## Appendix III: Group-mean FMOLS Estimator

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---

### Motivation and conceptual summary
- Conventional panel regressions typically difference consumption and GDP to achieve stationarity and thus measure risk sharing on transitory/business-cycle frequency; welfare gains at business-cycle frequency have been found small in the literature.
- If consumption and output are I(1), differencing removes permanent components that drive long-run risk-sharing welfare gains; the long-run (permanent) component can yield much higher welfare gains than transitory shocks.
- Nonstationary panel (cointegration) approach isolates long-run steady-state relationship between idiosyncratic consumption per capita and idiosyncratic GDP per capita using the levels model:
  - cit − c∗t = αi + βi (yit − y∗t) + uit, where βi is interpreted as the long-run (steady-state) cointegrating slope and used as the measure of long-run risk sharing.
- Group-mean Fully Modified OLS (FMOLS) and group-mean Dynamic OLS (DOLS) allow:
  - Heterogeneous short-run dynamics Ψi(L).
  - Heterogeneous cointegrating slopes βi (group-mean pooling rather than pooled-common-β).
  - Nonparametric correction for serial correlation and endogeneity using long-run covariance adjustments and internal instruments (differences of y as instruments).
- Advantages emphasized:
  - Isolates long-run risk sharing from short-run nuisance dynamics (intertemporal smoothing, tastes shocks).
  - Permits study of cross-country distribution of βi (heterogeneity).
  - Cross-sectional dimension helps overcome low power of pure time-series FMOLS in finite samples.

### Econometric recipe for group-mean FMOLS (operational steps)
- Notation: y1it = cit − c∗t, y2it = yit − y∗t; model y1it = αi + βi y2it + εit with y2it ∼ I(1), εit ∼ I(0).
- Step 1: Estimate country-specific OLS cointegration regression and collect residuals ε̂it.
- Step 2: For each country compute long-run covariance matrix Ω̂i of ξit = (ε̂it, Δy2it)′ (e.g., Newey-West with bandwidth Ki such as Ki = 4(Ti/100)2/9).
- Step 3: Compute adjustment terms γ̂i = Γ̂21i + Σ̂21i − (Ω̂21i / Ω̂22i)(Γ̂22i + Σ̂22i) and construct y∗1it = (y1it − y1i) − (Ω̂21i / Ω̂22i) Δy2it to correct endogeneity.
- Step 4: Compute country FMOLS: β̂∗FMi = [∑T (y2it − y2i)2]−1 [∑T (y2it − y2i) y∗1it − T γ̂i] and associated t-statistic using Ω̂11i.
- Step 5: Group-mean FMOLS estimator: β̂∗GFM = N−1 ∑N β̂∗FMi; group t-stat: tβ̂∗GFM = N−1/2 ∑N tβ̂∗FMi (≈ N1/2 tβ̂∗FM).
- Step 6: Compare t-statistic to N(0,1) critical values under H0: βi = β0 (for all i).

### Econometric assumptions and properties
- Key assumptions for FMOLS group-mean asymptotics:
  - Assumption 1.1: Multivariate functional central limit theorem for each panel member: partial sums 1/√T ∑t≤[Tr] ξit → Bi(r, Ωi) (Brownian motion with covariance Ωi).
  - Assumption 1.2: Cross-sectional independence across i (E(ξit, ξjt) = 0 for i ≠ j) to obtain straightforward group-mean asymptotic normality.
- Properties:
  - FMOLS corrects second-order biases induced by serial correlation and endogeneity in the cointegrating regression; after correction the estimator is asymptotically normal and free of nuisance parameters.
  - Group-mean FMOLS permits heterogeneous βi drawn from a distribution and is recommended over pooled FMOLS when heterogeneity is plausible.
  - FMOLS preserves superconsistency (fast convergence) of time-series FMOLS; panel group-mean FMOLS converges at rate T √N under the double-index asymptotics scenario discussed.

