An Econometric Model of U.S. Merchandise Imports Under Fixed and Fluctuating Exchange Rates, 1959-73

Econometric models of U. S. imports have usually emphasized the estimation of import demand relationships while paying little attention to the behavior of foreign suppliers and to the impact of this behavior on import prices.1 In forecasting import values, these models were generally not equipped to distinguish between volume and price changes. This was a relatively minor limitation under the conditions of substantial price and exchange rate stability which prevailed during the 1950s and 1960s, but it has become a major drawback in the more recent period.


Econometric models of U. S. imports have usually emphasized the estimation of import demand relationships while paying little attention to the behavior of foreign suppliers and to the impact of this behavior on import prices.1 In forecasting import values, these models were generally not equipped to distinguish between volume and price changes. This was a relatively minor limitation under the conditions of substantial price and exchange rate stability which prevailed during the 1950s and 1960s, but it has become a major drawback in the more recent period.

I. Introduction

Econometric models of U. S. imports have usually emphasized the estimation of import demand relationships while paying little attention to the behavior of foreign suppliers and to the impact of this behavior on import prices.1 In forecasting import values, these models were generally not equipped to distinguish between volume and price changes. This was a relatively minor limitation under the conditions of substantial price and exchange rate stability which prevailed during the 1950s and 1960s, but it has become a major drawback in the more recent period.

The present study analyzes the behavior of U. S. merchandise imports disaggregated by end-use categories over the period 1959–73. It extends the analysis to the more turbulent quarters of the era following the 1971 Smithsonian Agreement and must therefore deal extensively with the determination of the price as well as the volume of U. S. imports. Accordingly, it must take into account a number of complications that had not been dealt with formally in previous econometric studies of U. S. trade, such as the currency-denomination of import contracts and the possibility of absorption of exchange rate effects by foreign exporters.2

Some of the major problems in the empirical analysis of merchandise trade arise in moving from theoretical model building to statistical estimation. Therefore, a considerable effort was directed to the construction of variables that would approximate as closely as possible the concepts defined in the theoretical part of the study. Imports were adjusted to exclude categories largely determined by factors exogenous to the model (automotive imports from Canada because of the 1965 U. S.-Canadian Automotive Agreement, and fuels and lubricants because of the recent behavior of petroleum prices). The weights used to compute averages of foreign prices and exchange rates for aggregate imports and for each end-use category were also adjusted to exclude these commodities; and both domestic and import price indexes were adjusted to exclude the effect of oil prices.3 In addition, the authors relied heavily on the work of Parrish and DiLullo (1972a), Wilson (1973), Isard, Lowrey, and Swamy (1974), and Takacs (1974) for the specification and construction of tariff, strike, and cyclical variables.

The lack of reliable price data for traded goods has been a major obstacle to econometric work on U. S. foreign trade. The closest approximations to import price indexes now available for sufficiently long periods are the unit value indexes prepared by the U. S. Department of Commerce. Because a unit value index measures changes in the average value of imports per physical unit—and given the heterogeneity of the various categories used in import statistics—a change in the unit value index generally involves an unknown combination of price change and variation in the product mix. This problem, together with other difficulties that have been extensively documented in previous studies,4 have prompted many authors to use averages of partner-country wholesale prices instead of unit values in estimating import equations.5 However, this procedure may introduce serious specification errors in periods of wide exchange rate fluctuations. It will be shown that, in such periods, the magnitude and timing of changes can be substantially different for import prices and for foreign wholesale prices, even if the latter are “adjusted” for changes in exchange rates.6

In many situations, therefore, the use of unit value indexes may well be the lesser of two evils. Unfortunately, import unit values disaggregated by end-use categories are not available for sufficiently long periods.7 In some cases, this obstacle can be surmounted. For example, indexes for some end-use categories can be constructed by appropriate combinations of unit values based on the classification of imports by degree of processing, which are available beginning with 1959. In other cases, however, this cannot be done, and estimation must proceed without a satisfactory import price variable.

Against the background of these statistical difficulties, the analysis presented in this paper proceeds as follows: (1) a complete import model is specified as if an appropriate import price index were available; (2) using this complete model, an alternative specification is derived, which does not require specific information on import prices; (3) the complete model is estimated by using a unit value index in lieu of import price if a reasonably appropriate unit value index can be constructed; otherwise, the alternative model described under (2) is estimated by using information on the prices of main trading partners.

The theoretical structure of the model is presented in Section II, while Section III deals with the estimation of aggregative import volume and import price equations. The econometric results for the various end-use categories are reported in Section IV, and Section V summarizes the main results of the study.

II. Structure of the Model

demand function for volume of imports

The first ingredient of the complete import model is an aggregative relationship expressing the real demand for imports as a function of domestic real expenditure (Yd), the price of domestic goods (Pd), and the price of imported goods (Pm). Furthermore, it is recognized that domestic goods markets are cleared not only by movements in nominal (quoted) prices but also by changes in a set of nonprice variables, such as waiting times for delivery, discounts, credit terms, and quality standards. These variables should be considered as arguments in the demand function, since they influence the allocation of expenditures between domestic and foreign goods.8 Using the symbol Wd to summarize the impact of these nonprice variables, the demand function for imports in volume terms can be written as:


where M and M¯ stand for the nominal and real value of imports, respectively. The partial derivatives of f are assumed to be positive with respect to Yd, Pd, and Wd; and negative with respect to Pm.

Because of the existence of delivery lags and adjustment delays, each independent variable in equation (1) should be interpreted as representing a weighted sum of current and lagged values of that variable. To simplify the notation, it is assumed that the only variables that affect imports with a distributed lag are Pd and Pm. (The empirical analysis presented in Sections III and IV allows, however, for the possibility of lagged effects from all the explanatory variables.) It is also assumed that the function f can be approximated by a log-linear expression. If lower case letters stand for natural logarithms—for example, x = loge(X)—equation (1) can be rewritten as:


where m¯=mpm. The ajis are non-negative weights, and their sum for i = 0, … l is unity. Each α represents a sum of lagged coefficients and can be interpreted as a long-run demand elasticity; α3 is expected to be negative, while α1, α2, and α4 are expected to be positive.

