Estimation of the models reported in this section was conducted using LeSage's MATLAB routines, which are available online

^{6 }and incorporate efficient treatments for situations with a large number of spatial units [

25]. Some of these routines have been recently updated by Elhorst [

41].

The results of models considering the variables mentioned before are displayed in Table and Table . In order to reduce the scale of the dependent variable, IMR are transformed using the natural logarithm operator. This transformation also ensures that estimated values are not negative when converted back to the original scale. For comparative purposes, two models are presented in Table , namely a pooled model (i.e. it does not consider regional fixed effects) and a fixed effects panel model.

| **Table 3**Results for the pooled and the spatial fixed effects models for the log of infant mortality rate |

| **Table 4**Results for the fixed effects spatial error model (SEA) and the fixed effects spatial autocorrelation model (SAR) for the log of infant mortality rate |

As seen in Table , the pooled model gives the lowest coefficient of determination R^{2 }and suffers from several deficiencies. In this model, both sanitation and public health institutions are not significant. Also, the sign of water access is counterintuitive. The adolescent fertility rate has a negative sign. The strongest effects over the IMR are found with the female illiteracy rate (1.8) and the spatially lagged number of private health care institutions (-1.2).

The panel with fixed effects leads to some changes in the main results compared to the pooled one. Now, sanitation is highly significant and has the expected sign. However, average income and income inequality lose their significance. The coefficient of illiteracy rate is three times smaller than the one in the pooled framework, and the variable measuring public health care now has the expected sign, even if it is still not significant. Spatially lagged variables are all significant and have a strong effect over the IMR, suggesting that there is a spatial spillover in local health and infrastructure policies.

It is important to notice, however, that the spatial dependence tests performed for both models indicate the presence of significant spatial dependence, in the form of spatial error autocorrelation or a spatially lagged dependent variable. Spatial dependence is a serious issue that leads to inefficient and/or biased coefficients. Therefore, in Table we present the results of the estimation of the models accounting for spatial effects.

Two models are estimated that explicitly consider, in addition to the spatially lagged independent variables, spatial error autocorrelation (SEA) and spatial autocorrelation of the dependent variable (SAR), respectively. A general observation is that these models provide better goodness of fit indicators. Furthermore, examination of the residuals indicates that these are homoscedastic. Finally, the spatial effect is highly significant in both cases, and inference becomes more reliable, as the spatial structure corrects for the inconsistency of the coefficients. In spite of their structural differences, all variables have the same sign in the two models, and only the variable for public health care institutions loses its significance in the SAR model. This first aspect means that our model seems to be robust to the form of spatial structure adopted.

As previously mentioned, model selection can be based on statistical or theoretical criteria. In the present case, the SAR model returns a slightly higher *R*^{2}. The two models are comparable in terms of their AIC, squared-R, and variance, and therefore, on purely statistical grounds, both seem to provide very similar levels of goodness of fit. Conceptually, our view is that a model with a spatially lagged dependent variable is less appealing, because at the level of aggregation considered in this analysis (municipalities) there is no plausible mechanism to explain why IMR should be high when it is high in neighbouring regions, other than through the operation of common variables or spillovers. For this reason, we concentrate our discussion on the SEA model.

The fixed effects SEA model provides a number of valuable insights. Income is widely acknowledged to have a strong impact on the reduction in IMR. Our analysis, after controlling for a large number of confounding factors, is able to confirm the significance of income, even if the effect observed for this variable is relatively small (-0.001 - which means that an increase of 1 p.p. in average income reduces the level of infant deaths by 0.001 over 1,000 born alive). The results also indicate that as income inequality and women illiteracy rate grow, IMR also tend to increase by factors of 0.245 and 0.185, respectively. In the final analysis, we find that adolescent fertility rate associates positively with IMR (0.168), suggesting that family planning can help to reduce infant mortality.

In terms of health infrastructure, the results indicate that the number of private health care institutions exerts a stronger effect than public institutions (-0.165 and -0.048, respectively). This result aligns with the suggestion that poorer families have to resort to private health care, as specialized public services were often offered to and accessed by high and medium income individuals, and basic public services supply was neglected, resulting in a congested and low quality provision. Consequently, if private health care institutions are offered, the options of treatment for the poor are increased. Furthermore, water access and sanitation have a strong negative impact over the IMR. The first one is almost universally spread in the country, but even nowadays sanitation services cover only around 50% of the Brazilian households.

Finally, spatially lagged variables have almost twice the impact of their local counterparts. Any policy aiming to improve life conditions must take this result into consideration. Hence, there are indirect regional effects of improvements in health care infrastructure or the level of public services access. Further policy recommendations relate to the reduction of income inequality and illiteracy rate, and spreading information regarding fertility control.

While the models reported in Table correlate the levels of IMR to their cotemporaneous variables, they fail to provide a sense of the temporal trends. An alternative perspective can be gleaned by redefining the dependent variable as the first difference between time periods. The independent variables, on the other hand, are for the initial time period, and thus enter exogenously the model. This procedure results in *T *= 2 time periods (1980 to 1991 and 1991 to 2000) and reduces the size of our sample. It is important to note that the objective of this model is to identify the effect of initial conditions over the change in the infant mortality rate. The modeling strategy followed for this analysis mirrors the one described above. Once again, the spatial dependence is identified in the fixed effects model, requiring the estimation of a spatial panel model. For the sake of brevity, only the results of the spatial models (SEA and SAR) are reported in Table .

| **Table 5**Results for the fixed effects spatial error model (SEA) and the fixed effects spatial autocorrelation model (SAR) |

In order to interpret the coefficients of the model, it bears noting that a negative value of the dependent variable means that the IMR, as was the case countrywide, decreased in the implied period of time. The signs of the coefficients are the opposite of the models reported in Table . This provides an interesting, and intuitive, insight. For instance, a positive sign for the number of health care institutions in the previous period of time means that the dependent variable is *less *negative as this variable increases. Negative values for the coefficients associated with illiteracy rates and adolescent fertility rates imply *more *negative values of the difference, and therefore greater gains in IMR. Clearly, since IMR is a zero-bounded variable, the trend indicates that greater gains were obtained in regions with the worst initial conditions (e.g. lower levels of infrastructure provision, higher levels of illiteracy). The implication therefore is that further gains in IMR become increasingly difficult as conditions improve. This apparently simple conclusion nonetheless suggests that allocation of resources should try to efficiently target programs with potential to generate greater reductions of IMR.