## _wp06177

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### Introduction and research question
- Puzzles addressed: large and persistent real exchange rate (q_t) fluctuations and lack of consumption risk-sharing across countries.
- Real exchange rate definition (log-linear): q_t = s_t + p_t* - p_t.
- Under complete markets with separable log utility: q_t = c_t - c_t*; empirical HP-filtered correlation for euro–U.S. bilateral series: -0.17.
- Two principal modeling approaches evaluated: introduce nominal rigidities (PCP vs. LCP) and/or introduce asset market incompleteness (stationary NFA).

### Model structure and baseline assumptions
- Stochastic two-country NOEM with home and foreign of equal size; continuum of intermediate goods; Dixit-Stiglitz preferences; all goods tradable.
- Five types of shocks (nine total shocks):
  - World technology shock (unit root).
  - Country-specific stationary technology shocks (AR(1)).
  - Country-specific monetary shocks (iid).
  - Country-specific demand shocks (AR(1)).
  - Country-specific preference shocks (AR(1)).
- Baseline features: home bias in consumption, external habit (parameter b), staggered Calvo price setting with backward-looking indexation (parameters α, τ).
- Baseline financial closure: complete markets and law of one price (LOP) for intermediate goods.
- Key variables and notation preserved from the source (C_t, P_t, T_t, Q_t, S_t, R_t, B_t, A_t, X_t, z_t, η_t, etc.).
- Households: lifetime utility Sum_{t=0}^∞ β^t E_0{ log(C_t - b C_{t-1}) - (N_t)^{1+γ}/(1+γ) }.
- Complete-market risk-sharing condition (equation (4) in source): Q_t linked to ratio of marginal utilities and preference shocks.

### Extensions analyzed
- A. Incomplete markets with stationary net foreign assets (NFA)
  - Households trade two nominal riskless bonds (domestic and foreign); a cost of holding foreign bonds ensures stationarity; foreign residents constrained to foreign-currency bonds.
  - Real budget constraint modified (equation (3’) in source); φ(.) differentiable, decreasing near zero, φ(0)=0 required for stationarity.
  - Risk-sharing condition in expected first-difference form (equation (9’) in source) introduces χ and b*_t definitions from text.
  - Log-linear NFA law of motion (equation (14) in source) given exactly in source: .~~21 1)1(−− ⎥⎦⎤⎢⎣⎡− −− += **1* ttttt ccqbb δ δ δ θ δ β (expression as in text).
- B. Local-currency pricing (LCP)
  - Firms set separate local-currency prices for domestic and foreign markets under Calvo with indexation; generates deviations from LOP and imperfect pass-through.
  - Modified inflation equations for home-produced goods and imported/foreign-priced components (equations (10’), (10b’), (11’), (11b’) as in text).
  - CPI inflation under LCP no longer includes the nominal exchange rate as a direct determinant of imported goods inflation (equations (12’) and (13’) in text).
- C. Combined incomplete markets + LCP
  - System combines modifications from III.A and III.B; equations in text follow from combining those sections.

### Linearized dynamics and implications for RER
- Variables stationarized by dividing by world technology A_t (growth Γ).
- Risk-sharing condition linearized (equation (9)) links q_t to marginal utilities, habit, preference shocks, and world growth innovations.
- Inflation dynamics for domestically produced goods (equations (10) and (11) in source): domestic inflation driven by unit labor costs, productivity shocks, terms of trade.
- Relation under LOP between q_t and terms of trade preserved as in text.
- Role of home bias δ and elasticity θ:
  - With symmetric home bias positive comovement between RER and terms of trade arises.
  - With no home bias (δ = 1/2), RER is constant (PPP holds).
  - Larger home bias (smaller δ) → larger RER volatility.
  - Under incomplete markets, low θ can make a real depreciation reduce the value of net exports if volumes respond little.

### Data, transformations, and sample
- Home = euro area; foreign = United States.
- Sample period: 1973:1 to 2003:4, quarterly.
- Observables and transformations:
  - Per capita output and consumption growth rates and inflation: natural logs and first differences; short-term interest rate divided by four; real exchange rate: natural logs and first differences; all variables demeaned.
- Data summary (raw-data quarterly growth rates, 1973:1–2003:4):
  - Consumption Euro: Mean 0.47; Std. dev. 0.57
  - Output Euro: Mean 0.47; Std. dev. 0.58
  - Consumption USA: Mean 0.53; Std. dev. 0.67
  - Output USA: Mean 0.48; Std. dev. 0.85
  - Real Exch. Rate: Mean -0.14; Std. dev. 4.59
- HP-filtered series statistics (selected):
  - Consumption Euro Std. dev. 0.91; Output Euro Std. dev. 1.01; Consumption USA Std. dev. 1.28; Output USA Std. dev. 1.58; Real Exch. Rate Std. dev. 7.83
  - Correlation with RER: Consumption Euro -0.26; Consumption USA -0.02; Relative Cons., RER -0.17
  - First autocorr.: Consumption Euro 0.84; Output Euro 0.86; Consumption USA 0.87; Output USA 0.87; Real Exch. Rate 0.83
- Key empirical observations preserved:
  - U.S. per capita consumption mean 0.53 percent vs euro area 0.47 percent.
  - Real exchange rate HP-filtered standard deviation: 7.83 percent vs U.S. output and consumption HP-filtered std. dev. 1.58 percent and 1.28 percent respectively.
  - Correlation between consumptions across countries: 0.33; correlation between RER and relative consumptions: -0.17.

### Bayesian estimation setup and priors
- Observables vector: {∆q, ∆p_H, ∆p_F, ∆c_H, ∆c_F, r_H, r_F, y_H, y_F} (exact ordering preserved in source).
- Stationarity: first differences for real variables; ∆q first-differenced; inflation and nominal interest rates in levels.
- Fixed parameters:
  - Γ = 0.5 percent (implying world growth of per capita variables ≈ 2 percent per year).
  - β = 0.995 (steady-state real interest rate ≈ 4 percent per year).
- Selected priors (from Table 5; parameter distributions, mean, Std. Dev. preserved):
  - Habit b, *b : Normal, Mean 0.70, Std. Dev. 0.05
  - γ, *γ : Normal, Mean 1.00, Std. Dev. 0.25
  - Average duration between optimal price changes (1/(1−α), 1/(1−α*) ): Gamma, Mean 3.00, Std. Dev. 1.42
  - τ, *τ : Uniform(0,1), Mean 0.50, Std. Dev. 0.29
  - Fraction of imported goods δ : Normal, Mean 0.20, Std. Dev. 0.03
  - θ : Normal, Mean 1.50, Std. Dev. 0.25
  - χ : Gamma, Mean 0.02, Std. Dev. 0.014
  - Taylor-rule inflation coefficient pγ : Normal, Mean 1.50, Std. Dev. 0.25
  - Taylor-rule output coefficient yγ : Normal, Mean 1.00, Std. Dev. 0.20
  - Taylor-rule smoothing rρ : Uniform(0,1), Mean 0.50, Std. Dev. 0.29
  - AR coefficients uniform(0,0.96), Std. dev. 0.28 (mean 0.48)
  - Shock std. dev. priors: technology xσ, *xσ, aσ Gamma, Mean 0.007, Std. Dev. 0.003; preference gσ Gamma, Mean 0.010, Std. Dev. 0.005; monetary zσ Gamma, Mean 0.004, Std. Dev. 0.002; demand ησ Gamma, Mean 0.010, Std. Dev. 0.005
- Estimation implementation:
  - Likelihood via state-space and Kalman filter; Metropolis-Hastings with 250,000 posterior draws; marginal likelihood via modified harmonic mean estimator.

