# Incorporating Spatial Structures in Ecological Inference: An Information Theory Approach

## Abstract

**:**

## 1. Introduction

## 2. Ecological Inference Assuming Heterogeneity and Dependence across Space

#### 2.1. Theoretical Framework of Ecological Inference

#### 2.2. Ecological Inference and Heterogeneity across Space

**y**i, as a weighted geometric mean of the latent sub group/region indicator ${y}_{ij}$ in group/region i: ${y}_{i}={\displaystyle \prod _{j=1}^{{J}_{i}}{\left({y}_{ij}\right)}^{{\mathsf{\theta}}_{ij}}}$, that is:

_{i}denotes the number of sub groups/regions in i.

**x**

_{i}, an observed vector of explanatory variables for sub group/ sub region j in group/region i,

**z**

_{ij}, the latent sub-group indicators (values) are specified in a multiplicative form as follows:

**z**

_{ij,k}(k = 1, K) are the covariates observed at the level of sub group/ sub region j within the group/region i,

**x**

_{i,h}(h = 1,..H) are the covariates observed only at the level of group/region i, ${\mathsf{\alpha}}_{ij}$ are unobserved fixed effects, and ${\mathsf{\epsilon}}_{ij}$ are error terms.

#### 2.3. Ecological Inference and Spatial Dependence

_{ij}=1 for d

_{ij}≤δ and i≠j, where d

_{ij}is the distance function chosen, and δ is the critical cut-off value. More specifically, a spatial weights matrix

**w*** is defined as follows:

**w**(with elements of a row sum to one) result:

## 3. An Information Theoretic Approach

- (i)
- data consistency conditions:$$\mathrm{ln}{y}_{i}={\displaystyle \sum _{j=1}^{{J}_{i}}\left({s}_{\alpha}\prime {p}_{\alpha ,ij}+{\displaystyle \sum _{k=1}^{K}\left({s}_{\beta}\prime {p}_{\beta ,ij}\right)\mathrm{ln}{z}_{ij,k}}+{\displaystyle \sum _{h=1}^{H}\left({s}_{\gamma}\prime {p}_{\gamma ,ij}\right)\mathrm{ln}{x}_{i,h}}+\left({s}_{\rho}\prime {p}_{\rho ,ij}\right)\mathrm{ln}w{y}_{i}+\left({s}_{\epsilon}\prime {p}_{\epsilon ,ij}\right)\right)}{\mathsf{\theta}}_{ij}$$
- (ii)
- adding-up constraints for probabilities.$$\sum {p}_{\alpha ,ij}}={\displaystyle \sum {p}_{\beta ,ij}}={\displaystyle \sum {p}_{\gamma ,ij}}={\displaystyle \sum {p}_{\rho ,ij}}={\displaystyle \sum {p}_{\epsilon ,ij}}=1\begin{array}{cc}& \forall i,j\end{array$$

- (i)
- data consistency conditions:$$\mathrm{ln}{y}_{i}={\displaystyle \sum _{j=1}^{{J}_{i}}\left({s}_{\alpha}\prime {p}_{\alpha ,ij}+{\displaystyle \sum _{k=1}^{K}\left({s}_{\beta}\prime {p}_{\beta ,ij}\right)\mathrm{ln}{z}_{ij,k}}+{\displaystyle \sum _{h=1}^{H}\left({s}_{\gamma}\prime {p}_{\gamma ,ij}\right)\mathrm{ln}{x}_{i,h}+\left({s}_{\lambda}\prime {p}_{\lambda ,ij}\right)w\left({s}_{\epsilon}\prime {p}_{\epsilon ,ij}\right)+\left({s}_{\tau}\prime {p}_{\tau ,ij}\right)}\right)}{\mathsf{\theta}}_{ij}$$
- (ii)
- adding-up constraints for probabilities:$$\sum {p}_{\alpha ,ij}^{}}={\displaystyle \sum {p}_{\beta ,ij}^{}}={\displaystyle \sum {p}_{\gamma ,ij}^{}}={\displaystyle \sum {p}_{\lambda ,ij}^{}}={\displaystyle \sum {p}_{\tau ,ij}^{}}=1\begin{array}{cc}& \forall i,j\end{array$$

## 4. An Empirical Application

**Table 1.**Estimates of the value added of Emilia Romagna’s provinces disaggregated by six sectors for the year 2005.

Provinces (Emilia Romagna) | Agriculture | Industry | Construction | Transport, Hotels, Telecom. | Financial Services | Other Services |
---|---|---|---|---|---|---|

