The gravity model: an illustration of structural estimation as calibration.

Balistreri, Edward J. ; Hillberry, Russell H.

Dawkins, Srinivasan, and Whalley ("Calibration," Handbook of Econometrics, 2001) propose that estimation is calibration. We illustrate their point by examining a leading econometric application in the study of international and interregional trade by Anderson and van Wincoop ("Gravity with Gravitas: A Solution to the Border Puzzle," American Economic Review, 2003). We replicate the econometric process and show it to be a calibration of a general equilibrium model. Our approach offers unique insights into structural estimation, and we highlight the importance of traditional calibration considerations when one uses econometric techniques to calibrate a model for comparative policy analysis. (JEL F10, C13, C60)

I. INTRODUCTION

Contemporary economic analysis includes two broad traditions of fitting models to data. Many estimate stochastic, theory-based, reduced forms with few parameters, while others calibrate models by an extensive collection, and computation, of consistent fitted values. Although the first technique is called estimation and the second calibration, these exercises are identical under consistent identifying assumptions. Both calibration and estimation fit a model to data.

Dawkins, Srinivasan, and Whalley (2001) make this point succinctly: "Calibration is estimation, estimation is calibration." The point is widely recognized in the macroeconomic real business cycle literature (Hoover 1995). Our purpose is to demonstrate that it is equally relevant to micro-based general equilibrium models. In our view, there is too little communication between calibrators and estimators of such models and the lack of communication impedes research.

It is important to clearly delineate the processes of data fitting and the subsequent model analyses. In analysis of fitted models, the degree of concentration on counterfactual simulation versus hypothesis testing around specific parameters often cleaves with our respective notions of calibration and estimation, but counter examples are easily found. Sensitivity analysis found in computable general equilibrium (CGE) studies can be specifically designed to generate higher order moments to facilitate hypothesis testing. (1) Many econometric studies are specifically focused on model identification for the purpose of counterfactual simulation. (2)

A more informative distinction might be drawn between testing models and calibrating models. Hoover (1995) places this issue at the heart of the macroeconomic debate surrounding real business cycle models. We see a direct extension of the macroeconomic debate into any empirical methodology that involves general equilibrium systems. In the testing paradigm, stochastic measures of fit provide a critically important benchmark for evaluating alternative structural assumptions or analytical results derived from a particular set of assumptions. When the objective is to provide a quantitative context for counterfactual analysis, traditional measures of fit are little more than indicators of parsimony. One example of the macroeconomic debate spilling over to the empirical literature is forwarded by Learner and Levinsohn (1995) who advise us to "Estimate, don't test." The point being that falsifiable theories are informative--conceptually and empirically.

In the context of estimating, as opposed to testing, some researchers find it preferable to exhaust the degrees of freedom available in the data by expanding the set of structural parameters. This results in a calibrated model that perfectly fits the benchmark observation but requires external information on how it will respond to shocks. Model responses are usually summarized in a set of elasticity assumptions. Econometric estimation does not exhaust the degrees of freedom and may focus on internal measurements of response parameters.

There also seems to be convergence in the empirical international trade literature; some calibrators of traditional general equilibrium simulation models attempt to identify response parameters (e.g., Liu, Arndt, and Hertel 2004 or Francois 2001), and some gravity model estimations include numerous dummy variables to isolate cross-sectional fixed effects (e.g., Hummels 2001 or Feenstra 2004 [Chapter 5]). Hillberry et al. (2005) contend that idiosyncratic calibration parameters in traditional CGE applications operate in a way that is similar to econometric residuals; calibration parameters allow the model to fit the data exactly. Another way to think about it is that idiosyncratic calibration parameters in CGE applications are analogous to fixed effects in an econometric model that includes a fixed effect for each observation.

The model proposed and applied by Anderson and van Wincoop (2003) (henceforth AvW) is exceptionally well suited for communicating with both estimators and calibrators. In their two-country application, AvW fit a structural gravity model to observed trade patterns among Canadian provinces and U.S. states. It is our contention that estimating and calibrating the AvW model are equivalent methods for fitting the model to data. The model is, at its core, a general equilibrium that can be fitted with standard calibration techniques. One of these techniques is the econometric procedure proposed by AvW. Our illustration that the econometric procedure is a calibration entails a structural interpretation of the parametric estimates and an itemization of those remaining structural parameters that are implied by the estimable model's identifying assumptions.

We show that the AvW calibration is consistent with a broad class of first-order approximations of the general equilibrium system that are equally efficient statistically (as measured by the sum of squared residuals on observed trade flows). Within this class of models, there is ambiguity regarding the size of cross-border trade frictions and thus ambiguity regarding the welfare effects of trade frictions. Our approach offers unique insights into structural estimation, and we highlight the importance of traditional calibration considerations when one uses econometric techniques to calibrate a model for comparative policy analysis.

We propose alternative estimations of the core general equilibrium: one that eliminates a bias in the structurally consistent fitted trade flows and one that yields an R-squared of one. Like the AvW estimation, these procedures are incomplete--they cannot identify the trade costs or price response parameters without ad hoc identifying assumptions (or additional data). We conclude that the theoretic gravity model in question is useful as a descriptive tool, but its ability to match trade flows closely in calibration is not novel. Richer theories of trade (dependent on traditional comparative advantage or scale and variety effects) are routinely fit--exactly--to observed trade flows. (3)

It is not our intent to critique the theory behind the gravity model. We also do not advocate any particular approach to calibration. Our contribution is a bridge between the methods and the language used by modelers who calibrate systems regularly and those who use econometric methods.

In the following section, we outline the general theoretic framework for the illustrative example and also outline a general calibration strategy in terms of identifying the necessary parameters. In Section III, we recast the AvW econometric method as a calibration, showing that it generates a complete set of structural parameters under particular identifying assumptions. A comparison of AvW's restrictive case and the general set of equally efficient, or superior, approximations are examined in Section IV. Concluding remarks are offered in Section V.

II. FOUNDATION FOR THE ILLUSTRATIVE EXAMPLE

A. The General Equilibrium Model

Following AvW, consider a model in which representative consumers in each region have preferences over goods differentiated by region of origin. Goods are aggregated to a single region-specific variety. Each region is given an endowment of its variety, which it trades for foreign varieties. Ad valorem trade frictions induce substitution into varieties that are from home and nearby regions.

We formulate the multiregion competitive exchange equilibrium as a mixed complementarity problem (MCP). The MCP is essentially the Mathiesen (1985) formulation of an Arrow-Debreu general equilibrium (also see Rutherford 1995b for an introduction to MCPs). The key advantage of using the Mathiesen formulation here is that it represents the entire multiregion general equilibrium compactly, as a set of 4n conditions in 4n unknowns (where n is the number of regions). (4)

Income for a region, [Y.sub.i], is given by the product of its exogenous endowment, [q.sub.i], and the net (of trade cost or f.o.b.) price of output, [P.sub.i]:

(1) [Y.sub.i] = [P.sub.i][q.sub.i].

The second set of conditions require market clearing for each region-specific product such that the quantity endowed equals the sum of demands by each region j:

(2) [q.sub.i] = [summation over j][X.sub.ij](P, [Y.sub.j], [t.sub.j], [y.sub.j]).

Demand for commodity i by region j ([X.sub.ij]) is a function of P, the vector of f.o.b, prices across the regions; [Y.sub.j], the income level in region j; [t.sub.j], the vector of trade cost wedges faced by region j; and [[gamma].sub.j], a vector of structural parameters that identify the preferences of region j.

