Aggregate Data

For all countries combined, the proportion of reserve assets denominated in U.S. dollars has declined from nearly 80 percent at the end of 1976 to 66 percent at the end of 1986, and this shift has been matched by a rise in the proportions held as deutsche mark (from 8 percent to 15 percent) and yen (from 3 percent to nearly 8 percent).² However, these changes in the aggregate currency composition of foreign exchange reserves have encompassed somewhat contrasting behavior on the part of industrial and developing countries. For the industrial countries, the erratic but significant decline in the share of reserves denominated in U.S. dollars—from 87 percent at the end of 1976 to 69 percent at the end of 1986—was accompanied by sharp increases in the shares of the deutsche mark (from 4 percent to 18 percent) and the yen (from 2 percent to 8 percent). For the developing countries, the pattern of currency diversification has been much more uneven. After an initial sharp decline in the dollar’s share between the end of 1976 and the end of 1980 (by 12 percentage points), this share was relatively stable between the end of 1981 and the end of 1986. In contrast, although the share of reserves denominated in yen more than doubled over the period between 1976 and 1986, the share of developing country reserves denominated in deutsche mark reached a peak of nearly 16 percent at the end of 1980 before declining to about 11 percent at the end of 1986.

Previous Studies

Previous analyses of the behavior of the currency composition of foreign exchange reserves have argued that the proportion of foreign exchange reserves denominated in a particular currency should be related either to the currency composition of the authorities’ exchange market activities (transaction approach) or to the risks and returns associated with holding reserve assets denominated in different currencies (the mean-variance approach). The transaction approach postulates that the desired currency composition of reserve assets is independent of the optimal distribution of net wealth across currencies. Instead, the holdings of reserve assets denominated in a particular currency would be related to transaction costs associated with the government’s purchases and sales of foreign exchange. The alternative mean-variance approach suggests that a rational government would not put all of its wealth in the one currency that has the highest expected yield, because of the possibility of highly variable outcome for its wealth depending on future exchange rates changes. Moreover, countries producing or consuming different goods and services would optimally hold different portfolios of financial assets.³

Heller and Knight (1978) provided evidence in support of the view that transaction needs played a major role in determining the currency composition of reserves. This study explained variations in the proportions of a country’s foreign exchange reserves held as assets denominated in the U.S. dollar, pound sterling, deutsche mark, French franc, and other reserve currencies as a function of the country’s exchange rate arrangements, which influenced both the foreign currencies bought and sold to defend such arrangements and the share of its trade with a particular reserve-currency country.⁴ Their results indicated that countries increased the proportion of their foreign exchange reserves held as a given reserve currency if they pegged their exchange rate to that currency or if the reserve center was an important trading partner.

In the late 1970s and early 1980s, several studies applied optimal portfolio theory to the selection of official reserve portfolios. For example, Ben-Bassat (1980, 1984) argued that a country’s optimal reserve- portfolio composition depends on three factors: (1) a country’s motivations for holding reserves, (2) the risk and return on the various reserve currencies, and (3) a country’s interest in maintaining international currency stability. Establishing the optimal reserve portfolio would first involve identifying the combinations of reserve positions that are on the efficiency frontier (that is, those positions which yield the highest rate of return for a given level of risk as measured by the variance of yields) and then allowing the authorities to select their preferred risk-return combination. Using data from the 1970s, Ben-Bassat (1984) compared actual and optimal reserve portfolios for both developing and industrial countries in 1976 and 1980. For the developing countries, actual and optimal portfolios in 1976 were regarded as similar. Moreover, this group reduced the U.S. dollar share in its actual portfolio between 1976 and 1980, which further reduced the gap between the actual and optimal reserve portfolios. The gap between the actual and optimal reserve portfolios for the industrial countries, however, increased between 1976 and 1980. In addition, the proportion of reserves denominated in the U.S. dollar held by industrial countries that were not part of the Snake arrangements (members of the European System of Narrower Exchange Rate Margins) increased, although the calculated optimal portfolio implied that this proportion should decline.⁵

