## Bayesian Analysis

### Spatial Panel Data Model with Error Dependence: A Bayesian Separable Covariance Approach

#### Abstract

A hierarchical Bayesian model for spatial panel data is proposed. The idea behind the proposed method is to analyze spatially dependent panel data by means of a separable covariance matrix. Let us indicate the observations as $y_{it}$, in $i=1,\ldots,N$ regions and at $t=1,\ldots,T$ times, and suppose the covariance matrix of $\mathbf{y}$, given a set of regressors, is written as a Kronecker product of a purely spatial and a purely temporal covariance. On the one hand, the structure of separable covariances dramatically reduces the number of parameters, while on the other hand, the lack of a structured pattern for spatial and temporal covariances permits capturing possible unknown dependencies (both in time and space). The use of the Bayesian approach allows one to overcome some of the difficulties of the classical (MLE or GMM based) approach. We present two illustrative examples: the estimation of cigarette price elasticity and of the determinants of the house price in 120 municipalities in the Province of Rome.

#### Article information

Source
Bayesian Anal., Volume 11, Number 4 (2016), 1035-1069.

Dates
First available in Project Euclid: 29 October 2015

Permanent link to this document
https://projecteuclid.org/euclid.ba/1446124569

Digital Object Identifier
doi:10.1214/15-BA979

Mathematical Reviews number (MathSciNet)
MR3545473

Zentralblatt MATH identifier
1359.62189

#### Citation

Leorato, Samantha; Mezzetti, Maura. Spatial Panel Data Model with Error Dependence: A Bayesian Separable Covariance Approach. Bayesian Anal. 11 (2016), no. 4, 1035--1069. doi:10.1214/15-BA979. https://projecteuclid.org/euclid.ba/1446124569

