The Annals of Applied Statistics

Modeling US housing prices by spatial dynamic structural equation models

Pasquale Valentini, Luigi Ippoliti, and Lara Fontanella

Full-text: Open access


This article proposes a spatial dynamic structural equation model for the analysis of housing prices at the State level in the USA. The study contributes to the existing literature by extending the use of dynamic factor models to the econometric analysis of multivariate lattice data. One of the main advantages of our model formulation is that by modeling the spatial variation via spatially structured factor loadings, we entertain the possibility of identifying similarity “regions” that share common time series components. The factor loadings are modeled as conditionally independent multivariate Gaussian Markov Random Fields, while the common components are modeled by latent dynamic factors. The general model is proposed in a state-space formulation where both stationary and nonstationary autoregressive distributed-lag processes for the latent factors are considered. For the latent factors which exhibit a common trend, and hence are cointegrated, an error correction specification of the (vector) autoregressive distributed-lag process is proposed. Full probabilistic inference for the model parameters is facilitated by adapting standard Markov chain Monte Carlo (MCMC) algorithms for dynamic linear models to our model formulation. The fit of the model is discussed for a data set of 48 States for which we model the relationship between housing prices and the macroeconomy, using State level unemployment and per capita personal income.

Article information

Ann. Appl. Stat., Volume 7, Number 2 (2013), 763-798.

First available in Project Euclid: 27 June 2013

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

House prices Bayesian inference dynamic factor models spatio-temporal models cointegration lattice data


Valentini, Pasquale; Ippoliti, Luigi; Fontanella, Lara. Modeling US housing prices by spatial dynamic structural equation models. Ann. Appl. Stat. 7 (2013), no. 2, 763--798. doi:10.1214/12-AOAS613.

