Compensatory Financing: The Cyclical Pattern of Export Shortfalls
  • 1 0000000404811396https://isni.org/isni/0000000404811396International Monetary Fund

Compensatory financing may be considered as an insurance scheme allowing members to borrow at low interest rates when their export earnings fall below trend and to repay when their export earnings rise above trend. An insurance company can forecast the sum of the claims likely to be presented by all its customers during a given year although it may not be able to predict which of its customers will present a claim. The Fund is in a similar position regarding drawings made under the compensatory financing facility.

Abstract

Compensatory financing may be considered as an insurance scheme allowing members to borrow at low interest rates when their export earnings fall below trend and to repay when their export earnings rise above trend. An insurance company can forecast the sum of the claims likely to be presented by all its customers during a given year although it may not be able to predict which of its customers will present a claim. The Fund is in a similar position regarding drawings made under the compensatory financing facility.

Compensatory financing may be considered as an insurance scheme allowing members to borrow at low interest rates when their export earnings fall below trend and to repay when their export earnings rise above trend. An insurance company can forecast the sum of the claims likely to be presented by all its customers during a given year although it may not be able to predict which of its customers will present a claim. The Fund is in a similar position regarding drawings made under the compensatory financing facility.

Under the facility, a Fund member experiencing a shortfall in its total export earnings can draw an amount that cannot exceed the amount of its export shortfall.1 In the Fund facility, the amount of the shortfall measures the downward deviation from a medium-term trend, which is defined as the five-year arithmetic average centered on the year when the shortfall occurs.2

For the last 20 years, nine tenths of the variance in the sum of the export shortfalls experienced by a sample of 71 Fund members is explained by variations in an indicator of the business cycles in the major industrial countries and variations in average world export unit values. By contrast, only one third of the variance in individual country shortfalls is explained by variations in these two world economic indicators. The effects of variations in these indicators on country shortfalls are overshadowed by events specific to each country, such as floods, political events, and changes affecting the markets for particular products. Such events differ from country to country, and they offset each other to a considerable extent when country shortfalls are added up. The sum of the export shortfalls reflects, therefore, the variation in the economic indicators and follows a highly cyclical pattern. Thus, for the 71 sample countries, that sum jumped from SDR 0.1 billion during the peak of 1974 to SDR 8.9 billion during the trough of 1975.3

The sum of the shortfalls experienced by the 71 sample countries is very closely correlated with the shortfall in, or the excess of, the aggregated export earnings of the 71 countries.4 In order to project the sum of the shortfalls likely to be experienced by a group of countries, it is therefore convenient, first, to project the aggregated earnings of that group of countries and, second, to derive the sum of country shortfalls from the shortfall in, or the excess of, the aggregated earnings of the group.

The scope of this paper is limited to the analysis of export shortfalls. It is, nevertheless, important to note that the amounts drawn under the compensatory financing facility are substantially lower than the sum of the country shortfalls. There are two main reasons for this. First, a country experiencing an export shortfall may not seek assistance under the facility because that country is able to offset the impact of its export shortfall by drawing either on its own financial reserves or on other financial sources, including other Fund credit facilities. Second, a country seeking assistance under the facility may not be able to draw an amount equal to its export shortfall because the compensatory drawing cannot exceed a given percentage of the country’s quota in the Fund; thus, when compensatory drawings reached the record level of SDR 2.3 billion in 1976, the amounts drawn by most members were limited by their quotas. Although the amount disbursed under the facility is lower than the sum of the country shortfalls mainly for these two reasons, the analysis of export shortfalls provides a, useful insight into the cyclical pattern of compensatory drawings.

The analysis is presented in five sections. The first describes the selection of the sample. The second investigates how export earnings and export shortfalls can be explained in relation to a time trend and two world economic indicators. The third analyzes the relationship between the shortfall in, or the excess of, aggregated earnings and the sum of country shortfalls. The fourth considers the effects of two possible modifications in the definition of the shortfall—namely, (1) replacing the arithmetic average by a geometric one, and (2) conducting calculations in real, instead of nominal, terms. The fifth section summarizes the findings. The Appendix describes the statistical analysis from which the results are drawn.

I. The Sample

Of the 132 Fund members, the 26 members classified as industrial countries or oil exporters are excluded from the sample.5 Among the other 106 Fund members, 35 members are excluded for lack of consistent data on annual export earnings for the period 1957-76. Thus, the sample consists of the 71 remaining Fund members. The 1976 export earnings for 25 of these countries are based on provisional estimates. The 1977 data are forecast values and the statistical sample covers the 21-year period 1957-77.6

Annual export earnings, which cover all merchandise exports, are expressed in SDRs. The aggregated earnings of the 71 countries and the individual earnings of each country are related to two world economic indicators. The first is the indicator of the business cycle, which is calculated as a weighted average of the ratios between actual and potential output in the manufacturing sectors of eight industrial countries.7 The second is the average world export unit value calculated as the ratio between value and volume of world trade.

II. Export Earnings and Export Shortfalls

Aggregated export earnings of the 71 sample countries can be explained extremely well in relation to three factors: time trend, business cycle indicator taken as a proxy for the import demand of industrial countries, and world export unit value taken as a proxy for inflation in world trade.8 Over the last twenty years, these three factors explain 99 per cent of the variance in aggregated earnings and 92 per cent of the variance in individual country earnings.9

When the regression coefficients are constrained to remain the same for all countries, the residual variance increases from 8 per cent to 27 per cent of the total variance in countries’ earnings. Covariance analysis shows that all regression coefficients differ significantly among countries, which suggests that simple extrapolation formulas are not reliable enough to project export earnings.10 Moreover, when the level of nominal export earnings is used as the dependent variable, the high value found for the correlation coefficients is somewhat misleading because a large part of the variance explained is owing to the trend factor. What ultimately matters is the amount of the shortfall on which the compensatory drawing is based. The amount of the shortfall is defined as the deviation from the time trend.

When the amount of the shortfall, instead of the level of export earnings, is used as the dependent variable, the part of the variance that remains unexplained becomes considerably higher. In the case of aggregated earnings, the part of the variance explained by the regression equation falls from 99 per cent to about 90 per cent.11 In the case of country earnings with country-specific coefficients, it falls from 94 per cent to 33 per cent.12 When the coefficients applied to the business cycle indicator are constrained to remain the same for all countries, only 23 per cent of the variance is explained. When the coefficients applied to world export unit value are also constrained to remain the same for all countries, only 12 per cent of the variance is explained. It is worth noting that the explanatory power of the equations is somewhat improved when shortfalls are defined as deviations from a geometric (instead of an arithmetic) trend value and when the regression equation is expressed in logarithmic (instead of arithmetic) form. With these changes, the part of the variance explained by the regression with country-specific coefficients increases from 33 to 44 per cent.13

The fact that country-specific events account for more than half of the variance in individual country shortfalls does not imply that the amount of the shortfall on which requests for compensatory drawings are based cannot be estimated with reasonable accuracy. At the time a request for compensatory drawing is made, the values of export earnings are known for the shortfall year and the two preceding years. The error made in estimating the amount of the shortfall is, therefore, owing only to the difference between the actual and forecast values of export earnings in the two post-shortfall years. In order to assess this error of estimation, shortfalls were calculated by taking export earnings in the shortfall year and in the two preceding years at their actual values, and by taking export earnings in the two post-shortfall years at their estimated values; the latter were taken as the deterministic parts of the regression equations relating the logarithmic value of the level of export earnings to the logarithmic values of the two world economic indicators and to time.

