## _wp09212

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---

### Introduction and central puzzle
- Observed real exchange rates are highly volatile; standard IRBC models calibrated conventionally cannot reproduce this fact.
- Heathcote and Perri (2002) find the model can explain less than a fourth of observed real exchange rate volatility for U.S. data.
- Empirical evidence suggests TFP processes for the U.S. and a sample of main industrialized trade partners have a unit root and are cointegrated.
- Motivated by these findings, the paper introduces technology shocks that follow a vector error correction model (VECM) into a standard two-country, two-good IRBC model.

### Empirical specification and VECM restriction for balanced growth
- When TFP series are integrated and cointegrated, a VECM is appropriate because it imposes short-run coefficient constraints that matter in small samples.
- Balanced growth in the two-country model requires the cointegrating vector relating TFP processes to be (1, -1); equivalently the ratio of TFP levels (or difference of log-levels) across countries is stationary.
- The VECM for log differences of TFP (equation (6)) used in the model:
  - [Δ log A_t ; Δ log A*_t]′ = [c ; c*] + Ξ1 [Δ log A_{t-1} ; Δ log A*_{t-1}] + Ξ2 [Δ log A_{t-2} ; Δ log A*_{t-2}] + Π (log A_{t-1} - φ log A*_{t-1} - log ζ) + [ε_a ; ε_a*].
- Restriction φ = 1 (cointegrating vector (1, -1)) is sufficient for balanced growth; likelihood tests do not reject φ = 1 (likelihood 743.33, p-value 0.19 relative to unrestricted 744.18).

### Estimation of the VECM — selected results and parameters
- Data and construction:
  - U.S. sample: 1973:1 to 2007:3; rest-of-world aggregate: 1980:1 to 2007:3.
  - TFP constructed with β = 0.36 via log A(s_t) = log Y(s_t) - (1-β) log L(s_t) / (1-β).
- Unit root and cointegration tests:
  - Univariate tests indicate log TFP series are well characterized by unit roots with drift; first differences are stationary.
  - Johansen tests for [log A, log A*] (1981:2-2007:3) strongly support one cointegrating vector (trace p-value 0.01; max-eigenvalue p-value 0.01).
- Likelihood-ratio tests (sequential):
  - Unrestricted likelihood: 744.18.
  - Restrict φ = 1: likelihood 743.33, p-value 0.19.
  - Restrict ϕ = -ϕ*: likelihood 741.71, p-value 0.09.
  - Restrict c = c*: likelihood 740.43, p-value 0.06.
  - Symmetry across VAR coefficients (Ψ constraints): likelihood 736.51, p-value 0.032 (marginal rejection at 5 percent).
- Restricted VECM point estimates (selected, Table 5):
  - c = 0.0071 (t-stat 5.83)
  - φ (cointegration adjustment coefficient) = -0.0045 (t-stat -2.65)
  - Ξ1(11) = 0.2041 (t-stat 2.97)
  - Ξ1(12) = 0.1026 (t-stat 1.54)
  - Ξ2(12) = 0.1035 (t-stat 1.55)
  - Ξ2(11) = -0.1497 (t-stat -2.40)
  - Estimated standard deviations of innovations: σ_ε ≈ 0.0082 (for ε and ε*); correlation between ε and ε* set to zero in simulations (null not rejected).

### Model structure and normalization
- Two-country, two-good IRBC model with CES aggregation of imperfectly substitutable home and foreign intermediates (parameter ζ controls elasticity; ω is home intermediate share).
- Intermediate goods produced with local capital and labor; final good consumed or invested locally.
- Households trade a single uncontingent international riskless bond denominated in home intermediate goods; small bond-holding cost κ[D(s_t)] = η/2 · (D(s_t)/A(s_t-1))^2 for stationarity.
- Normalization for stationarity divides trending variables by lagged domestic TFP A(s_t-1) and A*(s_t-1) following King-Plosser-Rebelo (1988). Appendix lists normalized variables and first-order conditions (equations (34)-(59)).

### Parameterization for simulations (baseline)
- Preference and technology parameters:
  - Discount factor β = 0.99.
  - Consumption share ξ = 0.34.
  - Coefficient of relative risk aversion γ = 2.
  - Bond holding cost η = 0.01.
  - Depreciation δ = 0.025.
  - Capital share β = 0.36.
  - Home bias ω = 0.9.
  - Elasticity of substitution ζ assumed two values: ζ = 0.85 and ζ = 0.62.
- TFP shock calibrations:
  - Stationary case: parameters as in Heathcote and Perri (2002).
  - Cointegrated case: use estimated VECM point estimates and σ ≈ 0.0082.

### Main quantitative findings from simulations
- Simulation design:
  - Simulate TFP shocks for 125 periods, feed into log-linearized model, HP-filter simulated series, compute second moments; repeat 5,000 times.
- Key empirical moments (full sample, data):
  - SD(Y) = 1.25
  - SD(RER) = 4.28
- Selected simulation outcomes (Table 6a, full sample):
  - Data: SD(Y) = 1.25; SD(RER) = 4.28; ϑ(RER) = 0.84.
  - Cointegrated TFP, ζ = 0.85: SD(Y) = 0.81; SD(RER) = 2.75; SD(RER)/SD(Y) = 1.75; ϑ(RER) = 0.72.
  - Cointegrated TFP, ζ = 0.62: SD(Y) = 0.70; SD(RER) = 4.26; SD(RER)/SD(Y) = 4.26; ϑ(RER) = 0.70.
  - Stationary TFP, ζ = 0.85: SD(Y) = 1.19; SD(RER) = 0.75; SD(RER)/SD(Y) = 0.75; ϑ(RER) = 0.77.
  - Stationary TFP, ζ = 0.62: SD(Y) = 1.12; SD(RER) = 1.41; SD(RER)/SD(Y) = 1.41; ϑ(RER) = 0.75.
- Main quantitative conclusions:
  - Models with cointegrated TFP shocks generate substantially higher relative volatility of the real exchange rate (SD(RER)/SD(Y)) than models with stationary TFP shocks.
  - With ζ = 0.62 and cointegrated TFP shocks, model SD(RER)/SD(Y) = 4.26 closely matches observed SD(RER) = 4.28.
  - With stationary TFP and ζ = 0.62, SD(RER)/SD(Y) = 1.41 (about 30 percent of observed).
  - Cointegrated shocks improve fit to REER volatility without materially affecting other unconditional moments (consumption, hours, investment volatilities relative to output remain similar).
  - The cointegrating error-correction coefficient is significant but quantitatively small (slow convergence), which is crucial for the results.