Export citation


  • Ahn, S. K. and Reinsel, G. C. (1990). Estimation for partially nonstationary multivariate autoregressive models. J. Amer. Statist. Assoc. 85 813–823.
  • Anselin, L. (1988). Spatial Econometrics: Models and Applications. Kluwer Academic, Dordrecht, The Netherlands.
  • Apergis, N. and Payne, J. E. (2012). Convergence in U.S. housing prices by state: Evidence from the club convergence and clustering procedure. Letters in Spatial and Resource Sciences 5 103–111.
  • Banerjee, S., Carlin, B. P. and Gelfand, A. E. (2004). Hierarchical Modeling and Analysis for Spatial Data. Chapman & Hall/CRC, Boca Raton, FL.
  • Box, G. E. P., Jenkins, G. M. and Reinsel, G. C. (1994). Time Series Analysis: Forecasting and Control, 3rd ed. Prentice Hall Inc., Englewood Cliffs, NJ.
  • Brown, P. J., Vannucci, M. and Fearn, T. (1998). Multivariate Bayesian variable selection and prediction. J. R. Stat. Soc. Ser. B Stat. Methodol. 60 627–641.
  • Cameron, G., Muellbauer, J. and Murphy, A. (2006). Was There a British House Price Bubble? Evidence from a Regional Panel. Mimeo. Oxford Univ. Press, London.
  • Capozza, D. R., Hendershott, P. H., Mack, C. and Mayer, C. J. (2002). Determinants of real house price dynamics. NBER Working Paper 9262.
  • Carter, C. K. and Kohn, R. (1994). On Gibbs sampling for state space models. Biometrika 81 541–553.
  • Case, K. E. and Shiller, R. J. (2003). Is there a bubble in the housing market? Brookings Papers on Economic Activity 2 299–362.
  • Cho, S. (2010). Inference of cointegrated model with exogenous variables. SIRFE Working Paper 10–A04.
  • Clayton, J., Miller, N. and Peng, L. (2010). Price-volume correlation in the housing market: Causality and co-movements. Journal of Real Estate Finance and Economics 40 14–40.
  • Cressie, N. A. C. (1993). Statistics for Spatial Data. Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics. Wiley, New York.
  • Dawid, A. P. (1981). Some matrix-variate distribution theory: Notational considerations and a Bayesian application. Biometrika 68 265–274.
  • Debarsy, N., Ertur, C. and LeSage, J. P. (2012). Interpreting dynamic space–time panel data models. Stat. Methodol. 9 158–171.
  • Di Giacinto, V., Dryden, I., Ippoliti, L. and Romagnoli, L. (2005). Linear smoothing of noisy spatial temporal series. J. Math. Stat. 1 299–311.
  • Durbin, J. and Koopman, S. J. (2001). Time Series Analysis by State Space Methods. Oxford Statistical Science Series 24. Oxford Univ. Press, Oxford.
  • Elhorst, J. P. (2001). Dynamic models in space and time. Geographical Analysis 33 119–140.
  • Engle, R. F., Hendry, D. F. and Richard, J.-F. (1983). Exogeneity. Econometrica 51 277–304.
  • ESRI. (2009). ArcMap 9.2. Environmental Systems Resource Institute, Redlands, California.
  • Frühwirth-Schnatter, S. (1994). Data augmentation and dynamic linear models. J. Time Series Anal. 15 183–202.
  • Gallin, J. (2008). The long run relationship between housing prices and income: Evidence from local housing markets. Real Estate Economics 36 635–658.
  • Gelfand, A. E. and Ghosh, S. K. (1998). Model choice: A minimum posterior predictive loss approach. Biometrika 85 1–11.
  • Gelman, A. (1996). Inference and Monitoring Convergence. In Introducing Markov Chain Monte Carlo.
  • George, E. I., Sun, D. and Ni, S. (2008). Bayesian stochastic search for VAR model restrictions. J. Econometrics 142 553–580.
  • Geweke, J. (1992). Evaluating the accuracy of sampling-based approaches to the calculation of posterior moments. In Bayesian Statistics, 4 (PeñíScola, 1991) (J. Bernardo, J. Berger, A. Dawid and A. Smith, eds.) 169–193. Oxford Univ. Press, New York.
  • Gilks, W. R., Richardson, S. and Spiegelhalter, D. J., eds. (1996). Markov Chain Monte Carlo in Practice. Interdisciplinary Statistics. Chapman & Hall, London.
  • Giussani, B. and Hadjimatheou, G. (1991). Modeling regional housing prices in the United Kingdom. Papers In Regional Science 70 201–219.
  • Gourieroux, C. S. and Monfort, A. (1997). Time Series and Dynamic Models. Cambridge Univ. Press, Cambridge.
  • Holly, S., Pesaran, M. H. and Yamagata, T. (2010). A spatio-temporal model of house prices in the USA. J. Econometrics 158 160–173.
  • Ippoliti, L., Valentini, P. and Gamerman, D. (2012). Space-time modelling of coupled spatio-temporal environmental variables. J. R. Stat. Soc. Ser. C. Appl. Stat. 61 175–200.
  • Jochmann, M., Koop, G., Leon-Gonzalez, R. and Strachan, R. (2013). Stochastic search variable selection in vector error correction models with an application to a model of the UK macroeconomy. J. Appl. Econometrics 28 62–81.
