Correcting the population of Brazilian municipalities using the Jackknife model *

Transcription

1 Correcting the population of Brazilian municipalities using the Jackknife model * Pedro Santos and Enlinson Mattos ** Abstract This paper proposes a method for identifying and correcting the observed distortion in the population distribution of Brazilian municipalities present in the Demographic Census data (Litschig, 2012 and Monasterio, 2014). This distortion is characterized by a high concentration of municipalities with populations close to the bands used as criteria for distributing funds to municipalities in Brazil (Municipal Participation Fund, MPF). The proposed method uses two steps. First, we follow Gabaix (1999) and use a model that follows Zipf s Law to estimate the population of cities (ZIPF, 1949), as in Monasterio (2014). Next, using the Jackknife method, which minimizes population deviations from the standard model, we identify the candidate municipalities for the adjustment and replace their population with the expected counterpart, so as not to present significant discontinuities in the population distribution (McCrary, 2008). We found no significant change in the MPF distribution after this correction, which suggests that the observed phenomenon of discontinuous population distribution is small and, more importantly, models using fuzzy RDD (Brollo et al 2011) cannot be completely invalidated. Keywords: Municipal Participation Fund; Demographic Census; Vertical fiscal transfers; Zipf s Law; manipulation of the running variable. JEL codes: H70; H80; C40. * Submitted on 16/09/2015. Revised on 17/07/2017. We are grateful for the comments of an anonymous reviewer, Paulo Arvate and Leonardo Monasterio. Any errors are the authors responsibility. ** Sao Paulo School of Economics. Getulio Vargas Foundation. pcostasantos@gmail.com and Enlinson.mattos@fgv.br. Brazilian Review of Econometrics v. 38, n. 1, pp May 2018

2 Pedro Santos and Enlinson Mattos 1. Introduction The main instrument for distributing federal government resources to municipalities in Brazil is the Municipal Participation Fund (MPF), which considers population the only criterion for distribution to those municipalities with fewer than 142,633 inhabitants. Moreover, the distribution itself is carried out discretely, in which population bands (up to 10,188, between 10,188 and 13,584, etc.) are used to determine the quantity of resources granted, with there being a discontinuous increase in transfers for each new band reached. This becomes an incentive for municipalities to seek strategies to increase their population, in light of the gains they would obtain once the minimum population of the next band was reached. This distortion was initially studied in Litschig (2012) and Monasterio (2014), in which evidence was found from 1991 onwards that there was significance in the discontinuity of the density of municipalities specifically at the MPF band changes. This paper uses a strategy to correct the distortion in the population distribution of Brazilian municipalities. This distortion is characterized by a concentration of municipalities with population values close to the band changes of the Municipal Participation Fund (MPF), present in the data from the Demographic Census. The proposed method uses two steps. First, we follow Gabaix (1999) and use a model that follows Zipf s Law to estimate the population of cities (ZIPF, 1949), which is similar to the strategy found in Monasterio (2014). Next, using the Jackknife model, which minimizes population deviations from the model, we identify the candidate municipalities for adjustment and replace their populations with the estimated counterpart so as not to present significant discontinuities in the population distribution (McCrary test, 2008). This question is highly relevant in the area of public finance. As the rule for distributing funds to small Brazilian municipalities is based discontinuously on population, various studies use this discontinuity as an identification strategy to estimate the impact of an increase in municipality resources over some economic variable such as corruption (Brollo et al, 2013), education and poverty (Litshig and Morrison, 2013), public spending (Arvate et al, 2011, 2015, Castro and Regattieri, 2014) etc. 2 Brazilian Review of Econometrics 38(1) May 2018

3 Correcting the population of Brazilian municipalities using the Jackknife model The main contribution from this study consists of using an outlier detection method (Jackknife) to iteratively identify and treat units in a sample that do not follow a particular distribution or rule of growth. It is worth highlighting the similarity of our strategy with the use of instrumental variables. Zipf s Law (Aipf, 1949) works here as our instrument for positioning the cities in the distribution. That is, it is an exogenous rule used as an instrument for the municipalities identified as outliers. The success of using this method is then measured by comparing the discontinuities before and after its application. After successfully reconstructing the population distribution, it is then estimated what the MPF distribution would be with this corrected population. The object of the study are municipalities with up to 40,000 inhabitants, which corresponds to 6 different population bands. As in Monasterio (2014), these were chosen because they have the greatest incentives to overestimate their populations. The data used came from the 2000 Demographic Census, from the 2007 Federal Court of Auditors (TCU), and the 2010 Demographic Census. 1 Our results derived from implementing the McCrary (2007) test suggest that the Jackknife and correction methods were effective in identifying and eliminating discontinuity in the population distribution. On the other hand, our results suggest that this municipal population correction does not appear to significantly affect the MPF distribution. This evidence indicates the use of the fuzzy RDD regression, where the running variable in this model is the municipality s population and the MPF instrument would be what the municipality should receive in accordance with the law. The paper is divided into four sections besides this introduction. In the next section, we present the institutional background regarding the MPF and Zipf s Law. In the following section, we discuss the methodology and the data. In section 4, we present the results, and then we present the conclusion. 1 Also see Eggers et al (2015) and Gerard et al (2015) for possible solutions for running variables with possible manipulation problems. Brazilian Review of Econometrics 38(1) May