In evaluating the use of the mean-variance and transaction approaches to explain the currency composition of reserves of developing countries, Dooley (1986) stressed that there were a number of difficulties associated with applying the mean-variance optimal portfolio approach.⁶ First, the mean-variance approach typically deals with allocation of wealth to net holdings of particular assets, whereas foreign exchange data refer to gross holdings of particular assets. Second, for the optimal portfolio approach to be useful it must encompass decisions regarding all of the financial positions, whereas reserve holdings constitute only a small subset of potential assets or liabilities. This means that both the yields on a variety of assets and liabilities other than reserve assets and the covariances of these yields with the returns on reserve assets would typically have to he included as determinants of optimal reserve positions. In addition, since reserve decisions are made by central banks that use their reserve instruments for transactions, it may be more efficient for the typical government to alter its net foreign currency position by changing the currency composition of its assets and liabilities that are held as reserves. As a result, the currency composition and level of foreign exchange reserves would most likely reflect the authorities’ transaction needs in the foreign exchange market. Finally, mean-variance portfolio considerations would be relevant for a single country, whereas data on the currency composition of foreign exchange reserves considered in most studies have referred to country groups rather than to individual countries. Dooley’s empirical analysis supported the view that reserve assets are held for transaction reasons, and the currency composition of such assets was determined by the consideration that they could be easily liquidated and used to make payments.

Recently, Horii (1986) examined reserve-currency diversification during the 1970s and 1980s. Although he noted that the currency composition of reserves should be studied on an individual country basis, Horii focused on country group data because individual country data were unavailable. After allowing for factors that altered the distribution of reserves across countries, he argued that, for all countries during the 1970s and 1980s, there was no large-scale diversification out of reserve assets denominated in the U.S. dollar. There was, however, some evidence of a movement out of reserves denominated in pounds sterling in 1972–76, a small shift from dollar- to deutsche-mark-denominated instruments in 1976–79 (mainly because of passive exchange rate changes), and a slight diversification out of dollar-denominated instruments in 1980–84. These results led Horii to conclude that the currency composition of reserve holdings could not be explained in terms of a transaction demand determined by trade flows alone, since capital flows have become increasingly important. Moreover, the stability of the currency composition of reserves during a period of significant changes in exchange rate arrangements did not support the view that such arrangements were important determinants of the currency composition, as suggested by Heller and Knight (1978). Finally, Horii calculated the reserve portfolios that should have existed under efficient portfolio theory for 1979 (using data from 1974 to 1979) and 1984 (using data from 1979 to 1984). In general, optimal holdings for reserves denominated in U.S. dollars were well below actual holdings in both years. In contrast, actual holdings of reserves denominated in deutsche mark, French francs, and pounds sterling were well below their calculated optimal levels.

Basic Model

In this section we consider a simple model that incorporates the role of the transaction costs into the traditional mean-variance approach to the determination of a country’s optimal portfolio of external assets and liabilities. It is argued that, although the mean-variance approach provides a description of the authorities’ net foreign asset (or liability) positions in each potential currency of denomination, the structure of transaction costs as well as the scale of a country’s anticipated foreign exchange market transactions are the principal determinants of each country’s gross asset positions, including its holdings of foreign exchange reserves.

In the standard optimal international portfolio model (see Roll (1977), Macedo (1980), and Horii (1986)), the optimal proportion of currency i in a country’s net foreign asset portfolio is determined by the expected real returns on holding positions in different currencies as well as by the covariances between the yields. If x_i represents the proportion of currency i in the authorities’ portfolio and X is the vector of the x_i, then the mean (m) and variance (σ²) of the return on the net foreign asset position will be given by

where R is the vector of expected real returns on maintaining a position in the various currencies and V is the covariance matrix of expected real yields.⁹

Under the assumption that the authorities’ utility is positively related to the expected return on their portfolio and negatively related to portfolio risk (for example, if $U=m-\left(b/2\right){\mathrm{\sigma }}^2$), the optimal vector of portfolio positions (X^*) will be given by (see Horii (1986)):

where e is the unit vector and b is the degree of relative risk aversion.