Differences between the shortfalls thus calculated and actual shortfalls reflect the impact of the country-specific events that occur during the two post-shortfall years. These differences account for only 14 per cent of the variance observed in individual country shortfalls when the latter are defined in relation to a geometric average.14 The simulation, therefore, suggests that shortfalls can be estimated with 86 per cent accuracy, although the results need to be qualified. On the one hand, the errors are overestimated because no account is taken in the simulation of any insights regarding the likely impact of country-specific events on the level of post-shortfall earnings. On the other hand, errors are substantially underestimated in the simulation because the forecaster is unable to predict the exact values of the two world economic indicators two years ahead, and because the residual errors of the regression equation will be greater outside the period used for estimating that equation than inside.

III. Sum of Country Shortfalls and Shortfall in (or Excess of) Aggregated Earnings

Because country-specific events are responsible for the major part of individual country shortfalls, the sum of the country shortfalls can be projected much more accurately than the shortfalls of individual countries. The sum of the country shortfalls can be derived from the shortfall in, or excess of, the aggregated earnings of the 71 sample countries, according to the relation represented by the upper branch of the hyperbola shown in Chart 1. In order to facilitate comparisons over time, all shortfalls or excesses are measured along the axes as percentages of current aggregated earnings. Since the aggregated earnings of the 71 sample countries amount to SDR 100 billion in 1975, one percentage point is equivalent to SDR 1 billion in 1975. The shortfall or excess in aggregated earnings xt, expressed as a percentage of current aggregated earnings, is shown along the horizontal axis. It appears as a positive number in years when aggregated earnings are below trend and as a negative number when aggregated earnings are above trend. The sum of country shortfalls yt+, expressed as a percentage of the current aggregated earnings of the same 71 countries, is shown along the vertical axis, always as a positive number. The sum of country excesses (yt) appears along the vertical axis, always as a negative number.

Chart 1.
Chart 1.

Sum of Country Shortfalls or Excesses as a Function of Shortfall in (or Excess of) Aggregated Export Earnings of 71 Primary Producing Countries, 1959-75.

Citation: IMF Staff Papers 1977, 003; 10.5089/9781451969450.024.A003

When the trend value is calculated as a five-year arithmetic average, as is presently done in assessing member entitlements under the Fund’s compensatory financing facility, the shortfall in, or the excess of, aggregated earnings is the difference between the sum of the country shortfalls and that of the country excesses

xt=yt+(yt).(Shortfallin,orexcessof,aggregatedearnings)=(Sumofcountryshortfalls)(Sumofcountryexcess).

For primary exporting countries, 1974 was an exceptionally good year, during which very few countries had an export shortfall. The sum of the country shortfalls y74+ was only 0.1 per cent of aggregated earnings, while the sum of country excesses (y74) was equal to 10.3 per cent of aggregated earnings. The excess in aggregated earnings (−x74), obtained by subtracting the former from the latter, was equal to 10.2 per cent of aggregated earnings; hence, the representative points for 1974 are at the extreme left of Chart 1. By contrast, 1975 was a bad year during which the sum of the country excesses was small (y75 = 1.2) and the sum of the country shortfalls was large (y75+ = 8.9). The shortfall in aggregated earnings was almost as large (x75 = 7.7) and the representative points are at the right side of Chart 1.

The sum of country shortfalls or excesses (yt+ or yt) should be related to the shortfall in, or the excess of, aggregated earnings xt by a function corresponding to the curve represented in Chart 1; the two asymptotes to such a curve should be the horizontal axis and the bisector of the first quadrant. The simplest mathematical formula with such properties is the hyperbola yt2xtytk2=0; Chart 1 shows that this hyperbola provides an excellent fit for the period of 1959-75. The two branches of the hyperbola intersect the vertical axis at + k and − k. Constant k increases proportionally with the sum of the standard deviations of the error terms of the country equations 15; its value characterizes the cyclical pattern of the sum of the country shortfalls and, therefore, the cyclical pattern of the amounts drawn under the facility.

If all country shortfalls were fully explained by variations in the values of the world economic indicators, and if the regression coefficients were the same for all countries, the value of k would be equal to zero, and the hyperbola would degenerate into its two asymptotes. Whenever aggregated earnings were above trend, which would occur in all good years, no country would have a shortfall. Because country shortfalls are only partly explained by the two world economic indicators, the value of k is positive and the hyperbola does not degenerate into its two asymptotes. When the share of the country shortfalls attributable to country-specific events increases, the value of k rises, the hyperbola becomes flatter, the representative points move closer to the vertical axis (xt becomes smaller), and their ordinates move closer to constant k.

Because the effects of country-specific events largely offset each other, the shortfall in, or the excess of, aggregated earnings can be projected accurately for given values of the two world economic indicators. If the values of these two indicators could have been forecasted without error for the last 20 years, the coefficient of determination between the actual and the projected sums of country shortfalls would have been equal to approximately 0.9.

IV. Two Possible Modifications in the Definition of the Shortfall

The results reported in Chart 1 were arrived at by conducting all calculations in nominal terms and by taking the trend value as the five-year arithmetic average centered on the shortfall year. This practice, which is in accordance with the 1975 decision, could be modified by conducting the calculations in real terms and by using a geometric average.

Since nominal export earnings tend to increase exponentially rather than linearly, it would be logical to calculate the medium-term trend as a geometric average rather than as an arithmetic average. The difference between shortfalls calculated in these two ways increases rapidly with the growth rate of export earnings; the difference is therefore greater when earnings are measured in nominal terms than when they are measured in real terms.16

In order to calculate shortfalls in real terms, the nominal value of export earnings has to be divided by a price index that is taken as unity in the shortfall year. When the trend value is defined as a geometric average, the shortfall calculated in real terms exceeds the shortfall calculated in nominal terms if, in the shortfall year, the price index used as deflator is above its trend value, and vice versa.17 Over a large number of years, differences between real and nominal shortfalls offset each other because the sum of the price shortfalls must be approximately equal to the sum of the price excesses. Nevertheless, if inflation rates 18 do not remain constant, the time pattern of export shortfalls corresponds more closely to the variations in the country’s needs when calculations are made in real terms than when they are made in nominal terms, assuming that the price index used as deflator reflects reasonably well the import prices paid by the country. If export shortfalls were calculated in real (instead of nominal) terms, the country would be eligible to draw more when it had to pay abnormally high prices for its imports and to draw less when it had to pay abnormally low prices.