### Data and sample selection (used in the analysis)
- Data sources: Penn World Table (PWT) version 6.2 (GDP and consumption up to 2004) extended to 2008 using World Economic Outlook (WEO) national accounts data.
- Chosen sample: 45 OECD and emerging market countries with data span from 1950 to 2008 (includes all 26 OECD countries and 22 emerging markets defined by FTSE/Economist except East European transition economies and Russia). These 45 countries account for more than 80 percent of world GDP as of 2008.
- Demeaned (idiosyncratic) variables: country-level per capita GDP and consumption minus world-wide averages (c∗t, y∗t).
- Sample refinement: initial unit-root and cointegration tests on individual countries; countries failing unit-root or individual cointegration tests were removed, yielding a 21-country subsample used for robustness checks.

### Diagnostic test results (panel unit root and cointegration)
- Panel unit-root test results (45 countries) reported in Table 1:
  - IPS ADF (large sample adjustment values): 3.21*** (GDP) and 1.09*** (Consumption).
  - IPS ADF (Bootstrapped): 0.84*** (GDP) and -0.01*** (Consumption).
  - MW (Bootstrapped): 84.73*** (GDP) and 89.42*** (Consumption).
- Cointegration test results (Table 1):
  - Group mean panel ADF/P P Rho: -2.71***, -4.24***, -3.74***.
  - Pooled panel: -1.16, -2.67***, -2.06***.
- Note: Lag truncation: K = 4.

### Conventional panel regression findings (stationary/differenced and levels)
- Conventional (differenced) panel estimates (Table 2) — business-cycle frequency risk sharing:
  - Pooled OLS GDP growth coefficient ≈ 0.680 − 0.809 across samples and specifications (1960-2008 pooled estimate ~0.680), implying roughly 32 percent of business-cycle frequency risk shared (text statement: "an estimate of about 32 percent of businesscycle frequency risk has been shared").
  - Results stable across pooled OLS and fixed effects (FE), and across pre-1990 and post-1990 splits.
- Conventional (levels) panel estimates (Table 3) — naive long-run panel results:
  - Pooled OLS GDP coefficient ≈ 0.912 (1960-2008), but pooled-OLS level regression has second-order bias concerns in finite samples.
  - Text narrative: pooled-OLS level regression indicates less than 9 percent of long-run risks shared for 1960-2008, while FE indicates around 18 percent.

### Nonstationary panel (group-mean FMOLS/DOLS) empirical findings (key results)
- Group-mean FMOLS estimates on 45-country panel (1950-2008):
  - Whole 1950-2008 panel: point estimate ≈ 14 percent of long-run risk shared.
  - Test statistic for null of full risk sharing: t = 112.92.
  - Period splits:
    - Pre-1990: long-run risk sharing ≈ 12 percent.
    - Post-1990 (financial integration period): long-run risk sharing ≈ 27 percent.
- Subgroup patterns (45-country panel):
  - OECD vs Emerging (whole sample): long-run risk sharing similar across OECD and emerging markets for whole sample.
  - Post-1990:
    - OECD long-run risk sharing ≈ 34 percent.
    - Emerging market long-run risk sharing ≈ 23 percent.
  - Within-group distribution: OECD countries appear to evenly enjoy benefits of increased risk sharing; emerging markets exhibit more heterogeneity (benefits concentrated in advanced emerging markets).
- Regional/intra-region observations:
  - EU long-run risk sharing ≈ 10 percent for whole sample, ≈ 6 percent pre-1990.
  - Example: risk sharing within EU15 ≈ 24 percent after 1990, while with rest of world ≈ 36 percent (reported illustration).
- Robustness — 21-country subsample:
  - Whole sample: long-run risk sharing ≈ 14 percent.
  - Post-1990: long-run risk sharing increases to ≈ 39 percent on the 21-country panel (increase driven entirely by OECD countries in this subsample).