To illustrate the relationship between net and gross asset positions implied by the solution to equation (3), it is convenient to focus on the situation in which there are only two currencies: U.S. dollars (currency 1) and deutsche mark (currency 2). Let A_i represent the holdings of assets in currency i, L_i be the issuance of liabilities in currency i, and N_i equal the net asset position in currency $i\left(=A_i-{\mathrm{L}}_{\mathrm{i}}\right)$. Moreover, let the country be a net debtor, where $W(<0)$ represents the overall size of its net debt position (taken as exogenous).¹⁰ Assume that the solution to equation (3) indicates that the optimal net portfolio implies that half of the country’s debtor position should be denominated in dollars and half in deutsche mark $(N_1=N_2)$. A country with A^* in gross assets can still maintain its desired net debt position in each currency $(N_1=N_2)$ by altering the currency denomination of its liabilities (L^*). Suppose, for example, that to minimize transaction costs the authorities chose to hold most of their gross assets in dollars, so that $A_1^{*}>A_2^{*}$ (Figure 1). To maintain their desired net positions in each currency, the country would have to issue dollar and deutsche mark gross liabilities such that $L_1^{*}-A_1^{*}=L_2^{*}-A_2^{*}$. This implies that the theory of optimal net foreign exchange positions places no obvious theoretical restrictions on the currency denomination of a country’s gross reserve position.¹¹

View Full Size

Gross and Net Foreign Asset and Liability Positions

Citation: IMF Staff Papers 1989, 002; 10.5089/9781451947045.024.A004

• Download Figure
• Download figure as PowerPoint slide

In this two-currency model, the roles of transaction costs and risk-return considerations in determining a country’s net and gross asset positions can be described more formally as follows. As noted earlier, let A_i be gross holdings of reserve assets denominated in currency i, and let L_i be the gross external liabilities issued in that currency. The country can hold reserve assets in currency i that yield a random real world interest rate that has a mean of r_i and a variance of ${\sigma }_{r_1}^2\mathrm{.}$ Alternatively, the country can borrow in currency i and must pay $r_i+d_i,$ where d_i is a positive constant that reflects the spread between the lending and borrowing rates. The net interest earned on a given net asset position is $r_i(A_i-L_i)-d_i{L}_i$.

The expected return on the country’s foreign asset and liability positions will he given by

where $N_i=A_i-L_i\mathrm{.}$ The variance of this return will therefore equal

where

${\mathrm{\sigma }}_{r_i}^2$ = the variance of yield r_i

${\mathrm{\sigma }}_{r_i{r}_j}$ = the covariance of yields r_i and r_i.

In addition to earning net interest income, the authorities also incur transaction costs associated with their exchange market transactions.¹² To simplify, assume that the amount of transactions the country undertakes in each currency in each time period can be described by three possible states of nature (Figure 2):

View Full Size

Reserve Holdings and Transaction Structure

Citation: IMF Staff Papers 1989, 002; 10.5089/9781451947045.024.A004

• Download Figure
• Download figure as PowerPoint slide

(t₁, t₂) occurs with probability π₁ (point B in Figure 2)

(t₁, T–t₁) occurs with probability π₂ (point C in Figure 2)

(T–t₂, t₂) occurs with probability π₃ (point D in Figure 2),

with $T>t_1+t_2\mathrm{a}\mathrm{n}\mathrm{d}{\mathrm{\pi }}_{\mathrm{1}}\mathrm{+}{\mathrm{\pi }}_{\mathrm{2}}\mathrm{+}{\mathrm{\pi }}_{\mathrm{3}}\mathrm{=}1.$

Given this transaction structure, the country faces transaction costs that are influenced by two factors. First, there is the cost of converting one currency into another. If the authorities hold sufficient reserves denominated in a given currency (A_i) to meet the exchange market transactions in that currency, then it is assumed that the authorities do not incur any transaction cost. For example, assume that the authorities’ holdings of reserves are represented by point A in Figure 2, with A, of currency 1 and A₂ of currency 2. If actual transactions t₁ and t₂ (point B in Figure 2), the country would not incur any transaction costs because its reserve holdings in each currency would be sufficient to meet all transactions in the respective currencies. When the transactions are such that the authorities exhaust their holdings of reserves denominated in one currency, however, the authorities must convert the holdings of reserves denominated in the other currencies into the first currency, and this results in transaction costs. In Figure 2, for example, if actual transactions were represented by either point C or D, the authorities would incur the costs of converting reserves from one currency to another currency because the amount of reserves in one of the currencies will be lower than the level of transactions in that currency.