### Mechanism and intuition
- Two forces raise relative RER volatility:
  - High persistence of country-specific TFP shocks.
  - Low spillovers of TFP shocks across countries (slow transmission).
- No-spillover channel:
  - Home productivity increase → households feel richer → output, consumption, investment increase and labor rises initially → demand for foreign intermediates increases → terms of trade deteriorate and RER depreciates.
  - Higher persistence → households supply less labor/capital over time → lowers initial output increase and its volatility, but increases demand pressure on foreign intermediates → larger terms of trade movement → higher relative volatility of RER versus output.
- Spillover channel:
  - With spillovers, foreign households anticipate gains → feel richer now → supply less labor/capital but demand more consumption and foreign intermediates → foreign intermediates become less scarce → RER depreciates less relative to no-spillover case.
  - Faster spillovers lower relative volatility of RER.
- VECM interpretation:
  - Simple VECM with κ speed of adjustment produces eigenvalues λ_1 = 1 and λ_2 = 1 - 2κ.
  - Estimated κ = 0.0045 → eigenvalues λ_1 = 1 and λ_2 = 0.99 (very slow convergence).
  - Higher persistence in stationary AR(1) analogs (ρ_a increasing) raises SD(RER)/SD(Y) sharply (examples: ρ_a = 0.91 → ratio 1.07; ρ_a = 0.975 → ratio 2.33 as reported in Table 7 entries).

### Great Moderation and time variation in cointegration
- Empirical facts (rolling 40-quarter windows, U.S. example):
  - U.S. output volatility (HP-filtered SD of real GDP) fell from 2.3 percent (window 1973:1-1982:4) to 0.8 percent (window 1997:3-2007:2).
  - U.S. real exchange rate (HP-filtered SD of REER) about 4.5 percent (1973:1-1982:4), rose above 7 percent (1980:1-1989:4), then declined to 4.3 percent (1997:3-2007:2).
  - Ratio SD(REER)/SD(Y) for U.S. increased non-monotonically from 1.96 to 4.5 over the sample.
  - Similar patterns for UK, Canada, Australia: dramatic declines in output volatility, erratic absolute REER volatility, and dramatic increases in relative REER volatility.
- Time-variation in VECM estimates and implications:
  - VECM estimated on two equal sub-samples (1980:1–1993:4 and 1994:1–2007:3) shows:
    - Sample 1 κ = -0.0077 (faster convergence); standard deviations ε = 0.010, ε* = 0.0081.
    - Sample 2 κ = -0.0029 (slower convergence); standard deviations ε = 0.0062, ε* = 0.0086.
  - Model simulations with these sub-sample VECM estimates generate increases in relative RER volatility of more than 50 percent across samples, consistent with data increases of about 30 percent.

### Remaining puzzles and model extensions (Backus–Smith and IST shocks)
- Backus–Smith puzzle:
  - Baseline model (stationary or cointegrated TFP) yields correlation between RER and consumption ratio very close to one; data show negative and near-zero correlation.
- Two extensions considered:
  1. Taste shocks (Heathcote and Perri (2008)) — generate negative correlation between relative consumption and RER.
  2. Investment-specific technology (IST) shocks (Greenwood, Hercowitz, Krusell (1997)) — modify capital accumulation with V^s_t shocks; assume log V^s_t cointegrated across countries and follow same VECM as TFP.
- IST shock scaling results (Table 8; scaling = standard deviation of IST innovation relative to TFP innovation):
  - Scaling = 0 (only TFP): CORR(RER, C/C*) = 0.97.
  - Scaling = 1: CORR(RER, C/C*) = 0.55.
  - Scaling = 2: CORR(RER, C/C*) = 0.19.
  - Scaling = 3: CORR(RER, C/C*) = -0.08.
  - As IST shocks gain weight, correlation between relative consumption and RER drops toward negative values; SD(RER)/SD(Y) declines mildly (example SD(RER)/SD(Y) drops from 4.26 to 3.82 when scaling = 3).
- Conclusion: IST shocks or taste shocks can help reconcile the Backus–Smith correlation without materially undermining the model’s ability to match high relative RER volatility.

### Asset market structure effects
- Under complete markets, relative REER volatility falls (example: from 1.75 to 1.11 when ζ = 0.85; from 4.26 to 1.35 when ζ = 0.62).
- Under financial autarky (large bond-holding cost), relative REER volatility increases relative to baseline incomplete-markets case.

### Concluding remarks and suggestions for future research
- Empirical documentation:
  - TFP processes of the U.S. and the “rest of the world” are cointegrated with cointegrating vector (1, -1).
  - Relative volatility of real exchange rate with respect to output has increased in the United States, the United Kingdom, Canada, and Australia during the last 20 years.
- Main results:
  - Introducing cointegrated TFP processes into a standard IRBC model increases the model’s ability to explain real exchange rate volatility without harming fit to other second moments.
  - Allowing the speed of convergence to the cointegrating vector to change (as in the data) enables the model to account for the observed increase in relative RER volatility.
- Suggestions for future research:
  - Introduce cointegrated TFP processes in medium-scale open economy models with more frictions to match a broader set of domestic and international variables.
  - Investigate whether investment-specific technology shocks are cointegrated across countries and assess their role in resolving quantity and Backus–Smith puzzles.

