## _wp15280 - 7. Additional material is gathered in a companion online Appendix.

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---

### II. STYLIZED FACTS — Commodity Export Shares
- Data and sample:
  - World Bank WDI panel covering 189 countries, 1960–2013.
  - Country-specific average shares of agricultural, fuel and metals in total exports.
  - Countries: 61 EMEs and 128 non-EMEs (74 advanced; 54 low-income).
- Key statistics:
  - Median commodity export share in EMEs: 25.7 percent.
  - Median commodity export share in advanced economies: 11.2 percent.
  - Mann-Whitney test: rejects equivalence between EMEs and advanced economies at 1 percent significance level.
  - Kruskal-Wallis tests: reject equivalence across EMEs, advanced, low-income groups.
- Stylized fact:
  - The share of commodities in total exports in the average EME is more than double that of advanced economies.
- Robustness:
  - Stylized Fact 1 robust to gross and net commodity export shares to GDP.

### II. STYLIZED FACTS — Cyclicality
- Data and construction:
  - Quarterly country-specific commodity price indexes for EMEs, 1980.Q1–2014.Q4; indices average 44 commodity international prices with country-specific constant weights (1999–2004).
  - θ_{i,n} from UN Comtrade; PCo_{t,i} deflated using US CPI from IMF Primary Commodity Price Database.
- Sample for cyclicality correlations:
  - EME13: Argentina, Brazil, Bulgaria, Chile, Colombia, Ecuador, Malaysia, Mexico, Peru, Russia, South Africa, Ukraine, Venezuela.
  - Median commodity export share in EME13: 28.7.
  - LAC4 subset: Brazil, Chile, Colombia, Peru.
- Empirical method:
  - Serial correlations ρ(X_{n,t}, PCo_{n,t+j}), j = −4,...,4 for X = real output, real private consumption, real investment, trade balance, real exchange rates, EMBIG and CEMBI spreads.
  - HP filter to extract cyclical components; results robust to growth-rate filtering.
  - Statistics are simple averages across EME13; confidence bands ±1.5 standard deviations.
- Stylized Fact 2 (cyclicality):
  - Country-specific commodity prices in EMEs are procyclical and lead the cycle of output, consumption and investment.
  - Commodity prices are countercyclical with real exchange rates and external risk premia (stronger for CEMBI).
- Key quantitative correlations:
  - Average contemporaneous correlation between commodity price index and real GDP: about 0.5.
  - Correlation slightly higher with one-quarter lag.
  - Correlation increases to 0.6 if real income computed by deflating GDP with CPI.
  - Lagged index values negatively correlated with trade balance; contemporaneous comovement weaker.
  - Commodity price indices negatively correlated with real exchange rate (price increases → real appreciations).
  - Negative comovement with interest rate spreads, stronger for corporate risk premia (CEMBI).
- Interpretation:
  - High (low) commodity prices → cost of issuing debt in foreign markets decreases (increases) for EMEs.

### II. STYLIZED FACTS — Common Factors
- Visual and formal evidence:
  - Strong comovement in cyclical indices across EMEs; PCA corroborates.
- Stylized Fact 3:
  - Preponderant role of common factors in commodity price index dynamics across EMEs; extends to real GDP dynamics.
- PCA quantitative results:
  - First principal component accounts for 78 percent of variance in commodity price indices across 13 EMEs.
  - PCA across 44 goods grouped into five SITC categories: first PC accounts for 55 percent of variance.
  - PCA on cyclical real GDP across 13 EMEs: first PC accounts for 76 percent of variance.
- Additional observation:
  - High comovement not mechanically due to similar export profiles (example: Colombia–Peru serial correlation 0.9 despite differing commodity composition).

### Summary implications of stylized facts
- Commodity price movements are:
  - Large, procyclical, commonly driven.
  - Influence EME business cycles via: direct household income effects from commodity revenue (exogenous in model), real exchange rate appreciation when prices rise, and lower external financing costs when prices rise.
- Motivates a dynamic general equilibrium model linking exogenous commodity price changes to real activity fluctuations.

### III. MODEL — Overview and agents
- Framework:
  - Multi-country extension of small open economy DSGE (Mendoza (1991); Schmitt-Grohè and Uribe (2003)) with five departures:
    1. Country-specific commodity endowment sector with exogenous international price fluctuations; commodity sold abroad; revenue to households.
    2. Home and foreign goods imperfect substitutes for consumption and investment.
    3. Investment goods sector combining home and foreign goods.
    4. Commodity prices modeled with dynamic factor: latent common factor + idiosyncratic shocks.
    5. N EMEs interacting with rest of world; comovement across EMEs arises solely from common commodity price factor shocks.
- Agents per EME:
  - Households: GHH utility, supply labor & capital, issue one-period foreign bonds, receive commodity revenue p_{Co,t} \bar{Co} and profits ξ_t, face capital adjustment costs.
  - Firms (home goods): rent labor & capital, neoclassical production Y_t = z_t K_{t−1}^{α} L_t^{1−α}.
  - Investment goods producers: CES aggregation of home and foreign inputs.
  - Rest of world: provides funding at stochastic premium, demands home goods C_{h,j,t}^* = (p_{h,j,t})^{−ε_{j,e}} Y_t^*, supplies foreign goods.

### III. MODEL — Household and market structure (formal)
- Lifetime utility:
  - E_0 sum_{t=0}^{∞} β^{t} U(C_t, L_t) with U(C_t, L_t) = (1/(1−σ)) [ C_t − ψ_c L_t^{1+γ_c}/(1+γ_c) ]^{1−σ}.
- Budget constraint:
  - p_{c,t} C_t + p_{x,t} X_t + R_{t−1} D_{t−1} = w_t L_t + r_{k,t} K_{t−1} + D_t + p_{Co,t} \bar{Co} + ξ_t.
- Consumption aggregator:
  - CES with η_c and share α_c; consumption price p_{c,t} = [ (1−α_c) (p_{h,t})^{1−η_c} + α_c ]^{1/(1−η_c)}; real exchange rate = (p_{c,t})^{−1}.
- Capital accumulation with adjustment cost s_t(·), Christiano et al. (2010) functional form s_t(·) = (1/2)( e^{√a(·−1)} + e^{−√a(·−1)} − 2 ).
- Market clearing and external demand:
  - Y_t = C_{h,t} + X_{h,t} + C_{h,t}^*.
  - Trade balance TB_t = p_{Co,t} Co + p_{h,t} C_{h,t}^* − C_{f,t} − X_{f,t}.
  - Real income measure used for data matching: GDP_t / p_{c,t} where GDP_t = p_{h,t} Y_t + p_{Co,t} Co.

### 1. Common Factor Structure in Commodity Prices (modeled)
- Common factor:
  - fCo_t = φCo fCo_{t−1} + σfCo εfCo_t, εfCo_t ∼ N(0,1).
- Country-specific price:
  - pCo_j,t = ωCo_j fCo_t + εCo_j,t.
- Idiosyncratic dynamics:
  - εCo_j,t = ρCo εCo_j,t−1 + σCo_j νCo_j,t, νCo_j,t ∼ N(0,1).
- Other driving processes (examples):
  - ln R*_t AR(1): ln R*_t = (1−ρR*) ln R* + ρR* ln R*_{t−1} + σR* εR*_t.
  - S_jt spread term specification includes zR_j,t with AR(1) dynamics.