Consistent with expenditure minimization in each region, the third set of equilibrium conditions set the true-cost-of-living index, [E.sub.i], equal to the unit expenditure function:

(3) [E.sub.i] = [e.sub.i](P, [t.sub.i], [[gamma].sub.i]),

where the utility function is assumed to be linearly homogeneous. The final set of conditions require balance between the nominal value of utility and income:

(4) [U.sub.i][E.sub.i] = [Y.sub.i].

Together, Conditions (1) through (4) are a complete multiregion general equilibrium that can be solved numerically for relative prices ([P.sub.i] and [E.sub.i]), regional welfare levels ([U.sub.i]), and income levels ([Y.sub.i]). (5) As an artifact of the equilibrium, we can recover the individual bilateral trade flows using the demand functions evaluated at the equilibrium:

(5) [X.sup.*.sub.ij] = ([P.sup.*], [Y.sup.*.sub.j], [t.sub.j], [[gamma].sub.j]).

A convenient property of the equilibrium defined by Equations (1) through (4) is that it is fully identified by endowments, trade frictions, and an explicit representation of the unit expenditure functions for each region. (6) Rarely are equilibrium systems dependent on such a modest set of unknown parameters.

B. Identifying the Functions

As a starting point for identifying the expenditure functions, first consider a set of consistent utility functions for each region:

(6) [U.sub.j] = [U.sub.j]([X.sub.j]/[t.sub.j], [[gamma].sub.j]).

Utility in a region is a function of the vector of f.o.b, imports to that region scaled by the trade cost factors ([X.sub.j]/[t.sub.j]); this operationalizes the formulation of iceberg transportation costs. (7) The other argument in the utility function is the vector of structural parameters, [[gamma].sub.j].

AvW (2003) assume that the utility function takes on a constant elasticity of substitution (CES) functional form. The CES form is popular because of its curvature properties and empirical tractability. Typically, a CES representation of Equation (6) would include a distribution parameter ([[alpha].sub.ij], which indicates the preference of consumer j for goods from region i) and a substitution elasticity ([[sigma].sub.j], which indicates how responsive consumer j is to relative price changes):

(7) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Notice that we designate an element that will be informed by the data with a hat (e.g., [??]); these are estimates.

Rutherford (1995a) introduced a more convenient form of the CES function, the calibrated share form, which describes the function in terms of benchmark (fitted) values:

(8) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where the benchmark value shares are given by

(9) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The calibrated share form is a simple algebraic transformation that decomposes [[??].sub.ij]. It is important to notice our use of notation in the terms that enter Equations (8) and (9). The superscript 0 indicates the level a variable takes at the benchmark equilibrium. For example, the [[??].sup.0.sub.ij] are the estimated reference export quantities at the benchmark equilibrium. In our empirical equations, (8) and (9), and those that follow, we are careful to distinguish the estimated parameters ([[??].sup.0.sub.i], [[??].sup.0.sub.ij], and [[??].sup.0.sub.ij]), the model variables ([P.sub.i] and [X.sub.ij]), and the exogenous instruments ([t.sub.ij]).

The calibrated share form is useful in that it allows us to separately classify the parameters in terms of their role in the order of approximation. Following tradition, we will refer to the [[??].sub.ij] as first-order parameters and the [[??].sub.j] as second-order parameters. (8) Adopting the separable CES form is consistent with a restricted second-order approximation to the true function because all the second-order curvature parameters (the cross-elasticities of substitution) are equal and invariant. (9) An exercise, such as AvW's estimation, which identifies the [[??].sub.ij] but not the [[??].sub.j] gives us a first-order approximation to the functions. (10)

The unit expenditure functions, under the CES approximation, are given by

(10) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The corresponding demand functions are

(11) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

In computational applications, the calibrated share forms of the expenditure and demand functions are preferable because they provide a simple parameter and functional check that is independent of second-order curvature.

With the expenditure function specified in Equation (10), and provided estimates of each parameter, the general equilibrium is operational as a comparative static or simulation tool. We simply substitute Equations (10) and (11) into Equations (3) and (2), respectively. Table 1 lists all the symbolic elements of the general equilibrium. Again, the equilibrium consists of 4n variables. There are n + [n.sup.2] exogenous instruments that characterize endowments and trade frictions. Parameter requirements include 3n + [2n.sup.2] primitives that represent the fitted benchmark and an additional n substitution elasticities. (11) Many of these parameters are routinely eliminated or identified by assumption. In the following section, we show how adopting AvW's simplifying assumptions and least squares procedure identifies the elements of [??].

III. ECONOMETRIC CALIBRATION AND GENERAL EQUILIBRIUM SIMULATION

This section illustrates how the estimation made by AvW (2003) can be interpreted as a calibration of the general equilibrium. First, we outline the assumptions and least squares procedure necessary to arrive at a full set of benchmark calibration parameters from the data. The data include trade flows among U.S. states and Canadian provinces, state and province incomes, and bilateral distances. (12) Second, once the parameters are established, we bring them to the original model to complete the calibration procedure. We purposefully return to the extensive form represented in Conditions (1)-(4). The final step is to perform the counterfactual of interest: the integration of the U.S. and Canadian economies by removing the effect of the border.

A. Parametric Identification

Conditions (1)-(4) are useful as an empirical tool only after the elements of [??] are estimated using data. What follows is an illustration of how one might arrive at a set of parameters consistent with the set derived by AvW, but from the perspective of a calibration exercise. The data are limited, so some of the elements must be identified directly through parametric or structural assumptions.

As a first step in reducing data requirements, we identify some parameters by simply making the benchmark price normalization explicit. Estimates of income at the benchmark, [[??].sup.0.sub.i], and exchange rates are widely published so it seems logical to adopt the convention that common nominal units equal real units at the benchmark. This allows us to set [q.sub.i] = [[??].sup.0.sub.i] for each region. Income and, therefore, endowment quantities are in units of U.S. dollars (at the benchmark). (13) Given our choice of units and by the first equilibrium condition [[??].sup.0.sub.i] = 1 for all i, which is a convenient normalization. Any other convention for measuring real units can be adopted without loss of generality. We remove the hat on income because these are assumed to be primary data rather than estimates (i.e., [[??].sup.0.sub.i] = [Y.sup.0.sub.i]). (14)

Another convenient simplification is to assume symmetric trade costs, that is, [[??].sup.0.sub.ij] = [[??].sup.0.sub.ji]. This assumption decreases the number of parameters that need to be identified. More importantly, for AvW, it enables them to arrive at a relatively simple benchmark reduced form. (15) Additionally, the trade cost is assumed to follow a log-linear form such that,

(12) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The estimated border cost factor, represented by [[??].sup.0.sub.ij], equals one plus the tariff equivalent. The observed bilateral distance is given by [d.ij], and [??] is the elasticity of the total cost factor with respect to distance. Although convenient for AvW's theory, the form of Equation (12) is another ad hoc approximation. (16) Notice that Equation (12) indicates that absolute trade costs will be affected by the units in which distance is measured, which is potentially problematic. The indeterminacy is resolved by choosing an ad hoc normalization.

A key identifying, and restrictive, assumption of the econometric calibration is one of identical tastes across regions. (17) This significantly reduces the number of parameters needed. AvW used this assumption to derive a reduced form that is free of share parameters. Essentially, under identical tastes, one can use the information that in a frictionless world, each region j would consume its income share of each regional commodity i. The estimates of the distribution parameters in the CES functions are, thus, replaced by information on local (benchmark) income shares.