PIACENZA | 227.4573 | 1568.474 | 366.4849 | 1522.572 | 1747.364 | 1183.74 |

PARMA | 277.8251 | 3317.61 | 692.2093 | 2278.186 | 3007.466 | 1704.97 |

REGGIO NELL' EMILIA | 314.3605 | 4576.695 | 917.9048 | 2351.606 | 3288.713 | 1552.768 |

MODENA | 362.4585 | 6376.475 | 1076.482 | 3439.923 | 4597.192 | 2428.439 |

BOLOGNA | 375.1509 | 7001.803 | 1361.174 | 6264.928 | 8143.771 | 5113.127 |

FERRARA | 402.0568 | 1641.202 | 522.8528 | 1658.157 | 2074.427 | 1445.826 |

RAVENNA | 341.7594 | 2037.979 | 592.185 | 2176.184 | 2394.936 | 1617.214 |

FORLI-CESENA | 307.3608 | 2395.311 | 609.7457 | 2168.655 | 2435.76 | 1623.719 |

RIMINI | 141.9309 | 1048.246 | 410.3823 | 2219.982 | 1991.04 | 1294.397 |

Explanatory variables: ${x}_{i,h}$ (in logs) | GME estimates: ${\widehat{\mathsf{\gamma}}}_{j,h}$ | |||||

