Parameter estimation in spatial econometric models with non-random missing data

ABSTRACT

This study examines the problem of parameter estimation in spatial econometric/social interaction models with non-random missing outcome data. First, we construct a sample selection model considering spatial lag (autoregressive) dependence. Then, we suggest a parameter estimation method for this model by slightly modifying the Bayesian Markov chain Monte Carlo algorithm proposed in an existing study. A simple illustration indicates that the proposed parameter estimation method performs well overall if the spatial autocorrelation is moderate (spatial parameter equals 0.5 or less), even under a relatively high missing data ratio (around 40%).

KEYWORDS:

• Sample selection
• spatial lag model (SLM)
• spatial autocorrelation
• social interaction
• Bayesian Markov chain Monte Carlo (MCMC)

JEL CLASSIFICATION:

• C40
• C50
• C13

Acknowledgments

This study was funded by JSPS KAKENHI Grant Numbers 17K14738 and 18H03628.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

¹ For the case of random missing data, a parameter estimation method was presented by Wang and Lee (2013). For spatial econometrics in general, see Arbia (2006). Our study focuses on spatial (auto)correlation among individuals, not among alternatives (De Grange et al. 2013). See Billé and Arbia for spatial discrete choice models (2019).

² Omori (2007) and Wiesenfarth and Kneib (2010) considered a sampler that imputes latent variables for the selection equation but uses only the observed responses from the outcome equation. By doing so, we can improve the mixing and convergence problem by integrating out missing values from a likelihood function. However, in spatial econometric/social interaction models, where connections or networks between samples are important, listwise deletion of missing data leads to the deletion of such connections or networks.

³ Alternatively, spatial autoregressive model.

⁴ It may be a strong assumption that ${\epsilon }_{1i}$ and ${\epsilon }_{2i}$ are governed by two-dimensional normal distribution, and they are thought to form a flexible distribution with copulas as in, for example, Wojtys, Marra, and Radice (2016). However, emphasis is placed on widespread (or more standard) assumptions, and this study assumes two-dimensional normal distribution.

⁵ MCMC convergence was confirmed by Geweke (1992)’s convergence diagnostic. Because our model structure was fairly simple, the shape of the posterior distributions of each parameter was obtained as unimodal distributions.

Additional information

Funding

This work was supported by the Japan Society for the Promotion of Science [17K14738,18H03628].