## _wp1213

## Source details

**Canonical URL:** [_wp1213](https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2012/_wp1213.pdf)

## Other formats

- [Markdown version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2012/_wp1213.pdf.md)
- [Structured JSON version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2012/_wp1213.pdf.json)

---

### Key findings and thesis
- A baseline international real business cycle (IRBC) two-country, two-good model can explain the standard deviation (area below the spectrum) of the U.S. dollar real exchange rate (RER) when analyzed at all frequencies, but it implies a higher persistence of the RER than in the data (the "excess persistence of RER" puzzle).
- Spectral analysis shows most of the variance of the RER in the data is at low frequencies: about 70 percent for the U.S. dollar and between 60 to 75 percent depending on the currency examined.
- The baseline IRBC model with cointegrated, non-stationary TFP processes across countries explains about 80 percent of the standard deviation of the RER in the data under a benchmark calibration (including an elasticity of substitution between intermediate inputs of 0:85) but places too much weight on low-frequency fluctuations (model: 85 percent; data: 70 percent).
- Introducing adjustment costs in the composition of domestic and imported intermediate inputs and portfolio adjustment costs increases the short-run impact response of the RER while reducing long-run response, helping to replicate the shape of the empirical spectrum (i.e., resolve the excess persistence puzzle) while retaining the model’s ability to match the standard deviation of the RER.

### Spectral analysis of the RER (empirical results)
- Data sources and sample periods:
  - U.S. dollar: Federal Reserve's Real Broad Trade-Weighted Value of the U.S. dollar. Sample period: 1973:Q1-2010:Q3.
  - Euro area: ECB Area Wide Model series (sample period 1973:Q1-2008:Q4), extended to 2010:Q3 using IFS data.
  - Canadian dollar and U.K. pound sterling: IFS measure (sample period 1975:Q2-2010:Q3).
  - Australian dollar: Reserve Bank of Australia REER (sample period 1973:Q1-2010Q4).
  - Japanese yen: Bank of Japan REER using BIS methodology (sample period 1973:Q1-2010:Q3).
- Methodology:
  - Spectrum estimated using the modified Bartlett kernel methodology (Hamilton, Section 6.4).
  - Variance decomposed into: low (more than 32 quarters), BC (8 to 32 quarters), and high (less than 8 quarters).
- Variance decomposition (percent of variance):
  - U.S. - Federal Reserve: Low 73.9  BC 21.1  High 5.0
  - U.S. - Our measure: Low 70.0  BC 24.3  High 5.7
  - Euro Area: Low 64.5  BC 29.8  High 5.7
  - U.K.: Low 62.2  BC 31.0  High 6.8
  - Japan: Low 71.3  BC 23.8  High 4.9
  - Australia: Low 75.7  BC 19.2  High 5.1
  - Canada: Low 74.1  BC 20.0  High 5.9
- Empirical implications:
  - For the U.S. dollar, 74 percent of variance concentrated at low frequencies (less than 32 quarters); 21.1 percent attributed to BC frequencies; 5 percent to high frequencies.
  - Cross-country comparison: low-frequency share ranges from 62.2 percent (U.K.) to 75.7 percent (Australia).
  - Conclusion: focusing only on BC-frequency fluctuations omits a large portion of RER variance.

### Relationship to existing literature
- Empirical and time-series literature typically analyzes exchange rates at all frequencies (Meese and Rogoff, Steinsson, Sarno); many DSGE/IRBC papers focus on BC-frequency (HP-filtered) data and struggle to match RER volatility and persistence.
- Prior model fixes and findings:
  - Heathcote and Perri (2002) highlight inability of standard IRBC to explain volatility and persistence of HP-filtered RER.
  - Chari, Kehoe and McGrattan (2002): monetary model with monopolistic competition and sticky prices can explain HP-filtered RER volatility under high risk aversion.
  - Corsetti, Dedola and Leduc (2008): introducing nontraded goods helps.
  - Rabanal, Rubio-Ramírez and Tuesta (2011): cointegrated TFP processes improve model’s ability to explain HP-filtered RER volatility.
- Contribution of this paper: compares properties of RER in the DSGE model and data without applying any filtering; shows that non-stationary, cointegrated TFP in an IRBC framework explains standard deviation but requires additional mechanisms (input and portfolio adjustment costs) to match the spectrum shape and persistence.

### Baseline model specification and calibration highlights
- Model type: two-country, two-good IRBC model with log TFP non-stationary but cointegrated (VECM).
- Key modeling assumptions:
  - Single final good produced using domestic and imported intermediate goods; intermediate goods are imperfect substitutes.
  - Intermediate goods producers use domestic capital and labor and face domestic TFP shocks.
  - Final good can only be consumed or invested domestically; all cross-border trade occurs at the intermediate goods level.
  - Financial markets: households trade an uncontingent international riskless bond denominated in units of the home-country intermediate good; no other financial assets.
  - A small cost of holding bonds, Φ(·), is included to ensure stationarity of bond holdings; functional form: Φ(D_t; A_{t-1}) = ϵ/2 A_{t-1} (D_t / A_{t-1})^2 (modified to include home-country TFP A_t for balanced growth).
- Calibration highlights:
  - Quarterly frequency; discount factor β = 0.99.
  - Utility: consumption share μ = 0.34; coefficient of risk aversion γ = 2.
  - Technology: depreciation rate δ = 0.025; capital share β = 0.36; ratio of intermediate inputs ω = 0.9; elasticity of substitution between intermediate goods θ = 0.85 (baseline).
  - Portfolio adjustment cost ζ = 1 basis point (0.01).
  - VECM parameters: ρ = 1, Θ = -0.007, c = 0.001, c' = 0.006, Σ = 0.0108, Σ' = 0.0088.