### Cross-country risk-sharing patterns and links to financial integration
- Empirical patterns described:
  - Pre-1990: positive correlation between long-run risk sharing and gross assets+liabilities to GDP.
  - Post-1990: weaker positive relation between long-run risk sharing and gross capital flows; capital flows (gross assets+liabilities) on average quadrupled post-1990 (tripled excluding Ireland as outlier), while long-run risk sharing only doubled — implying risk sharing did not scale proportionally with financial flows.
  - Decomposition: post-1990 increases in gross flows primarily driven by FDI and portfolio flows (FDI+portfolio as percent of GDP quadrupled post-1990), whereas debt-to-GDP roughly doubled (excluding Ireland).
  - Relationship strength: the positive association between long-run risk sharing and FDI/portfolio-to-GDP and debt-to-GDP is weaker in the post-1990 period than pre-1990.
- Interpretation from text: financial integration is necessary but not sufficient for proportionate increases in long-run risk sharing; potential threshold effects and other frictions (sovereign risk, moral hazard, non-insurability of nontradables, measurement issues) can limit translation of capital flows into state-contingent insurance.

### Interpretation, limitations, and recommended future directions
- Interpretation:
  - Group-mean FMOLS estimates provide a "de facto" measure of long-run risk sharing (βi), capturing both endogenous market incompleteness (sovereign risk, moral hazard) and exogenous incompleteness (nontradable goods, labor income non-insurability).
  - Long-run risk sharing across the 45-country set is limited: ≈ 14 percent overall for 1950-2008 but increased notably during the recent financial integration episode, disproportionately in OECD countries.
- Limitations and caveats:
  - Finite-sample size distortions can affect individual country FMOLS estimates; group-mean pooling mitigates but does not eliminate these concerns.
  - Cross-sectional dependence must be considered; the analysis uses demeaning (c∗t, y∗t) to remove common global components, an approach that works reasonably well when common stochastic trends or global cycles dominate cross-sectional dependence.
  - The interpretation as insurance over tradable goods requires additivity of period utility; without additivity estimates are proxies at best.
  - Measurement issues: GNI/BOP factor income and capital gains not fully captured in some measures can bias alternative approaches (GNI-based) to separating state-contingent insurance and intertemporal smoothing.
- Suggested future research directions:
  - Use nonstationary vector autoregressive models to analyze short-run (business-cycle frequency) risk sharing and impulse responses to cyclical disturbances.
  - Decompose GDP and consumption processes into trends and cycles using long-run risk sharing estimates to study interaction between long-run and short-run risk sharing.

*Italic source: Appendix III: Group-mean FMOLS Estimator (excerpt), _wp1064 - Appendix III: Group-mean FMOLS Estimator, source PDF content provided.*

### 1. Introduction.............................................................................................3

### 1. Introduction.............................................................................................3

### Document structure and major sections
- 1. Introduction.............................................................................................3
- 2. Theoretical Motivations................................................................................5
- 3. An Illustration on Conventional Panel and Nonstationary Panel Approaches..................9
  - 3.1 Conventional Panel..........................................................................9
  - 3.2 Nonstationary panel........................................................................11
- 4. Data and Sample Selection...........................................................................14
  - 4.1. Dataset........................................................................................14
  - 4.2. Sample Selection............................................................................15
- 5. Interpreting the Risk Sharing Relationship........................................................16
  - 5.1. FMOLS and DOLS........................................................................16
  - 5.2. Conventional Panel Regression Results................................................18
  - 5.3. Nonstationary Panel Regression Results...............................................19
- 6. Cross-country risk sharing patterns................................................................20
- 7. Conclusion.............................................................................................21

### Tables (listed)
- Table 1. Panel Unit Root and Cointegration Test Results (45 countries)...................23
- Table 2. Conventional Panel Regression Results under Different Specifications......24
- Table 3. Level Panel Regression Results under Different Specifications...............25
- Table 4A. Country Group Cointegration Coefficient Estimates..........................26
- Table 4B. Country Group Cointegration Coefficient Estimates (Countries passed 
  individual tests).................................................................................26

