Heterogeneous spatial dynamic panels with an application to US housing data

ABSTRACT

This paper proposes two models that incorporate both heterogeneity and multiple sources of spatial correlation for dynamic panels. One uses convex combinations of them to form a single weight matrix. The second one includes explicitly different spatial weight matrices to form a higher order model. We use a Bayesian scheme for model estimation by deriving the full conditional distributions of heterogeneous parameters. Our Monte Carlo experiments demonstrate their finite-sample performance relative to a baseline model. In our empirical study we find the importance of including both geographical and non-geographical information in capturing correlations in real house price growth in the United States.

KEYWORDS:

• heterogeneous spatial dynamic panels
• Bayesian
• full conditional distributions
• real house price growth

JEL:

• C13
• C23
• R21
• R31

ACKNOWLEDGEMENTS

The authors are grateful to two anonymous referees and Editor-in-Chief Paul Elhorst for helpful comments. The second author thanks Josh Chan for fruitful discussions. The authors are responsible for all remaining errors.

DISCLOSURE STATEMENT

No potential conflict of interest was reported by the authors.

Notes

1 Another prominent approach to modeling cross-sectional dependence is to assume a multifactor structure, where cross-sectional units are simultaneously affected by a limited number of common factors.

2 See, inter alia, Fingleton (2001), Ertur and Koch (2007) and Parent and LeSage (2012) for economic growth; Baicker (2005) and Han and Lee (2016) for public economics; Lin (2010) and Lin and Weinberg (2014) for social networks; and Liu et al. (2018) for real estate economics.

3 For QML, see Lee (2004), Lee and Yu (2010) and Qu et al. (2017); for GMM, see Lee (2007), Lee and Yu (2014) and Jin and Lee (2019); for IV, see Kelejian and Piras (2014) and Qu et al. (2016); for II, see Kyriacou et al. (2017) and Bao et al. (2020); and for Bayesian MCMC, see Han et al. (2017), Han and Lee (2016) and Parent and LeSage (2012).

4 In our simulations detailed in the third section, the highest (theoretical) degree of similarity, as defined by LeSage and Pace (2014), among the weight matrices is 0, and in our empirical study in the fourth section, it is 0.0533. Additional simulation results in the supplemental data online suggest that model (2) may be subject to identification concerns when we have weight matrices that are highly similar. We thank a referee for bringing the issue of identification to our attention and the editor-in-chief for suggesting the additional simulations.

5 Although each ${\mathit{\gamma }}_i$ contains $q$ elements, there are only $q-1$ parameters that need to be estimated given the constraint $\sum _{s=1}^q{\gamma }_{is}=1$. Similarly, there are $q-1$ parameters in each ${\mathit{\delta }}_i$.

6 The large-$T$ asymptotic result may be restrictive for many empirical researchers when dealing with micro-level data. Intuitively, large-$T$ asymptotics are needed since there are so many parameters arising from unit-level heterogeneity. We thank a referee for pointing this out. Simulation results given in the third section and additional results in the supplemental data online suggest that the performance of the Bayesian estimator is not that sensitive to $N$.

7 When implementing the Gibbs sampler, we actually sample ${\mathit{\eta }}_i$, ${\mathit{\gamma }}_i$ and ${\mathit{\delta }}_i$ together in a single block to increase computational efficiency. This would allow us to evaluate the stationarity condition only once instead of three times for each $i$ within each simulation, although it may take larger number of simulations for the sampler to converge. Moreover, unlike the other sets of parameters, we do not need to update ${\mathit{y}}_{\circ 0}$ for each $i\in \left\{1,2,\dots ,N\right\}$ in each simulation because it does not vary across different units.

8 When simulating data, the values of exogenous regressor for each spatial unit $i$ and time $t$ are independently sampled from a standard normal distribution.

9 We set ${\mathit{\xi }}_i=\mathbf{0}$, $v_i^2=100$ and $a_i=b_i=0$ for each $i\in \left\{1,\cdots ,N\right\}.$

10 It would be less intuitive if we choose a third set $\{{\mathit{W}}_{\mathrm{g}2},{\mathit{W}}_n\}$ since ${\mathit{W}}_n$ is based on the residuals from the baseline model using ${\mathit{W}}_{g1}$. There are 11 MSAs that have no neighbours as defined in the second geographical weight matrix ${\mathit{W}}_{g2}$. For these 11 MSAs, we set the corresponding weights ${\gamma }_{i2}={\delta }_{i2}=0$ in model B and ${\psi }_i^{\left(2\right)}={\varphi }_i^{\left(2\right)}=0$ in model C. Similarly, we did the same for the 15 MSAs that have no neighbours as defined in ${\mathit{W}}_n$ when the second set $\left\{{\mathit{W}}_{g1},{\mathit{W}}_n\right\}$ is used for models B and C.

11 Figures for the heterogeneous estimates in this section are all generated from the R code of Aquaro et al. (2021). The sum ${\psi }_i+{\varphi }_i$ is of particular interest because it represents the net spatial effect from both contemporaneous and temporal components. Furthermore, the category ‘No-Neigh’ in the figure legends consists of the 39 MSAs without any neighbour as defined in ${\mathit{W}}_{g1}$.

12 When model B is a restricted version of model C, it implies that for each $i$, ${\psi }_i^{\left(s\right)}$, $s=1,\cdots ,q$, are of the same sign, and that for each $i$, ${\varphi }_i^{\left(s\right)}$, $s=1,\cdots ,q$, also have the same sign. We may use the post-convergence posterior draws under model C to see what proportion of the draws having ${\psi }_i^{\left(s\right)}$ and ${\varphi }_i^{\left(s\right)}$ satisfying these constraints for all $i$. For our empirical data, regardless of which of the two different sets of weight matrices is used, we get a p–value of 0 (in the sense that 0% of the draws meet these constraints). This is also consistent with the DIC result, which favours model C. As a referee pointed out, the DIC approach is more general and useful for comparing many different aspects of model specifications.

13 Given $q=2$ and the requirement that weights sum up to 1, ${\gamma }_i$ and $1-{\gamma }_i$ are the weights attached to ${\mathit{W}}_{g1}$ and ${\mathit{W}}_n$, respectively, for the $i$-th MSA’s composite term $y_{it}^{\star }$ on the right-hand side of (2). Similarly, ${\delta }_i$ and $1-{\delta }_i$ are those for $y_{it-1}^{\star }$.