Time Series Analysis of Export Demand Equations: A Cross-Country Analysis
Author: Mr. Claudio Montenegro1 and Mr. Abdelhak S Senhadji
  • 1 0000000404811396https://isni.org/isni/0000000404811396International Monetary Fund

Contributor Notes

Authors’ E-Mail Address: asenhadji@imf.org and cmontenegro@worldbank.org
Publication Date:
01 Oct 1998
eISBN:
9781451923582
Language:
English
Keywords:
WP; exchange rate; exports deflator; estimates of the export demand function; unit-price elasticity; long-run elasticity; export equation
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The paper estimates export demand elasticities for a large number of developing and developed countries, using time-series techniques that account for the nonstationarity in the data. The average long-run price and income elasticities are found to be approximately -1 and 1.5, respectively. Thus, exports do react to both the trade partners’ income and to relative prices. Africa faces the lowest income elasticities for its exports, while Asia has both the highest income and price elasticities. The price and income elasticity estimates have good statistical properties.

Abstract

The paper estimates export demand elasticities for a large number of developing and developed countries, using time-series techniques that account for the nonstationarity in the data. The average long-run price and income elasticities are found to be approximately -1 and 1.5, respectively. Thus, exports do react to both the trade partners’ income and to relative prices. Africa faces the lowest income elasticities for its exports, while Asia has both the highest income and price elasticities. The price and income elasticity estimates have good statistical properties.

I. Introduction

In many developing countries, with relatively limited access to international financial markets, exports play an important role in their growth process by generating the scarce foreign exchange necessary to finance their imports of energy and investment goods, both of which are crucial to their capital formation. In his Nobel prize lecture, Lewis (1980) pointed out that the secular slowdown in developed countries will inevitably reduce the development speed of developing countries unless the latter find an alternative engine of growth. That engine, he believed, was trade among developing countries. Riedel (1984) challenges Lewis’s conclusions by arguing that most developing countries face a downward export demand function and therefore could expand their exports, despite the slowdown in developed countries, by engaging in price competition. However, Faini et. al. (1992) empirically show that Riedel’s reasoning suffers from the fallacy of composition argument in the sense that a country alone can increase its market share through a real devaluation but all countries can not. A central element in this controversy is the size of the price and income elasticities of developing countries’ export demand. Similarly, export and import demand elasticities are critical parameters in the assessment of real exchange rate fluctuations on the trade balance.

The higher the income elasticity of the export demand, the more powerful will exports be as an engine of growth.2 The higher the price elasticity, the more competitive is the international market for exports of the particular country, and thus the more successful will a real devaluation be on promoting export revenues.

The recent literature is divided on the ability of a real devaluation in affecting imports and exports. Rose (1990, 1991) and Ostry and Rose (1992) find that a real devaluation has generally no significant impact on the trade balance, while Reinhart (1995) finds that a real devaluation does affect the trade balance. Using much larger samples, this paper and its companion paper on import demand elasticities (Senhadji, 1998) offer new evidence on this issue.

The high demand from policy makers for precise estimates of these elasticities generated an extensive empirical research into the issue during the last thirty years.3 The traditional export demand function is a log-linear function of the real exchange rate and an activity variable, generally defined as the weighted (by the trade shares) average of the trade partners’ GDP.

Because of data constraints and the empirical success of this specification, it has dominated the empirical literature for more than a quarter century. A problem that has been largely ignored in the literature is that of nonstationarity, present in most macroeconomic variables, which invalidates classical statistical inference. Thus, if the variables that enter the export demand equation are found to contain a unit root, ignoring nonstationarity in these variables may lead to incorrect inferences.

This paper tackles this problem by deriving a tractable export demand equation that can be estimated, within the data constraints, for a large number of countries using recent time series techniques that address the nonstationarity of the data. The same methodology has been used to estimate the import demand elasticities, (see Senhadji, 1998). A related paper is Reinhart’s (1995) which provides estimates of import and export demand elasticities for twelve developing countries, using Johansen’s cointegration framework.

