Time–Series Estimation of Structural Import Demand Equations:A Cross-Country Analysis
Author:
Mr. Abdelhak S Senhadji
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Publication Date:
01 Jan 1998
eISBN:
9781451974164
Language:
English
Keywords:
SP; paper number; EU country; monetary policy; income elasticity; import demand equation; long-run elasticity; standard deviation; elasticity estimate; Price elasticity; Imports; Real exchange rates; Personal income; Import prices; elasticities wi11
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A Structural import demand equation is derived and estimated for a large number of countries, using recent time-series techniques that address the problem of nonstationarity. The average price elasticity is close to zero in the short run hut is slightly higher than one in the long run. A similar pattern holds for income elasticities: the short-run income elasticities are on average less than 0.5, while the long-run income elasticities are close to 1.5. The paper also analyses the small-sample properties of both the ordinary-least-squares (OLS) and the fully modified (FM) estimators of the short- and long-run elasticities, using Monte Carlo methods.

Abstract

A Structural import demand equation is derived and estimated for a large number of countries, using recent time-series techniques that address the problem of nonstationarity. The average price elasticity is close to zero in the short run hut is slightly higher than one in the long run. A similar pattern holds for income elasticities: the short-run income elasticities are on average less than 0.5, while the long-run income elasticities are close to 1.5. The paper also analyses the small-sample properties of both the ordinary-least-squares (OLS) and the fully modified (FM) estimators of the short- and long-run elasticities, using Monte Carlo methods.

THE EMPIRICAL investigation of the import demand function has been one of the most active research areas in international economics. This is evidenced by the many surveys on this topic, although most focus on industrial countries. 1 Perhaps one of the main reasons for its popularity is its application to a wide range of important macroeconomic policy issues such as the impact of expenditure-switching through exchange rate nuinagemeni and commercial policy on a country’s trade balance: the international trans mission of domestic disturbances, where these elasticities are a crucial link between economies: and the degree to which the external balance constraint affects a country’s growth.

The traditional import demand function is specified as a log-linear function of the relative price of imports and real income. Because of data constraints and the empirical success of this specification, it has dominated the empirical literature for more than a quarter-century. But questions about its microeconomic foundation arise since it has not been derived from utility maximization. Another issue that has been largely ignored in the literature is the problem of nonstationarity. which is found in most macroeconomic variables and which invalidates classical statistical inference. Thus, if the variables that enter the import demand equation contain a unit root, ignoring nonstationarity in these variables may cause serious inference problems.

Empirical researchers are generally interested in two statistical properties of their estimates of import elasticities. First, they are interested in the magnitude of these elasticities. A relevant question, then, is how close the estimates are to their true value in small samples. The systematic deviation of the estimates from their true value is measured by the bias of the estimates. Second, they are interested in inference, that is hypotheses testing, about these estimates. For example, are the price and income elasticities significantly different from one? Testing such hypotheses requires knowing the distribution of the t-statistic (defined as the coefficient estimate divided by its standard deviation). The asymptotic distribution of this statistic is unknown for the long-run elasticities as these elasticities are nonlinear transformations of the import demand coefficients. In addition, the definition of the long-run elasticities includes the lagged dependent variable whose t-statistic follows a nonstandard distribution in the nonstationary case. In light of this, using the critical values of the asymptotic t-distribution for hypothesis testing may be misleading.

The objective of this paper is twofold. First, the paper seeks to address the two problems discussed above by deriving an empirically tractable import demand equation that can be estimated for a large number of countries, using recent time-series techniques that address the issue of nonstationarity present in the data. Second, the paper computes the exact bias as well as the exact distribution of the t-statistic, which is crucial for inference as discussed above, for the short- and long-run elasticities.

The derived aggregate import demand equation is log-linear in the relative price of imports and an activity variable defined as GDP minus exports.2 An important insight from the explicit derivation of the aggregate import demand equation is that the definition of the activity variable depends on the aggregation level.3The model predicts a unique cointegrat-ing vector among imports, the relative price of imports, and the activity variable. This prediction is not rejected by the data, and the cointegrating vector is estimated both by ordinary least squares (OLS) and by the Phillips-Hansen fully modified (FM) estimator. Two recent papers follow a similar methodology. Clarida (1994) derives a similar import demand function for U.S. nondurable consumption goods from explicit intertemporal optimiza-tion, carefully taking into account data nonstationarity. Similarly. Reinhart (1995) estimates both structural import and export demand functions for 12 developing countries using Johansen’s coinlegration approach.

