The Viability of Global Optimization for Parameter Estimation in Spatial Econometrics Models

This paper addresses parameter estimation of spatial regression models incorporating spatial lag. These models are very important in spatial econometrics, where spatial interaction and structure are introduced into regression analysis. Because of spatial interactions, observations are not truly independent, and traditional regression techniques fail. Estimation techniques include maximum likelihood estimation, ordinary least squares, and the method of moments. However, parameters of spatial lag models are difficult to estimate due to the simultaneity bias (Ord, 1975). These estimation problems are generally intractable by standard numerical methods, and, consequently, robust and efficient optimization techniques are needed. In the case of simple general spatial regressive models (GSRMs), standard local optimization methods, such as Newton-Raphson iteration (as suggested by Ord) converge to high-quality solutions. Unfortunately, a good initial guess of the parameters is required for these local methods to succeed. In more complex autoregressive spatial models, an analytic expression for good initial guesses is not available, and, consequently, local methods generally fail. In this paper, global optimization (specifically, particle swarm optimization, or PSO) is used to estimate parameters of spatial autoregressive models. PSO is an iterative, stochastic population-based technique that is increasingly used in a variety of fields to solve complex continuous- and discrete-valued problems. In contrast to genetic algorithms and evolutionary strategies, PSO exploits cooperative and social behavior among members of a population of agents, or particles, which represent a point in the search space. This paper first motivates the need for global methods by demonstrating that GSRM parameters can be estimated with PSO even without a good initial guess, while the local Newton-Raphson and Nelder-Mead approaches have a greater failure rate. Next, PSO was tested with an autoregressive spatial model, for which no analytic initial guess can be computed, and for which no analytic parameter estimation method is known. Simulated data were generated to provide ground truth values to assess the viability of PSO. The global PSO method was found to successfully estimate the parameters using two different MLE approximation techniques for trials with 10, 20, and 40 samples (R2 > 0.867 for all trials). These results indicate that global optimization is a viable approach to estimating the parameters of spatial autoregressive models, and suggest that future directions should focus on more advanced global techniques, such as branch-and-bound, dividing rectangles, and differential evolution, which may further improve parameter estimation in spatial econometrics applications.