Analytical Studies: Methods and References
Estimating the Effect of Changing Canada-US Border Costs on North American Trade Patterns and Expenditures: Detailed Methodology

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Abstract

This paper shows how to estimate the effect of the Canada-United States border on non-energy goods trade at a sub-provincial/state level using Statistics Canada’s Surface Transportation File (STF), augmented with United States domestic trade data.  It uses a gravity model framework to compare cross-border to domestic trade flows among 201 Canadian and United States regions in year 2012. It shows that some 25 years after the Canada-United States Free Trade Agreement (the North American Free Trade Agreement’s predecessor) was ratified, the cost of trading goods across the border still amounts to a 30% tariff on bilateral trade between Canadian and United States regions. The paper also demonstrates how these estimates can be used along with general equilibrium Poisson pseudo maximum likelihood (GEPPML) methods to describe the effect of changing border costs on North American trade patterns and regional welfare.

1 Introduction

World trade has expanded since the Second World War, facilitated by the ratification of multilateral and regional trade agreements. One of the earliest such agreements was the 1988 Canada $—$ United States Free Trade Agreement, to which Mexico acceded in 1994 to create the North American Free Trade Agreement (NAFTA). It was renegotiated in 2018. These agreements originally focused on disciplining tariffs and quotas applied to goods crossing the border. The most recent negotiations, while addressing market access in certain industries, also focused more on other burdens associated with administrative borders.^Note 

At the same time, Canadian provinces have been negotiating to further reduce trade barriers between them by updating the 1995 Agreement on Internal Trade (AIT) with the signing of the Canadian Free Trade Agreement (CFTA) in 2017. There are no tariffs or quotas levied at the borders separating provinces. The main impediments are thought instead to be caused by differences in regulatory frameworks and government procurement practices, and amount to about a 7% tariff equivalent (Bemrose, Brown and Tweedle 2017).

Frictions caused by red-tape and delays at borders, regulatory differences, or uncertainty over future policies, are commonly referred to as non-tariff barriers (NTBs). Unlike traditional trade policies like tariffs and quotas that are easy to identify and measure directly, frictions that range from red tape and border-related delivery delays, to divergent regulations and firms’ uncertainty about how policies will evolve, are not. The average bilateral costs of both tariff and non-tariff barriers can, however, be estimated indirectly by applying the gravity model framework to suitable data.

Trade frictions generated by the Canada-United States border have already been the subject of extensive research using the gravity model. The seminal work by McCallum (1995) on Canada and United States trade pointed to an extraordinarily large border effect. But the literature’s subsequent estimates of border frictions have declined as researchers have taken advantage of refinements to theory, measurement and estimation methods. These refinements include Anderson and van Wincoop’s (2003) incorporation of “multilateral resistance” terms to accommodate regional differences in access to and competition from global markets. The “structural” gravity model imbues the estimation with a general equilibrium structure that summarizes the ability of a region to generate and absorb trade from all other potential trading partners.

More recent work has suggested that further refinements are needed to accurately estimate border costs. Much of the emphasis has been on how to deal with the paucity of detail in available data. Trade flows are often measured at an aggregated level, between nations or large sub-national units, like provinces or states. But the level of geographic aggregation used in estimation matters, with higher levels of aggregation often biasing border estimates upwards. Hillberry and Hummels (2008) uses geographically detailed (i.e., zip code level) United States data to show that when the size of regions is reduced to the sub-state level the state borders’ effects on inter-state trade found with more aggregate data vanish entirely. Similarly, Bemrose, Brown and Tweedle 2017 found that smaller and more geographically uniform regions are associated with smaller estimated provincial border effects in Canada, although they are not eliminated.

This methodological note describes how to estimate the costs that administrative borders impose on Canada-United States trade using recently constructed data at Statistics Canada. A structural gravity model is applied to data from the Surface Transportation File (STF). The STF was built from shipping records to measure domestic trade between detailed locations within Canada and between the United States and Canada. It is augmented for this paper with the 2012 United States Commodity Flow Survey (CFS) to produce a data set that measures trade among 201 comparable Canadian and United States regions in year 2012. The combined data can measure the average costs of Canada-United States and provincial borders simultaneously. We are unaware of any work estimating the Canada-United States border effect at such a fine-grained geography.

The average tariff equivalents are useful to illustrate the effect of trade frictions on Canada-United States bilateral trade. But changing international border costs also affect domestic trade patterns as firms shift between domestic and international markets. Together these first- and second-order impacts give a fuller picture of the economic significance of changing international trade costs.^Note   Anderson, Larch and Yotov (2018) show how, using counterfactual scenarios, the structural gravity model allows for the direct and indirect impacts of a change in bilateral trade costs to be quantified. We present the application of the structural gravity model in this paper and use it to illustrate the aggregate outcomes of counterfactual changes to Canada-United States border costs.

The following section discusses the gravity model. Section 3 covers the construction of the data and highlights the advantages and limitations of using detailed shipment data from different sources. Section 4 discusses the model’s empirical implementation, including the need to adjust the estimating equation to allow for features of the data. Results for Canada-United States and provincial border effects are presented in Section 5. A methodology for elaborating on the general equilibrium effects with an example of results are discussed in Section 6. For a more detailed description of these results, see Brown, Dar-Brodeur and Dixon (2019).

