Bonus vetus OLS: A simple method for approximating international trade-cost effects using the gravity equation☆
Introduction
For nearly a half century, the gravity equation has been used to explain econometrically the ex post effects of economic integration agreements, national borders, currency unions, immigrant stocks, language, and other measures of “trade costs” on bilateral trade flows. Until recently, researchers typically focused on a simple specification akin to Newton's Law of Gravity, whereby the bilateral trade flow from region i to region j was a multiplicative (or log-linear) function of the two countries' gross domestic products (GDPs), their bilateral distance, and an array of bilateral dummy variables assumed to reflect the bilateral trade costs between that pair of regions; we denote this the “traditional” gravity equation.
However, the traditional gravity equation has come under scrutiny, partly because it ignores that the volume of trade from region i to region j should be influenced by trade costs between regions i and j relative to those of the rest-of-the-world (ROW), and the economic sizes of the ROW's regions (and prices of their goods) matter as well. While two early formal theoretical foundations for the gravity equation with trade costs — first Anderson (1979) and later Bergstrand (1985) — addressed the role of “multilateral prices,” Anderson and van Wincoop (2003) refined the theoretical foundations for the gravity equation to emphasize the importance of accounting properly for the endogeneity of prices. Two major conclusions surfaced from the seminal Anderson and van Wincoop (henceforth, A-vW) study, “Gravity with Gravitas.” First, traditional cross-section empirical gravity equations have been misspecified owing to the omission of theoretically-motivated endogenous multilateral (price) resistance terms for exporting and importing regions. Second, to estimate properly the general equilibrium comparative statics of a national border or an EIA, one needs to estimate these multilateral resistance (MR) terms for any two regions with and without a border, in a manner consistent with theory. Due to the nonlinearity of the structural relationships, A-vW applied a custom nonlinear least squares (NLS) program to account for the endogeneity of prices and estimate the general equilibrium comparative statics.
Another — and computationally less taxing — approach to estimate unbiased gravity equation coefficients, which also acknowledges the influence of theoretically-motivated MR terms, is to use region-specific fixed effects, as noted by A-vW, Eaton and Kortum (2002), and Feenstra (2004). An additional benefit is that this method avoids the measurement error associated with measuring regions' “internal distances” for the MR variables. Indeed, van Wincoop himself — and nearly every gravity equation study since A-vW — has employed this simpler technique of fixed effects for determining gravity-equation parameter estimates, cf., Rose and van Wincoop (2001) and Baier and Bergstrand (2007a). Yet, without the structural system of nonlinear equations, one still cannot generate region- or pair-specific general equilibrium (GE) comparative statics; fixed effects estimation precludes estimating MR terms with and without EIAs. Empirical researchers can use fixed effects to obtain the key gravity-equation parameter estimates, and then simply construct a system of nonlinear equations to estimate multilateral price terms with and without the “border.” But they don't.
Consequently, the empirical researcher faces a tradeoff. A customized NLS approach can potentially generate consistent, efficient estimates of gravity-equation coefficients and comparative statics, but it is computationally burdensome relative to ordinary least squares (OLS) and subject to measurement error associated with internal distance measures. Fixed-effects estimation uses OLS and avoids internal distance measurement error for MR terms, but one cannot retrieve the multilateral price terms necessary to generate quantitative comparative-static effects without also employing the structural system of equations. Is there a third way to estimate gravity equation parameters using exogenous measures of multilateral resistance and “good old” (“bonus vetus”) OLS and/or compute region-specific resistance terms that can be used to approximate MR terms for comparative statics or other purposes (to be discussed later) without using a nonlinear solver? This paper suggests a method that may be useful.
Following some background, this paper has three major parts: theory, estimation, and comparative statics. First, we suggest a method for “approximating” the MR terms based upon theory. We use a simple first-order log-linear Taylor-series expansion of the MR terms in the A-vW system of equations to motivate a reduced-form gravity equation that includes theoretically-motivated exogenous MR terms that can be estimated potentially using OLS. However — unlike fixed-effects estimation — this method can also generate theoretically-motivated general equilibrium comparative statics without using a system of nonlinear equations or assuming symmetric bilateral trade costs.
