Trade Elasticities in Aggregate Models Estimates for 191 Countries

The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent.


3
Liechtenstein. The results are generally consistent with macro or aggregate estimates and recent research findings. The more detailed results below show that trade elasticities typically rise with income. And consistent with the theoretical results behind the 1-2-3 model, the average elasticity is less than one (about 0.65) for developing countries and higher than one (about 1.4) for high-income countries.
In the next section, we briefly review the literature on estimating trade elasticities to inform our choice of specification and technique. Section III presents the 1-2-3 model, which serves as the specification of our estimates of trade elasticities in Section IV. Section V contains some concluding remarks.

II. Previous Estimates of Trade Elasticities
We survey the vast literature on estimates of trade elasticities selectively to highlight three points: 1) There are few estimates of trade elasticities for developing countries, not just the Armington elasticity but also the export supply elasticity; 2) past estimates vary substantially but appear lower in recent studies; and 3) some issues, such as trends and fluctuations, are especially relevant to developing countries.
Fifty years after Armington's contribution, Bajzik et al. (2020) counted 3,524 estimates of the elasticity of substitution, varying widely, mainly for developed countries and many at the sectoral level. The authors attributed the substantial differences in magnitude to differences in data: aggregation, frequency, size, and dimension. After correcting for biases against weak results and study quality, their meta-regression analysis implied a median Armington elasticity of 3.8 with a range of 2.5-5.1; the few developing countries in the survey were predominantly upper-middle-income countries by World Bank classification (such as former Soviet Republics). Anderson and Wincoop (2004) and Head and Mayer (2014) reviewed the variation in past studies from the lenses of trade costs and the gravity framework, respectively. The elasticity estimates tend to support a high value of 3 to 7, which was the conventional wisdom in the past. Within the EU market, Zofío et al. (2020) estimated the foreign trade elasticity at 2.2, lower than the national trade elasticity. Nevertheless, the estimates vary widely, and the median values are as low as 0.9 (Gallaway et al. 1997) and 0.97 (Reinert and Roland-Holst 1992) and as high as 6.5 (Hertel et al. 2007). Moreover, many estimates pertain to the sectoral level and primarily higher-income countries.
Although estimates have varied widely, common patterns have emerged from reviews of past studies. Ahmad, Montgomery, and Schreiber (2020), McDaniel and Balistreri (2003), and Imbs and Mejean (2015) found trade elasticity estimates to decrease with the level of aggregation. In particular, commodities exhibit a high Armington elasticity, while differentiated products (like manufactures) tend to have a low elasticity. McDaniel and Balistreri (2003) also observed that long-run elasticities are higher than their shortrun counterparts and that reduced-form time-series analyses have a lower magnitude than cross-sectional studies. Imbs and Mejean (2015) argued that aggregation restricted sector elasticities to be homogeneous and suggested using a weighted average of sector elasticities for aggregated ones to avoid heterogeneity bias. Although worth exploring further, data constraints in developing countries will make comparing macro estimates with averages of econometric estimations of sector values challenging.
But are macro elasticities consistently lower than those from micro elasticity studies of more specific sectors or commodities? Feenstra et al. (2018) found the results mixed: for between two-thirds and three-quarters of sample goods, there is no significant difference between the macro-and micro-elasticities, but the micro elasticity is significantly higher for the rest. There are also differences in findings at the disaggregated levels as well. Brenton and Winters (1992) do not assume separability between home and foreign goods and find low import price elasticities. In contrast, Panagariya, Shah, and Mishra (1996) employ better data, such as explicit competitors' prices (not proxies), and find high elasticities. These elasticities apply to more specific groups of commodities, i.e., not at the level of aggregation of the 1-2-3 model. At the aggregate level, foreign and domestic goods are composites of many different goods, and the substitution possibility (e.g., as found in Devarajan, Go, and Li, 1999) will lead to smaller elasticity values and ranges than more disaggregated cases (e.g., in Hillberry and Hummels, 2013;Bajzik et al. 2020).
Recent estimates tend to point towards elasticities that are lower in magnitude. For example, Boehm, Levchenko, and Pandalai-Nayar (2023) compared the differential effects on imports of countries that changed Most Favored Nations (MFN) tariffs with their trading partners relative to a control of countries not subject to the MFN scheme. The authors identified the trade elasticity for the short and long horizons. They found the short-term elasticity (one year after the exogenous tariff change) to be 0.76 and the long-run elasticity ranging from 1.75 to 2.25 (typically after 7-10 years). The sample covered trade flows of goods at the disaggregation level of HS6 (Harmonized System at the 6-digit codes). Each panel of countries had over 80 countries, most of them high-income; the developing countries included were mainly upper-middle-income or large countries in global trade like China and India. The method did not differentiate by country, so the paper did not provide elasticities for developing countries.
An earlier study by Whalley (1985) likewise inferred the value of the Armington elasticity from trade liberalization episodes and yielded an estimate in the neighborhood of 1.5 over 5-10 years.
Past studies concentrated on the Armington elasticity between imports and domestic goods, ignoring much of the export supply side because of estimation issues. In CGE models, the export supply side is usually defined as a function with a constant elasticity of transformation between exports and domestic goods, a symmetric formulation to the Armington formulation (see further discussion below). Hillberry and Hummels (2013) suggested that future research uses firm-level heterogeneity to uncover export behavior and elasticity, which affect trade flows jointly with the demand side.
