to our analysis, which concerns nominal bonds. While an unbalanced equity portfolio might have risk that is correlated with nominal exchange rates, equities are predominantly real assets, and so that risk is an equilibrium outcome rather than written into the security. Expanding our analysis to equities and other real assets will be a component of future research.


Developed economies follow the definition of Benetrix, Lane, and Shambaugh (2015), namely: Australia, Austria, Belgium, Canada, Denmark, Finland, France, Germany, Greece, Israel, Ireland, Italy, Japan, Netherlands, New Zealand, Norway, Portugal, Spain, Sweden, Switzerland, the United Kingdom, and the United States.


Home bias in bond holdings is a also component of the broader home bias in assets, which is puzzling in standard international RBC models; see Lewis (1999) for an early summary, and Coeurdacier and Rey (2013) for a more recent one. When asset returns are uncorrelated with labor income, as in Lucas Jr. (1982), households fully diversify. When domestic asset returns are positively correlated with labor income, households bias their portfolios towards foreign assets, as in Baxter and Jermann (1997). In our model, domestic bond returns are negatively correlated with labor income, so households bias their portfolios towards domestic bonds.


Stationarity holds because on average, interest rates are less than the discount rate.


This algorithm joins the large literature using projection methods to solve macroeconomic models, which Fernndez-Villaverde, Rubio-Ramrez, and Schorfheide (2016) survey.


The Devereux-Sutherland algorithm is presented in Devereux and Sutherland (2011), was developed independently and concurrently by Tille and van Wincoop (2010), and is closely related to the method of Evans and Hnatkovska (2012). Papers that use Devereux-Sutherland to solve endogenous portfolio problems with nominal bonds include Rahbari (2009), Berriel and Bhattarai (2013), Coeurdacier and Gourinchas (2016)


“Currency” here serves the role only of a unit of account, not a means of exchange nor a store of value. Given that we are most interested in wealth effects induced by relative fluctuations in competing units of account, this is an appropriate simplification for our purposes.


Indeed, in the calibrated solution of Section 3, we allow for a richer shock structure, with within-country correlation of the real and nominal shock, both contemporaneously and through the lag structure.


All approximations in this section are derived in appendix A.


Consistent with our critique of linearization solutions, this argument is not dependent on finding a steady state. It is an approximation around the conditional expectation in the next period give the current state.


Looking ahead to the solution techniques we discuss later, we note that equation (26) is precisely that used by Devereux and Sutherland (2011) to generate linearized solutions to the portfolio problem.


This holds even for methods which produce accurate levels of asset holdings, such as Tille and van Wincoop (2010) and Devereux and Sutherland (2011), because they too rely on linearizing near the steady state interest rate.


Our model is estimated as if two copies of the United States traded with one another. We select this symmetric case for three reasons. First, we choose not to estimate a pair of similar economies (e.g. the US and the Euro-zone) because we want to match the aggregate trade shares, and no two large economies have bilateral flows that are nearly as large as their total trade flows. Second, we choose not to estimate the US versus the rest of the world, because tradable/nontradable shares and prices are not available for the world as a whole. Finally, in this stylized model, we want to be clear that our results are generated by the few economic ingredients, rather than an asymmetry across countries.


We follow Stockman and Tesar (1995) and define tradable sectors as: agriculture, mining, manufacturing, and transportation.


This interpolation is, of course, not unique. But it is if we insist that the quarterly autocorrelations of produtivity and prices are, like their annual counterparts, positive.


Stabilizing the model with an EDF gives answers that vary from −1.0 to 0.1. This is included in Table 4 for completeness, but not discussed in detail until Section 3.3.


This is the same effect as studied in Benigno and Thoenissen (2008), and similar to Corsetti, Dedola, and Leduc (2008), who instead suppose that home and foreign goods are complements. Mukhin, Itskhoki, and others (2016) criticize these mechanisms as being inconsistent with the Meese-Rogoff puzzle and the purchasing power parity puzzle. This criticism at least partially applies to our model; nominal exchange rates are not a random walk, and less correlated with real exchange rates than in the data (Table 5). Engel (1999) also criticizes this channel, finding that tradable prices account for most of the changes in relative price levels at short horizons. Alternative mechanisms through which productivity shocks can produce consumption growth and a real exchange rate decline for some asset market structures include nonseparable utility with news shocks (Colacito and Croce, 2013), nonseparable utility with labor wedge shocks (Karabarbounis, 2014), and financial shocks (Mukhin, Itskhoki, and others, 2016).


