External Review of the Quota Formulas - Statistical Appendix (Part A)
Author:
International Monetary Fund
Search for other papers by International Monetary Fund in
Current site
Google Scholar Close
Publication Date:
05 Jan 2000
eISBN:
9781498327282
Language:
English
Keywords:
PP; regression No.; moving average; financial market; dummy variable; linear equation; short-term debt; significance level; market price; current account
Download PDF (2 MB)

External Review of the Quota Formulas - Statistical Appendix (Part A)

Abstract

External Review of the Quota Formulas - Statistical Appendix (Part A)

Section I. Existing And Alternative Data Sets For Use In Quota Formulas

A. Introduction

1. This statistical appendix presents the data used in the estimation and analysis of alternative quota formulas suggested by the QFRG. Annual data up to 1994 have been compiled for 183 countries who participated in the Eleventh Quota Review to form the statistical database used in the estimation of alternative quota formulas. The IMF International Financial Statistics, and Balance of Payments databases comprise the major sources of data, supplemented by the IMF World Economic Outlook and IMF staff estimates. All data have been converted, where applicable, into millions of SDRs. A general description of the data, organized by major economic concepts, is presented below. This is followed by statistical data tables showing data for individual countries, sorted by actual (proposed) Eleventh Review quotas in descending order, along with a correlation matrix of certain variables.

B. Eleventh Review of Quotas

2. The dependent variable (Q) used in all quota formula estimations is the proposed or actual quota from the Eleventh Quota Review. Please refer to Financial Organization and Operations of the IMF, Pamphlet Series No. 45, Fifth Edition, pages 24-27, for an overview of the methodology used in the determination of quotas. The then-existing quota (QL) would be, in most cases, the proposed or actual quota from the Ninth Review. The calculated quota (CQ) is derived as the higher of the results of: (1) the reduced Bretton Woods formula and (2) the average of the lowest two results of four other formulas containing the same variables but with larger weights, for external trade and variability of exports.

C. Measures of Output and Population

3. Five different measures of GDP/GNP were considered in the estimation of quota formulas. The "standard" GDP measure used in the Eleventh Quota Review is 1994 GDP (Y) measured at market exchange rates. Alternative national output measures include:

  • Five-year (1990-94) average of GDP (YAVG),

  • GDP valued at purchasing power parities (PPP) in a recent year (1994),

  • Five-year (1990-94) average of GDP using a centered, five-year moving average of annual exchange rates as a conversion factor (YM5X), and

  • GNP converted using the World Bank Atlas Method (YATL).

4. The major source for GDP and exchange rate data is the IMF International Financial Statistics database. The data source for PPP-based GDP is the IMF World Economic Outlook database, and the World Bank's Global Development Indicators database is the source for 1994 GNP converted using the World Bank Atlas Method. In cases where there is a lack of reliable alternative measures of GDP, the 1994 GDP (Y) measure is usually substituted. Population (POP) is a country's 1994 population measured in millions of persons.

D. Measures of Reserves and Current Account Data

5. The standard reserves variable (R) used in the Eleventh Quota Review is a twelvemonth average of gold (valued at SDR 35 per fine ounce) and foreign exchange reserves, including SDR holdings, reserve positions in the Fund, and ECUs for 1994. An alternative measure used is reserves with gold measured at market prices (RESM). This is computed as a twelve-month average of gold (valued at market prices in 1994, with monthly prices ranging from SDR 261.02 per fine ounce to SDR 273.82 per fine ounce) and foreign exchange reserves, including holdings, reserve position in the Fund and ECUs for 1994.

6. Current Receipts (C) is the 1990-94 average of the sum of goods (exports f.o.b.), services (credit), income (credit), and private current transfers (credit) divided by the average SDR value for the same years. Current Payments (P) is the 1990-94 average of the sum of goods (imports, f.o.b.), services (debit), income (debit), and private current transfers (debit); divided by the average SDR value for the same years. Trade (TRADE) is measured as the average of current receipts and current payments.

E. Variability Measures, Capital Flows, and Debt

7. Several measures of variability were considered in the estimation of alternative quota formulas. The standard variability measure in the Eleventh Review uses the variability of current receipts (VC) which is defined as one standard deviation of current receipts from its five-year moving average centered on the third year, for the period 1982-94. Another measure captures the variability of both current receipts and capital and financial account credits (VCK), also defined as one standard deviation from its five-year moving average centered on the third year, for the period 1982-94. The variability of real effective exchange rates (VREC) (Source: IMF Information Notice System) is defined in terms of the deviation of the real effective exchange rate from a normal level, represented by a five-year moving average centered on the middle year.

8. Normal net capital flow is proxied by a four-year average of net private capital flows and includes errors and omissions for the period 1991-94.