### Baseline results: matching the RER spectrum
- Baseline model outcome:
  - The baseline model with benchmark calibration closely replicates the standard deviation (area under spectrum) of the RER but places too much variance at low-frequency movements.
- Key statistics (U.S. data vs. model variants):
  - U.S. Data:
    - Standard deviation RER: 10.56
    - Output Growth: 0.87
    - Frequency shares: Low 70.0, BC 24.3, High 5.7
  - θ = 0.85:
    - Standard deviation RER: 8:33
    - Output Growth: 0:75
    - Frequency shares: Low 88:4, BC 8:8, High 2:8
  - θ = 0.62:
    - Standard deviation RER: 16:02
    - Output Growth: 0:68
    - Frequency shares: Low 88:2, BC 8:9, High 2:9
  - θ = 1.5:
    - Standard deviation RER: 3:55
    - Output Growth: 0:85
    - Frequency shares: Low 89:0, BC 8:4, High 2:6
- Main model problem: baseline assigns almost 89 percent of RER variance to low-frequency versus 70 percent in the data — the "excess persistence of RER" puzzle.

### Robustness checks (alternative shock specifications)
- Variations considered:
  - "Stationary": Heathcote and Perri (2002) VAR(1) in levels (stationary TFP shocks).
  - "TFP and IST": cointegrated TFP and cointegrated investment-specific technology (IST) shocks.
- Selected results:
  - Stationary; θ = 0.85:
    - RER SD: 4:03; Output Growth: 0:92; Frequency shares: Low 85:3, BC 11:3, High 3:3
  - TFP and IST; θ = 0.85:
    - RER SD: 8:58; Output Growth: 0:76; Frequency shares: Low 88:5, BC 8:7, High 2:8
- Conclusion: stationary TFP or adding IST shocks do not solve the excess persistence puzzle; cointegrated common unit root and slow shock transmission are crucial to explain large RER volatility.

### Mechanisms and model extensions that address the excess persistence puzzle
- Objectives for extensions:
  1. Increase short-run RER responsiveness (shift variance from very low frequencies to BC and high frequencies).
  2. Preserve the model’s ability to match the RER standard deviation.
- A. Input adjustment costs in imported intermediate inputs
  - Production specification:
    - Y_t = [ ω^{1/θ} Y_{H;t}^{(θ-1)/θ} + (1 - ω)^{1/θ} (ϕ_t Y_{F;t})^{(θ-1)/θ} ]^{θ/(θ-1)} with long-run θ = 2 in this extension.
  - Input adjustment index:
    - ϕ_t = [ 1 - (κ/2) ( (Y_{F;t}/Y_{H;t}) / (Y_{F;t-1}/Y_{H;t-1}) - 1 )^2 ]  (equation (18))
  - Calibration and quantitative implications (selected κ values):
    - U.S. Data: RER SD 10.56; Output Growth 0.87; Frequency shares: Low 70.0, BC 24.3, High 5.7
    - κ = 0:
      - RER SD 2:50; Output Growth 0:88; Frequency shares: Low 89:3, BC 8:2, High 2:5
    - κ = 125:
      - RER SD 4:98; Output Growth 0:86; Frequency shares: Low 84:8, BC 11:7, High 3:5
    - κ = 250:
      - RER SD 8:10; Output Growth 0:81; Frequency shares: Low 83:3, BC 12:8, High 3:9
    - κ = 330:
      - RER SD 10:66; Output Growth 0:77; Frequency shares: Low 82:5, BC 13:1, High 4:0
    - κ = 500:
      - RER SD 20:64; Output Growth 0:61; Frequency shares: Low 81:7, BC 13:9, High 4:3
  - Findings:
    - Increasing κ (input adjustment costs) increases RER volatility and raises the share of variance at BC- and high-frequencies.
    - κ = 330 matches RER volatility and output growth volatility better but still assigns 82:5 percent to low frequencies versus 70 percent in data — puzzle not fully solved.
  - Mechanism:
    - Adjustment costs reduce short-run sensitivity of relative quantities to relative prices (low short-run elasticity), increasing short-run terms-of-trade and RER volatility; long-run elasticity returns to θ = 2 reducing low-frequency volatility.
- B. Portfolio adjustment costs and interaction with input adjustment costs
  - Baseline ζ = 0.01; consider ζ = 0.05 with θ = 2 and varying κ.
  - Selected results:
    - U.S. Data: RER SD 10.56; Output Growth 0.87; Frequency shares: Low 70.0, BC 24.3, High 5.7
    - κ = 0; ζ = 0.05:
      - RER SD 2:43; Output Growth 0:84; Frequency shares: Low 88:7, BC 8:7, High 2:6
    - κ = 6; ζ = 0.05:
      - RER SD 2:5; Output Growth 0:81; Frequency shares: Low 76:0, BC 18:8, High 5:2
    - κ = 125; ζ = 0:05:
      - RER SD 11:13; Output Growth 0:70; Frequency shares: Low 72:9, BC 21:1, High 6:0
  - Findings:
    - Interaction of large long-run θ = 2, input adjustment costs κ, and higher portfolio adjustment costs ζ = 0.05 can replicate both the standard deviation and the shape of the RER spectrum.
    - Specifically, κ = 125 and ζ = 0:05 replicate volatility and persistence almost perfectly in the model.
  - Mechanism:
    - Input adjustment costs increase short-run NFA volatility and RER depreciation.
    - Larger portfolio adjustment costs amplify initial RER depreciation but accelerate return to balanced growth (faster decline of D_t), reducing long-run persistence.
  - Numerical interpretation:
    - κ = 125 ⇒ 1 percent deviation in input ratio implies output 0:0625 percent smaller.
    - ζ = 0:05 ⇒ 1 percent increase in NFA costs household 0:0025 percent units of consumption.

### Concluding remarks
- Two objectives when matching RER spectrum: match area (volatility) and shape (frequency composition/persistence).
- Baseline two-country IRBC with cointegrated TFP can explain area under spectrum for low θ but places too much weight on low frequencies (excess persistence of RER puzzle).
- Modeling TFP as stationary or adding IST shocks does not solve puzzle.
- Successful solution requires interaction of three ingredients:
  1. Large steady-state elasticity of substitution θ = 2 (long-run).
  2. Adjustment costs in intermediate inputs (κ = 125 in calibration), delivering low short-run elasticities and higher short-run RER volatility (κ = 125 ⇒ 1% input-ratio deviation → output 0:0625% smaller).
  3. Higher portfolio adjustment costs in range of 5 basis points (ζ = 0:05), which raise short-run volatility but reduce long-run persistence.
- Combined, these ingredients allow the model to match both the area and shape of the RER spectrum and thereby solve the excess persistence puzzle.