4 Pedro Santos and Enlinson Mattos 2. Background regarding the MPF distribution and Zipf s Law (1949) According to Mendes (2008), the MPF is a redistributive transfer. That is, resources are sent to each municipality in accordance with a previously determined distribution formula, which has no relationship with the amount of taxes raised in the municipality itself. [...] The distribution is mainly carried out in accordance with municipal population. Its first definition occurred on December 1 st 1965 in Constitutional Amendment n. 18, in article 21, applied to the 1946 Constitution. It consisted of 10% of receipts from IPI (tax on industrialized products) and from IR (income tax). Its distribution began in 1967, after regulation via the National Tax Code (October 25 th 1966). At the time, the only criterion for calculating the portion sent to each municipality was its respective population. On February 28 th 1967, via Complementary Act of the President of the Republic n. 35, the municipalities were categorized into Capitals (these would receive 10% of the MPF) and Interior (these would receive the remaining 90%). And on August 27 th 1981, in Decree-Law n. 1,881, a new category of municipality was created, called Reserve, which would consist of those with populations greater than 156,216 inhabitants. Thus, the division of the MPF amount was changed (Capitals: 10%, Reserve: 3.6%, and Interior: 86.4%). In the 1988 Constitution an increase was declared in the amount (at the time) of IPI and IR raised, from 17% to 22.5%, valid from 1993 onwards. Later, on September 20 th 2007, via Constitutional Amendment n. 55, this value rose to 23.5%. Specifically regarding the distribution of resources of Interior municipalities, the methodology takes the form described in Decree-Law n. 1,881 (August 27 th 1966) and ratified on December 28 th 1989 by Complementary Law n. 62. Two main criteria are taken into consideration: the Share of the States (re-released annually) and the Coefficients of the Municipalities. Thus, the distribution takes the following form: of the total MPF amount (100%), 86.4% is sent to the Interior municipalities. From this value the percentage corresponding to the Share (Table 1) of the State in question is extracted. Finally, to determine the value sent to the Municipalities, the coefficient of each of them is divided by the sum of the coefficients of all the Municipalities in the State (Table 2). In general, the payment made to each Municipality i and State j is given by: 4 Brazilian Review of Econometrics 38(1) May 2018

5 Correcting the population of Brazilian municipalities using the Jackknife model Table 1 Share of the States in the MPF State Share % State Share % Acre Paraíba Alagoas Paraná Amapá Pernambuco Amazonas Piauí Bahia Rio de Janeiro Ceará Rio Grande do Norte Espírito Santo Rio Grande do Sul Goiás Rondônia Maranhão Roraima Mato Grosso Santa Catarina Mato Grosso do Sul São Paulo Minas Gerais Sergipe Pará Tocantins Source: TCU Resolution n o 242/90, January 2 nd 1990 apud STN (2012) Table 2 Coefficient of Interior Municipalities in the MPF Population Coef Population Coef Up to 10, From to From 10,189 to 13, From to From 13,585 to 16, From to From 16,981 to 23, From to From 23,773 to 30, From to From 30,565 to 37, From to From 37,357 to 44, From to From 44,149 to 50, From to From 50,941 to 61, Over Source: Decree Law n o 1.881/1981 apud STN (2012) Brazilian Review of Econometrics 38(1) May

6 Pedro Santos and Enlinson Mattos Payment i,j = FPM 86.4 Payment j Coefficient i i j Coefficient i Note that as we are considering municipalities with up to 40,000 inhabitants, from Table 2 this is equivalent to considering 6 different population levels, and therefore 6 coefficients (0.6, 0.8, 1.0, 1.2, 1.4, and 1.6). (1) 2.1 Problems resulting from the MPF payment calculation Applying this methodology to distribute resources implies pros and cons (in the fiscal gap, in fiscal responsibility, in efficient management, among others) discussed in Mendes (2008). As the focus of this paper, one of these points becomes relevant: how municipalities with small populations are encouraged to behave strategically in relation to the MPF distribution rules. Two examples are (i) small municipalities subdividing in order to increase the income per capital derived from this source (Mendes, 2008, Mattos and Ponczeck, 2013, and Arvate et al, 2015) and (ii) encouraging population increases, whether via migratory policies, mobilization of the population for recensusing, or deliberate frauds in the population censuses (Monasterio, 2014) with the aim of reaching the next population band and thus increasing the income per capital obtained. This second example can be observed in the population distribution of the 2010 Demographic Census, in which there is a high concentration of municipalities immediately after MPF band changes, represented in Graph 1 by the vertical lines. 2.2 Exogenous instrument: Zipf s Law As it is difficult to find the (exogenous) determinants of the population of a municipality in order to identify what the population of that unit really should be, our hypothesis is to follow Zipf s Law (Zipf, 1949), which consists of the relationship between the size of an observation unit and its position in relation to the other units in the same sample (ranking), and this relationship becomes linear when taking into consideration the logarithm of the two variables, represented by the model below: log(r i) = a + b log(p i) + ε i (2) 6 Brazilian Review of Econometrics 38(1) May 2018

7 Correcting the population of Brazilian municipalities using the Jackknife model Figure 1 Histogram of the 2010 population census Source: IBGE (2011), Monasterio (2014), and the authors calculations where R i is the ranking of observation i in relation to the other units in the sample, P i is the value of the variable of this observation i, and [25B?] i is the error. Also in Zipf (1949), one of the demonstrated applications of this law is in the population of cities. 2 In Gabaix (1999) this evaluation goes one step further, by explaining why this phenomenon repeats in different countries with different structures and antecedents. Graph 1 thus reveals the subsequent concentration of band changes and suggests the influence of non-natural factors in the distribution of the population of Brazilian municipalities. We follow the authors above for calculating the estimated population, as shown below. 3. Methodology Our empirical strategy consists of four steps. The first concerns the use of a linear model that seeks to estimate the positioning of the cities within the distribution when it follows Zipf s Law (Zipf, 1949). Next, we use the Jackknife method to identify the municipalities with the greatest population distortions in relation to that estimated by the model, in order to adjust their populations and thus eliminate the discontinuity in the population distribution (McCrary, 2008). 2 Monasterio (2014) also adopts a similar procedure to the one adopted by us in this stage. Similarly, we use the fourth degree polynomial to obtain an adequate adjustment. Brazilian Review of Econometrics 38(1) May