The results underscore the presence of nonstationarity in the data and the adverse consequences of neglecting it. Both price and income elasticities generally have the expected sign and are precisely estimated. The average price elasticity is close to zero in the short run but is slightly higher than one in the long run. It takes five years for the average price elasticity to achieve 90 percent of its long-run level. A similar pattern holds for income elasticities in the sense that imports react relatively slowly to changes in domestic income. The short-run income elasticities are on average less than 0.5. while the long-run income elasticities are close to 1.5. Industrial countries have both higher income and lower price elasticities than do developing countries. On average, these estimates are relatively close to Reinhart’s.4

The analysis shows that the OLS bias is significantly higher than the FM bias for both the short- and long-run elasticity estimates. The FM bias reaches its minimum when the relative price of imports and the activity variable are exogenous. Strong endogeneily of the explanatory variables (that is, high correlation between the import demand innovations and the explanatory variables innovations) may induce substantial bias. But for most countries—being “small” relative to the rest of the world—the relative price of imports and the activity variable are only weakly endogenous, leading to a relatively small bias. For the benchmark case in which both explanatory variables are assumed to be exogenous, the t-statistics of the short-run elasticities are symmetric around zero but are Hatter than the asymptotic t-distribution. This implies that inference based on the usual t-or F-“-statistic may be misleading. For example, the exact confidence intervals are significantly wider than those based on the asymptotic t-distribution. The distribution of the t-statistic for the short-run elasticities becomes skewed and flatter when the relative price of imports and/or the activity variable is allowed to be endogenous. The stronger the endogeneity. the larger is this departure from the asymptotic t-distribution.

I. The Model

Assume that the import decision in each country is made by an infinitely lived representative agent who decides how much to consume from the domestic endowment (dt)and from the imported good (mt),5 The home good is the numeraire. The intertemporal decision can be formalized by the following problem:

max { d t , m t } t = 0 ∞ E 0   Σ t = 0 ∞ ( 1 + δ ) - 1 u ( d t , m t )                                                       ( 1 )

subject to:

b t + 1 = ( 1 + r ) b t + ( e t - d t ) - p t m t                                             ( 2 )
e t = ( 1 - ρ ) e ¯ + ρ e t - 1 + ξ t , ξ t ∼ ( 0 , σ 2 )                                                   ( 3 )
lim T → ∞ b T + 1 Π t = 0 T ( 1 + r ) - 1 = 0 ,                                 ( 4 )

where δ is the consumer’s subjective discount rate: ris the world interest rate: bt+1is the next period stock of foreign bonds if positive, and the next period’s debt level if negative; etis the stochastic endowment, which follows an AR(1) process with unconditional mean ēand an unconditional variance σ2/(1 - ρ2). where σ2is the variance of the i.i.d. innovation ξt, and p determines the degree of persistence of the endowment shocks; and ptis the relative price of the foreign good, that is, the inverse of the usual definition of terms of trade. In this two-good economy, ptis also the real exchange rate. Equation (2) is the current account equation, equation (3) is the stochastic process driving the endowment shock, and equation (4) is the transvcrsality condition that rules out Ponzi games. The first-order conditions of this problem are

u 1 d = λ 1                                                           ( 5 )
u 1 m = λ t P t                                                           ( 6 )
λ t = ( 1 + δ ) - 1 ( 1 + r ) E t λ t + 1 ,                                               ( 7 )

where λt is the Lagrange multiplier on the current account equation. From equation (5), λtis the marginal utility of the domestic good. Following Clarida (1994) and Ogaki (1992). it is assumed that the instantaneous utility function uis addilog:6

u ( d t , m t ) = A t d t 1 - α ( 1 - α ) - 1 + B t m t 1 - β ( 1 - β ) - 1       α > 0 , β > 0                             ( 8 )
A t = e a 0 + ε A , t                                                                   ( 9 )
B t = e b 0 + ε b , t                                                             ( 10 )                        

where At and Bt are exponential stationary random shocks to preferences, εAt εB.tare stationary shocks and α and β are curvature parameters. Substituting equation (8) into equations (5) and (6) yields

d t = λ t - 1 α A t 1 α                                         ( 11 )
m t = λ t - 1 β B t 1 β p t - 1 β                                     ( 12 )

Substituting equations (9)-(l 1) into equation (12) and taking logs yields

m ∼ t = c - 1 β p ∼ t + α β d ∼ t + ∊ t                   ( 13 )

where c0(1/β)(b0-a0, and ε= (l/b)(εBt- εAt). A tilde indicates the log of the corresponding variable. In this model, xt - et - dt=GDP1 - d1where xtis exports. Consequently, dt= GDPt -x1.Thus, the model yields an equation for import demand that is close to the standard import demand function

except (hat the correct activity variable is GDPt -xt. rather than GDPt.Equation (13) can be rewritten as

m ∼ t = c - 1 β p ∼ t + α β ( G d P ∼ t - x t ) + ε t                   ( 14 )

