## _wp04128

## Source details

**Canonical URL:** [_wp04128](https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2004/_wp04128.pdf)

## Other formats

- [Markdown version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2004/_wp04128.pdf.md)
- [Structured JSON version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2004/_wp04128.pdf.json)

---

### Overview and research questions
- Focus: Empirical examination of purchasing power parity (PPP) through time-series analysis of persistence of shocks to real effective exchange rates for 90 developed and developing countries in the post-Bretton Woods period.
- Research questions:
  - Do real exchange rates display parity-reverting behavior?
  - Is PPP an appropriate very long-run benchmark for real exchange rate developments?
  - Does real exchange rate behavior differ between developed and developing countries?
  - How do fixed versus flexible nominal exchange rate regimes affect parity reversion?
  - Can cross-country heterogeneity in parity reversion be explained by structural characteristics, and which characteristics are important?

### Data and sample
- Sample: Real effective exchange rates (REER) of 90 industrial and developing countries.
- Sample period and frequency:
  - Monthly REER series from IMF IFS, sample 1973:3 to 2002:3; total observations = 348 (source text also cites total observations = 358 in one table note).
  - CPI-based REER indicator (base 1990=100 or base 1995=100 in differing descriptions in source).
- Country groups: 20 industrial (advanced) and 70 developing countries; further classifications by region, export type, HIPC, net debtor status, and income-per-capita bands.
- Key data series definitions (sample period 1973–2002 unless denoted):
  - REER: trade-weighted, seasonally adjusted, CPI-based.
  - INF: Rate of change of consumer prices (percent per annum).
  - GGDP: General government final consumption spending as a share of GDP.
  - PCGDP: Growth of per capita GDP (percent per annum).
  - TGDP: Exports and imports as a share of GDP.
  - VPARER: Volatility of the parallel market exchange rate (std. dev. of monthly log changes), March 1973–December 1998.
  - VOFFER: Volatility of the official exchange rate (std. dev. of monthly log changes), March 1973–December 1998.

### Methodology and identification
- Critique of standard methods:
  - Unit root (UR) tests and least squares AR estimators are biased downward when the AR parameter α is near unity; this produces biased (shorter) half-life estimates.
  - Simulation evidence: Orcutt and Winokur (1969) for T = 40 and true α = 1 — least squares mean bias E(αˆ) – α = 0.129; Andrews (1993) for T = 40 — least squares median bias = 0.107.
- Estimation strategy:
  - Use median-unbiased estimator of Andrews and Chen (1994) to estimate α in AR(p) (Augmented Dickey-Fuller regression) specifications with lag selection per Ng and Perron (1995) / Hall (1994) (maximum lag = 12).
  - Apply Andrews (1993) unbiased model-selection rule (level 0.50 unbiased test) to decide between finite half-life (mean reversion) and permanent shocks (unit root).
  - Compute impulse response functions IR(h) = f_{11}(h) from AR(p) companion matrix and define half-life l_h as IR(l_h) ≤ 1/2 (time for a unit impulse to dissipate by one half).
  - Monte Carlo / simulation procedures: iterative median-unbiased correction with numerical simulation (2,500 iterations) for T = 348 observations.

### Main empirical findings — persistence and half-lives
- Biased (least-squares) estimates:
  - Average (median) half-life across all countries: 4.04 years.
  - Replicates Rogoff’s (1996) consensus (36 to 60 months) and Cheung and Lai (2000b) median of 3.3 years for industrial countries.
- Median-unbiased estimates:
  - Cross-country average (median) half-life across all countries: 8.17 years (compared with least-squares 4.04 years).
  - Implied rate of parity reversion: 8 percent per year (median-unbiased) versus 16 percent per year (least squares).
  - Range across countries: as little as one month (Bolivia) to infinity (Belgium, Sweden, and others); in group medians, infinity (∞) is operationalized as 30 years.
  - Using Andrews unbiased model-selection rule: 50 countries classified as subject to finitely-persistent REER shocks; 40 countries classified as experiencing permanent shocks.
  - For a given country, the unbiased rule implies a 50 percent probability that the interval from zero to the estimated median-unbiased half-life contains the true half-life and 50 percent that the interval from the estimated half-life to infinity contains the true half-life.
- Representative country behaviors (IRF patterns):
  - Sweden and Indonesia: exhibit permanent shocks to REER (unbiased half-life = ∞).
  - United States and Nigeria: exhibit hump-shaped IRFs with initial magnification lasting up to 1½ years and IRF taking about four years to return to unity.
  - United Kingdom and Mexico: exhibit monotonic decay and finite half-lives.
  - Post-magnification half-lives: dissipation after peak can be much faster; post-magnification half-life = unbiased half-life minus time-to-peak.

### Cross-country heterogeneity and regime differences
- By country group (medians and counts):
  - All countries median: Biased 4.04; Unbiased 8.17; Time to Peak 0.33 (years/months as reported in Table 1).
  - Advanced countries: median unbiased ≈ 7.67 years; 18 of 20 have finite half-lives.
  - Developing countries: median unbiased often operationalized as permanent (30.00 years in tables where ∞ set to 30); 32 of 70 developing countries have finite half-lives.
  - Nonfuel primary-product exporters: average half-life ≈ 22 years (finite but very slow reversion).
  - Fuel exporters: average half-life ≈ 22 years (Figure labels).
  - HIPC countries: average half-life = permanent (Figure label).
- By nominal exchange rate regime:
  - IMF de jure classification (mode 1975–98): pegged regimes show dominance of permanent shocks; flexible regimes show majority finite half-lives typically between two to ten years.
  - Reinhart-Rogoff de facto classification (mode 1973–98): de facto pegs typically experience permanent deviations; freely floating/falling countries average half-life: 7 years (Figure 3D label).
  - Table 2 summary:
    - IMF (de jure) Peg: Average Persistence 30.00 (n = 49).
    - IMF (de jure) Flexible: Average Persistence 7.25 (n = 35).
    - Reinhart-Rogoff de facto Peg: Average Persistence 30.00 (n = 22).
    - Reinhart-Rogoff Free floating/falling: Average Persistence 5.58 (n = 12).
- Frequency distributions (selected figure labels):
  - All Countries: Average half-life: 8 years.
  - Advanced Countries: Average half-life: 22 years (Figure label).
  - Developing Countries: Average half-life: 6 years (Figure label).
  - Regional labels: African, Western Hemisphere, Asian developing groups often labeled “permanent” in figure panels for average half-life.

