Chapter 1: Preliminaries 1
1.1 Stochastic Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Concepts of Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.1 Almost sure (a.s.) convergence . . . . . . . . . . . . . . . . . . 3
1.2.2 Convergence in Probability . . . . . . . . . . . . . . . . . . . . . 4
1.2.3 Convergence in Lq-norm. . . . . . . . . . . . . . . . . . . . . . . 6
1.2.4 Convergence in Distribution . . . . . . . . . . . . . . . . . . . . 7
1.3 Time Series Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 Law of Large Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.4.1 Dependent and Identically Distributed Observations . . . . . 14
1.4.2 Dependent and Heterogeneously Distributed Observations. 15
1.4.3 Martingale Difference Process . . . . . . . . . . . . . . . . . . . 16
1.5 Central Limit Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.5.1 Dependent and Identically Distributed Observations . . . . . 17
1.5.2 Dependent Heterogeneously Distributed Observations . . . . 18
1.5.3 Martingale Difference Observations . . . . . . . . . . . . . . . . 18
1.6 Elements of Spectral Analysis . . . . . . . . . . . . . . . . . . . . . . . 19
Chapter 2: DSGE Models, Solutions and Approximations 27
2.1 Few useful models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.1.1 A basic Real Business Cycle (RBC) Model . . . . . . . . . . . 28
2.1.2 Heterogeneous agent models . . . . . . . . . . . . . . . . . . . . 35
2.1.3 Monetary Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.2 Approximation methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.2.1 Quadratic approximations . . . . . . . . . . . . . . . . . . . . . . 45
2.2.2 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.2.3 Log linear Approximations . . . . . . . . . . . . . . . . . . . . . 51
2.2.4 Second order approximations . . . . . . . . . . . . . . . . . . . . . . 60
2.2.5 Parametrizing expectations . . . . . . . . . . . . . . . . . . . . . 62
2.2.6 A Comparison of methods . . . . . . . . . . . . . . . . . . . . . 65
Chapter 3: Extracting and Measuring Cyclical Information 67
3.1 Statistical Decompositions . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.1.1 Traditional methods . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.1.2 Beveridge-Nelson (BN) decomposition . . . . . . . . . . . . . . 69
3.1.3 Unobservable Components (UC) decompositions . . . . . . . 72
3.1.4 Regime shifting decomposition . . . . . . . . . . . . . . . . . . . 75
3.2 Hybrid Decompositions . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.2.1 The Hodrick and Prescott (HP) Filter . . . . . . . . . . . . . . 79
3.2.2 Exponential smoothing (ES) filter . . . . . . . . . . . . . . . . . 86
3.2.3 Moving average (MA) filters . . . . . . . . . . . . . . . . . . . . 88
3.2.4 Band Pass (BP) filters . . . . . . . . . . . . . . . . . . . . . . . . 89
3.3 Economic Decompositions . . . . . . . . . . . . . . . . . . . . . . . . . . 95
3.3.1 Blanchard and Quah (BQ) Decomposition . . . . . . . . . . . 95
3.3.2 King, Plosser Stock and Watson (KPSW) Decomposition . 97
3.4 Time Aggregation and Cycles . . . . . . . . . . . . . . . . . . . . . . . 99
3.5 Collecting Cyclical Information . . . . . . . . . . . . . . . . . . . . . . 100
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