8 Pedro Santos and Enlinson Mattos 3.1 Linear model We calculated a linear model with the following format (similar to Monasterio, 2014): ln (population i ) = β 0 + β 1 X 1 i + β 2 X 2 i + β 3 X 3 i + β 4 X 4 i + D a i + D p i + ε i (3) where population i is the population of municipality i in the observed sample, X i is the position ( rank ) of municipality i with relation to its population within the observed sample, β j are the linear coefficients, D a i and D p i are the dummy variables attributed to the municipalities, and ε i are the errors. Unlike Monasterio (2014) 3, the decision was taken to attribute two dummies to each discontinuity of the band, one before (D a i) for municipalities with a population within 97.5% to 100.0% of the MPF cut-off value and another after (D p i) for those within 100.0% to 102.5%. 4 The results of the regression of the model (2) are presented in Table 3 below, using the year All of the variables were statistically significant, suggesting that there is in fact an impact on the concentration of municipalities around the MPF band changes and that this effect may be different depending on whether the municipality is to the right or left of each threshold 6 Once the coefficient is obtained for all of the variables, the estimated population is calculated for each municipality, considering only the Intercept, Rank, Rankˆ2, Rankˆ3, and Rankˆ4 variables. By removing the dummies from the calculation, an approximation is expected to the population of each municipality that would be observed in the absence of distortions The author considers a single dummy variable for each discontinuity. Two dummies (before and after) were considered, since we allowed for different intercepts in relation to the average ( β 0. With this we are able to observe not only whether there is a difference in average at the distribution band transition threshold, but whether this is different before and after this frontier. The estimation with a single dummy does not alter our results and this is available from the authors on request. Results are robust to the choice of other bands. 1.5% and 3.5% were tested and the results are statistically similar. They are available from the authors on request. The results for the other years can be made available by the authors on request. They are all submitted as supplementary material. Monasterio (2014) had found similar results, but with an absence of significance in the fifth and sixth dummies. His evaluation covered municipalities with up to 50,000 inhabitants. 8 Brazilian Review of Econometrics 38(1) May 2018

9 Correcting the population of Brazilian municipalities using the Jackknife model Table 3 Linear model of the dependent variable ln(population) of 2010 Coefficients Estimation Standard deviation Intercept *** Rank *** Rankˆ *** Rankˆ *** Rankˆ *** dummy previous *** dummy previous *** dummy previous *** dummy previous ** dummy previous ** dummy previous ** dummy posterior *** dummy posterior *** dummy posterior *** dummy posterior *** dummy posterior *** dummy posterior *** N = ,000 < pop <40,000 R 2 adj = 0,94 Note: *** 1% significant; ** 5% significant; and * 10% significant. Standard deviation in the second column. Brazilian Review of Econometrics 38(1) May

10 Pedro Santos and Enlinson Mattos The obtainment of the estimated population (all of the estimated betas above) of municipality i takes the format below: estimatedpopulation i = e (β 0+β 1 X 1 i +β 2 X 2 i +β 3 X 3 i +β 4 X 4 i ) (4) Graph 2 below shows the difference between the observed and estimated population. A gradual increase in the deviations (or residuals) is clearly noted, consistently reaching their highest local values where the population lies on the margins of each MPF transfer discontinuity (vertical lines). Figure 2 Deviations by municipality, 2010 Source: IBGE (2011) and the authors calculations Thus, the quantity of municipalities in each of the bands can be identified and this can be compared with which band they should rightly be in, in accordance with the estimate without distortions. This comparison can be seen in Table 4. It is important to highlight that the example given above is the one that presents the greatest number of municipalities indicated outside of their estimated band, totaling 267. This corresponds to slightly more than 7% of all the municipalities with fewer than 40,000 inhabitants. A similar strategy was used in Monasterio (2014). This article aims to identify which municipalities contribute more to the 10 Brazilian Review of Econometrics 38(1) May 2018

11 Correcting the population of Brazilian municipalities using the Jackknife model Quantity of observed municipalities by band Table 4 Quantity of municipalities by MPF band Quantity of estimated municipalities by band Source: IBGE (2011) and the author s calculations deviation (from the estimated position) than others, and will have their population replaced with the one estimated in the model, as shown below. 3.2 Jackknife Method The Jackknife method aims to estimate estimation bias by dividing a sample into different subsamples and it is generally used to construct variance estimators and because of its versatility it has been applied for various purposes. Its use follows a simple process. The statistic of interest is calculated using the whole sample. Next, one element is eliminated from the sample (obtaining a new dataset, n-1) and the same statistics are recalculated. individually. This is done for all the elements in the sample The difference between the initial statistic of the sample and that obtained by removing each element explains the individual influence on the value of the statistic of the entire sample. 7 With this magnitude of influence, it can be determined which elements have the most impact on the original statistic and then they can be treated. Thus, this method has also come to be used in identifying outliers, enabling their adequate treatment. For example, in an efficient frontier context, Sousa (2004) and Sousa and Stosic (2005) identify municipalities in which there could be errors in the collection and storage of the data or even scale effects of those municipalities. Detecting and possibly excluding outliers enabled the researchers to obtain a more consistent 7 This procedure has already been used to correct outliers in other economic studies. See, for example, Souza and Stocic (2005). Brazilian Review of Econometrics 38(1) May