### Mechanism of a commodity price shock (qualitative log-linearized)
- Home goods market clearing decomposition: ŷ_t = (C^h/Y) ĉ^h_t + (X^h/Y) ẋ^h_t + (C^h*/Y) ĉ^h*_t.
- Three effects:
  - Effect 1 — Demand channel: higher commodity revenue raises household demand for home goods; raises p_h and domestic production; depends on η_c and commodity sector size.
  - Effect 2 — Investment goods channel: higher production raises rental return and demand for investment goods; depends on η_x and investment share.
  - Effect 3 — External demand offset: foreign demand for home goods falls as p_h rises; depends on ε_{j,e} and non-commodity export size.
- Net outcome:
  - If Effects 1 and 2 dominate Effect 3 → net expansion, higher labor demand, real appreciation; productivity shocks operate oppositely.

### Estimation preliminaries and calibration
- Sample selected for estimation: four EMEs — Brazil, Chile, Colombia, Peru.
  - Median commodity export share of these four: 35.4.
  - First principal component explains 81 and 67 percent of variance in commodity price indices and real output across the four economies, respectively.
  - Commodity export shares: Brazil (17.9); Chile (69.7); Colombia (28.6); Peru (60.5).
  - Low bilateral trade linkages: Brazil’s exports to Chile/Colombia/Peru are 0.2/0.1/0.1 percent of output; Brazil’s exports to main trading partners = 10.1 percent of output.
- Calibration highlights:
  - σ = 2; 1/γ_c = 1.72; δ = 10 percent; ε_{j,e} = 1.18; η_c = η_x = 0.43; Ω_u = 0.001.
  - Steady state labor normalized to 0.3; R* = 1.01; ̄z_j and Y* = 1.
  - D and ̄Co set to match long-run external debt and commodity export shares; z_R calibrated to match long-run country risk premium; β set as inverse of country real interest rate.
- Model fit:
  - Matches relative size of commodity sector by construction; reproduces historical interest rate premia and debt-to-output ratios with some deviations in consumption/imports/exports.
- Bayesian estimation (short-run dynamics):
  - Estimated parameters include persistences {φCo, ρCo_j, ρzR_j, ρR*, ρY*, ρz_j}, standard deviations {σfCo, σCo_j, σR_j, σzR, σY*, σz_j}, loadings {ωCo_j}, capital adjustment cost {a_j}.
  - Observables (quarterly): real private consumption, real income, real investment, trade balance/GDP, EM-BIG spread, real effective exchange rate, commodity price index.

### Estimation results — posteriors and common factor
- Posterior highlights:
  - φ_Co posterior mean = 0.73.
  - σ_fCo = 6.31 percent (common factor shock standard deviation).
  - Idiosyncratic AR parameters ρ_Co^j reported as high.
  - Country loading factors (Colombia normalized to 1): Chile posterior mode = 1.53; Peru = 0.89; Brazil = 0.53.
- Time series of f_Co (Kalman smoother at posterior mean):
  - Large, short-lived deviations post-2005; deviations reach near 30 percentage points; pattern: increase pre-crisis, sharp fall in crisis, recovery, fall late sample.

### Business cycle drivers — FEVD and historical decompositions
- Unconditional FEVD (median shares):
  - Commodity price shocks median share = 42 percent.
  - Country range for commodity shocks: Brazil 27.5 percent to Chile 77.1 percent; Colombia 43.5 percent; Peru 40.4 percent.
  - Much of these shares related to common factor εfCo.
- Horizon dependence:
  - At one-quarter horizon the share of commodity shocks falls to between 1/10 and 1/2 of unconditional share; domestic productivity shocks dominate short horizons.
- Historical decompositions:
  - Commodity and productivity shocks preponderant across four countries.
  - Commodity shocks important in second half of sample; Chile notable for persistent large influence.
- Counterfactual volatility (when commodity shocks turned off — S.D. of real income as percent of observed S.D.):
  - Peru 69.6 percent.
  - Chile 72.1 percent.
  - Colombia 75.4 percent.
  - Brazil 91.1 percent.
  - When only commodity shocks remain, Chile’s counterfactual volatility can exceed observed (other shocks smoothed volatility).

### Quantifying the mechanism — impulse responses (IRFs)
- Impulse: unexpected 1 S.D. common commodity shock = common factor increases by 6.3 percentage points.
- Key IRF magnitudes (percentage deviations from steady state; cross-country ranges reported):
  - Home goods production Y_t: hump-shaped; peak between 0.1 and a little over 0.5 percentage points.
  - Real income (GDP_t / p_{c,t}): increases on impact ranging from 3 to 0.5 percentage points across four countries, then declines.
  - Total private consumption increases; home goods consumption increases between 0.5 and 1.0 percentage point.
  - Investment: pronounced, hump-shaped; peaks ~4–5 quarters after shock; peak deviations between 0.5 and 3.5 percentage points.
  - Real rental return increases on impact between 10 and 100 basis points.
  - Non-commodity exports fall between 0.5 and 3 percentage points.
  - Real exchange rate appreciates between 0.5 and 2 percentage points.
- Net effect: expansion via consumption and investment dominates export contraction; labor demand and real wages rise.
- Cross-country comovement: common factor shock generates substantial comovement.

### Model performance: fit to moments and comovement
- Cross-country real income correlations:
  - Data average = 0.45.
  - Benchmark model with common factor (CF): average = 0.32.
  - No common factor model (NCF, ω_Co^j = 0 ∀ j): average = 0.02.
  - Marginal likelihood: NCF = 4445.3; CF = 4530.8 (large likelihood fall without common factor).
- Serial correlations within countries:
  - Model reproduces procyclicality of commodity prices with income, consumption, investment; captures trade balance to GDP and countercyclicality of real exchange rates.
- Second moments:
  - Commodity price volatility ≈ four times income volatility.
  - Investment and RER volatility ≈ twice income volatility.
  - Consumption and trade balance share volatility < income volatility.
  - Example exact numbers (selected, Table 8):
    - PCo S.D. Model = 11.08; Data = 12.14.
    - GDP/CPI S.D. Model = 3.80; Data = 2.86.
    - Investment S.D. Model = 7.40; Data = 5.14.

