## _wp0811

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### Key empirical concepts and equations
- Gibrat’s law: lack of correlation between a variable’s initial stock and its growth rate; when present, indicates proportional growth dynamics. Expressed as "the expected growth rate of reserves is independent of the initial size of reserves."
- Zipf’s law (linear representation): size realizations Si inversely related to ranking Ri with slope α = 1:
  - ln(Ri) = ln(A) – α ln(Si)
- Reported empirical regression (Zipf plot example):
  - y = -0.9877x + 10.43
- Historical attributions:
  - Gibrat (1931); Zipf (1949); link between laws: Gabaix (1999).

### Data and descriptive statistics
- Data source: IMF International Financial Statistics (IFS).
- Table 1 (as presented):
  - Mean2152062548262,9765,7179,07628,015
  - Standard error5342572045731,1191,8068,164
  - Median737691913343536822,367
  - Standard deviation4483565232,1736,61113,56624,030108,921
  - Minimum001011015
  - Maximum3,0991,8104,06214,62248,59278,501219,6481,068,493
  - Sum15,23414,82921,61593,380395,760840,4161,606,3754,986,624
  - Count717285113133147177178
  - Source: International Financial Statistics (Washington: IMF).
- Key descriptive findings:
  - "The mean reserve holdings increased 131 times since 1948 and almost 5 times since 1990."
  - "The maximum value went up during the same period by 355 times, 14 times since 1990."
  - Coverage expanded from "71 countries reporting to the IMF in 1948, to 147 in 1990, to 178 in 2006."
- Tables and Figures inventory (selected):
  - Tables: 1. Descriptive Statistics for International Reserves; 2. Holders of 90 Percent of Total Reserves; 3. Gibrat Coefficients for Reserve Holdings (Scaled by GDP); 4. Gibrat Coefficients for Reserve Holdings (Absolute Sizes); 5. Zipf Coefficients for Reserve Holdings
  - Figures: 1–7 (Gibrat’s Law and Zipf’s Law plots, including WEO-projected reserves 2008–12)

### Main empirical findings
- Reserve accumulation patterns:
  - Recent large accumulation of reserves, especially by developing countries, with a substantial share of this growth taking place in developing economies—most notably in a very rapidly growing China.
  - Most of the reserve assets are issued by the United States.
- Tests of relationship between reserve levels and growth:
  - Testing relationship between level of reserves (scaled by GDP) and reserve growth rates finds weak evidence for the inverse correlation predicted by an optimal-reserve convergence hypothesis.
  - Tests for Gibrat’s law using absolute levels of reserves (not scaled by GDP) find that evidence for its presence is not as significant.
  - Growth processes that follow Gibrat’s law give rise to Zipf’s law; the paper shows Zipf’s law applies to international reserve sizes.
- Interpretation:
  - Lack of statistically significant correlation between reserve growth rates and reserve-to-GDP ratios constitutes an empirical puzzle given GDP's correlation with other reserve-determining variables.
  - The presence of Zipf’s law implies limited convergence of reserves to optimal levels across countries.
  - Robustness of Zipf’s law over time suggests it can be used as a predictor of the distribution of projected international reserves.

### Gibrat’s law — empirical results
- Testing approach:
  - Growth of reserves related to initial size scaled by GDP and to absolute size.
  - Data split between Bretton Woods period and post-Bretton Woods period; initial years used include 1948 (1960 for reserves/GDP), 1971, 1997, and 1971–2006.
- Results for reserves scaled by GDP (Table 3):
  - "The results show that the evidence for existence of a relationship between an initial size of reserves and their subsequent growth (i.e., against Gibrat’s law) is weak."
  - For the population, t-statistics approach conventional levels of significance only for periods 1960–71 and 1997–2006—indicating growth rate may depend on initial period sizes scaled by GDP for those periods.
  - For sub-samples accounting for 75 and 90 percent of total world stock of reserves: "Gibrat’s law holds—the slopes are not significantly different from zero."
  - "All of the equations fit the data poorly."
  - Note: "Only a regression for the full sample is run. Data for GDP are not available for Germany—a holder of the largest share (16 percent) of reserves. Without Germany, the remaining observations account for 100 percent of usable data for all sample selections."
  - Significance annotation: "*** significant at 5 percent."
- Results for absolute reserve sizes (Table 4):
  - "Overall, the results are mixed but tilted more against Gibrat’s law than the results for reserves scaled by GDP."
  - For the population, t-statistics indicate that Gibrat’s law does not hold for all the periods tested—growth rate depends on initial sizes.
  - For 75 and 90 percent sub-samples evidence is mixed:
    - Slopes are not significantly different from zero for 1997–2006.
    - For other periods some results approach conventional (although low) significance levels.
  - "All of the equations fit the data (75, 90, and 100 percent) poorly."
  - Significance annotation in Table 4: "*** significant at 5 percent; ** significant at 10 percent."
- Graphical examples (Figures 1 and 2 — average growth rate on log initial size):
  - 1948/1971 (full sample): y = -0.1101x + 0.7301, R2 = 0.2545
  - 1997/2006 (full sample): y = -0.1407x + 1.2653, R2 = 0.1376
  - 1948/1971 (90 percent): y = -0.0129x + 0.1962, R2 = 0.0072
  - 1971/2006 (90 percent): y = -0.0231x + 0.3092, R2 = 0.1254

