مدل های تعامل فضایی با داده های سطح فردی برای توضیح جریان کار و توسعه بازارهای کار محلی

As a result of increased mobility patterns of workers, explaining labor flows and partitioning regions into local labor markets (LLMs) have become important economic issues. For the former, it is useful to understand jointly where individuals live and where they work. For the latter, such markets attempt to delineate regions with a high proportion of workers both living and working. To address these questions, we separate the problem into two stages. First, we introduce a stochastic modeling approach using a hierarchical spatial interaction specification at the individual level, incorporating individual-level covariates, origin (O) and destination (D) covariates, and spatial structure. We fit the model within a Bayesian framework. Such modeling enables posterior inference regarding the importance of these components as well as the O–D matrix of flows. Nested model comparison is available as well. For computational convenience, we start with a minimum market configuration (MMC) upon which our model is overlaid. At the second stage, after model fitting and inference, we turn to LLM creation. We introduce a utility with regard to the performance of an LLM partition and, with posterior samples, we can obtain the posterior distribution of the utility for any given LLM specification which we view as a partition of the MMC. We further provide an explicit algorithm to obtain good partitions according to this utility, employing these posterior distributions. However, the space of potential market partitions is huge and we discuss challenges regarding selection of the number of markets and comparison of partitions using this utility. Our approach is illustrated using a rich dataset for the region of Aragón in Spain. In particular, we analyze the full dataset and also a sample. Future data collection will arise as samples of the working population so assessing population level inference from the sample is useful.

Population mobility plays a key role both in the performance of an economic system and in the daily life of individuals. Commuting is increasing in absolute volume and in number of destinations. So, it has become a key element in defining organization with regard to economic geography (Ball, 1980, Simpson, 1992, Henley, 1998 and Kaufmann, 2000). Urban factors such as housing availability together with expanded automobile and public transport use, encourage the growth of commuting and result in consequential discrepancies between where individuals work and where they live (Jansen, 1993). Hence, it becomes useful to explain these daily mobility patterns. A frequent approach for dealing with this problem is to analyze the displacement flow matrices using spatial interaction models, see, e.g., the review paper of Roy and Thill (2003). These models seek to describe the processes by which entities located in different locations interact with each other, either for migratory movements, labor displacement or other reasons. At the individual level, they reflect personal cost–benefit decisions associated with the displacement. The most commonly used models include the so-called gravitational models (Alonso, 1978, De Vries et al., 2001, De Vries et al., 2002, De Vries et al., 2009, Ding and O’Kelly, 2008, Fotheringham, 1983, Hua, 2001, Roy and Thill, 2003 and Wilson, 1967) that try to explain flows observed through origin and destination explanatory variables. Early work introduced model fitting through utility maximization, with connections to entropy and likelihood maximization (see, e.g., Wilson (1967) and the review paper of Wilson (1975)). More recent work implements model fitting through least squares (De Vries et al., 2009, Ding and O’Kelly, 2008 and Sen and Sööt, 1981) or the use of instrumental variables (De Vries et al., 2002). In addition, these mobility patterns suggest formation of functional economic areas with strong internal flows of commuting. This phenomenon was originally conceived in terms of big urban agglomerations but is now applied to finer scales (Giuliano and Gillespie, 1997 and Lowe, 1998). Furthermore, customary administrative demarcations do not capture the limit of space in the daily life of populations (Gaussier et al., 2003 and Van Ham et al., 2001). Rather, the flows of daily mobility have become a primary identifier of territorial aspects and can contribute to better articulation of public policies with regard to land management, transport, housing, economic activity and work (Amedeo, 1969 and Cörvers et al., 2009). Unfortunately, administrative areas are often used as a surrogate for labor market areas in terms of statistical, analytical, and policy-making purposes though, again, these areas usually do not reflect functional reality (Ball, 1980, Coombes, 2002 and Smart, 1974) and may compromise the effectiveness of resulting policies (Coombes et al., 