2 The gravity model

In the gravity model framework, bilateral trade is analysed by analogy with Newtonian gravity: trade flows between two regions are assumed to be an increasing function of their economic sizes and inversely related to the distance between them. Trade flows not explained by these variables are attributed to other trade “frictions”. When applied to measuring administrative borders, the model estimates the degree to which interregional trade frictions must exceed domestic ones.

Anderson and van Wincoop’s (2003, 2004) showed that in addition to size and proximity, gravity models should also incorporate regions’ relationships with third parties. These models are often referred to as “structural” gravity, of which there are many varieties in the literature.^Note   Each imposes different conditions on consumers or producers, leading to different interpretations of the parameters, they all result in common expressions for the equilibrium value of bilateral exports, $X_{ij}$ , from geographical location $i$ to location $j$ :

$$X_{ij}={\left(\frac{t_{ij}}{{\Pi }_i{P}_j}\right)}^{-ϵ}{Y}_i{E}_j,\begin{array}{ccccccc}& & & & & & (1)\end{array}$$

where

$$P_j^{-ϵ}={\displaystyle \sum _i{\left(\frac{t_{ij}}{{\Pi }_i}\right)}^{-ϵ}}{Y}_i,\begin{array}{cccccccc}& & & & & & & (2)\end{array}$$

and

$${\Pi }_i^{-ϵ}={\displaystyle \sum _j{\left(\frac{t_{ij}}{P_j}\right)}^{-ϵ}}{E}_j.\begin{array}{cccccccc}& & & & & & & (3)\end{array}$$

The variables $Y_i(={\displaystyle \sum _j{X}_{ij}})$ and $E_j(={\displaystyle \sum _i{X}_{ij}})$  are the values of location $i$ ’s total output and location $j$ ’s total expenditure (on goods and services to/from all locations, including themselves), respectively.  The variable t_ij is the bilateral trade costs between i and j. The parameter $-ϵ$ is the elasticity of trade with respect to trade costs (or more commonly, the trade cost elasticity).^Note   

Depending on the underlying model, the trade cost elasticity reflects one or more of the heterogeneity of consumer preferences, scale economies, or the fixed costs of exporting. In the case of supply-side models, it reflects the (Frechet or Pareto) dispersion of firms’ productive efficiency.

The key feature distinguishing the structural gravity model from its earlier forms is the introduction of “multilateral resistance” (MR) terms, variables ${\Pi }_i$  and $P_j$ , which are the solutions to the non-linear system of 2N Equations (2) and (3), subject to a normalization.^Note 

The first term is referred to as region $i$ ’s outward multilateral resistance (OMR) and the second term as region $j$ ’s inward multilateral resistance (IMR). These terms account for the fact that bilateral trade flows are not only affected by trade frictions between the two regions, but also by their relative positions with respect to all other potential trading partners. As described by Anderson and Yotov (2010), the OMRs can be interpreted as measure of the trade cost incidence on sellers as they bring goods to a hypothetical world market. Similarly, the IMRs summarize the incidence of trade costs on consumers receiving goods from the world market. In models derived from the constant elasticity of substitution (CES) utility function, the IMRs are also the regional consumer price indexes. Economic geographers often refer to OMRs as an index of exporting regions’ market access and the IMRs as indexes of competition in the destination region. Together, the multilateral resistance terms account for the indirect effect that multilateral trading relationships have on bilateral trade flows.

3 Data

The literature has identified two data-related issues that may influence border effect estimates. Due to data limitations, many papers are confined to analysing trade between highly aggregated/heterogeneous regions like countries or provinces/states. The distance goods travel are approximated by the great circle distance between administrative or commercial capitals, or the regions’ economic or demographic centroids. Gravity researchers have observed that estimated border frictions can be influenced by heterogeneity in internal regional trade costs associated with trading regions of differing sizes (Coughlin and Novy, 2016). In addition, standard distance measures may bias the estimation of border frictions, because trade flows tend to be short distance and so are likely overestimated (Head and Mayer 2009).

The two different data sources used in this paper address these concerns by using a relatively homogeneous set of sub-provincial/state regions and a more accurate measure of the distance that goods travel between them. The following subsections discuss the construction of the data, their advantages and potential limitations.

3.1 Trade data

The data are derived from two different sources. Canadian domestic and cross-border trade flows are derived from Statistics Canada’s Surface Transportation File (STF). United States internal trade comes from the United States 2012 Commodity Flow Survey (CFS). They include shipments measured in terms of commodity, tonnage, value, network distance shipped, and by detailed origin and destination. The precision of the origins and destinations allows for the measurement of trade flows between 201 sub-provincial/state areas.