Only a few studies investigated export supply explicitly. Diewert and Morrison (1988) employ a production-based approach initially developed in Kohli (1978) to obtain export supply and import demand. Faini (1994) directly estimates transformation elasticities from a CET function and considers adjustment lags, factor prices, and the importance of capacity utilization. He finds that the CET elasticity is less than one for Morocco but much greater than one for Türkiye. It would be difficult to replicate these studies for many countries without extensive microdata. One reason is the measurement problems of factor accumulations and their returns. Another issue is the adjustment lags in supply that may require measuring capacity utilization. The impact of lagged variables may also require time-series estimation that accounts for nonstationarities, such as vector autoregression (VAR) or its restricted form, the vector error correction (VEC) model, which we use in our analysis below.
The few studies examining developing countries' exports indirectly modeled exports as import demand from major trade partners, e.g., Reinhart (1995) and Senhadji and Montenegro (1999). Reinhart (1995) calculated the import demand for 12 developing countries and the corresponding export demand (from industrial countries). Senhadji and Montenegro (1999) estimated export price and income elasticities as import demand from trading partners for 53 industrial and developing countries. As a salient feature, both studies used time-series estimation to handle nonstationarity issues and the lack of cointegration that might result in spurious relationships. Devarajan, Go, and Li (1999) also used time series and other techniques but estimated the Armington and CET elasticities directly for many developing countries. They found some export elasticities have the wrong sign, possibly because of the short data series and potential identification issues affecting export supply (see below). 5 These elasticities were excluded from the final estimates.
In addition to world prices, domestic prices affect imports and exports. However, the literature is divided on the impact and significance of the real exchange rate, or the ratio of domestic to world prices, on the trade balance. Earlier papers such as Branson (1972), Khan (1974), Rittenberg (1986), Bond (1987, and Marquez and McNeilly (1988) found that trade flows respond significantly to changes in relative prices. They are criticized today for inference problems associated with time-series variables with unit roots. Empirical work that considered the time-series properties of trade flows and prices, such as Rose (1990 and1991) and Ostry and Rose (1992), found little evidence that relative prices affect trade flows. Yet, the lack of theory in time-series techniques makes the estimates difficult to interpret. Marquez (1994), for example, stressed the importance of optimizing behavior and simultaneity in determining expenditures on domestic and foreign goods. For developing countries, Faini, Pritchett, and Clavijo (1988) discussed the importance of trade policy and restrictions, which are likely to understate the structural demand elasticities. Reinhart (1995) uses dynamic optimizing behavior with time-series techniques and finds significant trade relationships. Our analysis follows this approach by combining the optimizing behavior of the 1-2-3 model with time-series techniques.
Nonetheless, recent open-economy macroeconomic models use a trade elasticity below or around unity, often assumed, calibrated, or estimated for high-income countries. For example, Corsetti, Dedola, and Leduc (2008), Gust, Leduc, andSheets (2009), andJustiniano andPreston (2010) have elasticity estimates between 0.8 and 0.86, while Galí and Monacelli (2005) used unity. Among textbooks, Végh (2013) examined the impact of devaluation under alternative elasticity of substitution between tradables and nontradables from 0.2 to 1.0; 6 Uribe and Schmitt-Grohé (2017) set the elasticity to unity but tested the economic impact of termsof-trade shocks of alternative values of 0.75 and 1.5. 7 Another issue is the assumption of homotheticity of the Armington or CET function, which is violated by the time trends observed in trade shares. Import and export shares in GDP for many countries appear to be increasing, independently of relative price movements. For example, Alston et al. (1990) note that while the implicit assumption of homotheticity in the CES and CET formulations is theoretically appealing, it is highly restrictive in CGE modeling. The standard correction usually employs a scale variable, such as an income term, to denote aggregate income activity. Alternative formulations like the almost ideal demand system (AIDS) or one of the flexible functional forms are often suggested.
While it is plausible that the capacity to import among countries rises with income, Petri (1984) and Ho and Jorgenson (1997) believe that the estimated high-income elasticities are probably spurious. Trade shares seem to increase over time for rich and poor countries, as would be the case with increasing 5 Several estimates in Devarajan, Go, and Li (1999) used OLS (ordinary least squares) with simple time trends. While that approach could capture rising trade shares over time, it might not alleviate autocorrelation in the residuals, making the coefficient estimates inefficient or the standard t-tests improper. Seemingly unrelated regression (SUR) that assumes a joint distribution of error terms across countries (with a small sample) was also used to improve the efficiency of the variance. That approach is not necessary with a larger sample size now available for each country. 6 p. 295, Table 6.1. 6 globalization. A natural breakpoint was the 1970s when the international monetary and trading system changed substantially. Even for large industrial countries like the United States, there was a sharp acceleration in the import share in the 1970s. For developing and transitional economies, periods of rapid economic and trade liberalization (particularly in the late 1980s) are crucial factors. Compared to the earlier periods of inward orientation, changes in trade policy in the latter periods often led to structural breaks in the trade shares. We use a time trend to account for the shifts in trade shares or the ratios of factors (equations 7 and 8 below).
Moreover, trade ratios might not rise steadily; they may fluctuate erratically due to policy reversals, crises, or conflicts, particularly in developing countries. For example, Arbache et al. (2008) found that growth collapses in Sub-Saharan Africa were frequent before 1995. Weather conditions could also affect exports of developing countries when they are mainly agricultural products. In these cases, the trade shares in output are likely co-dependent variables with no exogenous trends.