It may be noted that pm and pd are included as separate explanatory variables on the right-hand side of equation (2). In many trade models, the inclusion of a single relative price variable (pmpd) is justified by postulating the absence of “money illusion.” Although the present model does not appeal to this postulate, it is worth noting that this justification is quite dubious on several counts. First, in a world in which imported and domestic goods differ in terms of both price and nonprice characteristics, different reactions to opposite but equal changes in pm and pd do not necessarily represent symptoms of “money illusion.” 9 Second, the use of a single relative price variable can be justified by the assumption of no money illusion if consumers allocate expenditures among two goods only (say, domestic and imported goods). If, however, consumers face n goods—and hence n prices—the demand function for each of these n goods must include n — 1 relative price variables, even in the absence of money illusion. Finally, there is a practical problem stemming from differences in the construction of unit value and wholesale price indexes. In the United States, unit value indexes are based on Fisher’s formula, while wholesale price indexes are based on Laspeyres’ formula, and it has been shown that these two types of indexes describe different time paths even if the commodity composition remains unchanged.10 Therefore, even a model that constrains the theoretical elasticities with respect to the “true” prices pd and pm to be equal in absolute value should allow the estimated elasticities with respect to the price indexes p¯d and p¯m to differ because of the methodological differences between the two indexes.

The elements of the vector Wd are not subject to direct measurement, although it is shown in Section IV that an adequate proxy for domestic delivery lags of industrial supplies and materials can be specified.11 More generally, it is hypothesized that the intensity of the rationing mechanisms—that is, the magnitude of excess demand at the given (nominal) price—can be expressed as a function of the discrepancy between actual and desired rates of domestic resource utilization12 denoted by R and R^, respectively.


Chart 1 shows three possible ways to approximate the function g subject to the conditions g > 0 and g(0) = 0. The identity function is the most simplified but not necessarily the most realistic from the economic standpoint. A somewhat more attractive alternative is the nonlinear expression


where ψ is a positive scalar. If ψ = 1, the variable is symmetric with respect to the origin (Chart 1). Such a variable was used by Parrish and DiLullo (1972a) in estimating aggregate import equations for the United States. R was taken to be the ratio of actual to potential gross national product (GNP), and R^ was estimated at 0.97. If ψ > 1, the variable Wd is nonlinear and asymmetric (dotted line in Chart 1). This asymmetry reflects the assumption that, ceteris paribus, the stimulating impact on imports of capacity bottlenecks is larger than the depressive impact resulting from excess capacity. This is likely to occur when, for example, outstanding contracts impose a limit on how far importers can turn from foreign to domestic suppliers in response to an improvement in the nonprice characteristics of domestic goods.

Chart 1.
Chart 1.

Alternative Versions of the VariableWd

Citation: IMF Staff Papers 1975, 003; 10.5089/9781451969382.024.A007

the import price function

The determination of the import price (pm) is analyzed in terms of a two-stage procedure, which has practical as well as expositional advantages. The first step deals with the transition from the import price (more specifically, in the U. S. case, from the import unit value) to the foreign export price expressed in foreign currency. The second stage deals with the specification of the foreign export price in terms of the exchange rate and of domestic prices in both the importing and the exporting country.

(1) Let px denote the foreign export price expressed in foreign currency, and let k stand for the exchange rate expressed in units of foreign currency per U. S. dollar.13 As before, pm is the U. S. import price expressed in U. S. dollar terms, and all variables are defined in terms of natural logarithms. If λ denotes the proportion of contracts denominated in U. S. dollars, θi the proportion of current imports contracted in period ti, and the subscript i indicates the delivery lag, the unit value may be expressed as 14


Rearranging terms and noting that Σθi = 1 by definition, equation (5) can be written more compactly as


where θ0=1λ(1θ0)0θi=λθiVi0andΣi=0nθi=1.

Equation (6) expresses pm as a linear combination of current and lagged values of foreign export prices and exchange rates. It could be estimated by standard regression techniques if an appropriate weighted average of foreign export prices could be constructed. Since this is not a feasible alternative, it is necessary to go a step further and examine the price-setting behavior of foreign exporters.15

(2) Given an optimal long-run export price p^x, the problem of short-run price determination can be viewed as an attempt by the exporting firm to achieve the best possible compromise between several competing targets. Obviously the firm will strive to minimize the cost of being out of equilibrium; at the same time it will try to prevent the emergence of a large discrepancy between its export price measured in U. S. dollars and the price charged by its U. S. competitors—one which could lead to a serious and perhaps irreversible loss in its share of the U. S. market; and finally it will attempt to avoid any sharp widening in the gap between actual and desired levels of resource utilization. One simple and yet reasonably realistic solution to this optimization problem is to set px so as to minimize the (positive definite) quadratic loss function:


where yf and ŷf are output levels corresponding to actual and desired levels of foreign capacity utilization, respectively.

As an illustration, it may be noted that a positive value for h1 implies that, faced with an exogenous decrease in k, the foreign exporter will attempt to “absorb” part of the dollar devaluation impact by reducing the local currency value of its export price (px).

The foreign producer’s optimal long-run export price (p^x) is assumed to be related to its prevailing home market price (pf) by the simple log-linear relationship p^x=ε+pf.16 Then, denoting the demand elasticity of yf with respect to the export price px by η = ∂yf/∂px < 0 and setting ∂L/∂px = 0 yields the first-order minimum condition:


where ξ0 = εh0/h > 0, ξ1 = h0/h > 0, ξ2 = h1/h > 0,

and ξ3 = —ηw′h2/h > 0

where h = h0 + h1 and w′ = ∂wf/∂yf > 0.

Substituting for px from equation (8) in equation (6) then gives


where ξ2=(1ξ2)<0.

reduced form expression for nominal imports

As was pointed out in Section I, the complete import model composed of equations (2) and (9) cannot be estimated for all end-use categories because import prices (even in the deficient form of import unit values) are not available for a sufficiently long period. For some categories—foods, industrial supplies, and materials—it is possible to construct alternative unit value indexes that are fairly adequate in terms of coverage. But this is not feasible in other cases (for example, automotive imports and capital goods). Even then, however, the model presented above in this section can be used to derive an equation which can be estimated, provided there is information concerning prices in the exporting countries.