While the STF covers the major modes of transportation used by most commodities, it does not cover air, marine and goods moved by pipeline, or own-account trucking. To account for these limitations, shipment values are weighted to ensure they add to inter-/intra-provincial trade totals by commodity for domestic trade reported by the provincial supply-use (input-output) tables. Similarly, cross-border shipments are weighted to add to provincial-United States exports/imports by commodity, again using the provincial supply-use tables.^Note  Hence, domestic and cross-border flows add to known totals. This approach assumes that the patterns of trade followed by modes not covered follow those generated by the for-hire trucking and rail.^Note 

The exception is commodities produced by the energy industry. Although energy products are also shipped by rail and truck, they are primarily moved by pipeline. Since pipelines are not included in the STF, there is not enough information in the file to accurately attribute energy flows down to the sub-provincial level. The energy sector is therefore excluded from this analysis.

The STF by itself is insufficient to measure the Canada-United States border effect. The appropriate comparison group to cross-border trade is domestic trade in both countries. The United States domestic trade flows are not included in the STF, and so comparable shipment-level data from the 2012 CFS are used to fill the gap.^Note  CFS trade flows are scaled to United States gross output as in Anderson and van Wincoop (2003), and the oil and gas industry is excluded.^Note 

The CFS is a quinquennial survey, and only has publicly available microdata for 2012, while the STF is based on annual data from 2004-2012. The combined data is restricted to the year 2012 to generate a shipment-level file for North America. The shipments are aggregated to generate intra- and inter-regional trade on a North America-wide basis, taking advantage of the (relatively) detailed origins and destinations reported on each file.

While the STF and CFS are broadly comparable, there are some important differences to be kept in mind when using the combined information for analytical purposes. First, domestic STF flows are benchmarked to the provincial supply-use tables estimating origin and final use of commodities across provinces. As a result, the domestic portion of the STF is converted from a ‘logistics file’ that measures where goods are picked up and dropped off to a quasi-‘trade file’, tracking where they are made and used. The CFS and cross-border trade from the STF remain logistics-based. Bemrose, Brown and Tweedle (2017) show that benchmarking stretches trade flows across space, relative to un-benchmarked logistics flows.

Second, both the STF and CFS derive trade flows from shipping records, but the STF obtains them from trucking firms (‘carrier-based’ survey), while the CFS obtains them from establishments that generate the shipments (‘shipper-based’ survey), leading to potential differences in coverage. The truck-based shipments in STF are derived from the Trucking Commodity Origin Destination (TCOD) Survey, which does not include private trucking. The CFS, on the other hand, does. If private trucking is a significant proportion of short distance flows, these flows will be better represented in the CFS than the STF. Finally, the CFS measures distances between zip codes, while the STF uses much more spatially detailed postal codes.

One additional factor complicates the calculation of a full set of regional flows and their respective distances below the provincial/state level. While truck shipments to and from the United States are reported by zip code on the STF file, only the state of destination/origin are reported for rail shipments. Interregional rail shipments between Canada and the United States were allocated to sub-state areas using the value-weighted share of corresponding truck exports/imports. While the pattern of rail and truck shipments at the sub-state level are likely not perfectly correlated, there should be a strong correspondence.^Note  Hence, any error introduced is expected to have a relatively small effect on the aggregate flows.

3.2 Geography

The gravity model, by construction, measures inter-regional relative to average intra-regional trade costs. This construction presents no issues when regions are homogeneous (i.e., they have roughly the same size and geography). However, in most data, the units of analysis have different sizes and thus different, usually unobservable, internal trade costs. Larger regions naturally have higher internal costs than smaller ones, and yet estimation often treats them both as dimensionless points in space. Coughlin and Novy (2016) show that the size-related heterogeneity can bias border effect estimates.^Note 

The data allow for two sub-national geographies to define trading units. The first is the province/state geography that the literature has relied upon. However, in the light of Coughlin and Novy (2016)’s findings, a second more fine-grained geography is also used. It is based on the CFS’s division of states into metropolitan and non-metropolitan (MA/non-MA) areas. While there is no perfect analogue in Canada, STF-based flows can be aggregated by Economic Regions (ERs) that are comparable in size to the United States MA/non-MA geography (see Map 1).^Note  Furthermore, ERs and MA/non-MA boundaries respect provincial and state borders in the same way.

There are some differences between ERs and MA/non-MAs. ERs are not purely metropolitan-based. Hence, while there is a tendency for them to roughly follow metropolitan boundaries (e.g., Edmonton and Calgary), other metropolitan areas (e.g., the Toronto metropolitan region) are composed of several ERs. Furthermore, the United States geography treats the rest of each state as a non-metropolitan residual, while the “non-metropolitan” portions of provinces may be split into multiple ERs (see Map 1). Still, there is enough commonality to treat them as compatible. To simplify the discussion both geographies will hereafter be referred to as ERs.

3.3 Network distance

Another factor potentially biasing border effect estimates is the mis-measurement of shipping distances (Head and Mayer, 2009). Most research is limited to using great circle distances between arbitrary points within trading areas. These measures often mischaracterize the distance goods traded between regions actually travel. The STF and CFS allow the derivation of the network distances that more accurately reflect the origins, destinations and journeys goods can be expected to take.^Note  However, the network distances used here are constructed using two sources that differ in their methodologies, and the differences between them may bias border effect estimates. It is thus important to understand how much they differ when comparing cross-border to domestic trade. 