12 Pedro Santos and Enlinson Mattos estimate of the efficiency in the provision of services by some of the Brazilian municipalities. Our next step is to obtain the impact that each of the municipalities is causing on the population distribution. For this, the Jackknife method will be used, which identifies the magnitude of the impact of a municipality in the adequacy of the linear model proposed in the previous step and shows which ones should be prioritarily adjusted in order to obtain a distribution with fewer deviations (McCrary, 2008). The steps for applying it are (1) define a J function that will be applied to the whole sample and (2) for each element in the sample, calculate the value that the J i function takes when the element in question is exclusively eliminated. For this use, the standard deviation function was chosen. Thus, each municipality i would have a related value J i in the format: J i = N j=1;i j (residual j residual ) 2 N 1 J i is the value resulting from the Jackknife of the residual of the i-th municipality in the sample, N is the sample size, and the numerator corresponds to the deviation (residual = observed estimated) in relation to the average of deviations ( residual ) when observation i is excluded. Note that the deviation attributed to element i is found in the previous step, and notice that j covers all of the elements from 1 to N (except i). Thus, the value obtained for each J i is the standard deviation of the sample without element i. The value obtained for each municipality can be observed in Graph 2. There is similarity between the municipalities with greater deviations in Graphs 2 and 3. However, the second enables a comparison of how much each element is distorting the statistics of the sample as a whole, and we are thus able to attribute different weights to each municipality depending on how far they are from their estimated population. 3.3 Elimination of the impact After obtaining the impacts of each municipality, regarding their contribution to the deviations in relation to Zipf s Law (Aipf, 1949) we have to eliminate which treatment should be given to each of these points. One of the options would be to (5) 12 Brazilian Review of Econometrics 38(1) May 2018

13 Correcting the population of Brazilian municipalities using the Jackknife model Figure 3 Standard deviation (Jackknife) Source: IBGE (2011) and the authors calculations remove these municipalities from the sample (donut hole strategy, see Barreca, et al, 2011). However, this method can affect the continuity of the data exactly at the most important points: the MPF band changes (see Arvate, Mattos, and Rocha, 2015). Thus, the decision was taken to replace their observed populations with the populations estimated using the linear model without the dummies. The process for choosing the municipalities was the following: First, (1) determine which municipalities had a population within the intervals [MPF i - d, MPF i + d], where MPF i is the value of each band shift and d is the value of the interval chosen to contain some of the municipalities that could be affected by the distortions. We choose 2.5% as the benchmark. 8 Next, (2) determine the accumulated density distribution of the municipalities ordered by the standard deviations (Jackknife) of the residuals (Graph 4). With this, we can (3) calculate the standard deviation of the residuals of the original sample. This will be called the initial reference sample. Within the iterative process below, this will be the first value that the variable will take. 8 Results robust to other choices, 1.5% and 3.5%. Brazilian Review of Econometrics 38(1) May

14 Pedro Santos and Enlinson Mattos Figure 4 Accumulated density of the standard deviations (Jackknife) by municipality Source: IBGE (2011) and the authors calculations Then, (4) a value λ {0, 1} is established in the accumulated density distribution. All of the municipalities whose accumulated density is less than λ will have their population adjusted; that is, the municipalities which after their exclusion generated low accumulated densities of deviations will have their population adjusted. After this, (5) the municipality data corrected to the level of λ are saved. This is (6) iterated by adding the value of α (0.05) to λ, and we return to step 4 above. For the calibration, the following variable values were chosen: d = 1500 λ (initial) = 0.00 λ (final) = 0.50 α = 0.05 After the iterations, the result obtained is a table of data for each year (2000, 2007, 2010) with the population distributions corrected to each value attributed to λ (from 0 to 0.50, α = 0.05). 14 Brazilian Review of Econometrics 38(1) May 2018

15 Correcting the population of Brazilian municipalities using the Jackknife model 3.4 McCrary Method McCrary (2008) develops a method for identifying the manipulation of a variable by evaluating its density function. It is based on a discontinuity estimator measured at a particular point of interest, in order to ultimately verify whether there is a distortion there. The process for defining this estimator follows two steps. The first consists of constructing a histogram of the sample to be verified so that each one of its classes (bins) contains points from only one of the sides of the discontinuity. That is, one of the histogram cut-offs should occur exactly at the evaluated points of interest (called c). The second step concerns the linear smoothing of the histogram (LOWESS - locally weighted scatterplot smoothing), where the average points of each class are the regressors (X j ) and the normalized count of the number of observations of each class is the variable of interest (Y j ). The normalization is calculated in the following way: Y j = 1 nb n 1(g (R i)) = X j (6) i=1 where Y j is the value of the normalized variable of interest, b is the length of each class of the histogram, n is the quantity of classes, the sum provides the count of the number of observations within the j-th class, and R i is the running variable, which in this case is population. In order to obtain clarity in the visualization of the data, two estimated smoothings (f) are calculated, one for each side of the point (c) where the discontinuity is being investigated. Thus, in order to obtain the estimated value of the discontinuity, the difference is calculated between the value that each smoothed function (to the right and to the left) would take at point c. 9 Finally, there is the definition of the length of each class (b) and of the width of the band (h). McCrary (2008) suggests different possibilities for obtaining these values numerically. Note that this topic enables it to be evaluated whether there has been a specific effort to concentrate elements of a sample to one side of a particular 9 See McCrary (2009). Brazilian Review of Econometrics 38(1) May

