A BAYESIAN INTERPOLATION METHOD TO ESTIMATE PER CAPITA GDP AT THE SUB-REGIONAL LEVEL: LLMS IN SPAIN

Transcription

1 A BAYESIAN INTERPOLATION METHOD TO ESTIMATE PER CAPITA GDP AT THE SUB-REGIONAL LEVEL: LLMS IN SPAIN Domenica Panzera University of Chieti-Pescara, Italy Ana Viñuela Departamento Economía Aplicada, University of Oviedo, Spain AREA TEMATICA Métodos de Análisis Regional RESUMEN Although there is increasing availability of economic data for almost any topic, there is still a dearth of data at very high levels of spatial disaggregation. GDP data (absolute or per capita) is offered at European level for the NUTS II regions and, the Spanish National Statistics Office (INE) publishes regularly such information at the NUTS III level, which in the Spanish case corresponds to provinces. In this paper we undergo to a higher level of spatial disaggregation estimating the per capita GDP for the 804 Local Labor Market Areas (LLMs) in which the Spanish territory can be divided applying the Bayesian Interpolation Method (BIM) introduced by Palma and Benedetti (1998) and including the possible existence of spatial dependence between observations. Our objective is to reconstruct the realizations of the original process (per capita GDP values for LLMs) when having information on the aggregated process (GDP values for the NUTS III regions). Before proceeding with the estimations, first we test the methodology estimating per capita GDP at NUTS

2 III level assuming that information on that variable is only known at a more aggregated level (NUTS III regions). Results for LLMs show a high level of internal heterogeneity on economic welfare within the Spanish regions. PALABRAS CLAVE Income disparities, Bayesian Interpolation Method, Local Labor Market Areas, Missing data, Spain

3 1. Introduction There has been a significant volume of research that has explored the underlying causes of income disparities across regions within a country and the concomutant spatial heterogeneity in economic outcomes. Nowhere is this more apparent than in Spain. When dealing with these issues, regional economists are constrained by the use of administrative data that are at the most available at NUTS III level (provinces in the Spanish case). However, given the internal heterogeneity within the regions (at whatever level), the challenge becomes one of trying to estimate income values for regional systems based on economic rather than administrative criteria. Differences in economic outcomes within administrative regions reflect differences in productivity, employment levels and wages (Moretti, 2011). While in Spain there are employment data are available at the local level (municipalities), even by industry level or by level of qualification, that confirm the internal heterogeneity of the Spanish regions, no data exists on GDP (or productivity) at provincial or local levels. Based on the idea that workers and firms interact only in local labor markets, whose size is much smaller than that of the national market, and that few people move from one market to another (Armstrong and Taylor, 1993; Bartik, 1996; Hughes and McCormick, 1994; Topel, 1986), the Local Labor Markets (LLMs) present themselves as a more suitable economic unit of analysis. In most countries, perhaps only a relatively small number of LLMs account for most of the country's output.. However, economic researchers still face the problem of the lack of GDP figures, either total or per capita, at this level of disaggregation. The purpose of this paper then is to estimate (per capita) GDP for the 806 Local Labor Markets (LLMs) into which the Spanish territory is divided. In the next section (2), we briefly explain the methodology used to delimitate the LLMs and describe the Spanish division into LLMs.. Section 3 provides the details on the Bayesian Interpolation Methodology (hereafter, BIM) used to disaggregate (per capita) GDP for the 50 Spanish provinces (NUTS III regions) into (per capita) GDP for the Spanish LLMs for a single year. The solution of the problem requires formulating a hypothesis for the probability distribution of the original process (LLM data) and exploits information on the spatial dependence between observations as well as auxiliary information such as population and other socio economic variables, that can be available at LLM level. In Section 4, the performance of the proposed method is evaluated by comparing the available data on per capita GDP for provinces with the values obtained by the application of the BIM. Results of the disaggregation of provincial data at the LLMs level are then presented and discussed. Finally, Section 5 suggests future extensions of this work and research agenda on this topic. 2. Local Labor Market Areas A number of researchers, initially in the United States from the 1960s (Fox and Kumar 1965), and then in Europe from the 1970s (Smart 1974), have designed quantitative techniques for identifying regions consistent with the theoretical framework of regional economics. Different names have been applied to these regions, such as Functional Economic Areas and Labor Market Areas (or Local Labor Market), but they all referred to a region that internalizes the home-to-work daily journeys of its residents. Thus, Local Labor Markets (LLM) reflect functional relationships between workers and jobs. Based on commuting patterns, LLMs identify the borders of labor catchment areas. In practice, this means that at least 75% of residents work in the area and that 75% of those who work in the area also live there, with a minimum size of 3,500 resident workers. The regionalization procedure to establish the borders of a LLM is based on an algorithm originally developed by Coombes et al. (1986). Starting out from the municipal administrative unit and combining the data on the resident employed population, total employed population and displacements from the place of residence to