### Posterior parameter estimates (selected posterior means and Std. Dev. from Table 6)
- Habit formation b across models: 0.57 (0.04), 0.78 (0.02), 0.72 (0.03), 0.62 (0.03), 0.64 (0.03) for Columns I–V respectively.
- (1−α)^{-1} (home average price duration) examples: 5.94 (0.89), 4.77 (0.46), 6.29 (0.62), 4.28 (0.39), 4.12 (0.47).
- (1−α)^{-1}* (foreign average price duration, starred value): 7.09 (0.92), 14.74 (1.68), 12.66 (1.36), 5.74 (0.63), 4.95 (0.63).
- τ (home indexation): 0.06 (0.07), 0.93 (0.06), 0.41 (0.17), 0.94 (0.06), 0.84 (0.13).
- δ (open models): 0.13 (0.02), 0.23 (0.02), 0.12 (0.01), 0.06 (0.01) across open columns.
- θ estimates (open models): 0.04 (0.03), 0.01 (0.01), 0.44 (0.00), 0.91 (0.01) (for Columns II–V as reported).
- χ (incomplete markets): 0.007 (0.004) under PCP (Column IV), 0.013 (0.008) under LCP (Column V).
- Shock std. dev. examples (in percent, posterior means and Std. Dev.):
  - xσ: 1.93 (0.45), 4.45 (0.52), 4.27 (0.48), 4.00 (0.55), 4.23 (0.54)
  - gσ: 2.32 (0.29), 5.72 (0.40), 4.78 (0.31), 3.64 (0.45), 3.41 (0.49)
- Log-Marginal likelihoods (Table 6 last row, open-economy models reported): 3981.6, 4033.2, 4106.8, 4070.9 (Columns II–V respectively).

### Model comparison and fit (marginal likelihoods, Bayes factors, and second moments)
- Marginal likelihood ranking:
  - Highest: incomplete markets with PCP (Column IV) — log-marginal 4070.9.
  - Second: incomplete markets with LCP (Column V) — log-marginal 4106.8 (note ordering preserved as in source; difference interpretation in text).
  - Source interpretation: models with incomplete markets outperform complete-market models by Bayes factors; reported difference between log marginals implies decisive evidence favoring Column IV when interpreted via Kass and Raftery (1995) and numeric comparison in text.
- Model-implied standard deviations for real exchange rate ∆q (selected):
  - Data ∆q: 4.59 (raw-data) and HP-filtered 7.83 (as reported earlier).
  - Complete, PCP: ∆q 4.96
  - Complete, LCP: ∆q 5.32
  - Incomplete, PCP: ∆q 4.90
  - Incomplete, LCP: ∆q 6.83
- Cross-correlations (HP-filtered) relevant to consumption–RER anomaly:
  - Data: c-c*, q = -0.17
  - Complete, PCP: c-c*, q = 0.03
  - Incomplete, PCP: c-c*, q = -0.37
  - Incomplete, LCP: c-c*, q = 0.04
- Fit implications:
  - Complete-market models can match RER volatility and persistence but at cost of excessive volatility/persistence in output, consumption, interest rates, and inflation (especially euro area).
  - Incomplete markets with PCP best match volatilities of real variables and the negative RER–relative consumption correlation while requiring smaller shock volatilities.
  - Incomplete + LCP over-predicts RER volatility (∆q 6.83 percent vs data 4.59 percent), reducing its Bayes-factor ranking.

### Preferred model dynamics and shock contributions (Incomplete markets with PCP)
- Contribution to RER variance (Table 8, percent):
  - Demand shock: 49.2 percent
  - Country-specific technology shocks: 35.5 percent
  - Monetary shocks: 9.0 percent
  - Preference shocks: 5.8 percent
  - World technology shocks: 0.6 percent
- Persistence measures (model-based autocorrelations for RER by shock):
  - Monetary shocks: 0.45
  - Country tech. shocks: 0.82
  - World tech. shocks: 0.92
  - Preference shocks: 0.97
  - Demand shocks: 0.87
- Interpretation:
  - Demand and country-specific technology shocks together explain most RER variability; monetary shocks play a minor role.
  - Only world technology and preference shocks generate high persistence but they contribute little to RER variance.
  - To jointly account for RER dynamics and the consumption–RER anomaly requires technology, preference, and demand shocks; incomplete markets help generate negative c-c*, q correlations when θ is sufficiently low.

### Impulse responses (selected qualitative points preserved)
- U.S. transitory technology shock:
  - Raises U.S. output and consumption, lowers U.S. inflation and interest rates; dollar depreciates (euro appreciates); euro area consumption expands while euro area output falls; euro area NFA improves and reverts slowly.
- U.S. demand shock:
  - Raises U.S. output and inflation, raises U.S. interest rates, reduces U.S. consumption; higher U.S. rates cause euro depreciation, temporarily boosting euro output and reducing euro consumption; persistent euro NFA decumulation follows.

### Forecasting performance (one-step ahead, in-sample MSEs; Table 9 highlights)
- All NOEM variants perform poorly forecasting the RER relative to VAR benchmarks.
- Representative RER one-step ahead MSEs (percent):
  - Complete, PCP: 4.55
  - Complete, LCP: 4.90
  - Incomplete, PCP (preferred): 4.52
  - Incomplete, LCP: 4.76
  - VAR(1): 4.27
  - VAR(2): 4.04
  - VAR(4): 3.73
  - VAR(6): 3.29
- Preferred model (incomplete, PCP) is the only NOEM variant among the four that beats a random walk with drift for the RER (preferred model MSE reported as 4.52 percent vs random walk with drift MSE 4.59 percent as noted in text).
- Overall: VARs with sufficient lags outperform DSGE models for RER forecasting; some DSGE variants outperform VAR(6) for other variables (consumption, interest rates, U.S. inflation).

### Main conclusions, caveats, and policy/modeling implications
- Main findings:
  - Incomplete markets with stationary NFA and PCP is the preferred specification by Bayes factor and matches the negative correlation between relative consumption and RER.
  - Demand and technology shocks are major drivers of RER volatility; monetary shocks play only a minor role.
  - Complete-market specifications imply near-zero θ and attribute little role to international trade/expenditure-switching.
- Caveats:
  - Model omits capital, sticky wages, and richer shock decompositions; technology-shock importance is thus an upper bound.
  - The LCP failure to improve fit may stem from not using import price series explicitly.
  - Using observed quarterly U.S. NFA as an observed variable produced broadly similar estimates and did not improve in-sample forecasting (exercise referenced in text).
- Suggested avenues for future research:
  - Explore other deviations from LOP and richer two-sector structures (tradable/nontradable, distribution services) to capture border deviations and exchange-rate pass-through.
  - Analyze out-of-sample forecasting performance of competing NOEM models.
  - Exploit available NFA estimates to inform model identification and policy analysis.

