*3.1. Data Description*

The data we have used is provided by Eurostat programs EU-SILC and ESSPROS and embeds EU-28 countries in 2018 (i.e., it is included Great Britain). The data have annual periodicity and comprise the period 2014–2018. From the database, we directly obtain observations on the following variables for every country and year. Concretely:


Likewise, our analysis also needs the observations on the proportion that each kind of social benefit supposes over whole social expending according to EU-SILC classification. These items are defined over the basis of eight protection functions linked with a set of needs [60]:


From the variables indicated above, we derivate for each country and year the diminution of poverty and the productivity that SE has reached in such diminution. Following [10], we measure poverty reduction in relative terms. Hence, for the *i*th country at year *t* we obtain:

$$RRP\_{i,t} = \frac{ARPR(0)\_{i,t} - ARPR(1)\_{i,t}}{ARPR(0)\_{i,t}}, \; i = 1, 2, \ldots, 28 \tag{8}$$

Hence, *RRPi,t* ranks from 0 (and so *ARPR*(0)*i,t* = *ARPR*(1)*i,t*), to 1 (if poverty is completely eliminated).

When analyzing the productivity of SE in reducing poverty risk, we seek to determine to what extent the diminution of poverty (the assessed output) is adequate to the initial situation of poverty and SER (inputs). To measure the efficiency of public spending in achieving poverty reduction for the *i*th country in a year *t* we follow [10] that quantifies efficiency by means of a Debreu–Farrell coefficient, *DFi,t*. Hence, we consider SE and, more concretely, its quantification by means of its ratio with GPD (SER) as the main input. We also use the GI before transfers as second input variable to reflect the social status of the population before executing the SE. Hence, GI is not strictly an input, but a contextual variable. Likewise, GI is clearly linked to economic context, social and demographical structure and public policy priorities of every state. A greater retired people supposes a larger population that depends on pensions. Likewise, higher unemployment rates imply a greater number of citizens with small (or null) personal income. The method used to evaluate the productivity of PPP in [10] is based on fitting the efficient productive frontier by mixing corrected least-squares method (CLS) and logit regression. Hence, for a year *t* it is estimated:

$$\log \text{it}(RRP\_{i,t}) = \beta\_{0,t} + \beta\_{1,t} SER\_{i,t} + \beta\_{2,t} GI\_{i,t} + \varepsilon\_{i,t} \tag{9}$$

where the error term accomplishes *<sup>ε</sup>i*,*<sup>t</sup>* ≥ 0, *i* = 1, 2, ... , 28. After adjusting the value of *β*0,*t*, *β*1,*t* and *β*2,*t* with corrected least squares, *β<sup>F</sup>*0,*t* , *β<sup>F</sup>*1,*t* , *β<sup>F</sup>*2,*t*, the estimate of the productive frontier value of *RRPi*.*t*, *RRP<sup>F</sup> i*,*t* is:

$$RRP\_{i,t}^F = \frac{1}{1 + \exp\left(-\beta\_{0,t}^F - \beta\_{0,t}^F SER\_{i,t} - \beta\_{0,t}^F GI\_{i,t}\right)}\tag{10a}$$

Hence, Debreu–Farrell efficiency measure for *i*th country in the year *t*, *DFi*,*t*, is the ratio between its attained RRP (*RRPi*.*<sup>t</sup>*) in (8) and frontier value of RRP in (10a), *RRP<sup>F</sup> i*,*t*:

$$DF\_{i,t} = \frac{RRP\_{i,t}}{RRP\_{i,t}^F} \tag{10b}$$

where 0 ≤ *DFi*,*<sup>t</sup>* ≤ 1. Hence, *DFi*,*<sup>t</sup>* = 1 imply full efficiency and *DFi*,*<sup>t</sup>* = 0, complete nonefficiency.

Eurostat database provides the values of the variables related to the social protection benefits and poverty of EU-28 countries with an annual periodicity. Therefore, the variables RRP and DF are calculated with this periodicity. To evaluate the results of social policies within a period of more than one year (e.g., a quinquennium), a usual practice consists of taking for the variables the average of their annual observations [8,10,12]. Other papers reduce the analysis to a concrete year [7,11]. A complete analysis consists of repeating it in every year of the period of interest as it is done in Lefevre et al. (2010). Alternatively, we propose quantifying the variables in a period of multiple years by means of TFNs that are built up from observed longitudinal point values of those variables.

Our analysis is done by using SER, the proportion that each kind of social expense suppose over whole SE, the relative reduction of poverty RRP (8) and Debreu–Farrell efficiency index (10) of all countries throughout 2014–2018. We fit for the *i*th country the value of any variable " *A*" (e.g., *SER*) for the whole period 2014–2018 as an FN *Ai* = *Ai*, *lAi* ,*rAi i* = 1, 2, ... , 28. They are fitted from the point observations in each year of the period that we are analyzing, {*<sup>a</sup>*2014, *a*2015, *a*2016, *a*2017, *a*2018} by following the method in Section 2.3. Figure 2a summarizes all the process followed to fit TFNs to the observations on variables embedded in the study. Table 6 shows the fuzzy estimates of SER (*SER i*), RRP (*RRP i*) and efficiency index DF, (*DF* # *i*) in EU-28 countries. Fuzzy values of the proportion that each item of social spending (Sick, Dis, Old, ... ) suppose in overall SE are in Table A1 of annex.

**Figure 2.** Flowcharts for the analysis of poverty policies in UE-28. (**a**) Fitting fuzzy estimates of variables for the while period 2014–2018. (**b**) Methodology used to rank PPPs. (**c**) Methodology followed to measure the influence of SER and its composition in the productivity of public poverty policies (PPPs).