16 Pedro Santos and Enlinson Mattos limit. For this case, the verification will concern the various MPF band changes, checking what the distortion is before and after the adjustment of the populations. This distortion is calculated via the difference in the logarithm of the f function calculated at the point of discontinuity c at the two sides (to the right and to the left). We chose to follow Monasterio (2014) and b was chosen as 283 because it is prime and a divider of all the MPF shift points (Monasterio, p. 185) and for h the automatic procedure proposed by the author (McCrary, 2008) was used, which follows Fan and Gijbels (1996). 10 The next step was to apply the method in two different samples, whose only difference would be the adjustment carried out in the previous steps. Thus, in the example below the 2010 census, the first change in MPF band (10,188 inhabitants), and a λ of 0.50 were chosen. Graphs 5 and 6 visually show how the adjustment influenced the distortion. For the sample prior to the adjustment, the discontinuity estimate (difference between the functions to the right and to the left of point c) was 1.43 with a standard error of After the adjustment, it decreased to 0.10 with a standard error of With this, the statistic ceased to be statistically significant and the evidence of discontinuity disappeared. This phenomenon can also be visually verified by comparing the graphs. 10 The automatic procedure used by McCrary (2008) can be summarized in the following steps: A). Calculate an initial histogram, which will be used as a basis for the calculations, using a bin size estimated by the algorithm or provided by the programmer. In the case of this study, the value provided was 283. B). Using the initial version of the histogram, the fourth order polynomial is estimated separately for both sides of the discontinuity. Based on this, for each side the value of the expression θ [ σ 2 (b a) / f (X j) 2] 1/5 is calculated and the h value is determined as its average. θ is equal to 3.348, σ 2 is the average square error of the regression, b a is equal to X j c for the right side of the regression and c X 1 for the left side and f (X j) is the estimate of the second order derivative of the polynomial found for the overall distribution. C). The second step of this algorithm is based on a bandwidth selection rule developed by Fan and Gijbels. With the bandwidth determined, the initial histogram (calculated with the pre-determined bin and the f (r) curve based on the value of h provides a detailed overview of the distribution of the variable around the point of discontinuity. We tested the Cross-Validation option proposed by Stone (1974) and the results are similar. 16 Brazilian Review of Econometrics 38(1) May 2018

17 Correcting the population of Brazilian municipalities using the Jackknife model Figure 5 Initial database (2010 census, first change in band) Source: IBGE (2011) Figure 6 After adjustment, λ to 0.50 (2010 census, first change in band) Source: IBGE (2011) and the authors calculations Brazilian Review of Econometrics 38(1) May

18 Pedro Santos and Enlinson Mattos 4. Results After applying our methods to the population samples from the 2000 and 2010 Demographic Censuses and the 2007 TCU data, we obtained the alterations in the discontinuity estimates for each municipality whose exclusion considerably reduced the standard deviation of the deviations from the estimated population. Their respective values can be observed in Tables 5, 6, and 7, as well as the standard error found and its significance. It can be observed that for each band, the λ needed to remove the significance of the discontinuity is different, and whose value is highlighted in each column. Moreover, depending on the increase in the λ the significance of the difference in the statistic of the functions to the right and to the left reduces more and more, in a monotonic way. This only does not repeat in the sixth MPF income band and for the year 2000 (λ = 0). However, for all of the λ values after this same band, monotonicity is shown. More importantly, except for the fourth 2010 MPF band, we can highlight that with a λ = 0.30 we would eliminate all of the population discontinuities observed in the different MPF bands, via the t test. Moreover, we detect which municipalities can be considered as outliers via this λ. By attributing the population estimated by Zipf s Law (Zipf, 1949) for these municipalities in accordance with that predicted by equation (3) we can note, also graphically, that the population distributions represent continuity again in those MPF bands. 11 Tables 5, 6, and 7 also indicate that with λ = 0.30, we could correct all of the population distortions observed in the small municipalities, except close to band 4 of the 2010 Census, when our procedure does not appear to be efficient. Note that in these tables for each band we highlight the λ that guarantees the absence of statistical difference in the municipality density histogram between the two sides of the population bands. One possible interpretation of our method is that we use a completely exogenous variable from the point of view of the municipalities (Zipf, 1949) to estimate the esti- 11 Graphs contained in the supplementary file. 18 Brazilian Review of Econometrics 38(1) May 2018

19 Correcting the population of Brazilian municipalities using the Jackknife model mated (instrumentalized) population which, in turn, is replaced in the municipalities whose population can be considered endogenous by the Jackknife method. Finally, Table 8 indicates how many municipalities should have their population corrected for each λ for each population band in 2000, 2007, and As can be observed, we would correct the sample of 302, 319, and 337 municipalities respectively for 2000, 2007, and 2010, for a λ of 0.30, which would correct all of the distortions found (except in band 4 of the 2010 census). This strategy would indicate the upper limit of alterations, since we could allow different λ s for each band and year MPF Distribution per capita With both the original and corrected distributions, we investigated what happens with the MPF distribution per capital granted to the municipalities in the two cases. The value obtained corresponds to the share that each municipality would have in the total MPF distributed in the year. 13 Once the MPF variables were calculated for the two populations, we considered the population variable as independent and MPF per capita itself as a dependent variable in a fourth degree linear regression. For each MPF band a specific polynomial was calculated in order to obtain the best smoothing possible of the values of each group of municipalities. A confidence interval of one standard deviation was added to the graphs for comparison. As a database for the calculation the two population samples from each year (2000, 2007, and 2010) were used, the original one with λ = 0.00 and the corrected one with λ = 0.30, which as verified is a sufficient value to remove the discontinuities. The resulting graphs, with the MPF per capita per municipality and their respective values expected by the exponential smoothing, can be observed below. It can be visually observed that their does not appear to be any difference between the two strategies for calculating the MPF, especially at the margins of the population bands, where our procedure acted; that is, the effect of correcting the distortion in the discontinuity of the population distribution does not appear to cause 12 By allowing for different λ s, the municipalities that would have their populations readjusted would be 154, 184, and 260, respectively. 13 The value was multiplied by 10ˆ6 for a better representation of the data. Brazilian Review of Econometrics 38(1) May