We calculate correlations of log quantities for the United States at a quarterly frequency. Income and consumption are taken from the national accounts. Real and nominal exchange rates use the BIS narrow effective exchange rate, which we seasonally adjusted at the quarterly frequency. When reporting correlations of levels, we first remove a linear time trend. Finally, the levels and growth rates are absolute, versus relative to the other country, which is why the consumption-real exchange rate correlation differs from that reported in Table 4.


Hence the average home bias figure in Table 4. Average home assets plus average foreign borrowing gives a GDP-hom bias ratio of 0.78.


A related issue is that of perfect substitutability of assets at the steady state. But this is a solved problem - Devereux and Sutherland (2011) show how to compute the correct approximation to asset holdings in such a model - and distinct from the one we address here. However, that technique also relies on approximating near a steady state interest rate of


This is also the only one of Schmitt-Grohe and Uribe‘s modifications that doesn’t alter the exact mechanism that we are trying to investigate: their ad hoc debt elastic demand curve is an endogenous feature of our model; portfolio adjustment costs would alter the portfolio decision that we want to study; and complete markets are precisely the paradigm that we know fails to produce empirically plausible Backus-Smith correlations.


When we solve the model using a local linearization, we make sure to approximate around the true asset level using the approach of (Devereux and Sutherland, 2011).


There is no commonly accepted strategy for this calibration. Different papers choose values in a wide range, on the order of 10−4 (Rabitsch, Stepanchuk, and Tsyrennikov, 2015) to 10−3 (Devereux and Yetman, 2010) to 10−2 Yao (2012).


There is one alternative method to our gloabl approach, suggested by Devereux and Sutherland (2010): guess a mean interest rate, linearize nearby, simulate, update the interest rate guess, and iterate to convergence. As noted by Rabitsch, Stepanchuk, and Tsyrennikov (2015), this will fail because the analytic solution for average asset holdings computed by local linearization formulae of Devereux and Sutherland (2011) do not hold away from the riskless interest rate.


The basis functions we use for this are the full set of quadratic functions of the states, although this choice is a matter of accuracy rather than anything more fundamental. There are six states in the model: real and nominal shocks in each country, plus stocks of each asset. This means that for each model variable, we solve for coefficients defining the projection onto the intercept, linear, and quadratic iterations for each state, i.e. 1 + 6 + .5 × 6 × 7 = 28 coefficients. With 13 model variables for each country, plus real and nominal exchange rates, this results in some (2 + 2 × 13) × 28 = 784 projection coefficients in total.


Because equilibrium assets holdings and endowments are highly correlated, the conditional distributions of BHH and BFF are much narrower than the unconditional distributions shown in Figure 2. The x-axis ranges of the panels in Figure 4 cover more than twice the conditional range of BHH and BFF observed in a ten-million period simulation. Observations outside these ranges essentially never happen.


Each is labeled with a variable name, because the (Maliar and Maliar, 2015) method associates each equation with a rules for one variable. Here, the Euler equations are satisfied by adjusting the rules for BH,tH, εtH,F, RH,t and RF,t


Macroeconomic papers estimate this parameter in a broad range: Taylor (1993) estimates .4, Heathcote and Perri (2002) estimate .9, Whalley (1985) estimates 1.5, and Corsetti, Dedola, and Leduc (2008) estimate 3.3.


A long literature following Bilson (1981) and Fama (1984) tests UIP and typically finds that the correlation is zero or negative. Engel (1996) surveys this evidence and many papers such as Burnside, Eichenbaum, Kleshchelski, and Rebelo (2011) and Menkhoff, Sarno, Schmeling, and Schrimpf (2012) demonstrates that the puzzle still holds in recent data.


The same notation can be easily extended to vector-valued functions y(X,Y) and c(X,Y) by stacking the coefficients into the vectors δy and δc.


Regularized polynomials often have superior performance for approximations of order three and higher. This is because near zero, xn looks very similar for n ≥ 2. As we approximate only to second order, though, we use “raw” unsclaed polynomials.

Why are Countries’ Asset Portfolios Exposed to Nominal Exchange Rates?
Author: Jonathan J. Adams and Mr. Philip Barrett