Book Chapter

# Appendix III. Statistical Analysis

International Monetary Fund
Published Date:
January 1980 Show Summary Details

The statistical analysis is presented in two sections. In the first section, export earnings are assumed to fluctuate randomly from an exponential trend and shortfalls are defined as the downward deviations from that trend. This definition is used to analyze (a) the expected value of the shortfall, (b) the value of the shortfall in terms of its volume and price components, and (c) the shortfall in total export earnings in terms of its commodity components. In the second section, shortfalls are defined as the downward deviation from a five-year average centered on the shortfall year, which is the definition used in the compensatory financing facility.

## Statistical Model

Export earnings are assumed to fluctuate randomly from an exponential trend where the error term is normally distributed The trend value of export earnings in year t is given by When random variable єt takes negative values (єt < 0), export earnings fall short of the trend value $\left({\stackrel{^}{X}}_{t}\right)$ and the amount of the shortfall $\left({\stackrel{^}{X}}_{t}-X\right)$ is related to the error term (єt) by which, for єt small, can be approximated by When random variable єt is negative and sufficiently small, its absolute value measures the amount of the shortfall divided by the value of export earnings in the shortfall year.

It should be noted that equation (1) is a simplified representation of the true world. First, coefficient c (which measures the growth rate of export earnings) does not remain constant over time. When export earnings are measured in nominal terms, the value of c is greater in periods of high inflation than in periods of low inflation. Second, error terms єt are not serially independent, and export earnings often follow some cyclical pattern. Cycles are, however, far from regular and they are not the same for all countries. The main virtue of the model selected is its simplicity.

### (a) Expected value of shortfalls

When random variable єt is positive, there is an excess which can be treated as a negative shortfall. Since the expected value of random variable єt is assumed to be zero, that of the algebraic shortfall is also equal to zero. The expected value of positive shortfalls is, however, positive and can be calculated in relation to its density function f(єt) as If variable єt is normally distributed with mean zero and σ2 as variance, its density function is given by The expected value of positive shortfalls can be derived by integration as  The expected value of shortfalls expressed as a fraction of the level of export earnings is, therefore, approximately 40 per cent of the standard deviation of random variable єt when this variable is normally distributed.

Similarly, the expected value of excesses would be given by The expected value of shortfalls (—єt > 0) is the same as that of excesses (єt > 0), and the expected value of algebraic shortfalls is equal to zero because fluctuations are assumed to be symmetrically distributed in relation to the trend value.

Calling αit the share of country i in the aggregate earnings of all the countries having access to the facility the expected value of the sum of the positive country shortfalls is given by41 The contribution of country i in the sum of the positive country shortfalls depends, therefore, on two sets of parameters. The first αit is the share of country i’s earnings in the aggregate earnings of the group. The second σi is the index of export earnings instability, which is measured by the standard deviation of fluctuations in the normalized variable. These results are not surprising since the amount a country may be expected to draw from a compensatory financing scheme obviously depends on the size of the country and the degree of instability of its export earnings.

Since shortfalls and disbursements are proportional to the standard deviations of export earnings fluctuations, shortfalls and disbursements can conveniently be analyzed in terms of the variance of these fluctuations.

### (b) Volume and price components

The value of export earnings can be decomposed into its volume and price components as Shortfalls in volumes and prices can be defined as downward deviations from a logarithmic trend in the same way as was the shortfall in export earnings. Assume that the volume, price, and value equations can be written as    It follows from (4) that where b is the elasticity of volume fluctuations in relation to price fluctuations, and Rpq the correlation coefficient calculated from regression equations   For selected values of the correlation coefficient, equation (8) can be simplified as follows: If volume and price fluctuations are independent, the variance of value fluctuations is the sum of the variances of volume and price fluctuations, and the relative role of each can be measured unambiguously. If volume and price fluctuations are dependent, the variance of value fluctuations exceeds that of volume fluctuations if elasticity coefficient b is greater than—0.5. The meaning of this elasticity coefficient is, however, ambiguous. It would represent the price elasticity of demand if the demand curve was stable, while the supply curve was unstable; but it would represent the price elasticity of supply, if the supply curve was stable while the demand curve was unstable. In order to remove the ambiguity, it is necessary to specify the structural model by writing the demand and supply equations as where bd and bs are, respectively, the price elasticities of demand and supply, while Zdt and Zst are, respectively, the exogenous factors affecting demand and supply. The exogenous variable affecting demand for imports Zdt could be the gross domestic product or the level of industrial activity in importing countries, while the variable affecting supply Zst could be the production capacity in the commodity sector concerned in exporting countries. Although the values of these variables often increase over time, their fluctuations from the time trend may be independent. In that event, the effects of the exogenous variables on demand and supply may be expressed in the form of a time trend and of independent fluctuations from this trend.42 The demand and supply equations may then be rewritten as   The demand and supply curves defined by (10) and (11) are subject to two types of shifts. The first is a uniform shift over time characterized by time-trend coefficients cd and cs. The second is a random shift characterized by variances σd2 and σs2.