*Source: _wp09212 - References*

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

### References

### Introduction and central puzzle
- Observed real exchange rates are highly volatile; standard IRBC models calibrated conventionally cannot reproduce this fact.
- Heathcote and Perri (2002) simulate a two-country, two-good economy with TFP shocks and find the model can only explain less than a fourth of the observed volatility in real exchange rates for U.S. data.
- Many prior studies consider stationary TFP shocks that follow a VAR process in levels; this paper presents evidence that TFP processes for the U.S. and a sample of main industrialized trade partners have a unit root and are cointegrated.
- Motivated by empirical findings, the paper introduces technology shocks that follow a vector error correction model (VECM) process into a standard two-country, two-good model.

### Empirical specification and key restriction for balanced growth
- If the system includes integrated variables and cointegrating relationships, a VECM is more appropriate than a VAR in levels because it imposes constraints on coefficient matrices that matter in small samples.
- The presence of cointegrated TFP shocks requires restrictions on preferences, production functions, and the law of motion of shocks to achieve balanced growth.
- In a two-country model an additional restriction on the cointegrating vector relating the TFP processes is needed: the cointegrating vector to be(1;1);which means the ratio of TFP levels (or the difference of log-levels of TFP) across countries is stationary.
- The authors present evidence for this additional restriction and show that the VECM specification for TFP processes substantially improves the model’s ability to match real exchange rate volatility.

### Main quantitative findings from model simulations
- A model with the estimated VECM TFP processes can generate a real exchange rate volatility more than four times larger than an equivalent model with stationary shocks calibrated as in Heathcote and Perri (2002).
- The VECM parameter estimates imply:
  - Higher persistence of TFP shocks.
  - Lower spillovers of TFP shocks across countries.
- Higher persistence and lower spillovers jointly imply:
  - Higher volatility of the real exchange rate relative to output.
  - Lower volatility of output.

### Mechanism and intuition
- Without spillovers: home productivity increases → home households feel richer → output, consumption, investment increase and labor rises initially → demand for foreign intermediate goods increases → terms of trade deteriorate and real exchange rate depreciates.
- As persistence of TFP shocks increases:
  - Home households supply less labor and capital over time → lowers initial increase in home output and its volatility.
  - Greater demand pressure on foreign intermediate goods → larger terms of trade deterioration and larger real exchange rate depreciation.
  - Net effect: higher persistence → higher relative volatility of real exchange rate vs. output.
- With spillovers:
  - Foreign households anticipate future productivity gains → feel richer now → supply less labor and capital but demand more consumption and foreign intermediate goods.
  - The foreign intermediate good becomes relatively less scarce → real exchange rate depreciates less versus the no-spillover case.
  - Faster spillovers lower relative volatility of the real exchange rate.
- The mechanism requires:
  - High persistence of each country’s TFP process.
  - High persistence in their difference (slow transmission of shocks across countries).
- Estimated VECM shows a very slow speed of convergence to the cointegrating relationship, implying the second largest root is very close to, but inside, the unit circle.

### Great Moderation and time variation in cointegration
- The Great Moderation: substantial decline in the volatility of most U.S. macroeconomic variables during the last 20 years.
- For most industrialized countries, the Great Moderation has not affected the real exchange rate as strongly as it has affected output; consequently the ratio of real exchange rate volatility to output volatility has increased.
- The increase in the relative volatility of the real effective exchange rate of the U.S. dollar coincides with a weakening of the cointegrating relationship of TFP shocks between the U.S. and the “rest of the world.”
- Allowing for a fading in the cointegrating relationship of the size estimated in the data, the model can jointly account for:
  - The observed increase in relative volatility of the real exchange rate.
  - The substantial decline in output volatility.

### Remaining puzzles and model extensions
- A persistent problem in IRBC models is the mismatch between the co-movement of real exchange rates and the ratio of consumption levels across countries (Backus and Smith, 1993): models tend to generate a correlation close to one, while in the data the correlation is negative and close to zero.
- Even with cointegrated TFP shocks the baseline model still generates a near-one correlation between real exchange rate and consumption ratios.
- Two model extensions considered to improve the fit for this correlation without affecting relative real exchange rate volatility:
  - A taste shock as in Heathcote and Perri (2008).
  - An investment-specific technology shock as in Raffo (2009).
- As in stationary-environment studies, both types of shocks help reconcile the model with the observed consumption-real-exchange-rate co-movement.

### Literature connections
- Connects to literature stressing the importance of stochastic trends (e.g., King, Plosser, Stock, and Watson (1991); Lastrapes (1992); Engel and West (2005); Nason and Rogers (2008); Aguiar and Gopinath (2007); Alvarez and Jermann (2005); Corsetti, Dedola, and Leduc (2008a); Lubik and Schorfheide (2006); Rabanal and Tuesta (2006); Justiniano and Preston (2008)).
- Differs from many earlier studies by formalizing a VECM, testing for cointegration, and estimating both the cointegrating vector and the short-run dynamics of the TFP system.
- Also relates to literature analyzing mechanisms for real exchange rate fluctuations (monetary shocks and nominal rigidities, pricing-to-market, non-tradable goods, distribution costs), but the model here includes only tradable goods with home bias as the source of real exchange rate fluctuations, guided by evidence that relative prices of tradable goods have large and persistent fluctuations explaining most of real exchange rate volatility.