Table 1 shows the distribution of distances between ERs for domestic Canadian and cross-border trade derived from the STF and domestic United States trade derived from the CFS. For domestic trade, the percentile distribution of distances are reported for intra-ER, distances between ERs within the same province/state, and between different provinces/states. Regardless of whether distances are intra- or inter-ER, the two files generate the same distribution of shorter distance flows. It is only at the 90^th and especially the 99^th percentile of ER pairs that we observe much longer distances shipped in Canada, reflecting its larger size and more dispersed population. Otherwise, the STF and CFS generate broadly comparable distributions of within and between ER distances.

Cross-border distances tend to be longer than domestic flows. There are no cross-border pair distances below 150 kilometres. Since cross-border flows are much more heavily weighted towards longer distance shipments (greater than 400 km), any difference in the effect of distance on trade flows stemming from the construction of the data may bias the estimated border effect derived from the model. This potential bias can be compensated for in the specification of the econometric model. One possible remedy is discussed in Section 4.

4 Estimating the structural gravity model

The estimation equation corresponding to the model for bilateral trade described in Section 2 is often expressed as:

$$X_{ij}=\exp (T_{ij}B+{\pi }_i+{\chi }_j){\xi }_{ij}.\begin{array}{cccccccc}& & & & & & & (4)\end{array}$$

The link between the estimating equation to Equation (1) can be seen by defining the natural log of bilateral trade costs, $\ln (t_{ij}^{-ϵ})$ , as the multiplication of the logarithmically transformed determinants of trade costs by their trade elasticities, $T_{ij}B$ (boldface denotes vectors), ${\pi }_i=-\ln \left(\frac{{\Pi }_i^{-ϵ}}{Y_i}\right)$ and ${\chi }_j=-\ln \left(\frac{P_j^{-ϵ}}{E_j}\right)$ . In cross-sectional data, the last two terms can be incorporated in the estimation as directional exporter and importer fixed effects accounting for the multilateral resistance terms and the relative economic size of regions $i$  and $j$ .^Note  The ${\xi }_{ij}$ is a random error term that captures un-modelled trade costs and allows for errors in data construction. When implemented, the estimation must exclude both the intercept and a fixed effect (importer or exporter) or two fixed effects (an importer and an exporter) to avoid perfect collinearity. To make interpretation of the fixed effects in the Section 5 simpler, we forgo an intercept and exclude the importer dummy for Ontario in the province/state regressions, and for the Greater Toronto area when the ER geography is used. We define bilateral trade costs as:

$$T_{ij}B=ln{d}_{ij}{B}_d+{\beta }_{BR}CUB{R}_{ij}+{\beta }_{PBR}PB{R}_{ij}+{\beta }_{SBR}SB{R}_{ij}.\begin{array}{cccccccc}& & & & & & & (5)\end{array}$$

The first term on the right-hand side captures the non-linear effect of distance, discussed in more detail at the end of this sub-section. The main focus of this paper is the trade frictions induced by the intercession of administrative borders. In Equation (5) the binary variables take a value of one if regions lie on either side of the Canada-United States ( $CUBR$ ), a provincial ( $PBR$ ) or a state border ( $SBR$ ), respectively, and zero otherwise. The exponentiated coefficient for the Canada-United States border dummy $\exp ({\beta }_{CUBR})$ and the trade elasticity can be used to calculate a tariff equivalent for the border’s impact on bilateral trade. The tariff equivalent summarizes the impact of tariffs and “tariffies” any border-related non-tariff barriers.^Note 

In much of the literature, trade costs of exporting from region $i$  to region $j$  are assumed to be a log-linear function of distance between them. While this assumption is likely innocuous when modelling trade over long distances, it may be problematic when using a fine geographic breakdown, as trade involves fixed logistical and search costs that must be covered to move goods any significant distance. These outlays inflate per km costs at short distances, but their impact declines rapidly as they are spread over longer distances (see Behrens and Brown (2018)). This non-linearity in trade costs may be more pronounced in estimations that include a large number of short distance flows within regions and between near neighbours (i.e., between ERs within provinces/states).^Note 

In order to accommodate potential non-linearities, as well as compensating for differences in the underlying data sources that make up our sample, the estimating Equation (4) includes a spline over distance, ( $d_{ij}$ ):

$$ln{d}_{ij}{B}_d=1\{d_{ij}<\overline{d}\}{\beta }_{\overline{d}}\ln (d_{ij})+{\displaystyle \sum _{d_{ij}=d}{\displaystyle \sum _{f_{ij}=f}{\beta }_{df}}}\ln (d_{ij}),\begin{array}{ccccccc}& & & & & & (6)\end{array}$$

where for $d_{ij}\ge \overline{d}$ parameter estimates are allowed to vary across trade types $f$  such that:

$$f\in \{\text{within Canada, Canada-US, within US}}\text{.}$$

For regressions at the provinces and state levels, $\overline{d}$ equals150 kilometres, while for regressions on ER flows, $\overline{d}$ equals 50 kilometres $—$ below these distances, trade flows are exclusively intra-regional. Constraining the effect of the shortest distances to be the same across regions essentially imposes the same distance-related intra-regional trade costs across the two countries. The other spline segments are represented by distance class set $d$ :  

$$d\in \left\{d_{min}\text{, 500 - 999, 1,000 - 1,999, 2,000 or greater}\right\},\begin{array}{ccccccc}& & & & & & (7)\end{array}$$

where

$$d_{min}=\{\begin{array}{ll}150-\text{499},\hfill & \text{for provinces and states},\hfill \\ 50-\text{149, 150 }-\text{ 499}\hfill & \text{for ERs}.\hfill \end{array}$$

