Common factors and spatial dependence: an application to US house prices

Abstract

This article considers panel data models with cross-sectional dependence arising from both spatial autocorrelation and unobserved common factors. It proposes estimation methods that employ cross-sectional averages as factor proxies, including the 2SLS, Best 2SLS, and GMM estimations. The proposed estimators are robust to unknown heteroskedasticity and serial correlation in the disturbances, unrequired to estimate the number of unknown factors, and computationally tractable. The article establishes the asymptotic distributions of these estimators and compares their consistency and efficiency properties. Extensive Monte Carlo experiments lend support to the theoretical findings and demonstrate the satisfactory finite sample performance of the proposed estimators. The empirical section of the article finds strong evidence of spatial dependence of real house price changes across 377 Metropolitan Statistical Areas in the US from 1975Q1 to 2014Q4.

Keywords:

• Cross-sectional dependence
• common factors
• spatial panel data models
• generalized method of moments
• house prices

JEL CLASSIFICATIONS:

• C13
• C23
• R21
• R31

Acknowledgments

I acknowledge valuable comments and suggestions from the editor and two anonymous reviewers. I also appreciate helpful comments from Cheng Hsiao, Yu-Wei Hsieh, Hyungsik Roger Moon, M. Hashem Pesaran, Wenguang Sun, and participants at the 2017 China Meeting of the Econometric Society, the Singapore Economic Review Conference 2017, the Third Annual Conference of International Association for Applied Econometrics (IAAE), the 2017 J-WEN Mentoring Event, and the Federal Reserve Board Macroeconomics and Monetary Policy Seminar. I would like to thank Natalia Bailey for helpful email correspondence regarding her approach. This article supersedes an earlier draft circulated under the title “Identification and Estimation of Spatial Autoregressive Models with Common Factors.”

Notes

1 For overviews of the literature on panel data models with error cross-sectional dependence, see Sarafidis and Wansbeek (2012) and Chudik and Pesaran (2015b).

2 Comprehensive reviews of spatial econometrics can be found in books including Anselin (1988) and Elhorst (2014). Also see the survey article by Lee and Yu (2010b) for the latest developments in spatial panel data models.

3 Much is written on estimating the number of unobservable factors. See, for example, Bai and Ng (2002, 2007), Kapetanios (2010), and Stock and Watson (2011).

4 Cohen, Ioannides, and Thanapisitikul (2016) also use a house price index different from ours. Specifically, the authors adopt the consolidated house price index by the Office of Federal Housing Enterprise Oversight that covers 363 MSAs over the period of 1996–2013.

5 In general, the spatial weights and variables in the model may dependent on N, but we suppress subscript N throughout the article for ease of notation.

6 The heterogeneity in factor loadings may arise, for example, from differences in endowment, technical rigidities, or innate ability.

7 See Remark 2 of Pesaran (2006).

8 See Lemma A.1 of Bai and Li (2014).

9 All lemmas are provided in the Supporting Information.

10 In practice, it may also worth including ${\overline{y}}_t^{*}=N^{-1}{\sum }_{i=1}^N{y}_t^{*}$ as factor proxies if ${\overline{y}}_t^{*}$ is not highly correlated with ${\overline{y}}_t.$

11 See Assumption 5 in Pesaran (2006).

12 An exception is made by Kapetanios, Pesaran, and Yamagata (2011), who show that the CCE approach can be extended to allow latent factors to follow unit root processes.

13 This model can be further extended to accommodate spatial correlations in the errors at the expense of more involved notations.

14 See Karabiyik, Reese, and Westerlund (2017) for a recent investigation on the role of the rank condition in CCE estimations.

15 Pesaran and Yang (2020b) made a recent effort to relax this assumption for cross-section SAR models.

16 The highest order of the power of W, denoted by R, is a preselected finite constant, and in practice is usually chosen to be at least two.

17 We do not use subscript “0” to refer to the true values of ${\mathit{\gamma }}_{\mathrm{i}},$ ${\mathbf{A}}_i,{\sigma }_i^2,$ and ${\mathbf{\Sigma }}_{v,i}$ in the main text in order to avoid notational complications.

18 Karabiyik, Reese, and Westerlund (2017) recently point out a common problem with the CCE approach and show that in the degenerate case when $m<k+1$ and when T/N tends to a finite positive constant as $(N,T)\stackrel{j}{\to }\infty ,$ there will be an additional bias term. They assume independently and identically distributed errors and only focus on $T/N=O(1).$ Similar additional bias term is expected to occur for our model under these conditions. A more general extension of Theorem 1 and the investigation of bias correction for the case of $T/N=O(1)$ is left for future work.

19 See the Supporting Information for a proof that ${\widehat{\mathbf{\Sigma }}}_{2sls}$ is consistent for ${\mathbf{\Sigma }}_{2sls}.$

20 Although our interest lies in the parameters $\mathit{\delta },$ we can gain insight into the variability of the factor loadings by running these regressions.

21 The bias correction approach suggested by Westerlund and Urbain (2015) need to be reexamined in light of the latest contribution by Karabiyik, Reese, and Westerlund (2017), who show that there could be an additional bias term when $T/N=O(1)$ and $m<k+1$ but do not investigate bias correction.

22 We use the aggregated moment conditions over time instead of a moment condition for each period separately, since the latter approach may induce the many-moment bias problem and is beyond the scope of the current article. See Lee and Yu (2014) for a discussion of this issue for spatial models.