_Italic source attribution: Content unit: _wp09212 - References_

### Section VI for concluding remarks.

### _wp09212 - Section VI for concluding remarks.

### The Great Moderation and Real Exchange Rate Volatility
- Empirical evidence (rolling 40-quarter windows) for USA, UK, Canada, and Australia:
  - U.S. output volatility (HP-filtered SD of real GDP) fell from 2.3 percent (window 1973:1-1982:4) to 0.8 percent (window 1997:3-2007:2) — the “Great Moderation.”
  - U.S. real exchange rate (HP-filtered SD of REER) was about 4.5 percent (1973:1-1982:4), rose above 7 percent (1980:1-1989:4), then declined to 4.3 percent (1997:3-2007:2).
  - Ratio of volatilities (SD(REER)/SD(Y)) for the U.S. increased non-monotonically from 1.96 percent to 4.5 percent over the sample — i.e., the real exchange rate volatility more than doubled relative to output volatility.
  - Similar patterns observed for the United Kingdom, Canada, and Australia: dramatic declines in output volatility, erratic absolute REER volatility, and dramatic increases in relative REER volatility.

- Focus of the paper:
  - Build and analyze a two-country, two-good IRBC model (U.S. vs. “rest of the world”) calibrated with standard IRBC parameters and estimated VECM parameters for TFP processes.
  - Show that changes in estimated VECM parameters can account for observed increases in relative REER volatility.

### The Model (two-country, two-good IRBC with cointegrated TFP)
- Key modeling innovation:
  - (Log) TFP processes for home and foreign are assumed cointegrated of order C(1,1) (integrated of order one, linear combination stationary).
  - By the Granger representation theorem, this implies a VECM for log differences of TFP (detailed in section B.3).

- Structure and markets:
  - One final good per country produced from imperfectly substitutable home and foreign intermediate goods (CES aggregator with parameter ζ controlling elasticity of substitution and ω the share of home intermediates).
  - Intermediate goods produced with local capital and labor; final good can only be locally consumed or invested.
  - Trade occurs at intermediate-good level. Consumers trade a single uncontingent international riskless bond denominated in home intermediate goods; no other financial assets.
  - Time is discrete with histories of events s_t and probabilities π(s_t).

- Households:
  - Representative household maximizes expected discounted utility:
    - Utility functional form as in equation (1) with discount factor β in (0,1), labor share and consumption arguments preserved.
  - Budget constraint and capital law of motion:
    - Budget constraint (equation (2)) with P(s_t), P_H(s_t), Q(s_t), D(s_t).
    - Capital accumulation: K(s_t) = (1-δ) K(s_t-1) + X(s_t) (equation (3)).
  - Small bond-holding cost κ[·] introduced for stationarity; κ functional form κ[D(s_t)] = η/2 · (D(s_t)/A(s_t-1))^2 (ensures stationarity of D/A along balanced growth path).

- Firms:
  - Final goods technology (home):
    - Y(s_t) = [ ω^{1/ζ} Y_H(s_t)^{(ζ-1)/ζ} + (1-ω)^{1/ζ} Y_F(s_t)^{(ζ-1)/ζ} ]^{ζ/(ζ-1)} (equation (4)).
  - Intermediate goods producers:
    - Production: Y_H + Y_H^* = A(s_t)^{1-β} K(s_t-1)^β L(s_t)^{1-β} (equation (5)).
  - TFP processes:
    - VECM for log differences:
      - [Δ log A_t ; Δ log A*_t]′ = [c ; c*] + Ξ1 [Δ log A_{t-1} ; Δ log A*_{t-1}] + Ξ2 [Δ log A_{t-2} ; Δ log A*_{t-2}] + Π (log A_{t-1} - φ log A*_{t-1} - log ζ) + [ε_a ; ε_a*] (equation (6)).
    - Π and cointegrating vector (1, -φ) drive error-correction dynamics. If A/A* > ζ, appropriate signs on adjustment coefficients drive both series toward equilibrium.
    - VECM residuals ε_a and ε_a* normal with variance-covariance as specified.

- Market clearing and equilibrium:
  - Final goods market clearing: C + X = Y and C* + X* = Y* (equations (25)-(26)).
  - Bond market clearing: D + D* = 0 (equation (31)).
  - Definitions of relative prices and RER: e_P_H = P_H/P, e_P*_F = P*_F/P*, RER = P*/P. Law of one price holds for intermediate goods.
  - First-order conditions and equilibrium conditions yield:
    - Marginal utilities, intertemporal Euler for capital (equation (9)), bond price expression (equation (15)), risk-sharing condition (equation (16)), factor price formulas (equations (17)-(20)), and intermediate-good demands (equations (21)-(24)).
  - Normalization for stationarity:
    - Divide trending variables by lagged domestic TFP A(s_t-1) (home) and A*(s_t-1) (foreign) per King-Plosser-Rebelo (1988). Full normalized system in Appendix (equations (34)-(59)).

- Balanced growth restriction on cointegrating vector:
  - Balanced growth requires the ratio A(s_t-1)/A*(s_t-1) to be stationary.
  - Sufficient condition: φ = 1, i.e., cointegrating vector = (1, -1).
  - Explanation: if ratio non-stationary, normalized demands (e.g., bY_F/bY) would be non-stationary and balanced growth would not exist (references to normalized equations (49), (50), (54), (55), (58), (59)).

### Estimation of the VECM
- Data:
  - U.S.: quarterly output (BEA) and employment (Payroll Survey) 1973:1 to 2007:3.
  - “Rest of the world” aggregate: nominal output and employment for 12 Euro Area countries (Eurostat / Area Wide Model), UK, Canada, Japan, Australia (national sources). Sample for rest-of-world: 1980:1 to 2007:3.
  - Aggregation: convert national nominal outputs to current U.S. dollars using PPP exchange rates, then convert to constant U.S. dollars using U.S. output deflator. Employment aggregated by summing national employee counts.
  - Constructed TFP series:
    - log A(s_t) = log Y(s_t) - (1-β) log L(s_t) / (1-β)
    - log A*(s_t) analogous, with β = 0.36.