1986). The dominant concept in defining functional regions is that of local labor markets (LLMs) (see, for instance Goodman, 1970, Smart, 1974, Coombes et al., 1986, Tolbert and Killian, 1987 and Coombes, 1992). See also Casado Díaz and Coombes (2011) for a full critical review. LLMs identify the areas within which there is a close relationship between labor supply and demand. Qualitatively, such a market is characterized as an area in which a large proportion of the workers both live and work. Some early approaches based on numerical taxonomy principles and statistical objective were proposed in Brown and Holmes (1971), Masser and Brown (1975), Fischer (1980), Masser and Scheurwater (1980) and Baumann et al. (1983). Brown and Holmes (1971) make a distinction between functional and nodal regions. They apply a Markov chain analysis to the interaction commuting flow matrix that transforms the matrix between basic spatial units (BSUs) into a mean first passage time matrix (MFPT). Those regions are delineated through hierarchical and non-hierarchical clustering techniques, applied to a distance matrix built from the MFPT. Masser and Brown (1975) describe two algorithmic clusterings based on flux. They use separate procedures for aggregating BSUs called Intramax and Intramin. The former defines subsystems to maximize the proportion of total interaction within the BSUs which do not cross the borders. The latter maximizes the proportion of border crossings between units. A more recent methodology to construct LLMs is the algorithm developed by Coombes et al. (1986) which was accepted by the UK Department of Employment to produce travel-to-work areas (TTWA), based on 1981 Census data (see also Coombes, 1992). Subsequently, this methodology has been adopted, with minor modifications, by many countries including Italy (Sforzi et al., 1991), Spain (Alonso et al., 2008 and Casado-Díaz, 2000), New Zealand (Newell and Papps, 2002), Denmark (Andersen, 2002), and Australia (Watts, 2004). This methodology uses the daily labor matrix of flows between a collection of BSUs, usually districts, counties or municipalities. LLMs are created by attempting to maximize the interaction level of each BSU within its LLM, subject to restrictions on the self-containment level and on the size of each LLM. A sophisticated version of this methodology is due to Flórez-Revuelta et al. ( Flórez-Revuelta et al., 2008a and Flórez-Revuelta et al., 2008b), who view the problem as one of optimization and propose a genetic algorithm to achieve a solution. A key remark is that none of these algorithmic approaches can obtain a “best”solution. The number of partitions of a collection of administrative units into potential LLMs for a region is enormous. With any criterion, there will be a very large number of local optima. Any solution will, at best, be one of these. Furthermore, these methodologies lack explicit probability modeling, precluding uncertainty in comparing partitions. Our contribution is to propose, using individual-level data, Bayesian hierarchical Poisson spatial interaction models for joint origin–destination modeling. These models employ spatial random effects and regression coefficients that allow us to incorporate an origin function, a destination function, a worker attributes component, and spatial structure to explain the variation in the origin–destination flows. To facilitate computation and search, over the partitioned space, the model is applied to a minimum market configuration (MMC), as described at the end of Section 2. The Bayesian approach lets us make a comparison between nested and non-nested models with regard to the explanation of the observed flows. It enables us to make an exact inference about the model parameters given the data we have observed, without relying on asymptotic theory (Banerjee et al., 2004). The model allows learning regarding systematic population mobility patterns. It allows comparison of patterns across varying individual-level characteristics. In fact, we obtain full posterior inference regarding the matrix of flows. Once the model is chosen, we turn to delineation of local labor markets. In fact, by introducing worker attributes, we can work at the sub-group level, e.g., delineate LLMs for males vs. females or for different age groups. In any event, a specification of a map providing LLMs can be viewed as a partition of the set of units in the MMC. As noted above, the number of partitions is enormous and we can only hope to find a good set of partitions. We offer a novel stochastic search approach to do this. In particular, we introduce a utility function that rewards concentration, which quantifies the foregoing notion of a subregion where a large proportion of the workers both live and work. Concentration is captured through approximate block diagonalization of the spatial interaction matrices P=(pj|i)P=(pj|i) and Q=(qi|j)Q=(qi|j) where pj|ipj|i denotes