### Determinants of persistence (statistical associations)
- Nonparametric tests used: Wilcoxon-Mann-Whitney (WMW) for medians; Brown-Forsythe (BF) for variances; Spearman rank correlation for associations.
- Group test results:
  - WMW: reject equal median half-lives at the 5 percent level for advanced vs developing countries (WMW statistic 2.18, p-value 0.029); reject at 5 percent for pegged vs flexible regimes (e.g., IMF de jure WMW 2.73, p-value 0.006).
  - BF: reject equal variances at 5 percent for industrial vs developing (BF = 6.11, p-value 0.015); cannot reject equality of variances across pegged vs flexible regimes.
- Spearman rank correlations (Table 3; rank correlation and approximate p-value):
  - Inflation (INF): rank correlation = -0.281; approximate p-value = 0.007. Interpretation: higher inflation associated with shorter half-lives (statistically significant).
  - Official nominal exchange rate volatility (VOFFER): rank correlation = -0.379; approximate p-value = 0.002. Interpretation: greater official exchange rate variability associated with shorter half-lives (significant).
  - Parallel-market exchange rate volatility (VPARER): rank correlation = -0.282; approximate p-value = 0.02. Interpretation: greater parallel-market volatility associated with shorter half-lives (significant).
  - Trade openness (TGDP): rank correlation = 0.180; approximate p-value = 0.10. Interpretation: insignificantly different from zero in this sample.
  - Productivity growth (PCGDP): rank correlation = -0.038; approximate p-value = 0.77. No significant correlation.
  - Government spending (GGDP): rank correlation = -0.073; approximate p-value = 0.50. No significant correlation.
- Overall inference:
  - Monetary and exchange-rate sources (inflation and nominal rate volatility) are inversely related to persistence: higher inflation or greater nominal volatility → faster parity reversion.
  - Little evidence that differential productivity growth (Balassa-Samuelson channel proxied by PCGDP) or government spending explains cross-country heterogeneity in persistence.
  - Sample-selection bias in prior literature: emphasis on Latin American high-inflation cases biased earlier findings that developing countries revert faster; broader sample shows developing countries typically revert more slowly.

### Contributions, robustness, and methodological implications
- Novelty:
  - First application (to authors’ knowledge) of median-unbiased estimation methods to developing-country real exchange rate data.
  - Use of Andrews’ unbiased model-selection rule provides an explicit, objective decision rule between finite and permanent persistence.
- Robustness and interpretation:
  - Median-unbiased correction substantially lengthens estimated half-lives relative to least squares (median-unbiased 8.17 years vs least-squares 4.04 years).
  - AR(p) specifications allow nonmonotonic (hump-shaped) impulse responses; AR(1) would force monotonic decay and understate half-lives when nonmonotonicity exists.
  - Confidence intervals for half-lives are typically very wide; the null that half-life is permanent is rarely rejected in two-sided 90 percent or one-sided 95 percent intervals.
- Practical implication for researchers:
  - Median-unbiased estimation and Andrews’ model-selection reduce downward bias and yield a more impartial decision framework for assessing PPP persistence.

### Key statistics and selected table values (as reported)
- Sample size: 90 countries (20 advanced, 70 developing).
- Median results (All countries):
  - Biased median half-life = 4.04 years.
  - Unbiased median half-life = 8.17 years.
  - Time to peak median = 0.33 (as labeled in Table 1).
- Counts from model-selection:
  - 50 countries with finite half-lives.
  - 40 countries with permanent shocks.
- Representative country entries (Table 1, selected):
  - Belgium: Biased Half-life 7.00; Unbiased Half-life ∞; Time to Peak 3.00.
  - Sweden: Biased Half-life 5.83; Unbiased Half-life ∞; Time to Peak 0.33.
  - United States: Biased Half-life 4.25; Unbiased Half-life 7.83; Time to Peak 1.00; Time to Unity 4.58.
  - Mexico: Biased Half-life 1.75; Unbiased Half-life 1.92; Time to Peak 0.50; Time to Unity 0.50.
  - Nigeria: Biased Half-life 4.25; Unbiased Half-life 5.58; Time to Peak 1.50; Time to Unity 3.83.
- Table 2 group averages:
  - All countries average persistence: 8.17 years (n = 90).
  - Advanced countries average persistence: 7.67 years (n = 20).
  - Developing countries average persistence: 30.00 years (n = 70) (note: ∞ set equal to 30 years in group medians/averages).
  - IMF (de jure) Peg: Average Persistence 30.00 (n = 49); Flexible: Average Persistence 7.25 (n = 35).
  - Reinhart-Rogoff de facto Peg: Average Persistence 30.00 (n = 22); Free floating/falling: Average Persistence 5.58 (n = 12).

### Policy-relevant implications
- PPP as a benchmark:
  - PPP provides a long-run benchmark for equilibrium exchange rates and for evaluating competitiveness of real exchange rates, but reversion to parity is often slow.
- Exchange-rate regime and inflation:
  - Fixed regimes and low nominal exchange rate variability are associated with slower parity reversion (greater persistence or permanence).
  - High inflation and greater nominal exchange rate volatility are associated with faster parity reversion.
- Sample-selection caution:
  - Prior results emphasizing rapid reversion in developing countries were affected by sample-selection bias (overweighting high-inflation Latin American cases); policymakers and researchers should use broad samples and median-unbiased methods when assessing long-run PPP and competitiveness.
- Methodological recommendation:
  - Use AR(p) models to allow nonmonotonic IRFs and apply median-unbiased estimation with an explicit unbiased model-selection rule to reduce downward bias in half-life estimates and to produce more impartial inference about persistence.

*Source: IMF Working Paper content unit _wp04128 (excerpts provided).*

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

### References

### Overview
- Focus: Empirical examination of purchasing power parity (PPP) through time-series analysis of persistence of shocks to real effective exchange rates for 90 developed and developing countries in the post-Bretton Woods period.
- Key methodological advance: Application of median-unbiased estimation methods (Andrews and Chen (1994); Andrews (1993)) to real exchange rate data, including developing countries.

### Research questions addressed
- Do real exchange rates display parity-reverting behavior?
- Is purchasing power parity (PPP) an appropriate very long-run benchmark for real exchange rate developments?
- Does real exchange rate behavior differ between developed and developing countries?
- How do fixed versus flexible nominal exchange rate regimes affect parity reversion?
- Can cross-country heterogeneity in parity reversion be explained by structural characteristics, and which characteristics are important?