There are no international network distances of less than 150 kilometres in the data, so the effects of the shorter unconstrained distances only vary between within-country Canada and US flows.^Note 

4.1     Estimation method

Equation (4) has traditionally been estimated by taking the natural log and using ordinary least squares (OLS). However, Silva and Tenreyro (2006) show that this approach leads to biased estimates in the presence of heteroskedasticity, because the conditional mean of the log error term will generally not be independent of the covariates (a requirement of OLS estimation) unless strong assumptions are made about the distribution of ${\xi }_{ij}$ .

Intuitively, as the value of trade flows approach zero, they can only vary in one direction (trade flows cannot be negative), whereas larger trade flows can vary upwards or downwards. Smaller regions that are farther apart being more likely to record flows that are near or at zero, where the log specification is undefined. The bias is thus a greater problem when considering smaller, more distant geographic aggregations. Silva and Tenreyro (2006) suggest dealing with heteroskedasticity by estimating (4) in its non-linear form using pseudo-maximum likelihood estimation. This approach has the added benefit of incorporating zero-valued flows as a matter of course. Among this class of estimators, they find that Poisson pseudo maximum likelihood (PPML) often yields the best results.^Note  Furthermore, PPML estimation has desirable properties that permit us to recover the multilateral resistance terms directly from the fixed effects (Fally 2015). These properties are important to accessing the model’s general equilibrium information, as discussed in Appendix A.

5 Gravity model estimates

The results presented in Table 2 replicate specifications used in the existing literature, with trade aggregated to the province/state level and the effect of distance on trade constrained to be the same across trade types ( $f$ ) and distance classes ( $d$ ). Estimating the model using all trade [column (1)], including intraregional flows, produces a distance parameter of -1.182, which is somewhat stronger than the standard of -1 found in the literature (see Head and Mayer (2014)) and a Canada-United States border parameter estimate of -0.626, which is much weaker than generally found: Anderson and van Wincoop (2003) estimated a border coefficient border using 1993 data to be -1.65. Column (2) repeats the same model but with intra-province/state flows excluded. There are no qualitative differences in parameters (see Table 2) when intraregional flows are excluded. Due to differences with the literature in the underlying data, it is unclear what inferences should be drawn regarding the initial border effect estimate.

The distance estimates presented in columns (1) and (2) are constrained across trade types. Because the STF benchmarks Canada’s domestic flows to intra- and inter-provincial trade totals, it is expected that the effect of distance on Canadian domestic trade to be weaker than cross-border or United States domestic trade. Benchmarking has the effect of stretching flows by tracing the trade in goods from where they are made all the way to where they are used, rather than to a point along the logistics chain prior to a good’s final destination, as is the case with the United States data. This stretching gives the appearance that distance-related trade costs are lower in Canada. Columns (3), (4) and (5) of Table 2 show gravity regressions for Canada, United States and cross-border trade separately. The trade elasticity of distance for United States domestic trade is -1.140, while for Canadian domestic trade it is -0.773. Cross-border trade’s distance elasticity, at -0.96, is close to unity, as is often found in the international trade literature.

Since distance has a stronger estimated negative effect on United States trade than in the full model [see Column (2)], there will be a tendency for internal United States flows to be underestimated, biasing the border effect upwards (i.e., towards zero). Of course, the opposite holds true for Canadian trade, but because United States flows account for the vast majority of North America’s domestic trade, their effect will dominate border estimates. These differing distance parameter estimates suggest that the effect of distance needs to be allowed to vary across trade flow types.

The fully specified version of Equation (4) is presented in Table 3. Given the larger number of variables in the model, for ease of exposition we present the border and distance estimates separately. The first table presents the border coefficients from all regression models; the second presents the distance coefficients for our preferred model. The full model allows the effect of distance to vary by trade type ( $f$ ) and distance ( $d$ ) at the province/state level [columns (1) and (2)] and the ER level of spatial aggregation [columns (3) and (4)]. Relaxing the constraints addresses the error that results from differences in the construction of the STF and CFS portions of the data, as well as allowing for non-linearities in trade costs, which are expected to be particularly pronounced at lower levels of geographic aggregation.

In contrast to the results in Table 2, the province-state regressions with the unconstrained distances yield a stronger Canada-United States border effect. The border effect estimates with and without accounting for internal flows changes to -1.435 and -1.555, respectively. Comparing Tables 2 and 3 show that constraining the effect of distance across categories and trade types has a significant influence on the border effect.