9. Total external debt (DEBT) is defined as debt owed to non-residents repayable in foreign currency, goods, and services. Short-term external debt (STDEBT) for most developing countries was obtained from the BIS-IMF-OECD-World Bank (BIOW) database. The World Bank's Global Development Finance (GDF) database was used for developing countries for which data were not available from the BIOW. For the industrial countries, the Bank for International Settlements database was used.

F. Qualitative and Dummy Variables

10. A.capital market accessibility (KMACC) classification was used to derive an Openness Index. This classification (KMACC) ranks each country's ability to access capital. markets on a 1 to 4 scale, with a 1 given to countries with the easiest access, and a 4 given to those with the least access. The Openness Index (OPEN) has the reverse scale (computed as 1+(5-KMACC)).

11. Some dummy variables were not used directly as explanatory variables but were created in order to select sub-samples for estimation. NOTBW is a 0-1 dummy variable, which equals 1 if a country's calculated quota is based on a variant of the Bretton Woods Formula. Similarly, MEM20 equals 1 if the member joined the Fund in the past twenty years. Also, DDEV is a 0-1 dummy variable, which equals 1 if a country is classified as either a developing country or a transitional economy.

12. According to the WEO country classifications, advanced economies include the industrial countries of North America and Europe, Japan, and two newly industrialized Asian economies (Korea and Singapore). The countries in transition include the 15 members that were formerly part of the Soviet Union, the successor countries to the former Yugoslavia and Czechoslovakia, Albania, Hungary, Poland, Romania, and Mongolia. The rest of the members are classified as developing countries. Furthermore, San Marino, Palau, and the Marshall Islands are not currently classified by the WEO. We have classified San Marino as an industrial country, and Palau and Marshall Islands as developing countries.

13. The next section presents statistical data tables for the 183 countries that participated in the Eleventh Review of Quotas, organized along the same economic concepts presented above.

Table I.1.

Eleventh Review Quotas

article image
article image
article image
article image
article image
article image
Table I.2.

Measures of Output

article image
article image
article image
article image
article image
article image

Source; IMF World Economic Outlook database.

Average of 1990-93 GDP.

19:94 GDP.

1995 data.

Table I.3.

Measures of Output and Population

article image
article image
article image
article image
article image
article image
article image

Source: 1999 World Bank Development Indicators data base.

1994 GDP.

1995 data.

Table I.4.

Measures of Reserves

article image
article image
article image
article image
article image
article image

Source: International Financial Statistics; twelve-month average of gold (valued at SDR 35 per fine ounce) and foreign exchange reserves, including SDR holdings, reserve positions in the Fund, and ECUs for 1994.

Source: International Financial Statistics; twelve-month average of gold (valued at market prices in 1994, with monthly prices ranging from SDR 261.02 per fine ounce to SDR 273.82 per fine ounce) and foreign exchange reserves, including SDR holdings, reserve position in the Fund and ECUs, for 1994.

Reserves are calculated by subtracting gold reserves times $35 and adding gold reserves times the average market price (in SDRs) for 1994 to the reserves data shown in the first column.

Same as reserves shown in first column.

Table I.5.

Current Account Data

article image
article image
article image
article image
article image
article image

Source: Balance of Payments database; 1990-94 average of the sum of goods(exports f.o.b.), services(credit), income(credit), and private current transfers(credit) divided by the average SDR value for the same years.

Source: Balance of Payments database; 1990-94 average of the sum of goods(imports f.o.b.), services(debit), income(debit), and private current transfers(debit); divided by the average SDR value for the same years.

Table I.6.

Variability Measures

article image
article image
article image
article image
article image
article image

Defined as one standard deviation of current receipts from its five-year moving average centered on the third year, for the period 1982-94.

Variability of current receipts plus capital and financial account credit; defined as one standard deviation from the five-year moving average centered on the third year, for the period 1982-94

Source: IMF Information Notice System (1982-94). Defined in terms of the deviation of the real effective exchange rate from a normal level, represented by a five-year moving average centered on the middle year. It is calculated as one standard deviation of the data from the normal level thus defined. The index shown is equal to I + V/A where V=variability and A=average value of the real effective exchange rate over 1982-94, The index is 1.0 for countries that do not have real effective exchange rate data.

The variability of capital account receipts for each of the former Soviet Union Republics is calculated as a share of the total variability for the group of the 15 republics, with weights being the ratio of each republic's capital and financial account.

Simple standard deviation of estimated capital account receipts based on the average ratio of capital and financial account debit to current account payments for the group of net debtor countries with diversified financing.