4 the workplace, boundaries of the LLMs are defined through a multi-stage aggregation process. 1 This methodology, slightly modified to meet the specific characteristics of countries, has been applied in several European countries. The Department of Employment has defined the LLMs (so-called Travel-To-Work-Areas or TTWAs) for Great Britain, Sforzi et al. (1997) for Italy, Andersen (2002) for Denmark and Boix and Galletto (2006) for Spain. The 806 LLMs in which the Spanish territory is divided (table 1) show high disparities in size either considering the number of municipalities or population included. The larger ones in population are the Madrid and Barcelona LLMs, which comprise 20.51% of the total Spanish population. More than 85% of the LLMs can be considered rural (i.e., less than 50,000 inhabitants) but only 23.23% of the population live in them. The larger ones can be found in areas around Madrid while the smaller ones are typically located in not very accessible areas, that are not well connected by road and/or are very hilly, like those in the northern part of the country. Table 1: Distribution of LLMs by population size (2001) Name/Number of LLMs Number of municipalities % of total population > 2,500,000 inhabitants Madrid 153 Barcelona % < 2,500,000 and Valencia Sevilla Bilbao Malaga % > 500,000 inhabitants Zaragoza Palmas de Gran Canaria Sabadell Sta. Cruz Tenerife < 500,000 and 60 2, % >100,000 inhabitants < 100,000 and % > 50,000 inhabitants < 50,000 inhabitants 686 4, % TOTAL 806 LLMs 8,106 municipalities 40,847,371 inhabitants Based on the idea that workers and firms interact only in LLMs, the LLMs could be an adequate spatial unit of analysis for studying topics such as the underlying causes of spatial income disparities -or many other economic outcomes- across a country. As expected, the delimitation of LLMs based on objective criteria does not respect the limits of the administrative regions, so there are a few cases where municipalities included in the same LLM belong to different NUTS III regions. 2 In Europe, economic researchers face the problem of the lack of GDP figures, either total or per capita, at this level of disaggregation. Some attempts have been made by the European Union to provide economic indicators (including GDP figures) for some European cities. The Urban Audit Project provides comparable statistics collected every three years for 321 cities in the 27 countries of the EU along with 36 additional cities in Norway, Switzerland and Turkey (for more details see In Spain there are population or employment figures at the local level (municipalities), even by industry level or by level of qualification, but no data exists on GDP (nor productivity) at that level 1For a description of the previous method, see Smart (1974). For a discussion of problems that arose with that method, see Ball (1980) and Coombes and Openshaw (1982). 2As it will be explained in Section 4, this fact has been taken into account when defining the aggregation matrix.