The implications of defining the trend as a geometric (instead of an arithmetic) average and of conducting the calculations in real (instead of nominal) terms have been assessed for the 71 sample countries for the period 1959-75. For the purpose of making calculations in real terms, the UN price index of manufactures exported by industrialized countries was used as deflator.

The four different ways of calculating export shortfalls are shown in Table 1. In the four cases, the sum of the country shortfalls averaged over the 17-year period (row 1) exceeds the sum of country shortfalls which would be experienced in a normal year (row 2) by approximately 70 per cent. Such a difference reflects the curvature of the upper branch of the hyperbola shown in Chart 1. Because of this curvature, the average ordinate of the dots corresponding to the 17 observation years exceeds the ordinate of the point at which the hyperbola intersects the vertical axis.

Table 1.

Seventy-One Countries: Alternative Ways of Measuring Shortfalls, 1959-75

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For 1975, one percentage point is equivalent to SDR 1 billion.

A normal year is defined as a year when the shortfall in the aggregated earnings of the sample countries is equal to zero. The representative point for such a year would be along the vertical axis of Chart 1.

The sum of country shortfalls is always greater with an arithmetic average than with a geometric average (rows 1 and 2); the average difference is twice as large when calculations are made in nominal terms as when they are made in real terms (row 1). With nominal earnings, redefining the trend value as a geometric average would have reduced shortfalls by 37 per cent on the average. Countries with slow growth of export earnings, which may be in greatest need of Fund assistance, would have been penalized less by this change than countries with rapid growth of export earnings.

When the trend value is defined as a geometric average, whether calculations are conducted in nominal or real terms, the sum of shortfalls averaged over the 17-year period remains the same, but the time pattern of the shortfalls does not remain the same. With real earnings, the sum of country shortfalls is equal to 6.5 per cent of aggregated earnings in 1972 and 11.8 per cent in 1975.19 With nominal earnings, the figures become 10.1 per cent in 1972 and 6.5 per cent in 1975. The relative importance of the shortfalls is reversed owing to an acceleration of inflation in 1973-74 and a deceleration in 1976-77. As shown in Chart 2, the cyclical pattern of export shortfalls reflects more closely the variations of the business cycle indicator when calculations are made in real terms than when they are made in nominal terms. In the latter case, nominal shortfalls were at their peak in 1972 because world export unit values increased very sharply both in 1973 and in 1974.

Chart 2.
Chart 2.

Time Patterns of Sum of Country Shortfalls and of Shortfalls in (or Excesses of) Business Cycle Indicator for 71 Countries, 1959-75

Citation: IMF Staff Papers 1977, 003; 10.5089/9781451969450.024.A003

When shortfalls are calculated in real terms and the trend value is taken as a geometric average, exactly half of the countries experience an export shortfall in a normal year, while the other half have an export excess (Table 1, row 3). The proportion of countries experiencing a shortfall in a normal year increases from 50 per cent to 52 per cent when the trend value is taken as an arithmetic (instead of a geometric) average. It further increases to 53.2 per cent when, in addition, shortfalls are calculated in nominal (instead of real) terms.20

V. Conclusion

The analysis of export shortfalls presented in this paper has the following implications:

(1) The sum of export shortfalls experienced by primary producing countries can be derived very precisely from the shortfall in, or the excess of, the aggregated earnings of all primary producers, as shown in Chart 1. The shortfall in aggregated earnings can, in turn, be derived accurately from variations in the level of the business cycle indicator in major industrial countries and average world export unit values. The sum of country shortfalls, which is indicative of the total amount likely to be drawn under the Fund compensatory financing facility, can therefore be derived from the expected values of the two world economic indicators.

(2) The time pattern of the sum of country shortfalls, and consequently of compensatory drawings, follows a highly cyclical pattern that is closely linked to the business cycle in the major industrial countries, as illustrated in Chart 2.

(3) More than half of the variance in individual country shortfalls is owing to country-specific events, but only 14 per cent of the variance is owing to the country-specific events which occur after the shortfall year. At the time a request for compensatory drawing is made, the amount of the shortfall can be estimated with reasonable accuracy because the impact of country-specific events on export earnings is known for the shortfall year and the preceding two years.

(4) Defining shortfalls as deviations from a five-year arithmetic average tends to overestimate the size of shortfalls because the export earnings of many countries increase exponentially rather than linearly, especially when earnings are measured in nominal terms. Between 1959 and 1975, the average size of export shortfalls would have been reduced by about one third if a geometric average had been used instead of an arithmetic average.

(5) When shortfalls are defined as deviations from geometric averages, whether they are calculated in nominal or real terms does not affect the sum of the shortfalls experienced during a large number of consecutive years. The distribution of shortfalls between years will, however, tend to reflect more closely changes in balance of payments needs when calculations are made in real terms than when they are made in nominal terms, assuming the price deflator is representative of the import prices paid by the country experiencing the export shortfall. Real shortfalls are greater (lower) than nominal shortfalls when import prices are abnormally high (low) in the shortfall year; this implies a deceleration (acceleration) in the rate of inflation after the shortfall year.

APPENDIX: Statistical Analysis

The background to the results previously described are presented in this Appendix, which is divided into seven sections. The first defines the basic notation. The second and the third, respectively, deal with the statistical analysis of export earnings and export shortfalls. The fourth provides the rationale for the relation between shortfalls in aggregated earnings and the sum of country shortfalls. The fifth and sixth deal with two possible modifications in the definition of shortfalls—namely, taking the trend value as a geometric, instead of an arithmetic, average, and conducting calculations in real (instead of nominal) terms. The seventh consists of statistical tables.

1. Basic Notation

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Negative shortfalls are generally referred to as excesses.

2. Export Earnings

The import demand by industrial countries and the export supply of primary exporting countries can be represented by the following equations:

ln X/p = a1 + b1 ln Bc1 ln p/P + d1t,

ln X/p = a2 + d2t,

where p is the price index of primary exports, and parameters b1 and c1 are positive constants. On the one hand, the volume of the import demand by industrial countries X/p is taken as a function of the relative prices of primary exports p/P and of the level of activity in the manufacturing sector, which, in turn, is characterized by a trend factor and the level of the business cycle indicator B. On the other hand, the volume of the export supply of primary producing countries is assumed to follow a time trend. The equality between demand and supply gives the price equation

ln p = a′ + blnB + clnP + dt,

which, added to the supply equation, gives export earnings equation

lnXt=a+blnBt+clnPt+dt+εt,(1)

where εt is the error term.