III.
Methodology: The Model, Data, and Specification The 1-2-3 Model The 1-2-3 model became a practical economic tool at the World Bank and for teaching more complex models due to its transparent algebraic and conceptual results and accessible numerical implementation using only national accounts and popular spreadsheet software (see Devarajan, Lewis, andRobinson 1990 and1993;Devarajan, Go, Lewis, Robinson, and Sinko 1997). Since its inception, the 1-2-3 model has addressed various policy issues in developing countries. The most common application has been the real exchange rate effects (including Dutch disease effects) of commodity price shocks or changes in capital flows (such as foreign aid and transfers). The algebraic and numerical spreadsheet solutions anticipate the relationship between external shocks and policy responses of more complex models. For example, the model was used to determine the pre-1994 overvaluation of the CFA franc (Devarajan 1997(Devarajan , 1999, which informed the discussion about the magnitude of the 1994 devaluation. Since the new millennium, the authors have conducted similar exercises as part of World Bank operational work in CFA countries, the Arab Republic of Egypt, Zambia, and other African countries. Another productive application has been the economic effects of trade reform, especially in the 1990s when the issue was crucial for many developing countries. Devarajan, Go, and Li (1999) quantify the fiscal effects of trade reform and show how the results depend on the substitution elasticity between foreign and domestic goods. Devarajan and Go (1998) incorporate rational-expectations dynamics to capture the intertemporal effects of trade reform and import price shocks. Relatedly, de Melo and Robinson (1992) use the framework to examine export externalities in developing countries. Taking advantage of the model's simplicity and minimal data requirements, Auriol and Warlters (2012) use the 1-2-3 model to calculate the marginal welfare cost of public funds in 38 African countries. Devarajan and Go (2003) link the framework to growth and poverty modules to examine the implications of growth and poverty reduction strategies, especially in those classified as Heavily Indebted Poor Countries (HIPCs) in Africa. The model has also been used to study the macroeconomic dynamics of scaling up foreign aid (Devarajan, Go, Page, Robinson, and Thierfelder, 2008). Extending the regional integration application in Devarajan, Go, Suthiwart-Narueput, and Voss (1997b), a global version called the R23 model exploits its parsimonious structure to link, through trade flows, over 200 1-2-3 models (McDonald, Thierfelder, and Walmsley 2012). Finally, Devarajan, Dissou, Go, and Robinson (2017) developed a dynamic stochastic general equilibrium version to examine budget rules when the export price of a resource-rich country is uncertain.