*Source: _wp06177*

### References .............................................................................................................

### _wp06177 - References

### Introduction and research question
- Most puzzles in international macroeconomics are related to real exchange rate dynamics: fluctuations in real exchange rates can be very large and persistent compared to other real variables, and there is clear evidence of lack of consumption risk-sharing across countries.
- The real exchange rate (q_t) between two currencies is defined as the ratio of the two countries’ price levels expressed in a common currency. In log-linear terms: q_t = s_t + p_t* - p_t.
- Under complete markets, with separable preferences and log utility, the equilibrium condition implies q_t = c_t - c_t*, yielding a correlation of one between the real exchange rate and the ratio of consumption levels; empirically, for the bilateral euro-U.S. dollar exchange rate the HP-filtered correlation is -0.17.
- Two primary modeling approaches to address empirical regularities: incorporate nominal rigidities (PCP vs. LCP pricing of imported goods) and/or introduce asset market incompleteness or preference shocks.

### Literature context and puzzles
- CKM (Chari, Kehoe, and McGrattan (2002)) construct a model with sticky prices and LCP and find that monetary shocks and complete markets, together with a high degree of risk aversion and price stickiness of one year, can account for real exchange rate volatility and partly for persistence; but their model struggles to account for the observed negative correlation between real exchange rates and relative consumption (the consumption-real exchange rate anomaly).
- Conventional modeling choices such as asset market incompleteness and habit persistence did not eliminate the CKM consumption-real exchange rate anomaly.

### Approach of the paper summarized in this content unit
- Uses a Bayesian approach to estimate and compare two-country NOEM models under alternative assumptions about imported goods pricing and asset market structure.
- The model includes a rich set of shocks (nine shocks because nine variables are being explained) and omits uncovered interest-rate parity (UIP)-type shocks to focus on the role of “traditional” shocks: technology, demand, monetary, and so on.
- Novel evaluation of incomplete markets assumption with stationary net foreign assets in a two-country NOEM model.
- Performs an in-sample forecast exercise and compares model forecasting performance to a vector autoregression (VAR) benchmark.

### Key empirical and methodological findings
- Monetary policy shocks have a minor role in explaining real exchange rate volatility; both demand and technology shocks have had some importance.
- Using the Bayes factor for model comparison, the crucial ingredient for explaining real exchange rate dynamics and the exchange rate-consumption anomaly is the introduction of incomplete markets with stationary net foreign asset positions.
- In a complete markets setup, introducing Local Currency Pricing (LCP) improves model fit.
- When incomplete markets are allowed, LCP lowers overall fit: it overestimates real exchange rate volatility and implies a lower correlation between the real exchange rate and the ratio of relative consumptions than in the data.
- The preferred model (with incomplete markets and stationary net foreign assets) does a good job in forecasting relative to other NOEM models, but still underperforms a VAR model in forecasting.

### Contributions relative to existing literature
- Focus on the relationship between relative consumption and the real exchange rate by introducing consumption data for the United States and the euro area.
- Excludes UIP-type shocks to concentrate on roles of traditional structural shocks in open economy settings.
- First evaluation (stated) of incomplete markets with stationary net foreign assets in a two-country NOEM model.
- Adds to the growing Bayesian estimation literature for NOEM models and complements prior work estimating small and medium-scale open economy models.

*Source: _wp06177 - References*

### Section V. First, we present the parameter estimates of the baseline model. Then, we analyze

### _wp06177 - Section V. First, we present the parameter estimates of the baseline model. Then, we analyze

### Overview and purpose
- Presents a stochastic two-country NOEM model to analyze real exchange rate dynamics.
- Sequence in the paper: parameter estimates of the baseline model; parameter estimates and second moments for extensions; model selection by Bayes factors; impulse response analysis of preferred model; variance decompositions to evaluate shock importance; comparison of forecasting performance of DSGE models vs VAR models.

### Model structure (baseline)
- Two countries: home and foreign, of equal size.
- Continuum of intermediate goods in each country indexed on ]1,0[.
- Preferences: Dixit-Stiglitz type implying monopolistic competition.
- All goods are internationally tradable.
- Nine shocks in the model:
  - World technology shock (unit root).
  - Country-specific stationary technology shocks (AR(1)).
  - Country-specific monetary shocks (iid).
  - Country-specific demand shocks (AR(1)).
  - Country-specific preference shocks (AR(1)).
- Baseline features that improve empirical properties: home bias in consumption, habit formation in consumption, staggered price setting with backward-looking indexation.
- Baseline assumption: complete markets and law of one price (LOP) for each intermediate good.

### Key variables and parameters (as defined)
- Consumption aggregate: C_t, C*_t.
- Consumer price indices: P_t, P*_t.
- Imports, domestically produced, intermediate goods price indices: notation in Table 1 and Table 2.
- Terms of trade: T_t.
- Real exchange rate: Q_t; nominal exchange rate: S_t.
- Interest rates: R_t, R*_t.
- Bonds: B_t, B*_t.
- Shocks: A_t (world technology), X_t, X*_t (country technology), G_t, G*_t (preference), z_t, z*_t (monetary), η_t, η*_t (demand).
- Preference and technology parameters:
  - Discount factor: β ∈ [0,1].
  - Habit persistence: b ∈ [0,1] (external habit; habit stock = last period’s aggregate consumption).
  - Inverse elasticity of labor supply: γ > 0.
  - Home-bias parameter: δ with symmetric home bias assumed so δ = δ*.
  - Elasticity of substitution between domestic and imported goods: θ.
  - Elasticity of substitution between types of intermediate goods: ε > 1.
  - World technology growth rate: Γ.
  - Calvo price-setting parameters: α (reoptimization frequency in home country; reoptimization probability α−1), τ (fraction indexing to last period’s inflation), and equivalent *α, *τ for foreign country.
  - Indexation parameters: b = (1/((1/β)τ + 1))?? (definitions in text: b ≡ (1/((1/β)τ + τ?)) — preserve original expressions where present).
  - Inflation slope: κ ≡ ])1/[()1)(1(αβτααβκ+−−≡ (expression as in text).

### Households and risk sharing (baseline complete markets)
- Consumers have access to complete markets at country and world level; wealth is insured across country-specific and world shocks.
- Households’ lifetime utility (home country): Sum from t=0 to ∞ of β^t E_0 { log(C_t - b C_{t-1}) - (N_t)^{1+γ}/(1+γ) } (utility with external habit formation).
- Complete-market risk-sharing condition: real exchange rate equals ratio of marginal utilities (equation (4)):
  - Q_t = ν ( (C_t - b C_{t-1}) / (C*_t - b C*_{t-1}) )^{ -1 } times terms including preference shocks — see equation (4) for exact expression.
- Under complete markets, unique stochastic discount factor 1,+t ξ exists; households maximize utility subject to the budget constraint (3) with state-contingent nominal payoffs 1+t B in domestic currency.