The second type of cost is associated with the possibility that the authorities may exhaust their reserve holdings. As Figure 2 is drawn, points C and D represent situations in which the country’s total reserves are inadequate. Such an outcome could force the authorities to engage in emergency borrowing, which is assumed to be relatively costly.

The costs of converting currencies or of engaging in emergency borrowing to offset reserve shortages are taken as being represented by quadratic functions of the amounts involved. In terms of Figure 2, if the outcome of transactions is B, there is no transaction cost because the level of reserves held in each currency is higher than the level of transactions in the respective currency, A₁ > t₁ and A₂ > t₂. If the outcome is C, holdings of currency 1 are higher than transactions in that currency, A₁ > t₁, but holdings of currency 2 are lower than transactions in that currency, A₂ > T – t₁. In addition, total holdings of reserves A₁ + A₂ are lower than total transactions T. Therefore, there is a cost associated with the conversion of the excess of currency 1 into currency 2, A₁ – t₁, and a cost associated with the overall shortage of reserves, $T-A_1-A_2$. The same type of reasoning applies to outcome D. As a result, the expected conversion and reserve-shortage costs for holdings of reserves A₁ and A₂ are given by¹³

where c and p are parameters associated with the conversion of reserves from one currency to another and with reserve shortages, respectively.

In determining their holdings of foreign assets and issuance of foreign liabilities, the authorities are assumed to maximize a utility function that is a positive function of the expected return (m) on their net foreign asset portfolio (net of expected transaction and emergency borrowing costs) and is negatively related to the variance (σ²) of the yield on that portfolio. In particular, the authorities select A₁, A₂, L₁, and L₂ subject to the constraint imposed by the size of their overall net foreign asset position, $W=A_1+A_2-L_1-L_2=N_1+N_2$(where W can be either positive or negative). Thus,

where

and where E(tc) is as defined in equation (6).

As shown in Appendix II, the first-order conditions yield

where $D={\mathrm{\sigma }}_{r_1}^2+{\mathrm{\sigma }}_{r_2}^2-2{\mathrm{\sigma }}_{r_1{r}_2},$ and A₁ and A₂ are given by

Equations (8) and (9) imply that the country’s net foreign asset positions in each currency are determined by the expected yields (or borrowing costs), the variances and covariances of these yields, and the degree of relative risk aversion; these positions, however, are independent of the structure of transaction costs or the likely volume of exchange market transactions. In contrast, equations (10)—(13) imply that gross holdings of reserve assets will be influenced by transaction costs associated with currency conversion and reserve shortages, and by the minimum and maximum levels of potential exchange market transactions. As a result, the currency composition of foreign exchange reserves will also reflect these transaction considerations.

Empirical Model

The hypothesis that transaction needs are the principal determinants of the currency composition of foreign exchange reserves can be examined in terms of the behavior of the proportion of reserves denominated in each of the major reserve currencies. In particular, equations (10) and (11) imply that the currency composition of reserves (as represented by A_i/A) would he sensitive to the scale of transactions in a given currency relative to total transactions (as well as other variables). But the scale of exchange market transactions undertaken by the authorities would also be influenced by the nature of the exchange rate arrangements they select. For example, maintaining a fixed exchange rate might require a higher level of exchange market intervention in a particular currency than maintaining a floating exchange rate. This relationship between the currency composition of reserves, the relative scale of exchange market transactions in different currencies, and exchange rate arrangements can be represented empirically by

where

t = 1, …, T (number of periods)

i = 1, …, n (number of countries)

k = 1, …, 5 (number of reserve-currency countries)

s = 1, …, 5 (number of exchange rate arrangements)