Specification of Trade Elasticities in the 1-2-3 Model
The 1-2-3 model is specified for one country producing two commodities: an export good (E) traded and sold only to foreigners and a domestic good (D) that is nontraded and sold domestically. The third commodity is a traded import (M) sold in the domestic market. One representative consumer receives all income and allocates it according to preferences for the two goods sold on the domestic market, D and M. The country is small in world markets, facing exogenous world prices for exports and imports. The two traded goods (E and M) and the nontraded good (D) are imperfect substitutes, a feature found in most CGE models that follows the distinction of "tradable" (imports and exports) and "nontradable" (the domestic good) of Salter (1959) and Swan (1960). The consumer's utility function consists of the Armington CES function of D and M. Production is specified by a CET production possibility frontier of D and E. There is no need for separate production functions for D and E-the transformation function is all that is needed for the analysis. 8 The two elasticities in the model characterize the trade substitution possibilities. These are the key parameters to be estimated.
The CES and CET functions have the same algebraic form and are distinguished by their parameters (convex for the CES and concave for the CET). Equation 1 represents the common CES and CET functions. Equation 2 is its corresponding dual price equation. (1) where X is the CES or CET composite or ̅ ; ̅ is the shift parameter, α is the share parameter, and  is the exponent: In the CES case, 1 ≡ and 2 ≡ ; and in the CET case, 1 ≡ and 2 ≡ . The P's are the corresponding domestic prices of the inputs, , , . The CES substitution elasticity σ and CET transformation elasticity Ω are given by = 1 (1 − ) ⁄ ; −∞ < < 1 in the CES case and Ω = 1 ( − 1); 1 < < ∞ ⁄ in the CET case.
Both the CET and CES functions exhibit constant returns to scale. The allocation of the composite good into its components depends on the relative prices of the individual components. Noting that = = in equilibrium, the corresponding export supply and import demand functions are expressed as ratios from the first-order conditions for profit and utility maximization (equations 3 and 4): export supply, and where and are the corresponding share parameters in the CET export transformation and CES import aggregation functions. 9 The CET function describes the production transformation frontier between D and E for a fixed level of ̅ or real GDP (since there are no intermediate inputs). The assumption that ̅ is fixed is equivalent to assuming full employment of all primary factors. The composite good price P x corresponds to the GDP deflator. The composite price of Q corresponds to the consumer price index. Following the numerical implementation in Devarajan et al. (1997), GDP not sold to the rest of the world (i.e., E, exports of goods and services) defines the domestic good D. Given price indices, P x and P e , the implicit price for the domestic good, P d , can be derived from the GDP identities: where Q is aggregate demand. The model can therefore be implemented using national data for macro aggregates (see Devarajan, Go, Lewis, Robinson, and Sinko 1997).

Estimating Equation
The log-linear transformation of the supply and demand equations (3) and (4) provides a convenient way to estimate the elasticities: Note that equations 5 and 6 extend beyond the 1-2-3 model. As mentioned in the literature review, the import equation derived from the CES function is also known as the Armington import demand in the trade literature. Exports are also often expressed as Armington import demand from the rest of the world. However, in our case, the CET function and the estimation of Ω complete the country-specific model.