Equation (1), which may be considered as the reduced form of a demand/supply model, is one of the two regression equations tested in Table 2. The other regression equation is written in arithmetic form as

Xt/X.=a+bBt/B.+cPt/P.+dt+εt;(2)

variables Xt, Bt, and Pt are divided by their means over time (X., B., and P.) so that coefficients b and c measure elasticities at mean values.

Table 2.

Seventy-One Countries: Aggregated Export Earnings, 1957-77

lnXt=a+blnBt+clnPt+dt+εt(1)1
Xt/X.=a+bBt/B.+cPt/P.+dt+εt(2)1
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X = aggregated earnings of the 71 countries, B = indicator of the business cycle, P = world export unit values, t = year, and the dot (.) subscript denotes the mean over time.

Figures in parentheses are t-ratios.

Regression 21 equations (1) or (2) explain approximately 99 per cent of the variance in the aggregated export earnings of the 71 sample countries between 1957 and 1977. As shown in Table 2, equation (1) provides a somewhat better fit than equation (2), although the error terms are positively serially correlated in both cases.

Regression equation (2) is also fitted to the export earnings of each of the 71 countries for the same 21-year period. In order to test whether the regression coefficients differ significantly among countries, the 71 country regressions are pooled and each of the three regression coefficients (b, c, and d) is constrained alternately to remain the same for all countries. In row 1 of Table 3, all regression coefficients are determined independently for each country; consequently, the number of estimated parameters is at its maximum while the residual variance is at its minimum value. In row 2, time trend coefficients di remain country-specific but coefficients b and c—which are applied, respectively, to the business cycle indicator and to world export unit values—are constrained to be the same for all countries; from row 1 to row 2, the residual variance almost doubles (from 0.030 to 0.058). In row 3, the price coefficients ci remain country-specific, while the trend coefficients d and the business cycle coefficients b are constrained to be the same for all countries; from row 2 to row 3, the residual variance doubles again (from 0.058 to 0.116). From row 3 to row 4, and from row 3 to row 5, the residual variance increases very little.

Table 3.

Seventy-One Countries: Covariance Analysis of Export Earnings, 1957-77 1

Xit/Xi.=ai+biBt/B.+ciPt/P.+dit+εit(2i)2
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Pooled country regressions.

Xit = export earnings of country i in year t, Bt = indicator of the business cycle, Pt = world export unit values, t = year, and the . subscript denotes mean over time.

Residual sum of squares divided by the corresponding number of degrees of freedom.

Degrees of freedom and sums of squares in row 6 (7, 8, 9) are obtained by taking the difference between those shown in rows 2 and 1 (3 and 1, 4 and 1, 5 and 1).

Ratio to the residual variance, calculated without constraint on the parameters, as given in row 1.

Variances owing to differences between country-specific coefficients are given in rows 6-9 of Table 3. When coefficients b and c are constrained to remain the same for all countries, 140 degrees of freedom are saved, but the residual sum of squares increases by 42. The variance owing to differences between the bi coefficients and between the ci coefficients is ten times larger than the residual variance from unconstrained country equations (0.3 in row 6 compared with 0.03 in row 1). Hence, differences between the values calculated for the bi and ci coefficients cannot reflect sampling errors only. None of the three coefficients should be assumed to remain the same for all countries. Nevertheless, if only one coefficient has to be taken as country-specific, this coefficient should be the one applied to the time trend.

The existence of country-specific trends is recognized in the extrapolation formula given in the 1975 decision. This formula22

X^i,t+1+X^i,t+2Xi,t1+Xi,t2=Xi,t+Xi,t1+Xi,t2Xi,t3+Xi,t4+Xi,t5

implicitly assumes that export earnings increase at the same yearly growth rate ri in period t-5 through t and period t-2 through t + 2, so that the left and right-hand sides of the equation are both equal (1 + rit)3. With this assumption, export earnings in the two post-shortfall years (X^i,t+1+X^i,t+2) are obtained by multiplying the known value of earnings in the two pre-shortfall years (Xi,t−1 + Xi,t−2) by the ratio on the right-hand side, which is also known at the time the compensatory request is made.

The value of the trend factor rit may differ substantially from the trend coefficient di estimated in regression equation (li) (in Section 3) or (2i) (in Table 3), in particular because the former (rit) is calculated without taking into account the effects of the two world economic indicators (Bt and Pt) on export earnings during the period of estimation. Moreover, post-shortfall earnings are calculated in the extrapolation formula without taking into consideration the likely values of the two world economic indicators in the two post-shortfall years. For these reasons, all country shortfalls tend to be overestimated or underestimated at particular phases of the business and inflation cycles.

This defect could be corrected by multiplying 1 + rit by an adjustment factor At calculated as

AtΣi(1+rit)3(Xi,t1+Xi,t2)=X^t+1+X^t+2

where X^t+1 and X^t+2 are forecasted aggregated earnings of all countries i during the two post-shortfall years. In the absence of forecasting errors on aggregated earnings, the sum of country shortfalls calculated from the adjusted extrapolation formula would be exact, but considerable difference would remain between actual and calculated individual country shortfalls. Because regression coefficients ai, bi, ci and di have to be taken as country-specific, it does not seem possible to find a simple extrapolation formula providing a satisfactory estimation of individual country shortfalls.

3. Export Shortfalls

In the Fund facility, the shortfall year is taken as the latest 12-month period for which export data are available. A forecast of export earnings during the following 24-month period has to be made in order to calculate the amount of the shortfall, because the latter is defined as the excess of the 5-year average centered on the shortfall year over the value of export earnings in that year:

SXt=X¯tXt(3)(Shortfallinexportearningsinyeart)=(5yearaveragecenteredonyeart)(Exportearningsinyeart)

In a similar manner, shortfalls in the business cycle indicator Bt and in world export unit values Pt can be defined in relation to their trend values as

SBt=B¯tBt,SPt=P¯tPt.

Since the best fit was obtained with the export earnings equation expressed in logarithmic form (see Table 2), let us start from equation (1)

ln Xt = a + b lnBt + c lnPt + dt + εt.

Adding up the five equations (1) relating, respectively, to years t−2 through t+2 and dividing the sum by five gives the following relation between the geometric trend values:

lnX¯t=a+blnB¯t+clnP¯t+dt+ε¯t.

Subtracting equation (1) from the above leads to

lnX¯tXt=blnB¯tBt+clnP¯tPt+ε¯tεt,

with εt − εt = (0.2)(εt−2 + εt−1 + εt+1 + εt+2) − (0.8)εt.