Because each of the three variables in the import demand equation (14) can either be trend stationary (TS) or difference stationary (DS). four cases need to be considered (Table 1). In Section II. results from unit root tests show that the first case is the most common one, with some countries falling into the second category. The prime interest is the estimates of the standard price and income elasticities of the import demand (-1/β and α/β, respectively). Note that mtand ptare, in general, endogenously determined by import demand and import supply (not modeled here). Therefore, ptis likely to be correlated with the error term εt in equation (14). Thus. OLS would yield biased estimates of the price and income elasticities. As shown in the Appendix, the Philips-Hansen FM estimator corrects within a cointegra-lion framework, for this potential simultaneity bias as well as for autocorrelation in εt.

Table 1.

The Four Possible Model Specifications

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p, is either exogenous under a perfectly elastic import supply or is endogenously determined by the interaction of the demand and supply of imports. The supply of imports is not modeled explicitly here.

See Slock and Walson (1988)

Equation (14) will be estimated in a dynamic form (that is, with the lagged dependent variable included as an explanatory variable) which proved to be more successful in the estimation stage, and also to keep the specification as close as possible to the literature where this autoregressive distributed lag (ARDL) specification has been widely used:7

m ∼ t = γ 0 + γ 1 m t - 1 + γ 2 P ∼ t + γ 3   ( G d P ∼   t - x t ) + ε t                   ( 15 )

While the lagged dependent variable enriches the dynamics of the import demand equation, its introduction into the reintegration framework outlined above is not innocuous as equation (15) resembles the error correction form of equation (14) except that the dependent variable and the two explanatory variables are in levels and not in first difference. Pesaran and Shin (1997) show that this ARDL specification is well specified and retains the usual interpretation under stationarity even if the variables are I(1). The authors also show that the FM estimator yields efficient estimates of the short- and long-run elasticities.

II. Estimation Results

The data come from the World Bank database BESD. The sample includes 77 countries for which the required data are available for a reasonable time span (the list of countries is given in Table 2). In general, the data are available from 1960 to 1993.8 The usual problem is of course the choice of the corresponding proxies for the variables in the model, since the model is usually a crude simplification of reality—which is the case here. Data constraints highly restrict this choice. Total imports and exports of goods and services will be used for mtand xt in equation (14). The relative price of imports pt will be computed as the ratio of the import deflator to the GDP deflator.9 The activity variable will be computed as the difference between GDP and exports.10

Table 2.

Augmented-Dickey-Fuller Test for Variables Entering the Import Demand Equation

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Notes: Variables are real imports of goods and nonfactor services (m), the real exchange rate (p), computed as the ratio of imports deflator to GDP deflator, and GDP minus exports (gdpx). These three variables are tested for the existence of a unit root using the Augmented-Dickey-Fuller (ADF) test. The optimal lag selected by the Schwarz criterion in the ADF regression is given by k. Critical values are a linear interpolation between the critical values for T=25, and T=50 given in Hamilton (1993, Table B.6, case 4), where T is the sample size. Significance levels at 1 percent and 5 percent are indicated by two aster-isks and one asterisk, respectively.

Unit Root Test

The FM procedure assumes that some of the variables entering the cointegrating equation (15) have a unit root and that there exists a stationary linear combination of these variables. We testod for the existence of a unit root in all three variables in the import demand equation (15). namely, real imports of goods and services (m). the relative price of imports (p), and the activity variable GDP minus exports (gdpx).The unit root hypothesis is tested using the Augmented-Dickey-Fuller (ADF) test. The lag length (k)in the ADF regression is selected using the Schwarz criterion (Table 2). For mtonly 4 of the 77 countries reject the unit root at 5 percent or less (Australia at 1 percent, Nicaragua, Peru, and Philippines at 5 percent). Similarly, the null of a unit root in ptis rejected only for 3 countries (China at 1 percent, and Papua New Guinea and Uruguay at 5 percent). Finally, as far as gdpx1is concerned, the unit root is rejected for 10 countries (Burundi, Central African Republic, Iceland, Switzerland, and Trinidad and Tobago at 1 percent: Korea, Rwanda, Togo, Tunisia, and Zaire at 5 percent). Thus for most of the countries, the unit root hypothesis cannot be rejected at conventional significance levels. This finding, of course, may reflect to a certain extent the low power of the ADF.