### Intermediate goods producers and price-setting
- Production linear in labor with two technology shocks: world (unit root) and country-specific (AR(1)).
- Firms face modified Calvo pricing: probability of being able to reoptimize = α−1; if not, a fraction τ index to last period’s inflation, fraction 1−τ index to steady-state inflation (to incorporate trend inflation).
- Real marginal cost = ω_t / A_t (or expressed as (X_t A_t)?? — cost minimization implies real marginal cost is )/( ttt XAω as in text).
- Demand for an intermediate good produced in h depends on home and foreign consumption and price ratios; expression given in text.
- Reoptimizing firm maximizes discounted expected sum of profits (equation (5)); optimal reset price appears in aggregate price evolution (equation (6)).

### Market clearing and closure
- For each good h: y_{h,t} + c_{h,t}^* = ... (market clearing for each individual good); aggregate real GDP definition in text.
- Home-country goods market clearing condition (equation (7)) and analogous foreign condition (equation (8)).
- Model includes exogenous demand shock η_t per country (interpreted as government purchases and/or trade with third countries).
- Monetary policy: each country follows a Taylor-type rule (specified in Table 3 and discussed in II.F).

### Stationarization and symmetric equilibrium
- World-wide technology shock grows at rate Γ; output, consumption, real wages, and exogenous demand grow at Γ.
- To render variables stationary, divide by level of world technology A_t; hours, inflation, interest rates, real exchange rate, and terms of trade are stationary.
- Symmetric home bias: δ = δ*.
- Lowercase variables denote percent deviations from steady state; tilde variables normalized by technology.

### Linearized dynamics (key relations affecting real exchange rate)
- Transformation between normalized variables and first differences involves world technology a_t: relationships shown in text (first-difference definitions).
- Risk-sharing condition linearized (equation (9)): real exchange rate depends on ratio of marginal utilities, habit stocks, preference shocks, and world growth innovations when effects differ across countries due to habit parameters.
- Euler equations, labor supply, goods market clearing, production functions, Taylor rules, terms of trade dynamics summarized in Table 3 (linearized equations).
- Inflation dynamics for domestically produced goods (home and foreign) given by equations (10) and (11): domestic inflation driven by unit labor costs (real wage), productivity shocks, and terms of trade.
- When law of one price holds, relation between real exchange rate and terms of trade: q_t = ( )21( t δ− = ) — exact expression in text: .)21( tt t qδ−= (text indicates q_t and terms of trade relation).
- CPI inflation combines domestic inflation and imported goods; under LOP nominal exchange rate has direct impact on CPI inflation: equations (12) and (13) in baseline.

### Role of home bias and elasticity of substitution
- With symmetric home bias, positive comovement between real exchange rate and terms of trade arises, consistent with data.
- With no home bias (δ = 1/2), real exchange rate is constant and PPP holds.
- Degree of home bias crucial for real exchange rate volatility: larger home bias (smaller δ) → larger real exchange rate volatility.
- Under incomplete markets, the effect of a real depreciation on net exports depends critically on elasticity of substitution θ: with low θ, a real depreciation can reduce value of net exports because volumes increase less than prices decline.

### Extensions to the baseline
A. Incomplete markets with stationary net foreign assets (NFA)
- Asset structure: home households can trade two nominal riskless bonds (domestic and foreign currency); home households face a cost of holding foreign bonds. Foreign residents constrained to foreign-currency bonds.
- Firms remain domestically owned; profits distributed equally.
- Real budget constraint under incomplete markets: equation (3’) in text; φ(.) depends on real holdings of foreign assets in the economy and is taken as given by households. φ(.) must be differentiable and decreasing near zero and φ(0)=0 to achieve stationarity.
- Risk-sharing condition under incomplete markets (expected first difference form): equation (9’) in text. Introduces χ = H Y0'φ− and b*_t = (P_t Y_t)/(S_t B_t^* ) as defined in text (exact definitions included there).
- NFA becomes a state variable with law of motion given by equation (14) in text (log-linearized): .~~21 1)1(2 **1* ttttt ccqbb−− ⎥⎦⎤⎢⎣⎡− −− += δ δ δ θ δ β (preserve exact expression as in text).
- Stationarity of NFA requires transaction cost specification and φ(.) properties.

B. Local-currency pricing (LCP) by intermediate goods producers (imperfect pass-through)
- Assumption: price stickiness for import prices in local currency; firms set a price for domestic market and a price for foreign market under modified Calvo with indexation.
- Generates deviations from law of one price at the border; nominal exchange rate movements generate ex-post deviations from LOP.
- Under LCP, even without home bias, real exchange rate fluctuations can be generated.
- Demand expressions for an intermediate good produced in h include price ratios defined in local-currency terms (text defines new terms of trade: p_t^f − p_t^h etc.).
- New inflation equations (home-produced goods) under LCP:
  - Home domestic inflation: equation (10’) (same as baseline for domestic prices).
  - Home imported/foreign-priced component: equation (10b’) includes q_t terms: ()],1 ~ [**1, *1, *, ttttt H t f H b t H qtxpEpp −−−−+∆+∆=∆ +− δωκγγ (exact as in text).
  - Analogous equations for foreign country: (11’) and (11b’).
- CPI inflation under LCP does not include nominal exchange rate as direct determinant of imported goods inflation because pass-through is low; CPI inflation equations become (12’) and (13’) in text:
  - tFtHt p p p ,, )1( ∆ + ∆ − = ∆ δ δ  (12’)
  - * , * , * )1( tFtHt p p p ∆ − + ∆ = ∆ δ δ  (13’)
- Market-clearing conditions adjust as in text (expressions provided).

C. Combined extension: incomplete markets + LCP
- Equations under both assumptions are given by combining the respective modified relations described above (text indicates these equations follow from combining sections III.A and III.B).

### Model solution and estimation approach (brief)
- Dynamics obtained by linear approximation around steady state at zero inflation.
- Observables in estimation include first-differenced real variables consistent with linearization and normalization by technology.
- Extensions (incomplete markets, LCP) change the linkages between real exchange rate, terms of trade, inflation, and net foreign asset dynamics — affecting second moments and impulse responses used for model comparison and selection (Bayes factors) and for forecasting comparisons with VAR models.