the probability of working in unit jj given living in unit ii and qi|jqi|j denotes the probability of living in unit ii given working in unit jj. We argue that increased concentration in this sense leads to better LLM delineation. We also note the need for penalizing delineations with too few markets. In the end, for any given partition, posterior samples of PP and QQ enable a posterior distribution for the utility of the partition. Partitions are compared under this utility, using these posterior distributions. Lastly, for reasons noted above, we also cannot obtain a best partition; we cannot assert that a partition created by our method is better or worse than one created through one of the other approaches referred to above. There is no notion of a true partition to compare with. What we can claim is that our approach avoids ad hoc iterative repairs and corrections in the optimization, that it only insists on contiguity and minimum size with regard to each local market, and that it can be routinely automated. Our flows model is related to that proposed in LeSage and Llano (2008). They use a simultaneous auto-regressive (SAR) specification to model the spatial effects parameters while we appeal to a conditionally auto-regressive (CAR) choice. The latter is better suited to hierarchical model fitting since the full conditional distributions that need to be sampled in the MCMC for updating any of the spatial random effects are trivial under the CAR model because that is how the CAR model is created within the theory of Markov random fields. This is not the case for the SAR model. (See Banerjee et al., 2004 in this regard.) Furthermore, we disaggregate the data to the individual level, we introduce interactions between worker attributes and spatial locations and we introduce multiplicative spatial origin–destination interactions to enrich the specification. The methodology is illustrated by means of an application to extract LLMs in Aragón, a Spanish region characterized by low population density in many areas and a single dominant city, elements that make the model estimation and the delineation of LLMs complicated due to the very large number of zeros in the observed flux matrix. We analyze both the full dataset and also a stratified sample. It will often be the case that the data will only consist of a sample of the workforce, perhaps due to cost concerns. In fact, this is the case for future data from Aragón where roughly a 15% sample will be collected. So, here we perform population level inference but also sample-based inference, for comparison. The paper is organized into 5 sections. Section 2 gives a description of the dataset under study and our clustering algorithm to construct an MMC. Section 3 develops our hierarchical model, clarifying model choice, and suggesting inference on labor flow using the fitted model. Section 4 presents our utility-based approach for LLM determination. Section 5 presents the data analysis using the methodology described in Sections 3 and 4. Comparison between full and sampled data is offered. Section 6 concludes with a summary and possible future work.

We have proposed a flexible Bayesian hierarchical modeling approach, applied to a minimal market configuration, to develop posterior flow analysis across this configuration, with associated inference. We then introduced a utility function to create LLMs. This function rewards concentration, which quantifies the notion of a subregion where a large proportion of the workers both live and work. It turns the problem of LLM creation to one of partitioning the rows of a concentration matrix obtained through approximate diagonalization. After a stochastic search, using posterior samples from the hierarchical model fitting, we computed the posterior distribution of the utility for a top collection of partitions of the configuration. Each partition provides a map of LLMs. We have applied this approach, working with individual level data which provides the origin unit (where the worker lives) and the destination unit (where the worker works), driving time, along with individual level characteristics as well as origin unit and destination unit characteristics. Using both the full dataset and a sample of the data, we have created LLMs in the province of Aragón in Spain, and, for each dataset, supplied the top map we obtained. Perhaps, most importantly, we have clarified that it is not possible to develop a best map. Any approach to this problem will deal with exploration of an enormous map space within which a large number of maps will be indistinguishable under any utility function. Future work will find us adding a temporal aspect to the modeling. Then, we may study evolution and change in market structure over time as well as forecast change in market structure as a function of change in population demographics, change in characteristics of localities, and change in driving distance.