*Source content provided from the PDF "_wp1213 - References ............................................................................................................."*

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

### _wp1213 - References .............................................................................................................

### Key findings and thesis
- A baseline international real business cycle (IRBC) two-country, two-good model can explain the standard deviation (area below the spectrum) of the U.S. dollar real exchange rate (RER) when analyzed at all frequencies, but it implies a higher persistence of the RER than in the data (the "excess persistence of RER" puzzle).
- Spectral analysis shows most of the variance of the RER in the data is at low frequencies: about 70 percent for the U.S. dollar and between 60 to 75 percent depending on the currency examined.
- The baseline IRBC model with cointegrated, non-stationary TFP processes across countries explains about 80 percent of the standard deviation of the RER in the data under a benchmark calibration (including an elasticity of substitution between intermediate inputs of 0:85) but places too much weight on low-frequency fluctuations (model: 85 percent; data: 70 percent).
- Introducing adjustment costs in the composition of domestic and imported intermediate inputs and portfolio adjustment costs increases the short-run impact response of the RER while reducing long-run response, helping to replicate the shape of the empirical spectrum (i.e., resolve the excess persistence puzzle) while retaining the model’s ability to match the standard deviation of the RER.

### Spectral analysis of the RER (empirical results)
- Data sources and sample periods:
  - U.S. dollar: Federal Reserve's Real Broad Trade-Weighted Value of the U.S. dollar. Sample period: 1973:Q1-2010:Q3.
  - Euro area: ECB Area Wide Model series (sample period 1973:Q1-2008:Q4), extended to 2010:Q3 using IFS data.
  - Canadian dollar and U.K. pound sterling: IFS measure (sample period 1975:Q2-2010:Q3).
  - Australian dollar: Reserve Bank of Australia REER (sample period 1973:Q1-2010Q4).
  - Japanese yen: Bank of Japan REER using BIS methodology (sample period 1973:Q1-2010:Q3).
- Methodology:
  - Spectrum estimated using the modified Bartlett kernel methodology (Hamilton, Section 6.4).
  - Variance decomposed into: low (more than 32 quarters), BC (8 to 32 quarters), and high (less than 8 quarters) frequencies.
- Variance decomposition (Table 1 — Variance Decomposition of the RER (in percent)):
  - U.S. - Federal Reserve: Low 73.9  BC 21.1  High 5.0
  - U.S. - Our measure: Low 70.0  BC 24.3  High 5.7
  - Euro Area: Low 64.5  BC 29.8  High 5.7
  - U.K.: Low 62.2  BC 31.0  High 6.8
  - Japan: Low 71.3  BC 23.8  High 4.9
  - Australia: Low 75.7  BC 19.2  High 5.1
  - Canada: Low 74.1  BC 20.0  High 5.9
- Empirical implications:
  - For the U.S. dollar, 74 percent of variance concentrated at low frequencies (less than 32 quarters); 21.1 percent attributed to BC frequencies; 5 percent to high frequencies.
  - Cross-country comparison: low-frequency share ranges from 62.2 percent (U.K.) to 75.7 percent (Australia).
  - Conclusion: focusing only on BC-frequency fluctuations omits a large portion of RER variance.

### Relationship to existing literature
- Empirical and time-series literature typically analyzes exchange rates at all frequencies (Meese and Rogoff, Steinsson, Sarno); many DSGE/IRBC papers focus on BC-frequency (HP-filtered) data and struggle to match RER volatility and persistence.
- Prior model fixes and findings:
  - Heathcote and Perri (2002) highlight inability of standard IRBC to explain volatility and persistence of HP-filtered RER.
  - Chari, Kehoe and McGrattan (2002): monetary model with monopolistic competition and sticky prices can explain HP-filtered RER volatility under high risk aversion.
  - Corsetti, Dedola and Leduc (2008): introducing nontraded goods helps.
  - Rabanal, Rubio-Ramírez and Tuesta (2011): cointegrated TFP processes improve model’s ability to explain HP-filtered RER volatility.
- Contribution of this paper: compares properties of RER in the DSGE model and data without applying any filtering; shows that non-stationary, cointegrated TFP in an IRBC framework explains standard deviation but requires additional mechanisms (input and portfolio adjustment costs) to match the spectrum shape and persistence.

### Baseline model specification and calibration highlights
- Model type: two-country, two-good IRBC model similar to Backus, Kehoe and Kydland (1994) and Heathcote and Perri (2002) with the log of TFP assumed non-stationary but cointegrated across countries (VECM).
- Key modeling assumptions:
  - Single final good produced using domestic and imported intermediate goods; intermediate goods are imperfect substitutes.
  - Intermediate goods producers use domestic capital and labor and face domestic TFP shocks.
  - Final good can only be consumed or invested domestically; all cross-border trade occurs at the intermediate goods level.
  - Financial markets: households trade an uncontingent international riskless bond denominated in units of the home-country intermediate good; no other financial assets.
  - A small cost of holding bonds, (·), is included to ensure stationarity of bond holdings; functional form: (D_t; A_{t-1}) = /2 A_{t-1} (D_t / A_{t-1})^2 (modified to include home-country TFP A_t for balanced growth).
- Calibration note:
  - Benchmark elasticity of substitution between intermediate inputs in final-good production: 0:85.
  - With the benchmark calibration, the model explains about 80 percent of the standard deviation of the RER while closely matching volatility of output growth, but places 85 percent of variance at low frequencies (data: 70 percent).