*Source: _wp06177 - Section V. First, we present the parameter estimates of the baseline model. Then, we analyze*

### Section II.F, Table 3, and modified by those in Section III.B. The additional change is that

### _wp06177 - Section II.F, Table 3, and modified by those in Section III.B. The additional change is that

### Model modification: NFA position dynamics and real exchange rate
- The behavior of the real exchange rate is the same as under incomplete markets (Equation 9’ in Section III.A).
- The net foreign asset (NFA) position dynamics substitute equation (14) in Section III.A with equation (14’):
  - ().
  - ~~
  - ))(1)(1(
  - ***
  - 1
  - *
  - ttttttt
  - ccqttbb−−+−−−+=
  - −
  - δδθδδβ                         (14’)            

### IV. Estimation and model comparison — Data (Section IV.A)
- Countries and series used:
  - United States (Haver USECON mnemonics): quarterly real GDP (GDPH), the GDP deflator (DGDP), real consumption (CH), 3-month T-bill interest rate (FTB3).
  - Euro area: Fagan, Henry, and Maestre (2001) dataset — real consumption (PCR), real GDP (YER), the GDP deflator (YED), short-term interest rates.
  - Population: total population of 16 years and over (LN16) for per capita conversions; euro zone population from Eurostat (annual data linearly interpolated to quarterly frequency).
- Home/foreign convention:
  - Home country = euro area; foreign = United States.
  - Real exchange rate = nominal exchange rate in euros per U.S. dollar × U.S. CPI ÷ euro area CPI.
  - Synthetic euro/U.S. dollar exchange rate prior to 1999 from Eurostat; U.S. CPI from Haver USECON (PCU); euro area CPI from Fagan, Henry, and Maestre (HICP).
- Sample period:
  - 1973:1 to 2003:4, quarterly frequency.
- Transformations and measurement:
  - Per capita output and consumption growth rates and inflation: natural logs and first differences of per capita output and consumption, and of the GDP deflator, respectively.
  - Short-term interest rate divided by four to obtain quarterly equivalent.
  - Natural logs and first differences of the euro/dollar real exchange rate.
  - All variables expressed as deviations from their sample mean.

### Data properties and descriptive findings (Table 4 and discussion)
- Raw-data quarterly growth rates (1973:1–2003:4) — sample means (percent) and standard deviations (percent) preserved as presented:
  - Consumption Euro: Mean 0.47; Std. dev. 0.57
  - Output Euro: Mean 0.47; Std. dev. 0.58
  - Consumption USA: Mean 0.53; Std. dev. 0.67
  - Output USA: Mean 0.48; Std. dev. 0.85
  - Real Exch. Rate: Mean -0.14; Std. dev. 4.59
- Raw-data quarterly rates:
  - Interest Rate Euro: Mean 2.08; Std. dev. 0.83; First autocorr. 0.96
  - Inflation Euro: Mean 1.44; Std. dev. 0.93; First autocorr. 0.89
  - Interest Rate USA: Mean 1.59; Std. dev. 0.73; First autocorr. 0.94
  - Inflation USA: Mean 1.00; Std. dev. 0.67; First autocorr. 0.90
- HP-filtered series statistics:
  - Consumption Euro Std. dev. 0.91; Output Euro Std. dev. 1.01; Consumption USA Std. dev. 1.28; Output USA Std. dev. 1.58; Real Exch. Rate Std. dev. 7.83
  - Correlation with RER: Consumption Euro -0.26; Output Euro -0.06; Consumption USA -0.02; Output USA -0.08; Real Exch. Rate 1.00
  - First autocorr.: Consumption Euro 0.84; Output Euro 0.86; Consumption USA 0.87; Output USA 0.87; Real Exch. Rate 0.83
  - Other correlations: Consumption Euro, USA 0.33; Output Euro, USA 0.47; Relative Cons., RER -0.17; Relative Outputs, RER 0.04
- Key empirical observations:
  - Per capita output growth rates in the United States and the euro area are similar: 0.48 percent versus 0.47 percent.
  - Per capita consumption and output in the euro area grow at the same rate: 0.47 percent.
  - Consumption growth in the United States has a higher sample mean (0.53 percent) than in the euro area.
  - Growth rates in the euro area are less volatile than in the United States.
  - Real exchange rate displays a small appreciating trend mean and is much more volatile than other series.
  - Bilateral real exchange rate HP-filtered standard deviation: 7.83 percent, versus U.S. output and consumption HP-filtered standard deviations of 1.58 percent and 1.28 percent, respectively.
  - Only consumption in the euro area displays a nonzero correlation of -0.26 with the real exchange rate; correlations of euro area output, and U.S. output and consumption with the real exchange rate are essentially zero.
  - Correlation between consumptions across countries (0.33) is smaller than between outputs (0.47).
  - Correlation between the real exchange rate and relative consumptions across countries is negative (-0.17), which is at odds with efficient risk-sharing.

### Bayesian estimation: likelihood, priors, and implementation (Section IV.B and Table 5)
- Observables vector used in estimation:
  - {}',,,,,,,,*
  - ,***
  - ,ttFttttHtttt
  - qprcyprcyz∆∆∆∆∆∆∆=.
- Stationarity and detrending:
  - World technology shock with a unit root makes real variables stationary in first differences; consumption and output growth per country used.
  - Real exchange rate is first-differenced; inflation and nominal interest rate enter in levels.
  - Variables expressed as deviations from sample mean.
- Fixed parameters:
  - Steady-state growth rate set Γ=0.5 percent (implying world growth rate of per capita variables about 2 percent per year).
  - Discount factor set β=0.995 (to match a real interest rate in steady state of about 4 percent per year).
- Prior distributions (Table 5) — parameter, distribution, mean, Std. Dev.:
  - Habit formation b, *b : Normal, Mean 0.70, Std. Dev. 0.05
  - Labor supply γ, *γ : Normal, Mean 1.00, Std. Dev. 0.25
  - Average duration between optimal price changes 1)1(−−α, 1* )1(−−α : Gamma, Mean 3.00, Std. Dev. 1.42
  - Indexation τ, *τ : Uniform(0,1), Mean 0.50, Std. Dev. 0.29
  - Fraction of imported goods δ : Normal, Mean 0.20, Std. Dev. 0.03
  - Elasticity of substitution between home and foreign goods θ : Normal, Mean 1.50, Std. Dev. 0.25
  - Elasticity of the real exchange rate to the NFA position χ : Gamma, Mean 0.02, Std. Dev. 0.014
  - Taylor rule: inflation p γ, * p γ : Normal, Mean 1.50, Std. Dev. 0.25
  - Taylor rule: output growth y γ, * y γ : Normal, Mean 1.00, Std. Dev. 0.20
  - Taylor rule: smoothing r ρ, * r ρ : Uniform(0,1), Mean 0.50, Std. Dev. 0.29
  - AR coefficients of shocks x ρ, * x ρ, g ρ, * g ρ, η ρ, * η ρ : Uniform(0,0.96), Mean 0.48, Std. Dev. 0.28
  - Std. dev. technology shocks x σ, * x σ, a σ : Gamma, Mean 0.007, Std. Dev. 0.003
  - Std. dev. preference shocks g σ, * g σ : Gamma, Mean 0.010, Std. Dev. 0.005
  - Std. dev. monetary shocks z σ, * z σ : Gamma, Mean 0.004, Std. Dev. 0.002
  - Std. dev. demand shocks η σ, * η σ : Gamma, Mean 0.010, Std. Dev. 0.005
- Additional prior comments:
  - Habit formation truncated between 0 and 1.
  - Prior mean for average duration of price contracts: 3 (implying D=1/(1-α)-1 specification).
  - Prior on indexation fraction: Uniform(0,1).
  - Open-economy priors: δ mean 0.2 (std. dev. 0.03); θ centered at 1.5 with substantive standard deviation; χ gamma mean 0.02 std. dev. 0.014.
  - Taylor-rule coefficients centered on Rabanal (2004): p γ prior mean 1.5, y γ prior mean 1.
  - AR(1) shock priors uniform between 0 and 0.96 to allow persistence exploration.
  - Gamma priors for standard deviations to avoid negative values; prior means chosen to match previous studies (e.g., 0.007 for technology shocks).
- Estimation implementation:
  - Likelihood evaluated via state-space representation and Kalman filter.
  - Metropolis-Hastings algorithm used to draw from the posterior.
  - Results based on 250,000 draws from the posterior distribution.
- Marginal likelihood and model comparison:
  - Marginal likelihood defined as the integral of likelihood times prior over parameter space (equation (17) form given).
  - Marginal likelihood approximated using the modified harmonic mean estimator.
  - Posterior odds ratio between models equals Bayes factor when prior model probabilities are equal.

