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The properties of tests for spatial effects in discrete Markov chain models of regional income distribution dynamics

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Discrete Markov chain models (DMCs) have been widely applied to the study of regional income distribution dynamics and convergence. This popularity reflects the rich body of DMC theory on the one hand and the ability of this framework to provide insights on the internal and external properties of regional income distribution dynamics on the other. In this paper we examine the properties of tests for spatial effects in DMC models of regional distribution dynamics. We do so through a series of Monte Carlo simulations designed to examine the size, power and robustness of tests for spatial heterogeneity and spatial dependence in transitional dynamics. This requires that we specify a data generating process for not only the null, but also alternatives when spatial heterogeneity or spatial dependence is present in the transitional dynamics. We are not aware of any work which has examined these types of data generating processes in the spatial distribution dynamics literature. Results indicate that tests for spatial heterogeneity and spatial dependence display good power for the presence of spatial effects. However, tests for spatial heterogeneity are not robust to the presence of strong spatial dependence, while tests for spatial dependence are sensitive to the spatial configuration of heterogeneity. When the spatial configuration can be considered random, dependence tests are robust to the dynamic spatial heterogeneity, but not so to the process mean heterogeneity when the difference in process means is large relative to the variance of the time series.

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  1. It should be noted that although the majority of attention on spatial effects in discrete distribution dynamics has been at the regional (i.e., subnational) level, spatial dependence and spatial heterogeneity are likely to hold implications for international studies (e.g., Quah 1996; Jones 1997).

  2. We thank an anonymous referee for suggesting this possibility.

  3. Since all the results suggest that they have very similar properties, we will only present and discuss results concerning \(\chi ^2\) tests. Full results are available from the authors.

  4. In this paper we always adopt the same discretization strategy on the spatial lags as that on regional time series.


  • Aghion P, Howitt P (1997) Endogenous growth theory. MIT Press, Cambridge

    Google Scholar 

  • Anselin L (1990) Some robust approaches to testing and estimation in spatial econometrics. Reg Sci Urban Econ 20(2):141–163

    Article  Google Scholar 

  • Anselin L, Rey S (1991) Properties of tests for spatial dependence in linear regression models. Geograph Anal 23(2):112–131

    Article  Google Scholar 

  • Barro RJ, Sala-i Martin X (1991) Convergence across states and regions. Brook Pap Econ Act 1:107–182

    Article  Google Scholar 

  • Bickenbach F, Bode E (2003) Evaluating the Markov property in studies of economic convergence. Int Reg Sci Rev 26(3):363–392

    Article  Google Scholar 

  • Bosker M, Krugell W (2008) Regional income evolution in South Africa after apartheid. J Reg Sci 48(3):493–523

    Article  Google Scholar 

  • Fingleton B (1997) Specification and testing of Markov chain models: an application to convergence in the European Union. Oxf Bull Econ Stat 59(3):385–403

    Article  Google Scholar 

  • Floden M (2008) A note on the accuracy of Markov-chain approximations to highly persistent AR (1) processes. Econ Lett 99(3):516–520

    Article  Google Scholar 

  • Galindev R, Lkhagvasuren D (2010) Discretization of highly persistent correlated AR(1) shocks. J Econ Dyn Control 34(7):1260–1276

    Article  Google Scholar 

  • Hammond GW (2004) Metropolitan/non-metropolitan divergence: a spatial Markov chain approach. Papers Reg Sci 83(3):543–563

    Article  Google Scholar 

  • Jones CI (1997) On the evolution of the world income distribution. J Econ Perspect 11(3):19–36

    Article  Google Scholar 

  • Kopecky KA, Suen RM (2010) Finite state Markov-chain approximations to highly persistent processes. Rev Econ Dyn 13(3):701–714

    Article  Google Scholar 

  • Kullback S, Kupperman M, Ku HH (1962) Tests for contingency tables and Markov chains. Technometrics 4(4):573–608

    Google Scholar 

  • Le Gallo J (2004) Space-time analysis of GDP disparities among European regions: a Markov chains approach. Int Reg Sci Rev 27(2):138–163