Second, we show that our first-order log-linear approximation method provides virtually identical coefficient estimates for gravity-equation parameters to those in A-vW. For tractability, we apply our technique first to actual trade flows using the same context and Canadian–U.S. data sets as used by McCallum (1995), A-vW, and Feenstra (2004). However, the insights of our paper have the potential to be used in numerous contexts, especially estimation of the effects of tariff reductions and free trade agreements on world trade flows — the most common usage of the gravity equation in trade. Using Monte Carlo techniques, we show that the linear approximation approach works in the context of regional (intra-continental) and world (intra- and inter-continental) trade flows.
Third, we demonstrate the economic conditions under which our approximation method works well to calculate comparative-static effects of key trade-cost variables… and when it does not. We compare the comparative statics generated using our approach versus those using A-vW's approach both for the Canadian–U.S. context and for world trade flows using Monte Carlo simulations. We find that the largest comparative static changes in multilateral price terms (and largest approximation errors) tend to be among — not just small GDP-sized economies (and consequently those with large trading partners) as emphasized in A-vW — but small countries that are physically close. Using a fixed-point iterative matrix manipulation, the approximation errors can be eliminated using an N × N matrix of GDP shares relative to bilateral distances, that is, measures of economic “density.” Since our approximation method can generate MR terms even when trade costs are bilaterally asymmetric, our method can yield lower average absolute biases of comparative statics than the A-vW method (which only addresses average border barriers under asymmetry).
The remainder of the paper is as follows. Section 2 reviews the A-vW analysis. Section 3 uses a first-order log-linear Taylor-series expansion to motivate a simple reduced-form gravity equation. In Section 4, we apply our estimation technique to the McCallum–A-vW–Feenstra data set and compare our coefficient estimates to these papers' findings. Section 5 examines the economic conditions under which our approach approximates the comparative statics of trade-cost changes well and under which it does not. Section 6 concludes.
Section snippets
The A-vW theoretical model
To understand the context, we initially describe a set of assumptions to derive a gravity equation; for analytical details, see A-vW (2003). First, assume a world endowment economy with N regions and N (aggregate) goods, each good differentiated by origin. Second, assume consumers in each region j have identical constant-elasticity-of-substitution (CES) preferences. Maximizing utility subject to a budget constraint yields a set of first-order conditions that can be solved for the demand for the
Theory
In this section, we apply a first-order log-linear Taylor-series expansion to the system of price equations ∏i and Pj in Eqs. (4), (5) above to generate a reduced-form gravity equation — including theoretically-motivated exogenous multilateral resistance (MR) terms — that can be estimated using OLS. A first-order Taylor-series expansion of any function f(xi), centered at x, is given by f(xi) = f(x) + [f′(x)](xi − x).2
Estimation
The goal of this section is to show that one can generate virtually identical gravity equation coefficient estimates (“partial” effects) to those generated using the technique in A-vW but using instead OLS with exogenous multilateral-resistance terms suggested in the previous section. While the approach should work in numerous contexts, we apply it first empirically in Section 4.1 to McCallum's U.S.–Canadian case, a popular context. In Section 4.2, to avoid measurement and specification biases,
Comparative statics
In this section, we examine the potential usefulness of our approximation approach for conducting comparative statics.12
Conclusions
Six years ago, theoretical foundations for the gravity equation in international trade were enhanced to recognize the systematic bias in coefficient estimates of bilateral trade-cost variables from omitting theoretically-motivated endogenous “multilateral (price) resistance” (MR) terms. This paper has attempted to make four potential contributions. First, we have demonstrated that a first-order log-linear Taylor series expansion of the system of nonlinear price equations suggests an alternative
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The authors are grateful for financial support under National Science Foundation grants SES-0351018 (Baier) and SES-0351154 (Bergstrand). The authors are also grateful for excellent comments on previous drafts from Eric van Wincoop, Jim Anderson, Tom Cosimano, Peter Egger, two anonymous referees, and participants at presentations at the University of Munich (CESifo), University of Nottingham (Leverhulme Center), University of Groningen, Vanderbilt University, and the 2006 Southern Economic Association annual meeting in Charleston, SC.
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