### V. Results — overview of estimation stages and findings
- Five-stage reporting structure:
  1. Posterior estimates for closed economy vis-à-vis four open-economy specifications.
  2. Model comparison via marginal likelihood for each model.
  3. Computation of standard deviations and correlations of each model at posterior mode values.
  4. Analysis of dynamics of preferred model: importance of structural shocks for real exchange rate fluctuations.
  5. One-step ahead in-sample forecast performance of all models and comparison to VARs.
- Posterior distributions and benchmark comparisons:
  - Closed-economy benchmark: each country estimated as closed economy with δ=θ=0 and no international bond trade; real exchange rate dropped from observables. Estimates align with literature (e.g., Galí and Rabanal (2005); Rabanal and Rubio-Ramírez (2005); Lubik and Schorfheide (2005)).
  - Benchmark open-economy model (complete markets and PCP) results (Column II) differ from closed economy:
    - Proportion of firms indexing prices to lagged inflation increases to almost one in the euro area; inflation remains almost purely forward looking in the United States.
    - Average duration of price contracts: decreases for the euro area to 4.77 quarters and increases significantly for the United States (value cut off in supplied text).

*Source: _wp06177 - Section II.F, Table 3, and modified by those in Section III.B. The additional change is that; Source PDF filename: _wp06177 - Section II.F, Table 3, and modified by those in Section III.B. The additional change is that; Canonical source URL: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2006/_wp06177.pdf*

### 14.74 quarters, which is a fairly large number.

### _wp06177 - 14.74 quarters, which is a fairly large number.

### Posterior distributions and key parameter estimates
- Table 6 reports posterior means and standard deviations for a set of model specifications: I. Two Closed Economies; II. Complete Markets, PCP; III. Complete Markets, LCP; IV. Incomplete Markets, PCP; V. Incomplete Markets, LCP.
- Selected posterior means (Std. Dev. where reported) from Table 6:
  - b: 0.57 (0.04), 0.78 (0.02), 0.72 (0.03), 0.62 (0.03), 0.64 (0.03)
  - b* : 0.61 (0.04), 0.69 (0.04), 0.68 (0.04), 0.55 (0.04), 0.54 (0.03)
  - γ: 1.09 (0.23), 1.25 (0.21), 1.47 (0.20), 1.18 (0.20), 0.79 (0.23)
  - γ* : 1.00 (0.21), 0.86 (0.25), 1.02 (0.23), 1.16 (0.20), 1.03 (0.23)
  - (1−α)^{-1}: 5.94 (0.89), 4.77 (0.46), 6.29 (0.62), 4.28 (0.39), 4.12 (0.47)
  - (1−α)^{-1}* (starred value highlighted in Table): 7.09 (0.92), 14.74 (1.68), 12.66 (1.36), 5.74 (0.63), 4.95 (0.63)
  - τ: 0.06 (0.07), 0.93 (0.06), 0.41 (0.17), 0.94 (0.06), 0.84 (0.13)
  - τ* : 0.09 (0.08), 0.04 (0.04), 0.06 (0.05), 0.09 (0.07), 0.09 (0.08)
  - δ (only in open models): 0.13 (0.02), 0.23 (0.02), 0.12 (0.01), 0.06 (0.01)
  - θ (intratemporal elasticity of substitution): 0.04 (0.03), 0.01 (0.01), 0.44 (0.00), 0.91 (0.01) for columns II–V (not reported for closed economy)
  - χ (real exchange-rate elasticity wrt foreign debt, incomplete markets): 0.007 (0.004) under PCP (Column IV), 0.013 (0.008) under LCP (Column V)
  - Autocorrelation parameters (examples): rρ 0.87 (0.02), 0.88 (0.01), 0.87 (0.01), 0.89 (0.01), 0.88 (0.01)
  - Shock standard deviations (examples, in %): xσ 1.93 (0.45), 4.45 (0.52), 4.27 (0.48), 4.00 (0.55), 4.23 (0.54); gσ 2.32 (0.29), 5.72 (0.40), 4.78 (0.31), 3.64 (0.45), 3.41 (0.49)
  - Log-Marginal reported row: 3981.6, 4033.2, 4106.8, 4070.9

### Open-economy vs closed-economy estimation: changes and drivers
- Taylor-rule estimates:
  - For the United States: estimates are similar to closed economy specification.
  - For the euro area: the estimated coefficient on inflation rises from 1.59 to 2.24 and the coefficient on output decreases from 1.08 to 0.07 (open economy, complete markets, PCP specification).
- Persistence and volatility of shocks increase markedly in open-economy complete-market specifications:
  - Standard deviations of shocks (except for monetary policy and demand shocks) are two or three times larger than in the closed economy case.
  - Autocorrelation of technology shocks in the United States and of preference shocks in the euro area increases to 0.96 (the upper bound allowed in the estimation) as the model matches a highly volatile real exchange rate with less volatile output and consumption.

### Home bias, elasticity θ, law of one price assumptions (PCP vs LCP)
- Baseline NOEM estimates:
  - Implied degree of home bias toward home goods is 0.87 in the benchmark two-country model (below 0.984 used by CKM (2002) and Heathcote and Perri (2002)).
  - Baseline two-country model delivers a very small estimate for θ; prior centered at 1.5 but data indicate a much smaller value.
- Under LCP (deviations from the law of one price):
  - Implied home-bias drops from 0.87 to 0.77 in the euro area (Column III).
  - Proportion of firms indexing prices to inflation drops from 0.93 to 0.41.
  - LCP adds endogenous volatility and persistence to the real exchange rate (equations (17) and (18) describe decomposition with lopt denoting deviations from the law of one price).
- Interpretation:
  - Equation (16) shows volatility of relative outputs increases with θ for a given volatility of the real exchange rate; hence low θ helps match observed low relative-output volatility together with a volatile real exchange rate.
  - Under PCP, a strong expenditure-switching effect would occur after devaluation; θ near zero neglects this effect.

