One interesting area of future research will be to determine whether these smaller municipalities are simply more idiosyncratic and, thus, harder to predict, or whether they differ structurally in observables ways. If the latter is true, different types of data or methods might improve estimation for the smallest municipalities. In other words, is it possible to improve the estimation of these municipalities, or are there simply weaker systematic relationships between geospatial data and labor market statistics for these areas?

# **B. Area-level model**

An alternative method for combining survey data and geospatial data is to obtain municipal estimates is an area-level model (Fay and Herriot, 1979). This method is convenient and wellknown, but suffers from a few limitations in this context. The main shortcoming is that it requires aggregating the auxiliary data to the municipal level, which discards variation across AGEBs within municipalities. In addition, the method faces challenges to accurately generate estimates of municipal variance, which is an important input into the model. This is particularly true when there is a significant number of zero values, as is the case for unemployment rates in this context.

To better understand whether these potential limitations are important in practice, we examine the results of two area-level modeling exercises. Fay-Herriot models can either be assumed to be linear models or the dependent variable can be transformed prior to estimation. For ratios such as labor force participation and unemployment rates, it is common to transform the variable using an arcsine transformation (Halbmeier et al, 2019). Use of such a transformation adds additional complexity to the procedure, but ensures that predicted participation or unemployment rates lie between zero and one. In addition, using the arcsine transformation is more consistent with the AGEB-level model discussed above. For these reasons, we use the transformed model as our main results, but display the results of a linear model in Table A4.

We select a model for each outcome by applying LASSO to the full set of municipal candidate geospatial variables, again selecting lambda to minimize BIC. When using the arcsine transformation, the dependent variable is the survey-weighted estimate of the transformed mean outcome variable of interest in each municipality. When using the linear model, no transformation is applied prior to the LASSO. We then calculate the sampling variance of each outcome using the Horvitz-Thompson approximation, as recommended by Halbmeier et al (2019), and fit the model using the restricted information maximum likelihood option of Stata’s fayherriot package. The R<sup>2</sup> s of the transformed models are quite low: 0.06 and 0.19 for male and female LFP, and only 0.10 and 0.06 for male and female unemployment. Even after reducing the noise associated with the unemployment measure by aggregating to the municipality level, the geospatial variables are weak predictors of unemployment rates and male LFP, although they predict cross-municipal variation in female LFP somewhat better.

Table 9 displays the results from the transformed Fay-Herriot (FH) area-level model.<sup>25</sup> We estimate the FH using an arcsine transformation of the target variable, just like with the sub-area

> 25 The table omits MSE because the State fayherriot command used to estimate the FH model does not output standard errors. Instead, it outputs confidence intervals derived from a bootstrap procedure. Because the confidence intervals are asymmetric, calculation of the MSE is not straightforward.

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