Data
Data are from the World Bank's World Development Indicators (WDI) 10 and the United Nations National Accounts database. 11 WDI is used where available. The two sources are combined to extend series or fill in missing observations. We retain the WDI levels and base year for constant prices where the two 9 sources are spliced. We use the series measured in current and constant U.S. dollars to derive the implicit price indices for the quantities. The base year is 2015, and WDI uses that year to calculate the world totals. 12 Using uniform dollar units affords two innovations in the estimation. First, it provides consistent units across countries to estimate trade elasticities. Second, having the world's aggregate GDP and its components in comparable dollar units yields global demand variables for a country's exports, potentially correcting an identification issue in the CET estimation for some cases (see below).
Where available, we obtain time series for 1970-2018, potentially having 49 data points for each country. We avoid the 1950s and 1960s because many developing countries had just become independent, which is a reason why the sample size was small for many countries in the previous estimation of Devarajan, Go, and Li (1999). Seventeen or more Sub-Saharan African countries became independent only in 1960 and after (e.g., Cameroon, Kenya, Madagascar, Senegal, Tanzania, Uganda, Zambia, etc.). Nascent developing countries often have limited statistical capacity in the early years after independence, with issues of measurement reliability. Moreover, we exclude the years after 2018 because of the coronavirus of 2019 (COVID-19) and the disruptive effects of the pandemic on supply chains and global trade.
In cases where the series is shorter, we flag them. For example, for countries belonging to the former Soviet Union, data can only begin from about 1990. As a general rule, we omit countries with fewer than 20 observations.

Nonstationarity
Macroeconomic series and aggregate price indices (such as , , , , ) are known to be nonstationary. It typically takes differencing twice to make the series stationary, implying an integration order of two, i.e., I(2). Without further transformation, the significant time trend present in the data could lead to spurious relationships and estimates.
Fortunately, the log ratios of the variables in the estimating equations above reduce the order of integration. Using the Augmented Dickey-Fuller (ADF) unit root test of each transformed variable , we cannot reject nonstationarity in the null hypothesis. However, differencing the series just once makes each case stationary, implying an integration order of one, I(1). We confirm this finding to be true for each variable for each country.

Identification
For some countries, the CET equation (5) might have identification problems as export supply could co-depend on global demand, expressed as an Armington import demand condition similar to equation 4: The demand for as a ratio to the global domestic good is a function of their relative price ; is the underlying CES share parameter. The log transformation of equation 7 is the familiar export demand equation in the literature, similar to Reinhart (1995) and Senhadji and Montenegro (1999): Previous literature often uses just the aggregation of industrial countries for and with the bilateral trade flows as weights. However, the trade weights are shifting significantly over time, difficult to derive, or unavailable consistently for each country's entire 1970-2018 period. For this reason, we use the global aggregation of national accounts already available in the WDI database to derive and , which are consistent with the specification of the 1-2-3 model. Since the global GDP and its components are also expressed in current and constant U.S. dollars, the global variables are consistent with the country variables.
However, for the 1-2-3 model, we are interested in the CET elasticity Ω from equation 5 and not the CES elasticity linked to global demand in equation 8. Whenever the identification issue arises (usually if there is an incorrect sign for the CET coefficient), we consider including the variables of equation 8 as additional cointegrating or exogenous variables in the long-term cointegration equation or part of the error correction of the VEC.
Equations 5 and 8 could also be solved simultaneously, yielding equation 9 as another option for estimating Ω. We apply a time series technique like VEC to it.

Fluctuations and Breaks
For many developing countries, the variables are often characterized by fluctuations rather than a single break, mainly due to policy reversals, crises, conflicts, or exogenous shocks from the weather (e.g., drought, hurricanes, etc.). Figure 1 shows the frequent fluctuations of relevant variables in Benin in contrast to the smoother movements in the United States. In these situations, we find that the growth rates of output, real exports, and real imports, as well as the appropriate share of trade in GDP, work well as extra exogenous variables in the cointegration equation or the error correction part. The growth rates are stationary, so they do not add to the number of cointegration equations for estimation if introduced as endogenous variables in rare instances. We adopt this approach for most countries. An episodic structural change could lead to a false unit root in a stationary series with a structural break. In this possible situation, we use a breakpoint unit root test to consider a dummy variable and its timing and ensure that the null hypothesis of a unit root is not rejected by the Dickey-Fuller t-test (with or without a dummy). The dummy break often pertains to the intercept but could also involve intercept and trend. The innovation associated with a structural change could be gradual (versus a one-time break) after the event; changes could also build up in the years before a significant historical event; and the type and timing could differ among the cointegration equation variables. Because of the many factors and different possible dates for the variables in the cointegration equation, we minimize using dummy variables.
Moreover, the statistical determination of a break could benefit from knowledge of the economic history of a country. However, except for well-known events, an in-depth understanding of the timing of actual events, shocks, or crises is beyond this study's scope (given the number of countries). In the estimation, we also consider a dummy when the time patterns of the right-and left-side variables begin to diverge over their past behavior, combining breakpoint unit root tests, visual inspection, and, if available, relevant economic information 13 to determine possible timings. 13 Such as country reports from the World Bank, IMF, Economist Intelligence Unit (EIU), or Wikepedia. Take two well-known cases, Germany and South Africa. Figure 2A  Conflicts, regime changes, and crises could also affect data quality. In this regard, we follow WDI's data vetting process about the first year to use while allowing for data splicing from the U.N. national accounts for one or two data series of a country. We only include countries with at least 20 years of time series data. 14 Unlike former Soviet republics and the allied communist states, Germany has data prior to the unification in 1990 and that goes back to 1970.

