A spatial Hausman test R. Kelley Pace a, ⁎ , 1 , James P. LeSage b a LREC Endowed Chair of Real Estate, Department of Finance, E.J. Ourso College of Business Administration, Louisiana State University, Baton Rouge, LA 70803-6308, United States b Fields Endowed Chair in Urban and Regional Economics, McCoy College of Business Administration, Department of Finance and Economics, Texas State University - San Marcos, San Marcos, Texas 78666, United States abstract article info Article history: Received 20 October 2007 Received in revised form 2 September 2008 Accepted 16 September 2008 Available online 23 September 2008 Keywords: Spatial autoregression Specification test Spatial econometrics SAR SEM JEL classification: C11 C13 Often, authors report materially different OLS and spatial error model estimates. However, under the null of correct specification, these estimates should be similar. We propose a spatial Hausman test and conduct a Monte Carlo experiment to examine its performance. © 2008 Elsevier B.V. All rights reserved. 1. Introduction Both ordinary least squares (OLS) and the spatial error model (SEM) have been widely applied to spatial data. Often, authors provide both sets of estimates along with standard errors, allowing a pairwise comparison. This type of comparison reveals cases where OLS and SEM estimates are quite similar (Pace, 1997; Cohen and Coughlin, 2006), other indeterminate cases where various, but not obviously significant, differences exist (Neill et al., 2007; Theebe, 2004), and cases where differences appear to be statistically significant. For example, Brasington (2007), in a study on the willingness to pay for public schools, found OLS and SEM coefficients with different signs on variables representing educational attainment and owner occupied housing. In a study on retail location, Lee and Pace (2005), report an OLS estimate relating store size to sales that was negative and significant, while the SEM estimate was positive and significant. In fact, in Ord's seminal paper on spatial regression models (Ord, 1975), he reports OLS and SEM estimates from a univariate model (with intercept) where the slope coefficient differs by two standard errors. Under the SEM model assumptions, OLS and SEM regression parameter estimates should be unbiased (Anselin, 1988, p. 59). This suggests that significant differences in regression parameter estimates will arise only from misspecification. We formalize this result with a spatial Hausman specification test for significant differences between OLS and SEM estimates. In a Monte Carlo experiment, we show that the spatial Hausman test has the correct size. 2. Spatial Hausman test The linear model where the disturbances are independent identically distributed (iid) represents a simple data generating process that we label the iid DGP, shown in 1. The n observation vector y represents the regressand, the matrix X contains n observations on k exogenous explanatory variables, β is a k by 1 vector of regression parameters, and ε is a n by 1 vector of iid disturbances. y ¼ Xβ þ ɛ: ð1Þ The canonical estimator for the iid DGP is β ˆ =(X′X) -1 X′y, or OLS. The iid error model has been widely used with spatial data samples where the observations represent points or regions located in space. As an alternative, assume the disturbances follow a spatial auto- regressive process, labeled the spatial error DGP in (2), y ¼ Xβ þ I-ρW ð Þ -1 ɛ ð2Þ Economics Letters 101 (2008) 282–284 ⁎ Corresponding author. Tel.: +1 225 578 6256 (OFF); fax: +1 225 578 9065. E-mail addresses: kelley@pace.am (R. Kelley Pace), jlesage@spatial-econometrics.com (J.P. LeSage). 1 The authors would like to thank David Brasington, Dek Terrell, Donald Lacombe and Jennifer Zhu for their valuable comments. In addition, the author would like to acknowledge support from NSF SES-0729259, 0729264 as well as the Louisiana and Texas Sea Grant Programs. 0165-1765/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2008.09.003 Contents lists available at ScienceDirect Economics Letters journal homepage: www.elsevier.com/locate/econbase