### Data and scope
- Sample: Real effective exchange rates of 90 industrial and developing countries.
- Sample period: Post-Bretton Woods (implicit in text as beginning after floating of developed-country exchange rates in the early 1970s; figures reference 1973:3–2002:3).

### Methodology
- Critique of standard UR/AR testing:
  - Unit root (UR) tests vs stationary autoregressive (AR) alternatives often fail to reject a unit root, but such tests can be uninformative about the speed of parity reversion.
  - Least squares estimators of AR models are biased downward, especially when the AR parameter is near unity, producing biased (shorter) half-life estimates.
- Estimation strategy:
  - Use median-unbiased estimator of Andrews and Chen (1994) to estimate the AR parameter in real exchange rate data.
  - Use these median-unbiased estimates and associated impulse response functions to calculate unbiased scalar measures of average duration (half-lives) of typical real exchange rate shocks.
  - Apply Andrews’ (1993) unbiased model-selection rule to determine whether finite half-lives (mean reversion) or permanent shocks better describe each country’s real exchange rate process.

### Main empirical findings
- Replication using least squares:
  - Using least squares estimation of unit root models, the study replicates Rogoff’s (1996) consensus estimate of the half-life of deviations from PPP of between three to five years.
- Median-unbiased estimates (robust to serial correlation):
  - Cross-country average (median) half-lives of deviations of real exchange rates from PPP are about eight years.
  - For 40 countries in the sample, deviations of the real exchange rate from parity are best viewed as being permanent.
  - The majority (50) of countries have finite half-life estimates, consistent with PPP holding in the post-Bretton Woods period.
- Heterogeneity by country group and regime:
  - Median half-life for industrial countries: eight years.
  - Median half-life for developing countries: permanent.
  - Median half-life for countries with fixed nominal exchange rate regimes: permanent.
  - Median half-life for countries with flexible nominal exchange rate regimes: six years.
- Determinants of persistence:
  - Two key structural characteristics affecting persistence of shocks to real exchange rates are:
    - Differences in a country’s inflation experience.
    - The extent of nominal exchange rate volatility.
- Sample-selection bias in prior literature:
  - Existing literature on parity reversion in developing countries often focused on Latin American experience (dominated by high inflation countries), producing a sample-selection bias that led to the (erroneous) finding that parity reversion in developing countries is faster than in developed countries.

### Contributions to the literature
- First application (to the authors’ knowledge) of median-unbiased estimation methods to developing-country real exchange rate data.
- Use of median-unbiased estimation and Andrews’ model-selection rule provides a more definitive framework for assessing whether real exchange rate shocks are mean-reverting or permanent, and for producing unbiased measures of half-lives.

### Structure of the paper (as presented)
- Section II: Median-unbiased procedure for estimation of AR/UR models.
- Section III: Description of the data.
- Section IV: Main empirical findings of biased and median-unbiased point estimates of persistence.
- Section V: Examination of key determinants of cross-country heterogeneity in duration of deviations from PPP.
- Section VI: Concluding comments.

*Source: _wp04128 - References*

### 1. Using simulation methods, Orcutt and Winokur (1969) find that, for T = 40 and true α = 1,

### _wp04128 - 1. Using simulation methods, Orcutt and Winokur (1969) find that, for T = 40 and true α = 1,

### Downward bias in least squares estimates of autoregressive parameters
- Orcutt and Winokur (1969) simulation result: for T = 40 and true α = 1, the least squares mean bias is E(αˆ) – α = 0.129.
- Andrews (1993) simulation result: for T = 40 and true α = 1, the least squares median bias of the AR(1) model is 0.107.
- General findings:
  - "The larger is the true value of α, the larger is the least squares bias, and so the bias is largest in the unit root case."
  - "This bias shrinks as the sample size grows, as the estimate converges to its true population value."
- Source of bias: asymmetry in the distribution of estimators for autoregressive parameters in AR models:
  - Distribution is skewed to the left, resulting in the median exceeding the mean.
  - Consequence: the median is a better measure of central tendency than the mean in least squares estimates of AR models.

### AR(p) model specification and persistence measurement
- AR(p) (Augmented Dickey-Fuller regression) form used:
  - Observed series q_t: t = – p,...,T.
  - Model components: μ (intercept); α (autoregressive parameter, where α ∈ (–1,1]); ε_t (innovations).
  - Lagged first differences included to control for serial correlation.
- Lag selection for serial correlation control:
  - General-to-specific lag selection procedure of Ng and Perron (1995) and Hall (1994) with maximum lag set to twelve.
  - Procedure: starting with maximum lag, sequentially remove first-differences of log(REER) (q_t) until last lag is statistically significant at the 5 percent level; include all lag lengths ≤ p – 1 in AR(p) regression.
- Alternative methods:
  - Phillips and Perron (1988) unit root procedures allow a wider class of serial correlation and heterogeneity of errors; benefit for exchange rate data with heterogeneous errors.
  - Cashin and McDermott (2003) find half-lives from Dickey-Fuller and Phillips-Perron regressions are broadly similar.
  - Trade-off: parametric models are easier for calculating persistence and comparing with previous studies; neglecting serial correlation leads to serious over-estimation of persistence.