Simple standard deviation of estimated capital account receipts based on the average ratio of capital and financial account debit to current account payments for the group of net debtor countries with official financing

The average covariance between current and capital account receipts for the countries with available data in the Net Debtor developing country group with diversified financing is applied to the formula for the variance of current and capital account receipts to obtain the variability of external receipts.

The average covariance between current and capital account receipts for the countries with available data in the Net Debtor developing country group with official financing is applied to the formula for the variance of current and capital account receipts to obtain the variability of external receipts.

Table I.7.

Capital Flows and Debt

article image
article image
article image
article image
article image
article image

Proxied by a four-year average of net private capital flows and including errors and omissions for the period 1991-94.

Source: 1996 World Bank World Development Indicators; defined as debt owed to non-residents repayable in foreign currency, goods, and services. It is the sum of public, publicly guaranteed, and private non-guaranteed debt, use of IMF credit, and short-term debt.

Sources: For Industrial Countries, the Bank of International Settlement database; for Developing countries, the Joint BIS-IMF-OECD-World Bank Statistics on External Debt, except where noted.

Source: World Bank Global Development Finance database.

Table I.8.

Qualitative Variables

article image
article image
article image
article image
article image
article image

Source: World Economic Outlook; based on WEO classifications where countries with the most access to capital markets are classified as "1" while those with the least access are classified as "4". This suggested classification corresponds approximately with those used in EB/CQuota/94/2, except to reflect recent (since the early 1990s) changes in some countries' abilities to access the market; and a reclassification of Canada and Italy to class "1". The variable was used to compute the openness index shown in the next column.

Source: IMF Secretary's Department, Records Division. Membership of the International Monetary Fund (Date of entry into force: December 27. 1945), December 16, 1997.

Table I.9.

Correlation of Variables (all members)

article image

Section II. Estimation Results Of Statistical Quota Formulas

This section presents the results of regressions requested by QFRG and analyzed in Chapter V of the report. Detailed results of the regression equations and the resulting quota distributions for members are presented in Statistical Appendix, Part B, Section I.

A. List of Regression Equations

1. Re-estimated Bretton Woods Formula for the Whole Membership.

2. Re-estimated Bretton Woods Formula Using PPP-based GDP Instead of GDP at Market Exchange Rates.

3. Linear Bretton Woods Formula With PPP-based GDP Replacing GDP at Market Exchange Rates.

4. Re-estimated Bretton Woods Formula With a Multiplicative Term, Which Includes a Dummy Variable Distinguishing Between Industrial and Developing Countries.

5. Re-estimated Bretton Woods (BW) Formula for Members with Calculated Quotas Based on the Variants of the BW Formula.

6. Re-estimated Bretton Woods Formula for Members Representing Developing Countries.

7. Re-estimated Bretton Woods Formula for Members with Actual Quota Shares Equal to or Less Than 1 Percent.

8. Re-estimated Bretton Woods Formula for Members Who Joined in the Past Twenty Years.

9. Linear Bretton Woods Formula Without the Multiplicative Factor.

10. Linear Bretton Woods Formula With Current Receipts.

11. Linear Bretton Woods Formula with an Openness Index.

12. Nonlinear Bretton Woods Formula with an Openness Index.

13. Nonlinear Bretton Woods Formula Without the Reserves Variable.

14. Nonlinear Bretton Woods Formula With Gold Reserves Valued at Market Prices.

15. Nonlinear Bretton Woods Formula With a Five Year Average of GDP Replacing the Existing One-year GDP.

16. Nonlinear Bretton Woods Formula With Population.

17. Nonlinear Bretton Woods Formula with Short Term Debt.

18. Nonlinear Bretton Woods Formula With the Vanability of External Receipts Replacing the Variability of Current Receipts.

19. Nonlinear Bretton Woods Formula With the Then-Existing Quota As a Multiplicative Explanatory Variable.

20. Nonlinear Bretton Woods Formula with the Then-Existing Quota As an Additive Explanatory Variable.

21. Regression of Actual Quotas on Variables Indicative of Ability to Contribute Financial Resources to the Fund.

22. Nonlinear Bretton Woods Formula with a Five-Year Average of GDP, where the Conversion Factors are Centered Five-Year Moving Averages of the Annual Exchange Rates, Replacing the Existing One-Year GDP.

23. Nonlinear Bretton Woods Formula with GNP Converted with the World Bank Atlas Method.

24. Linear Formula with the Then-Existing Quota, Short-Term Debt, Population, and Trade added, and Reserves and Current Payments dropped.

25. Nonlinear Bretton Woods Formula with the Then-Existing Quota As an Additive Explanatory Variable for Countries with Calculated Quotas Based on the Variants of the Bretton Woods Formula.

26. Nested Model Where a Regression of Vulnerability Variables (Represented by the Variability of Current Receipts and Population) is Estimated First.