Suppose, for instance, that an average px of foreign export prices in foreign currency is available. Then, taking the value of pm from equation (6) and substituting for this value in equation (2) yields the expression


It may be noted that the lag length for both px and k is equal to (l + n)—that is, to the lag with which the demand for imports adjusts fully to a change in import prices (l) plus the lag with which the import price reacts fully to a change in exchange rates and foreign prices (n). It may also be pointed out that, in the general case, the distribution of the cis will be humped, even if the lag distributions for ai and θi are monotonically declining.17

Since m¯=mpm, adding pm on both sides of equation (10) gives


Equation (11) requires data on px and k, but not on the import price (pm).

Suppose now that px is not available, but that there is information on the domestic prices of major trading partners, which can be used to construct the weighted average foreign price (pf). Then, taking the value of px from equation (8) and substituting it in equation (11) yields


where ζ0 = α0 + ξ0(1 + α3), ζ1 = ξ1(1 + α3), ζ2 = α2 + (1 + α 3)ξ2

ζ3 = (1 + α33 and ζ4=(1+α3)ξ2.

If α3 < -1 (an assumption for which considerable empirical evidence will be offered in the following sections), ζ1 and ζ2 will be unambiguously negative, and ζ4 unambiguously positive. Since 0 ≤ ξ2 < 1, a sufficient condition for ζ2 to be positive is 1 + α2 > — α3.

The reduced form (12) is several steps removed from the original equations of the model, and linear least-squares will not produce estimates of all of the original (structural) parameters. If, however, there is no information on import prices, equation (12) provides an operationally attractive specification of the import function, and one which is consistent with the original assumptions of the model. If these assumptions—including the existence of absorption and of import contracts denominated in foreign currency—are correct, then it is clear from (12) that the widespread procedure of collapsing the price and exchange rate variables into a single “relative price variable” [pd — (pfk)] is incorrect. And while this type of misspecification is likely to be rather benign as long as there are no sharp fluctuations in exchange rates, it will have serious consequences if the sample includes a period such as 1971–73.

III. Estimation Result: Aggregate Imports

This section presents the econometric results of aggregate import demand and import price equations based on the model specified in Section II. It also introduces some of the key variables and procedures used throughout the remainder of the paper. The estimation results for end-use categories will be presented in Section IV.

aggregate import equation

Table 1A summarizes the results obtained from estimating the aggregate import demand equation (2) from the first quarter of 1960 to the fourth quarter of 1973. The dependent variable (m¯=mpm) was defined as the logarithm of U. S. merchandise imports (census basis), net of fuels and lubricants and automotive imports from Canada, minus the logarithm of the aggregate import unit value adjusted to exclude petroleum prices. The price of import-competing goods (pd) was approximated by the U. S. wholesale price index, which seemed appropriate since imported goods carry a very small weight in the wholesale price index.18 All variables were seasonally adjusted, except the unit value pm for which there was no indication of seasonality.

Table 1.

Import Volume and Import Price Equations for Total Imports, Excepting Fuels and Lubricants and Automotive Imports from Canada

(Based on quarterly data)

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Note: The variable pm is the unit value of U.S. general imports, adjusted to exclude changes in petroleum prices;m¯ is the logarithm of U.S. merchandise imports (less fuels, lubricants, and automotive imports from Canada) deflated by pm. The symbol yd stands for real personal consumption expenditure in equation (1.1) and for real personal consumption expenditure on goods in equations (1.2) through (1.5). For definitions of other variables, see the text and Appendix, Table 10. All lower case Latin letters refer to natural logarithms.

Columns (1) through (5) in Table 1A give the estimated coefficients of the constant term and the four explanatory variables included in equation (2), while columns (6) through (9) show the coefficients of various strike variables explained below. Column (10) shows the value of the estimated first-order autocorrelation coefficient ρ^ which applies only to the autoregressive equation (1.5). Finally, column (11) gives the R¯2 and Durbin-Watson statistics. Figures in parentheses below the coefficients are t-ratios.

In the first equation of Table 1A, yd is defined as personal consumption expenditures at constant prices. The more narrow concept of personal consumption expenditures on goods (durable plus nondurable) is used in the other four equations. While the coefficients and standard errors of the price variable are not dramatically affected by these changes, the equations using expenditure on goods only have a slight advantage in terms of standard error and freedom from autocorrelation.

The consequences of using alternative versions of the nonprice rationing variable are somewhat more visible. In equation (1.3), Wd is the gap between actual and potential GNP, while in the last two equations this variable was generated according to the formula of equation (4), R being equal to the ratio of actual to potential GNP, and R^ being equal to 0.97. The standard errors associated with this second version of Wd proved to be consistently lower. The value of ψ (estimated through a scanning procedure) was not found to be significantly different from unity.19 Equations (1.3) and (1.4) performed somewhat better than equation (1.2), which does not include a nonprice rationing variable.

In Table 1, the coefficients for pm and pd are sums of lagged coefficients. (This is indicated by the symbol Σ, with the number of lagged terms given above the summation sign.) These price coefficients—which can be interpreted as long-run elasticities—correspond to α1 and α2, respectively, in the symbolism of equation (2). They were quite robust with respect to minor changes in specification, and they passed the t-test quite comfortably at the 0.01 level. The individual lagged coefficients, shown in Table 2, were constrained to lie upon a second-degree polynomial of length l and constrained to zero at t - l - 1.20 These are severe constraints, but they are far less rigid than those that would have resulted from using an (infinite) exponentially declining lag structure of the Koyck type.21 The length of the polynomial was determined by minimizing the standard error of the sum of lagged coefficients, subject to the constraint that none of the weights (the a’s) should be significantly smaller than zero.22

Table 2.

Distributed Lag Coefficients for Aggregate Import Volume and Import Price Equations

(t-statistics in parentheses)

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Less than 0.001.