### Zipf’s law — empirical results
- Testing approach:
  - Examine distribution of reserve holdings for population and sub-samples; regress log rank on log size (Zipf regressions).
- Summary (Table 5):
  - For the population, "reserve holdings are not distributed according to Zipf’s law."
    - "The slope is less than one for all (Bretton Woods and post-Bretton Woods) years, statistically significant, and the regression fit is very good."
  - For sub-samples covering up to 90 percent of total reserves:
    - "Distribution of reserves for a sample of countries accounting for a share of total reserves as high as 90 percent is fairly well approximated by Zipf’s distribution."
    - For 90 percent samples: "all slopes are very close to 1, very strongly significant, and the fit of regressions is very good (Table 5)."
- Selected numeric snippets from Table 5 (as presented):
  - 19487514-1.1-16.41.0
  - 9028-0.9-27.01.0
  - 10070-0.4-13.70.7
  - 20067517-0.9-24.01.0
  - 9040-0.9-55.21.0
  - 100176-0.4-30.70.8
- Graphical examples (Figures 3–6 — Zipf plots):
  - Full sample examples:
    - 1948 full: y = -0.4032x + 4.9147, R2 = 0.735
  - 90 percent sample examples:
    - 1960 90%: y = -1.1289x + 9.3377, R2 = 0.9829
    - 1997 90%: y = -1.1756x + 14.705, R2 = 0.9806

### Practical application and projections
- Zipf’s law as predictor:
  - "As the evidence of Zipf’s law is robust over time, it can be viewed as a useful predictor of the distribution of projected international reserves going forward, for instance the one embedded in the WEO projections."
  - Suggestion: "Estimates based on an application of Zipf’s law could usefully benchmark projections of reserve accumulation for countries accounting for 90 percent of the world reserve stock."
- Comparison with WEO projections (Figure 7):
  - The paper plots Zipf’s law for reserves in 2008, 2010, and 2012 and finds projections progressively diverge from Zipf-implied distribution.
  - Reported slopes for 90 percent of total reserves:
    - 2008 slope is "0.86—lower than the estimated slope of 0.92 for 2006."
    - 2010 slope "0.82."
    - 2012 slope "0.79."
  - Fitted lines (from WEO plots):
    - 2008: y = -0.8453x + 12.296, R2 = 0.9959
    - 2010: y = -0.8119x + 12.053, R2 = 0.9907
    - 2012: y = -0.7943x + 17.446, R2 = 0.986

### Policy-relevant implications and recommendations
- Continued value of:
  - Reserve management funds.
  - Collective insurance arrangements (such as IMF), to spread perceived benefits from high reserve holdings by a few countries to a larger number of countries.
- Practical use:
  - Use of Zipf’s law as a useful predictor for the distribution of projected international reserves going forward; particularly as a benchmark for countries accounting for 90 percent of world reserves.

### Conclusion
- Core conclusion: "I showed that [international reserve holdings] corresponds reasonably well to Zipf’s law and Gibrat’s law, both of which apply to size distributions of cities and countries (when ranked by population)."
- Framing: "This result is presented as an empirical puzzle and informed speculations as to what kind of process might be responsible for this phenomenon are left for future work."