With the simplifications established (and assuming a common elasticity across all regions, [[??].sub.i] = [sigma], AvW derived the reduced form for the local unit expenditure functions,

(13) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];

and the local nominal demand functions

(14) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Note that given our benchmark price normalization, the nominal flow ([[??].sup.0.sub.i][[??].sup.0.sub.ij]) equals the real flow ([[??].sup.0.sub.ij]) at the benchmark. Thus, we have simplified Equation (14) by removing [[??].sup.0.sub.i] from the left-hand side. With some slight rearranging and in log form, Equation (14) becomes the structural gravity equation presented by AvW:

(15) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where k is a regression constant that should approximate -ln [[summation].sub.m] [Y.sup.0.sub.m]. The dummy variable [a.sub.ij] is one if the shipment from i to j does not involve a border. The estimate [[??.sup.0] - 1 is interpreted as the ad valorem tariff equivalent of the border cost (where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]).

Direct measures of some of the trade flows ([X.sup.0.sub.st]) are observed. The regional pair (s, t) is a member of the set of potential bilateral pairs, {(s, t)|s [member of] I, t [member of] j}. (18) The [X.sup.0.sub.st] are assumed to be primary data but rampant with well-behaved measurement error. To reconcile Equation (15) with these observations, AvW define an objective function that is the sum of squared errors between the direct observations and the model prediction. Calibration involves the minimization of this objective subject to the reduced form of the unit expenditure functions, Equation (13). The nonlinear programming problem is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

subject to:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The coefficient [[beta].sub.1] equals (1 - [sigma])[??] and the coefficient [[beta].sub.2] equals (1 - [sigma])ln [[??].sup.0]. Estimation requires observations on distance for each (i, j) pair and the construction of the border dummy for each pair. Given the set of assumptions necessary to arrive at the reduced form and the solution values for 131, 132, and the [[??].sup.0(1 - [sigma]).sub.j] from the programming problem, we can recover all the elements of [??]. Comparative static experiments can then be simulated via the equilibrium conditions.

A key insight our calibration perspective offers is the absence of variables (the [Y.sub.i], [E.sub.i], and [X.sub.ij]) in the reduced form model, characterized in Equations (13) and (14). Rather, only the estimated levels of these variables at the benchmark equilibrium appear (the [Y.sup.0.sub.i], [E.sup.0.sub.i], and [[??].sup.0.sub.ij]). Once the nonlinear program (NLP) is solved, the first moments of these estimates are exogenous, and they enter the system of equations as parameters. The estimation (calibration) procedure brings the data to the economic model and is, therefore, devoid of endogenous elements. The reduced system, Equations (13) and (14), does not constitute an operational general equilibrium model, despite being locally consistent with the benchmark general equilibrium.

AvW (2003) conducted comparative static calculations using the reduced system, Equations (13) and (14). Despite sharing common data and parameters, these calculations are not consistent with our general equilibrium results--for two reasons. First, AvW treat regional incomes as exogenous in the trade equation. (19) While endowment quantities are appropriately held fixed in a counterfactual exercise, the purchasing power of the endowment will change as border costs are removed. Border removal induces significant terms of trade effects. AvW miss these effects when they control country size in the counterfactual using measured incomes (AvW 2003, 183). Our extensive form method (which distinguishes the parameter, benchmark income, from the endogenous variable, income) accommodates the direct impact income changes have on trade flows.

Second, the system of Equations (13) and (14) is only locally consistent with the proposed general equilibrium (Balistreri 2006). AvW's normalization (AvW 2003, Footnote 12) determines the geographic pattern of preferences conditional on the observed [Y.sup.0.sub.i], the estimated [[??].sup.0.sub.i], and an arbitrary choice of endowment units. Adopting Equation (14) in a counterfactual implicitly changes either preferences or the unit of measure. Balistreri (2006) shows that if preferences are held fixed (and incomes calculated correctly), AvW's measure of trade flows will deviate from the theory consistent measure by an endogenous scalar (which might be interpreted as a shift in the numeraire). (20) In contrast to AvW's method, our system requires the choice of a numeraire, and we hold this numeraire fixed across the counterfactual experiment.

B. Completing the Calibration

Although all the information has been processed and each identifying assumption itemized, calibration is only complete once the data have been brought back to the extensive form theory. Our goal is to identify each of the parameters listed in Table 1. Following AvW's adoption of [sigma] = 5, the benchmark price indexes are recovered from the direct measures of [[??].sup.0(1-[sigma]).sub.j] as they are estimated in the least squares problem. (21) The [[??].sup.0.sub.ij] are re. covered using Equation (12) and the estimated distance and border coefficients. Following Balistreri and Hillberry (2006), distance is normalized to minimize the resources devoted to melt, subject to no negative trade costs (the minimum estimated [[??].sup.0.sub.ij]. is normalized to one). The scale on distance has no relative implications, but it does change the level of the estimated benchmark price indexes and measures of resources devoted to trade costs. (22)

The elements on the right-hand side of Equation (14) are now determined, so we can use Equation (14) to recover the benchmark flows, [[??].sup.0.sub.ij]. Notice that we cannot use Equation (15) to recover the [[??].sup.0.sub.ij] if k is inconsistent with observed incomes. If k is not consistent with the theory, then the sum of exports will not add up to the total endowment. (23) Finally, we recover the [[??].sub.ij] from Equation (9) and the scale parameter [[phi].sub.j] = [[??].sub.j]/[[??].sup.0.sub.j]. All the items in the parameter vector, [??], presented in Table 1 are thus identified. The system is calibrated and available for counterfactual analysis.

C. Counterfactual Results

Counterfactual simulations are computed by evaluating the system with different endowments or trade costs. (24) Specifically, we compute the general equilibrium with border frictions eliminated, such that [t.sub.ij] = [d.sup.[??].sub.ij]. This simulates integration of the Canadian and U.S. economies. Table 2 presents the numeric benchmark equilibrium and percent changes in each of the variables using AvW's central elasticity ([sigma] = 5), AvW's coefficient estimates ([[beta].sub.1] = -0.79 and [[beta].sub.2] = -1.65), and choosing Alabama output as the numeraire. Utility is numeraire independent, but changes in income and the prices depend explicitly on our arbitrary choice of numeraire (the model is homogeneous degree zero in prices).

To analyze the effect of border removal on international versus intranational trade, we recover the disposition of each region's output ([q.sub.i]) using the demand equations. The endowed quantity for each region is either consumed in the United States, consumed in Canada, or it melts in transit. Table 3 presents these results.

IV. IMPLICATIONS FOR THE CALIBRATED MODEL AND ALTERNATIVE CALIBRATIONS

In the context of the econometric application, we highlight three lessons illuminated by our calibration perspective. First, structural integrity must be maintained throughout the calibration. We show that a lack of discipline concerning the regression intercept generates a bias in the structural fitted flows and taints the structural interpretation of other regression coefficients. Second, we take a partial step toward traditional calibration techniques by operationalizing AvW's proposed inclusion of home bias in tastes. This consumes at least one degree of freedom (outside the regression analysis) and generates an unidentified first-order approximation. Information on trade costs must be added for an operational system. Our third point is that if some estimates of trade costs are required (because the regression does not identify these), the econometric method uses the available information inefficiently. A more efficient calibration is achieved using the traditional calibration method to achieve an exact fit on observed trade flows.