27. Nested Model Where a Regression of Strength Variables is Estimated First.

28. Linear Estimation of Both Strength and Vulnerability Variables.

29. Re-estimated Bretton Woods Formula With Normal Net Capital Flows as an Additional Variable.

30. Re-estimated Bretton Woods Formula With Real Effective Exchange Rate Variability Times Current Receipts as an Additional Variable.

31. Re-estimated Bretton Woods Formula With Debt as an Additional Variable.

32. Members with Quota Shares of Equal to or Less Than 1.0 Percent.

Re-estimated Bretton Woods Formula With Normal Net Capital Flows as an Additional Variable.

33. Members with Quota Shares of Equal to or Less Than 1.0 Percent.

Re-estimated Bretton Woods Formula With Real Effective Exchange Rate Variability Times Current Receipts as an Additional Variable.

34. Members with Quota Shares of Equal to or Less Than 1.0 Percent.

Re-estimated Bretton Woods Formula With Financial Market Accessibility Times Current Payments as an Additional Variable.

35. Members with Quota Shares of Equal to or Less Than 1.0 Percent. Re-estimated Bretton Woods Formula With Debt as an Additional Variable.

36. Re-estimated Bretton Woods Formula With Financial Market Accessibility Times Current Payments as an Additional Variable.

37. Bretton Woods Formula for Schedule A Members Using 1934-43 Data.

Table II.1

Summary Statistics of Equations Fitted to Actual Quotas

(T-ratios in parentheses)

article image
article image
article image
article image
Note: The variables in the equations are defined as follows: Q = estimated quota, in millions of SDRs Y = GDP at market exchange rates in 1994 R = average monthly reserves in 1994 P = annual average current payments overthe 1991-1994 (five years) period C = annual average current receipts overthe 1991-1994 (five years) period VC = variability of current receipts, defined as one standard deviation from a five-year moving average over the 1982-1994(13 years) period VREC = real effective exchange rate variability times current receipts POP = population in 1994 DDEV = dummy variable distinguishing between industrial and developing countries. It is equal to 1 if a country is a developing or a transitional economy NCF = normal capital flow proxied by a four-year moving average of actual net private capital flows (inclusive of errors and omissions) DEBT = total external debt owed to non-residents repayable in foreign currency, goods and services. It is the sum of the public, publicly guaranteed and private non-guaranted debt, use of IMF credit, and short-temi debt YM5X = is the five-year average of GDP from 1990 to 1994 where the conversion factor is a centered five-year moving average of the annual exchange rate STDEBT = short term debt at the end of 1994 OPEN = openness index defined as 1 +(5 - KMACC); KMACC = capital market accessibility, which is based on the WEO classification RM = average monthly reserves with gold valued at market rates in 1994 NNKFL = four-year moving average of net private capital flows YPPP = PPP-based GDP in 1994 YAVG = five year averages of GDP from 1990-94 VCK = variability of external receipts (the sum of current receipts and capital and financial account credits), defined as one standard deviation from a five-year moving average over the 1982-1994 (13 years) period YATL = 1994 GNP converted using the World Bank Atlas Method QL = the then-existing quota FMP = finacial market accessibility times current payments; financial market accessibility is proxiedby a variable wluch takes values of4 for developmg countries wim limited access to private financial markets, 3 for the rest of developing countries, 2 for industrial countries with easy access to borrowing, and 1 for France, Germany, Japan, the United Kingdom, and the United States TRADE = average of current payments and receipts over a recent five year period (1991-94)

As numbered in the List of Regression Equations of this section.

Table II.2.

Relative Contributions of Variables to Calculated Quotas 1/

(In percent)

article image
article image
article image
article image
Note; the variables in the equations are oeiined in tableII.l.

The relative contribution of a variable to the calculated quota is the ratio between the variable times its coefficient and the members' calculated quota. The contribution of the ratio of current receipts to GDP is the contribution of the nonlinear element to the calculated quotas, i.e., the extent to which the calculated quota is revised by the application of the multiplier (unity plus the ratio of current receipts to GDP).

As numbered in the List of Regression Equations in the beginning of this section.

It includes the contribution of the dummy variable.

Table II.3

Estimated Quota Shares by WEO Classification 1/

(In percent)

article image
article image
article image
article image

According to the WEO country classifications, advanced economies include the industrial countries of North America and Europe, Japan, and two newly industrialized Asian economies (Korea and Singapore). The countries in transition include the 15 members that were formerly part of the Soviet Union, the successor countries to the former Yugoslavia and Czechoslovakia, Albania, Hungary, Poland, Romania, and Mongolia. The rest of the members are classified as developing countries.

