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

Abstract

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.

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

Given a stable first-order VAR of dimension n:

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

(16)

with:

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

(17)

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

(18)

and:

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

(19)

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:

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

(20)

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

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

(21)

The contemporaneous variance–covariance matrix is obtained when $h=0$:

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

(22)

and by the Yule–Walker equations, when $h>0$:

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

(23)

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

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

(24)

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

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

(25)

Using the vec operator this can be expressed as:

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

(26)

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:

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

(27)

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:

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

(28)

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

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

(29)

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.