Import Demand Equations

The results underscore the presence of nonstationarity in the data and the adverse consequences of neglecting it. Table 2 shows that most countries—60 of the 77—fall into the first case of Table 1 (the unit root hypothesis cannot he rejected for all three variables in the import demand equation) and the remaining countries—17 of the 77—into the second case (the unit root hypothesis can be rejected for only one of the three variables). In the first case, the model predicts a reintegrating relationship between the three I(1) variables, and in the second case between the two I(1) variables. No country belongs to either the third or fourth cases.

Table Al in the Appendix shows both the OLS and FM estimates of the import demand equation. Only countries with the right sign for the price and income elasticities are reported (66 of the 77 countries). The columns of Table Al labeled x1p, and gdpxgive, respectively, the coefficient estimates of the lagged dependent variable (log of imports of goods and non-factor services), the short-term price elasticity (that is, the coefficient of the log of relative price of imports), and the short-term income elasticity (the coefficient of the log of gdpx).The long-run price and income elasticities are defined, respectively, as the short-run price and income elasticities divided by one minus the coefficient estimate of the lagged dependent variable. They are given by Ep and Eyfor the FM estimates.11 The variance, and hence the t-statistic, for Ep and Ey are computed using the delta Method, which consists of taking the Taylor approximation of var(Ep) and var(Ey):

var ( E p ) = ( 1 1 - γ 1 ) 2 var ( γ 2 ) + [ γ 2 ( 1 - γ 1 ) 2 ] 2 var ( γ 1 ) + 2 ( 1 1 - γ 1 ) [ γ 2 1 - γ 1 ] cov ( γ 1 , γ 2 )                             ( 16 )

where γ1 is the coeffient of the lagged dependent variable and γ1 is the short-run price elasticity; var(Ey)is obtained by substituting γ2 by γ3 in var (Ep), where γ3 is the short-run income elasticity, The column labeled serreports the standard error of the regression. Finally, the column labeled ACgives Durhin’s autocorrelation test. It amounts to estimating an AR(1) process on the estimated residuals of the import equation. Durbin’s test is simply a significance test of the AR(1) coefficient using the usual t-test. For the OLS regressions, AR(1) autocorrelation is detected (at 10 percent or less) for 17 of the 66 countries.

Even though Table Al reports both the OLS and FM estimates of the import demand equation, the discussion will focus only on the latter since

both estimation methods yield relatively close results. The short-run price elasticities vary from -0.01 (Algeria) to -0.86 (Malawi), with a sample average (over the 66 countries) of-0.26. a median of-0.22, and a standard deviation of 0.19. Therefore, imports appear to be quite inelastic in the short term. The long-run price elasticities vary from -0.02 (Chile) to -6.74 (Benin). The sample average is -1.08. the median is-0.80, and the standard deviation is 1.08. As expected, imports are much more responsive to relative prices in the long run than in the short run. The short-run income elasticities vary from 0.0 (Zaire) to 1.36 (Haiti). The sample average is 0.45. the median is 0.32, and the standard deviation is 0.34. Thus, the average short-run income elasticity is significantly less than I. The long-run income elasticities vary from 0.03 (Zaire) to 5.48 (Uruguay). The sample average is 1.45. the median is 1.32, and the standard deviation is 0.93. Thus, imports respond much more to both relative prices and income in the long run than in the short run. Epc and Eyc give the long-run price and income elasticities corrected for bias. The correction is generally small. As will be discussed in Section III. the bias is negligible when the relative price of imports and the activity variable are either exogenous or weakly endogenous, as is the case for most countries. Since unit-price and unit-income elasticities are widely used as benchmark values, a formal test for long-run unit-price and unit-income elasticities is provided in the columns labeled Ep=-1 and Ey=1, respectively. This test uses exact critical values of the t-statistic (given in Table 5 and discussed in Section III). Fifteen of the 66 countries reject a long-run unit-price elasticity and 27 countries reject a long-run unit-tncome elasticity at 10 percent or less. The fit as measured by R2is good.

Table 1 showed that in the first two cases, estimates of the price and income elasticities wi11 be meaningful only if the I(1) variables are cointegrated. A cointegration lest is therefore required. The results of the Phillips-Ouliaris residual test of cointegration is given in Table AI under the heading P-O. Even with a relatively small sample size (and therefore low power), the null of noncointegration is rejected for 49 of the 66 countries (at 1 percent in most cases).