The quantity, price, and value equations (5), (6), and (7) can be rewritten in relation to the structural parameters of demand and supply equations (10) and (11) as   Equations (5′), (6′), and (7′) can be used to derive the variances of volume, price, and value fluctuations σq2, σp2, and σx2) from the variances of the fluctuations of the demand and supply curves σd2 and σs2) and the price elasticities of demand and supply (bd and bs): Elasticity coefficient b which was calculated in (9) by regressing quantity fluctuations against price fluctuations becomes: Coefficient b is, therefore, the average of price elasticities of demand and supply weighted, respectively, by the variances of supply and demand. It is identical to the supply elasticity when the demand curve is stable and to the supply elasticity when the supply curve is stable: When supply is price inelastic, which is a reasonable approximation in the short term for many commodities, volume fluctuations are identical to supply fluctuations: Equation (13), derived in the context of a world commodity model, can be rewritten in the context of country i as Fluctuations in average export unit values received by country i for a given commodity closely reflect fluctuations in spot prices recorded on a representative world market for that commodity. Consequently, it is reasonable to assume that the variance of price fluctuations at the country level and at the world level are the same (σpi = σp). In contrast, the variance of supply fluctuations is generally much higher at the country level than at the world level. This occurs because supply fluctuations in a country are generally due to factors specific to that country, and because the supply fluctuations of different countries tend to offset each other when the supplies of each country are added to a world total.

These properties are illustrated for coffee and cocoa for the period 1961 through 1977 (Table 22). For coffee, the variance of fluctuations in export unit values is almost the same at the world level as at the country level: the variance of average world export unit values is equal to 98 per cent of the one which would have been calculated from country data by assuming that fluctuations of average export unit values are perfectly correlated among countries (993 compared with 1,018). In contrast, the variance of volume fluctuations at the world level is only 36 per cent of the one which would be calculated from country data by assuming that volume fluctuations are perfectly correlated among countries (85 compared with 235). Volume fluctuations are not, however, strictly independent among countries since the variance at the world level is about twice the one which would have been calculated from country data by assuming no correlation between volume fluctuations among countries (85 compared with 42). For cocoa the results are similar: fluctuations in unit values are strongly correlated among countries, but volume fluctuations are not.

Table 22.Variances of World Export Volume, Unit Values, and Values for Coffee and Cocoa
Actual Variance and Hypothetical

Variances Under Two Extreme

Assumptions
Variances of Fluctuations
Volume

σq2
Unit value

σp2
Total value

σx2
Coffee
Actual85993595
Assuming
no correlation between country
fluctuations14217399
perfect correlation between
country fluctuations22351,018716
Cocoa
Actual130909469
Assuming
no correlation between country
fluctuations165212135
perfect correlation between
country fluctuations2319988686

Calculated as σ2 = Σiαi2σi2 where αi, is the share of country i in world exports.

σ = Σiαiσi.

Calculated as σ2 = Σiαi2σi2 where αi, is the share of country i in world exports.

σ = Σiαiσi. For most of the countries accounting for a small share of world exports, fluctuations in the volumes exported are not correlated with fluctuations in the volume of world exports and, consequently, are not correlated with the fluctuations in world prices. Volume and prices generally fluctuate independently of each other, and the price elasticity coefficient bi generally does not differ significantly from zero. In that case, value fluctuations exceed volume fluctuations (σx2 = σq2 + σp2), and price stabilization contributes to stabilizing export earnings.

For the country which accounts for the major share of world exports, the situation is different. Fluctuations in the volume exported by the major producers are positively correlated with fluctuations in the volume of world exports and, consequently, are negatively correlated with world prices. For the major exporters, price fluctuations are, therefore, partly offset by volume fluctuations. In Brazil for coffee and in Ghana for cocoa, price stabilization would have reduced the fluctuations of export earnings only to a limited extent and to a much more limited extent than for the minor exporting countries.43

### (c) Commodity components

Suppose a country exports only two commodities, A and B. The variance of fluctuations in total earnings (σ2A+B) can be derived from the variance in earnings from A and B (σ2A and σ2B) as where RAB is the correlation coefficient between value fluctuations for commodity A and value fluctuations for commodity B.