### Median-unbiased estimation of α and unbiased model-selection rule
- Andrews and Chen (1994) present approximately median-unbiased estimation for autoregressive parameters in Augmented Dickey-Fuller regressions.
  - Benefit: "The bias correction delivers an impartiality property to the decision making process, because there is an equal chance of under- or overestimating the AR parameter."
  - An unbiased estimate of α permits calculation of an unbiased scalar estimate of persistence—the half-life of a unit shock.
- Definition of median-unbiased estimator u_{αˆ}:
  - If αˆ has median function m(α) uniquely defined ∀α ∈ (-1,1], then the median-unbiased estimator is given by the inverse mapping m^{-1} applied to the observed median value (as formalized in equation (2) in the source).
  - Example interpretation: if least squares estimate αˆ = 0.8, use the value of α such that the least squares estimator has median 0.8 (i.e., m^{-1}(0.8)).
  - m(-1) = lim_{α→-1} m(α); m^{-1}:(m(-1), m(1)] → (-1,1] with m^{-1}(m(α)) = α for α ∈ (-1,1].
- Unbiased model-selection procedure (Andrews 1993):
  - Choose model I_m if u_{αˆ} ∈ I_m, where models partition parameter space (-1,1] (example: I_a = (-1,1), I_b = {1}).
  - This rule is a level 0.50 (unbiased) test of H_0: α ∈ I_a versus H_1: α ∈ I_b.
  - Property: u_{αˆ} are the lower and upper bounds of the two one-sided 0.5 confidence intervals for α when m(.) is strictly increasing; these intervals have probability one-half of containing the true α.
- Monte Carlo evidence (Andrews and Chen 1994, p.194):
  - Probability of correctly selecting the unit-root model when true α = 1 is about 0.5 (equal chance of accepting or rejecting the unit root null).
  - Comparison with conventional tests:
    - Unbiased model-selection rule has a greater size than conventional two-sided level 0.10 or one-sided level 0.05 tests.
    - For true α < 1, the probability of a type II error is smaller for Andrews’ rule than for conventional tests, especially in near unit root cases.
- Practical notes:
  - Bayesian estimation with flat or noninformative priors attempts impartiality but there are doubts about how noninformative such priors really are (Phillips, 1991).
  - For AR(p) models the unbiased selection procedure is approximate because the distribution of u_{αˆ} depends on true ψ_i terms (unknown); Andrews and Chen (1994) find approximately median-unbiased point estimates are very close to median-unbiased.

### Calculating half-lives of shocks (scalar measure of persistence)
- Impulse response for AR(p):
  - IR(h) = f_{11}(h) for h = 0,1,2,... where f_{11}(h) denotes the (1,1) element of F^h and F is the (p × p) companion matrix as given in equation (4).
- Half-life l_h definition:
  - Duration of time it takes for a unit impulse to dissipate permanently by one half from the occurrence of the initial shock (Cheung and Lai, 2000b).
  - Formal condition: IR(l_h) ≤ ½ such that l_{h+k} < ½ for k = 1,2,...
- Median-unbiased interpretation for half-lives:
  - Using Andrews unbiased model-selection rule, there is a 50 percent probability that the confidence interval from zero to the estimated median half-life contains the true half-life, and a 50 percent probability that the confidence interval from the estimated median half-life to infinity contains the true half-life.
- Confidence intervals and testing:
  - Two-sided 90 percent (or one-sided 95 percent) intervals for half-lives can be constructed (Cashin, Liang and McDermott 2000; Cheung and Lai 2000b; Murray and Papell 2002).
  - Such intervals are typically very wide; the null hypothesis that the half-life is permanent is rarely rejected, leaving high uncertainty about persistence.
  - Empirical practice: failure to reject unit root often leads researchers to proceed as if the null is false, despite unsatisfactory implications.

*Italic: Source content from the provided IMF PDF excerpt.*

### conclusions from different researchers, because each is now using an arbitrary decision

### _wp04128 - conclusions from different researchers, because each is now using an arbitrary decision

### Model selection and estimation approach
- Use of Andrews (1993) unbiased-model selection rule and median-unbiased estimation.
  - The authors use a 0.5 (unbiased) test of the null hypothesis of a finite half-life versus the alternative of a permanent half-life.
  - Rationale: provides an explicit, objective, and verifiable decision rule versus undisclosed/implicit rules.
- Estimation framework:
  - AR(p) specifications for REER dynamics to allow nonmonotonic impulse responses and control serial correlation.
  - Median-unbiased correction following Andrews and Chen (1994) iterative procedure and numerical simulation (2,500 iterations) to obtain median-unbiased half-lives for T=348 observations.

### Data
- Sample and series:
  - Monthly real effective exchange rate (REER) series from IMF IFS, sample 1973:3 to 2002:3, total 348 observations.
  - CPI-based REER (Fund’s CPI-based REER indicator, base 1995=100) for 90 countries: 20 industrial and 70 developing.
  - REER constructed as weighted geometric average across trade partners using W_ij competitiveness weights (based on 1988–90 manufacturing/non-oil primary commodities/tourism composition).
- Key data features:
  - Most countries’ REER series display drift or nonstationarity and substantial sustained deviations from PPP.
  - Sharp REER movements in the 1980s and 1990s are frequent for many developing countries (examples: Indonesia, Nigeria, Mexico).

### Empirical results — biased least squares versus median-unbiased
- Biased (least squares) estimates:
  - Average (median) half-life of parity reversion across all countries: 4.04 years.
  - Consistent with Rogoff (1996) consensus of 36 to 60 months and Cheung and Lai (2000b) median of 3.3 years for industrial countries.
- Median-unbiased estimates:
  - Median-unbiased average (median) half-life across all countries: 8.17 years (compared with least-squares 4.04 years).
  - Implied rate of parity reversion: 8 percent per year (median-unbiased) versus 16 percent per year (least squares).
  - Range across countries: as little as one month (Bolivia) to infinity (Belgium, Sweden, and others).
  - Using Andrews unbiased model-selection rule: 50 countries classified as subject to finitely-persistent REER shocks; 40 countries classified as experiencing permanent shocks.
  - Interpretation of the rule: for a given country there is a 50 percent probability that the interval from zero to the estimated median-unbiased half-life contains the true half-life, and 50 percent probability that the interval from the estimated half-life to infinity contains the true half-life.
  - Examples: Iceland — estimated median-unbiased half-life 1.1 years (finite); Togo — infinite half-life (permanent). Permanent deviations are operationalized as at least as long as the span of the data; with sample span 29 years, permanent deviation set equal to 30 years in duration.

### Impulse response functions and half-lives
- Half-life definition: time for a unit impulse to dissipate permanently by one half (IR(l_h) = 1/2) from the impulse response of equation (3).
- AR(p) permits nonmonotonic (hump-shaped) impulse responses; AR(1) would force monotonic decay and understate half-lives when nonmonotonicity exists.
- Representative country behaviors (least squares and median-unbiased IRFs):
  - Sweden and Indonesia: exhibit permanent shocks to REER.
  - United States and Nigeria: exhibit hump-shaped IRFs with initial magnification of shocks lasting up to 1½ years and IRF taking about four years to return to unity; magnification extends measured half-life.
  - United Kingdom and Mexico: exhibit monotonic decay and finite half-lives.
- Post-magnification half-lives (following Cheung and Lai, 2000b): once initial magnification completes, dissipation can be much faster; post-magnification half-lives can be derived as difference between unbiased half-life and time-to-peak.