An interesting question is whether the long-run income and price elasticities differ significantly between industrial and developing countries. The answer is given by the following two regressions:

E y = 0.96 + 13.010 d u m d c R 2 = 0.66 ( 17.49 ) ( 0.59 )                                           ( 17 )
| E p | = 0.73 - 0.316     d u m d c   R 2 = 0.63 ( 12.03 ) ( -3.25 )                                         ( 18 )

where Ep and Ey, are, respectively, the long-run price and income elasticities and dumdcis a dummy variable that takes the value of one for an industrial and zero for a developing country. The two equations are generalized least squares (GLS) regressions of the long-run income and price elasticities on dumdc.12Industrial countries have significantly higher long-run income elasticities than developing countries (equation 17) and face much more inelastic import demand than do developing countries (equation 18).

III. Small-Sample Properties of the OLS and FM Estimators

This section analyzes the small-sample properties of the OLS and FM estimators by deriving the exact bias of the short- and long-run elasticities as well as the distribution of their t-statistic, which are crucial for conducting inference. The Monte Carlo derivation of these small sample properties are described in detail in each table.

Bias of the Short- and Long-Run Elasticities

Table 3 shows the bias for the OLS and the FM estimates of the short-and long-run price and income elasticities varies significantly with the degree of endogeneity of the explanatory variables (see Table 3). that is. with the correlation between the innovations in the import demand equation and the innovations in the relative price of imports (R12), and the correlation between the innovations in the import demand equation and the innovations in the activity variable (R13), The FM bias reaches its minimum when R12=R13=0 and equals -0.37 percent. 0.99 percent, and -1.14 percent for the dependent variable and for the short-run price and income elasticities, respectively. This implies that both the short-run price and income elasticities are underestimated, the former by 0.99 percent and the latter by 1.14 percent.13 The corresponding OLS figures are -3.79 percent and 3.69 percent. The OLS bias is generally much higher than the corresponding FM one. Note that lor this benchmark case (where R12=R13=0). OLS elthers from FM both in the magnitude and in the direction of the bias. Negative values of R12 tend to bias both the price and the income elasticities downward while positive ones induce an upward bias. Negative values of R13tend to bias both the price and the income elasticities upward while positive values induce a downward bias. The bias becomes substantial for high values of R13 and R13

Table 3.

Bias for Short- and Long-Run Elasticities for OLS and FM Estimators

(in percent)

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Notes: The bias is generated by simulating the import demand model: m1= α1mi-1 + α2mi-1 + α2gdpx1+ εtr p1=pt-1+ε2r and gdpxt= gdpxt-1+ε3t(ε11t,ε2t,ε3t) ∼ N(0,Σ) and corr (ε11t,ε2t,ε3t) = Rij, i, j, = 1,2,3, where mt denotes imports, pt is the real exchange rate and gdpxt is the activity vari-able that is GDP-exports. All variables are in logs. The coefficients α1, α2, and α3 are set to 0.70, -1.00 and 1.00 respectively. The long-runelasticities are defined as Ep= α2/(l-α3) and Ey= α3/(l-α1). The empirical distribution of the elasticities is generated from 5,000 drawings of 34 observations each (the sample size in the data) from the import demand model. For each drawing, the import demand model is estimated.This yields 5,000 estimates of the short- and long-run elasticities. For each drawing, the bias is simply the difference between the elasticity estimate and its true value. The table reports the mean of these biases expressed in percent of the true elasticities. The bias is computed for 5 different values of Rl2 (the correlation between ε1t and ε2t) and 5 different values of R13 (the correlation between ε1t and ε3t). This yields 25 bias distributions for each elasticity.

Because long-run elasticities depend not only on the short-run elasticities (α1 and α2) but also on the adjustment speed as measured by the coefficient on the lagged dependent variable (α1). the bias in the short-run elasticities does not translate one-for-one to the long-run elasticities. The OLS bias is generally much higher than the corresponding FM one. When Rl2=Rl3=0, the FM bias still teaches its minimum for the price elasticity (0.27 percent) but is slightly higher than the minimum for the income elasticity (-0.41 percent). The corresponding OLS figures are 1.81 percent and -86 percent. These values imply that both the long-run price and income elasticities are underestimated. Negative values of R12 tend to bias both long-run elasticities upward, while positive values have the opposite effect. Negative values of R13 induce an upward bias in the long-run price elasticity and a downward bias in the long-run income elasticity. The reverse holds for positive values of R13. Interestingly, the bias on the long-run elasticities is generally lower than the bias on the short-run elasticities.