The border effects estimated using data at the ER level are, on the other hand, roughly the same in magnitude as the province/state level, but are estimated with greater precision.^Note  The Canada-United States border effect is -1.568 with and -1.434 and without intraregional flows. Assuming trade elasticity of $\epsilon =7$ , the tariff equivalent for the former estimate is 30% and for the latter 27%. These estimates are, broadly speaking, in line with the various estimates in the literature.^Note 

The economic region regressions are also able to simultaneously measure provincial border effects of -0.576 and state borders -0.387 when internal flows are included. The provincial results are close to those found in Bemrose, Brown and Tweedle (2017) using the Canadian data from the STF alone and the same level of geographic aggregation.^Note  The exclusion of intra-regional trade flows [see Table 3, Column (4)], raises the provincial border effect in absolute terms. The state border estimates are negative and significant as well, but these are likely an artefact of CFS data and the level of geographic aggregation (Hillberry and Hummels, 2008). Bemrose, Brown and Tweedle (2017), using the Canadian equivalent of zip codes, find provincial border effects that are not qualitatively different than those estimated in this paper.

Our preferred estimates use ERs as the trading units and include intra-regional trade [Table 3, Column (3)]. The provincial border effect is the closest among the specifications to that estimated by Bemrose, Brown and Tweedle (2017) and the Canada-United States border effect is estimated with the most precision.

The distance coefficients for the preferred specification can be found in Table 4; the results for the other specifications can be found in Appendix B. Over the 500-3000 KM range, all flows (both domestic and international) are around one in absolute value, although the coefficients on United States domestic flows (from the CFS data) are generally higher than the Canada and Canada-United States flows (from the STF), likely reflecting differences in the source data. Second, there is clear evidence of a non-linearity of the effect of distance on the value of trade for Canadian domestic flows: the trade elasticities are high for very short distance flows (0-50 KM), reflecting fixed costs, and drop off over the 50-500 km range, before rising again beyond 500 KM and gently declining over longer distances. A similar, although more muted, pattern is evident for the United States domestic flows. There is large drop in the effect of distance on the value of trade for very long distance domestic flows (3000 km +), but not for international flows. It is unclear the degree to which these differences indicate data issues versus features of the economy.^Note 

6 General equilibrium PPML

The previous section examined the average bilateral, first-order, estimates of Canada-United States border costs. But trade costs between two regions will change the opportunities for producers and consumers in their other potential trading partners, who will adjust accordingly. These general equilibrium, second order, adjustments will impact the relationships between all other trading regions. In other words, changes in, for example, costs at the Canada-United States border will also change the patterns of intra and inter-provincial/state trade. Welfare implications of any thinning or thickening of the border will depend not only on the bilateral impact on the two trading regions, but also on how patterns of trade respond across the continent.

The impact of a change in border costs can be illustrated by examining counterfactual scenarios. The most common counterfactuals explored in the literature are either frictionless borders or autarchy. These extreme scenarios can establish the total cost of the border or the benefits of liberalized trade. But the methodology can also be used to explore more plausible changes in trade costs. Two such plausible scenarios are discussed in Section 6.3 and the results are discussed in detail in Brown, Dar-Brodeur and Dixon (2019).

6.1 Fixed effects and multilateral resistances

Dekle, Eaton and Kortum (2008) show that, given data on output, expenditures and actual trade flows, the economic impact of counterfactual changes can be computed by recasting Equations (1) to (3) from levels to deviations from initial values and solving the transformed system for the anticipated adjustments in border costs. This method of calculating general equilibrium impacts is commonly referred to in the literature as exact hat algebra. Anderson, Larch and Yotov (2018) argue that another, equivalent, approach is to exploit the fixed effects ${\hat{\pi }}_{{}_i}$ and ${\hat{\chi }}_j$ , estimated in Section 4.5. In principle they should correspond to the MRs, but in practice they could include a variety of other importer/exporter-specific influences on bilateral trade. However, Fally (2015) observes that when Equation (4) is estimated by Poisson pseudo-maximum likelihood (PPML), estimates of the MR terms can be derived directly from the estimated fixed effects as follows:

$$\widehat{P_j^{-ϵ}}=\frac{E_j}{E_0}\exp (-{\widehat{\chi }}_j),\begin{array}{cccccccc}& & & & & & & (8)\end{array}$$

and

$$\widehat{{\Pi }_i^{-ϵ}}=E_0{Y}_i\exp (-{\widehat{\pi }}_i).\begin{array}{cccccccc}& & & & & & & (9)\end{array}$$

For their derivation, Equations (8) and (9) rely on the exclusion of an importer fixed effect ( ${\widehat{\chi }}_0=0$ ), the normalization of the corresponding IMR ( $P_0=1$ ), and the fact that the sum of the outward equals the sum of the inward fixed effects. Anderson, Larch and Yotov (2018) show that using MRs recovered from the fixed effects estimated by PPML under actual and counterfactual values for trade costs produce estimates for changes in trade and welfare that are equivalent to the ‘exact hat’ results.