Re-exports or Sudden Surges in Exports
Significant surges in re-exports are related to fluctuations and breaks, where foreign goods (imports, then exports) pass through from one country to another. Examples include Hong Kong SAR, China, after China's economic liberalization in 1978 and Ireland after the Good Friday Agreement for Northern Ireland in 1998. Singapore is another case with growing re-exports. With no good data yet to separate re-exports, the total value of exports will far exceed GDP, leaving a negative difference for an estimate of the domestic good. The log factor ratios of equations 7 and 8 will become indeterminate. In these cases, we avoid imputing domestic goods from historical shares since the proportions could fluctuate even before the surge in re-exports. Instead, we use the input demand and supply relative to output (GDP) for equations 7 and 8. In place of and , we use and , respectively, in both equations (using a country's aggregate output to approximate its aggregate demand on the import side). We use the same approach for equation (9), reasoning that a country's output is also a good approximation of domestic goods relative to the global variables. The solutions appear to work well for these few countries.
Sudden export surges or changes due to oil finds, mineral exports, or tourism could have the same effects. Examples include Antigua and Barbuda, Grenada, and Iraq.

Estimation Methods
As an estimation priority, we employ the vector error correction (VEC) model, a restricted form of the vector autoregression (VAR) model. 15 The method's output has two parts. The first part produces the cointegration equation (CE) or the long-term equilibrium (equations 5, 6, or 9), providing the desired elasticity estimates. It may include possible adjustments or exogenous variables like trends and global variables, which, as a general rule, we restrict to a relevant few. We use the Dickey-Fuller distribution that corrects for the fact that the p-value for the standard t-statistic is skewed to the left. The second part of the output is the error correction. The latter contains the impact of lagged variables that ensure perturbations or deviations will return to the long-term relationship estimated in the first part. We check the system for stability but only report the estimated elasticities and the corrected p-value and t-tests in the long-run cointegration relationship.
Alternatively, we try single cointegrated regressions using the Fully Modified OLS (FMOLS), which allows for various trends and additional regressors. The Wald test for simple linear restriction checks that the elasticities are not zeroes. For the CET case, we also apply VEC to equation 9. If VEC and FMOLS do not yield satisfactory results, we use the Generalized Method of Moments (GMM)). At least one of these methods almost always seems to produce reasonable estimates, so there was no need to look beyond them. 16 Each country presents unique circumstances suggesting a potential for self-contained estimation (for example, see discussion about Figures 2 and 3). However, we limit the interventions to a minimum consistent set across countries. In addition to possible intercepts, trends (linear or quadratic), and lag structure, we confine the introduction of other variables to those already defined in the equations or derived from them, such as trade shares, growth rates of underlying variables in real terms (like exports, imports, and outputs), or global demand and price ratios. If included, they are usually done as exogenous variables in CE or VAR in VEC; the CE deterministic regressors or additional deterministic regressors in FMOLS; or instrument variables in GMM.
Whatever the interventions, we further check the Johansen cointegration test summary and ensure that the chosen estimation has a cointegration equation in the case of VEC. In FMOLS, we look at tests such as Hansen Instability, Engle-Granger, etc., for confirmation. If the Ω estimation is the reduced form of the export system in equation 9, there could be up to two cointegration equations. No such proof is needed in the GMM case.