20 Pedro Santos and Enlinson Mattos Table 5 Value and significance of the MPF discontinuities (2000) Lambda 0% 5% 10% 15% 20% 25% 30% 35% 40% 45% 50% Statistics Year: Mean 0,500 0,284 0,491 0,570 0,440 0,097 Standard error 0,105 0,117 0,150 0,193 0,236 0,293 Test 4,748*** 2,427** 3274*** 2,947*** 1,862* 0,329 Mean 0,304 0,281 0,371 0,492 0,373-0,907 Standard error 0,099 0,116 0,144 0,188 0,240 0,324 Test 3,071*** 2,437** 2,568** 2,612*** 1,553 2,8*** Mean 0,209 0,285 0,186 0,204 0,231-0,575 Standard error 0,096 0,116 0,134 0,171 0,244 0,335 Test 2,18** 2,461** 1,389 1,193 0,95 1,717* Mean 0,150 0,289 0,103 0,139 0,156-0,567 Standard error 0,094 0,116 0,129 0,162 0,237 0,336 Test 1,597 2,487** 0,801 0,856 0,657 1,69* Mean 0,110 0,246 0,069 0,097 0,124-0,543 Standard error 0,093 0,115 0,126 0,155 0,232 0,338 Test 1,185 2,136** 0,549 0,624 0,535 1,608 Mean 0,091 0,192 0,016 0,048 0,114-0,511 Standard error 0,093 0,114 0,122 0,128 0,230 0,340 Test 0,987 1,687* 0,134 0,376 0,495 1,503 Mean 0,088 0,144-0,039 0,053 0,102-0,498 Standard error 0,093 0,112 0,116 0,122 0,229 0,341 Test 0,949 0,1292 0,34 0,433 0,446 1,458 Mean 0,089 0,128-0,048 0,059 0,098-0,498 Standard error 0,093 0,111 0,115 0,121 0,227 0,341 Test 0,953 1,151 0,419 0,485 0,431 1,457 Mean 0,081 0,096-0,038 0,021 0,077-0,493 Standard error 0,094 0,112 0,118 0,133 0,225 0,342 Test 0,858 0,861 0,318 0,161 0,341 1,439 Mean 0,078 0,080-0,029 0,022 0,119-0,493 Standard error 0,095 0,112 0,120 0,134 0,224 0,342 Test 0,826 0,714 0,245 0,167 0,532 1,439 Mean 0,077 0,037-0,021 0,032 0,279-0,493 Standard error 0,095 0,110 0,122 0,131 0,237 0,343 Test 0,809 0,334 0,175 0,245 1,18 1,438 Significance values based on the Student t test 20 Brazilian Review of Econometrics 38(1) May 2018

21 Correcting the population of Brazilian municipalities using the Jackknife model Table 6 Value and significance of the MPF discontinuities (2007) Lambda 0% 5% 10% 15% 20% 25% 30% 35% 40% 45% 50% Statistics Year: Mean 0,813 0,581 0,496 0,578 0,180 0,753 Standard error 0,111 0,122 0,144 0,191 0,213 0,279 Test 7,293*** 4,744*** 3,442*** 3,02*** 0,844 2,698*** Mean 0,119 0,267 0,105 0,586 0,180 0,713 Standard error 0,093 0,114 0,125 0,196 0,211 0,301 Test 1,284 2,347** 0,843 2,987*** 0,855 2,365** Mean 0,103 0,107-0,102 0,542 0,159-0,013 Standard error 0,092 0,106 0,125 0,195 0,212 0,289 Test 1,118 1,003 0,81 2,785*** 0,752 0,044 Mean 0,105 0,010-0,132 0,379 0,128-0,052 Standard error 0,092 0,104 0,128 0,184 0,211 0,295 Test 1,136 0,095 1,034 2,059** 0,607 0,175 Mean 0,112-0,012-0,118 0,312 0,138-0,027 Standard error 0,093 0,104 0,128 0,179 0,212 0,298 Test 1,204 0,111 0,924 1,743* 0,651 0,09 Mean 0,112-0,025-0,102 0,190 0,168-0,016 Standard error 0,093 0,105 0,127 0,171 0,216 0,301 Test 1,2 0,241 0,805 1,114 0,778 0,054 Mean 0,116-0,030-0,087 0,104 0,205 0,000 Standard error 0,093 0,105 0,125 0,164 0,219 0,303 Test 1,247 0,289 0,692 0,632 0,933 0 Mean 0,115-0,032-0,078 0,106 0,226 0,006 Standard error 0,093 0,105 0,124 0,164 0,221 0,303 Test 1,243 0,304 0,631 0,642 1,024 0,02 Mean 0,110-0,030-0,067 0,102 0,247 0,014 Standard error 0,094 0,105 0,123 0,164 0,223 0,303 Test 1,176 0,281 0,543 0,623 1,11 0,045 Mean 0,104-0,035-0,060 0,095 0,254 0,042 Standard error 0,094 0,107 0,123 0,164 0,224 0,300 Test 1,103 0,325 0,486 0,58 1,135 0,139 Mean 0,096-0,037-0,049 0,093 0,301 0,041 Standard error 0,095 0,108 0,123 0,164 0,227 0,300 Test 1,004 0,343 0,398 0,566 1,325 0,137 Significance values based on the Student t test Brazilian Review of Econometrics 38(1) May