  • Johansen, S. (1988). Statistical analysis of cointegration vectors. J. Econom. Dynam. Control 12 231–254.
  • Jones, G. L., Haran, M., Caffo, B. S. and Neath, R. (2006). Fixed-width output analysis for Markov chain Monte Carlo. J. Amer. Statist. Assoc. 101 1537–1547.
  • Kass, R. E. and Raftery, A. E. (1995). Bayes factors. J. Amer. Statist. Assoc. 90 773–795.
  • Kim, H., Sun, D. and Tsutakawa, R. K. (2001). A bivariate Bayes method for improving the estimates of mortality rates with a twofold conditional autoregressive model. J. Amer. Statist. Assoc. 96 1506–1521.
  • Koop, G., León-González, R. and Strachan, R. W. (2010). Efficient posterior simulation for cointegrated models with priors on the cointegration space. Econometric Rev. 29 224–242.
  • Koop, G. M., Strachan, R. W., Van Dijk, H. and Villani, M. (2006). Bayesian approaches to cointegration. In The Palgrave Handbook of Theoretical Econometrics 871–898. Palgrave Macmillan, Basingstoke, UK.
  • Kuethe, T. and Pede, V. (2011). Regional housing price cycles: A spatio-temporal analysis using US state-level data. Regional Studies 45 563–574.
  • Lopes, H. F., Salazar, E. and Gamerman, D. (2008). Spatial dynamic factor analysis. Bayesian Anal. 3 759–792.
  • Lopes, H. F. and West, M. (2004). Bayesian model assessment in factor analysis. Statist. Sinica 14 41–67.
  • Lütkepohl, H. (2005). New Introduction to Multiple Time Series Analysis. Springer, Berlin.
  • Malpezzi, S. (1999). A simple error correction model of housing prices. Journal of Housing Economics 8 27–62.
  • Mardia, K. V. (1988). Multidimensional multivariate Gaussian Markov random fields with application to image processing. J. Multivariate Anal. 24 265–284.
  • Mardia, K. V., Kent, J. T. and Bibby, J. M. (1979). Multivariate Analysis. Academic Press, London.
  • Meen, G. (1999). Regional house prices and the ripple effect: A new interpretation. Housing Studies 14 733–753.
  • Meen, G. (2001). Modelling Spatial Housing Markets: Theory, Analysis and Policy. Kluwer, Dordrecht, The Netherlands.
  • Moench, E. and Ng, S. (2011). A hierarchical factor analysis of U.S. housing market dynamics. Econom. J. 14 C1–C24.
  • Muellbauer, J. and Murphy, A. (1997). Booms and busts in the UK housing market. Econom. J. 107 1701–1727.
  • Osiewalski, J. and Steel, M. F. J. (1996). A Bayesian analysis of exogeneity in models pooling time-series and cross-sectional data. J. Statist. Plann. Inference 50 187–206.
  • Pesaran, M. H. (2006). Estimation and inference in large heterogeneous panels with a multifactor error structure. Econometrica 74 967–1012.
  • Pfeifer, P. E. and Deutsch, S. J. (1980). A three-stage iterative procedure for space-time modeling. Technometrics 22 35–47.
  • Pfeifer, P. E. and Deutsch, S. J. (1981). Space-time ARMA modeling with contemporaneously correlated innovations. Technometrics 23 401–409.
  • Primiceri, G. E. (2005). Time varying structural vector autoregressions and monetary policy. Rev. Econom. Stud. 72 821–852.
  • Rosenberg, B. (1973). Random coefficients models: The analysis of a cross-section of time series by stochastically convergent parameter regression. Annals of Economic and Social Measurement 60 399–428.
  • Sain, S. R. and Cressie, N. (2007). A spatial model for multivariate lattice data. J. Econometrics 140 226–259.
  • Sain, S. R., Furrer, R. and Cressie, N. (2011). A spatial analysis of multivariate output from regional climate models. Ann. Appl. Stat. 5 150–175.
  • Sims, C. A. and Zha, T. (1999). Error bands for impulse responses. Econometrica 67 1113–1155.
  • Spiegelhalter, D. J., Best, N. G., Carlin, B. P. and van der Linde, A. (2002). Bayesian measures of model complexity and fit. J. R. Stat. Soc. Ser. B Stat. Methodol. 64 583–639.
  • Strickland, C. M., Simpson, D. P., Turner, I. W., Denham, R. and Mengersen, K. L. (2011). Fast bayesian analysis of spatial dynamic factor models for large space time data sets. J. R. Stat. Soc. Ser. C. Appl. Stat. 60 1–16.
  • Sugita, K. (2009). A Monte Carlo comparison of Bayesian testing for cointegration rank. Economics Bulletin 29 2145–2151.
  • van Dijk, B., Franses, P. H., Paap, R. andvan Dijk, D. J. C. (2011). Modeling regional house prices. Applied Economics 43 2097–2110.
  • Vermeulen, W. and Van Ommeren, J. (2009). Compensation of regional unemployment in housing markets. Economica 76 71–88.
  • Wang, F. and Wall, M. M. (2003). Generalized common spatial factor model. Biostatistics 4 569–582.