The coefficients shown in columns (6), (7), and (8) of Table 1 refer to dummy variables for domestic dock strikes, and those in column (9) to a domestic steel and copper strike variable (S). The dock strike variable D was designed by Takacs (1974) and defined as


where the index π refers to various U. S. ports, Nπ, t to the number of days port π was on strike in quarter t, and Nπ,67/M67 is the proportion of the 1967 value of imports that entered the United States through port π. Following Takacs’ suggestion, D was used with a one-period lead to reflect anticipatory imports and with a one-period lag to account for the makeup of imports following each dock strike.23 The results presented in Table 1 are consistent with a priori expectations and suggest that, while dock strikes have seriously affected the timing of imports, their impact on the volume of imports has been largely offset on average by anticipatory and compensatory purchases.

The value of the Durbin-Watson statistic for equation (1.3) lies in the intermediate range, where the hypothesis of positive serial correlation cannot be conclusively accepted or rejected. The error term was then assumed to follow a first-order Markov scheme ut = ρut-1 + et (with values of et assumed to be serially independent).24 The equation was then re-estimated in its autoregressive form using a nonlinear least-squares technique. The results, shown in equation (1.5), Table 1A, indicate an estimated value of ρ which is fairly small and not significantly different from zero at the 0.05 level.

aggregate unit value equation

The estimation results for the import price equation (9) are presented in Table 1B. The dependent variable pm is defined as the unit value of total U. S. imports adjusted to exclude petroleum import prices. The average foreign price and exchange rate variables were defined as


where the pv’s and the kv’s are wholesale price and exchange rate indexes, respectively, for eight of the largest U. S. trading partners.25 The symbol μv denotes U. S. imports from country v as a proportion of total imports from the eight-country group, with figures adjusted to exclude petroleum products and automotive trade with Canada. The variable Wf is a foreign capacity utilization variable constructed by Parrish and DiLullo.26 Finally, the unit value pm is based on Fisher’s formula, while the components of pf are Laspeyres’ indexes. It was noted earlier that these two types of price indexes describe different rates of change. The linear time trend t was included in the regressions to account for this purely methodological difference.27 All the coefficients shown in Table 1 have the expected sign, and their significance levels are not seriously affected by changes in specification. The constraint that ξ2 = — (1 — ξ2) is not inconsistent with the results. Not surprisingly, the untransformed regression (1.6) is associated with low values of D-W. Adjustment for autocorrelation does not radically alter the magnitude of the coefficients, however, and it does not increase substantially their standard errors.

A domestic cyclical variable (Wd) was also introduced in the regressions to test whether the availability of locally produced goods affected the propensity of foreign exporters to raise prices. The results (not shown in Table 1) were unsatisfactory as the estimated coefficient of Wd had the wrong (negative) sign in most trials.

IV. Estimation Results: End-Use Categories


The end-use breakdown of import data provided by the U. S. Department of Commerce is designed to reflect the different demand forces stemming from various sectors of the economy. It includes six basic categories: (0) foods, feeds, and beverages; (1) industrial supplies and materials; (2) capital goods other than automotive vehicles; (3) automotive vehicles, parts, and engines; (4) consumer goods, excluding foods and automobiles; and (5) other imports. For the purpose of this study, imports of industrial supplies and materials are adjusted to exclude fuels and lubricants, while automotive imports from Canada under the Special Agreement are excluded from category (3). The behavior of these types of imports over the last 15 years is illustrated in Table 3.

Table 3.

Composition of U.S. Imports in Selected Years

(Percentage of total value of imports)

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Source: U.S. Department of Commerce, Survey of Current Business, various issues.

The relative importance in domestic demand of categories (0) and (1)—which together account for more than half the total value of imports—has not shown a noticeable tendency to increase during the past decade. At present, each represents less than 6 per cent of the domestic market.28 By contrast, imports of categories (2), (3), and (4) have increased rapidly in relation to domestic demand; the first two account for some 12 per cent and the last one for roughly 6 per cent of the domestic market.

The analysis of U. S. imports for the various end-use categories was confronted with the various difficulties, discussed earlier, stemming from the unavailability of appropriate import price data. A strategy designed to overcome these difficulties was outlined in Section I and formalized in Section II, providing the basis for the econometric results presented in the rest of this section.

foods, feeds, and beverages

The results reported in Table 4A were obtained by estimating the demand equation (2) for imports of foods, feeds, and beverages expressed in volume terms. A weighted average (pm) of unit value indexes for crude and manufactured foods was used to deflate the import value series. (In 1970, the proportion of value covered by the sample items was 76.2 per cent for the unit value index of crude foods, and 63.2 per cent for that of manufactured foods. The index pm is therefore not unsatisfactory in terms of coverage.) The import price used on the right-hand side of the equation was defined as Pm = Pm (1 + τ), where τ is an average tariff variable for imports of foods, feeds, and beverages.29 With regard to the domestic variables, the wholesale price index for processed foods and feeds was used for pd, and real disposable income was used for yd. Consumption expenditure on food was also tried as a scale variable, but it was substantially weaker than disposable income. Indeed, the more aggregative concepts of demand—for example, personal consumption expenditures and disposable income—performed consistently better than any sectoral demand variable for all of the end-use categories.

Table 4.

Import Volume and Import Price Equations for Imports of Foods, Feeds, and Beverages

(Based on quarterly data)

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Note: The variables pf1, pf2, and pf3 are dollar spot price indexes for coffee, sugar, and cocoa, respectively, in New York; pf is a weighted average of these three indexes; pm is a weighted average unit value index for imports of foods, feeds, and beverages; and m¯ denotes U.S. imports of foods, feeds, and beverages, deflated by pm. For definitions of other variables, see the text and Appendix, Table 10. All lower case Latin letters refer to natural logarithms.

Two types of dock strike variables are used in the equations of Table 4A. The Takacs strike variables—discussed in Section II—are used in equation (4.1), while a strike variable provided by Isard, Lowrey, and Swamy (1974) is used in the other three equations. This latter variable is defined as the difference between the logarithms of actual and normal imports during strike periods;30 the positive coefficient shown in equations (4.2) to (4.4) is therefore consistent with a priori expectations. The regressions using the Isard strike variable have a lower standard error than their counterparts which use the Takacs variables, but they also have a lower value for the Durbin-Watson statistic, which makes it difficult to evaluate the relative performance of these strike variables.

