Regression imputation with linear constraints on the variables

1. Regression imputation with linear constraints on the variables Jeroen Pannekoek Statistics Netherlands Work Session on Statistical Data Editing (Bonn, Germany, 25-27 September 2006)

2. Overview • Definition of the problem • Consistent linear regression predictions • Other models

3. Balance edits • Example of balance edits: 5 variables, 2 constraints

4. We need predictions that satisfy Constraints on missing values • Suppose that some part of y is missing • Partitioning of y and R gives:

5. Taking care of constraints is a minimum size adjustment obtained by: Minimize subject to => and so where Regression predictions and adjustments • Standard regression imputation

6. Consider the model and estimate the parameters simultaneously by OLS. This leads to normal equations: (1) (2) To be solved for α and β (2) Shows that the predictions are consistent A model incorporating the predictions

7. For records with missing values use: Parameter estimates • Estimates for αi and β in the simultaneous model:

8. Illustration Constraints: not a nuisance but a benefit !

9. This leads to predictions of the form: And this model can be estimated bij WLS Weighted adjustments • Suppose that and we want to make larger adjustments for variables with larger error variance: minimize subject to

10. WLS normal equations • Minimize w.r.t β and αi yields normaL equations The last equation shows consistency of the predictions

11. Estimate by WLS using covariance matrix Results in normal equations The last equation shows again consistency of the predictions Log transform • Model