ON THE SENSITIVITY OF GRAVITY MODEL FORECASTS*

In this paper we present a procedure for estimating small changes in gravity model forecasts in response to small changes in input values (trip end totals and/or parameter values). The procedure can be used to derive covariance matrices from which confidence intervals may be obtained and which may be used for tests of hypotheses. While this type of problem has been addressed by others, our approach is the only one that is conveniently applicable to the doubly constrained model and can accommodate the large numbers of origin and destination zones one typically encounters.     

THE GRAVITY MODEL AND HIERARCHICAL SPATIAL SYSTEMS: THEORETICAL AND EMPIRICAL OBSERVATIONS

The gravity model in its basic forms postulates that the volume of interaction between two places will be positively related to the populations (masses) of the two places and inversely related to their distance apart. Although it is not strictly speaking a hierarchical model, it is often used to explain flows among places that constitute a hierarchical spatial system. This paper is specifically concerned with the effect of hierarchical spatial structure on the performance of the gravity model. Accordingly, I will examine first the effect of hierarchical structure on the volume of interaction between places and second the degree to which the gravity model incorporates these hierarchical effects. Third, suggestions will be made for the improvement of the gravity model's performance by an explicit incorporation of a hierarchical element. The performance of the gravity model will be examined for both one-way and two-way flows between places.     

COMPUTATION OF MAXIMUM LIKELIHOOD ESTIMATES OF GRAVITY MODEL PARAMETERS*

ABSTRACT. This paper presents the derivations of several new algorithms for the computation of maximum likelihood estimates of the parameters of a very general form of the gravity model. The algorithms are then compared with previously available algorithms including GLIM and that given in Sen (1986). One of the new algorithms emerges as far superior in just about every way to its competitors. In particular, it is usually much more than an order of magnitude faster than the GLIM procedure and that given in Sen (1986). It is also not substantially affected by pitfalls such as multicollinearity and (unlike the GLIM procedure) is capable of comfortably handling large O-D matrices.     

PROPERTIES OF MAXIMUM LIKELIHOOD ESTIMATES OF GRAVITY MODEL PARAMETERS*

ABSTRACT. We give a very simple formula for obtaining covariance matrices of gravity model parameter estimates. We investigate bias and robustness of parameter estimates, as well as the convergence of the procedure given in Sen (1986) for obtaining these estimates.     

GRAVITY AND SPATIAL STRUCTURE: THE CASE OF INTERSTATE MIGRATION IN MEXICO*

ABSTRACT The estimation of gravity models of internal (aggregate) place‐to‐place migration is plagued with endogeneity (omitted‐variable) biases if the unobserved effects of spatial structure are not accounted for. To address this econometric problem, this paper presents a more general specification of the gravity model, which allows for (bilateral) parameter heterogeneity across individual migration paths—along with (unilateral) origin‐ and destination‐specific effects. The resultant “three‐way fixed‐effects” (3FE) model is applied for an analysis of interstate migration in Mexico based on cross‐sectional data. To overcome parameter‐dimensionality problems (due to limited or incomplete information), the 3FE model is estimated using the Generalized Maximum Entropy (GME) estimator. The empirical implications of this new modeling strategy are illustrated by contrasting the 3FE‐GME estimates with those for the traditional and two‐way fixed‐effects (2FE) models. The former are far more plausible and intuitively interpretable than their traditional and 2FE counterparts, with parameter estimates changing in expected directions. The (average) effect of the migrant stock is markedly smaller than usually estimated, providing a more realistic measure of network‐induced migration. Migration outflows from centrally located origins have significantly steeper distance decay. Path‐specific distance effects exhibit directional asymmetries and spatial similarities.     

UNDERSTANDING SPATIAL INERTIA: CENTER OF GRAVITY,POPULATION DENSITIES,THE WEBER PROBLEM,AND GRAVITY POTENTIAL*

ABSTRACT. In this paper we attempt to clarify the theoretical links between the concepts of “center of gravity” and “point of maximum population density” which describe the present, and the concepts of “minimum of the comprehensive Weber problem” and “maximum comprehensive gravity potential” which guide the future. Critical values of the characteristic parameters of the relevant functions are estimated. Implications for the understanding of spatial inertia are discussed.     

THE ECONOMICS OF AIR TRAVEL GRAVITY MODELS*

Starting with the aggregate demand model of economics, a model of demand for intercity air travel is developed which contains the gravity model as a less general submodel. The more general model is referred to as the alternative opportunities model since it takes account of alternative destinations open to travelers, not just origin and destination as does the gravity model. The demand model approach has the virtue of providing a theoretical basis for understanding and analyzing the gravity model. The gains from treating alternative locations and demand motivation variables are a substantial increase in explanatory power over that yielded by the gravity model, the identification of statistically significant determinants of air travel, and better measurement of the coefficients of population and distance by taking account of these other variables and somewhat better forecasts. A shortcoming of procedures used here is aggregation of air trips with different purposes and thus lack of clear specification of the size of effects of different variables on different types of travel. Overcoming this difficulty must await origin-destination data listed by trip purpose.     

REGIONAL GROWTH IN THE SOVIET ECONOMY A MODEL AND ANALYSIS*

A multiregional econometric model of the Soviet economy is presented. It consists of 1,391 equations, with time-series information for 1965-1980. In conformity with the number of the country's republics, 15 input-output tables are incorporated. The discussion includes the methodological framework, the structure and equations, theoretical justification for the interaction between the econometric and input-output components, data and estimation, and forecasting results.