23 See Lemma S.10 in the Supporting Information. The variance of the linear and quadratic form does not involve the third and fourth moments of the errors if P_l has zero diagonal.

24 See, for example, Assumption 5(b) in Lin and Lee (2010). It can be readily modified to allow for serial dependence in the errors without much difficulty. Also see Proposition 1 of Pesaran and Yang (2020b).

25 The proof that ${\widehat{\mathbf{\Sigma }}}_{\mathit{gmm}}$ is consistent for ${\mathbf{\Sigma }}_{\mathit{gmm}}$ is given in the Supporting Information.

26 Since G₀ involves unknown parameter ρ₀, it may be constructed from some initial consistent estimate of ρ₀ obtained in a preliminary estimation step.

27 As in the case of B2SLS estimation, under Assumption 6, ${\mathbf{G}}_0=\mathbf{W}{({\mathbf{I}}_N-{\rho }_0\mathbf{W})}^{-1}=\mathbf{W}+{\rho }_0{\mathbf{W}}^2+{\rho }_0^2{\mathbf{W}}^3+\dots ,$ and therefore including higher powers of W as P_l can be seen as an approximation for P*.

28 We have also examined the case where the errors are independent over time and heteroskedastic across individual units. The results are presented in the Supporting Information.

29 We have also considered ρ = 0.8, which represents high intensity of spatial dependence. The results are provided in the Supporting Information.

30 Bai and Li (2014) point out that one could switch the role of N and T if T is much smaller than N. We do not report results under this interchange, since it requires different strict assumptions on the disturbances and does not improve the performance of MLE under our Monte Carlo designs.

31 Bai and Li (2014) propose using an information criterion to estimate the number of factors in their Monte Carlo experiments.

32 We also considered $\lfloor T^{1/3}\rfloor$ as the window size. The results are close but using $\lfloor 2\sqrt{T}\rfloor$ has slightly better size properties.

33 Cohen, Ioannides, and Thanapisitikul (2016) and Bailey, Holly, and Pesaran (2016) focus on house price series itself and do not consider any explanatory variables.

34 The Office of Management and Budget periodically revises the MSA delineations to reflect the changes in population counts and commuting patterns. There are 381 MSAs in the US as of February 2013. The terms “area” and “MSA” are used interchangeably in the following discussions.

35 The 2SLS estimates are omitted to save space, since they are very close to the GMM estimates with slightly larger standard errors.

36 In the empirical analysis, ${\overline{y}}_t^{*}=N^{-1}{\sum }_{i=1}^N{y}_{it}^{*}$ is also included as factor proxies since it may potentially improve the small sample properties of the estimator. Nonetheless, whether ${\overline{y}}_t^{*}$ is included makes little difference to the results since ${\overline{y}}_t^{*}$ and ${\overline{y}}_t$ are highly correlated for most of the W matrices we considered.

37 Bailey, Holly, and Pesaran (2016) also consider regional effects, but the authors do not show the impact of eliminating regional factors to the estimated intensity of spatial dependence.

38 The eight BEA Regions are New England, Mideast, Great Lakes, Plains, Southeast, Southwest, Rocky Mountain, and Far West Regions.

39 ${\overline{R}}^2=1-{\widehat{\sigma }}_{\text{res}}^2/{\widehat{\sigma }}_{\text{tot}}^2,$ where ${\widehat{\sigma }}_{\text{res}}^2={\left[N(T-k_{cs}-m_d)-k_z\right]}^{-1}{\sum }_{i=1}^N{({\mathbf{y}}_{i.}-{\mathbf{Z}}_{i.}\widehat{\mathit{\delta }})}^{\prime }\overline{\mathbf{M}}({\mathbf{y}}_{i.}-{\mathbf{Z}}_{i.}\widehat{\mathit{\delta }}),$ ${\widehat{\sigma }}_{\text{tot}}^2={\left[N(T-1)\right]}^{-1}{\sum }_{i=1}^N{\sum }_{t=1}^T{(y_{it}-{\overline{y}}_{i.})}^2,$ ${\overline{y}}_{i.}=T^{-1}{\sum }_{t=1}^T{y}_{it},{\mathbf{y}}_{i.}={(y_{i1},y_{i2},\dots ,y_{iT})}^{\prime },$ m_d is the number of observed factors, and $\overline{\mathbf{M}}$ represents the de-factoring matrix of $T\times k_{cs}$ dimension. In specific, m_d = 4 since the observed factors consist of quarterly dummies and an intercept. The values of k_z and k_cs vary with detailed model specifications, that is, whether Durbin terms are included and regional factors are considered. This measure of model fit in the presence of unobserved factors is in accordance with the suggestion by Holly et al. (2010, p. 164).

40 Standard error in parenthesis.

41 See LeSage and Pace (2009) and Section 2.7 of Elhorst (2014) for detailed discussions on the computation.

42 Details on the construction of W_m and its characterization are given in Appendix C and the Supporting Information.

43 See Figure S.1 in the Supporting Information for the distribution of distance between the area of origin and the area of destination.

44 See Appendix C and the Supporting Information for a more detailed characterization and comparison of different spatial weights matrices.

45 We have also considered the Durbin terms, which are found to be insignificant.

46 See (S.11) in the Supporting Information for details.