5 of disaggregation precluding the calculation of (per capita) GDP or productivity at the LLM level. In this context, where data on GDP is available at NUTS III (provincial) level but not at the required LLM level, but where some economic and socio-economic data is available at the LLM, this paper proposes the use of the BIM to estimate the missing data (per capita GDP) for the Spanish LLMs. The next section introduces the BIM. 3. The Bayesian Interpolation Method The areal data conversion problem can be formalized by regarding spatial data as realizations of spatial stochastic processes or random fields. A random field is a collection of univariate or multivariate random variables indexed by their locations, that is. The set of indices identifies the topology of data and can refer to point level data or areal data. Denote by,, a finite set of areal units which forms a partition on the spatial domain. We refer to the set as the original process, that is the spatial stochastic process generating a set of data in its supposed original form. Assume that a partition on can be obtained by aggregating the areal units,, into the larger areal units,, with. We define the process as the derived process. Now suppose that the observations on the attribute variable are recorded on the partition, while we are interested in the disaggregation level defined by the partition. In the described framework, the areal data conversion will consist of reconstructing the original process starting from the knowledge, based on data, of the derived one. We propose as a possible solution to the described problem the application of the BIM, introduced by Palma and Benedetti (1998). The BIM assumes that the random vector is distributed according to a Gaussian Markov Random Field or Conditional Autoregressive (CAR) model (Besag, 1974), which is specified by a set of full conditional distributions as follows: where is the set of neighboring sites of the areal units, identified according to a specified proximity criterion, is the mean of the random variable, is its conditional variance that is assumed to be constant for all, and are spatial interaction coefficients. The spatial interaction coefficients in (1) can be defined as: where is a scalar parameter of spatial autocorrelation (or spatial dependence), and is the generic element of a proximity matrix such that: for. Furthermore we assume that the random vector is defined by two additive components, that is: where represents the variable of interest at different locations, and is a random vector of error terms. We assume that is distributed according to a Gaussian Markov Random Field, specified by the set of full conditional distributions: (5) where, are spatial interaction coefficients defined as in equation (4), and is the conditional variance of, that is assumed to be constant for all. From the formulated assumptions, it follows that the joint probability distributions of the random vectors and are respectively: (1) (2) (3) (4) (6)

6 where denotes a -dimensional identity matrix. The covariance structure defined above gives rise to additional issues that have to be considered. Specifically, the specification of the proximity matrix as in (5) may be not consistent when the number of neighbors varies for the areal units (Clayton and Berardinelli, 1992). In this case a more suitable weighting scheme is based on the matrix,, where and. When is used, the conditional variances and have to be inversely proportional to, for all, in order to ensure the symmetry condition required by the CAR specification (see e.g. Wall, 2004). Then, by setting the covariance matrix in equation (6) becomes: where is a -dimensional diagonal matrix with entries,, and is the -dimensional diagonal matrix with entries,. The same considerations hold for the covariance matrix of the random vector, which can be written as: where is the -dimensional diagonal matrix with entries, for. The above specification of the covariance matrix is that proposed in Gelfand and Vounatsou (2003) for the zero centered CAR, and a sufficient condition for its positive definiteness is. The transformation of the -dimensional random vector into the -dimensional random vector can be formalized by introducing a linear operator so that: The transformation operator in (10) is constructed as a aggregation matrix, whose elements can be specified according to any sum or averaging operations. Since the observed aggregated data derive from the unobserved disaggregated data, through the operator, Palma and Benedetti (1998) propose a solution to the areal data conversion problem based on the posterior probability distribution. By recalling the Bayes' theorem, we can derive this posterior probability distribution as follows: where denotes the prior probability distribution of, and is its likelihood on the basis of the available data. For the assumption in (6), the prior probability distribution is a multivariate Gaussian distribution. The probability distribution can be derived by recalling a well known result (Anderson, 1958, p. 26) which states that if and, where is a linear operator, then is distributed according to the law. It follows that: (7) (8) (9) (10) (11) The posterior distribution is, then, again Gaussian as it includes and. Under the additional hypothesis of known covariance matrix, the described Bayesian approach lead us to the following result (see Pilz,1991): where and are BIM estimates defined as follows: Any inference about the original process can be carried out by the posterior distribution defined by (12) and (13). Point estimates for can be obtained using, which is its Maximum A Posterior (MAP) estimate, as shown in Benedetti and Palma (1994). Confidence intervals and hypothesis tests can be performed as usually for multivariate normal distributions. (12) (13)