A. Calibration Bias and Stochastic Efficiency

Equations (14) and (15) highlight a key difference between the economic model and the econometric procedure proposed by AvW. The economic theory motivating the regression equation imposes a restriction on the regression constant: it should equal -ln [[summation].sub.i] [Y.sup.0.sub.i]. Failure to impose this restriction biases econometric estimates relative to the theory, which requires equality between the value of aggregate demand and aggregate income. The bias manifests itself in the different implied benchmark trade flows depending on whether we compute them using Equation (14) and Equation (15). The theory cannot accommodate a free intercept, k, without violating a key adding-up requirement. (25)

Figure 1 illustrates the tradeoffs between calibration bias and stochastic efficiency. We generate the log difference between the fitted and the actual trade flows for three sets of fitted flows. (26) Columns 1 through 3 show the error distributions for the econometric procedure, a calibration based on estimates generated by the econometric exercise, and a theory consistent estimation/calibration, respectively. (27)

The figure demonstrates two points. First, structurally consistent estimation can reduce econometric efficiency when theory imposes restrictions on the parameter estimates. The distribution of errors under structurally consistent estimation (Column 3) is larger than in econometric estimation (Column 1). It is notable, however, that the econometric efficiency loss that occurs when the regression constant is restricted to its theory consistent value is small.

[FIGURE 1 OMITTED]

Second, parameter estimates taken from estimation procedures inconsistent with the theoretic structure of an economic model generate bias in subsequent calibrations that use the parameter estimates as inputs. The distribution of error terms in the implied calibration (Column 2) is identical to that of the econometric estimation, it is simply shifted higher. By allowing the estimation model to overstate aggregate income, the econometric procedure understates the trade costs necessary to fit trade flows within the model. A calibration based on trade costs from the estimating procedure substantially overstates interregional trade.

In contrast, the error terms from theory consistent estimation (a constrained regression constant) are replicated in calibration. (28) The error distribution in Column 3 is identical, whether one uses a structurally consistent estimation procedure or a calibration using structural parameters generated by structurally consistent estimation. It is for this reason that we advocate estimation procedures that are fully consistent when the goal is parameterizing a model for subsequent counterfactual analysis. We also emphasize that structural estimation is not a new procedure, it is simply calibration renamed.

B. Systematic Taste Bias

AvW cautioned that we should not accept their estimates of [[??].sup.0.sub.i] in their literal interpretation as consumer price indexes (AvW 2003, p. 176 and Footnote 17) because measured trade resistance, attributed to borders and distance, need not indicate a price markup. (29) AvW proposed home bias in preferences as a source of trade resistance. We adopt AvW's suggestion of structural taste bias, which is consistent with their regression analysis. In this context, the first-order calibration is underidentified. The ambiguity in the system is only resolved through a direct measure of the trade frictions that identifies these frictions independent of geographic regularities in the CES distribution parameters.

To illustrate the importance of our assumption about taste bias, consider a new benchmark parameter, [??] [member of] [0, 1], that represents the proportion of measured border resistance that is not due to taste bias. (30) The only modifications are to our definition of the reference distortions. Equation (12) becomes

(16) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

At the benchmark, the instruments must be set to their estimated local values ([t.sub.ij] = [[??].sup.0.sub.ij]), and as before, the counterfactual involves a computation of the equilibrium under [t.sub.ij] = [[d.sup.[??].sub.ij]

Using Equation (16) to compute tlae benchmark distortions fully operationalizes the structural taste bias suggested by AvW (2003). The advantages of the calibrated share form should be noted: no modifications are required in our functional representation of the equilibrium (the benchmark value shares are unchanged and there is no need to change the scaling). The benchmark distortions and value of the instruments change, but the calibrated share form allows us to directly input these changes to the first-order information without the tedious recalibration of the scale-dependent distribution parameters in the standard form represented in Equation (8).

It is apparent from Equation (16) that [??] (or equivalently, a direct measure of the border cost) will determine the magnitude of the price changes embodied in the counterfactual, and [sigma] determines the responses to those price changes. As a method of second-order approximation, the AvW econometric method is silent on the two most important pieces of information for welfare analysis ([??] and [sigma]). Welfare analysis depends critically on these parametric assumptions, which cannot be informed by the proposed regression. (31)

We illustrate the importance of good measures of [??] in Figure 2. If one assumes no taste bias, the welfare effects of border removal are maximized conditional on a given [sigma]. At the values preferred by AvW (2002), of [sigma] = 5 and [??] = 1 (no taste bias), the tariff equivalent of the border charge is 51%. On the other extreme, if we simply measure duties collected divided by trade flows, the ad valorem rate in 1993 was less than 0.4%. (32) Using this rate as the border charge (and [sigma] = 5), the proportion of border resistance attributed to taste bias is 99% (marked in the figure as Measured Tariffs). Using measured tariffs, the welfare impact on Canada of removing the border is only 0.1%. Including nontariff barriers in the calculated border charge will likely place [??] in the middle of its range. Our point is that freeing [??] implies that trade flow data cannot be used in the gravity framework to infer trade costs.

Thus, we contend that the econometric calibration outlined in Section III does not complete even the first-order approximation to the general equilibrium because it fails to quantify key first-order information on the benchmark trade costs (or equivalently, it fails to quantify the taste bias). The reduced form estimation is consistent with the broad class of models covered on the taste-bias interval, [??] [member of] [0, 1]. All these calibrations are equally as efficient as the AvW procedure, when measured by the sum of squared residuals.

C. Improved Stochastic Efficiency in a Generalized Calibration

Once one admits that taste bias must be considered--in order to allow the model to better approximate a world with production specialization and nontraded goods, for example--there is no particular need to add the restrictions imposed by the stochastic form of the regression. The regression no longer uniquely identifies the [[??].sup.0.sub.ij]. If the taste parameters are free to be asymmetric, then we can fit the trade flows exactly. That is, we use the observed trade flows rather than the fitted trade flows to compute the structural parameters. Effectively, this reallocates the assumed cross-sectional error terms in the econometric model to an idiosyncratic taste bias. From an econometric perspective, this is akin to adding pairwise fixed effects. With the restriction of identical tastes (or structural taste bias) lifted, there is at least one taste parameter for each observed trade flow. With no degrees of freedom, it is trivial to calibrate the system to a consistent equilibrium that produces an R-squared of one for the observed trade flows.

[FIGURE 2 OMITTED]

To fit the data more efficiently, we simply compute the value shares based on the observed trade flows:

(17) [[??].sub.st] = [X.sup.0.sub.st]/[Y.sup.0.sub.t].

We make the nontrivial assumption that the trade flows contain no measurement error (as we did with income). This directly accommodates the observed pattern of trade. The cost of this direct approach is that summary measures of how trade reacts to distance and borders, on average, are not directly reported. The benefit of this approach is that the benchmark replicates the (first order) observations.

The first-order calibration is not complete, however, without trade flows for the unobserved pairs and a measure of the trade costs. For simplicity and comparability with the econometric calibrations, we assume that trade costs are those implied by the regression analysis (with [??] = 1). Thus, we use the regression coefficients as a source of descriptive information on average costs. (33) Obviously, it would be more appropriate to find information on each of the bilateral trade costs, however, using the regression coefficients to compute the [[??].sup.0.sub.ij] offers a consistent point of departure to compare the more efficient calibration with the results presented above.

The unobserved trade flows are problematic because there might be any number of combinations that are consistent with an equilibrium that includes the observed flows (satisfying the R-squared equals one criteria). To solve this problem, we again appeal to the descriptive properties of the original regression. Using the regression coefficients as given, we minimize the sum of squared deviations between the fitted value of the unobserved flows and the right-hand side of Equation (14), choosing values for the unobserved pairs and subject to the adding-up constraints. The programming problem is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

subject to:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (NLP2)

Any number of modifications might be added to this programming problem to accommodate additional information. For example, some of the unobserved pairs might be restricted to zero flow if we believe that there is actually no trade (this is no longer inconsistent with the theory because some of the idiosyncratic share parameters might be zero). Once the fitted values from the programming problem are used to compute the remaining value shares, the calibration is complete.