- Integration and cointegration:
  - Univariate unit root tests (ADF, DF-GLS, P_T-GLS, Ng-Perron class) indicate log TFP for U.S. and rest-of-world are well characterized by unit roots with drift; first differences are stationary.
  - Johansen cointegration tests (trace and max-eigenvalue) for [log A, log A*] over 1981:2-2007:3 strongly support one cointegrating vector (trace p-value 0.01; max-eigenvalue p-value 0.01).

- Likelihood-ratio tests and imposed restrictions:
  - Tested restrictions sequentially (likelihood values and p-values reported):
    - Unrestricted likelihood: 744.18.
    - Restriction φ = 1: likelihood 743.33, p-value 0.19 → φ = 1 not rejected.
    - Restriction ϕ = -ϕ* (speed of adjustment equal and opposite): likelihood 741.71, p-value 0.09 → not rejected.
    - c = c* (constant terms equal): likelihood 740.43, p-value 0.06 → not rejected.
    - Symmetry across VAR coefficients (Ψ constraints): likelihood 736.51, p-value 0.032 → marginal rejection at 5 percent.
  - Conclusion: cointegrating vector (1, -1) and symmetry restrictions on constants and speed of adjustment are acceptable for simulation.

- Estimated VECM (restricted) — selected parameters (Table 5):
  - c = 0.0071 (t-stat 5.83)
  - φ (cointegration adjustment coefficient) = -0.0045 (t-stat -2.65)
  - Ξ1(11) = 0.2041 (t-stat 2.97)
  - Ξ1(12) = 0.1026 (t-stat 1.54)
  - Ξ2(12) = 0.1035 (t-stat 1.55)
  - Ξ2(11) = -0.1497 (t-stat -2.40)
  - Estimated standard deviations of innovations: σ_ε ≈ 0.0082 (for ε and ε*). Correlation between ε and ε* set to zero in simulations (null not rejected).

### Results
A. Parameterization (baseline)
- Preference and technology parameters:
  - Discount factor β = 0.99 (implies annual return on capital of 4 percent).
  - Consumption share (ϕ?) ϖ? notation preserved: consumption share ξ = 0.34.
  - Coefficient of relative risk aversion γ = 2.
  - Bond holding cost η = 0.01 (1 basis point).
  - Depreciation δ = 0.025 (quarterly).
  - Capital share β = 0.36.
  - Home bias ω = 0.9 (implies observed import/output ratio in steady state).
  - Elasticity of substitution parameter ζ assumed two values: ζ = 0.85 (Heathcote and Perri (2002)) and ζ = 0.62 (Corsetti, Dedola, and Leduc (2008b)).
- TFP shock calibration:
  - For stationary case, parameters set as in Heathcote and Perri (2002).
  - For cointegrated case, use estimated VECM point estimates (Table 5) and σ ≈ 0.0082.

B. Matching Real Exchange Rate Volatility (simulations)
- Simulation procedure:
  - Simulate TFP shocks for 125 periods, feed into log-linearized model, HP-filter simulated series, compute second moments, repeat 5,000 times.
- Key empirical moments (Full sample, data):
  - SD(Y) = 1.25
  - SD(RER) = 4.28
  - SD(RER)/SD(Y) = 3.42? (note: table reports SD(RER) + and ϑ(RER) — preserved key numbers below from Table 6a)
  - Data: SD(Y) 1.25; SD(C)+SD(X)+SD(N)? preserved reporting in Table 6a.
- Selected simulation findings (Table 6a, full sample and subsamples):
  - Full sample:
    - Data: SD(Y) = 1.25; SD(RER) = 4.28; ϑ(RER) = 0.84.
    - Cointegrated TFP, ζ = 0.85: SD(Y) = 0.81; SD(RER) = 2.75; SD(RER)/SD(Y) = 1.75; ϑ(RER) = 0.72.
    - Cointegrated TFP, ζ = 0.62: SD(Y) = 0.70; SD(RER) = 4.26; SD(RER)/SD(Y) = 4.26; ϑ(RER) = 0.70.
    - Stationary TFP, ζ = 0.85: SD(Y) = 1.19; SD(RER) = 0.75; SD(RER)/SD(Y) = 0.75; ϑ(RER) = 0.77.
    - Stationary TFP, ζ = 0.62: SD(Y) = 1.12; SD(RER) = 1.41; SD(RER)/SD(Y) = 1.41; ϑ(RER) = 0.75.
  - Subperiods (1980-1993 and 1994-2007) show analogous patterns with cointegrated TFP increasing relative REER volatility in the latter period.
- Main quantitative conclusions:
  - Models with cointegrated TFP shocks generate substantially higher relative volatility of the real exchange rate with respect to output than models with stationary TFP shocks.
  - With ζ = 0.62 and cointegrated TFP shocks, the model closely matches observed relative volatility: model SD(RER)/SD(Y) = 4.26 versus data 4.28.
  - With stationary TFP and ζ = 0.62, SD(RER)/SD(Y) = 1.41 (only about 30 percent of observed fluctuation).
  - Cointegrated shocks substantially improve the model’s ability to match REER volatility without materially affecting other unconditional moments (consumption, hours, investment volatilities relative to output remain similar across stationary and cointegrated specifications).
  - The coefficient on speed of adjustment in the VECM is significant but quantitatively small (slow convergence), which is key for the results.
  - Estimated σ_ε and σ_ε* ≈ 0.0082 used in simulations.

- Cross-sectional and correlation results (Tables 6b and 6c):
  - Domestic correlations (Table 6b) and international cross-correlations (Table 6c) are broadly similar across stationary and cointegrated models; cointegrated model with ζ = 0.62 does better on some international correlations.
  - The model (both stationary and cointegrated) does not resolve the “quantity puzzle”: in the data, output series are more correlated across countries than consumption series, whereas the model produces higher cross-country consumption correlations than output correlations.