### Cross-country heterogeneity of half-lives
- Sample-wide distribution:
  - Out of 90 countries: 50 countries with half-lives from 1 to 14 years; 40 countries with permanent real exchange rate shocks (half-lives of 29 years or more, operationalized as 30 years).
- By country group:
  - Industrial countries: average (median) half-life ≈ 8 years (about twice the historically reported biased estimates).
  - Developing countries: half-life estimates more widely spread; most estimated deviations often permanent.
  - Developing countries: 32 of 70 have finite half-lives.
  - Developed countries: 18 of 20 have finite half-lives.
  - Nonfuel primary-product exporters: average half-life ≈ 22 years (finite but very slow parity reversion).
  - Heavily Indebted Poor Countries (HIPCs, mostly African): average half-life ≈ 11 years.
  - Regional patterns: African and Western Hemisphere developing countries—persistence evenly split between finite and permanent; Asian developing countries—permanent shocks dominate.
  - Western Hemisphere notable cases (Brazil, Mexico, Chile, Argentina): half-lives all less than four years.
- By nominal exchange rate regime:
  - Using IMF de jure classification and Reinhart-Rogoff de facto classification:
    - Pegged exchange rate countries typically experience permanent deviations from parity.
    - Flexible exchange rate countries exhibit dispersed half-lives, with majority finite and typically between two to ten years.
  - De jure pegged regimes show dominance of permanent shocks; de facto classification yields similar conclusions.

### What explains heterogeneous persistence? (cross-country correlates)
- Estimation strategy:
  - Nonparametric tests used due to nonnormality: Wilcoxon-Mann-Whitney for equality of medians, Brown-Forsythe for equality of variances, Spearman rank correlation for associations with structural characteristics.
  - Structural variables examined (averages/medians over the sample): INF (inflation), VOFFER (official nominal exchange rate volatility), VPARER (parallel-market nominal exchange rate volatility), TGDP (openness), PCGDP (per capita real GDP growth), GGDP (government spending share).
- Nonparametric group tests:
  - Wilcoxon-Mann-Whitney: reject equal median half-lives at the 5 percent level for advanced vs developing countries; reject at 5 percent for pegged vs flexible exchange rate countries (using either classification).
    - Conclusion: advanced (industrial) countries have shorter median half-lives than developing countries; pegged regimes have longer median half-lives than flexible regimes.
  - Brown-Forsythe: reject equal variances at 5 percent for industrial versus developing countries; cannot reject equality of variances across pegged vs flexible regimes.
    - Conclusion: developing countries’ half-lives exhibit greater variability than industrial countries’.
- Spearman rank correlations (rank correlation coefficient and approximate p-value; significance assessed):
  - Inflation (INF): rank correlation = -0.281, approximate p-value = 0.007.
    - Interpretation: higher inflation associated with shorter half-lives; inverse relationship is statistically significant (reject null of no correlation; significant at 1 percent level using critical values).
  - Official nominal exchange rate variability (VOFFER): rank correlation = -0.379, approximate p-value = 0.002.
    - Interpretation: greater official exchange rate variability associated with shorter half-lives (significant).
  - Parallel-market nominal exchange rate variability (VPARER): rank correlation = -0.282, approximate p-value = 0.02.
    - Interpretation: greater parallel-market volatility associated with shorter half-lives (significant).
  - Per capita GDP growth (PCGDP, proxy for productivity growth): rank correlation = -0.038, approximate p-value = 0.77.
    - Interpretation: no significant correlation between productivity growth and half-life.
  - Government spending share (GGDP): rank correlation = -0.073, approximate p-value = 0.50.
    - Interpretation: no significant correlation between government spending share and half-life.
- Overall inference:
  - Monetary and exchange-rate sources of variation (inflation and nominal exchange rate volatility) are inversely related to persistence: countries with higher inflation or more variable nominal exchange rates tend to have faster parity reversion.
  - Little evidence that differential productivity growth (Balassa-Samuelson channel proxied by PCGDP) or government spending (GGDP) explains cross-country heterogeneity in persistence.

### Key substantive conclusions and implications
- Median-unbiased inference substantially lengthens estimated half-lives relative to conventional least-squares estimates (median-unbiased average half-life 8.17 years vs least-squares 4.04 years).
- Using an unbiased model-selection rule, a substantial minority of countries (40 of 90) exhibit permanent REER shocks, while a slight majority (50 of 90) exhibit finite persistence — implying that PPP holds in the post-Bretton Woods period for the majority but often with slow reversion.
- Exchange rate regime and country income levels are systematically associated with persistence:
  - Pegged regimes and lower-income/developing countries tend to display more persistent or permanent deviations.
  - Flexible regimes and higher inflation/nominal volatility environments tend to display faster parity reversion.
- Methodological implication: use of AR(p) and median-unbiased estimation with an explicit unbiased model-selection rule reduces downward bias in half-life estimates and accommodates nonmonotonic impulse responses that materially affect inferred durations.

*Source: IMF Working Paper content unit _wp04128 (excerpts provided).*

### conclusions are consistent with that of Rogoff (1996).

### _wp04128 - conclusions are consistent with that of Rogoff (1996).

### Trade openness and goods-market arbitrage
- Rogoff, Froot and Kim (2001) propose that goods market arbitrage can accelerate parity reversion across goods by promoting more frequent price adjustment.
- Trade openness measure (TGDP): average (median) ratio of exports plus imports to GDP for each country over the sample period.
- Empirical correlation:
  - Rank correlation coefficient between the half-life of the real exchange rate and external trade (as a share of GDP) = 0.180, with an approximate p-value of 0.10.
  - This rank correlation differs insignificantly from zero.
- Additional observed correlation:
  - Strong negative rank correlation between openness and inflation = -0.397 (for the sample of countries).
  - Implication noted: greater openness is associated with lower inflation, which implies a slower speed of parity reversion.
- Conclusion: Little evidence that differences in trade openness are associated with the persistence of parity deviations across countries; thus, little support in this sample for the goods-arbitrage view of PPP.