Exact Distribution of the t -Statistic

For the benchmark case in which both explanatory variables are assumed to be exogenous (Rl2=Rl3=0). the small-sample t-distribution for the imports relative price (p) and the activity variable (gdpx) are symmetric but are wider than the asymptotic t-distribution (Table 4). For reference, the asymptotic critical values of the t -distribution at I percent. 5 percent, and 10 percent are -2.33, -1.65, and -1.28, respectively. The corresponding small-sample critical values are - 3.13, - 2.06, and - 1.61 forp), and - 3.20, -2.15. and -1.64 for gdpx.The t-distribution of the lagged dependent variable (m1) is skewed to the left, as expected, since mt, has a unit root. When pis allowed to be endogenous (that is. R12≠ 0), the distribution of its t-statistic becomes skewed, while the t-statistic distribution of gdpx becomes flatter. Similarly, when gdpx is allowed to be endogenous (that is, R13≠0).the distribution of its t -statistic becomes skewed, while the t-statistic distribution of p becomes flatter. The stronger the endogeneity of p or gdpx. that is, the larger (in absolute value) Rl2 or R13. the larger is this departure from the asymptotic t -distribution.

Table 4.

FM t-Statistic Critical Values for the Parameters of the Import Demand Equation

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Notes: This table provides exact critical values of the FM t-statistic at 1%, 5%, 10%, 90%, 95% and 99% significance levels. These critical values are generated by simulat-ing the import demand model: m1= α1mi-1 + α2mi-1 + α2gdpx1+ εtr p1=pt-1+ε2r and gdpxt= gdpxt-1+ε3t(ε11t,ε2t,ε3t) ∼ N(0,Σ) and corr (ε11t,ε2t,ε3t) = Rij, i,, = 1,2,3, mtdenotes imports, pt is the real exchange rate, and gdpxt is the activity variable that is GDP-exports. All variables are in logs. The coefficients a1, a2 and a3 are set to 0.70, -1.00and 1.00, respectively. For each of the coefficients, the critical values are computed by (1) setting the coefficient for which the critical values are computed to zero (restrictedmodel), (2) drawing 5,000 samples of 34 observations each (the sample size in the data) from the restricted model,(3) computing the usual t-statistic for each drawing, and (4)using the resulting vector of 5,000 t-statistic values to generate an empirical distribution from which the critical values can be computed. For each coefficient, the empirical t-distribution is computed for 5 different values of R12 (the correlation between 1t and e2t) and 5 different values of R13 (the correlation between 1t and e3t). This yields 25 empir-ical t-distributions for each of the three coefficients.

For the benchmark case Rl2 Rl3 =0, the t-distributions of Ep (the long-run price elaslicily) and Ey(the long-run income elasticity) are symmetric but flatter than the asymptotic t-distribution (Table 5). The 1 percent. 5 percent, and 10 percent critical values are -3.38, -2.14. and -1,60 for Ep > and -3.65, -2,22, and -1.69 for Ey As for the short-run elasticity, when p is allowed to be endogenous, the t-statistic distribution of Ey becomes skewed, while the t-statistic distribution of Ey, becomes flatter. Similarly, when gdpx is allowed to be endogenous, the t -statistic distribution of Ep becomes skewed, while the t -statistic distribution of Ep becomes flatter. The stronger the endogeneity. the larger is the deviation from the asymptotic t -distribution.

Table 5.

FM t-Statistic Critical Values for Long-Run Import Price and Income Elasticities

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Notes: This table provides exact critical values of the FM t-statistic at 1%, 5%, 10%, 90%, 95%, and 99% significance levels for long-runimport price and income elasticities (Ep and Ey, respectively). These critical values are generated by simulating the import demand model:m1= α1mi-1 + α2mi-1 + α2gdpx1+ εtr p1=pt-1+ε2r and gdpxt= gdpxt-1+ε3t(ε11t,ε2t,ε3t) ∼ N(0,Σ) and corr (ε11t,ε2t,ε3t) = Rij, i, j, = 1,2,3, mt denotesimports, pt is the real exchange rate and gdpxt is the activity variable that is GDP-exports. All variables are in logs. The coefficients α1,α2and α3 are set to 0.70, -1.00 and 1.00, respectively. The long-run elasticities are defined as Ep=α2/(1-α1) and Ey= 3/1-α1). Their respectivet-statistic critical values are computed by (1) setting a2 and a3 equal to zero (restricted model), (2) drawing 5,000 samples of 34 observationseach (the sample size in the data) from the restricted model, (3) computing the usual t-statistic for Ep and Ey using the delta method formulafor each drawing, and (4) using the resulting vector of 5,000 t-statistic values to generate an empirical distribution from which the criticalvalues can be computed. For both Ep and Ey the empirical t-distribution is computed for 5 different values of R12 (the correlation between eltand ε2t) and 5 different values of R13 (the correlation between εlt and ε3t). This yields 25 empirical t-distributions for both long-run elasticities.

IV. Conclusions

Despite the stringent constraints imposed by data availability, which dictated both the level of aggregation and the simplicity of the model, this analysis provides the applied researcher with some interesting insights.

