Estimating Trade Flows Between U.S. Counties

Nick Mata

In the absence of survey data, trade flows between regions are often estimated by a gravity equation. One form of such an equation specifies that for a given industry, trade flows between two regions is determined by the interaction of the industry’s output in the region of origin and of demand in the region of destination. The intensity of interaction is measured by the distance between the two regions and a distance decay parameter beta. Thus, for any given industry,

Tij = Ai Bj Qi Dj / dijbeta (1)

where Tij is the trade flow between the region of origin i and the region of destination j

Qi is the output in the region of origin i

Dj is the demand in the region of destination j

dij is the distance between region i and region j estimated from longitude and latitude data

beta is the distance decay parameter to be estimated

Ai, Bj are balancing factors to be estimated

Ai = (ΣBjDjdij-beta)-1 (2)

Bj = (ΣAiQidij-beta)-1 (3)

The Ai and Bj ensure the following constraints are satisfied,

Qi = ΣTij

j=1

Dj = ΣTij i=1

Methodology

The parameter beta and factors A and B are estimated simultaneously using an iterative procedure where a value of beta is found at the point where the value of the objective function is a minimum. A and B are evaluated at that optimum beta using equations (2) and (3) above.

The objective function is evaluated as the sum of squares of differences between actual changes in output in the region of origin and the estimated changes of that output. The estimated change of output is determined by the change in demand in the demanding regions modified by the distance and the distance decay parameter beta.

This iterative procedure does not require the use of actual trade flows to estimate beta and A and B.

The data set used is a panel data set of output Q and demand D for 3086 counties and years 1990 thru 2007. The Qs and Ds are normalized to the U.S. Qs and Ds. The Qs are adjusted for exports to get domestic output, and the Ds are adjusted for imports to get local demand that is satisfied by local supply.

With a starting value of beta, the optimizing routine is iterated until the beta converges to a value within its lower and upper bounds at the point that the value of the objective function is a minimum.

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