The derived aggregate export demand equation is log-linear in both relative prices and in the activity variable defined as the weighted (by the trade shares) average of GDP minus exports of the trade partners. An important insight from the explicit derivation of the aggregate export demand equation is that the definition of the activity variable depends on the aggregation level.4 The model predicts a unique cointegrating vector between exports, the real exchange rate, and the activity variable. This prediction is not rejected by the data, and the cointegrating vector is estimated efficiently by the Phillips-Hansen’s Fully-Modified (FM) estimator. The small sample properties of the price and income elasticities are also provided.

Section II derives the export demand function, Section III discusses the estimation strategy, Section IV presents the results. Concluding remarks are contained in Section V.

II. The model

Assume that the exporting country (the home country) has only one trading partner. The exporter’s export demand will thus be the same as the import demand of the trade partner (the foreign country). Hereafter, we will refer to the exporting country as the home country and the importing country as the foreign country. Assume further that the import decision of the foreign country is made by an infinitely-lived representative agent who decides how much to consume from his domestic endowment (dt*) and of the imported good (mt*).5 As noted above, the import demand of the foreign country (mt*) is identical to the export demand of the home country (xt). The intertemporal decision of the representative agent from the foreign country can be formalized as:6

Max{dt*,mt*}t=∩∞E0∑t=0∞(1+δ)−1u(dt*,mt*)(1)

subject to:

bt+1 *=(1+r)bt*+(et*−dt*)−ptmt*(2)
et*=(1−ρ)e¯*+ρet−1 *+ξt*,ξt*∼(0,σ2)(3)
limT→∞bT+1 *∏t=0T(1+r)−1=0(4)

where: δ is the consumer’s subjective discount rate; r is the world interest rate; b*t+1 is the next period stock of home bonds held by the foreign country if positive and the next period stock of foreign bonds held by the home country if negative; et* is the stochastic endowment which follows an AR(1) process with unconditional mean e¯* and unconditional variance σ2/(1-ρ2), where σ2 is the variance of the iid innovation ξt*, and ρ determines the degree of persistence of the endowment shock; and pt is the price of the home good in terms of the foreign good. Equations (2-4) are (respectively) the current account equation, the stochastic process driving the endowment shocks, and the transversality condition that rules out Ponzi-schemes. The first order conditions of this problem are:

utd*=λt(5)
utm*=λtpt(6)
λt=(1+δ)−1(1+r)Etλt+1(7)

where λt is the Lagrange multiplier on the current account equation (2). From equation (5), λt is the foreign consumer’s marginal utility for the domestic good. Following Ogaki (1992) and Clarida (1994), it is assumed that the instantaneous utility function u is addilog:

u(dt*,mt*)=At(dt*)1−α(1−α)−1+Bt(mt*)1−β(1−β)−1α>0,β>0(8)
At=ea0+ϵA,t(9)
Bt=eb0+ϵB,t(10)

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

dt*=λt−1αAt1α(11)
mt*=λt−1βBt1βPt−1β(12)

Substituting equations (9)–(11) into equation (12) and taking logs yields:

log(mt*)=log(xt)=c0−1βlog(pt)+αβlog(dt*)+ϵt(13)

where: c0 =(1/β) (b0 - a0) and ϵt =(1/β) (ϵB,t - ϵA,t). In this model, xt*= et*-dt*=GDPt*-dt*, where xt* is exports of the foreign country. Consequently, dt*= GDPt*-xt*. Thus, the model yields an equation for the export demand for the home country (x˜t) that is close to the standard export demand function except that the correct activity variable is GDPt*-xt*, i.e. GDPt minus exports of the foreign country, rather than GDPt*. Equation (13) can be rewritten as:

log(xt)=c0−1βlog(pt)+αβlog(GDPt*−xti)+ϵt(14)

III. Estimation strategy

Because each of the three variables in the export demand equation (13) can be either Trend-Stationary (TS) or Difference-Stationary (DS), four cases need to be considered. These are given in Table 1. In Section IV, results from unit root tests show that case 1 is the most common among countries, with some countries falling into the second case. Of prime interest is estimates of the standard price and income elasticities for the export demand defined, respectively, as the coefficients on the log of the real exchange rate (-1/β) and on the log of the activity variable (α/β). Note that x˜t and p˜t are in general endogenously determined by the export demand and export supply (not modeled here). Therefore p˜t is likely to be correlated with the error term ϵt in equation (14) and OLS would yield biased estimates of the price and income elasticities. Phillips-Hansen’s FM estimator corrects for this potential simultaneity bias, as well as for autocorrelation in the cointegration framework. The four cases are as follows:

Table 1.