#### References

• Anselin, L., Gallo, J. L., and Jayet, H. (2008). “Spatial panel econometrics.” In: Mátyás, L. and Sevestre, P. (eds.), The Econometrics of Panel Data, volume 46 of Advanced Studies in Theoretical and Applied Econometrics, 625–660. Springer, Berlin, Heidelberg.
• Aston, J. A. and Gunn, R. N. (2005). “Statistical estimation with Kronecker products in positron emission tomography.” Linear Algebra and Its Applications, 398: 25–36. Special Issue on Matrices and Mathematical Biology.
• Baltagi, B. H. (2008). Econometric Analysis of Panel Data. 1. John Wiley & Sons.
• Baltagi, B. H. and Levin, D. (1986). “Estimating dynamic demand for cigarettes using panel data: The effects of bootlegging, taxation and advertising reconsidered.” The Review of Economics and Statistics, 68(1): 148–155.
• Baltagi, B. H. and Levin, D. (1992). “Cigarette taxation: Raising revenues and reducing consumption.” Structural Change and Economic Dynamics, 3(2): 321–335.
• Baltagi, B. H., Song, S. H., Jung, B. C., and Koh, W. (2007). “Testing for serial correlation, spatial autocorrelation and random effects using panel data.” Journal of Econometrics, 140(1): 5–51. Analysis of Spatially Dependent Data.
• Baltagi, B. H., Song, S. H., and Koh, W. (2003). “Testing panel data regression models with spatial error correlation.” Journal of Econometrics, 117(1): 123–150.
• Brown, P., Le, N., and Zideck, J. (1994). “Inference for a covariance matrix.” In: Lindley, D., Smith, A., and Freeman, P. (eds.), Aspects of Uncertainty: A Tribute to D. V. Lindley, Wiley Series in Probability and Mathematical Statistics, 77–92. Wiley.
• Caliman, T. and di Bella, E. (2011). “Spatial autoregressive models for house price dynamics in Italy.” Economics Bulletin, 31(2): 1837–1855.
• Capozza, D. R., Hendershott, P. H., Mack, C., and Mayer, C. J. (2002). “Determinants of real house price dynamics.” NBER Working Papers 9262, National Bureau of Economic Research, Inc.
• Chib, S. (1995). “Marginal likelihood from the Gibbs output.” Journal of the American Statistical Association, 90(432): 1313–1321.
• Chiu, T. Y., Leonard, T., and Tsui, K.-W. (1996). “The matrix-logarithmic covariance model.” Journal of the American Statistical Association, 91(433): 198–210.
• Chudik, A., Pesaran, H., and Tosetti, E. (2009). “Weak and strong cross section dependence and estimation of large panels.” Working Paper Series, European Central Bank 1100, European Central Bank.
• Cressie, N. and Huang, H.-C. (1999). “Classes of nonseparable, spatio-temporal stationary covariance functions.” Journal of the American Statistical Association, 94(448): 1330–1340.
• Dette, H. (2002). “Strong approximation of eigenvalues of large dimensional Wishart matrices by roots of generalized Laguerre polynomials.” Journal of Approximation Theory, 118(2): 290–304.
• Elhorst, P. J. (2005). “Unconditional maximum likelihood estimation of linear and log-linear dynamic models for spatial panels.” Geographical Analysis, 37(1): 62–83.
• Elhorst, P. J. (2010). “Spatial panel data models.” In: Fischer, M. M. and Getis, A. (eds.), Handbook of Applied Spatial Analysis, 377–407. Springer, Berlin, Heidelberg.
• Gelman, A. (2006). “Prior distributions for variance parameters in hierarchical models (comment on article by Browne and Draper).” Bayesian Analysis, 1(3): 515–534.
• Genton, M. G. (2007). “Separable approximations of space-time covariance matrices.” Environmetrics, 18(7): 681–695.
• Gilks, W. and Wild, P. (1992). “Adaptive rejection sampling for Gibbs sampling.” Applied Statistics, 41(2): 337–348.
• Gneiting, T., Genton, M. G., and Guttorp, P. (2007). “Geostatistical space–time models, stationarity, separability and full symmetry.” In: Finkenstädt, B., Isham, V., and Held, L. (eds.), Monographs in Statistics and Applied Probability. Chapman & Hall CRC Press.
• Kapoor, M., Kelejian, H. H., and Prucha, I. R. (2007). “Panel data models with spatially correlated error components.” Journal of Econometrics, 140(1): 97–130.
• Kelejian, H. H. and Prucha, I. R. (2010). “Specification and estimation of spatial autoregressive models with autoregressive and heteroskedastic disturbances.” Journal of Econometrics, 157: 53–67.
• Kyriakidis, P. and Journel, A. (1999). “Geostatistical space–time models: A review.” Mathematical Geology, 31(6): 651–684.
• Lee, L.-F. and Yu, J. (2010). “Some recent developments in spatial panel data models.” Regional Science and Urban Economics, 40(5): 255–271.
• Lee, L.-F. and Yu, J. (2012). “Spatial panels: Random components versus fixed effects.” International Economic Review, 53(4): 1369–1412.
• Leonard, T. and Hsu, J. S. J. (1992). “Bayesian inference for a covariance matrix.” The Annals of Statistics, 20(4): 1669–1696.
• LeSage, J. P. and Pace, R. K. (2007). “A matrix exponential spatial specification.” Journal of Econometrics, 140(1): 190–214.
• Lindley, D. (1972). Bayesian Statistics, A Review. CBMS-NSF Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics.
• Mitchell, M. W., Genton, M. G., and Gumpertz, M. L. (2006). “A likelihood ratio test for separability of covariances.” Journal of Multivariate Analysis, 97(5): 1025–1043.
• Naik, D. and Rao, S. (2001). “Analysis of multivariate repeated measures data with a Kronecker product structured covariance matrix.” Journal of Applied Statistics, 28(1): 91–105.
• Pesaran, M. H. (2004). “Estimation and Inference in Large Heterogeneous Panels with a Multifactor Error Structure.” CESifo Working Paper Series 1331, CESifo Group Munich.
• Pesaran, M. H. and Tosetti, E. (2011). “Large panels with common factors and spatial correlation.” Journal of Econometrics, 161(2): 182–202.
• Rodrigues, E., Assunçao, R., and Dey, D. K. (2014). “A closer look at the spatial exponential matrix specification.” Journal of Spatial Statistics, 9: 109–121.
• Rowe, D. B. (2002a). “Jointly distributed mean and mixing coefficients for Bayesian source separation using MCMC and ICM.” Monte Carlo Methods and Applications, 8(4): 395–403.
• Rowe, D. B. (2002b). Multivariate Bayesian Statistics: Models for Source Separation and Signal Unmixing. Taylor & Francis.
• Sarafidis, V. and Wansbeek, T. (2012). “Cross-sectional dependence in panel data analysis.” Econometric Reviews, Taylor & Francis Journals, 31(5): 483–531.
• Shults, J. (2000). “Modeling the correlation structure of data that have multiple levels of association.” Communications in Statistics – Theory and Methods, 29(5–6): 1005–1015.
• Srivastava, M., von Rosen, T., and von Rosen, D. (2008). “Models with a Kronecker product covariance structure: Estimation and testing.” Mathematical Methods of Statistics, 17(4): 357–370.
• Wall, M. M. (2004). “A close look at the spatial structure implied by the CAR and SAR models.” Journal of Statistical Planning and Inference, 121: 311–324.
• Zellner, A. (1962). “An efficient method of estimating seemingly unrelated regressions and tests for aggregation bias.” Journal of the American Statistical Association, 57(298): 348–368.
• Zheng, Y., Zhu, J., and Li, D. (2008). “Analyzing spatial panel data of cigarette demand: A Bayesian hierarchical modeling approach.” Journal of Data Science, 6: 467–489.