As numbered in the List of Regression Equations in the beginning of this section.

Section III. Chow and Wald Tests for the Stability of Coefficient Estimates

This section presents the results of Wald and Chow tests for the stability of the estimated coefficients of the Bretton Woods formula, using data from the Sixth to the Eleventh Reviews, including explanatory notes.

Part A presents the Chow test results on the stability of the coefficient estimates of the Bretton Woods formula, using data from the Sixth to the Eleventh Reviews, including the methodology used in performing the Chow tests.

The results of these statistical tests suggest instability in the coefficients of the Bretton Woods formula. The best result seems to come from the pair-wise test of the Seventh and Eighth Reviews, which could perhaps be attributed to two factors: (1) the short time period in between these reviews because the Eighth Review was accelerated; and (2) the relatively large selective element of the Eighth Review, which allows the underlying economic variables to have a somewhat greater influence on the outcome of the Eighth Review.

Part B presents the results of tests for the stability of the coefficient estimates of the Bretton Woods formula, using data from the Sixth to the Eleventh Reviews, while taking into account the systematic tendency of actual quotas to fall over time in relation to GDP or external trade.

The significance of the tendency of actual quotas to fall in relation to GDP or external trade over the Sixth to the Eleventh Reviews is examined by pair-wise testing (F-tests) of whether the variances of the error terms from the estimated Bretton Woods formula over these reviews are equal. If the error variance is the same over a pair of reviews, then the Chow test remains an appropriate test. If not, we need to use other statistics, like the Wald test.

The formal tests for the equality of variances between reviews show that the error variances for the rolling (or pair-wise comparisons of) Sixth and Seventh, Seventh and Eighth, Ninth and Tenth, and Tenth and Eleventh Reviews are statistically different. This implies that the Chow test for these pairs is inappropriate. Nonetheless, the alternative Wald test indicates that the coefficients generated by the Bretton Woods formula are not stable for these pair-wise comparisons of quota review periods. For the pair of the Eighth and Ninth Reviews, however, the results of the Chow test in Part A are valid.

In sum, regardless of whether we can use the Chow test or have to use the Wald test, the statistical tests suggest instability of the coefficient estimates of the Bretton Woods formula. A detailed list of the regressions performed is presented in Statistical Appendix, Part B, Section II.

A. Chow Tests

1. To test the stability of coefficients in the reduced Bretton Woods formula over the Sixth to Eleventh Reviews, we used the Chow test for (1) rolling consecutive pairs of reviews, starting with the Sixth Review (Table III. 1.1), and (2) cumulative reviews, starting with the Sixth Review (Table III. 1.2).

2. To perform the Chow tests, we used a constant-membership sample of 121 members participating in the Sixth Review plus China. To test, for example, the stability of the coefficient estimates under the Sixth and Seventh Reviews, we run the following regressions:

article image

If al= a2   bl= b2 cl'— c2 d1= cf, then we can estimate a common relationship for the entire (pooled) data, i.e.

article image

These four linear restrictions on a, 6, c, and d can be tested using the F test. The F test is

F = ( S S R p − S S R s ) / ( 4 + 1 ) S S R s / ( 122 + 122 − 2 x 4 − 2 )

where

F = ( S S R p − S S R s ) / ( 4 + 1 ) S S R s / ( 122 + 122 − 2 x 4 − 2 )

which has an F distribution with degrees of freedom (4 + 1), (122 + 122 -2x4-2).

3. If the F value is less than the critical value of 2.25 from the F tables at the 5 percent significance level, i.e., the calculated F value is not significant at the 5 percent level, we do not reject the null hypothesis that the relationship is stable (see Madala, G. S., Econometrics, 1977, McGrow-Hill, pp. 198-199).

4. The results shown in Tables III.1.1 and III. 1.2 indicate, at the 5 percent significance level, rejection of the hypothesis that the coefficients of the Bretton Woods formula are stable over time.

Table III. 1.1 :

Chow Tests (Rolling)

article image
Note: The asterisk indicates significance of the 5% significance level.
Table III. 1.2 :

Chow Tests (Cumulative)

article image
Note: The asterisk indicates significance of the 5% significance level.

B. Wald Tests: Tests of Structural Change with Unequal Variances

In using the Chow test, an important assumption made is that the error variance is the same in

all regressions. If this is not true, the error variance for one quota review period is σ12, while

that for the next review period is σ22, and so on, in the restricted (two-reviews combined) model. The restricted model is, therefore, heteroscedastic, and the results from applying the non-linear Bretton Woods formula to such a model present problems of statistical inference.1-In this case, it has been argued that it is likely that we overestimate the significance level of our test statistic.2 In other words, the calculated F statistic is biased upward and indicates greater instability in the coefficient estimates than in fact exists.