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

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Note to table:Variables are as follows: real exports of goods and nonfactor services (x), a weighted (by the share of exports) average of the trade partners’ GDP minus exports (gdpx*) and the real exchange rate (p) computed as the ratio of the exports deflator to the world export unit values index. 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 table B.6, case 4, in Hamilton (where T is the sample size). Significance levels at 1 percent and 5 percent are indicated by ** and *, respectively.

(1) All three variables are Difference-Stationary (DS)

In the model, the following three assumptions are necessary to achieve DS for all three variables:

(I) For d˜t* to be DS, we need to assume r=δ. Then the Euler equation (7) becomes:

λt=Etλt+1(15)

In other words, λt follows a martingale process. Therefore, λt can be written as: λt =λt+1+et, where et is such that Etet =0. In other words, λt has a unit root; therefore, d˜t* will also inherit a unit root since A˜t is stationary. If d˜t* has a unit root then, from equation (13), x˜t will also have a unit root.

(ii) The log of the real exchange rate will be assumed to be DS (it will be seen that this assumption cannot be rejected statistically for most countries).7

(iii) p˜t and d˜t* are not cointegrated with cointegrating vector (1 -α).

Under assumptions (I)–(iii), equation (13) implies that x˜t, d˜t* and p˜t are cointegrated with cointegrating vector (1 1/β-α/β). Furthermore, this cointegrating vector is unique (up to a scale factor), since the export demand equation (13) has three I(1) variables and two common stochastic trends.8 If a cointegration relation between these three variables does not exist, estimation of the export demand equation (13) will result in a spurious regression. Hence, to detect this potential spuriousness, a residual-based cointegration test will be performed on equation (13).

(2) One among the three variables is Trend-Stationary (TS)

We have three cases depending on which variable is TS:

(I) x˜t is TS. The model will yield this case if p˜t is DS and if assumption (I) in case (1) is satisfied. For this case, the model predicts that p˜t and d˜t* are cointegrated with cointegrating vector (1 -α).

(ii) p˜t is TS. The model implies that x˜t and d˜t* are cointegrated with cointegrating vector (1 -1/β).

(iii) d˜t* is TS. From equation (7) and (11), d˜t* will be TS if δ > r. In this case, the model predicts that x˜t and p˜t are cointegrated with cointegrating vector (1 -α/β).

In all three cases, if a cointegration relation between these pairs of variables does not exist, attempts to estimate the export demand equation (13) will result in a spurious regression. Hence, to detect this potential spuriousness, two residual-based cointegration test will be performed on equation (13).

(3) Two of the three variables are TS. This case can be viewed as a rejection of the model, since there is no linear combination of the three variables that yields a stationary process.

(4) All three variables are TS. This is the only case in which classical inference is valid. The export demand equation (13) becomes a classical regression equation with population coefficients (1 1/β-α/β).

Equation (14) was estimated in a dynamic form (that is, with the lagged dependent variable included as an explanatory variable) to keep the specification as close as possible to the literature where this autoregressive distributed lag (ARDL) specification has been widely used.9 This dynamic form also proved to be more successful in the estimation stage:

log(xt)=γ0+γ1log(xt−1)+γ2log(pt)+γ3log(GDPt*−xti)+ϵt.(16)

