1. 1 USING A QUADRATIC PROGRAMMING APPROACH TO SOLVE SIMULTANEOUS RATIO AND BALANCE EDIT PROBLEMS Katherine J. Thompson James T. Fagan Brandy L. Yarbrough Donna L. Hambric

2. 2 Economic Census Editing Ratio edits of basic data items 1  Annual Payroll/1 st Quarter Payroll  4.4 30  Annual Payroll/Employment  70 Balance edits to ensure additivity Annual Payroll = 1 st Quarter Payroll + 2 nd Quarter Payroll + 3 rd Quarter Payroll + 4 th Quarter Payroll Items that appear in both ratio and balance edits may fail to satisfy the original ratio edit tests.

3. 3 Addressing the Problem Often not a problem Motivated two Economic Census Applications: –Assets Complex (Manufactures Sector – 1997) –Gross Margin/Gross Profit (Wholesale Trade – 2002) Quadratic programs replaced manual or existing systems

4. 4 General Set Up (Motivating Example) Five data items: A, B, C, D, and E. Two ratio relationships (edits): – L AB  A/B  U AB – L CD  C/D  U CD One balance edit: A + C +D = E Non-negativity edits on all five items Objective: Minimize change from input to output data while satisfying the ratio and balance constraints

5. 5 Quadratic Program min [(A – A*) 2 + (B – B*) 2 + (C – C*) 2 + (D – D*) 2 + (E – E*) 2 ] subject to (s.t.) A* + C* + D* – E* = 0Balance constraint L AB  A/B  U AB Ratio edit constraints L CD  C/D  U CD A*, B*, C*, D*, E*  0Non-negativity constraints [Note: an asterisk denotes the output value from the quadratic program solution.]

6. 6 Special Considerations Not all items equally reliably reported Preference for adjusting imputed values before (retained) reported values Solution: item-specific reliability weight example 1:  A = 10 for data item A  B = 100 for data item B (B more reliably reported than A) example 2:

7. 7 Revised Quadratic Program Min [  A (A – A*) 2 +  B (B – B*) 2 +  C (C – C*) 2 +  D (D – D*) 2 +  E (E – E*) 2 ] s.t. same constraints (ratio, balance, non-negativity) Comments Minimizes residual sums of squares (Deming, 1943) –least squares estimation solution –Data item Y i = E(Y i )+  i and  i   /  i We used subjectively-defined reliability weights Non-integer solutions (controlled rounding)

8. 8 Other Modifications “Fixing” variables –drop squared term from objective function –include values in constraint as constants No limit (found yet) on number of allowable constraints –remove redundant edits from program for speed

9. 9 Assets Complex (Manufactures) RowDescriptionTotalBuildingMachinery aBeginning year assetsTABBABMAB bCapital expendituresTCEBCEMCE cRetirementsTRTBRTMRT dEnding year assetsTAEBAEMAE Two dimensional balancing constraints: Row Constraints: TOTAL item = the BUILDING item + the MACHINERY item Column constraints: value of d = a + b - c. Ratio constraints on three marginal row totals: Ending Assets, Capital Expenditures, and Retirements.

10. 10 Production Application Ratio edits on marginal totals performed separately (first) Quadratic program on all data items –original ratio constraints –additional ratio constraints on some details to associated marginal totals –all balance constraints Same reliability weights for all items

11. 11 Gross Margin/Gross Profit (Wholesale) Two Derived Items: Gross Margin = Sales – Gross Selling Value – Beginning Inventories – Purchases + Ending Inventories Gross Profit = Gross Margin – Operating Expenses + Commissions Payroll, Employment, Receipts, Operating Expenses, Purchases, and Inventories collected from all establishments (ratio edited) Commissions and Gross Selling Value collected when available (no pre-editing)

12. 12 Goals Replace manual edit system with automated edit Retain analyst preferences used in manual system Include all ratio and balance constraints in minimization problem (not feasible with manual system)

13. 13 First Attempt Included all eight data items plus two derived items in objective function Included all ratio and balance constraints Two different item reliability weights for each item (one for reported value; one for imputed value)

14. 14 Results Too many items changed by a “small” amount One-pass approach not implementing analyst procedure –If Gross Margin failed its ratio test, then wanted to first adjust values of Purchases, then Sales. –If after correcting Gross Margin, Gross Profit still failed its ratio test, wanted to adjust values of Operating Expenses, then Purchases, and lastly Sales.

15. 15 Production Program: Second Attempt Split into two separate quadratic equations –Gross margin and associated constraints –Gross profit and associated constraints –Different reliability weights for each item in each program Dropped distinction between reported and imputed values of same item Eliminated Gross Selling Value, Commissions, and Inventories Items from objective functions (treated as constants in constraints)

16. 16 Comments In both applications, quadratic program did not find solutions for all records –generally very intractable (poorly reported) data Very disparate values of reliability weights needed to control outcome (Wholesale weights different by a factors of 10 6 and 10 9 ) Solution as good as constraints

17. 17 Comments Objective different from “traditional” balance edit objective –does not try to preserve distribution of details (e.g., raking) –can adjust certain item values by large amount without using statistically based model No generalized programs...yet