Using the implied trade costs from the original model (a 51% tariff equivalent), Table 4 shows a comparison between the welfare impacts of border removal from the econometric calibration and the welfare impacts of border removal from the direct calibration. The econometric calibration, based on AvW's proposed econometric model, has significant implications on the size of the welfare impact on Canada and masks many of the interprovincial distributional impacts of border removal.

As one might expect, the dispersion in welfare effects is greatly reduced in the econometric calibration because the model is calibrated to the fitted (or average) flows. When we calibrate to the observed trade flows, a rich story about the pattern of trade emerges. For example, the simulated impact on Quebec's welfare of removing the U.S.-Canada border drops from 29% to 19% when we use the actual data.

The differences are directly attributable to errors in the fitted values implied by the econometric calibration. In the fitted values, the proportion of Quebec's Gross Domestic Product (GDP) that is exported across the border is 25.9%, but in the actual observations, the proportion of Quebec's GDP that is exported across the border is 15.7%. Ontario in contrast has fitted border flows that are almost identical to the observed flows (30.3% vs. 30.4%). Therefore, the alternative approaches to calibration produce comparable welfare impacts for Ontario but not for Quebec. The general pattern in Table 4 is an overstatement of welfare impacts under the original calibration, and this is attributed to the biased fitted flows generated by the atheoretic intercept (illustrated in Figure 1).

A further point to take away from Table 4 is that the welfare impacts of economic integration are sizeable in the last column, despite our use of traditional calibration techniques. AvW (2002) contended that their approach is more plausible than traditional calibration methods because they estimate nontariff barriers to be many times larger than formal trade barriers (of course, this requires the identical-taste assumption). They go on to critique traditional calibrated models as understating the effects of the North American Free Trade Agreement. We find the explanation of the differences trivial: the critiqued computational studies focused on experiments that removed much smaller barriers. The traditional approach is to directly measure trade barriers (tariff and nontariff, if available) and attribute any unexplained border resistance to idiosyncratic taste parameters. In contrast, AvW assume all trade resistance at the border to be a border charge. Table 4 shows that given comparable border charges (and comparable response parameters), the traditional calibration technique yields comparable aggregate results.

V. CONCLUSIONS

Contemporary economic analysis includes two broad traditions of fitting economic models to data. While the analytical objectives of calibration and estimation have traditionally differed, recent applications highlight the need to consider them in a unified framework. Under consistent identifying assumptions, both approaches generate the same structural parameters necessary to relate exogenous changes to endogenous outcomes.

For many structural econometric studies, the core assumptions of the economic model are without question. The goal is to identify an estimable reduced form of the equilibrium system. We emphasize the importance of returning to the model's extensive form if reduced form estimates are to be given their structural interpretation. Operationalizing an economic model requires identification of a full set of exogenous structural parameters.

In contrast to the unified framework proposed by Dawkins, Srinivasan, and Whalley (2001), AvW (2002) argue that estimated models are superior to calibrated simulation models. Our illustration that estimation is calibration moves the focus of analysis onto the key identifying structural assumptions that make the fitting procedure possible and beyond the particular label placed on the fitting procedure. The econometric procedure proposed by AvW generates results that seem to contradict similar calibrated models, but we show that this is due to specific structural and parametric restrictions not found in traditional simulation models. Our contribution, however, is broader than our specific example. Viewing empirical investigations from the unified perspective, that structural estimation is calibration, will aid in applying structural assumptions consistently and in directing assumptions toward more efficient use of limited data.

ABBREVIATIONS

AvW: Anderson and van Wincoop

CES: Constant Elasticity of Substitution

CGE: Computable General Equilibrium

GDP: Gross Domestic Product

MCP: Mixed Complementarity Problem

NLP: Nonlinear Program

REFERENCES

Anderson, J. E., and E.van Wincoop. "Borders, Trade, and Welfare," in Brookings Trade Forum: 2001, edited by S. M. Collins and D. Rodrik. Washington, DC: Brookings Institution Press, 2002, 207-30.

--. "Gravity with Gravitas: A Solution to the Border Puzzle." American Economic Review, 93, 2003, 170-91.

Balistreri, E. J. "A Note on the Local and Global Consistency of Anderson and van Wincoop's Gravity Model." Working Paper, Colorado School of Mines, 2006.

Balistreri, E. J., and R. H. Hillberry. "Trade Frictions and Welfare in the Gravity Model: How Much of the Iceberg Melts?" Canadian Journal of Economics, 39, 2006, 247-65.

Brown, D. K., and R. M. Stern. "Computational Analysis of the U.S.-Canada Free Trade Agreement: the Role of Product Differentiation and Market Structure," in Trade Policies for International Competitiveness, edited by R. C. Feenstra. Chicago, IL: University of Chicago Press, 1989, 217-50.

Cheng, I.-H, and H. J. Wall. "Controlling for Heterogeneity in Gravity Models of Trade and Integration." Federal Reserve Bank of St. Louis Review, 87, 2005, 49-63.

Dawkins, C., T. N. Srinivasan, and J. Whalley. "Calibration," in Handbook of Econometrics, Vol. 5, Chapter 58, edited by J. J. Heckman and E. E. Learner. Amsterdam: Elsevier, 2001, 3655-705.

Deaton, A., and J. Muellbauer. Economics and Consumer Behavior. Cambridge: Cambridge University Press, 1980.

Feenstra, R. C. Advanced International Trade: Theory and Evidence. Princeton, NJ: Princeton University Press, 2004.

Francois, J. "Flexible Estimation and Inference Within General Equilibrium Systems." Conference Proceedings: The Fourth Annual Conference on Global Economic Analysis, GTAP, Vol. 1, 2001.

Harrison, G. W., R. C. Jones, L. J. Kimbell, and R. M. Wigle. "How Robust is Applied General Equilibrium Analysis?" Journal of Policy Modeling, 15, 1992, 99-115.

Harrison, G. W., and H. D. Vinod. "The Sensitivity Analysis of Applied General Equilibrium Models: Completely Randomized Factorial Sample Designs." Review of Economics and Statistics, 74, 1992, 357-62.

Hertel, T., D. Hummels, M. Ivanic, and R. Keeney. "How Confident Can We Be in CGE-Based Assessments of Free Trade Agreements?" National Bureau of Economic Research Working Paper No. 10477, 2004.

Hillberry, R. H., M. A. Anderson, E. J. Balistreri, and A. K. Fox. "Taste Parameters as Model Residuals: Assessing the 'Fit' of an Armington Trade Model." Review of International Economics, 13, 2005, 973-84.

Hoover, K. D. "Facts and Artifacts: Calibration and the Empirical Assessment of Real-Business-Cycle Models." Oxford Economic Papers, 47, 1995, 2444.

Hummels, D. "Toward a Geography of Trade Costs." Working Paper, Purdue University, 2001.

Learner, E. E., and J. Levinsohn. "International Trade Theory: The evidence," in Handbook of International Economics, Vol. 3, Chapter 26, edited by G. M. Grossman and K. Rogoff. Amsterdam: Elsevier, 1995, 1339-94.

Liu, J., C. Arndt, and T. W. Hertel. "Parameter Estimation and Measures of Fit in A Global, General Equilibrium Model." Journal of Economic Integration, 19, 2004, 626-49.

Lopez-de-Silanes, F., J. R. Markusen, and T. F. Rutherford. "The Auto Industry and the North American Free Trade Agreement," in Modeling Trade Policy." Applied General Equilibrium Assessments of North American Free Trade, Chapter 8, edited by J. F. Francois and C. R. Shiells. Cambridge: Cambridge University Press, 1994, 223-55.

Mathiesen, L. "Computation of Economic Equilibria by a Sequence of Linear Complementarity Problems." Mathematical Programming Study, 23, 1985, 144-62.

de Melo, J. "Computable General Equilibrium Models for Trade Policy Analysis in Developing Countries: A Survey." Journal of Policy Modeling, 10, 1988, 469-503.

