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The Annals of Regional Science

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24-12-2015 | Original Paper

Spatial lag dependence in the presence of missing observations

Author: Takahisa Yokoi

Published in: The Annals of Regional Science | Issue 1/2018

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Abstract

We explore the estimation effectiveness of spatial lag models in the presence of missing observations. Spatial lag models are used to measure interdependency between dependent variables. If there are no missing data, it is easy to interpret this spatial autocorrelation process. Very sparsely sampled data are sometimes used in empirical studies. For such data, we observe only a small part of a population containing possible mutual dependencies. Simulation studies based on artificial data confirm the relation between the sampling rate and selection ratio of spatial and non-spatial models. Our findings include the following: (1) Negative spatial autocorrelation of the data-generating process (DGP) may not be observed. (2) Positive spatial autocorrelation of the DGP may be observed, but it is downward-biased. (3) We obtain less-biased estimates if we use a non-row-standardized weight matrix. (4) Non-spatial models tend to be selected in preference to the correct model, the spatial lag model. (5) Estimates of regression coefficients remain almost unbiased.

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The edge (or border) effect described in Anselin (1988, pp. 175–176) has a close relation with the missing data problem, in that it affects the expected value of the disturbance term and introduces heteroscedasticity.

See also Little and Rubin (2002, ch.1) and Arbia et al. (2015) for a more detailed survey on missing data and some examples.

In this current study, every data unit is observable with a \(\gamma \) probability. Sampling rate is expressed as ‘PMP’ in Arbia et al. (2015). ‘PMP’ there is defined as the proportion of missing points. Their PMP \(=0.05\) and PMP \(\,=0.25\) correspond to our \(\gamma =0.95\) and \(\gamma =0.75\), respectively. It is expressed as ‘\(\alpha \)’ in Wang and Lee (2013a, b). ‘\(\alpha \)’ there is defined as a missing data percentage. Their \(\alpha =10, 20\), and 40 correspond to our \(\gamma =0.9, 0.8\), and 0.6, respectively.

See Anselin (1988, ch. 6) for details of these two methods.

We also estimated the spatial two-stage estimation (Kelejian and Prucha 1998) of Eq. (1). The results were nearly identical with those of ‘lag,’ and so are omitted from our description.

See Stakhovych and Bijmolt (2009, p. 393) for a survey of spatial weights matrices used in simulation analyses.

The current simulation study employs \(\alpha = 2\) and \(\bar{d} = 5\).

Note that there is no spatial autocorrelation in the DGP when the true \(\rho \) value is 0.

Arbia et al. (2015) also pointed out that spatial correlation disappears when some points are missing.

A referee kindly suggested that ‘the SAR parameter \(\rho \) loses its interpretation, and its relationship with the eigen values of weight matrix is lost’ in this setting. Even if there should be further discussions, we here just point out the possibility of the non-row-standardized weight matrix setting.

Arbia et al. (2015) performed experiments for the relatively dense sampling cases. They control the spatial intensity of data deletion with a parameter \(\psi \). Their \(\psi =0\) result corresponds to our experiments. A comparison between their results tells us several things. (1) There are similarities in decline patterns of spatial parameter efficiency. (2) There are differences in decline patterns of regression coefficients efficiency. In our sparse sampling experiments, the pattern does not depend on the true value of spatial parameter. In Arbia et al. (2015)’s dense sampling experiments, it depends on the true value of spatial parameter.

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Wang W, Lee L-F (2013a) Estimation of spatial autoregressive models with randomly missing data in the dependent variable. Econom J 16(1):73–102CrossRef

Wang W, Lee L-F (2013b) Estimation of spatial panel data models with randomly missing data in the dependent variable. Reg Sci Urban Econ 43(3):521–538CrossRef

Title: Spatial lag dependence in the presence of missing observations
Author: Takahisa Yokoi
Publication date: 24-12-2015
Publisher: Springer Berlin Heidelberg
Published in: The Annals of Regional Science / Issue 1/2018
Print ISSN: 0570-1864
Electronic ISSN: 1432-0592
DOI: https://doi.org/10.1007/s00168-015-0737-2

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