22 Pedro Santos and Enlinson Mattos Table 7 Value and significance of the MPF discontinuities (2010) Lambda 0% 5% 10% 15% 20% 25% 30% 35% 40% 45% 50% Statistics Year: Mean 1,427 0,653 0,792 0,877 1,460 0,964 Standard error 0,131 0,121 0,145 0,199 0,280 0,327 Test 10.,904*** 5,419*** 5,452*** 4,401*** 5,206*** 2,948*** Mean 0,242 0,728 0,573 0,894 1,417 1,000 Standard error 0,093 0,126 0,139 0,204 0,281 0,338 Test 2,614*** 5,784*** 4,128*** 4,39*** 5,049*** 2,958*** Mean 0,133 0,375 0,258 0,893 0,808 0,862 Standard error 0,088 0,115 0,123 0,203 0,238 0,321 Test 1,506 3,271*** 2,09** 4,39*** 3,395*** 2,686*** Mean 0,133 0,282 0,165 0,882 0,547 0,331 Standard error 0,088 0,111 0,115 0,201 0,225 0,287 Test 1,518 2,544** 1,438 4,395*** 2,43** 1,154 Mean 0,123 0,209 0,107 0,801 0,450 0,242 Standard error 0,088 0,108 0,112 0,197 0,225 0,284 Test 1,41 1,927* 0,95 4,07*** 1,994** 0,853 Mean 0,118 0,17 0,095 0,736 0,215-0,327 Standard error 0,088 0,107 0,114 0,195 0,228 0,281 Test 1,339 1,59 0,835 3,781*** 0,943 1,167 Mean 0,116 0,073-0,003 0,650 0,124-0,399 Standard error 0,09 0,104 0,119 0,191 0,229 0,285 Test 1,29 0,705 0,027 3,398*** 0,541 1,4 Mean 0,114 0,038-0,057 0,606 0,066-0,362 Standard error 0,091 0,102 0,123 0,190 0,230 0,293 Test 1,254 0,375 0,468 3,196*** 0,287 1,238 Mean 0,113 0,014-0,117 0,475 0,003-0,312 Standard error 0,091 0,101 0,127 0,183 0,230 0,299 Test 1,242 0,14 0,923 2,596*** 0,014 1,045 Mean 0,112 0,002-0,117 0,380 0,011-0,252 Standard error 0,092 0,101 0,127 0,177 0,228 0,301 Test 1,214 0,023 0,922 2,147** 0,049 0,839 Mean 0,11 0-0,119 0,297 0,014-0,211 Standard error 0,093 0,101 0,128 0,173 0,225 0,301 Test 1,186 0,001 0,925 1,716* 0,061 0,7 Significance values based on the Student t test 22 Brazilian Review of Econometrics 38(1) May 2018

23 Correcting the population of Brazilian municipalities using the Jackknife model Table 8 Number of municipalities with altered population, by MPF band and year 2010 Census - Number of municipalities with altered population by band Lambda Alterations Total Census - Number of municipalities with altered population by band Lambda Alterations Total Census - Number of municipalities with altered population by band Lambda Alterations Total Brazilian Review of Econometrics 38(1) May

24 Pedro Santos and Enlinson Mattos Figure 7 MPF per capita, original population, 2000 Source: IBGE and the authors calculations an impact on the distribution of the MPF per capita for the municipalities. This is due to the fact that only some municipalities (around 300) have their population corrected. Thus, our proposed method appears to be robust in that it effectively manages to correct an anomaly observed in the sample, without causing other undesirable effects due to the methodology for distributing the fund among the other municipalities. Another possible interpretation concerns the fact that the use of estimated populations to correct this anomaly observed in the population of Brazilian municipalities appears to be a promising strategy. More importantly, the strategies that use estimated MPF (Fuzzy RDD) should not necessarily be discarded. 6. Conclusion The existence of a distortion in the municipality populations close to the MPF band changes was verified in Litschig (2012) and Monasterio (2014). The authors also identify that there appeared to be an increase in the population of the municipalities, which could cause their displacement into the next bands, with the aim of increasing 24 Brazilian Review of Econometrics 38(1) May 2018

25 Correcting the population of Brazilian municipalities using the Jackknife model Figure 8 MPF per capita, corrected population, 2000 Source: IBGE and the authors calculations Figure 9 MPF per capita, original population, 2007 Source: IBGE and the authors calculations Brazilian Review of Econometrics 38(1) May

26 Pedro Santos and Enlinson Mattos Figure 10 MPF per capita, corrected population, 2007 Source: IBGE and the authors calculations Figure 11 MPF per capita, original population, 2010 Source: IBGE and the authors calculations 26 Brazilian Review of Econometrics 38(1) May 2018

27 Correcting the population of Brazilian municipalities using the Jackknife model Figure 12 MPF per capita, corrected population 2010 Source: IBGE and the authors calculations the transfer per capita obtained. This raises the discussion of how adequate the current criteria would be for distributing the Municipal Participation Fund (MPF) and whether population could be correctly used as a running variable in a Regression Discontinuity strategy. This study contributes in two ways. It uses an exogenous rule (Zipf s Law, 1949) to quantify the expected population of each municipality. Based on this expected population, we use the Jackknife method to identify the municipalities that contribute most to the deviations from what is expected. For these municipalities, we apply the expected population. By applying our method in the samples from the 2000 Demographic Census, from the 2007 TCU, and from the 2010 Demographic Census, a significant improvement was obtained in the population distortion of municipalities with up to 40,000 inhabitants around the MPF band changes. Originally, in 16 of the 18 band changes a statistically significant distortion was observed, showing some form of distortion of the variable. After applying the method for identifying and adjusting the municipalities, we determined in an exploratory way what the optimal adjustment point Brazilian Review of Econometrics 38(1) May