It is sometimes suggested that the rapid expansion in imports of consumer goods since the mid-1960s has resulted not only from the growth of income in the United States but also from a change in tastes induced by increased familiarity with foreign goods. To allow for this possibility, the variable T (equal to zero up to the fourth quarter of 1959 and to a quarterly time trend thereafter) was introduced in equation (4.3).31 Its coefficient turns out to be significantly greater than zero at the 0.01 level, but the magnitude of the estimated shift is not very large—the equivalent of an increase of less than 0.1 in the income elasticity of import demand.

The type of import price equation estimated in Section III cannot be reasonably used in the case of foods, feeds, and beverages. A very large share of total imports in this category is accounted for by a small number of agricultural commodities (especially coffee, sugar, and cocoa), and these commodities are typically produced by countries other than those included in the average price and exchange rate variables defined in Section III. Moreover, the export price model of Section II is probably not applicable to agricultural commodities. For these reasons, the import price equations in Table 4B simply relate the unit value index pm to spot prices of the three principal imported commodities, using distributed lags to capture the delays between contract and delivery.32 In equation (4.4), pf is a weighted average of coffee, sugar, and cocoa prices, while in equations (4.5) and (4.6), the components of pf are used separately.33 The variable pf1 is a weighted average of Brazilian, Colombian, and Salvadoran coffee prices; pf2 is the New York price for raw sugar (Carribean); and pf3 is the New York price for Brazilian cocoa. The estimated coefficients in equations (4.5) and (4.6) are in line with the relative importance of these three commodities in U. S. imports.

industrial supplies and materials

Table 5A summarizes the estimation results for imports of industrial supplies and materials expressed in real terms. The dependent variable in each of three equations excludes fuels and lubricants, and it is deflated by a weighted average of unit value indexes for crude materials and semimanufactures adjusted to exclude petroleum prices.34 Among the explanatory variables, yd is the aggregate index of industrial production in the United States, and pd is a U. S. wholesale price index constructed by the authors for nonfood, nonfuel crude, and intermediate materials. As in the case of foods, feeds, and beverages, the import price variable p˜m is adjusted for changes in tariffs.35

Table 5.

Import Volume and Import Price Equations for Industrial Supplies and Materials

(Based on quarterly data, first quarter of 1959 to fourth quarter of 1973)

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Note: The variable pm is a weighted average of unit value indexes for crude materials and semimanufactures; m¯ is the logarithm of U.S. imports of industrial supplies and materials (other than fuels and lubricants) deflated by pm. For definitions of other variables, see the text and Appendix, Table 10. All lower case Latin letters refer to natural logarithms.

Two nonprice rationing variables were used to approximate the vector wd. The first (wd1) is a nonlinear and asymmetric version of the Federal Reserve Board’s index of capacity utilization for major material producing industries. This variable is based on the formula of equation (4), R^ being determined experimentally at 0.88. The asymmetry parameter ψ is equal to 2. The second variable (wd2) is a proxy for the length of delivery lags and is defined as the percentage of firms reporting that the majority of their contracts for the purchase of materials are 60 days or longer.36 It is expected that, as delivery lags increase, producers turn to importers to satisfy their regular and precautionary demand for materials. The coefficient of wd2 is thus expected to have a positive sign. Finally, the three strike variables are the same as those used for the aggregate equations in Section III.

All the estimated coefficients in Table 5A have the expected sign. All are considerably larger than their standard errors, with the exception of the dock strike variable D+1. This result suggests that there was no consistent propensity to anticipate dock strikes during the sample period. As noted earlier, however, this does not rule out the possibility of large anticipatory imports in specific instances.

The results for the unit value equation are shown in Table 5B. The dependent variable pm is the same as that used to deflate the import value, and the form of the estimated equations follows the theoretical expression (9) of Section II. The variable wd was used to test for cyclically induced changes in import prices, with a moderate degree of success. The coefficient of the domestic price variable pd had a negative sign in all trials; since this result is contrary to theoretical expectations, this variable was removed from the equations. The adjustment for serial correlation reveals a fairly high value of ρ^; it lowers the coefficients of wd and wf, but otherwise does not affect the results drastically.

capital and consumer goods other than foods and automobiles

The unit values corresponding to end-use categories (2) and (4) are not available for the sample period 1959–73. However, since imports in these two categories consist mainly of finished manufactures, an obvious approach is to combine these imports into a single category and to use the unit value index for finished manufactures—which is available for the entire period—as the relevant import price variable. This approach is followed in the present section.37 Its major weakness is that the unit value for finished manufactures is based on items which, in 1970, represented only 43.4 per cent of the total value of imports in that economic class, thus failing to meet even the modest standards proposed by the U. S. Department of Labor to assess the representativeness of a sample (50 per cent minimum coverage). Accordingly, the results presented in this section should be interpreted with particular caution.

The equations for imports of capital and consumer goods in volume terms are shown in Table 6A. The variable yd is personal consumption expenditure on goods; pm is the unit value index for finished manufactures; pd is the U. S. wholesale price index for finished goods other than foods; and Wd is a nonlinear version of the actual to potential GNP ratio specified in equation (4), with ψ = 1. All the activity and price variables have the expected signs.38 However, the estimated coefficient of pd is surprisingly large, a result that will be given further consideration in Section V.

Table 6.

Import Volume and Import Price Equations for Consumer and Capital Goods Other Than Foods and Automobiles

(Based on quarterly data, first quarter of 1959 to fourth quarter of 1973)

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Note: The variable pm is the unit value index for U.S. imports of finished manufactures; m¯ is the logarithm of U.S. imports of consumer and capital goods (other than foods and automobiles) deflated by pm. For definitions of other variables, see the text and Appendix, Table 10. All lower case Latin letters refer to natural logarithms.

The results for the unit value equation—shown in Table 6B—are consistent with the theoretical structure of expression (9) in Section II. The first-order autoregressive transformation leaves the results practically unchanged.

automotive vehicles, parts, and engines

As in the case of nonautomotive consumer and capital goods, no unit value for automotive imports is available for the entire sample period. Furthermore, it is not possible to construct an appropriate index for this category by combining unit values based on the economic classification, as was done in previous sections. Hence, it is necessary to fall back on the second-choice specification of equation (11) in Section II. Fortunately, this is fairly straightforward, because indexes based on contract prices for car exports are available for both the Federal Republic of Germany and Japan, two countries which in 1970 accounted for more than 80 per cent of U. S. imports from areas other than Canada. The foreign export price variable px was therefore defined as a weighted average of automotive export price indexes for these two countries. Similarly, the exchange rate variable k was defined as a weighted average of exchange rate indexes for the deutsche mark and the Japanese yen.

