Aggregation rules
Growth rates
World Bank Atlas method
Alternative conversion factors
This section describes some of the statistical procedures used in preparing the World Development Indicators. It covers the methods employed for calculating regional and income group aggregates and for calculating growth rates, and it describes the World Bank’s Atlas method for deriving the conversion factor used to estimate GNP and GNP per capita in U.S. dollars. Other statistical procedures and calculations are described in the About the data sections that follow each table.
Because of missing data, aggregations of data for groups of economies should be treated as approximations to unknown totals or average values. The regional and income group aggregates at the end of most of the tables are based on the largest available set of data, including the values for the 148 economies shown in the main tables, the smaller economies shown in tables 1.1, 1.2, and 1.3, and Taiwan, China. The aggregation rules are intended to yield estimates for a consistent set of economies from one time period to the next and for all indicators. However, small differences between the values of subgroup aggregates and overall totals and averages may occur because of the approximations used. In addition, discrepancies due to data reporting and accounting practices may cause differences in such theoretically identical aggregates as world exports and world imports.
How group aggregates in the World Development Indicators are calculated depends on the nature of the indicator. In group and world totals (indicated in the tables by t) missing data are imputed using a suitable proxy variable in a benchmark year, usually 1987. The imputed value is calculated so that it (or its proxy) bears the same relationship to the total of available data as it did in the benchmark year. Imputed values are not calculated if missing data account for more than one-third of the total in the benchmark year. Proxy variables are selected from a set of variables for which complete data are available for 1987. The variables used as proxies are GNP in U.S. dollars, GNP per capita in U.S. dollars, total population, exports and imports of goods and services in U.S. dollars, and value added in agriculture, industry, manufacturing, and services in local currency.
Aggregates of ratios are generally calculated as weighted averages of the ratios (indicated by w) using the value of the denominator or, in some cases, another indicator as a weight. The aggregate ratios are based on the available data, including data for economies not shown in the main tables. Missing values are assumed to have the same average value as the available data. If missing data account for approximately one-third of the total value of the weights in the benchmark year, no aggregate is calculated. In a few cases the aggregate ratio may be computed as the ratio of group totals after imputing values for missing data according to the rules for computing totals. Aggregates calculated as medians of the available data are indicated by m.
Aggregate growth rates are generally computed as the weighted averages of growth rates. In a few cases growth rates may be computed from time series of group totals (see the discussion below on methods of computing growth rates). Growth rates are not calculated if more than one-third of the observations in a period are missing.
Exceptions to the rules occur throughout the book. Depending on the judgment of World Bank analysts, the aggregates may be based on as little as 60 percent of the available data. In other cases, where missing or excluded values are judged to be small or irrelevant, aggregates are based only on the data shown in the tables.
Growth rates shown in the World Development Indicators are calculated as annual averages and represented as percentages. Except where noted, growth rates of values are computed from constant price or real value series. Three main methods are used to calculate growth rates: the least squares, the exponential endpoint, and the geometric endpoint. Rates of change from one period to the next are calculated as proportional changes from the earlier period. Note, however, that the annual changes in the speed of integration indicators in table 6.1 are not proportional growth rates but average annual differences.
Least-squares growth rate. Least-squares growth rates are used wherever there is a sufficiently long time series to permit a reliable calculation. If more than one-half of the observations in a period are missing, no growth rate is calculated.
The least-squares growth rate, r, is estimated by fitting a linear regression trend line to the logarithmic annual values of the variable in the relevant period. The regression equation takes the form
which is equivalent to the logarithmic transformation of the compound growth equation,
In this equation X is the variable, t is time, and a = log Xo and b = log (1 + r) are the parameters to be estimated. If b* is the least-squares estimate of b, the average annual growth rate, r, is obtained as [antilog (b*) – 1] and multiplied by 100 for expression as a percentage.
The calculated growth rate is an average rate that is representative of the available observations over the period. It does not necessarily match the actual growth rate between any two periods.
Exponential growth rate. The growth rate between two points in time for labor force and population indicators is calculated from the equation
where pn and p1are the last and first observations in the period, n is the number of years in the period, and ln is the natural logarithm operator.
This growth rate is based on a model of continuous, exponential growth between two points in time. It does not take into account the intermediate values of the series.
Geometric growth rate. The geometric growth rate is applicable to compound growth over discrete periods, such as the payment and reinvestment of interest or dividends. Although continuous growth, as modeled by the exponential growth rate, may be more realistic, most economic phenomena are measured only at intervals for which the compound growth model is appropriate. The average growth rate over n periods is calculated as
In calculating GNP in U.S. dollars and GNP per capita for certain operational purposes, the World Bank uses a synthetic exchange rate commonly called the Atlas conversion factor. The purpose of this conversion factor is to reduce the impact of exchange rate fluctuations in the cross-country comparison of national incomes.
The Atlas conversion factor for any year is the average of a country’s exchange rate (or alternative conversion factor) for that year and its exchange rates for the two preceding years, after adjustment for differences between the inflation rate in the country and the inflation rate in the G-5 countries (France, Germany, Japan, the United Kingdom, and the United States). The country’s rate of inflation is measured by its GNP deflator. The inflation rate for G-5 countries is measured by changes in the deflator for the SDR (special drawing right, the International Monetary Fund’s unit of account). The SDR deflator is calculated as a weighted average of the G-5 countries’ GDP deflators in SDR terms. The weights are determined by the amount of each currency included in one SDR unit. Weights vary over time both because the IMF changes the composition of the SDR and because the SDR exchange rate for each currency changes. The SDR deflator is calculated in SDR terms first and then converted to U.S. dollars using the SDR to dollar Atlas conversion factor.
This three-year averaging smooths annual fluctuations in prices and exchange rates for each country. The Atlas conversion factor is applied to the country’s GNP. The resulting GNP in U.S. dollars is divided by the midyear population for the latest of the three years to derive GNP per capita. When official exchange rates are deemed to be unreliable or unrepresentative of the effective exchange rate during a period, an alternative estimate of the exchange rate is used in the Atlas formula (see below).
The following formulas describe the computation of the Atlas conversion factor for year t:
and for calculating GNP per capita in U.S. dollars for year t:
where et* is the Atlas conversion factor (national currency to the U.S. dollar) in year t, et is the average annual exchange rate (national currency to the U.S. dollar) for year t, pt is the GNP deflator for year t, ptS$ is the SDR deflator in U.S. dollar terms for year t, Yt$ is the Atlas GNP in U.S. dollars in year t, Yt is current GNP (local currency) for year t, and Nt is the midyear population for year t.
Alternative conversion factors
The World Bank systematically assesses the appropriateness of official exchange rates as conversion factors. An alternative conversion factor is used when the official exchange rate is judged to diverge by an exceptionally large margin from the rate effectively applied to domestic transactions of foreign currencies and traded products, the case for only a small number of countries (see Primary data documentation). Alternative conversion factors are used in the Atlas method and elsewhere in the World Development Indicators as single-year conversion factors.