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.     

NETWORK LOADING VERSUS EQUILIBRIUM ESTIMATION OF THE STOCHASTIC ROUTE CHOICE MODEL MAXIMUM LIKELIHOOD AND LEAST SQUARES REVISITED*

Daganzo (1977, 1979), Daganzo and Sheffi (1977), Sheffi (1985), and Sheffi and Daganzo (1980) have used one assumption about traveler behavior in developing estimation techniques for the stochastic route-choice problem and another assumption in predicting flows on networks by using the same model. In estimation, they calculate the congested travel costs of the network links from observed flows on the network, and the network is loaded based on these costs. In prediction, they follow their stochastic user-equilibrium assumption by which travelers evaluate costs using the mean of the observed flows (or equilibrium flows). The travel-cost coefficient obtained from the loading method systematically overestimates the true travel-cost coefficient from which the observed flow data (which must be used in loading) is generated. The estimates of the same coefficient, obtained in this paper, by constraining the estimation results to conform to the equilibrium conditions are unbiased, and only marginally less efficient (have larger standard deviations). The average percentage error and inefficiency of the link flow predictions based on the loading method increases as the level of congestion on the network rises. In contrast, the average percentage error of link flow predictions based on the equilibrium estimation method declines and their efficiency remains very high as the level of congestion on the network rises.     

SPATIAL CONVERGENCE OF REGIONS REVISITED: A SPATIAL MAXIMUM LIKELIHOOD PANEL APPROACH*

ABSTRACT This paper argues that one should account for the endogeneity of important explanatory variables and the initial differences in technological efficiency when analyzing spatial income convergence among regions. In addition, the approach of Wooldridge (2005), who proposes a convenient solution to the initial condition problem in dynamic panels, proves to be fruitful. In a panel of 211 European regions observed from 1980 to 2005, the estimated speed of convergence is substantially higher, on average, than the legendary 2 percent found in many cross‐section studies. Moreover, it exhibits pronounced variation across regions due to factor mobility and knowledge spillovers.     

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.     

DERIVATIVES OF FLOWS IN A DOUBLY CONSTRAINED GRAVITY MODEL

For many practical applications it is important to know how the flows in a doubly constrained gravity model react to slight variations in the predetermined marginal totals. The first-order approximation of these variations is a linear function on the set of feasible variations of marginal totals, i.e., the set of variations not violating the consistency constraint of the model. Several methods to find a matrix describing this linear function are developed and compared with former contributions to this issue. Finally, applicability of the methods to sensitivity and error propagation analysis is demonstrated.     

基于引力模型的关中城市间联系测度分析

在新一轮西部大开发进程中,为提升关中城市群整体实力,需要对关中城市间经济联系度进行科学测量,找出其中存在的问题、原因与解决对策。本文借鉴国内外学者分析城市间联系测度的方法,以标准引力模型为基础,在对关中城市的中心等级进行科学划分和可达性分析的基础上,测量了关中城市间的联系程度,找出关中城市群发展的现存问题,并提出从行政区划适当调整、发展规划与组织协调、财政体制完善、壮大县域经济发展等方面来促进关中城市群发展。     

基于引力模型的城际交通网络布局规划方法研究

交通网络的发展是城镇体系空间完善的重要条件。根据城镇体系空间特点和经济社会布局,制定与之相适应的城际交通网络布局规划,是路网规划的重要组成部分。本文在分析城际交通网络布局与经济及产业布局、城镇体系布局的关系基础上,结合交通网络的特征,提出了基于引力模型和拓扑结构的城际交通网络布局规划方法,并结合湖北省城际高速公路网布局规划开展了实证研究,对该省2015年城际高速公路网的布局模式进行了预测,并就高速公路网络的发展特征与城际经济社会空间联系的关系进行了分析。     

GRAVITY MODEL APPLICATIONS IN HEALTH PLANNING: ANALYSIS OF AN URBAN HOSPITAL MARKET*

ABSTRACT. In this paper a gravity model is formulated to model a hospital patient flow system. Both spatial and nonspatial measures of separation are used to estimate the model. Travel time was found to provide improved fits over linear distance in modeling patient flow behavior. Transformations of travel time (square root and log) were found to improve model performance further. Model diagnostics suggested additional spatial separation measurw, separation measure parameters demonstrated stability over time. The model is used to forecast the effects of health care financing reform and hospital closure on patient flows in an urban hospital market. Universal health coverage was found to dramatically improve access to hospital services for lower income patient populations. Hospitals at risk for closure were forecast to receive increased patient flows. Medical school hospitals are expected to lose patients after implementation of health reform, with important implications for graduate medical education. Hospital closures were shown to have negative effects on access to services in poor neighborhoods, but relatively little effect on access in the system as a whole.