While the lagged dependent variable enriches the dynamics of the export demand equation, its introduction into the cointegration framework outlined above is not innocuous. Indeed, equation (16) resembles but is not the error correction form of equation (14) since the dependent variable and the two explanatory variables are in levels and not in first differences. 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. In particular, the FM estimates of the short-run elasticities are T -consistent, and the covariance matrix of these estimators has a well-defined limit that is asymptotically singular, such that the estimators of γ2 and γ3 are perfectly collinear with the estimator of γ1. These results have the interesting implication that the FM estimators of the long-run price and income elasticities, defined respectively as Ep=γ2/(1-γ1) and Ey= γ3/(1-γ1), converge to their true value faster than the estimators of the short-run elasticities, that is γ2 and γ3. Indeed, the estimates of Ep and Ey are T-consistent. Despite the singularity of the covariance structure of the FM estimators of the short-run elasticities, valid inferences on the short- and long-run elasticities can be made using standard normal asymptotic theory. Indeed, Pesaran and Shin (1997) show that the short and long-run elasticities follow a mixture normal distribution. However, the asymptotic theory may be only a crude approximation for small samples. This issue will be examined by computing the small sample distribution of the elasticities using Monte Carlo simulation methods.

IV. Estimation results

The national account data come from the World Bank national accounts database. The data for the trade shares used to compute the activity variable were taken from UNSO COMTRADE, a United Nations disaggregated trade flow database. The sample includes 70 countries for which the required data are available for a reasonable time span. The list of countries is given in Table 1. In general the data are available from 1960 to 1993, with some exceptions.10 The usual problem is choosing the corresponding proxies for the variables in the model, since the latter is a crude simplification of reality. The variables in equation (16) will be proxied by the following: xt will be measured by total exports of goods and services in real terms. The activity variable (gdpxt*) is computed as the weighted average of the trade partners GDP minus their exports. The weights are given by the share of the home country exports to each of its partners:

gdpxt*=∑i=1NωtiGDPti, and ωti=xti/∑i=1Nxti(17)

where GDPti is real GDP of trade partner i of the home country in year t, xti refers to nominal exports of the home country to its trade partner i in year t.

The choice of a proxy for pt is not straightforward. In the model, since the only competing market to the home country’s exports is the domestic market of the foreign trade partner, pt is simply the ratio of the export price of the home country to the domestic price of the (unique) trade partner. In reality, the home country has many trading partners as well as non-trading partners competing for the same export markets. Ideally, a relative price should be included for all potential competitors of the home country exports, namely the export price of the home country relative to the domestic price of each importing country, as well as the export price of the home country relative to the export price of each potential competitor.

Obviously, this strategy cannot be implemented econometrically as the equation will contain many highly correlated relative prices leading to the usual multicollinearity problem. Instead, researchers have constructed one relative price that extracts most of the information contained in all the relative prices mentioned above.11 One possibility is to use the weighting scheme for the activity variable, described in equation (17), also for the construction of a composite price index that captures closely the potential competitive pressures facing the home country’s exports. However, the home country’s exports compete not only with the domestic market of each trading partner but also with other potential suppliers to these markets. The world export unit value, used in this paper, implies that the threat imposed by each country in the world to the home country’s exports is measured by each country’s share in world exports. The export unit value index has been retained not because it is necessarily the most appropriate one from a theoretical point of view, but because it is readily available.

1. Unit root test

To determine in which of the four categories (discussed in section III) each country falls, the three variables in the export demand equation must be tested for the presence of a unit root. The three variables in the export demand equation (16) are as follows: real exports of goods and services of the home country (x), the real exchange rate (p), and the activity variable computed as the weighted average of the trade partners’ GDP minus exports (gdpx*). The unit-root hypothesis is tested using the Augmented-Dickey-Fuller (ADF) test which amounts to running the following set of regressions for each variable:

xt=μ+γt+ϕ0xt−1+∑i=1k−1ϕiΔyt−i+ξt,k=1,…,5(18)

Note that for k=1, there are no Δyt-I terms on the right-hand side of equation (18). The lag length (k) in the ADF regression is selected using the Schwarz Criterion (SIC). The results are reported in Table 1. For xt, only 6 out of the 70 countries reject the unit root at 5 percent or less (Algeria, Burundi, Mauritania, Rwanda, and Senegal at 1 percent, Dominican Republic at 5 percent). Similarly, the null of a unit root in pt is rejected only for 1 country (Ecuador at 5 percent). Finally, as for gdpx*t, the unit root is rejected for 10 countries (Brazil, Cote d’Ivoire, Paraguay, and Zaire at 1 percent; Bolivia, Gambia, Greece, Malawi, New Zealand, and Pakistan at 5 percent). These results show that for a large number of countries, the unit root hypothesis cannot be rejected at conventional significance levels. This may simply reflect the low power of the ADF test, especially considering the small sample size.