The regression results for automotive imports show considerably more instability than those for other end-use categories.39 In the first two equations of Table 7, the domestic price pd is the wholesale price index for motor vehicles. Its estimated coefficient in equation (7.1) is quite significant, but its magnitude is implausibly large. In equation (7.2)—which includes the structural shift variable T, defined in Section IV—the estimated coefficient of pd is sharply reduced and becomes barely larger than its standard error. The other two equations of Table 7 show the results obtained by using alternative domestic price variables. In equation (7.3), pd is the price deflator of the gross automotive product, while in equation (7.4) it is the consumer price index for new cars. In both equations, the estimated coefficient of pd is close to 5 and is significant at the 0.01 level. In these equations, however, the coefficient of the foreign export price variable becomes insignificantly different from zero.

Table 7.

Equations for U.S. Imports of Automotive Vehicles, Parts, and Engines (Excluding Imports from Canada)

(Based on quarterly data, first quarter of 1961 to fourth quarter of 1973)

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Note: The dependent variable in all equations is m, the logarithm of U.S. imports of automotive vehicles and parts from countries other than Canada, expressed in value terms. The domestic price variable pd is the wholesale price index for motor vehicles in equations (7.1) and (7.2); the implicit price deflator of the gross automotive product in equation (7.3); and the consumer price index for new cars in equation (7.4). For definitions of other variables, see the text and Appendix, Table 10. All lower case Latin letters refer to natural logarithms.

The coefficients of yd, wd, and k and the strike variables are consistent with expectations and—unlike the price coefficients—are not vulnerable to changes in specification. Total personal consumption expenditure on goods performed substantially better than alternative demand variables, notably consumption expenditure on automobiles.

import equations with no information on traded goods prices: two alternative specifications

In Section II, the import equation (12) was specified to handle the case when no information on either import prices or foreign export prices is available. The estimation of this type of reduced form equation is illustrated below for end-use category (4)—consumer goods other than foods and automobiles


where pf and k are appropriately weighted averages of foreign prices and exchange rates, respectively, for eight major trading partners; and the three domestic variables are identical to those defined in Section IV. The sample period includes 60 quarterly observations ranging from the first quarter of 1959 to the fourth quarter of 1973. It may be noted that the total lag for pf in equation (13) is equal to the total lag for pm in Table 6A (four quarters), plus the total lag for pf in Table 6B (three quarters). This result—which agrees with the analysis of Section II—also holds for variables pd and k.

Equation (14) is a deliberately misspecified equation. It includes a distributed lag on the relative price variable log (PdK/Pf) instead of three separate distributed lags on pd, pf, and k, as required by the model of Section II.


The seriousness of this specification error is signaled quite clearly by the Durbin-Watson statistic. It is also interesting to note that the relative error of forecast for the value of imports was four times as large for equation (14) as for equation (13) in 1972, and more than 50 times as large in 1973.40

V. Concluding Remarks

The analysis of U. S. merchandise imports presented in this paper is based on a framework which integrates the demand-oriented features of traditional foreign trade models with a theory of import price determination. This framework can be reduced to a system of two equations—the first relating U. S. import prices to current and lagged values of foreign prices, domestic prices, and exchange rates; the second relating the volume of U. S. imports to import prices, domestic expenditure, and domestic prices. Unit value indexes for several categories of U. S. imports are used in estimating most of the equations, despite the well-known imperfections of these indexes. The results of the estimation suggest that these imperfections need not be a cause for concern, particularly in forecasting. Indeed, the degree of consistency which characterizes the performance of these unit value indexes is one of the most interesting features of the empirical results presented in Sections III and IV. These results also indicate that equations which are properly specified can explain U. S. imports in periods of fluctuating rates as well as in periods of fixed rates. It is remarkable that these equations do not exhibit unusually large residuals in the period following the 1971 Smithsonian realignment. This result contrasts sharply with those obtained from models that ignore the behavior of foreign suppliers and the impact of such behavior on the determination of U. S. import prices.

Table 8 summarizes some of the most important results obtained for total imports as well as for the various end-use categories. For each equation, the table shows the regression coefficients of the main explanatory variables as well as the value of the estimated autocorrelation coefficient (when applicable). Figures in parentheses are t-statistics and numbers followed by an asterisk represent sums of lagged coefficients. The equations were selected according to three criteria, in order of priority: (1) conformity with a priori expectations, (2) freedom from serial correlation, and (3) goodness of fit.

Table 8.

Summary of Regression Results

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Note: For results relating to other explanatory variables, see the regression tables in Sections III and IV.

Sum of lagged coefficients.

Excluding fuels and lubricants.

Excluding automotive imports from Canada.

Excluding foods and automotive products.

Autoregressive version.

In terms of the demand elasticity estimates for the various end-use categories, it is interesting to note that there is a similar pattern for the two price variables and for the expenditure variable. The smaller coefficients are those for foods, feeds, and beverages and for industrial supplies, while the larger coefficients are those for consumer and capital goods and for automotive products. A similar pattern was found, at least for the expenditure elasticities, by Hooper and Wilson (1974).

Another noteworthy result emerging from Table 8 is the very large size of some of the demand elasticities with respect to the domestic price variable (pd). It may be noted that, given the marginal rate of substitution between imported and import-competing goods, this elasticity can be expected to be larger, the lower is the ratio of imported to domestically produced goods. Since this ratio is fairly low for the United States, the estimated coefficient of pd can be expected to be on the high side.41 However, the fact that, at least in some equations, this coefficient is so much larger in absolute value than the coefficient of pm is puzzling. Although no attempt will be made here to provide a fully persuasive explanation, two possibilities may be mentioned. First, part of the discrepancy between the two price elasticities might be accounted for by methodological and compositional differences in the indexes used for pm and pd (see Section II). Second, the estimated elasticities with respect to pd might be subject to an upward bias which could result from the omission of a variable that is positively correlated with pd (the magnitude of the bias being proportional to the correlation coefficient between the two variables).