### Summary of empirical findings on PPP persistence
- Sample and data scope:
  - Number of countries in full sample = 90 advanced and developing countries.
  - Sample period = 1973–2002 (unless otherwise denoted).
- Previous consensus:
  - Univariate least squares studies yielded consensus estimates of the half-life of deviations of real exchange rates from PPP of about four years (Rogoff, 1996).
- Results using median-unbiased estimation:
  - Cross-country average of unbiased half-lives of deviations from parity ≈ eight years.
  - Confirms Rogoff’s (1996) ‘PPP puzzle’: PPP often holds but speed of reversion is frequently slow.
- Model-selection and inference:
  - Using median-unbiased estimates of half-lives and Andrews (1993) unbiased model-selection rule:
    - Number of countries with finite half-lives of parity reversion = 50 of the 90 countries.
      - Interpretation: for these 50 countries, there is a better than even chance that shocks to their real exchange rates are transitory; PPP holds in the post-Bretton Woods period for them.
    - Number of countries with little evidence of parity reversion = 40 of the 90 countries.
      - Interpretation: for these 40 countries, shocks to their real exchange rate are best viewed as being permanent.
- Heterogeneity findings:
  - Parity reversion is more likely for developed countries than for developing countries.
  - Parity reversion is more likely under flexible nominal exchange rate regimes than under fixed nominal exchange rate regimes.
  - Parity reversion tends to be faster in high-inflation countries than in low-inflation countries.
  - Parity reversion tends to be slower in countries with less nominal exchange rate variability.

### Policy-relevant implications
- PPP as benchmark:
  - PPP provides a long-run benchmark for equilibrium exchange rates and a criterion for evaluating competitiveness of real exchange rates.
  - The quality of policy advice based on exchange-rate-determination theories may depend on the empirical validity of PPP.
- Sample-selection caution:
  - Prior literature focus on developed countries and on high-inflation Latin American developing countries produced a biased view that developing countries revert faster; expanding the sample shows developing countries typically revert more slowly than developed countries.
- Exchange-rate regime and inflation considerations:
  - Policymakers should consider that fixed regimes and low nominal exchange rate variability are associated with slower parity reversion.
  - High inflation environments are associated with faster parity reversion, which affects interpretations of real exchange rate dynamics and policy responses.

### Data definitions and key series (sample period 1973–2002 unless denoted)
- REER:
  - Real effective exchange rate data: monthly frequency, March 1973–March 2002; total observations = 358.
  - Trade-weighted, seasonally adjusted, CPI-based REER (base 1990=100); source: IMF’s INS.
- INF:
  - Rate of change of consumer prices (percent per annum); period average of annual data 1973–2002; source: IMF’s IFS.
- GGDP:
  - General government final consumption spending as a share of GDP; period average of annual data 1973–2002; source: World Bank (2002).
- PCGDP:
  - Growth of per capita GDP (percent per annum); period average of annual data, 1973–2002; source: World Bank (2002).
- TGDP:
  - Exports and imports of goods and services (valued in current U.S. dollars) as a share of GDP (valued in current U.S. dollars); period average of annual data, 1973–2002; source: World Bank (2002).
- VPARER:
  - Volatility of the parallel market exchange rate; standard deviation of the monthly rate of change of the parallel market exchange rate (log of local currency per U.S. dollar), March 1973 to December 1998; source: Reinhart-Rogoff (2002).
- VOFFER:
  - Volatility of the official exchange rate; standard deviation of the monthly rate of change of the official exchange rate (log of local currency per U.S. dollar), March 1973 to December 1998; source: Reinhart-Rogoff (2002).

### Exchange rate regime classifications (periods and modes)
- IMF de jure classification:
  - Used between 1975–97 in AREAER; ten categories grouped into regimes 1–5 (fixed pegs), regimes 6–7 (intermediate), regimes 8–10 (flexible).
  - For each country, the de jure classification used = the mode of the annual IMF (AREAER) classification numbers over 1975–98.
- Reinhart-Rogoff (2002) de facto classification (mode over 1973–98):
  - Regime 1: de facto pegs (including no separate legal tender and currency boards).
  - Regime 2: limited flexibility (including crawling pegs and narrow crawling bands).
  - Regime 3: managed floating (including wider crawling bands).
  - Regime 4: freely floating.
  - Regime 5: freely falling (annualized inflation exceeds 40 percent).
- Note: The following countries had no Reinhart-Rogoff (2002) classification data: Ethiopia, Fiji, Papua New Guinea, Seychelles, Sudan, Samoa, Democratic Republic of Congo, Barbados, Rwanda, Sierra Leone, and Trinidad and Tobago.

### Country grouping taxonomy (overview of classifications used)
- Total countries listed in sample = 90.
- Major groupings and examples:
  - Advanced economies: Australia, Austria, Belgium, Canada, Finland, Germany, Iceland, Ireland, Italy, Japan, Korea, Netherlands, Norway, New Zealand, Portugal, Spain, Sweden, Switzerland, United Kingdom, United States.
  - Developing countries: Argentina, Brazil, Cameroon, Central African Republic, Chile, Côte d’Ivoire, Egypt, Ethiopia, Fiji, The Gambia, Guyana, Haiti, Honduras, Hungary, India, Indonesia, Islamic Republic of Iran, Kenya, Madagascar, Malawi, Malaysia, Mauritania, Mauritius, Morocco, Myanmar, Niger, Pakistan, Papua New Guinea, Paraguay, Philippines, Senegal, Seychelles, Sri Lanka, Sudan, Syria, Tanzania, Thailand, Togo, Tunisia, Turkey, Samoa, Democratic Republic of Congo, Zambia, Barbados, Bolivia, Burkina Faso, Chad, Colombia, Republic of Congo, Dominican Republic, Ecuador, El Salvador, Gabon, Ghana, Guatemala, Jamaica, Mexico, Malta, Nepal, Nigeria, Peru, Rwanda, Sierra Leone, Trinidad and Tobago, Uruguay, Venezuela, Costa Rica, Lesotho, Panama, Uganda.
  - Nonfuel primary-product exporters, fuel exporters, regional developing groups (African, Asian, Western Hemisphere, Middle East and Turkey), Heavily Indebted Poor Countries (HIPCs), net debtor classifications (private vs. official external financing), income-per-capita bands (specific $ ranges listed), and IMF AREAER pegged/intermediate/flexible country lists are provided as classification rules and member lists in the source.

