# Recovering Matrices of Economic Flows from Incomplete Data and a Composite Prior

## Abstract

**:**

## 1. Introduction

## 2. The Ce Solution for the Matrix Balancing Problem

${p}_{11}$ | … | ${p}_{1j}$ | … | ${p}_{1M}$ | ${y}_{1}$ |

… | … | … | … | ||

${p}_{i1}$ | … | ${p}_{ij}$ | … | ${p}_{iM}$ | ${y}_{i}$ |

… | … | … | … | ||

${p}_{N1}$ | … | ${p}_{Nj}$ | … | ${p}_{NM}$ | ${y}_{N}$ |

${v}_{1}$ | … | ${v}_{j}$ | … | ${v}_{M}$ |

## 3. A Composite Ce Method: The Dwp Estimation Technique

## 4. Testing the Dwp Estimation Technique with a Numerical Experiment

**y**. The information contained in the margin vectors of the matrix (total imports and export per region) is much more accessible, given that it can be obtained from the Regional Accounts regularly published by the Spanish Statistical Institute (see http://www.ine.es/en/inebmenu/mnu_cuentas_en.htm for more details on the Spanish Regional Accounts).

Technique (prior used) | Deviation measures | ||
---|---|---|---|

TSE | TAE | TKL | |

Adjusting from ${\mathit{Q}}^{a}$ | 0.014 | 0.936 | 0.052 |

Adjusting from ${\mathit{Q}}^{b}$ | 0.037 | 1.425 | 0.142 |

DWP (mixture of ${\mathit{Q}}^{a}$, ${\mathit{Q}}^{b}$) | 0.013 | 0.882 | 0.049 |

**Figure 1.**Absolute deviations between the target and estimates matrices under different levels of similarity between $\mathit{P}$ and ${\mathit{Q}}^{b}$.

## 5. An Empirical Application: Estimating the Interregional Trade Matrix in Spain, 2006

**Z**. For this purpose, we will apply an adjusting process to obtain the column-coefficients matrix

**P**from two different initial matrices. Moreover, we also assume that we have some information on the expected structure of this matrix, obtained from the observed matrices of interregional trade column coefficients for two consecutive years in the past, specifically 2004 and 2005 (${\mathit{Q}}^{04}$ and ${\mathit{Q}}^{05}$ respectively).

**P**: an adjustment considering from ${\mathit{Q}}^{04}$, from the prior ${\mathit{Q}}^{05}$ and the DWP estimator that takes both possible priors. Table 3 summarizes the results obtained in this study case, applying the same criteria as before for comparing the estimated and the target matrix:

Technique (prior used) | Deviation measures | ||
---|---|---|---|

TSE | TAE | TKL | |

Adjusting from ${\mathit{Q}}^{04}$ | 0.063 | 2.424 | 1.096 |

Adjusting from ${\mathit{Q}}^{05}$ | 0.083 | 2.428 | 0.783 |

DWP (mixture of ${\mathit{Q}}^{04}$, ${\mathit{Q}}^{05}$) | 0.047 | 1.999 | 0.672 |

## 6. Conclusions

## References and Notes

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## Share and Cite

**MDPI and ACS Style**

Fernández-Vázquez, E. Recovering Matrices of Economic Flows from Incomplete Data and a Composite Prior. *Entropy* **2010**, *12*, 516-527.
https://doi.org/10.3390/e12030516

**AMA Style**

Fernández-Vázquez E. Recovering Matrices of Economic Flows from Incomplete Data and a Composite Prior. *Entropy*. 2010; 12(3):516-527.
https://doi.org/10.3390/e12030516

**Chicago/Turabian Style**

Fernández-Vázquez, Esteban. 2010. "Recovering Matrices of Economic Flows from Incomplete Data and a Composite Prior" *Entropy* 12, no. 3: 516-527.
https://doi.org/10.3390/e12030516