### Mechanisms and model extensions that address the excess persistence puzzle
- Input adjustment costs:
  - Introducing adjustment costs in the use of imported intermediate inputs (and in composition of domestic vs. imported inputs) allows the model to combine:
    - Low short-run elasticity of substitution between imported and domestic intermediate goods (increasing RER volatility at BC frequencies).
    - Higher long-run elasticity (reducing excessive low-frequency volatility).
- Portfolio adjustment costs:
  - Including portfolio adjustment costs further helps increase short-run RER impact responses and reduce long-run responses.
- Net effect:
  - The combination of input adjustment costs and portfolio adjustment costs alters the spectrum of model-implied RER fluctuations to better match the empirical shape (reallocating variance from too-low to appropriate higher frequencies) while maintaining the model’s ability to match the RER’s standard deviation.

### Data construction and additional methodological points
- For consistency with model calibration, the authors compute an own U.S. dollar RER series vis-‡-vis Japan, Canada, the U.K., Australia, and the euro area; aggregate bilateral RERs using currency weights from the Broad Index of the Foreign Exchange Value of the dollar (U.S. Federal Reserve).
- The RER series examined show no evident time trend but are highly persistent (autocorrelogram decays slowly, correlation with 15th lag basically zero), consistent across major currencies.
- The paper uses the modified Bartlett kernel for spectral estimation as per Hamilton (1994).

*Italic: Source content provided from the PDF "_wp1213 - References ............................................................................................................."*

### 1.  Final goods producers

### 1.  Final goods producers

### Final goods production technology
- The final good in the home country, Y_t, is produced using home-country intermediate goods, Y_{H;t}, and foreign-country intermediate goods, Y_{F;t}, with the following technology:
  - Y_t = [ ω^{1/ζ} Y_{H;t}^{(ζ-1)/ζ} + (1−ω)^{1/ζ} Y_{F;t}^{(ζ-1)/ζ} ]^{ζ/(ζ−1)}  (equation (2))
- Parameters and interpretations:
  - ω denotes the fraction of home-country intermediate goods that are used for the production of the home-country final good.
  - ζ is the elasticity of substitution between home-country and foreign-country intermediate goods.

### Representative final goods producer problem
- The representative final good producer in the home country solves:
  - max_{Y_t, Y_{H;t}, Y_{F;t}} P_t Y_t − P_{H;t} Y_{H;t} − P'_{F;t} Y_{F;t}
  - subject to the production function (2).
- P'_{F;t} is the price of the foreign-country intermediate good in the home country.

### Intermediate goods producers: profit maximization
- The representative intermediate goods producer in the home country uses domestic labor and domestic capital to produce home-country intermediate goods and sells to both home and foreign final good producers.
- Taking prices of all goods and factor inputs as given, she maximizes profits by solving:
  - Max_{L_t, K_{t−1}} P_{H;t} Y_{H;t} + P'_{H;t} Y'_{H;t} − P_t (W_t L_t + R_t K_{t−1})

### Intermediate goods production function and notation
- Production function for intermediate goods:
  - Y_{H;t} + Y'_{H;t} = A_{t}^{1−β} K_{t−1}^{β} L_{t}^{1−β}  (equation (3))
- Notation:
  - Y'_{H;t} is the amount of home-country intermediate goods sold to the foreign-country final good producers.
  - P'_{H;t} is the price of the home-country intermediate good in the foreign country.

*Source: _wp1213 - 1.  Final goods producers*

### 3.  TFP processes

### 3.  TFP processes

### Cointegration and VECM for TFP processes
- Assume that logA_t and logA'_t are cointegrated of order C(1;1).
- Specify VECM for log first-differences of TFP processes (home and foreign):
  - (logA_t, logA'_t)' = (c, c')' + Θ (logA_{t-1} - ρ logA'_{t-1} - ξ)' + (ε_t, ε'_t)'  (equation (4) in source)
  - (1; -ρ) is the cointegrating vector, ξ is the constant in the cointegrating relationship.
  - ε_t ~ N(0; Σ) and ε'_t ~ N(0; Σ'), ε_t and ε'_t can be correlated.
  - Δ denotes the first-difference operator.

### Market clearing
- Final goods market:
  - C_t + X_t = Y_t  (equation (5))
- International bond market:
  - D_t + D'_t = 0  (equation (6))

### Equilibrium conditions (home country summary)
- Relative prices defined:
  - eP_{H;t} = P_{H;t} / P_t
  - eP'_{F;t} = P'_{F;t} / P'_t
  - RER_t = P'_t / P_t
  - Law of one price (LOP): P_{H;t} = P'_{H;t} and P_{F;t} = P'_{F;t}
- Household FOCs and labor supply:
  - U_{C;t} = ζ_t  (equation (7))
  - U_{L;t} / U_{C;t} = W_t  (equation (8))
- Capital intertemporal condition:
  - ζ_t = β E_t[ ζ_{t+1} (R_{t+1} + 1 - δ) ]
- Capital law of motion:
  - K_t = (1 - δ) K_{t-1} + X_t  (equation (9))
- Price of riskless bond:
  - Q_t = β E_t[ ζ_{t+1} / ζ_t ( eP_{H;t+1} / eP_{H;t} ) ] - Φ_0(D_t)  (equation (10))
- Optimal risk sharing across countries (using bond prices):
  - E_t[ ζ'_{t+1}/ζ'_t · eP_{H;t+1}/eP_{H;t} · RER_t/RER_{t+1} - ζ_{t+1}/ζ_t · eP_{H;t+1}/eP_{H;t} ] = -Φ_0(D_t)  (equation (11))
- Intermediate goods producers (marginal product pricing):
  - W_t = (1 - β) eP_{H;t} A_{t}^{1-β} K_{t-1}^{β} L_t^{1-β}  (equation (12))
  - R_t = β eP_{H;t} A_{t}^{1-β} K_{t-1}^{β-1} L_t^{1-β}  (equation (13))
- Demand for home and foreign intermediate goods:
  - Y_{H;t} = ω (eP_{H;t})^{-θ} Y_t  (equation (14))
  - Y_{F;t} = (1 - ω) (eP'_{F;t} RER_t)^{-θ} Y_t  (equation (15))
- Final good production aggregator and deflator:
  - Y_t = [ ω^{1/θ} Y_{H;t}^{(θ-1)/θ} + (1 - ω)^{1/θ} Y_{F;t}^{(θ-1)/θ} ]^{θ/(θ-1)}  (equation (16))
  - P_t = [ ω P_{H;t}^{1-θ} + (1 - ω) P_{F;t}^{1-θ} ]^{1/(1-θ)}
- RER expression (LOP used):
  - RER_t = P'_t / P_t = [ ω P_{F;t}^{1-θ} + (1 - ω) P_{H;t}^{1-θ} ]^{1/(1-θ)} / [ ω P_{H;t}^{1-θ} + (1 - ω) P_{F;t}^{1-θ} ]^{1/(1-θ)}
  - Note: only source of RER fluctuations is presence of home bias (ω > 1/2)
- Law of motion of level of bonds (using zero-profit conditions and earlier definitions):
  - eP_{H;t} Q_t D_t = eP_{H;t} Y'_{H;t} - RER_t eP'_{F;t} Y_{F;t} + eP_{H;t} D_{t-1} - eP_{H;t} Φ(D_t; A_{t-1})  (equation (17))
- TFP shocks follow the VECMs above; normalize real variables by lagged TFP to obtain stationarity and log-linearize for solution.
- Rabanal, Rubio-Ramírez and Tuesta (2011) estimate ρ = 1 as sufficient for balanced growth; normalize and log-linearize.