A_i,k,t = reserves of country i held as assets denominated in the currency of reserve country k at time t (converted to U.S. dollars at the end of the period)

D_i,v,t = debt service payments of country i denominated in the currency of reserve currency country v at time t

E_i,s,t = exchange rate arrangement of type s adopted by country i at time t

${\overline{A}}_{i,}t$ = total end-of-period foreign exchange reserves for country i at time t (measured in U.S. dollars)

TT_i,t = sum of exports, imports, and (in the case of developing countries) debt-servicing payments

TR_i,v,t = trade flows (exports plus imports between country i and reserve currency country v) at time t.

In this formulation, the proportion of a country’s reserves held in assets denominated in a particular currency is assumed to be influenced by the currency composition of both its trade flows and debt-servicing payments as well as by the nature of its exchange rate arrangements. Trade transactions denominated in a particular reserve currency are represented by the sum of imports and exports between country i and reserve-currency country v. The reserve-currency countries in this study are taken to be France, the Federal Republic of Germany, Japan, the United Kingdom, and the United States. Because the currency composition of a country’s import and export contracts need not correspond to the country pattern of its trade flows (for example, contracts for oil imports from a Middle East oil producer could be denominated in U.S. dollars), this measure can he taken as only an approximation to the true proportion of trade flows denominated in a particular currency. Despite these shortcomings, earlier studies (for example, by Heller and Knight (1978)) of the currency composition of foreign exchange reserves have found that this measure of the distribution of trade flows is a useful explanatory variable. However, the signs of the β_{i, v} are subject to some ambiguity. Although an increase in trade flows to a given reserve currency v would he likely to lead to increased holding of reserves denominated in that reserve currency $\left({\mathrm{\beta }}_{i,v}>0\right)$, the signs of the other ${\mathrm{\beta }}_{i,v}\left(i\ne v\right)$ could be either negative or positive. For example, larger imports from reserve-currency country A could imply the need for larger holdings of reserve country B’s currency if some of these imports were priced in terms of B’s currency.

Because data on financial flows are not available for most countries, the scale of financial flows was proxied by the level of interest payments associated with the country’s external debt denominated in the different currencies. External debt was measured in terms of a country’s public and publicly guaranteed debt as reported in the World Bank’s World Debt Tables (various issues). Because these data are reported by currency of denomination, the interest payments in the different currencies were approximated by multiplying the stock of debt denominated in each currency by the six-month Eurocurrency deposit rate for that currency. Obviously, this measure captures only one component of financial flows; but, for many of the developing countries in the sample, interest payments on external debts account for a significant portion of total financial flows.

The effects of a country’s exchange rate arrangements on the currency composition of its foreign exchange reserves were represented by a series of dummy variables corresponding to the type of arrangements used by a country during each year under consideration. The arrangements were classified into categories similar to those used in the Fund’s Annual Report on Exchange Rate Arrangements and Trade Restrictions (various issues). The categories include pegging to the U.S. dollar, pegging to the French franc, pegging to a composite or other currency membership in a cooperative arrangement, and maintenance of a flexible exchange rate.¹⁴ Although the placement of countries within this spectrum of arrangements is to some degree arbitrary, these categories do reflect the employment of greater or lesser degrees of exchange rate flexibility. In the regressions, a country’s exchange rate arrangement is represented by a series of zero-one dummy variables, one for each type of arrangement. Because there is a general intercept term in each regression (the β₀) and the set of exchange rate arrangements is exhaustive, one of the exchange rate dummies was excluded in each regression to avoid creating a linear dependency. The β₀, in each regression therefore reflect the effects of both the excluded exchange rate arrangement and other factors not represented by trade and capital flows.

Because earlier analyses found that the behavior of the currency composition of reserves of the industrial and developing countries differed, the present study examines separate regressions for these groups. Moreover, although the World Bank reporting system for debt contains extensive information on the currency composition of the external debt of the developing countries, no such comparable reporting system exists for industrial countries. Some industrial countries do not appear to have any external debt denominated in foreign currencies (for example, the Federal Republic of Germany and Japan), but many others (for example, certain Scandinavian countries) have significant amounts of such debt. The absence of comprehensive information on the currency composition of external debt for the industrial countries means, however, that the interest payments variable has been excluded from the industrial country regressions.