### Figures (listed)
- Figure: Cross-country Risk Sharing and Financial Assets................................27

### Appendix material (listed)
- Appendix Tables
  - Table A1. Individual and Panel Unit Root Test Results 1950-2008 (45 countries)...28
  - Table A2. Individual and Panel Cointegration Test Results 1950-2008................29
  - Table A3a. Cointegration coefficient estimates (45 countries)...........................30
  - Table A3b. Cointegration coefficient estimates (21 countries)............................31
- Appendix I: Studies using Conventional Panel Analysis 35
- Appendix II: Technically Illustration on Conventional and Nonstationary Panel............. 31

*Source: _wp1064 - 1. Introduction (PDF), https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2010/_wp1064.pdf*

### Appendix III: Group-mean FMOLS Estimator.......................................................35

### Appendix III: Group-mean FMOLS Estimator

### Motivation and conceptual summary
- Conventional panel regressions typically difference consumption and GDP to achieve stationarity and thus measure risk sharing on transitory/business-cycle frequency; welfare gains at business-cycle frequency have been found small in the literature (references in source).
- If consumption and output are I(1), differencing removes permanent components that drive long-run risk-sharing welfare gains; the long-run (permanent) component can yield much higher welfare gains than transitory shocks.
- Nonstationary panel (cointegration) approach isolates long-run steady-state relationship between idiosyncratic consumption per capita and idiosyncratic GDP per capita:
  - Model (levels): cit − c∗t = αi + βi (yit − y∗t) + uit, where βi is interpreted as the long-run (steady-state) cointegrating slope and used as the measure of long-run risk sharing.
- Group-mean Fully Modified OLS (FMOLS) and group-mean Dynamic OLS (DOLS) allow:
  - Heterogeneous short-run dynamics Ψi(L).
  - Heterogeneous cointegrating slopes βi (group-mean pooling rather than pooled-common-β).
  - Nonparametric correction for serial correlation and endogeneity using long-run covariance adjustments and internal instruments (differences of y as instruments).
- Advantages emphasized:
  - Isolates long-run risk sharing from short-run nuisance dynamics (intertemporal smoothing, tastes shocks).
  - Permits study of cross-country distribution of βi (heterogeneity).
  - Cross-sectional dimension helps overcome low power of pure time-series FMOLS in finite samples.

### Econometric recipe for group-mean FMOLS (operational steps)
- Notation used: y1it = cit − c∗t, y2it = yit − y∗t; model y1it = αi + βi y2it + εit with y2it ∼ I(1), εit ∼ I(0).
- Step 1: Estimate country-specific OLS cointegration regression and collect residuals ε̂it.
- Step 2: For each country compute long-run covariance matrix Ω̂i of ξit = (ε̂it, Δy2it)′ (e.g., Newey-West with bandwidth Ki such as Ki = 4(Ti/100)2/9).
- Step 3: Compute adjustment terms γ̂i = Γ̂21i + Σ̂21i − (Ω̂21i / Ω̂22i)(Γ̂22i + Σ̂22i) and construct y∗1it = (y1it − y1i) − (Ω̂21i / Ω̂22i) Δy2it to correct endogeneity.
- Step 4: Compute country FMOLS: β̂∗FMi = [∑T (y2it − y2i)2]−1 [∑T (y2it − y2i) y∗1it − T γ̂i] and associated t-statistic using Ω̂11i.
- Step 5: Group-mean FMOLS estimator: β̂∗GFM = N−1 ∑N β̂∗FMi; group t-stat: tβ̂∗GFM = N−1/2 ∑N tβ̂∗FMi (≈ N1/2 tβ̂∗FM).
- Step 6: Compare t-statistic to N(0,1) critical values under H0: βi = β0 (for all i).