It follows from equation (14) that the expected value of the net shortfall in total export earnings is always lower than the expected value of the sum of the commodity shortfalls, except when value fluctuations are perfectly correlated between commodities A and B: Because earnings from different commodities are never perfectly correlated, the net shortfall in earnings from all commodities (compensatory financing formula) is always smaller than the sum of commodity shortfalls (STABEX formula).

The expected value of the net shortfall remains, nevertheless, greater than the shortfall in any of its components, except when the fluctuations in the earnings of the components are strongly correlated negatively: Because earnings derived from different commodities are seldom strongly correlated negatively, and never perfectly correlated positively (—σA/2σB < RAB < 1), expanding the coverage of a compensatory financing scheme from A to A + B generally increases the size of the expected value of the net shortfall, but the cost of the expanded scheme is less than that of establishing two independent schemes for commodities A and B ## Shortfalls Under Compensatory Financing Facility

For the purpose of the Fund facility, the trend value of export earnings in year t is measured by the five-year geometric average centered on that year. Since the shortfall year is the latest 12-month period for which export earnings are known, earnings in the two post-shortfall years have to be forecast. The trend value and the amount of the shortfall, defined as the downward deviation from that trend, are therefore calculated as:  The five-year geometric average given in equation (15) is identical with the trend value which would have been calculated from a semilogarithmic equation (ln Xt = a + bt) adjusted by least squares for the five-year period centered on the shortfall year. Provided the same five years were used and the same logarithmic trend formula was applied, the amount of the shortfall calculated by least-squares regression would, therefore, be the same as the one calculated from equations (15) and (16).

With the extrapolation formula of the 1975 decision, earnings in the two post-shortfall years would be derived from earnings in the two pre-shortfall years by applying a three-year growth factor taken as The trend value given in equation (15) could, therefore, be derived from known values of export earnings by calculating post-shortfall earnings as: ### (a) Expected value of compensatory financing shortfalls

If export earnings were fluctuating randomly from an exponential trend as previously assumed in equation (1), the amount of the compensatory financing shortfall could be written as More generally, when the shortfall is defined as the downward deviation from an n-year moving average, its amount ηt becomes a linear combination of n error terms ϵt+τ: with coefficients (1—n)/n for τ = 0, and 1/n for τ ≠ 0. As all covariance terms are equal to zero, E(ϵtϵt+τ) = 0 for τ ≠ 0, the variance of ηt is given by: When ϵt is normally distributed, ηt is also normally distributed, as it is obtained as a linear combination of ϵt+τ. The expected value of positive shortfalls can therefore be derived from equation (3) as Approximating the trend by an n-year average has the effect of reducing the expected value of shortfalls, as the expected value of the shortfall defined in relation to the true trend is multiplied by $\sqrt{1-1/n}$ , which can be approximated by 1—1/2n when n is a large number. With a five-year average, the expected value of shortfalls is reduced by 10.6 per cent.

When a request under the compensatory financing facility is made, the values of the error terms are known for the shortfall year and the years before (ϵt+τ known for τ ≤ 0) but not for the years after. With the logarithmic model given in equation (1), the shortfall would be estimated at the end of year t as Consequently, the expected value of shortfalls would be given by If the shortfall year was excluded from the calculation of the trend value, the shortfall ζt would become a linear combination of ϵt+τ with coefficients—1 for τ = 0 and 1/n—1 for τ ≠ 0. Consequently, the variance of ζt would be given by when ϵt+τ are known for τ > 0, and when ϵt+τ are unknown for τ > 0.

Whether or not the shortfall year is included in the average, the variance of estimated shortfalls and, consequently, their expected value is reduced by taking the error terms in the post-shortfall years at their expected mean value. With a five-year moving average, shortfalls would be estimated on the average at 5 per cent below their true value.

The algebraic findings are summarized in Table 23, which also illustrates the significant estimation bias for a five-year average. Approximating the trend value by a moving average always reduces the expected value of shortfalls when the shortfall year is included in the calculation of the average, but always increases it when the shortfall year is excluded. The bias is significant when the number of years included in the average is small, but it is negligible when the number of years is large. These results may be understood intuitively by noting that a country presents a request under the compensatory financing facility precisely when its export earnings are abnormally low. Including a year with abnormally low exports in the calculation of the average leads to an underestimation of the trend value and, consequently, of the amount of the shortfall. By the same token, excluding years with abnormally low exports leads to an overestimation of the trend value and, consequently, of the amount of the shortfall.