  • The paper offers a wide range of income and price elasticities for both industrial and developing countries, estimated within a consistent framework using recent lime series techniques that address nonstationarity in the data.

  • The long-run price and income elasticities for a large majority of countries have the expected sign and, in most cases, are statistically significant. Imports react relatively slowly to movements in their relative price and in the activity variable. Industrial countries tend to have significantly higher income elasticities and lower price elasticities than developing countries.

  • The bias and the distribution of the t-statistic are shown to depend critically on the degree of endogeneity of (he explanatory variables. It is shown that inference conducted on the basis of the usual asymptotic t-or F-“-distributions may be very misleading.

A similar framework can be used for the estimation of the export demand elasticities. This is the object of ongoing research.

Appendix

The Fully Modified Estimator

Assume that the generating mechanism for v, is the following cointegrating system:

y 1 t = α 0 + α 1 + β ′ y 2 t + u 1 t = γ ′ z 1 + u 1 t                                                 ( A 1 )
Δ y 2 t = u 2 t                                                                     ( A 2 )
u t = ( u 1 t u 2 t ) = Ψ ( L ) ε t ,   E ( ε t ε t ′ ) = P P ′                                                 ( A 3 )

where y1t and y2t, are scalar and mx1vector of I(1) stochastic processes, γ = (α0,α1 β) and zt=(lty2t). Define

Ω = Ψ ( 1 ) P ,       Σ = Ω Ω ′ = ( Σ 11 Σ ′ 21 Σ 21 Σ 22 ) .                                                   ( A 4 )

Σ is the long-run euvartance matrix of ut.The FM estimator is an optimal single-equation method based on the use of OLS on equation (Al) with semi parametric corrections for serial correlation and potential endogeneity of the right-hand side variables. The method was developed in Phillips and Hansen (1940) and generalized to include deterministic trends by Hansen (1992),

Detinc the OLS estimator of the cointegrating equation (Al) by Ŷ=(Z,′Zt)-1Zt y1t where Ztand Ytare (respectively) Tx (m+ 2) and Tx1 matrices of observations on zt and y1t. In general, Ŷ is consistentbut biasedbecause of serial correlation in u1tand endogeneity of y2t, The idea in the FM procedure is to modify the OLS estimator Ŷto correct for serial correlation and endogeneity bias. The FM estimator is given by

γ ^ + + = ( α ^ 0 + + α ^ 1 + + β ^ + + ) = ( T ∑ t = 1 T ∑ t T ∑ t t = 1 T ∑ t y 21 ′ t = 1 T ∑ t = 1 T t 2 ∑ y 21 t = 1 T ∑ y 21 y 21 ′ t = 1 T ∑ y 21 t t = 1 T ) - 1 ( ∑ y ^ 1 t + t = 1 T ∑ t y 1 t + t = 1 T ∑ t = 1 T y 21 , y 1 t + - T V ^ T )                                               ( A 5 )          
y ^ 1 t + = y 1 t - Σ ^ 21 ′ Σ 22 - 1 Δ y 21                                                     ( A 6 )
Σ ^ = ( Σ ^ 11 Σ ^ 21 ^ Σ ^ 21 Σ ^ 22 ) = Γ ^ 0 + ∑ r = 1 q ( 1 - v q + 1 ) ( Γ ^ r + ( Γ ^ r + ( Γ ^ r ) ′ )                                             ( A 7 )
Γ ^ v = T - 1 ∑ t = v + 1 T ( u ^ 1 t u ^ 1 t - v u ^ 2 t u ^ 1 t - v   u ^ 1 t u ^ 2 t - v ′ u ^ 2 t u ^ 2 t - v ) = ( Γ ^ 11 1 Γ ^ 12 1 Γ ^ 21 1 Γ ^ 22 1 )                                     ( A 8 )
V ^ T + = ∑ v = 1 q ( 1 - v q + 1 ) [ ( Γ ^ 12 1 ) ′ + ( Γ ^ 22 1 ) ′ ) ] ,                                         ( A 9 )

where q is the bandwidth parameter in the Bartlett window used in the estimation of the long-run eovarianee matrix Σ by the method given in Andrews and Monahan (1991), The difference between the OLS estimator and the FM estimator is highlighted in the last vector of equation (A5) where y1tis replaced by y^1t (which corrects for the potential endogeneity of y2t) and the factor TVT+(which corrects for the potential autocorrelation of the error term). These two transformations are sufficient to remove the bias induced by autocorrelation of u1t, and the endogeneity of y2t The FM estimator γ^++ has the same asymptotic behavior as the full systems maximum likelihood estimators.14

Table A1.