This specification was estimated using data on the currency composition of foreign exchange reserves supplied to the International Monetary Fund on a regular basis by a large number of central monetary institutions. A series of five cross-section time-series regressions for each of the groups of industrial and developing countries were estimated. In these regressions, the dependent variables were the proportion of foreign exchange reserves held as instruments denominated in U.S. dollars, pounds sterling, deutsche mark, French francs, and yen. The explanatory variables were the exchange rate regime, the five variables representing trade with the reserve centers (as a proportion of total trade plus interest payments), and, in the case of developing countries, five variables reflecting the interest payments on external debt (as a proportion of total trade plus interest payments). Data from a total of 19 industrial countries and 39 developing countries were included.¹⁵ The data consist of annual observations for each country’s variables for the period 1976–85. Note that in this formulation the marginal effects of exchange rate arrangements, trade flows, and interest payments are assumed to be uniform across countries (that is, there are no country-specific parameters).¹⁶

Estimation Methods

Estimation of equation (13) using ordinary least-squares techniques would in general be inappropriate, since the dependent variable can only take values in the interval (0, 1). To illustrate the nature of the problems involved, let the standard regression model be represented by

where β is a k × 1 vector of unknown parameters; x_i is a k × 1 vector of known independent variables, and the u_i are residuals independently and normally distributed with a mean of zero and a common variance of σ². Because the u_i, are assumed to be distributed normally, this model implies that y_i may take any positive or negative value, which is inconsistent with the particular dependent variable in the problem.

More appropriate for the purposes here is the censored-regression model, also known as the tobit model.¹⁷ In this particular case, the tobit model can be represented by

if the right-hand side is greater than zero but less than unity,

if the right-hand side is less than or equal to zero, and

if the right-hand side is greater than or equal to unity, and where the β, and u_i are defined as above. This model implies that the authorities decide on the share of a particular currency in their foreign exchange reserves according to the linear function in equation (15), but they hold a share of either zero or unity when that linear function indicates a negative share or a share higher than unity, respectively.

Estimation of a censored-regression model by ordinary least squares results in biased and inconsistent estimators. Maximum likelihood estimators, on the other hand, are consistent and asymptotically normal (see Amemiya (1973)). Among the various procedures available to obtain maximum likelihood estimates, we used an iteration method suggested by Fair (1977).¹⁸ Although the problem examined in this paper is strictly described by the two-limit censored-regression model in equations (15)–(17), we estimated a one-limit model because there were virtually no observations on the upper limit,¹⁹ and observations on both limits are needed to estimate a two-limit rnode1.²⁰ Therefore, the estimated model was

if the right-hand side is greater than zero, and

otherwise.

As indicated by Dhrymes (1986), it is convenient to have goodness of fit statistics for observations with y_i > 0 that are separate from those for observations with y_i = 0. For the set of observations with y_i > 0, we have therefore used the square of the simple correlation coefficient between the predicted and actual values of the dependent variable $\left(R_{T1}^2\right)$.²¹ For the set of observations with y_i = 0, we calculated the proportion of observations correctly predicted to be zero by the model $\left(R_{T2}^2\right)$ and the mean error $\left({\overline{E}}_T\right)$ for the observations for which the model incorrectly predicted y_i > 0.²² To give an idea of the overall goodness of fit, we also calculated the square of the simple correlation coefficient between predicted and actual values of the dependent variable for the complete sample $\left(R_{T3}^2\right)$.²³

One difficulty with the tobit maximum likelihood estimator is that it is sensitive to violations of the assumptions that its error terms have a normal distribution and are homoscedastic. The presence of either nonnormality (see Goldberger (1983)) or heteroscedasticity (see Hurd (1979)) can result in inconsistent tobit estimates. However, Powell (1986) has recently developed estimators that are robust for a wide set of nonnormal or heteroscedastic disturbance distributions. As noted by Newey (1987), Powell’s symmetrically censored least-squares (SCLS) estimator not only provides estimates of the regression parameters that are robust to failure in the normality assumption but also make feasible Hausman (1978) tests for the presence of heteroscedasticity or non-normality on the basis of differences between the tobit estimates and the SCLS estimators.