### Econometric assumptions and properties
- Key assumptions for FMOLS group-mean asymptotics:
  - Assumption 1.1: Multivariate functional central limit theorem for each panel member: partial sums 1/√T ∑t≤[Tr] ξit → Bi(r, Ωi) (Brownian motion with covariance Ωi).
  - Assumption 1.2: Cross-sectional independence across i (E(ξit, ξjt) = 0 for i ≠ j) to obtain straightforward group-mean asymptotic normality.
- Properties:
  - FMOLS corrects second-order biases induced by serial correlation and endogeneity in the cointegrating regression; after correction the estimator is asymptotically normal and free of nuisance parameters.
  - Group-mean FMOLS permits heterogeneous βi drawn from a distribution and is recommended over pooled FMOLS when heterogeneity is plausible.
  - FMOLS preserves superconsistency (fast convergence) of time-series FMOLS; panel group-mean FMOLS converges at rate T √N under the double-index asymptotics scenario discussed.

### Data and sample selection (as used in the analysis)
- Data sources: Penn World Table (PWT) version 6.2 (GDP and consumption up to 2004) extended to 2008 using World Economic Outlook (WEO) national accounts data.
- Chosen sample: 45 OECD and emerging market countries with data span from 1950 to 2008 (includes all 26 OECD countries and 22 emerging markets defined by FTSE/Economist except East European transition economies and Russia). These 45 countries account for more than 80 percent of world GDP as of 2008.
- Demeaned (idiosyncratic) variables: country-level per capita GDP and consumption minus world-wide averages (c∗t, y∗t).
- Sample refinement: initial unit-root and cointegration tests on individual countries; countries failing unit-root or individual cointegration tests were removed, yielding a 21-country subsample used for robustness checks.

### Diagnostic test results (panel unit root and cointegration)
- Panel unit-root test results (45 countries) reported in Table 1:
  - IPS ADF (large sample adjustment values): 3.21*** (GDP) and 1.09*** (Consumption).
  - IPS ADF (Bootstrapped): 0.84*** (GDP) and -0.01*** (Consumption).
  - MW (Bootstrapped): 84.73*** (GDP) and 89.42*** (Consumption).
- Cointegration test results (Table 1):
  - Group mean panel ADF/P P Rho: -2.71***, -4.24***, -3.74*** (reported across columns in table).
  - Pooled panel: -1.16, -2.67***, -2.06***.
- Note: Lag truncation: K = 4 (as in Table 1 note).

### Conventional panel regression findings (stationary/differenced and levels)
- Conventional (differenced) panel estimates (Table 2) — interpreted as business-cycle frequency risk sharing:
  - Pooled OLS GDP growth coefficient ≈ 0.680 − 0.809 across samples and specifications (1960-2008 pooled estimate ~0.680), implying roughly 32 percent of business-cycle frequency risk shared (as described in paper discussion: "an estimate of about 32 percent of businesscycle frequency risk has been shared").
  - Results stable across pooled OLS and fixed effects (FE), and across pre-1990 and post-1990 splits (exact coefficients in Table 2).
- Conventional (levels) panel estimates (Table 3) — long-run naive panel results:
  - Pooled OLS GDP coefficient ≈ 0.912 (1960-2008), but these OLS estimates have second-order bias concerns in finite samples.
  - Narrative summary in text: pooled-OLS level regression indicates less than 9 percent of long-run risks shared for 1960-2008, while FE indicates around 18 percent (text emphasizes potential finite-sample second-order bias affecting pooled OLS).