- Asset market structures:
  - Under complete markets, relative REER volatility falls (example: from 1.75 to 1.11 when ζ = 0.85; from 4.26 to 1.35 when ζ = 0.62).
  - Under financial autarky (large bond-holding cost), relative REER volatility increases relative to baseline incomplete-markets case.

*Source: _wp09212 - Section VI for concluding remarks.*

### 2.05 when= 0:85, and from 4.26 to 5.41 when= 0:62).

### _wp09212 - 2.05 when= 0:85, and from 4.26 to 5.41 when= 0:62).

### C. Intuition
- Two forces drive relative volatility of the real exchange rate with respect to output in typical IRBC models: high persistence of TFP shocks and low spillovers of TFP across countries.
- VECM estimates imply higher persistence and slower spillovers than typical calibrations, enabling matching of high relative volatility of the real exchange rate.
- Simple stationary TFP processes used in simulation:
  - a_t = ρ_a a_{t-1} + ε^a_t and a*_t = ρ_a a*_{t-1} + ε^{a*}_t, with uncorrelated innovations and no spillovers.
  - Persistence parameter ρ_a ∈ [0.9; 0.95; 0.975]; calibration uses ϕ = 0:62.
- Table 7 (selected entries) shows as persistence ρ_a increases, SD(RER)/SD(Y) rises:
  - ρ_a = 0.91 → SD(RER) = 1.43, SD(Y) = 1.33, SD(RER)=SD(Y) = 1.07
  - ρ_a = 0.95 → SD(RER) = 1.96, SD(Y) = 1.21, SD(RER)=SD(Y) = 1.64
  - ρ_a = 0.975 → SD(RER) = 2.47, SD(Y) = 1.06, SD(RER)=SD(Y) = 2.33
- Effect mechanism when persistence rises:
  - Home households experience larger positive income effect → supply less labor and capital → initial increase in home output is lower but more persistent.
  - Demand for foreign intermediate goods increases → foreign intermediate goods become scarcer → terms of trade and real exchange rate rise (depreciate) more.
  - In calibration, rising persistence lowers standard deviation of HP-filtered output while raising RER volatility → relative volatility rises.
- Spillovers modeled via simple VECM with one unit root:
  - Δa_t = -κ (a_{t-1} - a*_{t-1}) + ε^a_t and Δa*_t = κ (a_{t-1} - a*_{t-1}) + ε^{a*}_t, where κ is speed of adjustment.
  - κ ∈ [0:005; 0:05; 0:25]; calibration uses ϕ = 0:62.
- Spillover effects:
  - Larger κ → faster response of foreign TFP to home TFP shocks → "news" channel induces foreign households to feel income effect sooner → they demand more consumption goods → demand for home intermediate goods increases and foreign intermediate goods become less scarce → terms of trade and RER depreciate less.
  - Table 7 confirms: as κ increases, volatility of HP-filtered output increases and volatility of HP-filtered RER decreases → relative volatility decreases.
  - Example: for κ = 0:25 relative standard deviation is 0:48 (much lower than under stationary TFP shocks), despite VECM having one unit root.
- VAR eigenvalue interpretation (VECM as VAR in levels, equation (33)):
  - Eigenvalues: λ_1 = 1, λ_2 = 1 - 2κ. Small κ implies λ_2 close to one.
  - Example VECM estimate: κ point estimate = 0:0045 → eigenvalues λ_1 = 1 and λ_2 = 0:99.
- Comparison to other calibrations:
  - Heathcote and Perri (2002): eigenvalues λ_1 = 0:995 and λ_2 = 0:945; correlation of innovations = 0:29 → faster spillovers and contemporaneous spillovers reduce relative RER volatility. Authors' estimated residual correlation = 0:07 (not significant).
  - Backus, Kehoe, and Kydland (1992): λ_1 = 0:994, λ_2 = 0:812, correlation = 0:26 → relative volatility = 0:65.
  - Heathcote and Perri (2008): eigenvalues both 0:91, uncorrelated innovations → relative volatility = 1:05.
- VAR in levels estimation on authors' data: eigenvalues 0:999 and 0:952, correlation of innovations 0:16 → relative volatility = 1:67 (with ϕ = 0:62).

### D. Matching the Increase in Real Exchange Rate Volatility
- Empirical fact: volatility of the real exchange rate relative to output increased in last decade for most industrialized economies; for the U.S. increase dated around early to mid 90’s.
- Data (U.S.): relative volatility rose from less than four times (1980:1–1993:4) to more than five times (1994:1–2007:3).
- VECM estimated on two non-overlapping sub-samples (equal observations):
  - Sample 1: 1980:1 to 1993:4
    - κ moves from -0:0045 (full sample) to -0:0077 → faster speed of convergence.
    - First own lag ζ_1 moves from 0:2041 to 0:2203 → higher autocorrelation.
    - Second crossed lag close to zero.
    - Standard deviations: ε (U.S.) = 0:010, ε* (rest of world) = 0:0081.
  - Sample 2: 1994:1 to 2007:3
    - κ estimate = -0:0029 → much slower catching up.
    - Second crossed-lag ζ_2 = -0:4124 (larger and negative); first own lag moves close to zero.
    - Standard deviations: ε = 0:0062, ε* = 0:0086.
    - Interpretation: post-1994 co-movement characterized by short-run negative co-movement and slow return to long-run.
- Simulation results (Tables 6a and 6b):
  - Data: relative volatility of RER increases by 30 percent across samples.
  - Model simulations: changes in VECM estimates across samples generate increases in relative volatility of more than 50 percent for both low and high values of ϕ.

