Learning in Economic Systems with Expectations Feedback

Contents......Page 7
1. Introduction......Page 9
2. Economic Systems With Expectations Feedback......Page 14
2.1 An Introductory Example......Page 15
2.2 The General Setup......Page 17
2.3 Forecasting Rules With Finite Memory......Page 19
2.4 Consistent Forecasting Rules......Page 21
2.5 Unbiased Forecasting Rules......Page 24
3. Adaptive Learning in Linear Models......Page 30
3.1 Linear Models......Page 31
3.2 Unbiased Forecasting Rules......Page 35
3.2.1 No-Updating Rules......Page 37
3.2.2 Linear MSV Predictors......Page 41
3.3 Approximating Unbiased Forecasting Rules......Page 44
3.4 Convergence of an AML-Based Learning Scheme......Page 46
3.5.1 On the Stability of Linear Systems......Page 52
3.5.2 Proofs of Theorem 3.2 and Theorem 3.3......Page 54
4. Economic Models Subject to Stationary Noise......Page 64
4.1 Forecasting Rules for Stationary Noise......Page 66
4.1.1 Consistent Forecasting Rules......Page 68
4.1.2 Unbiased Forecasting Rules......Page 71
4.2 Existence of Unbiased Forecasting Rules......Page 75
4.3 MSV Predictors......Page 80
4.4 Random Fixed Points......Page 82
4.5 Mathematical Appendix......Page 86
5. Nonparametric Adaptive Learning......Page 91
5.1 The Method of Stochastic Approximation......Page 92
5.2 A Basic Convergence Result......Page 96
5.3 Some Probabilistic Prerequisites......Page 105
5.4 Sufficient Conditions for Contractions......Page 108
5.5.1 Sobolev Imbedding Theorem......Page 114
5.5.2 Proofs of Section 5.2......Page 117
5.5.3 Proofs of Section 5.4......Page 124
6.1 A Model of Pure Exchange......Page 132
6.2 The Existence of Perfect Forecasting Rules......Page 134
6.3 Perfect Foresight Dynamics......Page 136
6.4 Adaptive Learning of Perfect Forecasting Rules......Page 139
6.5 Mathematical Appendix......Page 141
7. Heterogeneous Beliefs in a Financial Market......Page 143
7.1 The Model......Page 145
7.2 Heterogeneous Beliefs......Page 149
7.3 Risk Premia and Reference Portfolios......Page 151
7.4 Mean-Variance Preferences......Page 153
7.5.1 Perfect Forecasting Rules for First Moments......Page 156
7.5.2 Perfect Forecasting Rules for Second Moments......Page 157
7.6 Dynamics Under Rational Expectations......Page 160
7.7.1 The General Idea......Page 165
7.7.2 The Case With Two Mediators......Page 166
7.8 Mathematical Appendix......Page 172
B......Page 174
C......Page 175
H......Page 176
L......Page 177
R......Page 178
Y......Page 179