To deal with this problem, we estimate all separate regressions and examine the estimates of the error variances. To test for significant differences, we use pairwise F-tests.3 Without any significant difference, we proceed with Chow tests. If, however, there is evidence to suggest that the variances are actually different, we may explicitly estimate the model, accounting for the heteroscedasticity. However, if the sample is reasonably large, we may use the Wald test that is valid whether or not the error variances are the same. To set up this test, we suppose

that θ^1and θ^2 are two normally distributed estimators of a parameter based on independent samples,4 with variance matrices V1 and V2. Then, under the null hypothesis that the two estimates have the same expected value, i.e., there is no structural change between the two quota reviews,

θ ^ 1 − θ ^ 2 ⁢     has mean  0 ⁢     and variable  V 1 + V 2

and the Wald statistic,

W = ( θ ^ 1 − θ ^ 2 ) ′ ( V 1 + V 2 ) − 1 ( θ ^ 1 − θ ^ 2 ) ⁢     ,

has a chi-squared distribution with K degrees of freedom. A test that the difference between the parameters is zero can be based on this statistic. It is straightforward to apply this to our -test of common parameter vectors in our regressions. Large values of the statistic lead us to reject the hypothesis of no difference (or of stability in the coefficients). Note that we base such a test on estimates of V\ and Vi. The test is valid in large samples, so we may use our least squares estimates of the two covariance matrices to compute W.5

As shown in the attached tables, the F-test results indicate that the null hypothesis (that the error variance from the estimated Bretton Woods formula is equal over rolling pairs of reviews, from the Sixth through the Eleventh Reviews) is rejected for the Sixth and Seventh, Seventh and Eighth, Ninth and Tenth, and Tenth and Eleventh Reviews (Table III.2.1). Therefore, Chow tests are not appropriate for testing the stability of coefficients of the Bretton Woods formula over these review periods. Nonetheless, application of the Wald test suggests that the coefficient estimates of the Bretton Woods formula are not stable over these reviews (Wald test values exceed the critical x2 value (4 restrictions) of 9.49, at the 5 percent significance level, for all such pairs of reviews—Table III.2.2). However, the pair-wise comparison of the Eighth and Ninth Reviews suggests that the corresponding error variances are equal, and therefore the Chow test results are valid (Table III.2.3).

S See Greene, William, H., 1993, Econometric Analysis, 2d edition, Prentice Hall, Englewood Cliffs, N. J., pp. 215-6.

Table III.2.1 :

Test for Equality of Variances of Error Terms (F-test)

article image
Note: An asterisk indicates significance at the 5% level. The critical values of the F-statistic (120.120) are 1.53, 1.35, and 1.26 at the 1% level, 5%, and 10% significance levels, respectively.
Table III.2.2 :

WaldTest

article image
Note: An asterisk indicates significance at the 5% level. The critical values of the Chi-square statistic for 4 restrictions are 13.28, 9.49, and 7.78 at the 1%, 5%, and 10% significance levels, respectively.
Table III.2.3 :

Chow Tests

article image
Note: An asterisk indicates significance at the 5% level. The critical values of the F-statistic (5.200) are 3.11, 2.66, and 1.88 for the 1%, 5%, and 10% significance levels, respectively.

Section IV. Nested Formulas with Vulnerability and Strength Variables

This section presents the methodology and statistical results from applying the nested model to certain vulnerability and strength variables.

Part A presents the Davidson-MacKinnon J test used in estimating the relative weights of strength and vulnerability variables in a two-equation formula system.

Part B presents a summary table and regression results from nested models used to estimate the relative weights of strength and vulnerability variables in the determination of actual quotas.

The regression results for the nested model where the vulnerability model is estimated first (Regression No. 53) indicate that the relative weight for the vulnerability variables (a) is 0.54, while that for the nested model where the strength model is estimated first (Regression No. 54) indicate that the relative weight for the vulnerability variables (a) is 0.43. Since the t statistics for the relative weights in both regressions are statistically significant, neither regression can be rejected. The relative contributions of variables in these regressions, based on the weighted coefficients, indicate that the relative contribution of Y is around 20 percent on average, and that of VC is over 50 percent on average.

The regression equations and resulting quota distributions for members from the nested formulas is presented in Statistical Appendix, Part B, Section III.