    Article  Google Scholar 

  • Le Gallo J, Chasco C (2008) Spatial analysis of urban growth in Spain, 1900–2001. Empir Econ 34(1):59–80

    Article  Google Scholar 

  • LeSage JP, Cashell BA (2015) A comparison of vector autoregressive forecasting performance: spatial versus non-spatial Bayesian priors. Ann Reg Sci 54(2):533–560

    Article  Google Scholar 

  • Liao FH, Wei YD (2012) Dynamics, space, and regional inequality in provincial China: a case study of Guangdong province. Appl Geogr 35(1–2):71–83

    Article  Google Scholar 

  • López-Bazo E, Vayá E, Mora AJ, Suriñach J (1999) Regional economic dynamics and convergence in the European Union. Ann Reg Sci 33(3):343–370

    Article  Google Scholar 

  • Lütkepohl H (2005) New introduction to multiple time series analysis. Springer, Berlin

    Book  Google Scholar 

  • Monasterio LM (2010) Brazilian spatial dynamics in the long term (1872–2000): “path dependency” or “reversal of fortune”? J Geogr Syst 12(1):51–67

    Article  Google Scholar 

  • Quah D (1993a) Empirical cross-section dynamics in economic growth. Eur Econ Rev 37(2):426–434

    Article  Google Scholar 

  • Quah D (1993b) Galton’s fallacy and tests of the convergence hypothesis. Scand J Econ 95(4):427–443

    Article  Google Scholar 

  • Quah DT (1996) Empirics for economic growth and convergence. Eur Econ Rev 40(6):1353–1375

    Article  Google Scholar 

  • Rey SJ (2001) Spatial empirics for economic growth and convergence. Geograph Anal 33(3):195–214

    Article  Google Scholar 

  • Rey SJ (2015) Discrete regional distribution dynamics revisited. J Reg Urban Econ 1/2:83–103

  • Rey SJ, Gutiérrez MLS (2015) Comparative spatial inequality dynamics: the case of Mexico and the United States. Appl Geogr 61(July):70–80

    Article  Google Scholar 

  • Rey SJ, Le Gallo J (2009) Spatial analysis of economic convergence. In: Mills TC, Patterson K (eds) Palgrave handbook of econometrics, vol 2., Applied econometrics Palgrave Macmillan UK, London, pp 1251–1290

    Chapter  Google Scholar 

  • Schettini D, Azzoni CR, Paez A (2011) Neighborhood and efficiency in manufacturing in Brazilian regions: a spatial Markov chain analysis. Int Reg Sci Rev 34(4):397–418

    Article  Google Scholar 

  • Tauchen G (1986) Finite state Markov-chain approximations to univariate and vector autoregressions. Econ Lett 20(2):177–181

    Article  Google Scholar 

  • Tauchen G, Hussey R (1991) Quadrature-based methods for obtaining approximate solutions to nonlinear asset pricing models. Econometrica 59(2):371–396

    Article  Google Scholar 

  • Terry SJ, Knotek ES (2011) Markov-chain approximations of vector autoregressions: application of general multivariate-normal integration techniques. Econ Lett 110(1):4–6

    Article  Google Scholar 

  • Wolf LJ, Rey SJ (2015) On the lumpability of regional income convergence. Letters in spatial and resource sciences, pp 1–11

Download references


This research was supported in part by National Science Foundation Grant SES-1421935.

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Correspondence to Sergio J. Rey.