Replacing the trend value by its value from identity (3) gives the shortfall equation

ln(1+SXtXt)=bln(1+SBtBt)+cln(1+SPtPt)+ηt.(4)

Alternatively, starting from arithmetic regression equation (2) and taking trend values as arithmetic averages, the shortfall equation becomes

SXt=bSBt+cSPt+ε¯tεt,

which can be rewritten in a scale-free form as

SXtXt=bSBtBt+cSPtPt+ηt.(5)

Equations (4) and (5) are estimated for the 17 years (1959-75) using actual values for each of the two post-shortfall years. Whichever equation is used, the trend factor disappears. The error term becomes a moving average of the original error terms εt. Short of making the appropriate adjustment for moving averages, a correction was made for first-order serial autocorrelation. (See rows 1 through 3 in Table 4.)

Table 4.

Seventy-One Countries: Shortfall in Aggregated Export Earnings, 1959-75

ln(1+SXtXt)=a+bln(1+SBtBt)+cln(1+SPtPt)+ηt(4)
SXtXt=a+bSBtBt+cSPtPt+ηt(5)
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Figures in parentheses are t-ratios.

Owing to the elimination of the trend factor, the coefficients of determination are considerably lower for export shortfalls than for export earnings (about 0.90 in Table 4 compared with 0.99 in Table 2). The coefficients of elasticity of export shortfalls SX in relation to the shortfalls in the business cycle indicator SB and export unit values SP, however, are highly significant.23 As a matter of fact, the impact of the business cycle indicator is much more significant when export shortfalls (instead of the level of nominal export earnings) are used as the dependent variable; the t-ratio is more than twice as large.

Regression equation (5) was also used to analyze the export shortfalls experienced by each of the 71 sample countries over the 17-year period 1959–75. As was done for export earnings, country regressions were pooled to test whether the values of the elasticity coefficients differed significantly among countries. In row 1 of Table 5, all regression coefficients are country-specific. In row 2, the coefficients applied to the shortfall in the business cycle indicator are constrained to be the same for all countries; from row 1 to row 2, the residual variance increases from 97 to 114. In row 3, the coefficients applied to the shortfall in world export unit values are constrained to be the same for all countries; from row 1 to row 3, the residual variance rises from 97 to 117. In row 4, both coefficients are constrained to be the same for all countries and the residual variance reaches 130.

Table 5.

Seventy-One Countries: Covariance Analysis of Shortfalls, 1959–75 1

(100)SXitXit=ai+bi(100)SBtBt+ci(100)SPtPt+ηit(5)
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Pooled country regressions.

Residual sum of squares divided by the corresponding number of degrees of freedom.

Degrees of freedom and sums of squares in row 5 (6, 7) are obtained by taking the difference between figures shown in rows 2 and 1 (3 and 1, 4 and 1).

Ratio relates to the residual variance calculated without constraint on the parameters. (See row 1.) Thus, 3.7 is the ratio of 361 to 97.

When the country-specific coefficients bi are replaced by coefficient b that is common to all countries, 70 degrees of freedom are saved but the residual sum of squares rises by 25,244; hence, the variance owing to differences between bi coefficients, which is obtained by dividing the latter (25,244) by the former (70), is 361 (in row 5). Such a variance is 3.6 times greater than the residual variance from the regression with country-specific coefficients (97 in row 1). Differences between bi cannot be assumed to reflect sampling errors, even at the 1 per cent level of significance. The same result applies to the differences between the a coefficients (in row 6) and to the differences between the bi and ci coefficients (in row 7). The covariance analyses of country export earnings (shown, respectively, in Tables 3 and 5) both indicate that elasticities related to the two world economic indicators should be country-specific.

When shortfalls are defined as deviations from a geometric (instead of an arithmetic) trend value, and when the regression equation is expressed in a logarithmic (instead of a linear) form, the forecasting power of the equation is improved. Thus, when all coefficients are country-specific, the coefficient of determination increases from 0.33 in the arithmetic regression (in row 1 of Table 5) to 0.44 in the logarithmic regression 24

ln(1+SXitXit)=ai+biln(1+SBtBt)+ciln(1+SPtPt)+ηit,(4i)

and the error of estimation is given by

ln(1+SXitXit)ln[1+(SX^itXit)]=ηit.

Alternatively, it would have been possible to start from the level equation in logarithmic form

lnXit=ai+bilnBt+cilnPt+dit=lnX^it+εit,(1i)

and to derive shortfalls as

ln(SXitXit)=.2Στ=2τ=2(lnX^i,t+τ+εi,t+τ)lnX^itεit,(4i)

the error of estimation being 25

ln(1+SXitXit)ln[1+(SX^^itXit)]=(.2)(εi,t2+εi,t1+εi,t+1+εi,t+2)(.8)εi.t.

Estimating shortfalls indirectly from the level equation (1i) is less efficient than estimating them directly from the shortfall equation (4i), since the coefficient of determination falls from .44 in equation (4i) to .27 in equation (4′i). When the shortfall is estimated at the time a request for compensatory drawing is made, it is nevertheless useful to start from level equation (1i) because the actual values of export earnings (Xi,t−2, Xi,t−1, and Xi,t) are known for the first three years; only the values (X^i,t+1 and X^i,t+2) in the two post-shortfall years need to be estimated from equation (1i) Equation (4′i) can be rewritten as

ln(1+SXitXit)=(.2)(lnXi,t2+lnXi,t1)(.8)lnXit+(.2)(lnX^i,t+1+lnX^i,t+2)+(.2)(εi,t+1+εi,t+2),(4i)

and the error of estimation becomes

ln(1+SXitXit)ln(1+SX^itXit)=(.2)(εi,t+1+εi,t+2).

Because the three error terms εi,t−2, εi,t−1, and εi,t are known in equation (4′′i), the coefficient of determination rises from 0.26 in equation (4′i) to 0.86 in equation (4′′i).26 The gain in forecasting power would have been even greater in the absence of any serial correlation between the error terms. Thus, with

E(εitεi,t+τ)=0forτ0andE(εit2)=σi2,

the individual variance would have been reduced from 0.8 σi2 in equation (4′i) to 0.08 σi2 in equation (4′′i). Even if the deterministic elements (X^i,t+τ) explain a limited part of the variance in country shortfalls, the forecasting power of equation (4′′i) is high because three of the first error terms are known, particularly the error term in the shortfall year.