### Nonstationary panel (group-mean FMOLS/DOLS) empirical findings (key results)
- Group-mean FMOLS estimates on 45-country panel (1950-2008):
  - Whole 1950-2008 panel: point estimate ≈ 14 percent of long-run risk shared.
  - Test statistic for null of full risk sharing: t = 112.92 (indicates far from complete risk sharing).
  - Split by period:
    - Pre-1990: long-run risk sharing ≈ 12 percent.
    - Post-1990 (financial integration period): long-run risk sharing increased to ≈ 27 percent (more than doubled relative to pre-1990).
- Subgroup patterns (45-country panel):
  - OECD vs Emerging (whole sample): long-run risk sharing similar across OECD and emerging markets for whole sample.
  - During financial integration period (post-1990):
    - OECD long-run risk sharing ≈ 34 percent.
    - Emerging market long-run risk sharing ≈ 23 percent.
  - Within-group distribution: OECD countries appear to evenly enjoy benefits of increased risk sharing; emerging markets exhibit more heterogeneity (benefits concentrated in advanced emerging markets).
- Regional/intra-region observations:
  - EU (as reported): long-run risk sharing only ≈ 10 percent for whole sample, ≈ 6 percent pre-1990; intra-region analysis indicates EU15 used to share risks more among themselves but post-1990 share more with the rest of the world (example reported: risk sharing within EU15 ≈ 24 percent after 1990, while with rest of world ≈ 36 percent).
- Robustness check — 21-country subsample:
  - Whole sample: long-run risk sharing ≈ 14 percent (consistent with 45-country estimates).
  - Post-1990: long-run risk sharing increases to ≈ 39 percent on the 21-country panel (increase driven entirely by OECD countries in this subsample).

### Cross-country risk-sharing patterns and links to financial integration
- Visual/empirical patterns (described in text):
  - Pre-1990: positive correlation between long-run risk sharing and gross assets+liabilities to GDP.
  - Post-1990: weaker positive relation between long-run risk sharing and gross capital flows; capital flows (gross assets+liabilities) on average quadrupled post-1990 (tripled excluding Ireland as outlier), while long-run risk sharing only doubled — implying risk sharing did not scale proportionally with financial flows.
  - Decomposition: post-1990 increases in gross flows primarily driven by FDI and portfolio flows (FDI+portfolio as percent of GDP quadrupled post-1990), whereas debt-to-GDP roughly doubled (excluding Ireland).
  - Relationship strength: the positive association between long-run risk sharing and FDI/portfolio-to-GDP and debt-to-GDP is weaker in the post-1990 period than pre-1990.
- Interpretation: financial integration is necessary but not sufficient for proportionate increases in long-run risk sharing; potential threshold effects and other frictions (sovereign risk, moral hazard, non-insurability of nontradables, measurement issues) can limit translation of capital flows into state-contingent insurance.

### Interpretation, limitations, and recommended future directions
- Interpretation:
  - The group-mean FMOLS estimates present a "de facto" measure of long-run risk sharing (βi), capturing both endogenous market incompleteness (sovereign risk, moral hazard) and exogenous incompleteness (nontradable goods, labor income non-insurability).
  - Long-run risk sharing across the 45-country set is limited (≈ 14 percent overall for 1950-2008) but increased notably during the recent financial integration episode, disproportionately in OECD countries.
- Limitations and caveats:
  - Finite-sample size distortions can affect individual country FMOLS estimates; group-mean pooling mitigates but does not eliminate these concerns.
  - Cross-sectional dependence must be considered; the analysis uses demeaning (c∗t, y∗t) to remove common global components, an approach that works reasonably well when common stochastic trends or global cycles dominate cross-sectional dependence.
  - The interpretation as insurance over tradable goods requires additivity of period utility; without additivity estimates are proxies at best.
  - Measurement issues: GNI/BOP factor income and capital gains not fully captured in some measures can bias alternative approaches (GNI-based) to separating state-contingent insurance and intertemporal smoothing.
- Suggested future research directions (from text):
  - Use nonstationary vector autoregressive models to analyze short-run (business-cycle frequency) risk sharing and impulse responses to cyclical disturbances.
  - Decompose GDP and consumption processes into trends and cycles using long-run risk sharing estimates to study interaction between long-run and short-run risk sharing.

*Italic source: Appendix III: Group-mean FMOLS Estimator (excerpt), _wp1064 - Appendix III: Group-mean FMOLS Estimator, source PDF content provided.*

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### _wp1064 - References

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*Source: _wp1064 - References (PDF).*

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