6.2 Trade costs, output and expenditure

The general equilibrium impact of trade costs on trade and welfare will depend on the underlying model used to derive Equation (1). To establish a lower bound for welfare estimates, this paper uses a version of the canonical Armington endowment model used in Anderson and van Wincoop (2003). In this model, each region has an endowment of its own distinct good that generates utility in a CES function that is identical across regions. Welfare changes from trade stem from producers fetching a lower or higher price for their endowment of goods, and consumers having more or less favourable access to a greater variety of goods.^Note  The first effect is captured by changes in producer prices, while the second is measured by the changes in the consumer price index.

With CES demand, the consumer price index is simply the inward multilateral resistance term ( $P_j$ ). Producer prices in the Armington framework can be recovered from the aggregate CES demand for region $i$ ’s unique good. The demand for this good in region $j$ is given by

$$X_{ij}={\left(\frac{{\gamma }_i{p}_i{t}_{ij}}{P_j}\right)}^{1-\sigma }{E}_j,\begin{array}{cccccccc}& & & & & & & (10)\end{array}$$

where the parameter ${\gamma }_i>0$ reflects the geographic distribution of comparative advantage and $p_i$ is region $i$ producers’ prices. The trade elasticity in this model is $-ϵ=1-\sigma$ , or the consumers’ elasticity of substitution between regional varieties. Summing over all its consumers (including in the region itself), the market clearing condition for region $i$ ’s output is $Y_i={\displaystyle \sum _j{X}_{ij}}={({\gamma }_i{p}_i)}^{1-\sigma }{\displaystyle \sum _j{(\frac{t_{ij}}{P_j})}^{1-\sigma }}{E}_j$ .

Using Equation (10) and solving for producer prices yields:

$$p_i=\frac{Y_i^{\frac{1}{\sigma -1}}}{{\gamma }_i{\Pi }_i}.\begin{array}{cccccccc}& & & & & & & (11)\end{array}$$

Arriving at general equilibrium estimates for counterfactual trade costs involves calculating how producers’ prices ( $p_i$ ), output ( $Y_i$ ), and expenditures ( $E_j$ ) change in response to shifting market opportunities (as summarized by changes in ${\Pi }_i$ ). Note that in the endowment model, region $i$ ’s volume of goods remain unchanged, i.e., $\frac{Y_i^c}{p_i^c}=\frac{Y_i}{p_i}$ , where the superscript $c$ denotes counterfactual values. Following the literature, the model’s counterfactuals are derived assuming constant bilateral trade balances, or $\frac{E_j}{Y_j}={\varphi }_j=\frac{E_j^c}{Y_j^c}$ . Finally, welfare ( ${\widehat{W}}_j$ ) implications of trade policy changes in this model can be summarized by changes in the value of region $j$ ’s expenditures, deflated by the region’s consumer price index:

$${\widehat{W}}_j=\frac{E_j^c/P_j^c}{E_j/P_j}.\begin{array}{cccccccc}& & & & & & & (12)\end{array}$$

Anderson, Larch and Yotov (2018) propose an algorithm for calculating counterfactual trade, prices, output and expenditures after a policy change that exploits the MRs estimated by the fixed effects. They characterize their method as ‘estibration’, combining the advantages of PPML estimation and calibration. The algorithm consists of four steps:

1. Estimate fixed effects with PPML and use them to construct baseline indexes, ${\widehat{\Pi }}_i^k$ , ${\widehat{P}}_j^k$ and $p_i^k$ according to Equations (8), (9) and (11), where the iteration counter $k=0$ .
2. Impose counterfactual trade costs, $T_{ij}\widehat{B^c}$ , re-estimate Equation (5) and use the fixed effects to calculate “conditional” general equilibrium (GE) indexes, ${\widehat{\Pi }}_i^{k+1}$ , ${\widehat{P}}_j^{k+1}$ and ${\widehat{p}}_i^{k+1}$ in the first iteration.
3. Use the change in producer prices ( $\frac{{\widehat{p}}_i^{k+1}}{{\widehat{p}}_i^k}={[\frac{\exp ({\widehat{\pi }}_i^{k+1})}{\exp ({\widehat{\pi }}_i^k)}]}^{\frac{1}{1-\sigma }}$ ) and MRs to update the values of bilateral trade flows according to:

$${\widehat{X_{ij}}}^{k+1}=\frac{({\widehat{t_{ij}^{1-\sigma })}}^c}{\widehat{t_{ij}^{1-\sigma }}}\frac{{\widehat{Y_i}}^{k+1}{\widehat{E_j}}^{k+1}}{{\widehat{Y}}_i^k{\widehat{E}}_j^k}\frac{{(\widehat{{\Pi }_i^{1-\sigma }})}^k{(\widehat{P_j^{1-\sigma }})}^k}{{(\widehat{{\Pi }_i^{1-\sigma }})}^{k+1}{(\widehat{P_j^{1-\sigma }})}^{k+1}}{\widehat{X_{ij}}}^k.\begin{array}{cccccccc}& & & & & & & (13)\end{array}$$

• Note that all the ratios to the right of the trade cost terms can be expressed in terms of changes in fixed effects.

4. Using ${\widehat{X_{ij}}}^{k+1}$ , re-estimate the fixed effects. Iterate this estimation for $k=1,\dots ,K$ substituting the appropriate k-indexed terms into Equation (13) for each iteration until all regions’ producer prices change by less than some predetermined tolerance level, or $\Delta p_i^{k+1}<z$ for all $i$ .