*Source: _wp0811 - Section VII concludes.*

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

### _wp0811 - References

### Tables and Figures inventory
- Tables:
  - 1. Descriptive Statistics for International Reserves
  - 2. Holders of 90 Percent of Total Reserves
  - 3. Gibrat Coefficients for Reserve Holdings (Scaled by GDP)
  - 4. Gibrat Coefficients for Reserve Holdings (Absolute Sizes)
  - 5. Zipf Coefficients for Reserve Holdings
- Figures:
  - 1. Gibrat’s Law for Reserves—Full Sample Average Growth Rate on Log Initial Size—Initial Year/Final Year
  - 2. Gibrat’s Law for 90 Percent of Reserves Average Growth Rate on Log Initial Size—Initial Year/Final Year
  - 3. Zipf’s Law for Reserves—Full Sample Log Rank on Log Size, 1948–71
  - 4. Zipf’s Law for Reserves—Full Sample Log Rank on Log Size, 1980–2006
  - 5. Zipf’s Law for 90 Percent of Reserves—Log Rank on Log Size, 1948–71
  - 6. Zipf’s Law for 90 Percent of Reserves—Log Rank on Log Size, 1980–2006
  - 7. Zipf’s Law for 90 Percent of WEO Projected Reserves—Log Rank on Log Size, 2008–12

### Key empirical concepts and equations
- Gibrat’s law: lack of correlation between a variable’s initial stock and its growth rate; when present, indicates proportional growth dynamics.
- Zipf’s law (linear representation): size realizations Si inversely related to ranking Ri with slope α = 1:
  - ln(Ri) = ln(A) – α ln(Si)
- Reported empirical regression (Zipf plot example):
  - y = -0.9877x + 10.43

### Main findings and empirical results (as presented)
- Reserve accumulation patterns:
  - Recent large accumulation of reserves, especially by developing countries, with a substantial share of this growth taking place in developing economies—most notably in a very rapidly growing China.
  - Most of the reserve assets are issued by the United States.
- Empirical tests:
  - Testing relationship between level of reserves (scaled by GDP) and reserve growth rates finds weak evidence for the inverse correlation predicted by an optimal-reserve convergence hypothesis.
  - Tests for Gibrat’s law using absolute levels of reserves (not scaled by GDP) find that evidence for its presence is not as significant.
  - Growth processes that follow Gibrat’s law give rise to Zipf’s law; the paper shows Zipf’s law applies to international reserve sizes.
- Interpretation and implications:
  - Lack of statistically significant correlation between reserve growth rates and reserve-to-GDP ratios constitutes an empirical puzzle given GDP's correlation with other reserve-determining variables.
  - The presence of Zipf’s law implies limited convergence of reserves to optimal levels across countries.
  - Robustness of Zipf’s law over time suggests it can be used as a predictor of the distribution of projected international reserves (for example, WEO projections).

### Policy-relevant implications and recommendations (as presented)
- Continued value of:
  - Reserve management funds.
  - Collective insurance arrangements (such as IMF), to spread perceived benefits from high reserve holdings by a few countries to a larger number of countries.
- Practical application:
  - Use of Zipf’s law as a useful predictor for the distribution of projected international reserves going forward.

### Notable quotations and illustrative remarks (verbatim excerpts)
- Lawrence H. Summers: "with respect to the use of reserves in crisis prevention, that is, reducing the probability of a crisis, it is hard to demonstrate conclusively that there are any statistically significant benefits at all (Jeanne, 2007)."
- Richard Cooper (paraphrase in general discussion): reminded the panel that he had never met an official who thought in terms of optimal reserves; "it should not be surprising, then, that reserves accumulation in many developing economies is not in accord with the normative prescriptions of [...] [the] model (Jeanne, 2007)."
- On optimal level uncertainty (Jeanne, 2007, quoted): "depending upon certain assumptions that were difficult to pin down, the optimal level was somewhere between $20 billion and $2 trillion."

*Source: _wp0811 - References*

### Section VII concludes.

### _wp0811 - Section VII concludes.

### Data
- Table 1 includes descriptive statistics of the data—international reserve excluding gold—used in this paper. The data are sourced from the IMF International Financial Statistics (IFS).
- Table 1 (as presented):
  - Mean2152062548262,9765,7179,07628,015
  - Standard error5342572045731,1191,8068,164
  - Median737691913343536822,367
  - Standard deviation4483565232,1736,61113,56624,030108,921
  - Minimum001011015
  - Maximum3,0991,8104,06214,62248,59278,501219,6481,068,493
  - Sum15,23414,82921,61593,380395,760840,4161,606,3754,986,624
  - Count717285113133147177178
  - Source: International Financial Statistics (Washington: IMF).
- Key descriptive findings:
  - "The mean reserve holdings increased 131 times since 1948 and almost 5 times since 1990."
  - "The maximum value went up during the same period by 355 times, 14 times since 1990."
  - Coverage expanded from "71 countries reporting to the IMF in 1948, to 147 in 1990, to 178 in 2006."

