Geoscience Reference
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vector of census tract
i
0
s other time variant characteristics (such as vacancy rates
and median household income relative to the city median) in period
t
,Z
i
is a vector
of the census tract
i
0
s time invariant characteristics (such as its city and distance
from the city center). And,
α
,
ʲ
,
ʳ
and
ʴ
are parameters to be estimated, with
ʵ
i
as a
random error term.
When analyzing spatial data, spatial dependence in the outcomes should be
considered, as failure to appropriately account for a spatially dependent outcome
may result in biased and/or inefficient coefficient estimates. Although Table
19.2
indicates spatial autocorrelation in the locations of gay and lesbian households, the
need for spatially explicit estimation procedures is commonly assessed through
analysis of the residuals from an ordinary least squares (OLS) regression.
We test the residuals of the OLS models for each household type (gay men,
lesbian, and all households), nationally and within each region, using the simple
Lagrange Multiplier (LM) statistics for spatial error dependence and spatial
autoregressive dependence derived in Burridge (
1980
) and Anselin (
1988
), and
the robust LM statistics for either type of dependence derived in Bera and Yoon
(
1993
) and Anselin et al. (
1996
). The simple versions of these LM statistics test for
the presence of spatial dependence in the form of a spatial autoregressive process or
a spatial error process (assuming that neither is present), while the robust versions
test for a spatial autoregressive process when the actual data generating process is a
spatial error process, and vice versa. Based on the results described in Anselin
et al. (
1996
), we first assess the significance of the simple LM statistics. When only
one of the simple LM statistics is significant (either autoregressive or error), we
proceed with estimation of that type of model. In cases where both simple LM
statistics are significant, the robust LM statistics are used to determine the appro-
priate model.
12
When neither of the simple LM statistics is significant, a spatial
model is not appropriate.
We find evidence of spatially dependent residuals in each of the three models
(gay men, lesbians, all households), nationally and within each region, with the LM
statistics suggesting the presence of a spatial autoregressive process in census tract
shares of the city's gay and lesbian households and a spatial error process in total
household shares.
13
Therefore, the estimations of gay and lesbian household shares
12
When only one of the robust LM statistics was significant, that type of model was estimated.
When both of the robust LM statistics were significant, the model with the larger test statistic was
chosen.
13
The LM statistic tests for all household shares strongly suggested the presence of a spatial error
process. The LM statistic tests for gay male and lesbian household shares appeared weaker. The
spatial lag model was the most consistently “preferred”, although gay male and lesbian household
share in the Midwest and gay male household share nationally showed no evidence of a spatial
process. We estimated a spatial lag model for all regions nevertheless, to allow for easier
comparisons between regions. The estimation of a spatial lag model in cases where there is no
underlying spatial process should not unduly bias the results. We also tested a spatial error
specification for gay and lesbian household share, and the results from these estimations were
not substantively different than those from those shown. The LM statistics from each of the OLS
estimations are shown in Table
19.6
in Appendix.
