Updating Matrices: Biproportional and Cross-Entropy Methods

1. EC 936 ECONOMIC POLICY MODELLING LECTURE 2: PART I UPDATING MATRICES: i: BIPROPORTIONAL (RAS) METHOD [DEMING & STEPHAN, 1940; LEONTIEF, 1941; STONE, 1962; BACHARACH, 1970]

2. MATRIX BALANCING ISSUES • TYPE I & TYPE II PROBLEMS: • I: AMENDING MATRIX ENTRIES TO CONFORM TO NEW ACCOUNT TOTALS • II: AMENDING MATRIX ENTRIES WHEN ACCOUNT TOTALS ARE UNKNOWN, BUT ARE GOVERNED BY KNOWN ACCOUNTING CONSTRAINTS

3. RAS AND TYPE I PROBLEMS

4. PROBLEM: HOW TO UPDATE A TRANSACTIONS MATRIX WHEN ONLY THE GROSS OUTPUT FIGURES ARE KNOWN

5. SOME BASIC DEFINITIONS

7. For rectangular matrices (m≠n) min s.t. for i = 1,2,…,m for j = 1,2,…,n

10. MAD = MAPE =

11. RAS AND TYPE II PROBLEMS Use Diagonal Similarity Scaling (DSS) Method min s.t.

12. EC 936 ECONOMIC POLICY MODELLING LECTURE 2: PART I UPDATING MATRICES: ii: CROSS-ENTROPY METHOD [GOLAN ET AL, 1994; ROBINSON ET AL, 2001]

14. RAS vs CROSS ENTROPY METHODS(McDougall, 1999) • The RAS is an entropy optimization method, and has long been known to be so. • For the matrix filling problem, in general, the entropy optimization method of choice is proportional allocation. • For the matrix balancing problem, in general, the entropy optimization method of choice is the RAS. • If, following the GCE approach, we treat matrix elements as expected values of discrete random variables, the method of choice (in the absence of distributional data) is equivalent to the RAS. • Entropy theory may fruitfully be used, not in attempting to supplant the RAS, but in extending and adapting it to problems that do not well fit the traditional matrix balancing framework.

15. EC 936 ECONOMIC POLICY MODELLING LECTURE 2: PART II CALIBRATING SAMS: ERRORS, INCOMPLETE & INCONSISTENT INFORMATION [STONE, CHAPERNOWNE, MEADE 1942; STONE 1977; BYRON 1978]

22. EC 936 ECONOMIC POLICY MODELLING LECTURE 2: PART III INPUT-OUTPUT TECHNIQUES & CONSISTENCY MODELLING

23. since m = n

26. x = f = Z =

27. A = I = [ I – A ] x = f

28. ______ =