A. Davidson-MacKinnon J Test

To estimate the relative weights of strength and vulnerability variables in a two-equation formula system, we used the Davidson-MacKinnon J test. The J test proceeds as follows:

1. Assume the following two models explain actual quotas:

article image

Models A and B are nonnested if one cannot be derived as a special case of the other. To test whether the models are nonnested, we estimate the artificially nested model C, and then test one or both of the original models against it:

article image

2. Davidson-MacKinnon J test. Since model C is not estimable, because the parameters a, β, and γare not separately identifiable, Davidson and MacKinnon1suggested that model C be replaced by one in which the unknown parameters of the model that is not being tested are replaced by consistent estimates of those parameters. The idea is that if one model is the correct model, then the fitted values from the other model should not have further explanatory power when estimating that model. Thus, the J testing procedure follows the steps:

  • (a) Estimate (by OLS) model A and obtain the estimated (fitted) Yvalues, f*.

  • (b) Add Y4 of step 1 as an additional regressor to model B and estimate (by OLS) the following model D:

article image

Alternative Hypotheses," Econometrica, Vol. 49. pp. 781-93.

                     where S = coefficients of X, with d= fβ(l-α), and w1 = error term.

In this step, we obtain the estimate of cc, i.e., the relative weight of the vulnerability variables.

  • (c) Using the t test, test the hypothesis that a = 0.

  • (d) If the hypothesis that a = 0 is not rejected, (i.e., the t statistic on the a is not statistically significant), we can accept (i.e., not reject) model B as the true model because f* included in model D, which represents the influence of vulnerability variables not included in model B, has no additional explanatory power beyond that contributed by model B. In other words, model B encompasses model A in the sense that the latter model does not contain any additional information that will improve the performance of model B. By the same token, if the null hypothesis is rejected, model B cannot be the true model.

  • (e) Then, we reverse the order of estimation of models A and B. We now estimate model B first, use the estimated Y values from this model as regressor in model D, repeat step (d), and decide whether to accept model A over model B. More specifically, we estimate the following model E:

article image

In this step, we obtain the estimate of (1- a), i.e., the relative weight of the strength variables.

We now test the hypothesis that 6 = 0. If this hypothesis is not rejected, we choose model A over B. If the hypothesis that 0 = 0 is rejected, choose B over A, as the latter does not improve over the performance of B.

When one does a pair of nonnested tests, there are four possible outcomes, since each of model A and B may or may not be rejected. Furthermore, although it is intuitively appealing, the J test will not be able to provide a clear answer if it leads to the acceptance or rejection of both models. In case both models are rejected, neither model helps to explain the behavior of Y. Similarly, if both models are accepted, as Kmenta notes, the data are apparently not rich enough to discriminate between the two hypotheses [models]."2 Or, as Davidson and MacKinnon note, "when neither model is rejected, we must conclude that both models apparently fit the data about equally well and that neither

provides evidence that the other is misspecified. Presumably, either the two models are very similar, or the data set is not very informative."3

3.       Applying the J test to data for the Eleventh Review, we find that neither model A nor B can be rejected,4 and as noted by Davidson and MacKinnon, a possible interpretation is that the.traditional data set does not capture all available information. An important missing element could be the political agreements at the time of the Bretton Woods conference, whose influence has survived through the equiproportional element in subsequent quota increases.

B. Nested Models With Vulnerability and Strength Variables

The model with the vulnerability variables includes two variables: the variability of current receipts, VC, defined as one standard deviation from a five-year monthly average over a 13-year period (1982-94), and population, POP, in 1994. The model with the strength variables includes four variables: GDP, Y, in 1994; the average monthly reserves with gold valued at market prices, RM, in 1994; the annual average of current receipts over a five-year-period (1990-94), C; and the four-year moving average of net private capital flows (1991-94), NNKFL!

The attached table summarizes the regression results of two nested models and a linear equation used to estimate the relative weights of strength and vulnerability variables. The first nested model is estimated using fitted values for the vulnerability model (Regression No. 53), while the second nested model is estimated using fitted values for the strength model (Regression No. 54). The coefficient estimates of the linear equation of actual quotas on both strength and vulnerability variables (Regression No. 55) are OLS estimates. The coefficients shown in this table are weighted coefficients, equal to the product of the estimated coefficients from the respective regressions times their corresponding weights (a) and (1-a).

Table IV.

1. Comparison of Coefficients from Nested Models and Ordinary Least Squares to Estimate the Relative Weights of Strength and Vulnerability Variables

article image
Q is the actual quota; Y is GDP in a recent year (1994); RM is average monthly reserves with gold valued at market prices in a recent year (1994); C is the annual average current receipts over a recent five-year period (1990-94); NNKFL is the four-year moving average of net private capital flows (1991-94). VC is the variability of current receipts, defined as one standard deviation from a five-year moving average over a recent 13-year period (1982-1994); POP is population in 1994.

Coefficients shown have been multiplied by the estimated relative weights, for the strength and vulnerability variables, i.e., they represent the "net" effect of the variable on the estimated quota.

