1. DSGE Modeling

A Markov-switching DSGE in the modern toolbox is a forward-looking nonlinear system

\[E_t f(y_{t+1}, y_t, y_{t-1}, \varepsilon_t, p(r_t)) = 0,\]

with \(r_t = 1, 2, \dots, h\) a Markov-chain regime, transition matrix \(Q_{i,j}(I_t)\) that can be constant or endogenous, and a parameter vector \(p(r_t)\) that can switch on one or more chains. The constant-parameter case is the trivial \(h = 1\) special case of the same engine; there is no separate constant-parameter code path.

The modern toolbox builds a DSGE via the dsge_model factory. Once you have the object, solve, filter, estimate, forecast, simulate, irf and the decompositions operate on it directly. The rest of this chapter covers the DSGE-specific machinery, one section at a time:

• 1.1. Model file language
  • 1.1.1. Block keywords at a glance
  • 1.1.2. Declarations
  • 1.1.3. The @model block
  • 1.1.4. Steady state
  • 1.1.5. Markov chains
  • 1.1.6. Optimal policy
  • 1.1.7. Epilogue
  • 1.1.8. What does not go in the model file
• 1.2. Zero-th order approximation: steady state and balanced-growth path
  • 1.2.1. Anatomy of a steady-state file
  • 1.2.2. Incomplete analytical solution
  • 1.2.3. Steady-state loops
  • 1.2.4. Facilitators
  • 1.2.5. Multiple regimes
• 1.3. First-order perturbation
  • 1.3.1. The state vector
  • 1.3.2. Solvers for the quadratic matrix polynomial
  • 1.3.3. User-defined solver
• 1.4. Higher-order perturbation
  • 1.4.1. Choosing the derivatives
  • 1.4.2. Solving the system of generalized Sylvester equations
• 1.5. Solving
  • 1.5.1. The signature
  • 1.5.2. The retcode pattern
  • 1.5.3. Picking a solver
  • 1.5.4. Solver options
  • 1.5.5. Order of approximation
  • 1.5.6. Optimal policy
  • 1.5.7. Occasionally-binding constraints
  • 1.5.8. Perturbation strategy
  • 1.5.9. Other options worth knowing
  • 1.5.10. The diagnostic protocol when solve fails
  • 1.5.11. Return-code reference
  • 1.5.12. The retcode-aware pattern
• 1.6. Optimal policy
  • 1.6.1. The modern distinctive
  • 1.6.2. Declaring the planner’s problem
  • 1.6.3. Solve-time choices
  • 1.6.4. Solver selection
  • 1.6.5. Loose commitment and stochastic replanning
  • 1.6.6. A worked example: Tatiana’s monetary-fiscal game
  • 1.6.7. Nonstationary optimal policy: geometric multipliers
  • 1.6.8. Where to look next
• 1.7. Optimal (optimized) simple rules
  • 1.7.1. The loss function
  • 1.7.2. (Approximate) analytical loss
  • 1.7.3. Simulated loss
  • 1.7.4. OSR and indirect inference
  • 1.7.5. OSR with a user-defined objective
  • 1.7.6. Policy frontiers
• 1.8. Occasionally-binding constraints
  • 1.8.1. Form 1. Kinky functions
  • 1.8.2. Form 2. Regime-switching approximation
  • 1.8.3. Side constraints inside an optimization problem
  • 1.8.4. Simulation and IRFs under OBC
  • 1.8.5. Where to look next
• 1.9. Deterministic and quasi-deterministic solutions
  • 1.9.1. The perfect-foresight problem
  • 1.9.2. Setting the conditions
  • 1.9.3. Free terminal: commitment and optimal-policy models
  • 1.9.4. Choosing the horizon
  • 1.9.5. Solvers and performance
  • 1.9.6. Quasi-deterministic solutions: the extended path
  • 1.9.7. Regime switching
  • 1.9.8. See also
• 1.10. Heterogeneous agents (HANK)
  • 1.10.1. Heterogeneity axes
  • 1.10.2. Deterministic axes
  • 1.10.3. Individual variables
  • 1.10.4. The individual block
  • 1.10.5. Aggregation: @Agg
  • 1.10.6. The aggregate block
  • 1.10.7. Setting grid and transition values
  • 1.10.8. Composing with regime switching
  • 1.10.9. Worked example
• 1.11. Very large models
  • 1.11.1. Why RISE requires MATLAB R2023b
  • 1.11.2. Writing a large model
  • 1.11.3. Parsing, differentiation and memory
  • 1.11.4. Solving a large model
• 1.12. Automatic translation of files
  • 1.12.1. Dynare files
  • 1.12.2. Dynare steady state files
  • 1.12.3. AIM/YADA files