If the normality or homoscedasticity assumptions are violated, then the use of the SCLS estimator involves symmetrically censoring the dependent variable so that symmetry of its distribution about the regression $\mathrm{\left(}{\mathrm{\beta }}^{\mathrm{\prime }}{x}_i\right)$ is restored, and least-squares techniques will yield consistent estimators. The nature of the SCLS procedures can he illustrated by Figure 3.²⁴ As given in equations (18) and (19), $y_i=\max \left\{0,{\mathrm{\beta }}^{\prime }{x}_i+u_i\right\}$ $(i=1,\mathrm{...},T)$. In a censored sample, for data points with $u_i<-{\mathrm{\beta }}^{\prime }{x}_i,$ the value of y_i is taken to be zero. In contrast to the sided (lower-bound) censoring of the tohit model, however, the SCLS procedures set y_i equal to $2{\mathrm{\beta }}^{\prime }{x}_i$ when $u_i>{\mathrm{\beta }}^{\prime }{x}_i$. Then the observations would have terms in the interval $\left(0,2{\mathrm{\beta }}^{\prime }{x}_i\right)$. Because this approach assumes that the original errors were distributed symmetrically, the residuals of the “symmetrically censored” regressions will also be symmetrically distributed about zero, and the dependent variable will take on values between zero and $2{\mathrm{\beta }}^{\prime }x$ and will be symmetrically distributed about ${\mathrm{\beta }}^{\prime }x$ (Figure 3). For such a symmetric sample, Powell (1986) derived a series of “normal” equations that allowed for the estimation of the SCLS parameter (β_s) as well as their standard errors (see Appendix III for the normal equations).²⁵

View Full Size

Symmetrically Censored Least Squares

Citation: IMF Staff Papers 1989, 002; 10.5089/9781451947045.024.A004

Source: Powell (1986, p. 1439)

• Download Figure
• Download figure as PowerPoint slide

In addition, Newey (1987) has shown that one can use the tobit and SCLS estimates (and their variance-covariance matrices) to derive a Hausman test statistic (h) which tests to see if the normality and homoscedasticity assumptions of the tobit analysis are violated. If ${\hat{\mathrm{\beta }}}_t$ and ${\hat{\mathrm{\beta }}}_s$ are the vectors of tobit and SCLS estimated parameters, then $h=T{\left({\hat{\mathrm{\beta }}}_s-{\hat{\mathrm{\beta }}}_t\right)}^{\prime }{\left[V\left({\hat{\mathrm{\beta }}}_s-{\hat{\mathrm{\beta }}}_t\right)\right]}^{-1}\left({\hat{\mathrm{\beta }}}_s-{\hat{\mathrm{\beta }}}_t\right)$, where T is the number of observations, $V\left({\hat{\mathrm{\beta }}}_s-{\hat{\mathrm{\beta }}}_t\right)$ are estimates of the asymptotic covariance matrix for $T\left({\hat{\mathrm{\beta }}}_s-{\hat{\mathrm{\beta }}}_t\right)$ and h will be distributed as x² with k (number of exogenous variables) degrees of freedom. (See Appendix III for the estimate of $V\left({\hat{\mathrm{\beta }}}_s-{\hat{\mathrm{\beta }}}_t\right)$.) In the empirical analysis that follows the SCLS and tobit estimates are examined to see if the normality and homoscedasticity assumptions have been violated.

Empirical Results

The empirical results indicate that transaction variables and exchange rate arrangements have played important roles in determining the currency composition of foreign exchange reserves throughout the period 1976–85. The estimation results for both the developing and industrial countries for the period 1976–85 are reported in Tables 2–11, which appear together, for ease of comparison, following the appendices.