Section V. Hypothetical Quota Calculations for Past Quota Reviews According to the Relative Size of the Equiproportional and Selective Elements

This section presents a summary table on simulated quota shares from the Sixth through the Eleventh Quota Reviews had the distribution of quota increases been consistently either fully equiproportional, fully selective, or evenly divided between the equiproportional and selective elements. .

For this exercise, we use a constant sample of countries participating in the Sixth Review (121 members) plus China for the Seventh through the Eleventh Reviews, and take as given the size of the overall quota increases that were agreed. The simulation calculations are structured as follows: For the Seventh (initial) Review, in the fully equiproportional distribution scheme, the overall quota increase for the Seventh Review (50.9 percent) is applied to each member's actual (proposed) quotas under the Sixth Review to obtain the "fully equiproportional" quotas. In the fully selective distribution scheme, the overall percentage quota increase (50.9 percent) is first applied to the sum of the actual (proposed) quotas of the Sixth Review for the 122 members in the sample, and then distributed according to each member's calculated quota share in the total (Seventh Review) sample to obtain the "fully selective" quotas. The outcome of the evenly divided distribution scheme is generated as the average of the thus resulting "fully equiproportional" and "fully selective" quotas. For the Eighth through the Eleventh Reviews, the overall percentage increase in quotas for each review is applied in the same manner as described for the Seventh Review, except that calculations are based on quotas generated from the simulation of the previous review, instead of actual (proposed) quotas from the previous review.

The results indicate that if the distribution of past quota increases had been (1) fully equiproportional, the United States, the United Kingdom, India, and China would have ended up with higher quota shares under the Eleventh Review, while Japan, Germany, France, Italy, and Saudi Arabia would have ended up with lower quota shares; (2) fully selective, the reverse would have held true, except for the United States and the United Kingdom, which are marginally affected, and Saudi Arabia, which gains; and (3) evenly divided between the equiproportional and selective elements, the simulated quota shares under the Eleventh Review for the United States and the United Kingdom would be higher, while those for Saudi Arabia and Japan would be lower and for Germany would be unchanged, compared with the actual quota shares under the Eleventh Review.

Table V.l.

Summary of Simulated Quota Shares from Sixth Through Eleventh Reviews

(In percent of total)

article image
1

In particular, heteroscedasticity exists whenever the variance of the error term changes across different segments of the population, which are determined by the different values of the explanatory variables (Wooldridge, Jeffrey, M., 2000, Introductory Econometrics: A Modern Approach, South-Western College Publishing, Thomson Learning, United States, p. 248). In the case of the estimated Bretton Woods formula, heteroscedasticity is present if the variance of the error term increases with the factors affecting actual quotas, i.e., GDP, trade, reserves, and variability.

2

Toyoda, Toshihisa, 1974, "Use of the Chow Test Under Heteroscedasticity," Econometrica, Vol. 42, No. 3, May, pp. 601-8; and Schmidt, Peter and Robin Sickles, 1977, "Some Further Evidence on the Use of the Chow Test Under Heteroscedasticity," Econometrica, Vol. 45, No. 5, July, pp. 1293-98.

3

The F-test is used for variance equality tests with two subgroups (G = 2). We compute the variance for each subgroup and denote the subgroup with the larger variance as L and the subgroup with the smaller variance as S. Then me ^-statistic is given by F=sL2/sS2⁢  where   Sg2 where S is the variance in subgroup g=L, S. This F-statistic has an F-distribution with nL — 1 numerator degrees of freedom and ns - 1 denominator degrees of freedom under the null hypothesis of equal variance and independent normal samples.

4

Without independence, this test fails.

5

See Greene, William, H., 1993, Econometric Analysis, 2d edition, Prentice Hall, Englewood Cliffs, N. J., pp. 215-6.

1

1 Davidson, R. and J.G. MacKinnon, 1981, "Several Tests for Model Specifications in the Presence of Alternative Hypotheses," Econometrica, Vol. 49. pp. 781-93.

2

Kmenta, Jan, 1986, Elements of Econometrics, Macmillan, 2d ed., New York, p. 597.

3

Davidson, Russell and James G. MacKinnon, 1993, Estimation and Inference in Econometrics, Oxford University Press, New York and Oxford, p. 383.

4

See Part B of this note.

Same Series

Other IMF Content

Other Publishers

  • Collapse
  • Expand
Cover Policy Papers
External Review of the Quota Formulas - Statistical Appendix (Part A)
Author:
International Monetary Fund
Volume/Issue:
Volume 2000: Issue 001
Publisher:
International Monetary Fund
ISBN:
9781498327282
ISSN:
2663-3493
Pages:
86
DOI:
https://doi.org/10.5089/9781498327282.007