The first step, computing estimates of the multilateral resistances, is achieved in this paper by using the fixed effect estimates from Section 3 along with the predicted expenditures/output, in Equations (8) and (9). The second step involves generating counterfactual trade costs. The assumptions involved in generating our counterfactual scenarios (one in which the border is removed, and another where a revised Canada-United States trade agreement is sun-setted without replacement) are discussed in section 6.3.

6.3 Counterfactual scenarios: an illustration

To illustrate the ‘estibration’ methodology described above, we consider two scenarios for changing border costs. The first considers a world in which the cost of trading between Canada and the United States is assumed to be equivalent to trading across provincial borders. This amounts to reducing the estimated Canada-United States border effect from a 30% to a 10% tariff equivalent. The second scenario is one in which Canada and the United States withdraw altogether from a preferential trading agreement. In this case, tariffs would return to their Most-Favoured Nation (MFN) levels as the bilateral trading relationship would be governed by World Trade Organization rules, plus any additional non-tariff trade costs, such as heightened trade policy uncertainty for exporters, who face a less predictable trading environment. We use an approximation of the effect NAFTA had on reducing trade costs between the two countries beyond the explicit reduction in tariffs. This amounts to increasing trade costs by six percentage points to 36%.^Note 

Aggregate general equilibrium impacts are presented in Table 5. The effect of changing border costs is significant; a reduction increases trade from Canada to the United States by 82%, and from the United States to Canada by almost 72%. This comes at the expense of internal trade in Canada, both inter- or intra-provincial, falling by about half. The much larger United States economy, however, gets a boost in domestic trade. Welfare (total expenditures on domestic and imported goods) increases in both Canada (11.4%) and the United States (0.8%).

On the other hand, an increase in border costs reduces trade from Canada to the United States by almost a quarter, and from United States to Canada by 18%. There is a substitution towards internal trade in Canada, which rises by around 10%. Domestic trade in the United States rises slightly as well, by around 1%. Overall welfare declines by almost 2% in Canada and -0.2% in the United States.

7 Conclusion

The world economy has become increasingly integrated since the Second World War. It has been knit together by flows of goods and services facilitated, in part, by falling trade costs associated with administrative borders. Some of these costs take the form of explicit tariffs and quotas levied on imports. Others consist of difficult-to-enumerate and quantify non-tariff barriers, which are a diverse set of frictions ranging from red tape and delay at the border to regulatory differences across jurisdictions.

Canada and the United States share one of the most important trading relationships in the world. Over the past thirty years, this relationship has developed under the auspices of an agreement that ultimately became NAFTA. The uncertainty generated by the 2018 renegotiation of that agreement has highlighted the need to understand the impact of a changing Canada-United States border on firms and consumers within the two countries. This paper uses the gravity model framework on data from a combination of Canada’s Surface Transportation File and the United States’ 2012 Commodity Flow Survey to quantify indirectly the cost of the border on bilateral trade. It also shows how to use these results in combination with the GEPPML methodology to generate general equilibrium estimates of cross-border and domestic trade and welfare resulting from changing border costs.

8 Appendix

Appendix Table 1 presents the full set of estimates of Equation (4) for trade measured at province/state and ER levels of aggregation. The province-state level regressions constrain the effect of distance to be the same below 150 kilometres. As Table 1 in section 3.3 shows, most of these flows are still intra-province/state and so costs imposed by distances 0 to 150 kilometres is the minimum distance at the province/state regressions level of aggregation. On the other hand, the ER level regressions allow for a more detailed specification at shorter distances. The lowest distances, ranging from 0 to 50 kilometres correspond to the network length travelled exclusively by goods within ERs; the 50 to 150 kilometre range captures the distance covered by flows outside the regions, but still within the United States or Canada. For network distances above 150 kilometres, the effect of distance is allows to vary by distance class and trade type.

The spline captures the non-linearity of trade costs, at least for Canada and Canada-United States flows. These flows are captured in more detail by the STF portion of the data than is available for United States flows through the CFS. For the within Canada flows, the effect of distance is initially high, dropping over the 50 to 500 kilometres range, before rising above one in absolute value for distances up to 1,000 kilometres. The effect of distance for all flows is roughly comparable for all three types of flows over the middle distances. But for flows longer than 2,000 kilometres, the cost of distance drops for Canada and United States flows and is actually positive for United States flows. By contrast, the impact of distance for Canada-United States flows stays consistently high for all levels of the spline.

The drop in trade costs at longer distances is most likely due to how the Alchian-Allen effect manifests in models using iceberg transportation costs. Goods that are worth shipping very long distances are typically of much higher value than shorter distance flows, creating the impression that less “ice is melting” over these distances. It does not seem to be the case for Canada-United States flows, however. The source of the discrepancy is between long distance trade costs for intra- versus international flows. This pattern is consistent with higher transport rates over these longer distances resulting from a lack of cabotage rights for Canadian trucking firms. It is, however, beyond the scope of this paper to identify this effect.

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