2. Export demand equations

The results in Table 1 underscore the presence of nonstationarity in the data and the adverse consequences of neglecting it. Most countries (53 of the 70) fall into case 1 (for which the unit-root hypothesis cannot be rejected for all three variables in the export demand equation) and the remaining 17 countries fall into case 2 (for which the unit-root hypothesis can be rejected for only one of the three variables). No country belongs to case 3, which can been viewed as a rejection of the model, since the export demand equation becomes ill-specified in the sense that its estimation inevitably leads to a spurious regression. No country belongs to case 4 either (case 4 is the only case where all three variables are TS, and therefore classical inference would have been valid).

As shown in Section III, the model predicts a cointegrating relationship between the three I(1) variables in the first case and between the two I(1) variables in the second case. The export equation (16) has been estimated for the 70 countries in the sample (all falling into the first and second cases) using both the OLS and FM estimators.

Table 2 reports the results for the 53 countries that show the correct sign for both the income and price elasticities. Columns labeled x-1, p and gdpx* give, respectively, the coefficient estimates of the lagged dependent variable (log of exports of goods and nonfactor services in real terms), the short term price elasticity γ1 (i.e., the coefficient of the log of the relative price) and the short term income elasticity γ2 (i.e., the coefficient of the log of gdpx*). The long-run price and income elasticities are defined as the short term price and income elasticities divided by one minus the coefficient estimate of the lagged dependent variable. These are given by Ep and Ey for the FM estimates. Their variance, and hence their t-statistics, are computed using the delta method.12 The column labeled ser reports the standard error of the regression. Finally, column AC gives Durbin’s autocorrelation test. It amounts to estimating an AR(1) process on the estimated residuals of the export 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 6 of the 53 countries. Another potential problem with the OLS estimates is the possible endogeneity of pt. The FM estimator corrects for both autocorrelation and simultaneity biases.

Table 2.

Export Demand Equations

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Note to table:The dependent variable is real export of goods and nonfactor services (x). The explanatory variables are the lagged dependent variable (x-1), the real exchange rate (p) computed as the ratio of exports deflator to the world export unit value index and the weighted (by export shares) average of trade partners’ GDP minus exports (gdpx*). The export demand equation is estimated using both OLS and the Phillips-Hansen’s Fully Modified estimator. The long-run price and income elasticities are given by Ep and Ey, respectively. Epc and Eyc give the long run price and income elasticities corrected for bias (see Table 4). For each country, the estimated coefficients and their t-stat (below the coefficient estimates) are provided. The following statistics are also provided: Durbin’s test for autocorrelation (AC), R2, standard error of the regression (ser), and the number of observations for each country (nobs). Cointegration between the three variables in the export demand equation is tested using the Phillips-Ouliaris residual test given in column P-O. Finally, the columns labeled Ep=-1/and Ey=1 report the two-tailed test for unit-price and unit-income elasticities, respectively. The asymptotic critical values for the Phillips-Ouliaris test at 10 percent, 5 percent and 1 percent are, respectively, -3.84, -4.16 and -4.64. The letters a, b, c indicate significance at 1 percent, 5 percent and 10 percent. Exact critical values (from Table 8) are used to compute the significance level of Ep,Ey, Ep=-1 and Ey=1.
Table 3.