4. Sum of Country Shortfalls and Shortfall in (or Excess of) Aggregated Earnings

The analysis of export shortfalls presented in Tables 4 and 5 can be used to rationalize the relationship found in Chart 1 between the shortfall in, or the excess of, aggregated earnings xt and the sum of country shortfalls yt+. In order to facilitate comparison over time, shortfalls in year t are measured along the axes of Chart 1 as percentages of the aggregated earnings of the 71 sample countries in year t. The arithmetic regression equations (5) (in the text) and (5i) (in table 5) need therefore to be rewritten 27 as

xt=100SXtXt=a+b100SBtBt+c100SPtPt+ut,(6)

and

xit=100SXitXit=ai+bi100SBtBt+ci100SPtPt+uit,(6i)

which can be expressed in a compact form as

xt=x^t+ut,

and

xit=x^it+uit,

where ut and uit are error terms with zero means and with variances equal, respectively, to σ2 and σi2.

When shortfalls are defined as deviations from a moving arithmetic average, the shortfall in aggregated earnings is the sum of the country shortfalls,

SXt=ΣiSXit,whichimplies
xt=Σixit,a=Σiai,b=Σibi,c=Σici,x^t=Σix^it,

and

ut=Σiuit.

Variables yt+ and yt, shown along the vertical axis of Chart 1, relate, respectively, to the sum of the country shortfalls and to that of the country excesses; they are defined as

yt+=Σixitforxit>0,
yt=Σixitforxit<0.

The point at which the upper branch of the hyperbola intersects the vertical axis corresponds to a normal year t when the shortfall in aggregated earnings is equal to zero, which implies

E(xt)=E(x^t+ut)=E(x^t)+E(ut)=E(x^t)=E(Σix^it)=0.

The ordinate k of the intersection point is E(Σixit), where the summation is made for the positive values of xit. Constant k can, therefore, be expressed as

k=E(Σix^it+Σiuit)=E(Σix^it)+E(Σiuit)=E(Σiuit)

for uit > 0.

Given a random variable ui of density function f(ui), the expectancy of ui for ui > 0 is

E(ui\ui>0)=0uif(ui)dui,

which, in the case of a normal distribution, can be written

E(ui\ui>0)=1σi2π0uieui2/2σi2dui=σi2π.

If error terms uit were normally distributed, the value of constant k would be

k=12πΣiσi.(7)

For the 71 sample countries and the 17-year period (1959–75), the sum of the standard deviations is Σiσi = 6.478, which implies, for equation (7), that k = 2.58. The value of constant k thus calculated is almost identical to the one given in row (1) of Table 6 (2.58 compared with 2.5628). In establishing equation (7), it was therefore justified to assume that the 1,207 error terms uit were normally distributed.

Table 6.

Seventy-One Countries: Sum of Country Shortfalls and Excesses(yt)in Relation to Shortfall in (or Excess of) Aggregated Earnings(xt)

(yt2βxtytk2=0)
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Figures in parentheses are t-ratios.

For aggregated earnings, the residual variance calculated from equation (6) is σ2 = 3.9, which is only 9 per cent of (Σiσi)2 = (6.478)2 = 41.96, which would have been the value of σ2 if all error terms uit had been perfectly correlated among countries—that is, if E(uitujt) = σiσj for i ≠ j. On the other hand, if the country error terms had been un-correlated—that is, if E(uitujt) = 0 for i ≠ j—the residual variance of aggregated earnings (σ2) would have reduced to Σiσi2 = 2.22. Residual variance σ2 therefore exceeds the minimum value corresponding to a zero correlation between country error terms by σ2Σiσi2=2.22. The latter is only 6 per cent of (Σiσi)2Σiσi2=40.3, which measures the difference between the maximum and the minimum values of σ2 associated, respectively, with a unity and a zero correlation between country error terms. Because the error terms uit are loosely correlated among countries, the effects of country-specific events are largely independent, and the sum of the positive country shortfalls can be forecasted accurately while shortfalls experienced by individual countries cannot.

When the trend is taken as an arithmetic average, the shortfall in aggregated earnings xt is equal to the sum of the country shortfalls yt+ minus the sum of the country excesses yt,

xt=yt++yt=yt+(yt).(8)

The sum of the country shortfalls yt+ should be a decreasing function of the sum of the country excesses yt. The representative function should have the vertical and the horizontal axes as asymptotes. It could therefore be represented as a constant elasticity curve with a negative elasticity coefficient. The regression equation found for the period 1959–75 is

ln(yt)=ln(2.6)20.91ln(yt+),R¯2=0.60,DW=2.8.

The t-ratios are, respectively, 11.2 for the constant term and 5.0 for the coefficient of ln yt+.

As a first approximation, the regression coefficient could be taken as equal to unity, which would imply that the sum of the country shortfalls multiplied by the sum of the country excesses remains constant,

yt+yt=k2.(9)

When the algebraic shortfall in aggregated earnings is known, the sum and product of yt+ and yt are also known from equations (8) and (9). Hence, yt+ and yt are, respectively, the positive and negative roots of

yt2xtytk2=0,(10)

given by

yt+=xt2+xt24+k2,yt=xt2xt24+k2.

When the arithmetic average is replaced by a geometric one, the shortfall in, and the excess of, aggregated earnings becomes smaller than the difference between the sum of the country shortfalls and that of the country excesses,

xtyt++yt.(8)

In turn, the relation between xt and yt, becomes

yt2βxtytk2=0,withβ1,(9)

where β measures the tangent of the angle between the asymptote and the horizontal axis.

The effect of replacing the arithmetic average by a geometric one was tested by simultaneously estimating coefficients β and k of equation (9′). When the trend value is calculated as an arithmetic average, the estimated values of coefficients β and k remain practically the same, whether the regression is applied to the sums of country shortfalls yt+, the sums of country excesses yt, or to both simultaneously; the average value of β is 0.97. (See Table 6.) When a geometric average is used instead of the arithmetic average, the values of both k and β are lower when calculations are based on the sum of the country excesses.

The various steps required for estimating the sum of the positive country shortfalls in year t are summarized in Table 7. The first step is to estimate the annual values of the business cycle indicator and world export unit values for the period t−2 to t+2. The second is to calculate the trend value of these two indicators using the arithmetic average or the geometric average, as defined in row 1 of Table 8. The third step is to derive xt, the percentage of shortfall in aggregated earnings, from the regression coefficients b and c, given in rows 2 and 3 of Table 8. The fourth step is to calculate y^t+, the percentage of the sum of positive country shortfalls, from the regression coefficients β given in rows 4 and 5 of Table 8. Finally, the percentage yt+ has to be multiplied by the projected value of the aggregated export earnings of the countries for which the facility is designed.

Table 7.

Steps Required for Estimating the Sum of Country Shortfalls

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Table 8.

Numerical Values of Coefficients

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See row 2 of Table 4.

See row 1 of Table 4.

See row 1 of Table 6.

See row 4 of Table 6.