*Content derived from the IMF working paper text provided.*

### APPENDIX II

### APPENDIX II

### Country classifications (Reinhart-Rogoff (2002) classification)
- Pegged Exchange Rate Countries: Austria, Belgium, Netherlands, Cameroon, Central African Republic, Côte d’Ivoire, Haiti, Kenya, Niger, Senegal, Thailand, Togo, Burkina Faso, Chad, Ecuador, El Salvador, Gabon, Guatemala, Mexico, Venezuela, Lesotho, and Panama.
- Limited Flexibility and Managed Float Exchange Rate Countries: Canada, Finland, Iceland, Ireland, Italy, Norway, New Zealand, Portugal, Spain, Sweden, Switzerland, United Kingdom, Chile, Egypt, The Gambia, Guyana, Honduras, Hungary, India, Indonesia, Islamic Republic of Iran, Korea, Madagascar, Malawi, Malaysia, Mauritania, Mauritius, Morocco, Myanmar, Pakistan, Paraguay, Philippines, Sri Lanka, Syria, Tanzania, Tunisia, Bolivia, Colombia, Dominican Republic, Jamaica, Malta, Nepal, Nigeria, Costa Rica, and Uganda.
- Freely Floating/Falling Exchange Rate Countries: Australia, Germany, Japan, United States, Argentina, Brazil, Turkey, Zambia, Republic of Congo, Ghana, Peru, and Uruguay.

### Real Effective Exchange Rate (1973:3 - 2002:3)
- Figure 1 presents real effective exchange rate series (1990 = 100) for selected countries: Sweden, Indonesia, United States, Nigeria, United Kingdom, Mexico.
- Source: International Monetary Fund.

### Impulse Response Functions (IRFs) of a Shock to the Real Effective Exchange Rate (1973:3 - 2002:3)
- Figure 2 shows unbiased and biased IRFs for selected countries: Sweden, Indonesia, United States, Nigeria, United Kingdom, Mexico.
- The plots display:
  - Lag in Months (horizontal axis) with values indicated by month counts: 1, 2, 5, 4, 9, 7, 12, 11, 14, 5, 16, 9, 19, 3, 21, 7 (as labeled in figure).
  - Persistence of Derivations from PP (vertical axis) with scales varying by country (e.g., Sweden up to 1.6, Indonesia up to 3.5, United States up to 2.0, Nigeria up to 2.0, Mexico up to 1.2).
- Several panels annotate "Persistence = 0.5".
- Source: Authors' calculations.

### Frequency distributions of median-unbiased half-lives of deviations from Purchasing Power Parity (1973-2002)
- Figure 3A–3D present frequency distributions by country groups:
  - Panels include: All Countries; Advanced Countries; Developing Countries; Nonfuel Primary-Product Exporters; Fuel Exporters; African Developing Countries.
  - Additional panels (Figure 3B) include Western Hemisphere Developing Countries; Asian Developing Countries; Middle East and Turkey Developing Countries; Net Debtor Countries (Private External Financing); Net Debtor Countries (Official External Financing); HIPC Countries.
  - Figure 3C panels classify by annual income (real GDP per capita, 1995 U.S. dollars) with ranges: $102 - $280; $281 - $769; $770 - $2111; $2112 - $5792; $5793 - $15891; $15892 - $43600.
  - Figure 3D panels include IMF AREAER (de jure): Pegged Exchange Rate Countries; IMF AREAER (de jure): Intermediate Exchange Rate Countries; IMF AREAER (de jure): Flexible Exchange Rate Countries; and Reinhart-Rogoff (de facto): Pegged; Limited Flexibility and Managed Float; Freely Floating/Falling Exchange Rate Countries.
- Reported average half-lives by selected panels (as labeled on figures):
  - All Countries: Average half-life: 8 years (Figure 3A label).
  - Advanced Countries: Average half-life: 22 years (Figure 3A label).
  - Developing Countries: Average half-life: 6 years (Figure 3A label).
  - Nonfuel Primary-Product Exporters: Average half-life: 8 years (Figure 3A label).
  - Fuel Exporters: Average half-life: 22 years (Figure 3A label).
  - African Developing Countries: Average half-life: permanent (Figure 3A label).
  - Western Hemisphere Developing Countries: Average half-life: permanent (Figure 3B label).
  - Asian Developing Countries: Average half-life: permanent (Figure 3B label).
  - Middle East and Turkey Developing Countries: Average half-life: permanent (Figure 3B label).
  - Net Debtor Countries, Private External Financing: Average half-life: 6 years (Figure 3B label).
  - Net Debtor Countries, Official External Financing: Average half-life: 11 years (Figure 3B label).
  - HIPC Countries: Average half-life: permanent (Figure 3B label).
  - Income-group panels (Figure 3C) report averages such as: $102 - $280: Average half-life: 14 years; $281 - $769: Average half-life: 4 years; $770 - $2111: Average half-life: 6 years; other income groups labeled as "permanent" in respective panels.
  - IMF AREAER (de jure): Pegged Exchange Rate Countries: Average half-life: 8 years (Figure 3D label).
  - IMF AREAER (de jure): Intermediate Exchange Rate Countries: Average half-life: 8 years (Figure 3D label).
  - IMF AREAER (de jure): Flexible Exchange Rate Countries: Average half-life: 6 years (Figure 3D label).
  - Reinhart-Rogoff (de facto): Pegged Exchange Rate Countries: Average half-life: permanent (Figure 3D label).
  - Reinhart-Rogoff (de facto): Limited Flexibility and Managed Float Exchange Rate Countries: Average half-life: permanent (Figure 3D label).
  - Reinhart-Rogoff (de facto): Freely Floating/Falling Exchange Rate Countries: Average half-life: 7 years (Figure 3D label).
- Source: Authors' calculations.