Perroni, C., and T. F. Rutherford. "Regular Flexibility of Nested CES Functions." European Economic Review, 39, 1995, 335-43.

Rose, A. K., and E. van Wincoop. "National Money as a Barrier to International Trade: The Real Case for Currency Union." American Economic Review, 91, 2001, 386-90. Papers and Proceedings of the Hundred Thirteenth Annual Meeting of the American Economic Association.

Rutherford, T. F. "Constant Elasticity of Substitution Functions Some Hints and Useful Formulae." mimeo, University of Colorado, 1995a.

--. "Extensions of GAMS for Complementarity Problems Arising in Applied Economic Analysis." Journal of Economic Dynamics and Control, 19, 1995b, 1299-1324.

Shoven, J. B., and J. Whalley. "Applied General-Equilibrium Models of Taxation and International Trade: An Introduction and Survey." Journal of Economic Literature, 22, 1984, 1007-51.

Trefler, D. "Trade Liberalization and the Theory of Endogenous Protection: An Econometric Study of U.S. Import Policy." Journal of Political Economy, 101, 1993, 138-60.

(1.) See Harrison et al. (1992), Harrison and Vinod (1992), and Hertel et al. (2004).

(2.) See Trefler (1993) or Rose and van Wincoop (2001).

(3.) Myriad constant returns general equilibrium models appear in the trade and tax policy literature (the early work is surveyed by de Melo 1988 and Shoven and Whalley 1984). Studies that incorporate new theories of scale or variety effects include Lopez-de-Silanes, Markusen, and Rutherford (1994) and Brown and Stern (1989).

(4.) More generally, an MCP is a powerful numeric tool that directly accommodates complementary slack conditions that arise in economics (Rutherford 1995b).

(5.) The system (1) through (4) represents the equilibrium as it might be solved numerically. The 4n unknowns are the [Y.sub.i], [U.sub.i], [E.sub.i], and [P.sub.i]. Only relative prices are determined, however, so one of the market clearance conditions is removed (by Walras's law), and we assign the associated price as the numeraire.

(6.) The demand functions embedded in Equation (2) also need to be identified, but it is a routine calculus exercise to recover these from the unit expenditure function.

(7.) AvW (2003) define the nominal trade flow from i to j as the product of the f.o.b. price, the trade cost factor, and the net quantity consumed ([P.sub.i][t.sub.ij][c.sub.ij]). This also equals the product of the f.o.b. price and the export quantity ([P.sub.i][X.sub.ij]). The net (of iceberg melt) quantity consumed is therefore [X.sub.ij][t.sub.ij]. We simply make this substitution and directly define utility as a function of the bilateral export quantities scaled by their respective trade costs.

(8.) The zero-order scale parameter [[??].sub.j], is set equal to the benchmark utility level, which from Equation (4) is benchmark income divided by the benchmark consumer price index, [[??].sub.j] = [[??].sup.sub.j]/[[??.sup.0.sub.j].

(9.) Perroni and Rutherford (1995) formally characterized the orders of approximation approach to functional identification in their analysis of the (second order) flexibility of nested CES functions.

(10.) The second-order curvature parameters (the [[??].sub.j]) used by AvW (2003) are assumed, not informed by their data. We adopt the same approach in this analysis.

(11.) We do not include the familiar summary parameters in our count of primitive parameters. In general, theoretic and econometric exercises focus on these summary parameters, where as calibration exercises focus on their primitives (the estimated benchmark fitted values). There are 2n + [n.sup.2] summary parameters: the zero-order [[??].sub.i], the first-order [[??].sub.ij], and the second-order [[??].sub.i]. This obscures the fact that many more primitive estimates are needed to identify the summary parameters, and our count assumes that each of the substitution elasticities has only a single underlying primitive. Realistically, the count of primitives should be expanded to include multiple first-order fitted observations (a minimum of two different observations of the first-order information are needed to identify the [[??].sub.i]).

(12.) Our illustrative example follows AvW's two-country application, which assumes that the United States and Canada are the only countries and that states and provinces are the relevant geographic divisions for region specific varieties.

(13.) It is important to emphasize that the benchmark price normalization is only valid locally. Away from the benchmark, we normalize on the price of a single region's endowment commodity, this commodity serves as numeraire.

(14.) In making this seemingly innocent assumption about physical units and endowment values, we have implicitly asserted that there is no error associated with the measurement of income or its conversion into common nominal units. Our failure to account for uncertainty in these underlying procedures may not bias our subsequent analysis but seriously undermine the validity of statistical inference. After all, many degrees of freedom have been consumed in the generation of what we accept as primary data. Following the tradition in most of empirical economics, we acknowledge this problem and ignore it.

(15.) AvW explain that although symmetry is assumed, the econometric model cannot distinguish between this equilibrium and one in which there are asymmetries that produce the same average trade resistance (AvW 2003, Footnote 11).

(16.) For example, Hummels (2001) argues that an additive form is more sensible. As with many theories that support the gravity literature, the origin of Equation (12) is more likely the log-linear regression, not the most plausible microfoundations.

(17.) In deriving their reduced form, AvW assume homogeneity in both relative and absolute tastes across regions (the cardinalization of utility is maintained across regions). Subsequently, AvW suggest structural taste bias as an alternative to homogeneity: we explore this suggestion in Section IV. For the equations here to be consistent with absolute taste homogeneity, one could normalize the utility functions by a positive monotonic transformation. For example, multiplying Equation (8) by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

normalizes the scale of utility across regions. With this modification, the equilibrium Condition (4) becomes [U.sub.i][E.sub.i]/[[??].sub.i] = [Y.sub.i]. Relative homogeneity in utility is achieved by holding the distribution parameters constant across the regions. For example,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In subsequent analysis, we adopt AvW's reduced form for calibration, noting that their cardinalization is different than ours but the resulting demand systems and, therefore, the extensive forms are isomorphic.

(18.) AvW (2003) reduce the full set of bilateral pairs by: flows that do not have observations, flows that are observed to be zero, and any measured internal flows.

(19.) AvW calculate changes in relative income as they apply to their multilateral resistance indexes, Equation (13). They do not adjust the regional income terms as they appear in Equation (14).

(20.) For a full presentation of the local versus global consistency of AvW's model and analysis, see Balistreri (2006). Balistreri shows that the numeraire shift will be large if the original numeraire commodity is associated with a region largely affected by the shock. For example, in the border removal experiment, Balistreri reported that if Alberta's endowment commodity is used as the numeraire, AvW's gravity equation understates trade by 67%. In contrast, if Alabama's endowment commodity is used as the numeraire, AvW's measure understates trade by 4%. These calculations depend on [sigma] = 5 (and a proper account of income changes).

(21.) Our arguments in this article do not depend qualitatively on the value of [sigma], but the reader is warned that subsequent quantitative illustrations depend on [sigma] = 5. This is the value AvW (2003) prefer based on their reading of the literature.

(22.) Under an agnostic stance on absolute trade costs, the benchmark general equilibrium is unidentified and the regression loses its ability to inform the structure. The estimating system, Equation (15) subject to Equation (13), is consistent with the general equilibrium at the benchmark, but this system alone cannot inform us about the economic variables away from that benchmark. Moving from regression coefficients to an operational economic model requires a normalization that defines the absolute sizes of the [[??].sup.0.sub.ij] Our normalization is based on the illogic of negative distance-related costs.