Appendix: Contemporaneous cross-correlation in a VAR with temporally lagged spatial spillovers

Appendix: Contemporaneous cross-correlation in a VAR with temporally lagged spatial spillovers

Given a stable first-order VAR of dimension n:

$$\begin{aligned} Y_t = v + A Y_{t-1} + \epsilon _t \end{aligned}$$


$$\begin{aligned} E[\epsilon _t, \epsilon _t']= & {} \Sigma _{\epsilon }, \end{aligned}$$
$$\begin{aligned} E[\epsilon _t]= & {} 0, \end{aligned}$$


$$\begin{aligned} A = {\hat{\alpha }} + \hat{\rho }W, \end{aligned}$$

with \({\hat{\alpha }}\) and \(\hat{\rho }\) diagonal matrices of order n, the variance–covariance matrix at lag h, \(\Gamma (h)\) can be derived by following the approach in Lütkepohl (2005) by first re-expressing the VAR in mean adjusted form:

$$\begin{aligned} Y_t - \mu = A(Y_{t-1}-\mu ) + \epsilon _t \end{aligned}$$

where \(\mu = E[Y_t] = (I - A)^{-1}\nu \), post-multiplying by \((y_{t-h}-\mu )'\) and taking expectations:

$$\begin{aligned} E[(y_t - \mu )(y_{t-h} - \mu )'] = A E[(y_{t-1} - \mu )(y_{t-h} - \mu )'] + E[\epsilon _t(y_{t-h} - \mu )'] . \end{aligned}$$

The contemporaneous variance–covariance matrix is obtained when \(h=0\):

$$\begin{aligned} \Gamma (0) = A \Gamma (1)' + \Sigma _{\epsilon } \end{aligned}$$

and by the Yule–Walker equations, when \(h>0\):

$$\begin{aligned} \Gamma (h) = A \Gamma (h-1). \end{aligned}$$

For \(h=1\), Eq. (23) becomes:

$$\begin{aligned} \Gamma (1) = A \Gamma (0). \end{aligned}$$

and substituting for \(\Gamma (1)\) in Eq. (22) gives:

$$\begin{aligned} \Gamma (0) = A \Gamma (0)'A' + \Sigma _{\epsilon }. \end{aligned}$$

Using the vec operator this can be expressed as:

$$\begin{aligned} \hbox {vec}(\Gamma (0)) = (I_{n^2} - A \otimes A)^{-1} \hbox {vec}(\Sigma _{\epsilon }). \end{aligned}$$

As an example, consider the \(n=3\) VAR(1) system. Setting \(\alpha _1=\alpha _2=\alpha _3=0.5\), \(\rho _1=\rho _2=\rho _3=0.4\) and noting the spatial relations are such that regions 1 and 2 are neighbors, as are regions 2 and 3. With a simple row-standardized weights matrix we have:

$$\begin{aligned} A = \left[ \begin{array}{ccc} 0.50 &{} 0.40&{} 0.00 \\ 0.20 &{} 0.50&{} 0.20 \\ 0.00 &{} 0.40&{} 0.50 \end{array} \right]. \end{aligned}$$

Further assume the innovations are independent over time and space so that \(\Sigma _{\epsilon } = 0.5I_{n}\), and using Eq. (25), the contemporaneous variance–covariance matrix for \(Y_t\) is:

$$\begin{aligned} \Gamma (0) = \left[ \begin{array}{ccc} 1.37 &{} 0.80 &{} 0.71 \\ 0.80 &{} 1.31&{} 0.80 \\ 0.71 &{} 0.80&{} 1.37 \end{array} \right] . \end{aligned}$$

By contrast, if \(\rho =0.0\) and all other parameters are the same, the variance–covariance matrix takes a very different form:

$$\begin{aligned} \Gamma (0) = \left[ \begin{array}{ccc} 0.67 &{} 0.00 &{} 0.00 \\ 0.00 &{} 0.67&{} 0.00 \\ 0.00 &{} 0.00&{} 0.67 \end{array} \right] . \end{aligned}$$

Not only are the regional covariances all zero, but the process variances (\(V(Y_t)\)) become spatially homoscedastic. In contrast, when there are spillovers, the contemporaneous covariances (Eq. 28) between all pairs are nonzero and the process variances are spatially heteroscedastic even though the original innovations were spatially homoscedastic.

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Rey, S.J., Kang, W. & Wolf, L. The properties of tests for spatial effects in discrete Markov chain models of regional income distribution dynamics. J Geogr Syst 18, 377–398 (2016).

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