Import Demand Equations

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Notes: This table provides exact critical values of the FM t-statistic at 1%, 5%, 10%, 90%, 95%, and 99% significance levels for long-runimport price and income elasticities (Ep and Ey, respectively). These critical values are generated by simulating the import demand model:m1= α1mi-1 + α2mi-1 + α2gdpx1+ εtr p1=pt-1+ε2r and gdpxt= gdpxt-1+ε3t(ε11t,ε2t,ε3t) ∼ N(0,Σ) and corr (ε11t,ε2t,ε3t) = Rij, i, j, = 1,2,3, mt denotesimports, pt is the real exchange rate and gdpxt is the activity variable that is GDP-exports. All variables are in logs. The coefficients α1,α2and α3 are set to 0.70, -1.00 and 1.00, respectively. The long-run elasticities are defined as Ep=α2/(1-α1) and Ey= 3/1-α1). Their respectivet-statistic critical values are computed by (1) setting a2 and a3 equal to zero (restricted model), (2) drawing 5,000 samples of 34 observationseach (the sample size in the data) from the restricted model, (3) computing the usual t-statistic for Ep and Ey using the delta method formulafor each drawing, and (4) using the resulting vector of 5,000 t-statistic values to generate an empirical distribution from which the criticalvalues can be computed. For both Ep and Ey the empirical t-distribution is computed for 5 different values of R12 (the correlation between eltand ε2t) and 5 different values of R13 (the correlation between εlt and ε3t). This yields 25 empirical t-distributions for both long-run elasticities.

References

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*

Abdelhak Senhadji is an Economist in the 1ME Institute. He holds an M.A. in Econometrics from the University of Brussels and a Ph.D. in Economics from the University of Pennsylvania. He is grateful to participants in the Summer Econometric Society Meetings (June 2 J—24. 1996) and in the Midwest Macroeconomic Conference at Ohio State University (September 13-15, 1996). colleagues at the IMP Institute, and the referees for helpful comments. He is particular) indebted to Mohsin Khan and Masao Ogaki for insightful discussions. Patricia Laverley and Veronique Calany provided excellent editorial assistance.

1

See Goldstein and Khan (1985) for a recent survey, and Faini. Pritchett. and Clavijo (1992) for an analysis of developing countries’ imports.

2

As will he shown in Section 1. the correct activity variable is GDP minus exports, not GDP as in the standard import demand equation.

3

’Because disaggregated import prices are not available for most developing countries, only aggregate import demand equations can be estimated, for highly disaggregated import demand equations for the United States, see Marston (1990) and Swagel (1995).

4

Hcr average long-run price and income elasticities arc -0,66 and 1.31, respectively. The corresponding estimates in this paper are - 1.08 and 1.45, The small discrepancy may simply reflect different samples (these averages are over 1 2countries in Reinhart and over 66 countries in this paper) as well as different estimation methods.

5

The strong assumptions arc necessary in order to derive an aggregate import demand equation that does not require more data than what is available. In particular, there is no production sector that implies that the model does not distinguish between intermediate and linal goods.

6

For the properties of the addilog utility function, see Houthakker (1960).

7

A specilicalion analysis by Thursby and Thursby (1984) shows that (his type of dynamic specilication outperforms the static ones. The lagged dependent variable can be introduced in the model by assuming some type of adjustment costs, see Goldstein and Khan (1985).

8

The following countries have a shorter data range: Cameroon (1965-93), Ecuador (1965-93), Sudan (1960-91), Tunisia (1961-93), and Yugoslavia (1960-90).

9

Feenstra (1994) points out the problem that these aggregate prices do not account properly for quality upgrade over time.

10

Even though the behavior of imports of consumption and intermediate goods may differ, data availability precludes us from analyzing disaggregated import demand equations.

11

The OLS estimates of Ep and EY are close to the FM estimates and were therefore omitted from Table A1.

12

The variance-covariance matrix in the GLS estimation is constructed using the FM estimates of the variances of the income and price elasticities for each country. The R2 is computed using Buse’s (1973) method, which yields a properly normalized statistic for (it.S residuals,

13

Because the price elasticity is negative while the income elaslicily is positive, these elasticities are underestimated if the price elasticity bias is positive and the income elaslicily bias is negative.

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Cover IMF Staff Papers
IMF Staff papers, Volume 45 No. 2
Author:
International Monetary Fund. Research Dept.
Volume/Issue:
Volume 1998: Issue 003
Publisher:
International Monetary Fund
ISBN:
9781451974164
ISSN:
1020-7635
Pages:
204
DOI:
https://doi.org/10.5089/9781451974164.024