(23.) AvW (2003) did not report information obtained on the estimated constant k, which is inevitably inconsistent with its structural interpretation, given direct measures of benchmark income. Empirical estimates of k from the least squares problem imply an aggregate income that is more than 3.5 times larger than observed (by summing across the [Y.sup.0.sub.i]). If one were interested in testing the theory rather than calibrating the theoretic model, the model is easily rejected based on a hypothesis test of the similarity between k and its theory consistent value, -ln [[summation].sub.i] [[Y.sup.0.sub.i]. The implications of these inconsistencies are further explored in Section IV.

(24.) The relatively small nonlinear system (of only 164 equations) is solved using PATH, a complementarity problem algorithm, available in the GAMS software. The code is available upon request.

(25.) Unlike AvW's method, some traditional econometric methods of estimating demand systems (e.g., the Linear Expenditure System or the Almost Ideal Demand System) automatically impose adding up (Deaton and Muellbauer 1980). Adding up is only achieved in the AvW estimation by restricting the regression intercept.

(26.) It is important to note that the log difference in trade flows is not the residual being minimized in the regression Equation (15). The log difference in trade flows is the more common residual in the broader gravity literature.

(27.) The plots follow the conventions of STATA software. The central line within each box represents the median value of the distribution. The box includes estimates within the interquartile range (those between the 25th and 75th percentile). The whiskers extend beyond the box in each direction at a distance of 1.5 times the interquartile range. Observations outside the whiskers are outliers represented as individual data points.

(28.) The regression coefficients change substantially when structure is imposed on the constant term: [[beta].sub.1] = -1.44 and [[beta].sub.2] = -1.85 under true structural estimation.

(29.) The [[??].sup.0.sub.i] are always interpreted as the price of the composite good (the composite good is regional units of utility), but its value relative to the benchmark f.o.b. prices of endowments depends on the arbitrary scale of utility. To interpret this measure literally as a consumer price index, the benchmark units of the composite good need to be comparable with endowment units (another special cardinalization of utility). In this case when trade frictions that cause an f.o.b. to c.i.f. price wedge (pecuniary costs as AvW call them) are removed, the consumer price indexes revert to unity. Like other issues relating to the cardinalization of utility, the interpretation of [[??].sup.0.sub.i] as a consumer price index, as opposed to the relative price of utils, is largely irrelevant in the more general discussion of regularities in trade resistance due to borders and distance. AvW's point is that some portion of the resistance em bodied in the measured [[??].sup.0.sub.ij] might be due to things other than transport or border policy that vary with distance and country, respectively. Our conclusions from an earlier article support this, more general, interpretation of trade resistance (Balistreri and Hillberry 2006).

(30.) We only examine border-related taste bias, but it is also reasonable, and prudent, to think that distance-related resistance also includes a taste component. We concentrate on the border bias because of its relevance to welfare effects in the counterfactual simulation of border removal. For this illustration, we also make a stark simplification that the portion of border costs that are due to taste bias are constant across regions.

(31.) The large welfare benefits of integration advertised by AvW (2002) follow from the assumption that all measured trade resistance at the border is a border charge.

(32.) Measured tariffs are calculated from the USITC Interactive Tariff and Trade DataWeb. This illustrative calculation only measures the ad valorem rate on U.S. imports from Canada.

(33.) An econometric analogue to this procedure is conducted by Cheng and Wall (2005). They estimate a model with country-pair fixed effects and then regress those fixed effects on distance and other geographic variables to measure the average effect of the geography variables on trade.

EDWARD J. BALISTRERI and RUSSELL H. HILLBERRY *

* We thank James E. Anderson and Eric van Wincoop for providing their data. This article was largely completed while the authors were employed with the U.S. International Trade Commission. The opinions and conclusions are solely those of the authors.

Balistreri: Assistant Professor, Division of Economics and Business, Colorado School of Mines, Golden, CO 80401-1887. Phone 303-384-2156, Fax 303-273-3416, E-mail ebalistr@mines.edu

Hillberry: Senior Lecturer, Department of Economics, Economics and Commerce Building, University of Melbourne, Parkville, VIC 3010, Australia. Phone +61 3 8344 5354, Fax +61 38344 6899, E-mail rhhi@ unimelb.edu.au

TABLE 1 Scope of the AvW General Equilibrium with CES Preferences

Variables

[Y.sub.i] = incomes
[E.sub.i] = unit expenditure index
[P.sub.i] = prices (f.o.b.)
[U.sub.i] = utility levels

Instruments

[q.sub.i] = endowments
[t.sub.ij] = trade frictions

Structural Parameters ([gamma])

[Y.sup.o.sub.i] = benchmark incomes
[E.sup.o.sub.i] = benchmark unit expenditure indexes
[[PHI].sup.o.sub.i] = [Y.sup.o.sub.i]/[E.sup.o.sub.i] (zero-order
 summary parameters)
[X.sup.o.sub.ij] = benchmark trade flows
[P.sup.o.sub.i] = benchmark prices (f.o.b.)
[t.sup.o.sub.ij] = benchmark trade frictions
[[theta].sub.ij] = ([P.sup.o.sub.i][X.sup.o.sub.ij])/[Y.sup.o.sub.j]
 (first-order summary parameters)
[[sigma].sub.i] = elasticities of substitution (second-order summary
 parameters)

TABLE 2 Benchmark Equilibrium and Simulated Border Removal
([sigma] = 5)

 Cost-of-Living
 Income ([Y.sub.i]) Index ([E.sub.i])

 Benchmark Percent Percent
 ($US Change Change
 billion) (%) Benchmark (%)

Province
 Alberta 56.3 14 2.1 -16
 British Columbia 62.9 10 1.9 -12
 Manitoba 16.7 19 2.1 -20
 New Brunswick 9.5 16 2.1 -18
 Newfoundland 6.4 19 2.4 -20
 Nova Scotia 12.4 16 2.1 -17
 Ontario 194.3 14 1.8 -16
 Prince Edward Island 1.6 17 2.2 -19
 Quebec 107.1 12 1.9 -14
 Saskatchewan 15.3 18 2.2 -19
State
 Alabama 83.0 0 1.6 -1
 Arizona 85.0 0 1.8 -1
 California 843.1 0 1.4 -1
 Florida 300.7 0 1.5 -1
 Georgia 170.9 0 1.5 -1
 Idaho 22.4 1 1.8 -2
 Illinois 312.3 0 1.5 -1
 Indiana 129.7 0 1.5 -1
 Kentucky 79.9 0 1.5 -1
 Louisiana 94.7 0 1.6 -1
 Massachusetts 174.0 0 1.5 -1
 Maryland 124.6 0 1.4 -1
 Maine 25.1 1 1.6 -2
 Michigan 217.3 0 1.5 -1
 Minnesota 114.6 0 1.6 -1
 Missouri 118.3 0 1.5 -1
 Montana 16.1 1 1.8 -2
 North Carolina 168.6 0 1.5 -1
 North Dakota 12.7 1 1.7 -2
 New Hampshire 27.2 I 1.6 -2
 New Jersey 243.9 0 1.4 -1
 New York 541.1 0 1.4 -1
 Ohio 256.6 0 1.5 -1
 Pennsylvania 283.1 0 1.4 -1
 Tennessee 116.7 0 1.5 -1
 Texas 453.0 0 1.6 -1
 Virginia 170.0 0 1.5 -1
 Vermont 13.0 1 1.6 -2
 Washington 136.4 1 1.7 -2
 Wisconsin 117.7 0 1.6 -1
 Rest of United States 988.6 0 1.8 -1

 Price of Output
 ([P.sub.i]) Utility ([U.sub.i])

 Percent Percent
 Change Change
 Benchmark (%) Benchmark (%)