IV. Estimates
Using the methodology described above, we estimated elasticities for as many as 191 countries. 17 Tables A1 and A2 in the appendix present the estimated elasticities, the method used, the t-tests, and a summary of the interventions introduced. Table A3 lists the data source and years covered for each country. The elasticities appear reasonable, and the t-tests (except for a handful) are significant. As the summary in Table 1 shows, one hundred twenty-six countries (66.0%) have the full sample size of 1970-2018; 153 countries (80.1%) have at least 30 years of observations in their data. Of the 38 cases (19.9%) with less than 30 observations, 58% (22 cases) are borderline, with 29 observations covering 1990-2018 and mostly former Soviet republics and allied communist states.
The estimates cover 45 of 48 countries in Sub-Saharan Africa, including low-income countries like Benin, the Republic of Congo, Niger, etc. Island economies of various sizes tally as many as 47, including Fiji, Kiribati, Marshall Islands, etc. The list includes seven microstates, economies with fewer than 1,000 residents, and 49 hectares of land, such as Nauru, Tuvalu, Liechtenstein, etc. Many of these countries have severe data constraints that would have made estimation difficult with more elaborate formulations and data requirements. 18   Table 2 presents the simple averages of the Armington (import) and CET Ω (export) elasticities for various income and regional groups according to World Bank classification. The income classification is based on GNI (gross national income) per capita in U.S. dollars (Atlas methodology) for 2015, the base year of the constant price series in the estimation. The table averages could provide "rules of thumb" estimates for missing countries. They also show interesting tendencies by broad income categories. The average and Ω of high-income countries tend to be higher than 1.0, about 1.4, while those of the lowerincome groups are less than 1.0. The average elasticities for low-income and lower-middle-income countries tend to be close to one another, around 0.7 for and slightly less than 0.6 for Ω. The uppermiddle-income countries tend to have slightly higher values but still less than 1.0, around 0.7 for , and

Group Classification
Armington ( somewhat more than 0.6 for Ω. For practical purposes, the elasticities in developing countries could be approximated as 0.65. Regarding regional averages, the North America region (NAR) and the Europe and Central Asia (ECA) groups have the highest elasticities. They are followed by the Middle East and North Africa (MENA), East Asia and the Pacific (EAP), Latin America and the Caribbean (LAC), South Asia region (SAR), and Sub-Saharan Africa (SSA). The order generally follows the average regional income. Where some regions have mixed incomes, the table shows the average differences between high-income and developing countries. The only high-income country in SSA in 2015 was Seychelles, while Mauritius is approaching this category. The pattern generally supports the hypothesis that trade elasticities increase with per-capita income. Figure 3 shows the scatter plots of the elasticities against GDP per capita in 2015 U.S. dollars, the constant price series base year in the estimation and income group classification. 19 The simple correlation of the variables is about 0.71 for both the and the Ω cases. The regression line in each plot confirms an approximately positive relationship. 20 The graphs could also easily be non-linear, flat for much of the lower income levels, then dispersing and rising rapidly at higher income at around $22,000 (log of 10). The dichotomy of elasticities at about 1.0 is consistent with the summary in Table 2.
The split of elasticities at 1.0 between low and high-income countries is also consistent with the trade theory behind the 1-2-3 model (see Devarajan et al. 1997 and other studies in the sub-section on the 1-2-3 model). When the world price of imports (say) rises in an economy, there are two effects: an income effect (as the consumer's real income is now lower) and a substitution effect (as the domestic good now becomes more attractive). The resulting equilibrium will depend on which effect dominates. When < 1, the income effect dominates. The economy contracts the output of the domestic good and expands that of the export commodity. To pay for the needed, imperfectly substitutable import, the real exchange rate depreciates. However, when > 1, the substitution effect dominates. The economy's long-term response is to contract exports (and hence also imports) and produce more of the domestic substitute. For most developing countries, it is likely that < 1, so that the standard policy advice to depreciate the real exchange rate in the wake of an adverse terms-of-trade shock is correct. 21 For developed economies, one might reasonably expect substitution elasticities to be high. In this case, the response to a terms-of-trade shock is a real appreciation, substitution of domestic goods for the more expensive (and non-critical) imports, and a contraction in the aggregate volume of trade. In all countries, one would expect substitution elasticities to be higher in the long run. The long-run effect of the real exchange rate will thus differ, and may be of the opposite sign from the short-run effect.
Another example relates to the revenue effects of tariff reforms. Devarajan, Go, and Li (1999) show how the fiscal impact of tariff liberalization also depends on the substitution elasticity between foreign and domestic goods. Unless there is an upward shift in output productivity, a reduction in tariffs will invariably involve losses in revenue for much of the plausible range of the trade elasticities unless compensated by increases in domestic taxes. There is no Laffer Curve for import tariffs.  Table 3 provides additional summary descriptive statistics for the two main income groups. The median trade elasticities are close to the simple averages in all cases. In developing countries, the upper range is also less but close to one for both the Armington and CET elasticities. Countries with elasticities near the value of one are usually those close to the boundaries of the high-income group, such as emerging economies like Brazil and South Africa and some Latin American countries like Costa Rica and Argentina. 22 Although the average and median elasticity of high-income countries is higher than one, the lower range is less than one, overlapping with the upper range of the developing countries. The reasons are many: the boundary is ad hoc, and the income range for high income is wide; at the lower range are new entrants, such as former communist countries like Latvia and Poland, and island and Latin American countries like Barbados, Bermuda, and Chile. If we restrict the group to early OECD countries, the lower value is greater than 1.10 for both elasticities. Our results are consistent with recent findings of lower elasticities for macro or aggregate elasticities than those of micro elasticity studies of specific sectors or commodities. Because a cointegration equation corresponds to a long-term equilibrium, the results are comparable to the range of 1.75 to 2.25 for the long-term elasticity estimates of Boehm, Levchenko, and Pandalai-Nayar (2023), especially for the high-income group. Past results that do not correct for nonstationarity issues in time-series estimation will also be spuriously high. The survey and analysis by Bajzik et al. (2020), including many past studies at sectoral levels, show a broader range in magnitude. Plausible explanations mentioned before include aggregated imports, exports, and domestic goods are composites of many goods; developing countries are less diversified, and the composition of these goods tends to differ; and the external balance of payments constraint at the macro level could limit substitution possibilities. Finally, this study provides countryspecific estimates of trade elasticities for many countries, 128 developing countries and 63 high-income countries. Not only were estimates for developing countries lacking in the literature, but estimates of the export supply side were also lacking.