### E. The “Backus-Smith Puzzle”
- Model implication: correlation between real exchange rate and ratio of consumption across countries is very close to one; data show negative but near-zero correlation → Backus-Smith puzzle.
- Existing fixes: adding non-tradable goods (Corsetti, Dedola, and Leduc (2008a); Benigno and Thoenissen (2007)) or taste shocks (Heathcote and Perri (2008)).
- Authors' exploration:
  - Introduce taste shocks: generates negative correlation between relative consumption and RER (as expected).
  - Consider investment-specific technology (IST) shocks per Greenwood, Hercowitz, and Krusell (1997):
    - Modify capital accumulation: K^s_t = (1 - δ) K^s_{t-1} + V^s_t X^s_t and foreign counterpart, with log V^s_t and log V^{s*}_t cointegrated of order C(1,1) and following same VECM process as TFP shocks.
    - Calibration: use same parameterization as TFP VECM (Table 5), vary standard deviation of IST shocks relative to TFP shock from 0 to 3 times.
  - Table 8 (selected outcomes) as IST shock scaling increases (Scaling = 0,1,2,3):
    - SD(RER): 4.26, 4.17, 4.13, 3.82
    - SD(Y): 4.26, 4.17, 4.13, 3.82  (table lists SD(RER) and SD(Y) on a single line; preserve entries as in source)
    - CORR(RER, C/C*): 0.97, 0.55, 0.19, -0.08
    - Scaling is ratio of standard deviation of IST shock innovation to that of TFP shocks (0 = only TFP shocks; 3 = IST shocks three times as volatile as TFP shocks).
  - Results:
    - As IST shocks gain importance, correlation between relative consumption and RER drops toward and into negative territory; with scaling = 3 the correlation becomes negative and similar to data.
    - Relative volatility of RER w.r.t. output declines mildly: when IST scaling = 3, SD(RER)/SD(Y) drops from 4.26 (only TFP) to 3.82.
  - Mechanism: IST shock raises home investment and lowers home consumption → price of foreign intermediate goods rises (because home investment uses foreign intermediates) → RER depreciates → foreign households feel richer and consume more → negative correlation between consumption ratio and RER emerges.

### VI. Concluding Remarks
- Empirical documentation:
  - TFP processes of the U.S. and the “rest of the world” are cointegrated with cointegrating vector (1; -1).
  - Relative volatility of real exchange rate with respect to output has increased in the United States, the United Kingdom, Canada, and Australia during the last 20 years.
- Main results:
  - Introducing cointegrated TFP processes in a standard IRBC model increases the model’s ability to explain real exchange rate volatility without harming fit to other second moments.
  - Allowing the speed of convergence to the cointegrating vector to change (as in the data) enables the model to account for the observed increase in relative RER volatility.
- Suggestions for future research:
  - Introduce cointegrated TFP processes in medium-scale open economy models with more frictions to match a broader set of domestic and international variables (reference to Adolfson et al., 2007).
  - Investigate whether investment-specific technology shocks are cointegrated across countries and assess their role in international business cycle models focusing on quantity and Backus-Smith puzzles.

*Source: Excerpt from the provided IMF working paper content unit.*

### References

### _wp09212 - References

### Key cited works
- Adolfson, M., S. Laseen, J. Lindé and M. Villani, 2007, “Bayesian Estimation of an Open Economy DSGE Model with Incomplete Pass-Through,” Journal of International Economics, Vol. 72 (2), pp. 481–511.
- Aguiar, M. and G. Gopinath, 2007, “Emerging Market Business Cycles: The Cycle Is the Trend,” Journal of Political Economy, Vol. 115 (1), pp. 69–102.
- Alvarez F. and U. Jermann, 2005, “Using Asset Prices to Measure the Persistence of the Marginal Utility of Wealth,” Econometrica, pp. 1977–2016.
- Backus, D., P. Kehoe and F. Kydland, 1992, “International Business Cycles,” Journal of Political Economy, Vol. 100, No. 41, pp. 745–75.
- Benigno, G., 2004, “Real Exchange Rate Persistence and Monetary Policy Rules,” Journal of Monetary Economics, Vol.51, pp. 473–502.
- Benigno, G. and C. Thoenissen, 2008, “Consumption and Real Exchange Rates with Incomplete Markets and Non-Traded Goods,” Journal of International Money and Finance, Vol. 27(6), pp. 926–48.
- Burstein, A., M. Eichenbaum, and S. Rebelo, 2006, “The Importance of Nontradable Goods’ Prices in Cyclical Real Exchange Rate Fluctuations,” Japan and the World Economy, Vol. 18(3), pp. 247–253.
- Chari, V.V., P. Kehoe and E. McGrattan, 2002, “Can Sticky Price Models Generate Volatile and Persistent Real Exchange Rates?,” Review of Economic Studies, Vol. 69, pp. 533–563.
- Corsetti, G., L. Dedola and S. Leduc, 2008, “International Risk Sharing and the Transmission of Productivity Shocks,” Review of Economic Studies, Vol. 75, pp. 443–473.
- Dickey, A. and W. Fuller, 1979, “Distribution of the Estimators for Autoregressive Time Series with a Unit Root,” Journal of the American Statistical Association, Vol. 74, pp. 427–431.
- Engle R. and C. Granger, 1987, “Co-Integration and Error Correction: Representation, Estimation, and Testing,” Econometrica, Vol. 55, No. 2, pp. 251–276.
- Fernandez-Villaverde, J. and J. Rubio-Ramírez, 2007, “Estimating Macroeconomic Models: A Likelihood Approach,” Review of Economic Studies, Vol. 74(4), pp. 1059–1087.
- Greenwood, J., Z. Hercowitz, and Per Krusell, 1997, “Long-Run Implications of Investment-Specific Technological Change,” American Economic Review, Vol. 87(3), pp. 342–62, (June).
- Johansen, S., 1991, “Estimation and Hypothesis Testing of Cointegration Vectors in Gaussian Vector Autoregressive Models,” Econometrica59, pp. 1551–80.
- Justiniano, A., and G. Primiceri, 2008, “The Time-Varying Volatility of Macroeconomic Fluctuations,” American Economic Review, Vol. 98(3), pp. 604–41.
- Kehoe, P. and F. Perri, 2002, “International Business Cycles with Endogenous Incomplete Markets,” Econometrica70:3, pp. 907–928.
- King, R., C., Plosser and S., Rebelo, 1988, “Production, Growth and the Business Cycle,” Journal of Monetary Economics, Vol. 21, pp. 195–232.
- Lastrapes, W., 1992, “Source of Fluctuations in Real and Nominal Exchange Rates,” Review of Economics and Statistics, Vol. 74 (3), 530-539.
- MacKinnon, J., 1996, “Numerical Distribution Functions for Unit Root and Cointegration Tests,” Journal of Applied Econometrics, Vol. 11 (6), pp. 601–18.
- McConnell, M. and G. Perez-Quiros, 2000, “Output Fluctuations in the United States: What Has Changed Since the Early 1980 ́s?,” American Economic Review, Vol. 90(5), pp. 1464–1476.
- Ng, S. and P. Perron, 1995, “Unit Root Tests in ARMA Models with Data-Dependent Methods for the Selection of the Truncation Lag,” Journal of the American Statistical Association, Vol. 90, pp. 268–281.
- Rabanal, P. and V. Tuesta, 2006, “Euro-Dollar Real Exchange Rate Dynamics in an Estimated Two-Country Model: What Is Important and What Is Not,” CEPR Discussion Paper No. 5957.
- Stock, J., and M. Watson, 2002, “Has the Business Cycle Changed and Why?,” NBER Macroeconomics Annual, MIT Press.
- Said, S. and A. Dickey, 1984, “Testing for Unit Roots in Autoregressive-Moving Average Models of Unknown Order,” Biometrika, Vol 71, Issue 3, pp. 599-607.
- (The list continues with additional cited works in the References section as provided in the source.)