Province
 Alberta 1.0 14 26.5 36
 British Columbia 1.0 10 32.9 26
 Manitoba 1.0 19 7.8 49
 New Brunswick 1.0 16 4.6 42
 Newfoundland 1.0 19 2.7 50
 Nova Scotia 1.0 16 5.9 39
 Ontario 1.0 14 107.3 35
 Prince Edward Island 1.0 17 0.7 45
 Quebec 1.0 12 57.5 29
 Saskatchewan 1.0 18 7.1 46
State
 Alabama 1.0 0 53.2 1
 Arizona 1.0 0 48.4 1
 California 1.0 0 595.1 0
 Florida 1.0 0 195.1 1
 Georgia 1.0 0 111.9 1
 Idaho 1.0 1 12.7 2
 Illinois 1.0 0 210.3 1
 Indiana 1.0 0 86.7 1
 Kentucky 1.0 0 52.8 1
 Louisiana 1.0 0 59.0 1
 Massachusetts 1.0 0 119.6 1
 Maryland 1.0 0 88.0 1
 Maine 1.0 1 15.2 3
 Michigan 1.0 0 142.6 2
 Minnesota 1.0 0 71.1 2
 Missouri 1.0 0 76.4 1
 Montana 1.0 1 9.0 3
 North Carolina 1.0 0 110.3 1
 North Dakota 1.0 1 7.4 2
 New Hampshire 1.0 1 17.5 2
 New Jersey 1.0 0 168.2 1
 New York 1.0 0 386.7 1
 Ohio 1.0 0 174.0 1
 Pennsylvania 1.0 0 196.7 1
 Tennessee 1.0 0 75.9 1
 Texas 1.0 0 283.7 1
 Virginia 1.0 0 114.1 1
 Vermont 1.0 1 8.2 3
 Washington 1.0 1 82.6 3
 Wisconsin 1.0 0 75.3 1
 Rest of United States 1.0 0 552.5 2

TABLE 3 Disbursement of Endowments ([sigma] = 5)

 Consumed in
 Canada
 Endowment
 ([q.sub.i] Benchmark Counterfactual

Province
 Alberta 56.3 19.3 5.7
 British Columbia 62.9 33.0 12.8
 Manitoba 16.7 4.6 1.0
 New Brunswick 9.5 3.0 0.8
 Newfoundland 6.4 1.5 0.3
 Nova Scotia 12.4 4.4 1.2
 Ontario 194.3 79.4 24.0
 Prince Edward 1.6 0.5 0.1
 Island
 Quebec 107.1 47.7 16.7
 Saskatchewan 15.3 4.3 1.0
State
 Alabama 83.0 0.4 1.6
 Arizona 85.0 0.4 2.0
 California 843.1 2.1 9.8
 Florida 300.7 1.2 5.2
 Georgia 170.9 0.8 3.4
 Idaho 22.4 0.2 0.8
 Illinois 312.4 1.5 6.8
 Indiana 129.7 0.8 3.4
 Kentucky 79.9 0.5 2.1
 Louisiana 94.7 0.4 1.8
 Massachusetts 174.0 1.3 5.7
 Maryland 124.6 0.7 3.1
 Maine 25.1 0.4 1.6
 Michigan 217.4 1.9 8.6
 Minnesota 114.6 0.8 3.6
 Missouri 118.3 0.6 2.7
 Montana 16.1 0.2 0.7
 North Carolina 168.6 0.9 4.2
 North Dakota 12.7 0.1 0.5
 New Hampshire 27.2 0.3 1.3
 New Jersey 243.9 1.6 7.4
 New York 541.1 3.9 17.7
 Ohio 256.6 1.8 7.9
 Pennsylvania 283.1 2.1 9.2
 Tennessee 116.7 0.6 2.6
 Texas 453.0 1.7 7.7
 Virginia 170.0 1.0 4.5
 Vermont 13.0 0.2 0.8
 Washington 136.4 1.7 8.1
 Wisconsin 117.7 0.8 3.6
 Rest of United 988.6 6.8 30.0
 States

 Consumed in the
 United States

 Benchmark Counterfactual

Province
 Alberta 4.1 15.7
 British Columbia 3.9 17.6
 Manitoba 1.7 5.4
 New Brunswick 0.9 3.0
 Newfoundland 0.6 1.8
 Nova Scotia 1.0 3.7
 Ontario 16.5 64.7
 Prince Edward 0.2 0.5
 Island
 Quebec 7.3 31.4
 Saskatchewan 1.4 4.7
State
 Alabama 37.7 36.3
 Arizona 32.9 31.4
 California 548.4 537.4
 Florida 147.0 142.4
 Georgia 81.1 78.2
 Idaho 8.4 7.8
 Illinois 153.8 148.1
 Indiana 61.5 58.8
 Kentucky 36.8 35.2
 Louisiana 42.3 40.8
 Massachusetts 105.6 100.2
 Maryland 75.9 72.9
 Maine 10.4 9.4
 Michigan 99.3 93.2
 Minnesota 48.6 45.9
 Missouri 53.6 51.4
 Montana 5.8 5.4
 North Carolina 80.6 77.2
 North Dakota 4.7 4.4
 New Hampshire 12.8 11.9
 New Jersey 132.3 126.0
 New York 323.3 307.4
 Ohio 126.8 120.6
 Pennsylvania 152.5 144.9
 Tennessee 52.6 50.5
 Texas 205.1 198.5
 Virginia 85.9 82.1
 Vermont 5.6 5.1
 Washington 69.3 63.1
 Wisconsin 51.4 48.7
 Rest of United 348.9 328.6
 States

 Transport
 Melt

 Benchmark Counterfactual

Province
 Alberta 32.9 34.9
 British Columbia 26.1 32.5
 Manitoba 10.5 10.3
 New Brunswick 5.6 5.7
 Newfoundland 4.3 4.2
 Nova Scotia 7.0 7.5
 Ontario 98.5 105.7
 Prince Edward 1.0 1.0
 Island
 Quebec 52.1 59.0
 Saskatchewan 9.6 9.6
State
 Alabama 45.0 45.1
 Arizona 51.6 51.6
 California 292.6 296.0
 Florida 152.6 153.1
 Georgia 89.0 89.3
 Idaho 13.8 13.7
 Illinois 157.0 157.5
 Indiana 67.5 67.5
 Kentucky 42.7 42.7
 Louisiana 52.0 52.1
 Massachusetts 67.2 68.1
 Maryland 48.0 48.5
 Maine 14.3 14.1
 Michigan 116.1 115.6
 Minnesota 65.2 65.1
 Missouri 64.2 64.2
 Montana 10.1 10.0
 North Carolina 87.0 87.2
 North Dakota 7.9 7.8
 New Hampshire 14.1 14.0
 New Jersey 109.9 110.5
 New York 213.9 216.1
 Ohio 128.1 128.2
 Pennsylvania 128.6 129.0
 Tennessee 63.5 63.5
 Texas 246.2 246.9
 Virginia 83.1 83.4
 Vermont 7.2 7.1
 Washington 65.4 65.1
 Wisconsin 65.5 65.4
 Rest of United 632.9 630.1
 States

TABLE 4 Welfare Impacts of Economic Integration under Alternative
Calibrations

 Equivalent Variation

 Econometric Exact-Fit
 Benchmark GDP Calibration Calibration
Province ($US billion) (%) (%)

Alberta 56 36 28
British Columbia 63 26 14
Manitoba 17 49 20
New Brunswick 9 42 24
Newfoundland 6 50 78
Nova Scotia 12 40 14
Ontario 194 35 36
Prince Edward Island 2 45 64
Quebec 107 29 19
Saskatchewan 15 46 19
GDP-weighted average 34 27