7 Additional issues are raised from the phycnophylactic (or mass preserving) property that consists in finding an estimate of such that, by applying the operator, the observed data are again obtained (Tobler, 1979). In order to preserve this property, the posterior distribution is conditioned to the linear constraint, so that the constraint BIM estimates are obtained as follows: Point estimates of can be obtained again by using ; confidence intervals and hypothesis tests can be performed as usual for multivariate normal distributions. (14) (15) 4. Applications of BIM In this section, the BIM is applied in order to spatially disaggregate data on per capita GDP. The aim is to estimate per capita GDP for LLMs, which is not available for Spain. However, first in subsection 4.1, the outcomes of the BIM are evaluated, disaggregating per capita GDP at the provincial level (NUTS II regions), for which real data on per capita GDP exists. Thus, estimated and real per capita GDP can be compared and the BIM tested. Assuming that same data generating process works from provinces to LLMs, then in subsection 4.2 we estimate and discuss per capita GDP for the 806 Spanish LLMs from the provincial data. 4.1 Estimation of per capita GDP for provinces in Spain Assume the focus is on estimating per capita GDP for Spanish provinces when data are available only for the autonomous communities. According to the BIM theory, we refer to the provincial data generating process as the original process. The probability distribution of this process is assumed to be a multivariate normal distribution. Values for provincial per capita GDP are drawn from the probability distribution of the original process, given the aggregated data, observed for autonomous communities. The computation of these posterior distribution parameters, that are the BIM estimates, requires the specification of the prior mean and variance for the original process. We assume a linear relationship between the unknown GDP at provincial level and some observed explanatory variables. Some economic figures for provinces, as population, foreign population, and active population, divided into employed and unemployed individuals, are collected in the Population and Household Census administrated by the INE (National Statistics Institute of Spain) whose most recent data are for Auxiliary variables related to (per capita) GDP can be derived from this database. Then, the relationship between per capita GDP for provinces and some selected explanatory variables can be written as: (16) where is a -dimensional vector of unobserved (per capita) GDP, is a matrix including a column of 1's and observed explanatory variables, is a vector including the intercept term and regression coefficients, and is a -dimensional vector denoting the error term of the model. Then, we propose to define the prior mean of the original process as follows: where is a vector of coefficients which are estimated by fitting the model in (16) on data observed for autonomous communities. The covariance matrix of the original process is defined as: where the scalar parameter can be estimated by the residual variance of the model (16) fitted on

8 data observed for autonomous communities. Table 2 shows results for different regression models estimated on the aggregated data. Data available at the level of autonomous communities have been divided by the total population in each region, in order to obtain per capita values. Table 2: Regression coefficients estimates at provincial level Coeffici Model 1 Model 2 Model 3 Model 4 ents Constant *** *** *** *** (pvalues) Employe d Populatio n (pvalues) Active Populatio n (pvalues) Foreign Populatio n (pvalues) Models (0.0001) (0.0003) (0.0002) (0.0002) *** *** (0.0000) (0.0000) *** *** (0.0000) (0.0000) * (0.1868) (0.0458) Adjusted Residual s Standard Error Degrees of freedom Significance Levels: *** 1%,**10%, * 5% The explanatory variables included in models 1 and 2 seem to perform reasonably well. The Employed Population is able to explain GDP with a high = A lower (0.7707) is reported for model 2. The regression coefficients in both the models are highly significant. The coefficient associated with the Foreign Population is negative for both models 3 and 4: the coefficients not significant in model 3 and significant, at level of 0.05, in model 4. The BIM estimates are carried out according to the alternative hypothesis of prior mean estimated by assuming model 1 and model 2. Three different values of the autocorrelation parameter, as, and, are, in turn, assumed in the specification of the covariance matrix of the process. The proximity