### Robustness and extensions — major findings
- A. SVAR comparison:
  - SVAR replacing terms of trade with p_Co_t: commodity shocks account for large output share.
  - SVAR median = 50 percent; DSGE median = 42 percent (unconditional).
  - Country differences: Brazil SVAR 15 vs DSGE 28; Chile SVAR 53 vs DSGE 77; Colombia aligned (47 vs 44); Peru SVAR 80 vs DSGE ~40.
  - Predicted series using only p_Co_t shocks highly correlated (median corr ≥ 0.9 except Chile).
- B. Countercyclical spreads extension:
  - Allows spreads to depend on expected TFP and commodity prices and adds working-capital effect.
  - Share of variance from commodity prices rises in Brazil, Colombia, Peru; Chile slightly reduced but remains largest.
- C. Correlated external forces:
  - Allow contemporaneous correlation between f_Co_t and Y*_t; posterior mean σ_{Co,Y*} = 0.021.
  - Share from external demand shocks increases to average 5.5 percent (from ~0.1 percent).
  - Marginal likelihood improves relative to benchmark.
- D. Absence of commodity shocks:
  - TFP shocks dominate output variance in three of four countries; Brazil exception with larger interest rate shock role.
  - Marginal likelihood drops from 4530.8 to 4410.6 without commodity shocks.
  - Reduced model fails to reproduce negative comovement between RER and real income.
- E. Alternative dynamic factor models (longer samples, up to 13 EMEs):
  - Common factors track DSGE benchmark; post-2005 volatility pronounced.
- F. Alternative filtering (growth rates vs HP):
  - Stylized facts robust; FEVD median share using growth rates = 68.1 percent; Chile’s share drops but remains >50 percent; others increase.
- G. Cyclical production of commodities:
  - Serial correlations between commodity volumes and international prices show no systematic comovement at business-cycle frequencies; Granger causality tests show no causality.
  - Supports price-taker assumption and exogenous commodity revenue shocks.
- H. Role of fiscal policy:
  - Government sector absent from model; empirical evidence suggests real government expenditure not positively correlated with commodity prices.
  - Removing public expenditure marginally increases comovement between real activity and commodity prices; suggestive evidence fiscal transmission via public spending is not first-order for episodes studied.

### Concluding remarks and policy-relevant implications
- Main conclusions:
  - Exogenous international commodity price fluctuations are major drivers of aggregate fluctuations in sampled EMEs.
  - Commodity prices exhibit a dominant common factor with large correlated swings affecting domestic business cycles ("roller coaster ride").
  - Commodity shocks often amplify or are amplified by contemporaneous movements in interest rates/spreads.
- Policy-relevant implications and open questions:
  - Understanding fiscal and monetary policy responses to commodity shocks is crucial and requires model extensions.
  - Important extensions: government behavior, optimal policy design, richer commodity-sector features (storage, inventories, cyclicality of production, ownership).

*Source: _wp15280 - 7. Additional material is gathered in a companion online Appendix.*

### 7. Additional material is gathered in a companion online Appendix.

### _wp15280 - 7. Additional material is gathered in a companion online Appendix.

### II. STYLIZED FACTS — Commodity Export Shares
- Data and construction:
  - Large (unbalanced) annual panel using World Bank’s World Development Indicators (WDI) covering 189 countries between the years 1960 and 2013.
  - For each country, the panel contains the share of three broad groups of commodities in total exports: agricultural, fuel and metals.
  - Country-specific commodity export shares obtained by averaging the sum of these three shares across time.
  - Countries separated into 61 EMEs and 128 non-EMEs (further subcategorized into 74 advanced and 54 low-income economies).
- Key descriptive statistics (reported in Table 1 in source):
  - Median commodity export share in EMEs: 25.7 percent.
  - Median commodity export share in advanced economies: 11.2 percent.
  - Mann-Whitney test: rejects equivalence between EMEs and advanced economies at 1 percent significance level.
  - Kruskal-Wallis tests: reject equivalence across the three groups (EMEs, advanced, low-income).
- Stylized Fact 1:
  - The share of commodities in total exports in the average EME is more than double that of advanced economies.
- Robustness:
  - Stylized Fact 1 is robust to two alternative measures of the relative importance of commodities: the gross and net share of commodity exports to GDP (see Appendix).

### II. STYLIZED FACTS — Cyclicality
- Data and construction:
  - Quarterly panel dataset of country-specific commodity price indexes for EMEs between 1980.Q1 and 2014.Q4.
  - Indices constructed by averaging international prices of 44 commodity goods using constant weights equal to country-specific average shares of each commodity in total exports between 1999 and 2004.
  - Formal definition: PCo_{n,t} = sum_{i=1}^{44} θ_{i,n} PCo_{t,i}, where θ_{i,n} is export share of commodity i for country n and PCo_{t,i} is the real USD spot price of commodity i in world markets.
  - θ_{i,n} computed from United Nations’ Comtrade data; PCo_{t,i} deflated using US CPI from IMF’s Primary Commodity Price Database.
- Sample for cyclicality correlations (reduced by data requirements):
  - Sample reduced to 13 EMEs (EME13) due to minimum time-series requirements.
  - The 13 countries: Argentina, Brazil, Bulgaria, Chile, Colombia, Ecuador, Malaysia, Mexico, Peru, Russia, South Africa, Ukraine and Venezuela.
  - Median commodity export share in this EME13 group: 28.7.
  - Subset LAC4: Brazil, Chile, Colombia, and Peru.
- Empirical method:
  - Compute serial correlations ρ(X_{n,t}, PCo_{n,t+j}) for j = −4,...,4 where X is sequentially replaced by: real output, real private consumption, real investment, trade balance, real exchange rates, and external interest rate premia (EMBIG and CEMBI spreads).
  - Hodrick-Prescott filter used to extract cyclical components; results robust to growth-rate filtering (Appendix).
  - Statistics reported are simple averages across the 13 countries (EME13); confidence bands denote +/-1.5 standard deviations.
- Stylized Fact 2:
  - Country-specific commodity prices in EMEs are procyclical and lead the cycle of output, consumption and investment.
  - They are countercyclical to real exchange rates and measures of external risk premia.
- Key quantitative correlations and observations:
  - Average contemporaneous correlation between the commodity price index and real GDP: about 0.5.
  - Correlation slightly higher when the index is lagged one quarter.
  - Correlation increases to 0.6 if real income is computed by deflating GDP with the CPI.
  - Lagged index values are negatively correlated with the trade balance but commove contemporaneously (correlations less tightly estimated).
  - Commodity price indices are negatively correlated with the real exchange rate (increases in indices accompanied by real appreciations).
  - Negative comovement between commodity price indices and interest rate spreads in EMEs, stronger for corporate risk premia proxied by the CEMBI.
- Interpretation:
  - When commodity prices are high (low), the cost of issuing debt in foreign capital markets decreases (increases) for EMEs.

### II. STYLIZED FACTS — Common Factors
- Visual and formal evidence:
  - Figure 3 plots time series of cyclical indices for each EME in the sample showing strong comovement.
  - Principal component analysis (PCA) results (presented in Appendix) corroborate the visual evidence.
- Stylized Fact 3:
  - There is a preponderant role of common factors when accounting for the dynamics of commodity price indices across EMEs; this extends to the dynamics of real gross domestic product.
- PCA quantitative results:
  - First principal component accounts for 78 percent of the variance in the commodity price indices across the 13 EMEs.
  - Extending PCA across the 44 commodity goods grouped into five SITC categories: first principal component accounts for 55 percent of variance in commodity prices across the five SITC categories.
  - PCA on cyclical real GDP across the 13 EMEs: first principal component accounts for 76 percent of the variance of economic activity.
- Additional observations:
  - High comovement across countries does not mechanically arise from similar commodity export profiles (example: Colombia and Peru serial correlation of commodity price indexes is 0.9 despite differing export composition: Colombia — crude oil 53 percent, coal 15 percent; Peru — gold 28 percent, copper 22 percent).
  - Results are robust to alternative filtering (growth rates) to address potential HP-filter artifacts.