### The Laws
- Gibrat’s law: "the expected growth rate of reserves is independent of the initial size of reserves."
  - Historical attribution: Robert Gibrat (1931).
  - Prior applications cited: Sutton (1997); Delgado and Godinho (2004); Rose (2007).
- Zipf’s law: "the number of occurrences of a certain variable greater than S is approximately proportional to 1/S." (exponent -1 on S; generalized with rank proportional to S^-α).
  - Historical attribution: George Kingsley Zipf (Zipf, 1949).
- Link between laws: "Gabaix (1999) shows that there is a strong link between Gibrat’s law and Zipf’s law."
- Empirical sampling note: when the largest possible sample (population) is included the laws hold only weakly; evidence is stronger for samples limited to the largest subjects accounting for the majority of population. To control for this, the paper estimates laws for sub-samples corresponding to 75, 90, and 100 percent coverage of total reserves.

### Reserves Size and Rank: Gibrat’s Law
- Testing approach:
  - Growth of reserves related to initial size scaled by GDP and to absolute size.
  - Data split between Bretton Woods period and post-Bretton Woods period.
  - Initial years used: 1948 (1960 for reserves/GDP due to GDP data limits), 1971 (collapse of Bretton Woods), 1997 (Asian crisis), and 1971 for whole post-Bretton Woods (till 2006).
- Findings using reserves scaled by GDP (Table 3):
  - "The results show that the evidence for existence of a relationship between an initial size of reserves and their subsequent growth (i.e., against Gibrat’s law) is weak."
  - For the population, t-statistics approach conventional levels of significance only for periods 1960–71 and 1997–2006—indicating growth rate may depend on initial period sizes scaled by GDP for those periods.
  - Two possible explanations posited: (1) a genuine relationship exists between reserves growth and initial stock; (2) peculiarities of the data—the lower tail of the population is not diversified enough.
  - For sub-samples accounting for 75 and 90 percent of total world stock of reserves: "Gibrat’s law holds—the slopes are not significantly different from zero."
  - "All of the equations fit the data poorly."
  - Note: "Only a regression for the full sample is run. Data for GDP are not available for Germany—a holder of the largest share (16 percent) of reserves. Without Germany, the remaining observations account for 100 percent of usable data for all sample selections."
  - Significance annotation: "*** significant at 5 percent."
- Findings using absolute sizes (Table 4):
  - "Overall, the results are mixed but tilted more against Gibrat’s law than the results for reserves scaled by GDP."
  - For the population, t-statistics indicate that Gibrat’s law does not hold for all the periods tested—growth rate depends on initial sizes.
  - For sub-samples (75 and 90 percent) evidence is mixed:
    - Slopes are not significantly different from zero for 1997–2006.
    - For other periods some results approach conventional (although low) significance levels.
  - "All of the equations fit the data (75, 90, and 100 percent) poorly."
  - Significance annotation in Table 4: "*** significant at 5 percent; ** significant at 10 percent."
- Graphical evidence:
  - Figures 1 and 2 present scatterplots and fitted lines for average growth rate on log initial size for full sample (Figure 1) and 90 percent sample (Figure 2). Examples of fitted lines and R2 in figures:
    - 1948/1971 (full sample): y = -0.1101x + 0.7301, R2 = 0.2545
    - 1997/2006 (full sample): y = -0.1407x + 1.2653, R2 = 0.1376
    - 1948/1971 (90 percent): y = -0.0129x + 0.1962, R2 = 0.0072
    - 1971/2006 (90 percent): y = -0.0231x + 0.3092, R2 = 0.1254

### Reserves Size and Rank: Zipf’s Law
- Testing approach:
  - Examine distribution of reserve holdings for population and sub-samples; check log rank on log size (Zipf regressions).
- Table 5 summary and main findings:
  - For the population, "reserve holdings are not distributed according to Zipf’s law."
    - "The slope is less than one for all (Bretton Woods and post-Bretton Woods) years, statistically significant, and the regression fit is very good."
  - For sub-samples covering up to 90 percent of total reserves:
    - "Distribution of reserves for a sample of countries accounting for a share of total reserves as high as 90 percent is fairly well approximated by Zipf’s distribution."
    - For 90 percent samples: "all slopes are very close to 1, very strongly significant, and the fit of regressions is very good (Table 5)."
- Selected numeric entries from Table 5 (illustrative rows as presented):
  - 19487514-1.1-16.41.0
  - 9028-0.9-27.01.0
  - 10070-0.4-13.70.7
  - 20067517-0.9-24.01.0
  - 9040-0.9-55.21.0
  - 100176-0.4-30.70.8
  - (Table 5 lists year / percent of total reserves / sample / slope / T-statistics / R2 for multiple years and sample coverages.)
- Graphical evidence:
  - Figures 3–6 present Zipf plots:
    - Full sample (Figures 3 and 4) show slopes substantially less than 1 but high R2 (e.g., 1948 full: y = -0.4032x + 4.9147, R2 = 0.735).
    - 90 percent sample (Figures 5 and 6) show slopes near -1 with very high R2 (e.g., 1960 90%: y = -1.1289x + 9.3377, R2 = 0.9829; 1997 90%: y = -1.1756x + 14.705, R2 = 0.9806).