Aside from the relative size of these elasticities, it is important to note that all the price coefficients shown in Table 8A exceed unity in absolute value, some by a substantial margin. There is no room in these results for “elasticity pessimism.” Indeed, the discussion of the possibility of an upward bias is itself indicative of how notions in this area have evolved since 1950—when the widespread view that foreign trade elasticities were low came under attack.

Comparing estimates obtained by different authors is a difficult task because even models constructed to tackle a similar set of problems may differ considerably in terms of specification, method of estimation, and definition of variables. For example, a comparison of the price elasticities in this paper with those reported in previous studies would not be very meaningful, because in most of the other studies the coefficients of the domestic and foreign price variables are constrained to be equal in absolute value. However, somewhat more meaningful comparisons are possible with respect to income elasticities.

The following comparison takes into account only models in which the structural import demand equation is specified in log-linear terms,42 and the comparison is also restricted to the expenditure (or income) elasticity of total U. S. merchandise imports. Moreover, the equations selected do not include nonprice rationing variables that could affect the magnitude of the estimated income elasticity because of their cyclical behavior. For this reason, equation (1.2) was selected from Table 1, although equation (1.4) has a lower standard error and a higher Durbin-Watson statistic. Despite these attempts to achieve comparability, the four models sketched in Table 9 differ substantially in terms of sample period, magnitude of the autoregressive transformation, and definition of both the import variable and the domestic demand variable. In addition, the four models use different methods to estimate distributed lags—an important point since the comparison focuses on estimates of long-run elasticities, which depend on the dynamic properties of each model. Houthakker and Magee use a continuous time flow-adjustment model adapted from a prior study by Houthakker and Taylor. Wilson reports both ordinary least-squares equations with no lags and distributed lag equations using Shiller’s method—a Bayesian procedure which constrains the dth differences (second differences in Wilson’s case) of the distributed lag coefficients to be close to zero. Hooper’s equations are based on a partial adjustment mechanism of the Koyck-Nerlove type, while equation (1.2)—like all the equations reported in the present paper—uses second-degree Almon polynomials.43 In spite of these methodological and definitional differences, there is considerable agreement among the four models. In particular, it is interesting to note that the four point estimates lie between 1.42 and 1.63—that is, they are all within the 95 per cent confidence interval of equation (1.2).

Table 9.

Estimates of Long-Run Income (or Expenditure) Elasticities for Total U.S. Imports

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Sources: (1) Houthakker and Magee (1969), Table 6, p. 121; (2) Hooper and Wilson (1974), Table 3, p. 26; (3) Ibid., Table 11, p. 55. Imports exclude automotive trade with Canada in models (2) and (4) and fuels in model (4).

Sum of gross national product components, weighted by their importance in total merchandise imports.


Table 10.

Description of Variables Used in the Regression Tables1

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Key: PCE = Personal consumption expenditures in the United States; WPI = U.S. wholesale price index; and CPI = U.S. consumer price index.

Details on the construction of weighted averages are given in a supplement, which is available from the authors on request.

Dependent variable is deflated by pm.

Excluding imports of fuels and lubricants.

Excluding automotive imports from Canada.

Adjusted to exclude petroleum prices.


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Ms. Ahluwalia, an economist in the North American Division of the Western Hemisphere Department is a graduate of the Delhi School of Economics. Currently on study leave from the International Monetary Fund, she is a Research Fellow at The Brookings Institution. Mr. Hernández-Catá, an economist in the North American Division of the Western Hemisphere Department, is a graduate of Yale University and the Graduate Institute of International Studies, Geneva, Switzerland.


Among the most important contributions to the econometric literature on the U.S. trade balance are the works of Hooper and Wilson (1974), Gregory (1970), and Magee (1971). The model developed in this article builds upon these studies and also relies extensively on the recent analysis by Artus (1974) of export prices in the Federal Republic of Germany and Japan.


The currency denomination problem has been extensively discussed outside the framework of econometric analysis by Magee (1973 and 1974). For a cross-sectional analysis of absorption phenomena in the context of the Smithsonian realignment, see Schwartz and Pérez (1974). Although important, these two issues do not exhaust the list of problems raised by exchange rate changes for the specification and interpretation of import functions. For example, Magee (1974) has carefully analyzed the errors introduced into the 1971–73 U.S. import statistics by the conventions followed by the U.S. Bureau of the Census with regard to exchange rates.


A comprehensive description of these adjustments is given in a mimeographed supplement, which is available from the authors on request.


See, in particular, U.S. Congress (1961). Another difficulty which has been particularly severe in the manufacturing categories is the low proportion of total value covered by the sample items. In addition to these “traditional” problems, the reliability of unit value indexes is also impaired by valuation errors related to exchange rate changes. These have been recently analyzed by Magee (1974).


More on this topic follows in Section II. Note that the problem may also arise in the absence of exchange rate changes if domestic and foreign inflation rates diverge substantially.


For some end-use categories, the available series begin as late as 1968 or even 1969—too small a sample if one is to investigate the lagged impact of prices on the demand for imports. A few simple equations with no lags (not shown in this paper) did, however, perform reasonably well.


For an extensive treatment of the impact of the nonprice elements of a contract on merchandise imports, see Gregory (1971). Gregory introduces the concept of “effective price” to denote the price and the various nonprice elements of the contract.


Even if the long-run impacts of pm and pd are equal in absolute value (α2 = - α3), this does not imply that the lag distributions of the two variables should be identical (α2i = α3i Vi).


For a comprehensive review of these differences and their implications for the estimation of price elasticities in international trade, see Walsh (1972).


See also Gregory (1971) for the construction of an analogous proxy variable based on the ratio of unfilled orders to current production.


Along the same lines, Gregory (1971, pp. 43–45) assumes that the rate of capacity utilization in manufacturing can serve as a proxy for variations in his “effective price.”


