A glance at further approaches in spatial panel data modeling
The literature on spatial panel data models has grown rapidly in recent years and the largest number of theoretical and applied papers published in spatial econometrics are now related to this topic (see Arbia, 2012 for a review). The basic frameworks presented here are only a small description of a much larger field of research and applications. Excellent survey papers in this area are provided by Bal tagi and Pesaran (2007), the review article by Lee and Yu (2010b) and the book by Elhorst (2014) to which we refer the reader for more details. Furthermore, we limit ourselves to what are referred to in the literature as static panel data models which do not include any time lag. When a time lag is included we deal with what are termed “dynamic spatial panel data models”. This large class of models are further distinguished into “stable” and “unstable” models and include the discussion of important issues such as spatial unit roots, spatial cointegration and explosive roots. Despite the interest in these models they are not included in this book where we want to keep the discussion as simple as possible. Furthermore, for this class of models, no predefined package is currently available in the language R. (For an exhaustive review of these topics see Lee and Yu, 2011; Baltagi ct ah, 2013; Bai and Li, 2018;,Li ct ah, 2019; OrwatAcedanska, 2019.)
Example 5.1: Insurance data
Although there is great interest in the current scientific debate on spatial panel data for microdata, it is still difficult to find freely available datasets to test the methods discussed in this chapter referring to the single individual agent, because most of the applications so far are still confined to regional aggregated data. However, just to give the reader a taste of the possible applications, in this example we will make use of regional data, but treating them as if they were individuals, considering the coordinates of the centroids of each area as point data. The dataset {splmjlnsurance contains data on insurance consumption in the 103 Italian provinces from 1998 to 2002 and contains different variables among which the “real per capita gross domestic product” and the “real per capita bank deposit”. In what follows we will build up a model where we explain the per capita bank deposit as a function of per capita GDP. The locations of the 103 centroids are given in Figure 5.1.
We start by estimating a pooled model with no spatial components with OLS obtaining the results in Table 5.1. We then estimate the different specifications of spatial panel data models. In all cases we used a кnearest neighbor setting к  3. We start considering the results of a random coefficient models. Remember that in the case of random coefficients, we have an extra parameter (p described in Equation 5.11 which is the ratio between the variance of the individual effect and the variance of the idiosyncratic error.
The first model is a panel model without spatial effects which leads to the results estimated with maximum likelihood in Table 5.2, where all parameters including (p are significant.
Figure 5.1 Centroids of the 103 Italian provinces.
The second model is a spatial lag random effects model estimated with maximum likelihood which leads to the results in Table 5.3. Notice that in this specification the spatial correlation parameter is positive and significant.
The third model is a spatial error random effects model with the specification of Baltagi et al. (2003) estimated with maximum likelihood which leads to the results in Table 5.4.
Table 5.1 Output of a pooled regression with no spatial components of the per capita bank deposit expressed as a function of per capita GDP in the 103 Italian provinces. Estimation method: maximum likelihood
Parameter 
Standard error 
ttest 
pvalue 

Intercept 
2088.1 
424.46 
6.436 
1.822e  010 
Real per capita GDP 
0.615 
0.0179 
34.235 
2.2c  16 
R^{2} = 0.69556
Table 5.2 Output of a panel data random effect model with no spatial components of the per capita bank deposit expressed as a function of per capita GDP in the 103 Italian provinces. Estimation method: maximum likelihood
Parameter 
Standard error 
ttest 
pvalue 

Intercept 
1346.5 
999.68 
13.4689 
2.2c  16 
Real per capita GDP 
0.272 
0.052 
5.217 
1.816e  07 
Table 5.3 Output of a spatial lag panel data random effect model of the per capita bank deposit expressed as a function of per capita GDP in the 103 Italian provinces. Estimation method: maximum likelihood
Parameter 
Standard error 
ttest 
pvalue 

Intercept 
2078.1 
756.98 
2.745 
0.0006 
Real per capita GDP 
0.1027 
0.0456 
2.541 
0.011 
X = 0.546 (pvalue < 2.2e  16)
Table 5.4 Output of a spatial error panel data random effect model of the per capita bank deposit expressed as a function of per capita GDP in the 103 Italian provinces. Specification of Baltagi et al. (2003). Estimation method: maximum likelihood
Parameter 
Standard error 
ttest 
pvalue 

Intercept 
186.733 
608.442 
0.306 
0.758 
Real per capita GDP 
0.506 
0.033 
15.219 
<2e  16 
The fourth model is a spatial error, random effects model with the specification of Kapoor et al. (2007) estimated with maximum likelihood which leads to the results in Table 5.5. In these last two models the spatial error correlation parameters are positive and significant and of comparable absolute values in the two specifications.
Remaining in the area of random effects we have a last case of spatial error, but using GMM as estimation method giving the results in Table 5.6. In all models except the last, the estimated variance of the individual effect is much bigger than that of the idiosyncratic error, the latter showing substantial spatial correlation. This is a strong evidence in favor of a spatial process in the errors.
We now move to consider the case of a fixed effect. Notice that in this case we do not have an intercept to the model due to the operation of demeaning described in Equation 5.12. We start with spatial lag, random effects model using maximum likelihood, which gives the results in Table 5.7.
Finally, for the case of a fixed effect, with spatial error, random effects model, using maximum likelihood we have Table 5.8. If we adopt a random effect approach notice that the estimates of the parameters ft are not too different in the
Table 5.5 Output of a spatial error panel data random effect model of the per capita bank deposit expressed as a function of per capita GDP in the 103 Italian provinces. Specification of Kapoor et al. (2007). Estimation method: maximum likelihood
Parameter 
Standard error 
ttest 
pvalue 

Intercept 
403.618155 
759.860 
0.531 
0.5953 
Real per capita GDP 
0.4659 
0.036 
12.84 
<2e  16 
ip = 5.452 (pvalue = 3.852e  07); p = 0.611333 (pvalue < 2.2e  16)
Table 5.6 Output of a spatial error panel data random effect model of the per capita bank deposit expressed as a function of per capita GDP in the 103 Italian provinces. Estimation method: GMM
Parameter 
Standard error 
ttest 
pvalue 

Intercept 
208.971 
716.670 
0.292 
0.771 
Real per capita GDP 
0.500 
0.034 
14.659 
<2e  16 
(p = 0.785; p = 0.623
Table 5.7 Output of a spatial lag panel data fixed effect model of the per capita bank
deposit expressed as a function of per capita GDP in the 103 Italian provinces. Estimation method: maximum likelihood
Parameter 
Standard error 
ttest 
pvalue 

Real per capita GDP 
0.3466 
0.049 
7.024 
2.152e  12 
X = 0.448 (pvalue < 2.2e  16)
Table 5.8 Output of a spatial error panel data random effect model of the per capita bank deposit expressed as a function of per capita GDP in the 103 Italian provinces. Estimation method: maximum likelihood
Parameter 
Standard error 
ttest 
pvalue 

Real per capita GDP 
0.274 
0.062 
4.393 
1.117e  05 
p = 0.477 (pvalue < 2.2c  
16) 
various specifications and from those obtained with a nonspatial specifications. Notice also, however, that looking at the standard errors, those related to the pooled estimators are the smallest so that we can think they are underestimating the true values and provide a less reliable estimate biased towards the rejection of the null hypothesis.
In the two cases of the fixed effects models, in contrast, the /5 coefficients become negative with similar absolute values. Spatial correlation is always positive and significant in all specifications confirming the need to introduce these corrections in the case examined.
Note
1 We are thankful to Giovanni Millo for collaborating in the writing of this chapter.
Part III