Between the percentage y^t+, calculated in the manner just described, and its actual value yt+, the adjusted coefficient of determination R¯2 is about 0.9 over the period 1959-75. (See row 6 of Table 8.) This would characterize the average forecasting accuracy if the values of the four regression coefficients were to remain the same outside as they were inside the sample period, and if the values of the business cycle indicator and world export unit values could be forecasted exactly.

Suppose that export earnings are known for the shortfall year and that the sum of country shortfalls is estimated on the basis of projected values of the business cycle indicator for the two post-shortfall years. A 5 per cent error on the average value of that indicator for the two post-shortfall years results in a 2 per cent error for the shortfall in the value of that indicator and, in turn, in an error of 0.026 for xt, which implies for yt an error of 0.025 in a very good year (xt = 10). The error in yt is reduced to 0.013 in a normal year (aft = 0), and 0.01 in a very bad year (xt = −10).29 Suppose that, at the end of 1975, the prospects for a rapid economic recovery had improved (worsened) considerably so that the average value of the business cycle indicator needed to be revised upward (downward) by 5 per cent for 1976 and 1977. Such a revision would have raised (reduced) the estimation of the sum of the country shortfalls in year t by SDR 2.5 billion for the 71 sample countries, and by SDR 3.0 billion for the 106 Fund members not classified as industrial countries or oil exporters.

5. Trend Value Defined as Arithmetic or Geometric Average

In the case of most countries, earnings derived from all merchandise exports tend to increase at a geometric rate rather than at an arithmetic rate, especially when earnings are measured in nominal terms. If the trend value were calculated as an arithmetic average, a country with export earnings growing at a constant geometric rate would experience a shortfall every year. For growth rates of 10, 20, and 30 per cent per year, shortfalls would be equivalent, respectively, to 0.9, 3.4, and 7.0 per cent of current earnings. On the other hand, if the trend value were to be defined as a geometric average, a country would experience a shortfall only if its export earnings were to increase at a faster rate after the shortfall year than before it. That country could not experience a shortfall every year, since the growth rate of its export earnings could not always be increasing.

If the arithmetic average were replaced by a geometric average, it would be easier to analyze the shortfall in export earnings in terms of its volume and price components. On the other hand, it would be more difficult to analyze the shortfall in total earnings in terms of its commodity components. The shortfall in total earnings would always be smaller than the sum of the shortfalls calculated for each commodity component. Moreover, in the case of commodities that may be unavailable for export in some years, commodity shortfalls calculated with a geometric average could become meaningless because the growth rate of commodity exports would become infinite in some years. Thus, after exports had fallen below a given point, the amount of the calculated shortfall would be reduced by a further decline in export earnings,30 which is the opposite of what should occur.

6. Shortfalls Calculated in Nominal or Real Terms

Real export earnings RX are defined as the ratio of nominal export earnings NX to a price index P used as deflator. The shortfall in year t is measured at the price level of that year by taking the value of the price index in the shortfall year as unity. Consequently, real earnings are derived from nominal earnings as

RXt+τ=NXt+τPtPt+τwithτ=2,1,0,1,2.

Shortfalls are defined as the downward deviations of real and nominal export earnings from their respective trend values. They are calculated as

SRXt=RX¯tRXt,

and

SNXt=NX¯tNXt

Real and nominal earnings are, by definition, identical in the shortfall year,

RXt = NXt for τ = 0;

consequently, the difference between real and nominal shortfalls is that between the real and nominal trend values,

SRXtSNXt=RX¯tNX¯t.(11)

Taking the trend value as a 5-year geometric average

RX¯t=(NXt2PtPt2NXt1PtPt1NXtPtPtNXt+1PtPt+1NXt+2PtPt+2)0.2,

the relation between real and nominal trend values is

RX¯t=NX¯tPtP¯t.(12)

Replacing RX¯t in equation (11) by its value from equation (12) gives

SRXtSNXtNX¯t=PtP¯tP¯t.(13)

The amount of the shortfall is, therefore, increased (or decreased) by conducting the calculations in real, instead of nominal, terms, if, during the shortfall year, the price index used as deflator is above (or below) its trend value.

Dropping subscript t for the shortfall year, yearly rates of inflation can be defined as

iττ+1=Pτ+1PτPτ,forτ=2,1,0,1.(14)

Alternatively, i can be defined as the average yearly rate of inflation from the average of the two pre-shortfall years to the shortfall year, and i+ as the average yearly rate of inflation from the shortfall year to the average of the two post-shortfall years. Rates i and i+ are derived from the price levels as

P0=(1+i)3/2(P1P2)1/2,(15)
(P1P2)1/2=(1+i+)3/2P0.(16)

If the right-hand side of equation (13) is expressed in terms of the inflation rates denned in equations (14), (15), and (16), equation (13) can be rewritten as

SRXSNXNX¯=(1+i1+i+)0.61=(1+i101+i01)0.4(1+i211+i12)0.21,(17)

hence SRX ≷ SNX ⇆ i- ≷ i+, which is equivalent to

1+i12(1+i21)(1+i101+i01)2.

The amount of the shortfall is therefore raised (reduced) by conducting calculations in real terms if inflation accelerates (decelerates) after the end of the shortfall year, and vice versa. Over a large number of years, periods of accelerating and decelerating inflation must alternate; consequently, the average amounts of the shortfalls, calculated in real and nominal terms for a large number of consecutive years, must ultimately converge.

When the trend is taken as an arithmetic average, the derivations made after equation (11) do not apply. With a positive and constant rate of inflation, the nominal shortfall exceeds the real shortfall, and the difference between the two increases with the rate of inflation. With rates of inflation fluctuating around a zero mean, differences between real and nominal shortfalls would be similar to those described when the trend value is taken as a geometric average.

7. Statistical Tables

Data are shown for individual years but not for individual countries. Table 9 gives the yearly values of the aggregated export earnings of the 71 sample countries and of the two world economic indicators. Tables 10 and 11 summarize yearly results in terms of shortfalls in aggregated earnings, sums of country shortfalls, and percentages of the sample countries experiencing a shortfall.

Table 9.

Basic Data

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Table 10.

Seventy-One Countries: Shortfalls in (or Excesses of) Aggregated Export Earnings and Sums of Country Shortfalls, 1959-75

(In per cent of current values of aggregated export earnings)

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Shortfalls are shown as positive numbers and excesses as negative numbers.

Table 11.

Percentage of 71 Sample Countries with Shortfalls, 1959-75

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BIBLIOGRAPHY

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  • Commission of the European Communities, The Convention of Lomé (Brussels, July 1976).

  • “Compensatory Financing of Export Fluctuations,” IMF Survey, Vol. 5 (January 5, 1976), pp. 56.