For simplicity, it may be assumed that the “rest of the world” can be treated as a single homogeneous unit. Alternatively, the variables px and k can be interpreted as weighted averages of the relevant variables for each of the exporting countries.


Equation (5) is similar to that used by Artus (1974, p. 590), except that the transition here is from the import unit value in dollar terms to the export price index in foreign currency terms. Of course, equation (5) is only a mathematically convenient approximation to the true relationship, which involves the variables Pm, Px, and K, rather than their logarithms.


Among major U.S. trading partners, only the Federal Republic of Germany and Japan have export price indexes based on contract prices.


This may be shown to be the Cournot solution to the problem of long-run price determination by profit-maximizing oligopolists in a multicountry framework. See Hernández-Catá and Sakakibara (1973).


ci is linked to the distributed lag parameters of equations (2) and (6) by the equation


An analogous expression relates the vector c′ to the vector α and θ′.


As of December 1972, the relative importance of imports in the aggregate index was only 1.35 per cent. See U.S. Department of Labor (1973).


The results were different in this respect for industrial supplies and materials (see Section IV).


No constraints were imposed at the “near” end of the polynomial. The choice of a second-degree polynomial for all of the equations presented in this paper was dictated by efficiency considerations. Amemiya and Morimune (1974) have shown that the optimal order of the polynomial is negatively related to the degree of multicollinearity and to the smoothness of the lag distribution. On both counts, the second-order polynomial appeared to be the most reasonable choice in the present context.


This would have imposed the constraint that the shape (and the length) of the lag distribution should be identical for all independent variables. In addition, the Koyck transformation leads to an expression in which the lagged value of the dependent variable appears on the right-hand side of the equation. In the present context, this would probably have raised the very serious estimation problems associated with the simultaneous presence of errors in variables and lagged dependent variables. On this point, see Grether and Maddala (1973).


This is an intuitively appealing procedure, but there is no evidence that it leads to determining the correct length of the lag. On this point see the results of a Monte Carlo study by Cohen, Gillingham, and Heien (1973).


A weakness of this approach is that it assumes a constant “propensity” to anticipate the impact of dock strikes. Nevertheless, it is considerably more flexible than alternative procedures, such as those used by Parrish and DiLullo (1972a) and Wilson (1973). The Takacs strike variables were used in all of the import equations presented in this paper with the exception of the foods, feeds, and beverages category, for which a specific strike variable was available (see Section IV).


The value of R¯2 given for equation (1.5), and for all other autoregressive equations presented in this paper, is the lower value associated with ut rather than the higher value associated with et.


Belgium-Luxembourg, Canada, France, the Federal Republic of Germany, Italy, Japan, the Netherlands, and the United Kingdom. These countries accounted for almost two thirds of total U.S. imports (other than fuels and lubricants and automotive imports from Canada) in the year 1970.


The variable is the inverse of a weighted average of capacity underutilization indexes for the countries listed in footnote 25, except Belgium-Luxembourg. For further details, see Parrish and DiLullo (1972b), pp. 32–33.


Suppose that the rate of growth of a Fisher index F is equal to the rate of growth of the Laspeyres index L plus a constant A. By integration, it follows that log F = log L + At.


For details on the concepts used to measure “domestic demand,” see U.S. Department of Commerce (1974), Table 6, p. 24.


This variable was taken from Isard, Lowrey, and Swamy (1974), Table A–4, and deflated by its base period value.


The variable includes estimates of anticipatory and compensatory imports. For further details, see Isard, Lowrey, and Swamy, pp. 21–24 and 43.


Wilson (1973) makes extensive use of this type of structural shift variable.


Since all prices are expressed in U.S. dollars, no exchange rate variable is introduced in the equations.


The value of ρ in equation (4.6) was estimated using the Hildreth-Lu scanning technique, as convergence was not achieved with the iterative procedure.


The coverage of the component series in 1970 was 77.2 per cent for crude materials and 58.1 per cent for semimanufactures.


The tariff variable used here was constructed by Wilson (1973), Table A.9, p. 309.


The variable is based on a monthly survey of purchasing agents for 225 companies (selected by Standard International Trade Classification categories and geographical areas) conducted by the National Association of Purchasing Management.


An alternative approach is illustrated in Section IV.


The current and lagged values of the dock strike variable D perform according to expectations. However, the coefficient of D+ 1 is negative in all equations, a result which is not consistent with the interpretation of D+1 as a measure of anticipatory imports. The most likely explanations for this result are that the variable is capturing the initial effects of certain dock strikes and that these negative effects outweigh the positive effects (if any) of anticipatory imports.


This is in line with a long tradition in the history of U.S. import equations. See, for example, Wilson (1973). At the least, however, all the coefficients shown in Table 8 have the expected sign; and the results are not seriously affected by autocorrelation.


Both equations (13) and (14) were estimated by including the three versions of the strike variable D as regressors.


Let ηij stand for the demand elasticity of good i with respect to the price of good j. By definition, ηij = (dqi/qi)/(dpj/pj). Multiplying and dividing the right-hand side of this expression by dqj/qj yields the relation ηij = (dqi/dqj) · ηji · (qj/qi).


This rules out models where the dependent variable is defined as the ratio of imports to GNP—for example, Gregory (1971)—or models specified in linear-arithmetic form—for example, Parrish and DiLullo (1972a).


In the Shiller and Almon models, the long-run elasticities are simply computed as the sum of the estimated lagged coefficients. In Hooper’s partial adjustment model, the long-run elasticity is obtained by dividing the estimated short-run income elasticity by the speed of adjustment coefficient (that is, unity minus the estimated coefficient of the lagged dependent variable). In the flow-adjustment model used by Houthakker and Magee, the long-run structural elasticity μ is derived from the estimated reduced form coefficients D1 and D2 through the formula μ = 2D2/(1 - D1). See Houthakker and Magee (1969), pp. 120-21. The upper and lower bounds of models (2) and (4) in Table 9 are 95 per cent confidence intervals. For models (1) and (3), the upper and lower bounds were calculated by taking into consideration only the standard error of the estimated short-run elasticity coefficient but assuming the estimated speed of adjustment coefficient to be a fixed number.

IMF Staff papers: Volume 22 No. 3
Author: International Monetary Fund. Research Dept.