Abstract
This study explores dynamic relationships among Chinese housing prices for the years 2010–2019. With monthly data from 99 major cities in China, we use the vector error correction model and directed acyclic graph to characterize contemporaneous causality among housing prices from different tiers of cities. The PC algorithm identifies the causal pattern and the LiNGAM algorithm further identifies the causal path, from which we perform innovation accounting analysis. Complex housing price dynamics are found in the price adjustment process following price shocks, which is not only dominated by the top tiers of cities. This suggests that policies on housing prices in the long run might need to be planned from a national perspective.
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Notes
The evidence for T6 is a little weaker as compared to other series as its Jarque–Bera p value is slightly above 5%.
In our case, \(p=6\) and \(X_{t}=\left( X_{1, t} X_{2, t} X_{3, t} X_{4, t} X_{5, t} X_{6, t}\right) ^{\prime }\), where \(X_{1, t}\), \(X_{2, t}\), \(X_{3, t}\), \(X_{4, t}\), \(X_{5, t}\), and \(X_{6, t}\) correspond to average housing price series of tier-1 (T1), tier-2 (T2), tier-3 (T3), tier-4 (T4), tier-5 (T5), and tier-6 (T6) cities, respectively.
Directed graph theory indicates that the off-diagonal elements of the scaled inverse of any correlation matrix are the negatives of the partial correlation coefficients between corresponding pairs of variables, given the remaining variables in the matrix (Whittaker 2009). In practice, Fisher’s z is used to test whether conditional correlations are significantly different from zero (Bessler and Akleman 1998; Awokuse and Bessler 2003; Yang 2003; Haigh and Bessler 2004), which is also employed here. For the PC algorithm, Spirtes et al. (2000) state: “In order for the method to converge to correct decisions with probability 1, the significance level used in making decisions should decrease as the sample size increases, and the use of higher significance levels (e.g., 0.2 at sample sizes less than 100, and 0.1 at sample sizes between 100 and 300) may improve performance at small sample sizes.” For the sample size of 108 in the current study, the 15% and 20% significance levels should be appropriate. We also try the conditional correlation independence (CCI) test (Ramsey 2014), and, as compared to Fisher’s z, the CCI test leads to one more undirected edge between T5 and T6. Following Moneta et al. (2013), bootstrap analysis with 100 iterations is performed to test the stability of results based on the PC algorithm. The top three most frequent outputs are the “pattern” shown in Fig. 4 (46 out of 100), a “pattern” with one more undirected edge between T5 and T6 as compared to Fig. 4 (21 out of 100), and a “patter” with one more undirected edge between T1 and T4 as compared to Fig. 4 (13 out of 100).
For the LiNGAM algorithm, the pruning method follows Shimizu et al. (2006), which is an algorithm that combines the Wald test for examining significance of edges, a Chi-Square test for evaluating the overall fit of the estimated model, and a difference Chi-Square test for model comparisons of nested models. The prune factor approach is a simplified version of bootstrapping and the prune factor indicates the number of standard deviations that can be away from the mean bootstrap values (Lai and Bessler 2015). More edges will be pruned out with the factor, between 0 and 1, increasing. However, there has not been conclusive research on the selection of the factor. Bizimana et al. (2015) use 0.5 when the sample size is 158. For the sample size of 108 here, the factors of 0.4 and 0.5 should be appropriate. Similar to the case for the PC algorithm, the top three most frequent outputs following the bootstrap analysis for the LiNGAM algorithm are the DAG shown in Fig. 5 (41 out of 100), a DAG with one more directed edge from T5 to T6 as compared to Fig. 5 (24 out of 100), and a DAG with one more directed edge from T1 to T4 as compared to Fig. 5 (11 out of 100).
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Xu, X., Zhang, Y. Contemporaneous causality among one hundred Chinese cities. Empir Econ 63, 2315–2329 (2022). https://doi.org/10.1007/s00181-021-02190-5
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DOI: https://doi.org/10.1007/s00181-021-02190-5