Bias for Short- and Long Run Elasticities for Both OLS and the Fully-Modified Estimator (in percent)

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Note to table: Table 3 provides the bias for both the short- and long-run elasticities. The bias is generated by simulating the export demand model: xt=α1xt-1+α2pt+α3gdpxt*+ε1t,pt=pt-1+ε2t and gdpxt*=gdpxt-1*+ε3t; (ε1t, ε2t, ε3t) ~N (0, Σ) and corr (ε1t, ε2t, ε3t) =Rij, i, j=1,2,3; xt denotes exports, pt, is the real exchange rate and and gdpxt* is the activity variable, i.e. GDP-exports of the trade partner. All variables are logs. The coefficients α1,α2 and α3 are set to .80, -1.00 and 1.00, respectively. The long run elasticities are defined as Ep= α2/(1-α1) and Ey= α3/(1-α1). The empirical distribution of the elasticities is generated from 5000 drawings of 34 observations each (the sample size in the data) from the restricted model. For each drawing, the export demand model is estimated. This yields 5000 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. Table 3 reports the mean of these biases expressed in percentage of the true elasticities. The bias is computed for 5 different values of R12 (the correlation between ε1t and ε2t) and R13 (the correlation between ε1t and ε3t). This yields 25 bias estimates for each elasticity.
Table 4.

Fully-Modified t-statistic Critical Values for the Export Demand Equation Parameters

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Note to table: Table 4 provides exact critical values of the Fully-Modified t-statistic at 1%, 5%, 10%, 90%, 95% and 99% significance levels. These critical values are generated by simulating the export demand model: xt=α1xt-1+α2pt+α3yt*+ε1t, pt=pt-1+ε2t and yt*=yt-1*+ε3t; (ε1t,ε2t, ε3t,) ∼ N (0, Σ) and corr (ε1t, ε2t, ε3t) =Rij, i, j=1,2,3; xt denotes exports, pt is the real exchange rate and yt is the activity variable, i.e. GDP-exports of the trade partner. All variables are in logs. The coefficients α1, α2 and α3 are set to .80, -1.00 and 1.00, respectively. For each of the coefficients α1, α2 and α3, the critical values are computed by (i) Setting the coefficient for which the critical values are computed to zero (restricted model), (ii) Drawing 5000 samples of 34 observations each (the sample size in the data) from the restricted model, (iii) Computing the usual t-statistic for each drawing, (iv) Finally, using the resulting vector of 5000 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 ε1t and ε2t).

Even though Table 2 reports both the OLS and FM estimates of the export demand equation, that is, the cointegrating equation (16), the discussion will focus only on the FM estimates, since both estimation methods yield relatively similar results. The short-run price elasticities vary from -0.0 (Peru) to -0.96 (Paraguay) with a sample average (over the first 53 countries) of-0.21, a median of -0.17, and a standard deviation of 0.19. The long-run price elasticities vary from -0.02 (Peru) to -4.72 (Turkey). The sample average is -1.00, the median is -0.76, and the standard deviation is 0.97. Exports are much more responsive to relative prices in the long-run than in the short-run. The short-run income elasticities vary from 0.02 (Ecuador) to 1.15 (Finland). The sample average is 0.41, the median is 0.33, and the standard deviation is 0.31. Thus, the average short-run income elasticity is significantly less than 1. The long-run income elasticities vary from 0.17 (Ecuador) to 4.34 (Korea). The sample average is 1.48, the median is 1.30, and the standard deviation is 0.85. Thus, exports respond significantly more to both relative prices and income in the long-run than in the short-run.

The columns Epc and Eyc give the long-run bias-corrected price and income elasticities. The correction is generally small. As will be discussed in the Appendix, the bias is negligible when the relative price 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 columns labeled Ep=-1 and Ey=1, respectively. This test uses exact critical values of the t-statistic given in Table 5. Twenty of the 53 countries reject a long-run unit-price elasticity and 18 countries reject a long-run unit-income elasticity at 10 percent or less. The fit as measured by R¯2 is good.

Table 5.

Fully-Modified t-statistic Critical Values for Long Run Export Price and Income Elasticities

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Note to table: Table 5 provides exact critical values of the Fully-Modified t-statistic at 1%, 5%, 10%, 90%, 95% and 99% significance levels for long run export price and income elasticities (Ep and Ey, respectively). These critical values are generated by simulating the export demand model: xt=