### Incomplete markets results and implications (PCP and LCP)
- θ estimates increase under incomplete markets:
  - Point estimates: θ = 0.45 under PCP (Column IV) and θ = 0.91 under LCP (Column V).
  - Intuition: to avoid explosive NFA dynamics after real depreciation, θ needs to be near 1/2 under PCP and near 1 under LCP; the law of motion of the NFA position implicitly pushes θ upward.
- χ estimates reported in incomplete markets:
  - χ = 0.007 (PCP), χ = 0.013 (LCP).
  - Comparison: these χ values are larger than Bergin (2004) G-7 estimate 0.0038 and smaller than Lane and Milesi-Ferretti (2001) panel OECD estimate 0.0254; other studies report ranges 0.004–0.071.
- Effect on shock volatilities:
  - Volatility of shocks affecting the U.S. economy becomes much smaller under incomplete markets (even productivity shock standard deviations roughly half the size of those under closed economy).
  - In the euro area, estimated volatility of shocks is smaller under incomplete markets, though the reduction is less pronounced than in the U.S.
- Conclusion: estimation results give support to an asset market structure with incomplete markets and a stationary net foreign asset position; incomplete markets improve internal dynamics by requiring smaller shocks.

### Model comparison: marginal likelihoods and Bayes factors
- Marginal likelihoods (last row of Table 6): 3981.6, 4033.2, 4106.8, 4070.9 (reported for the four open economy models; marginal likelihood for closed economies not computed).
- Ranking and interpretation:
  - Models with incomplete markets yield higher marginal likelihood than those with complete markets (either PCP or LCP).
  - Highest marginal likelihood: incomplete markets model with PCP (Column IV); second: incomplete markets model with LCP (Column V).
  - Difference between log marginals of Columns IV and V is about 36, implying decisive Bayesian evidence favoring Column IV; interpreting via Kass and Raftery (1995), accepting the second model would require a prior favoring it by a factor of 3.9*10^15.

### Second moments and model fit (Table 7 highlights)
- Data (selected standard deviations, in percent):
  - Euro Area: ∆c 0.57, ∆y 0.58, r 0.83, ∆p 0.93
  - United States: ∆c* 0.67, ∆y* 0.85, r* 0.73, ∆p* 0.67
  - Real exchange rate ∆q: 4.59
- Model-implied standard deviations (selected comparisons):
  - Closed Economy: Euro Area ∆c 0.73, ∆y 0.81; United States ∆c* 0.85, ∆y* 1.00; ∆q not reported.
  - Complete, PCP: ∆q 4.96
  - Complete, LCP: ∆q 5.32
  - Incomplete, PCP: ∆q 4.90
  - Incomplete, LCP: ∆q 6.83
- Autocorrelations (data vs models, examples):
  - Data: c 0.84, y 0.86, r 0.96, ∆p 0.89, q 0.83
  - Incomplete, PCP: c 0.87, y 0.85, r 0.94, ∆p 0.79, q 0.72
  - Incomplete, LCP: c 0.87, y 0.86, r 0.93, ∆p 0.78, q 0.70
- Cross-correlations (HP-filtered, data vs models):
  - Data: c,c* 0.33, c,y 0.81, y,y* 0.47, c*,y* 0.85, c-c*,q −0.17, y-y*,q 0.04
  - Complete, PCP: c,c* 0.32, c,y 0.97, y,y* 0.48, c*,y* 0.91, c-c*,q 0.03, y-y*,q 0.09
  - Incomplete, PCP: c,c* 0.53, c,y 0.86, y,y* 0.60, c*,y* 0.76, c-c*,q −0.37, y-y*,q 0.09
  - Incomplete, LCP: c,c* 0.53, c,y 0.90, y,y* 0.44, c*,y* 0.78, c-c*,q 0.04, y-y*,q 0.26
- Overall fit conclusions:
  - Baseline (complete-market) models can match real exchange rate volatility and persistence, but at the cost of excessive volatility and persistence in output and consumption and of interest rates and inflation in the euro area.
  - Allowing LCP yields slightly better fit for real variables but reduces fit to real exchange rate volatility.
  - Incomplete markets best match volatilities of real variables and reduce required shock sizes, but incomplete+LCP (Column V) over-predicts real exchange rate volatility (∆q rising from 4.90 percent to 6.83 percent while data is 4.59 percent), which explains why that richer model does not rank best by Bayes factor.

*Source: _wp06177 - 14.74 quarters, which is a fairly large number.*

### 0.20 respectively). CKM (2002) refer to this discrepancy between the models and the data as

### _wp06177 - 0.20 respectively). CKM (2002) refer to this discrepancy between the models and the data as

### Key findings on consumption–real exchange rate anomaly and incomplete markets
- CKM (2002) label the discrepancy between models and data the "consumption real exchange rate anomaly."
- The proposed incomplete markets extension replicates the data feature: correlation between relative consumption across countries and the real exchange rate is -0.37.
- Extending the incomplete markets model with deviations from the law of one price (LCP) yields a correlation close to zero, worsening fit relative to the incomplete markets with PCP model.
- Under complete markets the correlation between relative consumption ratio and the real exchange rate is 1; preference and world technology shocks disturb this perfect correlation.
- Under incomplete markets with the law of one price (PCP), the NFA position is given (as in source).
- Under financial autarky the NFA position is zero at all times and equilibrium condition becomes (as in source).
- The sign of the correlation between relative consumptions and the real exchange rate can be positive or negative depending on parameters θ and δ.
- When 1=θ the model mimics complete markets and qt = ct - c*t for all t.
- Low values of θ generate negative correlations between relative consumptions and the real exchange rate.
- The incomplete markets model is intermediate between financial autarky and complete markets and can generate negative correlation for sufficiently low θ.
- Preference shocks also contribute to explaining the lack of risk sharing.

### Model comparison and estimation approach
- Table 7 (referenced) shows each model matches particular data moments better than others.
- Bayesian (likelihood-based) model comparison via marginal likelihood ranks the model that fits most features of the data best.
- The preferred model is the incomplete markets model with PCP.

### Shocks and real exchange rate dynamics (preferred model: incomplete markets with PCP)
- Contributions to the standard deviation of the real exchange rate (from Table 8):
  - Demand shock: 49.2 percent
  - Country-specific technology shocks: 35.5 percent
  - Monetary shocks: 9.0 percent
  - Preference shocks: 5.8 percent
  - World technology shocks: 0.6 percent
- Note: the importance of technology shocks is an upper bound because the model lacks capital, sticky wages, and separates price/wage/temporary technology shocks only lumped in a composite shock.
- Only world technology and preference shocks generate a highly persistent real exchange rate response, but they do not explain real exchange rate variability.
- Monetary policy shocks alone yield first autocorrelation of the real exchange rate of 0.49 and cannot account for observed real exchange rate persistence.
- Models with only monetary or world technology shocks produce positive correlations between relative consumptions and the RER of 0.79 and 0.42, respectively.
- Conclusion: to account simultaneously for real exchange rate dynamics and the consumption–real exchange rate anomaly requires technology, preference, and demand shocks.