Table 23.Effect of Replacing Logarithmic Trend Value by a Moving Average on Expected Value of Shortfalls
Treatment of Shortfall

Year in Calculation

of Trend Value
Estimation of Shortfall
Ex postEx ante Expected values1 Included$\sqrt{1-\frac{1}{n}}$$\sqrt{\left(1-\frac{1}{n}\right)\left(1-\frac{1}{2n}\right)}$
Excluded$\sqrt{1+\frac{1}{n-1}}$$\sqrt{1+\frac{1}{2\left(n-1\right)}}$ Approximate relative change Included$-\frac{1}{2n}$$-\frac{3}{4n}$
Excluded$\frac{1}{2\left(n-1\right)}$$\frac{1}{4\left(n-1\right)}$ Percentage change for n=5 Included−10.6−15.2
Excluded11.86.1

As a ratio of the expected value of the shortfall for the case of logarithmic trend.

As a ratio of the expected value of the shortfall for the case of logarithmic trend. It has been assumed above that export earnings were fluctuating randomly from a logarithmic trend. Actual export data show that a logarithmic trend provides a better fit than an arithmetic one but that fluctuations from the trend are generally serially correlated for two reasons. First, the value of the trend coefficient does not generally remain constant over a long period. Second, fluctuations from the trend may follow a cyclical pattern.

As the purpose of the facility is to provide assistance to members adversely affected by temporary shortfalls in their export earnings, the amount of the shortfall has to be defined in relation to a medium-term trend. Apart from data problems and difficulties of projecting export earnings many years ahead, the use of a long-term trend would not be consistent with the purpose of the facility, as illustrated for Brazil for the period 1958 through 1978 (Chart 6). When the trend line is fitted by least squares over the entire 21-year period, shortfalls occur during 11 consecutive years (1962 through 1972) and in no other years. In contrast, shortfalls never occur during more than 2 consecutive years when the trend is calculated as a 5-year moving average.

### Chart 6.Brazil’s Export Earnings, 1958–78 ## (b) Volume and price components

When values are decomposed into volume and average export unit value shortfalls expressed as percentages of the shortfall year levels can be defined as It follows that and for pt and qt small, ## (c) Commodity components

Defining the trend value as a geometric average instead of an arithmetic average helps to analyze the value shortfall in terms of its volume and price components. However, it complicates the analysis of the shortfall in total export earnings into its commodity components. With an arithmetic average, the net shortfall in total export earnings is the sum of the commodity shortfalls minus the sum of the commodity excesses. With a geometric average, the shortfall in total export earnings is less than the algebraic sum of the commodity shortfalls unless the growth rate of earnings is the same for all commodities. If earnings from different commodities increase exponentially at different rates constant through time, total export earnings will rise at an increasing rate, since the growth rate of total earnings will rise asymptotically toward the highest of the growth rates. Consequently, a shortfall will be calculated for total export earnings, although no shortfalls will be calculated for any of the commodity components.

## (d) Calculations in real terms

For the purpose of the Fund facility, shortfalls are calculated in relation to the nominal value of export earnings measured in SDRs. It has often been proposed that the calculations be conducted in relation to the real value of export earnings. Real earnings (RXt) would be calculated by dividing nominal earnings (Xt) by an index of the purchasing power of exports (It) taken as unity in the shortfall year so that The trend value of real export earnings would then be calculated as Replacing RXt+τ by its value from (20), and calling $\overline{{X}_{t}}$ , and $\overline{{I}_{t}}$ the trend values of the nominal earnings and of the price index the trend value of real export earnings becomes As the price index is taken as unity in the shortfall year, nominal and real export earnings have the same value in that year (Xt = RXt). The difference between nominal and real shortfalls is, therefore, identical with the difference between nominal and real trend values: Equation (21) shows that the amount of the shortfall is raised by conducting calculations in real terms (SRXt > SXt) when the price index in the shortfall year is above its trend value $\left({I}_{t}>\overline{{I}_{t}}\right)$ , but reduced when the price index is below its trend value. In other words, the amount of the shortfall is raised (reduced) by using real terms when the price level is abnormally high (low) in the shortfall year.

Call i the average rate of inflation from the two pre-shortfall years to the shortfall year and i+ the rate from the shortfall year to the two post-shortfall years Equation (21) can be rewritten Conducting calculations in real terms raises the amount of the shortfall (SRXt > SXt) if the rate of inflation falls after the end of the shortfall year (i+ < i), but reduces it if the rate of inflation increases after the end of the shortfall year (i+ < i). Periods of higher and lower inflation alternate and, for many consecutive years, the average shortfalls calculated in nominal and real terms are about the same.