### Summary of Implications from Stylized Facts
- Commodity price movements are large, procyclical, commonly driven, and substantially influence EMEs’ business cycles via:
  - Direct income effects to households from commodity endowment receipts (modeled as exogenous revenue fluctuations).
  - Effects on real exchange rates (real appreciation when commodity prices rise).
  - Effects on external financing costs (lower spreads when commodity prices rise).
- These stylized facts motivate a dynamic general equilibrium model that formally links exogenous commodity price changes to fluctuations in real economic activity.

### III. MODEL — Setup and Agents
- Model overview:
  - Multi-country version of a small open economy framework (Mendoza (1991); Schmitt-Grohè and Uribe (2003)) with five departures:
    1. Country-specific commodity sector facing exogenous international price fluctuations; commodity good is an endowment entirely sold abroad; revenue accrues to households.
    2. Home (h) and foreign (f) goods that are imperfect substitutes for consumption and investment production.
    3. Investment goods sector producing investment using home and foreign goods.
    4. Commodity prices modeled with a dynamic factor structure incorporating a latent common factor plus idiosyncratic shocks to capture cross-country comovement.
    5. Multi-country structure: collection of N EMEs interacting with the rest of the world as small open economies; comovement across EMEs arises solely from shocks to the common commodity price factor.
- Agents in each EME:
  - Households: consume a bundle of home and foreign goods, supply labor and capital, issue non state-contingent one-period bonds in international markets, receive commodity sector income and firms’ profits, face capital adjustment costs.
  - Firms producing home goods: rent labor and capital from households; use neoclassical production technology.
  - Investment goods producers: combine home and foreign goods to produce investment goods for households.
  - Rest of the world: provides funding at stochastic premium over world interest rate, demands home goods as function of relative price and price elasticity, supplies foreign goods.

### III. MODEL — Households (Formal)
- Lifetime utility:
  - E_0 sum_{t=0}^{∞} β^{t} U(C_t, L_t).
  - U(C_t, L_t) = (1/(1−σ)) [ C_t − ψ_c L_t^{1+γ_c}/(1+γ_c) ]^{1−σ}, a GHH specification.
  - σ: constant relative risk aversion; γ_c: inverse Frisch elasticity parameter.
- Budget constraint:
  - p_{c,t} C_t + p_{x,t} X_t + R_{t−1} D_{t−1} = w_t L_t + r_{k,t} K_{t−1} + D_t + p_{Co,t} \bar{Co} + ξ_t
    - p_{c,t}: price of consumption good.
    - p_{x,t}: price of investment good.
    - R_{t−1}: gross external real interest rate.
    - D_t: stock of international debt at beginning of period.
    - w_t: real wage.
    - r_{k,t}: rent of capital; K_{t−1}: capital stock.
    - p_{Co,t} \bar{Co}: revenue from constant endowment flow of commodity quantities accruing to households.
    - ξ_t: profits from domestic production sector.
  - Commodity price shocks enter as exogenous revenue fluctuations via p_{Co,t} \bar{Co}.
- Consumption aggregation:
  - CES aggregator: C_t = [ (1−α_c)^{1/η_c} (C_{h,t})^{(η_c−1)/η_c} + α_c^{1/η_c} (C_{f,t})^{(η_c−1)/η_c} ]^{η_c/(η_c−1)}
  - Expenditure: p_{c,t} C_t = p_{h,t} C_{h,t} + C_{f,t}, with imported good price as numeraire.
  - Consumption price function: p_{c,t} = φ(p_{h,t},1) = [ (1−α_c) (p_{h,t})^{1−η_c} + α_c (1)^{1−η_c} ]^{1/(1−η_c)}.
  - Real exchange rate in model defined as (p_{c,t})^{−1}.
- Capital accumulation:
  - K_t = (1−δ) K_{t−1} + X_t (1 − s_t(X_t / X_{t−1})).
  - Adjustment cost s_t(·) with properties s_t(1) = s'_t(1) = 0 and s''_t(·) > 0.
  - Specific functional form (Christiano et al. 2010):
    - s_t(X_t / X_{t−1}) = (1/2)( e^{√a(X_t / X_{t−1} − 1)} + e^{−√a(X_t / X_{t−1} − 1)} − 2 ).

### III. MODEL — Production and Investment
- Domestic goods production:
  - Firms maximize profits ξ_t = p_{h,t} Y_t − w_t L_t − r_{k,t} K_{t−1}.
  - Production technology: Y_t = z_t K_{t−1}^{α} L_t^{1−α}, where z_t is productivity shock.
- Investment goods production:
  - Investment good X_t produced from domestic and imported inputs:
    - X_t = [ (1−α_x)^{1/η_x} (X_{h,t})^{(η_x−1)/η_x} + α_x^{1/η_x} (X_{f,t})^{(η_x−1)/η_x} ]^{η_x/(η_x−1)}.

### III. MODEL — Market Clearing and External Demand
- Home goods market clearing:
  - Y_t = C_{h,t} + X_{h,t} + C_{h,t}^*.
- External demand for home goods (for country j):
  - C_{h,j,t}^* = (p_{h,j,t})^{−ε_{j,e}} Y_t^*, where Y_t^* is exogenous aggregate demand in the rest of the world and ε_{j,e} governs price elasticity of foreign demand.
- Measures used for data matching:
  - Real income measure: GDP_t / p_{c,t}, where GDP_t = p_{h,t} Y_t + p_{Co,t} Co.
  - Trade balance TB_t = p_{Co,t} Co + p_{h,t} C_{h,t}^* − C_{f,t} − X_{f,t}.

### III. MODEL — Driving Forces and Modeling Choices
- Exogenous stochastic drivers:
  - Commodity prices: dynamic factor structure with common latent factor plus idiosyncratic shocks (captures empirical commonality).
  - International financial premia and world interest rate: exogenous stochastic processes.
- Modeling assumptions and simplifications:
  - Commodity sector modeled as a constant endowment whose entire revenue accrues to households (no domestic resource use in commodity production; no government receipts modeled).
  - Abstraction from trade linkages across EMEs; comovement across EMEs driven solely by common commodity price factor (empirical application provides evidence of relatively low trade linkages among chosen EMEs).
- Note on full equilibrium conditions:
  - Full set of equilibrium and optimality conditions included in the Appendix.

*Source: _wp15280 - 7. Additional material is gathered in a companion online Appendix.*

### 1.    Common Factor Structure in Commodity Prices

### 1.    Common Factor Structure in Commodity Prices

### Model specification: common and idiosyncratic components
- Latent common commodity factor:
  - fCo_t = φCo fCo_t−1 + σfCo εfCo_t,  εfCo_t ∼ N(0,1)  (Equation (14))
- Country-specific commodity price relation:
  - pCo_j,t = ωCo_j fCo_t + εCo_j,t  (Equation (15))
  - ωCo_j is the loading factor for economy j.
- Idiosyncratic component dynamics:
  - εCo_j,t = ρCo εCo_j,t−1 + σCo_j νCo_j,t,  νCo_j,t ∼ N(0,1)  (Equation (16))
- Interpretation:
  - fCo_t captures global demand shocks that simultaneously affect many commodity goods.
  - εCo_j,t captures supply-side, technology, or endowment shocks specific to economy j (e.g., new discoveries, productivity changes).

