Supplement to “ Estimating the Technology of Cognitive and Noncognitive Skill Formation ” : Appendix

A1. Deriving the Technology of Skill Formation From Its Primitives . . . . . . . . . . . . 2 A1.1. A Model of Skill Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 A1.2. How the Flourishing of Noncognitive Traits With the Stage of the Life Cycle Can Raise the Elasticity of Substitution for Noncognitive Skills at Later Stages Compared to Earlier Stages . . . . . . . . . . . . . . . . 7 A2. Investments as a Function of Inputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 A3. Proofs of Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 A3.1. Proof of Theorem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 A3.2. Proof of Theorem 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 A3.3. A More Explicit Discussion of the Analysis of Section 3.6.2 . . . . . . . . . . 10 A4. Identification of Factor Loadings for the Case With Unobserved Heterogeneity . . . 13 A5. The Likelihood in the Notation of the Text . . . . . . . . . . . . . . . . . . . . . . . . . 13 A6. Nonlinear Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 A6.1. The Update Step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 A6.2. The Prediction Step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 A6.3. Unscented Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 A6.4. Implementation of Nonlinear Filtering . . . . . . . . . . . . . . . . . . . . . . 20 A7. Implementation of Anchoring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 A7.1. Linear Anchoring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 A7.2. Nonlinear Anchoring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 A7.3. Adjusting the Likelihood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 A8. A Limited Monte Carlo Study of the Empirical Model of Section 4.2.5 . . . . . . . . 24 A9. Data Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 A9.1. Survey Measures on Parental Investments . . . . . . . . . . . . . . . . . . . . . 24 A9.2. Survey Measures on Child’s Cognitive Skills . . . . . . . . . . . . . . . . . . . 25 A9.3. Survey Measures on Child’s Noncognitive Skills . . . . . . . . . . . . . . . . . 27 A10. Estimated Parameters for Outcome and Investment Equations and Initial Conditions; Normalizations for the Measurement System . . . . . . . . . . . . . . . . 32 A11. Sensitivity Analyses for a One-Stage Model . . . . . . . . . . . . . . . . . . . . . . . . 37 A11.1. Sensitivity to Alternative Anchors . . . . . . . . . . . . . . . . . . . . . . . . . 37 A11.2. Relaxing the Normality Assumption . . . . . . . . . . . . . . . . . . . . . . . . 37 A12. Variance Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 A12.1. Cognitive and Noncognitive Skills . . . . . . . . . . . . . . . . . . . . . . . . . 40 A12.2. Parental Investment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 A13. Further Simulations of the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 A14. Analysis of One-Skill (Cognitive) Models . . . . . . . . . . . . . . . . . . . . . . . . . 44 A14.1. Implications of Estimates From a Two-Stage–One-Skill Model of Investment 44 A14.2. The Implications of Estimates From a Single-Stage–One-Skill (Cognitive) Model of Investment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 Author’s Addresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50