  • de Vries, Jos, “Compensatory Financing: A Quantitative Analysis,” World Bank Staff Working Paper No. 228 (mimeographed, December 1975).

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  • Erb, Guy F., “North-South Negotiations and Compensatory Financing” (unpublished, Overseas Development Council, June 2, 1977).

  • Goreux, L. M., “The Use of Compensatory Financing,” Finance and Development, Vol. 14 (September 1977), pp. 2024.

  • International Monetary Fund, International Financial Statistics, various issues.

  • International Monetary Fund, Selected Decisions of the International Monetary Fund and Selected Documents (Washington, Eighth Issue, 1976).

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  • Krishnamurty, K., and Ke-young Chu, “Export Earnings of Primary Commodity Exporting Countries and Business Cycles in Industrial Countries” (unpublished, International Monetary Fund, July 5, 1977).

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  • Morrison, Thomas K., and Lorenzo Perez, “Analysis of Compensatory Financing Schemes for Export Earnings Fluctuations in Developing Countries,” World Development, Vol. 4 (1976), pp. 68794.

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  • United Nations Conference on Trade and Development, Trade and Development Board, “An Integrated Programme for Commodities: Compensatory Financing of Export Fluctuations,” TD/B/C.1/195 (October 16, 1975).

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*

Mr. Goreux, Assistant Director in the Research Department, holds doctorates from the Universities of Paris, Louvain, and Chicago. Before joining the Fund staff, he held various positions in the Food and Agriculture Organization and in the World Bank. He is the author of Interdependence in Planning and Agricultural Productivity and Economic Development in France.

1

For reasons that will be explained later, the amount drawn is often lower than the amount of the shortfall.

2

Shortfall years are taken here as calendar years. Under the Fund facility, the shortfall year is based on the latest available data and can end in any month. For a brief description of the facility, see Goreux (1977).

3

See the second column of Table 10 in the Appendix.

4

When export earnings are below trend, the member experiences a shortfall. When exports are above trend, the member experiences an excess. The maximum amount that can be drawn under the facility is defined by the sum of the country shortfalls. When that sum is larger (or smaller) than the sum of the country excesses, the value of aggregated country earnings is below (or above) trend; the deviation from the trend measures the shortfall in (or excess of) the aggregated export earnings of the group of countries. (See Chart 1.)

5

None of the 26 countries classified in the Fund’s publication, International Financial Statistics, as either industrial countries or oil exporting countries has made use of the facility since it was liberalized by Executive Board Decision No. 4912-(75/207), adopted December 24, 1975 and reproduced in Selected Decisions of the International Monetary Fund and Selected Documents (Washington, Eighth Issue, 1976), pp. 62-66.

6

Forecasts for 1977 were required to calculate shortfalls experienced in 1975. The forecasts were made in February 1977.

8

This equation can be interpreted as the reduced form of a three-equation model defining the import demand by industrial countries and the export supply from the 71 sample countries. (See section II in the Appendix.)

9

See Tables 2 and 3 in the Appendix.

10

In an attempt to minimize the impact of subjective factors on the estimation of the shortfall, Executive Board Decision No. 4912-(75/207)—cited in footnote 5—provides an extrapolation formula. The decision, however, specifies that the extrapolated value of post-shortfall earnings should be replaced by a judgmental forecast if the extrapolated value is not considered reasonable. In practice, a judgmental forecast is always made.

11

Comparison between Tables 2 and 4 in the Appendix.

12

Comparison between Tables 3 and 5 in the Appendix.

13

Little was gained by replacing the two world economic indicators by country-specific indicators. In one case, the business cycle indicators relating to each of the eight industrial countries were weighted by their relative shares in the imports of the primary exporting country concerned. In the other case, commodity spot prices were weighted according to the shares of each commodity in export earnings of the country. A detailed analysis of export earnings based on semiannual data for 48 countries between 1957 and 1975 is given by Krishnamurty and Chu (1977).

14

When shortfalls are defined in relation to an arithmetic average and the arithmetic forecasting equation is used, the sum of squares of the forecasting errors increases from 14 to 42 per cent of the variance in shortfalls.

15

See Section 4 in the Appendix.

16

See Section 5 in the Appendix.

17

See Section 6 in the Appendix.

18

More precisely, the rates of change in the value of the deflator.

19

See Table 10 in the Appendix.

20

The results are presented on a yearly basis in Table 11 in the Appendix.

21

All equations in this paper are estimated by ordinary least-squares techniques.

22

A circumflex (^) above a variable denotes its estimated value.

23

The values of the business cycle coefficients are about the same whether export earnings or export shortfalls are used as dependent variable. However, the estimated price coefficients are lowered in the shortfall equation; the value estimated in the earnings equation was affected by the correlation between the price and trend variables.

24

When the regression coefficients are alternately constrained to remain the same for all countries, the coefficients of determination comparable to those shown in rows 2-4 of Table 5 become, respectively, 0.24 (for ai, b, ci), 0.18 (for ai, bi, c), and 0.17 (for a, b, c).

25

The symbol

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is used to denote the estimated value in order to distinguish it from the estimated value derived from equation (4i).

26

When the trend is taken as the deviation from an arithmetic trend and the regression equation is expressed in the arithmetic form, the coefficient of determination is reduced to 0.58.

27

The dependent variable of equation (6i) is the dependent variable of equation (50 multiplied by the percentage share of country i in aggregated earnings, namely 100 Xit/Xt.

28

The coefficient 2.56 was estimated by fitting the upper branch of the hyperbola shown in Chart 1 to equation (10).

29

Calling ΔB the error made in forecasting B, the resulting errors δSB, δx, and δy can be calculated as follows:

δ(Bt+1 + Bt+2)/2Bt = 0.05 → δSBt/Bt = 0.02 → δxt = 1.3 δSBt/Bt = 0.026.

For xt = 10, dy/dx = 0.95 → δy = 0.025

xt = 0, dy/dx = 0.50 → δy = 0.013

xt = −10, dy/dx = 0.05 → δy = 0.0013

30

Consider the five consecutive values 1, 1, x, 1, 1 so that the calculated shortfall is S = x0.2x. Its derivative dS/dx = 0.2x−0.8 − 1 is positive for 0 < x < (0.2) = 0.134. When the value of export earnings x declines from 0.134 to zero, the amount of the calculated shortfall also declines, from 0.535 to zero.

IMF Staff papers: Volume 24 No. 3
Author: International Monetary Fund. Research Dept.
  • View in gallery

    Sum of Country Shortfalls or Excesses as a Function of Shortfall in (or Excess of) Aggregated Export Earnings of 71 Primary Producing Countries, 1959-75.

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    Time Patterns of Sum of Country Shortfalls and of Shortfalls in (or Excesses of) Business Cycle Indicator for 71 Countries, 1959-75