### Other driving forces in the model
- Interest rate decomposition:
  - R_jt = R*_t S_jt  (Equation (17))
  - R*_t evolves as AR(1):
    - ln R*_t = (1−ρR*) ln R* + ρR* ln R*_t−1 + σR* εR*_t,  εR*_t ∼ N(0,1)  (Equation (18))
  - Country spread S_jt specification:
    - S_jt = zR_jt { exp [Ω_j,u (D_j,t − D_j)] − 1 }  (Equation (19))
  - Exogenous country risk spread zR_j,t:
    - zR_j,t = (1−ρzR_j) zR_j + ρzR_j zR_j,t−1 + σzR_j νR_j,t,  νR_j,t ∼ N(0,1)  (Equation (20))
- World foreign demand and country technology:
  - ln Y*_t = (1−ρY*) ln Y* + ρY* ln Y*_t−1 + σY* εY*_t,  εY*_t ∼ N(0,1)  (Equation (21))
  - ln z_j,t = (1−ρz_j) ln ̄z_j + ρz_j ln z_j,t−1 + σz_j εz_j,t,  εz_j,t ∼ N(0,1)  (Equation (22))

### Equilibrium definition
- Given initial conditions K_j,−1 and D_j,−1 and exogenous shock sequences {νCo_j,t, νR_j,t, εz_j,t, εfCo_t, εR*_t, εY*_t}, an equilibrium is:
  - State-contingent allocations {C_j,t, L_j,t, D_j,t, X_j,t, K_j,t, C^h_j,t, C^f_j,t, X^h_j,t, X^f_j,t, Y_j,t, C^h* _j,t}∞_t=0
  - Prices {p^c_j,t, p^x_j,t, R^*_j,t, w_j,t, r^k_j,t}∞_t=0
  - Such that:
    1. Allocations solve the consumer’s problem given prices and capital laws of motion.
    2. Allocations solve the firm’s problem given prices.
    3. Markets clear for capital, labor, investment goods, and home goods.
- Solution method: first-order Taylor approximation, log-linearized system solved using perturbation methods in DYNARE.

### Mechanism of a commodity price shock (qualitative)
- Market clearing for home goods (log-linearized):
  - ŷ_t = (C^h/Y) ĉ^h_t ︸︷︷︸ Effect1 + (X^h/Y) ẋ^h_t ︸︷︷︸ Effect2 + (C^h*/Y) ĉ^h*_t ︸︷︷︸ Effect3
- Three effects described:
  - Effect 1 — Demand channel:
    - Households increase demand for home goods in response to income shock.
    - Depends on intratemporal elasticity of substitution between h and f goods, η_c, and relative sizes of consumption and the commodity sector in steady state.
    - Raises relative price of home goods p_h and induces higher domestic production.
  - Effect 2 — Investment goods channel:
    - Increased domestic production raises rental price of capital and demand/price for new investment goods.
    - Further outward shift in demand for home goods.
    - Strength depends on η_x, relative size of investment, and revenue shock magnitude.
  - Effect 3 — External demand offset:
    - Foreign demand for home goods falls because home goods become relatively more expensive.
    - Strength depends on price elasticity of foreign demand, ε_j,e, and relative size of (non-commodity) exports.
- Net effect:
  - If Effects 1 and 2 dominate Effect 3, real economic activity increases.
  - Labor market accommodates higher labor demand.
  - Real exchange rate appreciates driven by increase in p_h.
  - Productivity shocks operate in the opposite direction (positive productivity → p_h decreases → real depreciation).

### Taking the model to the data: preliminaries
- Sample selection (A):
  - Selected four EMEs for estimation: Brazil, Chile, Colombia and Peru.
  - Reasons: tractability, heterogeneity in commodity production, data requirements for steady state calibration.
  - The four countries have a median commodity export share of35.4.
  - First principal component explains81and67percent of the variance in the commodity prices indices and real output across the four economies, respectively.
  - Commodity export shares by country:
    - Brazil (17.9); Chile (69.7); Colombia (28.6); and Peru (60.5).
  - Low trade linkages among the four:
    - Brazil’s exports to Chile, Colombia and Peru are, respectively,0.2,0.1, and0.1percent of output.
    - Brazil’s exports to its main trading partners are10.1percent of output.
- Calibrating the steady state (B):
  - Parameter choices and matches:
    - Risk aversion coefficient σ = 2.
    - Frisch elasticity 1/γ_c = 1.72.
    - Annual depreciation rate of capital δ = 10percent.
    - Price elasticity of exports ε_j,e = 1.18.
    - Elasticities of substitution η_c and η_x = 0.43.
    - Debt-elastic parameter Ω_u set to 0.001.
    - Steady state labor normalized to 0.3 via ψ_c.
    - Steady state (gross annual) real world interest rate R* = 1.01.
    - Scale parameters and steady states normalized: ̄z_j and Y* = 1.
    - Parameters D and ̄Co calibrated to match long run shares of external debt and commodity exports.
    - z_R calibrated to match long-run country risk premium; discount factor β set as inverse of country real interest rate.
  - Model fit:
    - Model matches the relative size of the commodity sector by construction.
    - The model reproduces historical interest rate premia and historic debt-to-output ratios for the four countries.
    - Some deviations remain in matching the relative size of consumption, imports or exports, but overall fit is satisfactory.
- Bayesian estimation (C):
  - Parameters estimated by Bayesian techniques (short-run dynamics):
    - Persistences: {φCo, ρCo_j, ρzR_j, ρR*, ρY*, ρz_j}
    - Standard deviations: {σfCo, σCo_j, σR_j, σzR, σY*, σz_j}
    - Loading factors: {ωCo_j}
    - Capital adjustment cost parameters: {a_j}
  - Observables used (quarterly) from each country:
    - Real private consumption, real income, real investment, trade balance to GDP ratio, the EM-BIG spread, the real effective exchange rate, and the commodity price index.

*Source: _wp15280 - 1.    Common Factor Structure in Commodity Prices*

### 2. Two additional observables are the 3-month real US TBills rate, and the United States’s real GDP,

### _wp15280 - 2. Two additional observables are the 3-month real US TBills rate, and the United States’s real GDP,

### Estimation setup
- Sample: balanced panel covering the period 2000.Q1 to 2014.Q3.
- Observables: 30 quarterly time series mapped to the model via measurement equations.
- Additional observables (as proxies): the 3-month real US TBills rate (world interest rate) and the United States’s real GDP (foreign aggregate demand).
- Measurement and transformations:
  - Data expressed in log-deviations from the Hodrick-Prescott trend and measured in percent.
  - Interest rates and EMBI measured in logs of gross rates.
  - Use of real income instead of real GDP (following Kehoe and Ruhl (2008)) for observables.
- Measurement errors added in: consumption, investment, the trade balance share, and the real effective exchange rate.
- Priors (Bayesian estimation, described as fairly agnostic):
  - AR(1) coefficients in six driving forces: Beta distribution with mean0.5 and standard deviation0.15; prior 90percent highest density region between0.25 and0.74.
  - Capital adjustment cost parameter a_j: Gamma prior with mean0.5 and standard deviation0.25.
  - Loading factor ω_Co^j prior: Normal centered at zero with standard deviation0.015; 90percent prior higher density region (−0.025,0.025).
- Identification: normalize loading factor associated with f_Co_t to1 in one economy (Colombia in empirical application).