### Table 1 — Half-lives of Reversion to Parity (years), Real Effective Exchange Rates, March 1973–March 2002 (selected entries and medians)
- Advanced countries (selected entries):
  - Belgium: Biased Half-life 7.00; Unbiased Half-life ∞; Time to Peak 3.00; Time to Unity ∞.
  - Sweden: Biased Half-life 5.83; Unbiased Half-life ∞; Time to Peak 0.33; Time to Unity ∞.
  - Portugal: Biased Half-life 4.92; Unbiased Half-life 8.42; Time to Peak 0.92; Time to Unity 3.17.
  - United States: Biased Half-life 4.25; Unbiased Half-life 7.83; Time to Peak 1.00; Time to Unity 4.58.
  - Iceland: Biased Half-life 1.00; Unbiased Half-life 1.08; Time to Peak 0.08; Time to Unity 0.08.
  - Advanced countries’ median: Biased 3.63; Unbiased 7.67; Time to Peak 0.79.
- Africa (selected entries and median):
  - Burkina Faso: Biased Half-life 5.58; Unbiased Half-life ∞; Time to Peak 0.08; Time to Unity NCU.
  - Chad: Biased Half-life 10.92; Unbiased Half-life ∞; Time to Peak 17.58; Time to Unity ∞.
  - Nigeria: Biased Half-life 4.25; Unbiased Half-life 5.58; Time to Peak 1.50; Time to Unity 3.83.
  - Zambia: Biased Half-life 0.67; Unbiased Half-life 1.50; Time to Peak 0.08; Time to Unity 0.08.
  - African median: Biased 4.21; Unbiased 22.08; Time to Peak 0.17.
- Western Hemisphere (selected entries and median):
  - Colombia: Biased Half-life 8.00; Unbiased Half-life ∞; Time to Peak 3.25; Time to Unity ∞.
  - Haiti: Biased Half-life 19.33; Unbiased Half-life ∞; Time to Peak 0.08; Time to Unity NS.
  - Panama: Biased Half-life 10.08; Unbiased Half-life ∞; Time to Peak 0.08; Time to Unity NCU.
  - Brazil: Biased Half-life 1.75; Unbiased Half-life 2.50; Time to Peak 0.08; Time to Unity 0.42.
  - Mexico: Biased Half-life 1.75; Unbiased Half-life 1.92; Time to Peak 0.50; Time to Unity 0.50.
  - Western Hemisphere median: Biased 4.46; Unbiased ∞; Time to Peak 0.54.
- Asia, Middle East and Europe (selected entries and median):
  - India: Biased Half-life 17.67; Unbiased Half-life ∞; Time to Peak 16.75; Time to Unity ∞.
  - Indonesia: Biased Half-life 11.92; Unbiased Half-life ∞; Time to Peak 0.75; Time to Unity NS.
  - Philippines: Biased Half-life 3.50; Unbiased Half-life 8.08; Time to Peak 0.42; Time to Unity 2.17.
  - Turkey: Biased Half-life 1.92; Unbiased Half-life 5.67; Time to Peak 0.08; Time to Unity 0.17.
  - Asia, Middle East and Europe median: Biased 5.46; Unbiased ∞; Time to Peak 0.54.
- All countries median: Biased 4.04; Unbiased 8.17; Time to Peak 0.33.
- Notes and legend (as provided):
  - Column (2): Biased half-life based on least squares estimation of the Augmented Dickey-Fuller regression.
  - Column (3): Median-unbiased half-life based on Andrews and Chen (1994).
  - Column (4): “Time to peak” in months following the unit shock when IRF reaches its peak.
  - Column (5): “Time to unity” in months following the unit shock when IRF crosses unity.
  - NCU denotes nonstationary series—the IRF never crosses unity.
  - NS denotes nonstationary series—the IRF is above unity, then crosses unity but does not decay.
  - ∞ denotes classic nonstationary series—the IRF rises above unity and stays there.
  - S denotes stationary series—the IRF never rises above unity and has a monotonic decay.
  - In calculating group and all-country medians, infinity (∞) is set to equal 30 years.
- Source: Authors’ calculations.

### Table 2 — Descriptive statistics of persistence of PPP deviations, country groups
- All countries:
  - Average Persistence of Deviations from PPP (years): 8.17
  - Number of observations: 90
- Advanced countries:
  - Average Persistence: 7.67
  - Number of observations: 20
- Developing countries:
  - Average Persistence: 30.00
  - Number of observations: 70
  - WMW test H0: Equality of Median Deviation from PPP: 2.18 (p-value 0.029)
  - BF test H0: Equality of Variance of Deviation from PPP: 6.11 (p-value 0.015)
- IMF (de jure) Classification:
  - Peg: Average Persistence 30.00; Number of observations 49
  - Flexible: Average Persistence 7.25; Number of observations 35; WMW: 2.73 (p-value 0.006); BF: 0.68 (p-value 0.413)
- Reinhart-Rogoff (de facto) Classification:
  - Peg: Average Persistence 30.00; Number of observations 22
  - Free floating/falling: Average Persistence 5.58; Number of observations 12; WMW: 3.22 (p-value 0.001); BF: 0.67 (p-value 0.418)
- Notes: Column definitions as provided; WMW denotes Wilcoxon-Mann-Whitney test; BF denotes Brown-Forsythe test.

### Table 3 — Rank correlation of half-life of deviation from PPP with country characteristics (All Countries)
- Rate of inflation: Rank correlation -0.281*; Approximate p-value 0.007 (number of observations 90).
- Volatility of the official nominal exchange rate: Rank correlation -0.379*; Approximate p-value 0.002 (number of observations 70).
- Volatility of the parallel-market nominal exchange rate: Rank correlation -0.282*; Approximate p-value 0.02 (number of observations 70).
- Productivity growth: Rank correlation -0.038; Approximate p-value 0.77 (number of observations 90).
- Government spending: Rank correlation -0.073; Approximate p-value 0.50 (number of observations 88).
- Trade openness: Rank correlation 0.180; Approximate p-value 0.10 (number of observations 89).

*Source: Authors’ calculations.

### Appendix I. Column (2): Spearman rank correlation coefficient—the null hypothesis

### Appendix I. Column (2): Spearman rank correlation coefficient—the null hypothesis

### Methodology: Spearman rank correlation test
- The null hypothesis of the Spearman test is that there is no correlation between the (tie-adjusted) rank of each country’s bias-corrected half-life of parity deviation and the rank of its period-average value of the country characteristic.
- The Spearman test uses tie-adjusted ranks for both variables.

### Measurement of persistence
- Persistence of deviations from PPP is calculated as the median-unbiased half-life (in years) of real exchange rate deviations from parity.
- These median-unbiased half-lives are listed in column (3) of Table 1 (as referenced in the source).

### Statistical inference and significance
- Column (3) (of the Appendix table) reports the approximate p-value taken from Zar (1972).
- An asterisk (*) denotes that the null hypothesis of no rank correlation is rejected at the 5 percent level of significance.
- The number of country observations used in each Spearman test is given in parentheses alongside results.

*Source: Appendix I. Column (2): Spearman rank correlation coefficient—the null hypothesis*

---


_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2004/_wp04128.pdf_