### Practical Application
- Zipf’s law as predictor:
  - "As the evidence of Zipf’s law is robust over time, it can be viewed as a useful predictor of the distribution of projected international reserves going forward, for instance the one embedded in the WEO projections."
  - Application suggestion: "Estimates based on an application of Zipf’s law could usefully benchmark projections of reserve accumulation for countries accounting for 90 percent of the world reserve stock."
- Comparison to WEO projections:
  - The paper plots Zipf’s law for reserves in 2008, 2010, and 2012 (Figure 7) and finds projections progressively diverge from Zipf-implied distribution.
  - Reported slopes:
    - For 90 percent of total reserves: 2008 slope is "0.86—lower than the estimated slope of 0.92 for 2006."
    - 2010 slope "0.82."
    - 2012 slope "0.79."
  - Figure 7 fitted lines (from WEO plots):
    - 2008: y = -0.8453x + 12.296, R2 = 0.9959
    - 2010: y = -0.8119x + 12.053, R2 = 0.9907
    - 2012: y = -0.7943x + 17.446, R2 = 0.986

### Conclusion
- Core conclusion: "I showed that [international reserve holdings] corresponds reasonably well to Zipf’s law and Gibrat’s law, both of which apply to size distributions of cities and countries (when ranked by population)."
- Framing: "This result is presented as an empirical puzzle and informed speculations as to what kind of process might be responsible for this phenomenon are left for future work."

*Source: _wp0811 - Section VII concludes.*

### REFERENCES

### REFERENCES

### Works on international reserves and reserve adequacy
- Ben-Bassat, Avraham, and Daniel Gottlieb, 1992, “On the Effect of Opportunity Cost on International Reserve Holdings,” Review of Economics and Statistics, Vol. 74 (May), pp. 329–32.
- Bird, Graham, and Ramkishen S. Rajan, 2003, “Too Much of a Good Thing? The Adequacy of International Reserves in the Aftermath of Crises,” World Economy, Vol. 26 (June), pp. 873–91.
- Beaufort Wijnholds, J.A.H. de, and Arie Kapteyn, 2001, “Reserve Adequacy in Emerging Market Economies,” IMF Working Paper 01/143 (Washington: International Monetary Fund).
- Jadresic, Esteban, 2007, “The Cost-Benefit Approach to Reserve Adequacy: the Case of Chile” (mimeo; Central Bank of Chile).
- Jeanne, Olivier, 2007, International Reserves in Emerging Market Countries: Too Much of a Good Thing?, Brookings Papers on Economic Activity:1, Brookings Institution, pp. 1–79.
- Summers, Lawrence H., 2006, “Reflections on Global Account Imbalances and Emerging Markets Reserve Accumulation,” paper presented at L.K. Jha Memorial Lecture, Reserve Bank of India, Mumbai, India, March. Available via the Internet: http://www.president.harvard.edu/speeches/2006/0324_rbi.html.

### Works on city, firm size distributions, and related empirical regularities
- Axtell, Robert L., 2001, “Zipf Distributions o f U.S. Firm Sizes,” Science, Vol. 293, pp. 1818–20.
- Delgado, Ana Paula, and Isabel Maria Godinho, 2004, “The Evolution of City Size Distribution in Portugal: 1864–2001,” paper presented at the World Conference of International Regional Science Association, Porto, Portugal, April.
- Gabaix, Xavier, 1999, “Zipf’s Law for Cities: An Explanation,” The Quarterly Journal of Economics, Vol. 114 (August), pp. 739–67.
- Krugman, Paul, 1996, “Confronting the Mystery of Urban Hierarchy,” Journal of the Japanese and International Economies, Vol. 10 (December), pp. 399–418.
- Rose, Andrew, 2005, “Cities and Countries,” NBER Working Paper No. 11762 (Cambridge, Massachusetts: National Bureau of Economic Research).
- Sutton, John, 1997, “Gibrat’s Legacy,” Journal of Economic Literature, Vol. 35 (March), pp. 40–59.
- Zipf, G.K., 1949, “Human Behavior and the Principle of Least Effort” (Cambridge, Massachusetts: Addison-Wesley Press).

*Source: _wp0811 - REFERENCES*

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