### Baseline model calibration (benchmark)
- Model frequency: quarterly.
- Discount factor β = 0.99 (implies annual real rate of 4 percent).
- Utility parameters:
  - consumption share μ = 0.34
  - coefficient of risk aversion γ = 2
- Technology parameters:
  - depreciation rate δ = 0.025
  - capital share of output β = 0.36
  - ratio of intermediate inputs in final good ω = 0.9
  - elasticity of substitution between intermediate goods θ = 0.85 (baseline; also consider other θ)
- Cost of bond holdings (portfolio adjustment cost) ζ = 1 basis point (0.01)
- VECM parameters (from Rabanal, Rubio-Ramírez and Tuesta (2011) estimates):
  - ρ = 1, Θ = -0.007, c = 0.001, c' = 0.006, Σ = 0.0108, Σ' = 0.0088
  - Zero lags necessary; cannot reject Θ = -Θ' (same speed of convergence)

### Baseline results: matching the RER spectrum
- The baseline model with benchmark calibration can closely replicate the standard deviation (area under spectrum) of the RER but fails on the shape of the spectrum (assigns too much variance to low-frequency movements).
- Key statistics (Table 2 reproduced as bullets; RER denotes log of RER; output is real GDP; growth rates are first differences of logs):
  - U.S. Data:
    - Standard deviation RER: 10.56
    - Output Growth: 0.87
    - Frequency shares: Low 70.0, BC 24.3, High 5.7
  - θ = 0.85:
    - Standard deviation RER: 8:33
    - Output Growth: 0:75
    - Frequency shares: Low 88:4, BC 8:8, High 2:8
  - θ = 0.62:
    - Standard deviation RER: 16:02
    - Output Growth: 0:68
    - Frequency shares: Low 88:2, BC 8:9, High 2:9
  - θ = 1.5:
    - Standard deviation RER: 3:55
    - Output Growth: 0:85
    - Frequency shares: Low 89:0, BC 8:4, High 2:6
- Main model problem: assigns almost 89 percent of RER variance to low-frequency in the baseline model versus 70 percent in the data — the "excess persistence of RER" puzzle.

### Robustness checks (alternative shock specifications)
- Two variations analyzed:
  - "Stationary": Heathcote and Perri (2002) VAR(1) in levels (stationary TFP shocks)
  - "TFP and IST": cointegrated TFP and cointegrated investment-specific technology (IST) shocks (Mandelman et al. (2011))
- Results (Table 3 summary; same units as above):
  - Stationary; θ = 0.85:
    - RER SD: 4:03; Output Growth: 0:92; Frequency shares: Low 85:3, BC 11:3, High 3:3
  - Stationary; θ = 0.62:
    - RER SD: 7:35; Output Growth: 0:87; Frequency shares: Low 85:5, BC 11:1, High 3:4
  - Stationary; θ = 1.5:
    - RER SD: 1:86; Output Growth: 0:97; Frequency shares: Low 85:8, BC 11:3, High 2:9
  - TFP and IST; θ = 0.85:
    - RER SD: 8:58; Output Growth: 0:76; Frequency shares: Low 88:5, BC 8:7, High 2:8
  - TFP and IST; θ = 0.62:
    - RER SD: 16:61; Output Growth: 0:65; Frequency shares: Low 88:2, BC 8:9, High 2:9
  - TFP and IST; θ = 1.5:
    - RER SD: 3:77; Output Growth: 0:82; Frequency shares: Low 89:1, BC 8:3, High 2:6
- Conclusion: stationary TFP or adding IST shocks do not solve the excess persistence puzzle; cointegrated common unit root and slow shock transmission are crucial to explain large RER volatility.

### Extensions to the baseline model — objectives
- Two additional ingredients introduced to help solve the puzzle while replicating RER variance:
  1. Adjustment costs in the use of intermediate imported inputs (input adjustment costs).
  2. Portfolio adjustment costs (costs of changing foreign asset positions).