V. Conclusion
This paper tries to fill a lacuna in the literature. While the Armington elasticity has been estimated at the sectoral level for a number of (mostly) developed countries, the equivalent elasticity in the widely used aggregate, 1-2-3 model of developing countries has typically been assumed. We provide empirical estimates of the import and export elasticities of the 1-2-3 model for 191 countries. Using data from 1970-2019 and the Vector Error Correction model as the dominant technique, we derive robust estimates that also square with intuition. Elasticities for high-income countries are generally greater than one, averaging around 1.4, while those of lower-income countries are below one, averaging around 0.65. Not only do these estimates confirm 22 Argentina became high-income in 2017.

High-income countries
Developing countries

Armington (σ) CET (Ω)
that most of the assumed parameters were broadly correct, but they provide the foundation for future, multicountry use of the 1-2-3 model, whose parsimony and data economy have been the hallmarks of its extensive application.
Shiells, Clinton R., David W. Roland-Holst, and Kenneth A. Reinert. 1993 Source: Authors' calculations Notes: C = constant CE = cointegration equation dummy = dummy variable egr = growth rate of real exports eshare = exports/GDP FMOLS = fully modified OLS for cointegration regression global demand and price ratios = variables in equation 9 GMM = generalized method of moments inst(s) = instrument variable(s) lyx = log of output (real GDP) index mgr = growth rate of real imports mshare = imports/GDP NOB = number of observations after adjustments T = linear trend T^2 = quadratic trend reduced eq of export system = equation 9 VEC = vector error correction VAR = error correction part in VEC or vector autoregression xgr = growth of output (real GDP) 31 Appendix GMM = generalized method of moments inst(s) = instrument variable(s) LIML = limited information maximum likelihood lyx = log of output (real GDP) index mgr = growth rate of real imports mshare = imports/GDP NOB = number of observations after adjustments T = linear trend T^2 = quadratic trend reduced eq of export system = equation 9 VEC = vector error correction VAR = error correction part in VEC or vector autoregression xgr = growth of output (real GDP)