Real Capital Stock: RC | 0.22** |

## 5. Conclusions

## Acknowledgements

## References

- Johnston, R.; Pattie, C. Ecological inference and entropy-maximizing: An alternative estimation procedure for split-ticket voting. Polit. Anal.
**2000**, 8, 333–345. [Google Scholar] - Judge, G.; Miller, D.; Cho, W.K.T. An information theoretic approach to ecological inference. In Ecological Inference: New Methodological Strategies; King, G., Rosen, O., Tanner, M.A., Eds.; Cambridge University Press: Cambridge, UK, 2004; pp. 162–187. [Google Scholar]
- Bernardini Papalia, R. Modeling mixed spatial processes and spatio-temporal dynamics in information-theoretic frameworks. In COMPSTAT 2006: Proceedings in Computational Statistics; Rizzi, A., Vichi, M., Eds.; Physica-Verlag: Heidelberg, Germany, 2006; pp. 1483–1491. [Google Scholar]
- Bernardini Papalia, R. Combining Incomplete Information and Learning in Microeconometric Models. In Correlated Data Modeling; Gregori, D., Mackenzie, G., Friedl, H., Corradetti, R., Eds.; Franco Angeli: Milano, Italy, 2007; pp. 137–143. [Google Scholar]
- Bernardini Papalia, R. A Composite Generalized Cross Entropy Formulation in Small Samples Estimation. Econom. Rev.
**2008**, 27, 596–609. [Google Scholar] [CrossRef] - Bernardini Papalia, R. Analyzing Trade Dynamics from Incomplete Data in Spatial Regional Models: A Maximum Entropy Approach. In Bayesian Inference and Maximum Entropy Methods in Science and Engineering; Springer-Verlag: New York, NY, USA.
- Bernardini Papalia, R. Data Disaggregation Procedures within a Maximum Entropy Framework. J. Appl. Statist.
**2010**, in press. [Google Scholar] - Golan, A.; Judge, G.; Robinson, S. Recovering Information from Incomplete or Partial Multisectoral Economic Data. Rev. Econom. Statist.
**1994**, 76, 541–549. [Google Scholar] [CrossRef] - Golan, A.; Judge, G.; Miller, D. Maximum Entropy Econometrics: Robust Estimation with Limited Data; Wiley: New York, NY, USA, 1996. [Google Scholar]
- Golan, A.; Judge, G.; Miller, D. The Maximum Entropy Approach to Estimation and Inference: An Overview. In Advances in Econometrics. Applying Maximum Entropy to Econometric Problems; Fomby, T.B., Hill, R.C., Eds.; Jai Press Ltd.: London, UK, 1997; pp. 3–24. [Google Scholar]
- Golan, A. Information and Entropy Econometrics: A Review and Synthesis. In Foundations and Trends® in Econometrics; Now Publishers: Hanover, MA, USA, 2008; Volume 2, pp. 1–145. [Google Scholar]
- Shannon, C.E. A mathematical theory of communication. Bell Syst. Techn. J.
**1948**, 27, 379–423. [Google Scholar] [CrossRef] - Jaynes, E.T. Information theory and statistical mechanics. Phys. Rev.
**1957**, 106, 620–630. [Google Scholar] [CrossRef] - Kullback, J. Information Theory and Statistics; Wiley: New York, NY, USA, 1959. [Google Scholar]
- Levine, R.D. An information theoretical approach to inversion problems. J. Phys. A
**1980**, 13, 91–108. [Google Scholar] [CrossRef] - Freedman, D.A.; Klein, S.P.; Ostland, M.; Roberts, M.R. Review of “A solution to the ecological inference problem”. J. Am. Statist. Assoc.
**1998**, 93, 1518–1522. [Google Scholar] [CrossRef] - Schuessler, A.A. Ecological inference. PNAS
**1999**, 96, 10578–10581. [Google Scholar] [CrossRef] [PubMed] - King, G.; Rosen, O.; Tanner, M.A. Ecological Inference: New Methodological Strategies; Cambridge University Press: Cambridge, UK, 2004; pp. 162–187. [Google Scholar]
- Achen, C.H.; Shively, W.P. Cross-level Inference; Chicago University Press: Chicago, IL, USA, 1995. [Google Scholar]
- Cho, W.K.T. Latent groups and cross-level inferences. Elect. Stud.
**2001**, 20, 243–263. [Google Scholar] - Rao, J.N.K. Small Area Estimation; John Wiley & Sons,Inc.: Hoboken, NJ, USA, 2003. [Google Scholar]
- Barker, T.; Pesaran, M.H. An introduction in disaggregation in econometric modelling. In Disaggregation in Econometric Modelling; Barker, T., Pesaran, H., Eds.; Routledge: London, UK, 1990; pp. 1–14. [Google Scholar]
- King, G. A Solution to the Ecological Inference Problem: Reconstructing Individual Behavior from Aggregate Data; Princeton University Press: Princeton, NJ, USA, 1997. [Google Scholar]
- Smirnov, O.; Anselin, L. Fast maximum likelihood estimation of very large spatial autoregressive models: A characteristic polynomial approach. Comput. Statist. Data Anal.
**2001**, 35, 301–319. [Google Scholar] [CrossRef] - Anselin, L. Spatial Econometrics: Methods and Models; Kluwer: Boston, MA, USA, 1988. [Google Scholar]
- Kelejian, H.H.; Prucha, I.R. A generalized moments estimator for the autoregressive parameter in a spatial model. Int. Econ. Rev.
**1999**, 40, 509–533. [Google Scholar] [CrossRef] - Lee, L.F. Asymptotic Distributions of Quasi-maximum Likelihood Estimators for Spatial Econometric Models: I. Spatial Autoregressive Processes; Ohio State University: Columbus, OH, USA, 2001. [Google Scholar]
- Lee, L.F. Asymptotic Distributions of Quasi-maximum Likelihood Estimators for Spatial Econometric Models: II. Mixed Regressive,Spatial Autoregressive Processes; Ohio State University: Columbus, OH, USA, 2001. [Google Scholar]
- Bell, K.P.; Bockstael, N.E. Applying the generalized-moments estimation approach to spatial problems involving microlevel data. Rev. Econ. Statist.
**2000**, 82, 72–82. [Google Scholar] [CrossRef] - Zellner, A. A Bayesian Method of Moments (BMOM): Theory and Applications. Adv. Econome.
**1997**, 12, 85–105. [Google Scholar] - Honoré, B.E.; Hu, L. Estimation of cross sectional and panel data censored regression models with endogeneity. J. Econom.
**2004**, 122, 293–316. [Google Scholar] [CrossRef] - Calvo, E.; Escolar, M. The local voter: A Geographically Weighted Approach to Ecological Inference. Am. J. Polit. Sci.
**2003**, 47, 189–204. [Google Scholar] [CrossRef]

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Bernardini Papalia, R.
Incorporating Spatial Structures in Ecological Inference: An Information Theory Approach. *Entropy* **2010**, *12*, 2171-2185.
https://doi.org/10.3390/e12102171

**AMA Style**

Bernardini Papalia R.
Incorporating Spatial Structures in Ecological Inference: An Information Theory Approach. *Entropy*. 2010; 12(10):2171-2185.
https://doi.org/10.3390/e12102171

**Chicago/Turabian Style**

Bernardini Papalia, Rosa.
2010. "Incorporating Spatial Structures in Ecological Inference: An Information Theory Approach" *Entropy* 12, no. 10: 2171-2185.
https://doi.org/10.3390/e12102171