### A. Input adjustment costs in imported intermediate inputs
- New production specification (following Erceg, Guerrieri and Gust (2006)):
  - Y_t = [ ω^{1/θ} Y_{H;t}^{(θ-1)/θ} + (1 - ω)^{1/θ} (ϕ_t Y_{F;t})^{(θ-1)/θ} ]^{θ/(θ-1)}
  - θ is long-run elasticity of substitution between home and foreign intermediate goods (set to 2 in this extension).
- Input adjustment index ϕ_t:
  - ϕ_t = [ 1 - (κ/2) ( (Y_{F;t}/Y_{H;t}) / (Y_{F;t-1}/Y_{H;t-1}) - 1 )^2 ]  (equation (18))
  - Interpretation: deviation of relative ratio by 1 percent → ϕ_t = 1 - κ/2 (0.01)^2.
  - Examples:
    - κ = 200 → ϕ_t = 0.99 (foreign intermediate input 1 percent less efficient); since foreign input is 10 percent of home final production, home output 0.1 percent smaller.
    - κ = 2000 → output cost in terms of output would be 1 percent.
- Final good producer problem (intertemporal due to adjustment cost depending on t-1):
  - Max E_t sum_{k=0..∞} β^k ξ_{t+k} ( P_{t+k} Y_{t+k} - P_{H;t+k} Y_{H;t+k} - P_{F;t+k} Y_{F;t+k} )
  - ξ_{t+k} is stochastic discount factor; first-order conditions (equations (19) and (20)) reflect intertemporal terms.
- Calibrate as in section V.A except set long-run θ = 2; vary κ and report implications.

- Key quantitative results (Table 4 summary):
  - U.S. Data: RER SD 10.56; Output Growth 0.87; Frequency shares: Low 70.0, BC 24.3, High 5.7
  - κ = 0:
    - RER SD 2:50; Output Growth 0:88; Frequency shares: Low 89:3, BC 8:2, High 2:5
  - κ = 125:
    - RER SD 4:98; Output Growth 0:86; Frequency shares: Low 84:8, BC 11:7, High 3:5
  - κ = 250:
    - RER SD 8:10; Output Growth 0:81; Frequency shares: Low 83:3, BC 12:8, High 3:9
  - κ = 330:
    - RER SD 10:66; Output Growth 0:77; Frequency shares: Low 82:5, BC 13:1, High 4:0
  - κ = 500:
    - RER SD 20:64; Output Growth 0:61; Frequency shares: Low 81:7, BC 13:9, High 4:3
- Findings:
  - Increasing κ (input adjustment costs) increases RER volatility and raises the share of variance at BC- and high-frequencies.
  - κ = 330 matches RER volatility and output growth volatility better but still assigns 82.5 percent to low frequencies versus 70 percent in data — puzzle not fully solved.
- Mechanisms:
  - Input adjustment costs reduce sensitivity of relative quantities to relative prices in short run → larger short-run terms-of-trade and RER volatility.
  - Larger short-run trade deficits and NFA deterioration amplify RER through risk-sharing condition linking RER to expected discounted sum of NFA (equation (23)).
  - Short-run pseudo-elasticity of substitution is low (near zero) and rises over time to long-run θ = 2; low short-run elasticity increases BC-frequency volatility while large long-run θ reduces low-frequency volatility — helps but not sufficient.

### B. Portfolio adjustment costs and interaction with input adjustment costs
- Baseline ζ = 0.01 (1 basis point). Consider ζ = 0.05 (5 basis points) while keeping θ = 2 and vary κ.
- Results (Table 5 summary):
  - U.S. Data: RER SD 10.56; Output Growth 0.87; Frequency shares: Low 70.0, BC 24.3, High 5.7
  - κ = 0; ζ = 0.05:
    - RER SD 2:43; Output Growth 0:84; Frequency shares: Low 88:7, BC 8:7, High 2:6
  - κ = 6; ζ = 0.05:
    - RER SD 2:5; Output Growth 0:81; Frequency shares: Low 76:0, BC 18:8, High 5:2
  - κ = 125; ζ = 0:05:
    - RER SD 11:13; Output Growth 0:70; Frequency shares: Low 72:9, BC 21:1, High 6:0
- Findings:
  - Interaction of large long-run θ = 2, input adjustment costs κ, and higher portfolio adjustment costs ζ = 0.05 can replicate both the standard deviation and the shape of the RER spectrum.
  - Specifically, κ = 125 and ζ = 0:05 replicate volatility and persistence almost perfectly in the model.
- Mechanism and interpretation:
  - Input adjustment costs increase short-run NFA volatility and RER depreciation.
  - Larger portfolio adjustment costs amplify initial RER depreciation but accelerate return to balanced growth (faster decline of D_t), reducing long-run persistence.
  - Numerical interpretation:
    - κ = 125 → 1 percent deviation in input ratio implies output 0:0625 percent smaller.
    - ζ = 0:05 → 1 percent increase in NFA costs household 0:0025 percent units of consumption.

### Concluding remarks (summary of findings)
- Two objectives when matching RER spectrum: match area (volatility) and shape (frequency composition/persistence).
- Baseline two-country IRBC with cointegrated TFP can explain area under spectrum for low θ but places too much weight on low frequencies (excess persistence of RER puzzle).
- Modeling TFP as stationary or adding IST shocks does not solve puzzle.
- Successful solution requires interaction of three ingredients:
  1. Large steady-state elasticity of substitution θ = 2 (long-run).
  2. Adjustment costs in intermediate inputs (κ = 125 in calibration), delivering low short-run elasticities and higher short-run RER volatility (κ = 125 ⇒ 1% input-ratio deviation → output 0:0625% smaller).
  3. Higher portfolio adjustment costs in range of 5 basis points (ζ = 0:05), which raise short-run volatility but reduce long-run persistence.
- Combined, these ingredients allow the model to match both the area and shape of the RER spectrum and thereby solve the excess persistence puzzle.