9 matrix is constructed according to the 5 nearest neighbors and is symmetrized. The accuracy of estimates is evaluated by computing the correlation coefficients between real and estimates values, and some indices from the forecasting literature, as the Root Mean Squared Error (RMSE), the Mean Absolute Error (MAE) and the Mean Absolute Percentage Error (MAPE). 3 Table 3 shows results for the unconstrained and the constrained estimates of per capita GDP for provinces, when the model 1 is assumed in the specification of the prior mean. As mentioned in section 3, the constrained solutions preserve the pychnopylactic property, which is not considered in the unconstrained version of the BIM estimates. Table 3: Accuracy of constrained and unconstrained estimates for per capita GDP at provincial level (prior mean specified by model 1) rho type correlation RMSE MAE MAPE 0.06 unconstrained constrained unconstrained constrained unconstrained constrained As expected, when results are constrained to the known total, the accuracy measures assume lower values. The constrained solutions also exhibit a higher correlation with real data. This result is due to the lack of the mass preserving condition characterizing the unconstrained solutions. The values assumed by correlation coefficients, RMSE, MAE and MAPE seem to be essentially independent from the hypothesis on. This result reveals the robustness of the estimates with respect to the autocorrelation parameter. In Table 4, provincial per capita GDP, estimated by assuming the model 1 in the specification of the prior mean and variance, are compared with both the real values and the estimates carried out by considering model 2. This comparison reveals an improvement of estimates when the prior mean is specified according to the model 1, as is also confirmed by the computed accuracy measures. Table 4: Disaggregation of per capita GDP (euro) at provincial level Prior mean specified by Model 1 Real values Prior mean specified by Model 2 Álava The considered accuracy measures are computed as follows: RMSE= MAE= MAPE= where and denote real and estimated values respectively. RMSE and MAE can range from 0 to

10 Albacete Alicante Almería Ávila Badajoz Balears Barcelona Burgos Cáceres Cádiz Castellón Ciudad Real Córdoba Coruña, A Cuenca Girona Granada Guadalajara Guipúzcoa Huelva Huesca Jaén León Lleida Rioja, La Lugo Madrid Málaga Murcia Navarra Ourense Asturias

11 Palencia Palmas Pontevedra Salamanca Sta Cruz Tenerife Cantabria Segovia Sevilla Soria Tarragona Teruel Toledo Valencia Valladolid Vizcaya Zamora Zaragoza Correlation RMSE MAE MAPE The results displayed in table 4 suggest a good performance of the BIM in converting data from the NUTS II to the NUTS III level, and can validate its application to higher levels of spatial disaggregation. 4.2 Estimation of per capita GDP for LLMs in Spain The BIM procedure is applied in order to disaggregate per capita GDP available for the 50 provinces at level of the 804 LLMs. 4 Figure 2 displays the source and the target zones. Figure 1: Provinces and LLMs in Spain 4

12 According to the BIM, a multivariate normal distribution is assumed for the original process, that is the data generating process at LLMs level. The prior mean of the process is specified as: where are explanatory variables available for LLMs and are coefficients estimated by fitting a regression model on provincial data. The variance of the regression residuals is used to estimate the prior variance of the process. From the Population and Household Census, total population, foreign population, active and employed population at municipal level, were aggregated into LLMs. Some explanatory variables, like those summarized in Table 5, can be derived from this database. (17) Table 5: Economic and socio-economic variables for Spanish LLMs classified by population size. Source: 2001 Spanish Census INE (2007) Type of LLM % active % employed % foreign population population population > inhabitants 15.63% 16.04% 33.03% < and > inhabitants 18.07% 17.67% 10.92% < and > inhabitants 33.61% 33.75% 22.48% < and > inhabitants 9.23% 9.18% 13.07% < inhabitants 23.45% 23.36% 20.49% TOTAL: 804 LLMs 17,499,182 14,976,512 1,572,013 Results for different specifications of the regression model on the provincial data are displayed in Table 6. Significant coefficients and a high are reported for model 1. As in the previous application, we focus on this model for estimating coefficients in (17) and the scalar parameter included in the covariance structure of the original process. When we deal with the conversion of provincial data into the LLMs level, there are additional issues that have to be considered. As mentioned in section 2, several LLMs go beyond the provincial limit, so that some part can belong to a given province and rest to one or more other provinces. In order to take into account this feature the aggregation matrix, which expresses the LLMs' aggregation into provinces, is constructed as follows. We denote by the weight which is assigned to the LLM, belonging to the province, proportionally to the active population in the municipality where the LLM is included, so that and. Then, we construct the matrix with elements:

13 for i and. The entries of the aggregation matrix are, thus, specified as follows: where and denote the Total Population in the i-th LLM and in the -th province respectively, for and j. Table 6: Regression coefficients estimates at LLM level Coefficients Model 1 Model 2 Model 3 Model 4 Constant *** *** *** *** (p-values) (0.0000) (0.0000) (0.0000) (0.0001) Employed Population *** *** (p-values) (0.0000) (0.0000) Active Population *** *** (p-values) (0.0000) (0.0000) Foreign Population (p-values) (0.1260) (0.6430) Models Adjusted Residuals Standard Error Degrees of freedom Significance Levels: *** 1%,**10%, * 5% The BIM estimates are then carried out in their constrained version, on the basis of the above specified prior mean, covariance matrix and aggregation matrix. Different values of the spatial autocorrelation parameter, as, and are considered, and the proximity matrix is specified according to the -nearest neighbors. The BIM estimates are used as parameters of the multivariate normal distribution from which the estimated per capita GDP for LLMS are drawn. The estimated per capita GDP for LLMs, obtained by setting different values for and classified by size, are displayed in Table 7. Table 7: Estimated per capita GDP (euros) for Spanish LLMs classified by population size Estimated per capita GDP Type of LLM Min Max Min Max Min Max >2,500,000 inhabitants 21,412 19,533 22,489 21,480 19,721 22,488 21,486 19,737 22,488 <2,500,000 and >500,000 inhabitants 15,901 12,124 22,058 15,839 12,108 21,093 15,833 12,104 21,015

14 <500,000 and >100,000 inhabitants 16,206 7,894 24,669 16,183 7,884 24,520 16,181 7,890 24,527 < 100,000 and > 500,000 inhabitants 16,272 8,303 25,455 16,278 8,462 24,873 16,279 8,446 24,829 <50,000 inhabitants 14,308 2,042 33,253 14,320 2,307 33,388 14,322 2,329 33,423 The results confirm the existence of internal spatial disparities on per capita GDP between the metropolitan areas, i.e. LLMs characterized by a city in the core and municipalities surrounding this core in such a way that their total population overcomes the 2.5 million inhabitants and the rest of the country (small and medium size cities and rural areas). The larger the size (quantified by population) of the functional area defined, the higher its estimated per capita GDP, suggesting the importance of agglomeration economies. However, as suggested by Polèse and Shearmur (2004) for Canada and by Polèse et al. (2006) for Spain, not only the larger cities will benefit from agglomeration economies but also those areas close to them (see figure 2). Figure 2: Real per capita GDP (provinces) and estimated per capita GD) for LLMs (w5r006) Spanish per capita GDP (2001) = 100 < > 120 Geographical location and spatial dependence seem to explain economic activity and welfare as well. As Polèse (2009) assessed, relative location matters. The position of a local area must be considered not only with regard to the national urban system, but also to the international connections. Proximity to international borders with important trade flows could be relevant, as it seems to be confirmed in the Spanish case. In Spain the North-East concentrates the higher per capita GDP values, either by provinces or by LLM. However, by working with disaggregated data, we realize about the internal spatial divergences and the importance of the large cities, and the dichotomy between the rural and the urban areas. 5 Conclusions and future research Regional economists use administrative units as a proxy of the Region in their empirical analysis, either because of the lack of alternatives or the impossibility of having a region consistent with the theoretical assumptions of regional economics. Such a basic economic variable as (per capita) GDP is only available in Spain at NUTS III level, which corresponds to provinces. However, we believe that the ideal unit of analysis are the Local Labor Markets Areas (LLMs). Defined from home-towork daily journeys of its residents, the larger LLMs (the larger cities and their surrounding areas)

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