28 Pedro Santos and Enlinson Mattos would be using the McCrary (2008) test and thus managed to (almost) eliminate the totality of the distortion in the population of the municipalities for the three observed Censuses. With this, the distortion was smoothed, enabling the obtained sample to be used for purposes that take into account the population order and relationship between Brazilian municipalities. As can be seen in Tables 5, 6, and 7, over the years the distortions only increased (visible via the increasing value of the discontinuities for λ = 0) - and this can concentrate the MPF distribution more and more in those municipalities identified in our strategy (Mendes, 2008). In particular, we estimate that around 300 municipalities are contributing to this practice, which suggests the need to review the way that the MPF is distributed to municipalities from the Interior group. On the other hand, when evaluating the MPF per capita of the municipalities before and after the proposed correction, no significant alteration is verified in the values obtained close to the band changes. Thus, the impact on the MPF per capita caused by our strategy does not appear to significantly impact the municipalities that should not be the object of correction. Our results suggest that the use of a sharp RD (regression discontinuity) in models that explain MPF by population may not be appropriate. This is due to the proven discontinuity of the population of the municipalities. However, our strategy leans towards the use of fuzzy RDD type regressions in which the MPF is instrumentalized by its theoretical counterpart in which the corrected population could be considered as a determinant of the amount to be received by each municipality. That strategy could capture the exogenous variation in the transfers for each economic variable of interest. References Arvate, P., Mattos, E. and Rocha, F. (2015). Conditional versus unconditional grants and local public spending in Brazilian municipalities. 35th Meeting of the Brazilian Econometric Society, Foz do Iguacu, Brazil. Arvate, P., Mattos, E. and Rocha, F. (2011). Flypaper effect revisited: Evidence for tax collection efficiency in Brazilian municipalities. Estudos Econômicos, v. 41, p Brazilian Review of Econometrics 38(1) May 2018

29 Correcting the population of Brazilian municipalities using the Jackknife model Barreca, A.; Guldi, M.; Lindo, J.; Waddell, G. (2011) Saving Babies? Revisiting the Effect of the Very Low Birth Weight Classification. Quarterly Journal of Economics, 126. Brollo, F.; Nannicini, T.; Perotti, R.; Tabellini, G. (2013). The Political Resource Curse, American Economic Review, 103(5), p Castro, M. and Regatieri, R. (2014). Impacto do Fundo de Participação dos Municípios sobre os gastos públicos por função e subjunção: analise através de uma regressão em descontinuidade. 42 Encontro Nacional de Economia, Natal, Rio Grande do Norte, Brasil. Eggers, A. C., Freier, R., Grembi, V., and Nannicini, T. (2015). Regression Discontinuity Designs Based on Population Thresholds: Pitfalls and Solutions. Working Paper IZA No. 9553, December. Funaro et al (2009). Diretrizes para apresentação de dissertações e teses da USP: documento eletrônico e impresso. Universidade de São Paulo. Available from: pdf. Accessed on July 10 th Gabaix, X. (1999). Zipf s Law for Cities: An Explanation. The Quarterly Journal of Economics, v.114, n.3, p Gerard, F., Rokkanen, M., Rothe, C. (2015). Identification and Inference in Regression Discontinuity Designs with a Manipulated Running Variable (working paper). IBGE - Instituto Brasileiro de Geografia e Estatística. Censo Brasília, IBGE: Available from: populacao/censo2010/default.shtm. IBGE - Instituto Brasileiro de Geografia e Estatística. Contagem da população Brasília, IBGE: Available from: Brazilian Review of Econometrics 38(1) May

30 Pedro Santos and Enlinson Mattos Litschig, S. (2012). Are rules-based government programs shielded from specialinterest politics? Evidence from revenue-sharing transfers in Brazil. Journal of public Economics 96.11: Litschig, S.; Morrison, K. (2013). The impact of intergovernmental transfers on education outcomes and poverty reduction. American Economic Journal: Applied Economics. Mattos, E. and V. Ponczek (2013). Efeitos da divisão municipal na oferta de bens públicos e indicadores sociais. Revista Brasileira de Economia 67 (3), Mccrary, J. (2008). Manipulation of the running variable in the regression discontinuity design: A density test. Journal of Econometrics 142(2): Mccrary, J. (2009). Codes for Manipulation of the Running Variable. Contém o código para uso no software STATA (DCdensity.ado), chamadas e saídas de um exemplo (DCdensity example.do, DCdensity example.log, DCdensity example.eps) e explicação do código implementado (DCdensity.pdf). Available from: jmccrary/dcdensity. Accessed on June 7 th Mendes, M.; Miranda, R. B.; Cossio, F. B. (2008). O Fundo de Participação dos Municípios precisa mudar. Constituição de 1988: O Brasil 20 anos depois - Estado e economia em vinte anos de mudanças, v.4, Monasterio, L. M. (2014). A estranha distribuição da população dos pequenos municípios brasileiros. Rev. Econ. NE, Fortaleza, v.45, n.4, p , R Core Team. (2014). R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. Available from: S Original, from StatLib and by Rob Tibshirani. R PORT by Friedrich Leisch. (2015). bootstrap: Functions for the Book An Introduction to the Bootstrap. Available from: 30 Brazilian Review of Econometrics 38(1) May 2018