*Source: _wp1213 - 3.  TFP processes*

### References

### _wp1213 - References

### References (bibliographic entries)
- Adolfson, M., S. Laseen, J. LindÈ and M. Villani, 2007. Bayesian estimation of an open economy DSGE model with incomplete pass-through. Journal of International Economics 72, 481-511.
- Backus, D., P. Kehoe and F. Kydland, 1992, International Business Cycles, Journal of Political Economy, Vol. 100, No. 41, 745 -775.
- Backus, D., P. Kehoe and F. Kydland, 1994, Dynamics of the Trade Balance and the Terms of Trade: The J-Curve?, American Economic Review, Vol. 84, No. 1, pp. 84-103.
- Baxter, M., 2011, International Risk Sharing in the Short Run and in the Long Run, NBER Working Paper 16789.
- Baxter, M. and M. Crucini, 1995, Business Cycles and the Asset Structure of Foreign Trade, International Economic Review, Vol. 36, No. 4, 821-854.
- Baxter, M. and R. King, 1999, Measuring Business Cycles: Approximate Band-Pass Filters for Macroeconomic Time Series. Review of Economics and Statistics, 81(4), November 1999: 575-93.
- Benigno, G., 2004. Real exchange rate persistence and monetary policy rules. Journal of Monetary Economics 51, 473-502.
- Bergin, P. and R. Feenstra, 2001, Pricing to Market, Staggered Contracts, and Real Exchange Rate Persistence. Journal of International Economics 54 (2), 333-359.
- Bouakez, H, 2005, Nominal rigidity, desired markup variations and real exchange rate persistence. Journal of International Economics, Vol. 61, 49-74.
- 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, 533-563.
- Cheung, Y., and K. Lai, 2000, On the Purchasing Power Parity, Journal of International Economics, Vol. 52, 321-330.
- Cheung, Y., M. Chinn and A. Garcia-Pascual, 2005, Empirical exchange rate models of the nineties: Are any Öt to survive?, Journal of International Money and Finance, Vol. 24(7), 1150-1175.
- Christiano, L. and T. Fitzgerald 2003, The Band Pass Filter. International Economic Review, Vol. 44, No. 2, 435-465.
- Clarida, R. and J. GalÌ, 1994, Sources of real exchange-rate áuctuations: How impor­tant are nominal shocks?, Carnegie-Rochester Conference Series on Public Policy, Vol. 41(1), 1-56.
- Corsetti, G., L. Dedola and S. Leduc, 2008, International Risk Sharing and the Trans­mission of Productivity Shocks, Review of Economic Studies, Vol. 75, 443-473.
- Corsetti, G., L. Dedola and F. Viani, 2011, The International Risk-Sharing Puzzle is at Business Cycle and Lower Frequency, CEPR Discussion Papers 8355.
- Eichenbaum, M. and C. Evans. 1995, Some Empirical Evidence on the E§ects of Shocks to Monetary Policy on Exchange Rates. Quarterly Journal of Economics, 110(4): 975ñ1009.
- Engel, C. and K. West, 2005, Exchange Rates and Fundamentals, Journal of Political Economy, Vol. 107, No. 3, 485-517.
- Erceg, C., L. Guerrieri and C. Gust, 2006, SIGMA: A New Open Economy Model for Policy Analysis, International Journal of Central Banking, vol. 2(1), March.
- Faust, J. and J. Rogers, 2003, Monetary Policyís Role in Exchange Rate Behavior, Journal of Monetary Economics, vol. 50 (November 2003), pp. 1403-1424.
- Hamilton, J., 1994, Time Series Analysis, Princeton University Press.
- Heathcote, J., and F. Perri, 2002, Financial Autarky and International Business Cycles, Journal of Monetary Economics, Vol. 49, 601-627.
- Hooper, P., Johnson, K. and J. Marquez, 2000, Trade Elasticities for the G-7 Coun­tries. Princeton Studies in International Economics, no. 87, August.
- Huizinga, J., 1987, An Empirical Investigation of the Long-Run Behavior of Real Ex­change Rates. Carnegie-Rochester Conference Series on Public Policy, Vol. 27, pp. 149-214.
- Johansen, S., 1991, Estimation and Hypothesis Testing of Cointegration Vectors in Gaussian Vector Autoregressive Models, Econometrica 59, 1551-1580.
- King, R., C., Plosser and S., Rebelo, 1988, Production, Growth and the Business Cy­cle, Journal of Monetary Economics, 21, 195-232.
- Mandelman, F., Rabanal, P., Rubio-RamÌrez, J.F. and D. Vil·n, 2011, Investment-SpeciÖc Technology Shocks and International Business Cycles: An Assessment, Review of Economic Dynamics 14, pp. 136-155.
- Meese, R. and K. Rogo§, 1983, Empirical exchange rate models of the seventies: do they Öt out of sample?, Journal of International Economics 14, pp. 3ñ24.
- Mumtaz, H. and P. Surico, 2009, The Transmission of International Shocks: A Factor-Augmented VAR Approach, Journal of Money, Credit & Banking 41, pp. 71-100.
- Murray, C. and D. Papell, 2002, The Purchasing Power Parity Persistence Paradigm. Journal of International Economics, 56(1): 1ñ19.
- Nason J. and J. Rogers, 2008, Exchange Rates and Fundamentals: A Generalization. Federal Reserve Bank of Atlanta Working Paper 2008-16.
- Obstfeld, M, and K. Rogo§, 2000, The Six Major Puzzles in International Macroe­conomics: Is There A Common Cause? NBER Working Paper 7777.
- Rabanal, P., J. Rubio-RamÌrez, and V. Tuesta, 2011, Cointegrated TFP Processes and International Business Cycles, Journal of Monetary Economics, Vol. 58, 156-171.
- Rabanal, P. and V. Tuesta, 2010, Euro-Dollar Real Exchange Rate Dynamics in an Estimated Two-Country Model: An Assessment, Journal of Economic Dynamics and Control, Vol. 34, pp. 780-797.
- Rogo§, K., 1996, The Purchasing Power Parity Puzzle, Journal of Economic Literature, vol. 34, 647ñ68.
- Sarno, L., 2003, Nonlinear Exchange Rate Models: A Selective Overview, IMF Working Paper 03/111.
- Steinsson, J., 2008, The Dynamic Behavior of the Real Exchange Rate in Sticky Price Models, American Economic Review, Vol. 98, pp. 519-533.
- Stockman, A., L. Tesar, 1995, Tastes and Technology in a Two-Country Model of the Business Cycle: Explaining International Comovements, American Economic Review, Vol 85(1), 165-185.