### Appendix A — Normalized Equilibrium Conditions: scope and definitions
- Purpose: normalize a system where both log A(st) and log A'(st) are integrated, to obtain a stationary system and impose additional restrictions on the VECM for balanced growth.
- Normalized variables (exact definitions preserved):
  - bYH(st) = YH(st) / A(st-1)
  - bY'H(st) = Y'H(st) / A(st-1)
  - bYF(st) = YF(st) / A'(st-1)
  - bY'F(st) = Y'F(st) / A'(st-1)
  - bK(st-1) = K(st-1) / A(st-1)
  - bK'(st-1) = K'(st-1) / A'(st-1)
  - bY(st) = Y(st) / A(st-1)
  - bY'(st) = Y'(st) / A'(st-1)
  - bC(st) = C(st) / A(st-1)
  - bC'(st) = C'(st) / A'(st-1)
  - bX(st) = X(st) / A(st-1)
  - bX'(st) = X'(st) / A'(st-1)
  - cW(st) = W(st) / A(st-1)
  - cW'(st) = W'(st) / A'(st-1)
  - bD(st) = D(st) / A(st-1)
  - bD'(st) = D'(st) / A'(st-1)
  - bφ(st) = φ(st) A(st-1)^(1-(1-ξ))
  - bφ'(st) = φ'(st) A'(st-1)^(1-(1-ξ))
  (variable notation and exponents preserved as in source)

### Appendix A — Stationary first order conditions and equilibrium identities
- Representative stationary first order conditions (equations preserved with numbering):
  - U_C(st) = bφ(st) ; (34)
  - U_L(st) / U_C(st) = cW(st) ; (35)
  - [A(st)/A(st-1)]^(1-(1-ξ)) bφ(st) = E_{st}[ bφ(st+1) / R(st+1) + 1 ] ; (36)
  - bK(st) = (1-δ) bK(st-1) A(st-1)/A(st) + bX(st) A(st-1)/A(st) ; (37)
  - U_C'(st) = bφ'(st) ; (38)
  - U_L'(st) / U_C'(st) = cW'(st) ; (39)
  - [A'(st)/A'(st-1)]^(1-(1-ξ)) bφ'(st) = E_{st}[ bφ'(st+1) / R'(st+1) + 1 ] ; (40)
  - bK'(st) = (1-δ) bK'(st-1) A'(st-1)/A'(st) + bX'(st) A'(st-1)/A'(st) ; (41)
  - Q(st) = E_{st}[ bφ(st+1)/bφ(st) [A(st-1)/A(st)]^(1-(1-ξ)) e^{P_H(st+1) - P_H(st)} - D(st) ] ; (42)
  - A more complex expectation identity involving bφ', prices, and RER appears as equation (43) and is preserved in its structural form in the source.
  - cW(st) and R(st) expressions linking wages, capital, labor, prices, and technology growth preserved as (44) and (45).
  - cW'(st) and R'(st) analogous expressions preserved as (46) and (47).
  - Goods market and aggregation identities:
    - bYH(st) = α e^{P_H(st)} bY(st) ; (48)
    - bYF(st) = (1-α) [ e^{P'_F(st)} / RER(st) ] bY(st) A(st-1)/A'(st-1) ; (49)
    - bY'H(st) and bY'F(st) relations preserved as (50) and (51).
    - bC(st) + bX(st) = bY(st) ; (52)
    - bC'(st) + bX'(st) = bY'(st) ; (53)
  - Aggregation formulas for bY(st) and bY'(st) combining domestic and foreign components preserved as (54) and (55).
  - Resource and production identities linking bYH, bY'H, bK, labor, and technology preserved as (56) and (57).
  - bD and bD' relation and a pricing identity involving Q, bD, bY'H, bYF, and lagged bD preserved as (58) and (59).

### Stationarity and normalization remarks
- Productivity shocks are not normalized.
- The functional form [D(st)] = κ^2 A(st-1) ( D(st)/A(st-1) )^2 (as expressed in the source) implies that E[bD(st)] = A(st-1) and E^0[bD(st)] are stationary; this ensures that normalized equations (42) to (43) are stationary.

*Source: _wp09212 - References*

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