### Posterior distributions and common factors
- Posterior densities differ considerably from priors, implying informative data.
- Commodity common factor:
  - AR(1) coefficient in common factor of commodity prices, φ_Co, posterior mean: 0.73.
  - Standard deviation of shocks to common factor, σ_fCo: 6.31percent.
  - Common factor shocks are at least an order of magnitude larger than other global shocks.
- Commodity idiosyncratic persistence and volatility:
  - AR parameters for idiosyncratic components ρ_Co^j reported as high.
- Productivity (country-specific) processes (posterior means; reported ranges):
  - ρ_z range between0.39/0.40/0.51in Brazil/Chile/Colombia to0.68for Colombia.
  - σ_z ranges between0.84/0.89percent in Peru/Colombia to1.32/3.39for Brazil/Chile.
- Loading factors (ω_Co^j) — degree of transmission from common factor to country-specific commodity price index (Colombia normalized to one):
  - Chile: posterior mode equal to1.53.
  - Peru: posterior mode equal to0.89.
  - Brazil: posterior mode equal to0.53.
- Time series of the common factor f_Co (Kalman smoother at posterior mean):
  - Tightly estimated with large, short-lived deviations from trend, particularly post 2005.
  - Deviations reaching near 30 percentages points.
  - Pattern: increase pre-financial crisis, sharp fall during crisis, vigorous recovery within next two years, fall again in last couple of years of sample.

### Business cycle drivers (FEVD and historical decompositions)
- Tools: forecast error variance decomposition (FEVD) of real income across four countries at various horizons; statistics reported using posterior means.
- Unconditional FEVD (median shares and ranges):
  - Commodity price shocks median share: 42percent.
  - Country range: 27.5percent in Brazil up to 77.1percent in Chile; Colombia 43.5percent; Peru 40.4percent.
  - Considerable portion of these shares related to the common factor in commodity prices.
- Other external shocks (foreign demand, world interest rate, spreads) generally account for small shares of unconditional FEVD, except interest rate shocks in Brazil (linked to Brazil’s relatively large stock of external debt).
- Horizon dependence:
  - At one-quarter forecast horizon the share of commodity price shocks decreases considerably, between1/10and1/2 of the unconditional share.
  - At short horizons domestic productivity shocks account for the highest share.
- Historical decompositions (grouping shocks into four groups: Productivity, Commodity, Spread, Foreign):
  - Commodity and Productivity shocks preponderant across four countries.
  - Commodity shocks more marked in second half of sample; Chile exception where influence is large throughout sample.
  - Commodity shocks contributed strongly in two-year boom preceding the global financial crisis, turned negative in 2009 recession, then positive in the recovery.
- Counterfactual volatility (Table 6): standard deviation (S.D.) of real income when commodity price shocks are turned off reported as percentage of observed S.D.
  - Peru: counterfactual S.D when commodity price shocks turned off = 69.6percent of observed.
  - Chile: 72.1percent.
  - Colombia: 75.4percent.
  - Brazil: 91.1percent.
  - When all shocks are turned off except commodity price shocks, similar qualitative results obtain; Chile notable where counterfactual volatility increases above observed (other shocks helped smooth commodity volatility).

### Quantifying the mechanism (IRFs and magnitudes)
- Impulse response to unexpected 1 S.D. common commodity shock (common factor increases by6.3percentage points above long run mean):
  - Numbers reported as percentage deviations from steady state values.
  - Home goods production (Y_t): hump-shaped, peaking between one tenth and a little over a half of a percentage point.
  - Real income (GDP_t / p_c_t): increases on impact, ranging from3to0.5percentage points across four countries, then steadily decreases.
  - Consumption:
    - Total private consumption increases, mainly driven by consumption of home goods.
    - Home goods consumption increases between half and one full percentage point relative to steady state.
    - Strong persistence in consumption response (inherited from commodity shock persistence).
  - Investment:
    - Pronounced, hump-shaped; peaks approximately 4 to 5 quarters after shock.
    - Deviates between0.5and3.5percentage points from long run value at peak.
    - Real rental return increases on impact between10and100basis points.
  - Non-commodity exports fall between one half and three percentage points.
  - Real exchange rate appreciates between half and two percentage points.
  - Net effect: expansion driven by consumption and investment dominates contraction from non-commodity exports; home good production expands; labor demand and real wages rise.
- Cross-country comovement: common factor shock generates sufficient comovement across countries.

### Model performance (fit to moments and correlations)
- Cross correlations of real income:
  - Average cross correlation in data: 0.45.
  - Average predicted by benchmark model (with common factor "CF"): 0.32.
  - Alternative model with no common factor ("NCF", ω_Co^j = 0 ∀ j): average cross correlation falls to 0.02.
  - Large fall in alternative model’s marginal likelihood relative to benchmark (reported in Table 7).
- Serial correlations within countries: model reproduces procyclicality of commodity prices with income, consumption, investment; captures trade balance to GDP and countercyclicality of real exchange rates.
- Second moments (Table 8):
  - Commodity price volatility close to four times that of income.
  - Investment and real exchange rate volatility nearly twice that of income.
  - Consumption and trade balance share volatility less than that of income.
  - Model replicates moderately high serial autocorrelations and comovement with real income.