### Table 8 — Selected quantitative moments from the preferred model (high-level points)
- Percent variance contributions (real exchange rate column highlighted above): Demand shocks 49.2 percent; Country tech. shocks 35.5 percent; Monetary shocks 9.0 percent; Preference shocks 5.8 percent; World tech. shocks 0.6 percent.
- Autocorrelations (data): real exchange rate autocorrelation = 0.96.
- Model-based autocorrelations for real exchange rate under individual shocks:
  - Monetary shocks: 0.45
  - Country tech. shocks: 0.82
  - World tech. shocks: 0.92
  - Preference shocks: 0.97
  - Demand shocks: 0.87
- Other correlations (data):
  - c,c* = 0.33
  - c,y = 0.81
  - c,c* (second listing) = 0.47
  - c*,y* = 0.85
  - c-c*,q = -0.17
  - y-y*,q = 0.04

### Impulse response analysis (focus on technology and demand shocks)
- U.S. transitory technology shock (Figure 1):
  - Expands U.S. output, reduces inflation, lowers U.S. interest rates, increases U.S. consumption.
  - Dollar depreciates (euro appreciates), euro area consumption expands but euro area output is reduced.
  - Real exchange rate appreciation of the euro is consistent with a decrease in U.S. domestic prices due to improved productivity.
  - Euro area net foreign asset (NFA) position improves and returns to steady state very slowly.
  - Euro area technology shocks produce similar responses with opposite sign and lower persistence.
- U.S. demand shock (Figure 2):
  - Positive U.S. demand shock increases U.S. output and inflation, raises U.S. interest rates, and reduces U.S. consumption.
  - U.S. interest rate increase causes the euro to depreciate, leading to short-lived boost to euro area output and decline in euro area consumption.
  - Persistent NFA decumulation in the euro area due to wealth effects; slow recovery of U.S. consumption and gradual euro area NFA increase to steady state.
  - Demand-shock-driven impulse responses imply negative comovement between the real exchange rate and relative consumptions, consistent with the data.

### Forecast performance (one-step ahead in-sample MSEs; Table 9 highlights)
- Table 9 shows Mean Squared Errors (MSE) of one-step ahead forecasts (in percent) for models and VARs for selected variables.
- All versions of the NOEM model perform poorly forecasting the real exchange rate.
- The preferred model (incomplete markets with PCP) is the only NOEM variant among the four that beats a simple random walk with drift for the real exchange rate (preferred model MSE for RER reported in table as 4.52 percent).
- Representative MSE entries from Table 9 (RER column, United States):
  - Complete, PCP: 4.55
  - Complete, LCP: 4.90
  - Incomplete, PCP (preferred): 4.52
  - Incomplete, LCP: 4.76
  - VAR(1): 4.27
  - VAR(2): 4.04
  - VAR(4): 3.73
  - VAR(6): 3.29

*Italic: Content derived from the provided IMF working paper excerpt.*

### 4.56 percent, while the random walk with drift has a MSE of 4.59 percent. Interestingly, a

### VI. CONCLUDING REMARKS

### Main findings on forecasting performance and model comparison
- A VAR with just one lag does not perform better than the DSGE models; only after including 4 lags in the VAR does the MSE for forecasting the real exchange rate drop significantly.
- Forecasting performance examples:
  - One model comparison reports a MSE of 4.56 percent, while the random walk with drift has a MSE of 4.59 percent.
- None of the DSGE models does a particularly good job in forecasting real exchange rates.
- Some versions of the NOEM model outperform the VAR that includes six lags in forecasting:
  - consumption in both areas,
  - interest rates in both areas,
  - inflation in the United States.
- The forecasting performance of the five DSGE models is quite similar; no single model stands out as best across all variables.

### Estimation results and interpretation
- Complete markets assumption:
  - Leads to attributing a very small role to international trade.
  - Estimated parameter for the elasticity of substitution between home and foreign goods is very low (near-zero), implying negligible expenditure-switching effect of a real devaluation.
- Incomplete markets assumption:
  - Receives empirical support from the Bayesian estimation.
  - The baseline model with complete markets and law of one price performs well for real exchange rate dynamics but implies volatilities too large for other real variables, especially in the euro area.
  - The addition of incomplete markets, where the law of one price holds, produces the best fit:
    - Simultaneously accounts for real exchange rate volatility and persistence,
    - Accounts for the negative correlation between the real exchange rate and relative consumptions.
  - A model with both incomplete markets and sticky import prices in local currency (LCP) does not perform as well as incomplete markets alone, but still outperforms models with complete markets.
- Shocks contribution:
  - Demand and technology shocks have played a major role in explaining the behavior of the real exchange rate.
  - Monetary shocks have not played a major role.

### Caveats and robustness notes
- Using net foreign asset position data could enrich real exchange rate dynamics (Lane and Milesi-Ferretti (2001)); an exercise using quarterly U.S. net foreign asset position as an observed variable produced broadly similar estimated parameters and did not improve in-sample forecast performance (results available upon request).
- The failure of the LCP assumption may be due to not explicitly using import price series.

### Policy and modelling implications / avenues for future research
- Explore other forms of deviations from the law of one price to assess implications for aggregate data.
- Incorporate distribution services in two-sector (tradable and nontradable goods), two-country models to capture border deviations from the law of one price and explain differential exchange rate pass-through (Campa and Goldberg (2004) evidence).
- Analyze out-of-sample forecasting performance of competing NOEM models (as in Del Negro et al. (2004) for closed economies).
- Exploit information content of available estimates of the net foreign asset position (Lane and Milesi-Ferretti, 2001) to clarify the role of these models for policy formulation and analysis.

---

### APPENDIX: THE METROPOLIS-HASTINGS ALGORITHM

### Algorithm implementation (summary of steps)
- Goal: obtain a random draw of size N from the posterior distribution via a random walk Markov Chain using the Metropolis-Hastings algorithm.
- Step 1:
  - Start with an initial value ( 0 ψ).
  - From that value, evaluate the product )()|}({ 00 1 ψψΠ = T tt xL.
- Step 2: For each i:
  - 1− = ii ψψ with probability 1-R
  - { ,*ii ψψ= with probability R
  - where iii v+= −1,* ψψ, i v follows a multivariate Normal distribution, and
  - } )()|}({ )()|}({ ,1min{ 11 1,*,* 1 −− = = Π Π = iiT tt iiT tt xL xL R ψψ ψψ .
- Rationale:
  - Regardless of the starting value, more draws will be accepted from parameter regions where the posterior density is high; tails are less represented but eventually visited.
  - The variance-covariance matrix of i v is proportional to the inverse Hessian of the posterior mode; the constant of proportionality is specified so the random draw has desirable time series properties.

### Implementation details and diagnostics
- Acceptance rates were between 20 and 30 percent.
- Autocorrelation functions of the parameters decay fairly fast.
- Two methods used to simulate the posterior delivered the same result (with very small numerical differences):
  1. Simulate the posterior 250,000 times taking as initial value the prior mean, updating the Hessian of the posterior every 25,000 draws; discard all values from that chain, and from the last value generate a second chain of size 250,000, updating the Hessian each 25,000 draws.
  2. Find the posterior mode using standard optimization algorithms to be used as initial value; then generate a chain of 250,000 draws, updating the Hessian every 25,000 draws.

*Excerpt from VI. CONCLUDING REMARKS and Appendix of the provided PDF content.*

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