### Figures (captions and displayed numeric labels)
- Figure 1. U.S. dollar: time series, autocorrelogram and spectral density.
  - Time axis labels: 19751980198519901995200020052010
  - Vertical labels: 4.4, 4.5, 4.6, 4.7, 4.8
  - Series labels: Log (Real Exchange Rate), lnrer, hptrend
  - Autocorrelogram axis: 051015202530 (Lag); values: -0.5, 0, 0.5, 1
  - Spectrum frequency axis: 00.511.522.53; Spectrum values: 0, 0.005, 0.01, 0.015, 0.02
  - Note: BC frequencies; label: spectrum

- Figure 2. Comparison between the model and the data.
  - Spectrum frequency axis: 00.511.522.53
  - Spectrum values: 0, 0.005, 0.01, 0.015, 0.02, 0.025, 0.03
  - Series: Spectrum of Log (Real Exchange Rate); BC; Model, θ =0.85; Data

- Figure 3. IRF to a home TFP shock whenchanges. Home-country variables.
  - Time horizon axis: 0102030
  - Panels and labels with values:
    - Y: values 0, 0.005, 0.01
    - C: values 0, 2, 4, 6, 8 x 10 -3
    - K: values 0, 2, 4, 6, 8 x 10 -3
    - L: values 0, 1, 2, 3 x 10 -3
    - YH: values 0, 0.005, 0.01
    - YF: values -5, 0, 5, 10 x 10 -3
    - YF-YF(-1)-(YH-YH(-1)): values -3, -2, -1, 0, 1 x 10 -3
    - YH+YH*: values 0, 0.005, 0.01
  - Parameter labels: ι =0; ι =125; ι =250; ι =330

- Figure 4. IRF to a home TFP shock whenchanges. Foreign-country variables.
  - Time horizon axis: 0102030
  - Panels and labels with values:
    - Y*: -1, 0, 1, 2, 3 x 10 -3
    - C*: 0, 1, 2, 3 x 10 -3
    - K*: -10, -5, 0, 5 x 10 -3
    - L*: -1, 0, 1, 2 x 10 -3
    - YH*: 0, 0.005, 0.01, 0.015
    - YF*: -2, -1, 0, 1, 2 x 10 -3
    - YH*-YH*(-1)-(YF*-YF*(-1)): -1, 0, 1, 2, 3 x 10 -3
    - YF+YF*: -5, 0, 5, 10, 15 x 10 -4
  - Parameter labels: ι =0; ι =125; ι =250; ι =330

- Figure 5. IRF to a home TFP shock whenchanges. Relative prices.
  - Frequency axis: 051015202530
  - Panels and labels with values:
    - RER: 0, 0.002, 0.004, 0.006, 0.008, 0.01, 0.012, 0.014, 0.016, 0.018
    - D: -0.035, -0.03, -0.025, -0.02, -0.015, -0.01, -0.005, 0
    - PH: -2.5, -2, -1.5, -1, -0.5, 0 x 10 -3
    - PFS: 0, 0.5, 1, 1.5, 2, 2.5 x 10 -3
  - Parameter labels: ι =0; ι =125; ι =250; ι =330

- Figure 6. Implied elasticity whenchanges.
  - Frequency axis: 051015202530
  - Elasticity values: 0, 0.2, 0.4, 0.6, 0.8, 1, 1.2, 1.4, 1.6, 1.8, 2
  - Parameter labels: ι =0; ι =125; ι =250; ι =330

- Figure 7. Comparison between the extended model and the data.
  - Spectrum frequency axis: 00.511.522.53
  - Spectrum values: 0, 0.005, 0.01, 0.015, 0.02, 0.025, 0.03
  - Series: Spectrum of Log (Real Exchange Rate); BC; Ext. Model; Data

- Figure 8. IRF to a home TFP shock whenchanges and= 125. Home-country variables.
  - Time horizon axis: 0102030
  - Panels and labels with values:
    - Y: 0, 0.005, 0.01
    - C: 0, 2, 4, 6, 8 x 10 -3
    - K: 0, 2, 4, 6, 8 x 10 -3
    - L: -1, 0, 1, 2 x 10 -3
    - YH: 0, 0.005, 0.01
    - YF: -0.01, -0.005, 0, 0.005, 0.01
    - YF-YF(-1)-(YH-YH(-1)): -3, -2, -1, 0, 1 x 10 -3
    - YH+YH*: 0, 0.005, 0.01
  - Parameter labels: φ =0.01; φ =0.02; φ =0.04; φ =0.05

- Figure 9. IRF to a home TFP shock whenchanges and= 125. Foreign-country variables.
  - Time horizon axis: 0102030
  - Panels and labels with values:
    - Y*: -2, 0, 2, 4 x 10 -3
    - C*: 0, 1, 2, 3 x 10 -3
    - K*: -5, 0, 5, 10 x 10 -3
    - L*: -1, 0, 1, 2 x 10 -3
    - YH*: 0, 0.005, 0.01, 0.015, 0.02
    - YF*: -1, 0, 1, 2 x 10 -3
    - YH*-YH*(-1)-(YF*-YF*(-1)): -1, 0, 1, 2, 3 x 10 -3
    - YF+YF*: 0, 0.5, 1, 1.5 x 10 -3
  - Parameter labels: φ =0.01; φ =0.02; φ =0.04; φ =0.05

- Figure 10. IRF to a home TFP shock whenchanges and= 125. Relative prices.
  - Time horizon axis: 0102030
  - Panels and labels with values:
    - RER: 0, 0.005, 0.01, 0.015, 0.02, 0.025, 0.03
    - D: -0.025, -0.02, -0.015, -0.01, -0.005, 0
    - PH: -3.5, -3, -2.5, -2, -1.5, -1, -0.5, 0 x 10 -3
    - PFS: 0, 0.5, 1, 1.5, 2, 2.5, 3, 3.5 x 10 -3
  - Parameter labels: φ =0.01; φ =0.02; φ =0.04; φ =0.05

*Source: _wp1213 - References (figures and bibliographic entries as listed in the provided PDF content).*

---


_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2012/_wp1213.pdf_