### Robustness and extensions — summary of major findings
- A. Comparison with a SVAR:
  - Strategy: estimate SVAR replacing terms of trade with country-specific commodity price indices; compare role of commodity shocks in SVAR vs DSGE.
  - SVAR results broadly aligned with DSGE: commodity price shocks account for a large part of output fluctuations.
  - SVAR unconditional variance share of shocks to p_Co_t (mean/median): 49/50percent across four countries.
  - DSGE unconditional variance share (mean/median): 47/42percent.
  - Country differences noted:
    - SVAR predicted share in Brazil: 15; Chile: 53.
    - DSGE counterparts: Brazil 28; Chile 77.
    - Colombia: SVAR 47, DSGE 44 (pretty much aligned).
    - Peru: SVAR predicted share 80percent, described as "twice that of the DSGE".
  - Predicted time series from both models using only p_Co_t shocks are highly correlated (median correlation ≥ 0.9 for all countries but Chile).
  - Mean standard deviation of SVAR simulated series relative to data: 67percent; DSGE: 62percent.
  - Overall: SVAR consistent with benchmark DSGE results that commodity prices are important drivers.
- B. Countercyclical spreads (extension allowing spreads to respond to fundamentals and working-capital effect):
  - Modified risk premia process allows dependence on expected TFP and commodity prices (parameters η_z^j, η_Co^j).
  - Working-capital constraint introduces parameter ψ_j affecting labor demand via spreads.
  - Estimation includes three new parameters per country.
  - Results: in Brazil, Colombia, and Peru the share of variance associated with commodity prices rises relative to benchmark; Chile’s share slightly reduced but remains largest.
  - Provides evidence linking commodity prices to spreads and spreads as a propagation mechanism.
- C. Correlated external driving forces:
  - Allow f_Co_t (commodity common factor) to be contemporaneously correlated with external demand Y*_t via parameter σ_Co,Y*.
  - Posterior mean of σ_Co,Y* estimated to be 0.021 (positive and statistically significant).
  - Share associated with external demand shocks increases to average 5.5percent (from about 0.1percent in benchmark).
  - Modified model shows improved fit (marginal likelihood increases relative to benchmark).
- D. Absence of commodity shocks (counterfactual estimation with commodity shocks turned off):
  - In three of four countries TFP shocks explain overwhelming share of output variance; Brazil exception where interest rate shocks take larger role.
  - Large fall in model’s marginal likelihood from 4530.8 to 4410.6 when commodity shocks are absent.
  - Reduced model fails to reproduce negative comovement between real exchange rate and real income observed in data (predicts counterfactual positive comovement).
- E. Alternative dynamic factor models (estimating dynamic factor model outside DSGE with longer series back to 1980 and up to 13 EMEs):
  - Estimated common factors across variants track benchmark common factor closely.
  - Post-2005 volatility remains pronounced in longer samples.
- F. Alternative filtering (use of simple growth rates instead of HP filter):
  - Stylized facts robust to detrending method.
  - FEVD results continue to point to preponderant role of commodity price shocks.
  - Median share of output explained by commodity shocks using growth rates: 68.1percent.
  - Chile’s share drops relative to benchmark but remains above half; other three countries’ shares increase.
- G. Cyclical production of commodities — empirical discussion:
  - Serial correlations between volumetric commodity production and international prices for key commodities show no systematic co-movement at business-cycle frequencies.
  - Granger-causality tests reveal no causality in either direction (appendix).
  - Evidence supports assumption that these countries are price takers and that commodity production is not cyclically responsive at business-cycle frequencies.
  - Non-commodity (or nontraded) GDP measures show considerable comovement with country commodity price indices, consistent with the model’s propagation mechanism.
- H. Role of fiscal policy — discussion and evidence:
  - Government sector (recipient of commodity revenue or taxes) absent from model; could matter for propagation if government expenditures respond to commodity revenue.
  - Cross-country serial correlations comparing activity measures with and without government expenditure indicate:
    - Real government expenditure is not positively correlated with commodity prices.
    - Removing public expenditure marginally increases comovement between real activity and commodity prices.
  - Suggestive evidence that fiscal transmission via public spending is not first-order for these episodes; transfers to households would be isomorphic in the model to direct household receipts.

### Concluding remarks and implications
- Main conclusions:
  - External forces, especially exogenous fluctuations in international commodity prices viewed as large income/revenue shocks, are major drivers of aggregate fluctuations in the studied EMEs.
  - Movements in commodity prices display a common factor with large, correlated swings (the "roller coaster ride") that materially influence domestic business cycles.
  - Commodity price shocks are often amplified by contemporaneous movements in interest rates/spreads, and they can both amplify and, at times, cushion the business cycle (e.g., recovery after the world financial crisis).
- Policy-relevant implications and open questions highlighted:
  - Understanding fiscal and monetary policy responses to commodity price shocks is crucial but requires further analysis beyond the paper’s scope.
  - Extensions to model government behavior, optimal policy design, and richer commodity-sector features (storage, inventories, production responses, ownership) are important avenues for future work.

*Italic line: Source — _wp15280 (PDF chapter/section) — estimation, results, robustness, and conclusions as contained in the provided content.*

### REFERENCES

### _wp15280 - REFERENCES

### Key datasets, models, and methodological notes
- Commodity prices: IMF; real RDP from Haver. Cyclical components computed with a Hodrick-Prescott filter with smoothing parameter of 1600.
- Estimation: DSGE benchmark uses quarterly unbalanced panel between 1990.Q1 and 2014.Q4 for serial correlations and 2000.Q1 to 2014.Q3 for model estimation. Dynare 4.4.2 and Metropolis-Hastings with 200,000 draws (first 100,000 discarded) used for posterior computations; proposal mean and covariance from maximum posterior and negative inverse Hessian via Nelder-Mead.
- FEVD, impulse responses, historical decompositions, and counterfactual experiments are computed at the posterior mean; shocks include idiosyncratic commodity price shock (νCo), common factor commodity shock (εfCo), spread shock (νR), domestic productivity shock (εz), world riskless interest rate shock (εR?), and world demand shock (εY?).

### Calibrated parameters (selected exact values)
- Common calibrated parameters (Brazil, Chile, Colombia, Peru shown identically where applicable):
  - Gross steady state external annual interest rate R? = 1.01 1.01 1.01 1.01
  - Interest rate elasticity of debt to GDP ratio Ωu = 0.001 0.001 0.001 0.001
  - Inverse of labor supply elasticity γc = 0.58 0.58 0.58 0.58
  - Relative risk aversion coefficient σ = 2.00 2.00 2.00 2.00
  - Elasticity of substitution - consumption bundle ηc = 0.43 0.43 0.43 0.43
  - Elasticity of substitution - investment bundle ηx = 0.43 0.43 0.43 0.43
  - Capital depreciation annual rate (%) δ = 10.00 10.00 10.00 10.00
  - Price elasticity of exports e = 1.18 1.18 1.18 1.18
  - Steady state aggregate demand of ROW Y* = 1.00 1.00 1.00 1.00
  - Steady state productivity level ̄z = 1.00 1.00 1.00 1.00
- Calibrated to match long run relations (selected):
  - Steady state level of external debt ̄D = 26.80 (Brazil), 3.64 (Chile), 9.41 (Colombia), 11.25 (Peru)
  - Governing import share in consumption αc = 0.36 (Brazil), 0.48 (Chile), 0.23 (Colombia), 0.09 (Peru)
  - Governing import share in investment αx = 0.13 (Brazil), 0.58 (Chile), 0.60 (Colombia), 0.82 (Peru)
  - Capital share in production α = 0.32 (Brazil), 0.39 (Chile), 0.34 (Colombia), 0.37 (Peru)
  - Steady state level of commodity price pCo = 0.85 (Brazil), 3.16 (Chile), 0.37 (Colombia), 1.36 (Peru)
  - Annual steady state spread 100(1−zR) = 5.39 (Brazil), 1.45 (Chile), 3.93 (Colombia), 3.56 (Peru)

### Long-run model vs data ratios (exact values)
- Consumption / GDP (Model / Data):
  - Brazil: 78.08 / 82.11
  - Chile: 76.66 / 72.67
  - Colombia: 77.06 / 82.00
  - Peru: 76.64 / 77.67
- Investment / GDP (Model = Data reported for several):
  - Brazil: 17.74 / 17.74
  - Chile: 23.16 / 23.16
  - Colombia: 21.00 / 21.00
  - Peru: 22.22 / 22.22
- Commodities Exports / GDP (Model = Data):
  - Brazil: 8.49 / 8.49
  - Chile: 26.19 / 26.19
  - Colombia: 6.40 / 6.40
  - Peru: 12.27 / 12.27
- External Real Interest Rate (%, annual): Brazil 6.44; Chile 2.46; Colombia 4.94; Peru 4.60
- External Debt / GDP: Brazil 66.39; Chile 7.54; Colombia 40.00; Peru 25.31

### Priors and posteriors (selected posterior means and 90% HPD intervals preserved)
- Global posteriors (mean and 90% HPD interval):
  - ρY? mean = 0.8048; 90% HPD interval = 0.7322–0.8833
  - ρR? mean = 0.6229; 90% HPD interval = 0.5471–0.7042
  - φCo mean = 0.7310; 90% HPD interval = 0.6558–0.8046
- Country-specific examples (posterior mean and 90% HPD interval):
  - Brazil ρCo mean = 0.7022; 90% HPD = 0.5923–0.8139
  - Chile ρzR mean = 0.6996; 90% HPD = 0.5213–0.8422
  - Colombia ρCo mean = 0.6741; 90% HPD = 0.5624–0.7846
  - Peru ρCo mean = 0.6945; 90% HPD = 0.5774–0.8051

### Forecast Error Variance Decomposition of Real Income (Table 5) — selected exact numbers (percent)
- Unconditional (panel a) — share of variance attributable to commodity price shocks:
  - νCo (idiosyncratic): Brazil 16.3; Chile 0.5; Colombia 17.6; Peru 4.9
  - εfCo (common factor): Brazil 11.1; Chile 76.6; Colombia 25.8; Peru 35.5
  - All other shocks combined (sum reported row): Brazil 72.5; Chile 22.9; Colombia 56.5; Peru 59.6
- Conditional (one quarter ahead, panel b):
  - νCo: Brazil 7.2; Chile 0.7; Colombia 10.0; Peru 3.3
  - εfCo: Brazil 4.3; Chile 5.2; Colombia 11.1; Peru 19.8
  - All other shocks combined: Brazil 88.5; Chile 44.1; Colombia 78.9; Peru 76.9
- Conditional (twelve quarters ahead, panel d):
  - νCo: Brazil 13.4; Chile 0.5; Colombia 13.5; Peru 3.4
  - εfCo: Brazil 9.0; Chile 74.3; Colombia 18.7; Peru 22.9
  - All other shocks combined: Brazil 77.7; Chile 25.2; Colombia 67.8; Peru 73.7

### Historical decomposition and counterfactual experiments (selected exact values)
- Counterfactual historical decomposition (Table 6):
  - S.D. of observed real income cycle (full sample 2000.Q1–2014.Q3): Brazil 0.0218; Chile 0.0418; Colombia 0.0184; Peru 0.0273
  - Counterfactual S.D. excluding νR (idiosyncratic spread) (percent of observed S.D): Brazil 92.7; Chile 98.8; Colombia 97.8; Peru 99.9
  - Excluding νCo (idiosyncratic commodity price): Brazil 103.1; Chile 99.7; Colombia 98.2; Peru 94.0
  - Excluding εz (domestic productivity): Brazil 53.0; Chile 114.9; Colombia 55.1; Peru 43.9
  - Excluding εfCo (common factor): Brazil 89.1; Chile 72.2; Colombia 80.3; Peru 75.4
  - Excluding both commodity shocks {νCo, εfCo}: Brazil 91.1; Chile 72.1; Colombia 75.4; Peru 69.6
  - Excluding all shocks except commodity price shocks {νR, εz, εR?, εY?}: Brazil 41.5; Chile 114.7; Colombia 47.9; Peru 44.1

### Impulse responses and model dynamics (qualitative and units)
- Impulse responses to a common factor shock in commodity prices (Figure 7):
  - Units: percentage deviations from steady state levels; horizontal axis unit is a quarter.
  - Shocks generate responses in Output (Y), Real Income (GDP/pc), Consumption (C), Domestic consumption (Ch), Investment (X), Real Rent of Capital (rk), Exports (Ch*), Home Price (ph), Real Exchange Rate (RER), Real Wage (w) as depicted in subplots.

### Model fit, cross-country comovements, and model comparison statistics
- Cross-correlations of Real Income across the four estimated countries (Table 7):
  - Data upper-diagonal and model lower-diagonal presented; marginal likelihood:
    - Model with no common factor (NCF) marginal likelihood = 4445.3
    - Model with common factor (CF) marginal likelihood = 4530.8
- Other second moments (Table 8) — selected exact values:
  - Commodity price index standard deviation (Model) PCo S.D. = 11.08; Data PCo S.D. = 12.14
  - GDP/CPI S.D. (Model) = 3.80; Data = 2.86
  - Investment S.D. (Model) = 7.40; Data = 5.14
- Comparison with SVAR (Figure 9 and Table 9):
  - Unconditional variance decomposition of output: share of Pcom shocks (median reported):
    - SVAR median = 50; DSGE median = 42
  - SD(Output-counterfactual)/SD(Output-data): Only Pcom shocks:
    - SVAR median = 69; DSGE median = 46
  - corr(Output-counterfactual, Output-data): Only Pcom shocks:
    - SVAR median = 0.63; DSGE median = 0.73
  - Correlation between SVAR and DSGE counterfactuals (only Pcom shocks) median = 0.89

### Robustness and alternative model specifications (Table 10 — Forecast Error Variance Decomposition of Output)
- Panel (a) Countercyclical spreads (marginal likelihood = 4,502.40):
  - νCo: Brazil 24.4; Chile 0.6; Colombia 18.0; Peru 5.6
  - εfCo: Brazil 17.1; Chile 74.0; Colombia 27.7; Peru 37.6
- Panel (b) Correlation between external forces (marginal likelihood = 4,535.70):
  - νCo: Brazil 16.5; Chile 4.5; Colombia 17.8; Peru 4.8
  - εfCo: Brazil 7.9; Chile 64.3; Colombia 18.5; Peru 27.0
- Panel (c) Basic RBC model (no commodities) (marginal likelihood = 4,410.60):
  - All variance explained by non-commodity shocks (100.0 across countries)
- Panel (d) Benchmark model first differences:
  - εfCo: Brazil 44.7; Chile 56.6; Colombia 71.2; Peru 76.4
  - Marginal likelihoods and shares reported in table for full comparison.

### Factor extraction and extended factor models
- Estimated common factor (Figure 5): latent variable obtained from Kalman filter smoothing evaluated at posterior mean; HPDinf and HPDsup stand for the 10th and 90th percentiles of the posterior distribution.
- Extended Dynamic Factor Models (Figure 10):
  - Common factors from DFM 13 (1980), DFM 4 (1980), DFM 4 (2000), and DFM+DSGE compared; DFM+DSGE reproduces the common factor from DSGE benchmark.

### Production and price correlations (Figure 11)
- Serial correlations between volumetric production of country-specific commodities and international market prices reported for multiple country-commodity pairs with exact lag structures (e.g., Brazil - Tons of Iron, Chile - Tons of Copper, Colombia - Barrels per Day of Oil, Peru - Fine Grams of Gold) and LAC-4 and EME-11 averages for non-commodity GDP correlations.

### Role of government expenditure in comovement (Figure 12)
- Panels report average serial correlations between real GDP (RGDP) at time t and country-specific commodity prices Pco at t+j for j = −4,...,4, for LAC4 and 12 EMEs; comparisons shown with RGDP minus government expenditure (RGDP−G) and with real government expenditure (RG).

*Source: _wp15280 - REFERENCES (PDF content unit).*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2015/_wp15280.pdf_