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. Deriving models with rise.microfound
  • 1.2.1. Write the economics, not the algebra
  • 1.2.2. A first example: the RBC model
  • 1.2.3. What is generated
  • 1.2.4. Multiple agents
  • 1.2.5. Regime switching
  • 1.2.6. Inequality constraints (Kuhn-Tucker)
  • 1.2.7. Output methods
  • 1.2.8. API reference
  • 1.2.9. Scope and roadmap
  • 1.2.10. Validation
• 1.3. Zero-th order approximation: steady state and balanced-growth path
  • 1.3.1. Anatomy of a steady-state file
  • 1.3.2. Incomplete analytical solution
  • 1.3.3. Steady-state loops
  • 1.3.4. Facilitators
  • 1.3.5. Steady states of growing variables in the dynamic equations
  • 1.3.6. Multiple regimes
• 1.4. First-order perturbation
  • 1.4.1. The state vector
  • 1.4.2. Solvers for the quadratic matrix polynomial
  • 1.4.3. User-defined solver
• 1.5. Higher-order perturbation
  • 1.5.1. Choosing the derivatives
  • 1.5.2. Solving the system of generalized Sylvester equations
  • 1.5.3. Higher orders when the steady state switches
• 1.6. Solving
  • 1.6.1. The signature
  • 1.6.2. The retcode pattern
  • 1.6.3. Picking a solver
  • 1.6.4. Solver options
  • 1.6.5. Order of approximation
  • 1.6.6. Reducing the size of the system
  • 1.6.7. Optimal policy
  • 1.6.8. Occasionally-binding constraints
  • 1.6.9. Perturbation strategy
  • 1.6.10. Other options worth knowing
  • 1.6.11. The diagnostic protocol when solve fails
  • 1.6.12. Return-code reference
  • 1.6.13. The retcode-aware pattern
• 1.7. Perturbation types for regime-switching models
  • 1.7.1. The types at a glance
  • 1.7.2. Perturbation M ('m')
  • 1.7.3. The Maih–Waggoner perturbation ('mw')
  • 1.7.4. The Foerster–Rubio-Ramírez–Waggoner–Zha perturbation ('frwz')
  • 1.7.5. Experimental types: 'mt' and 'mwt'
  • 1.7.6. Which type to use
  • 1.7.7. The evidence
  • 1.7.8. First-order solvers and the types
• 1.8. Optimal policy
  • 1.8.1. The modern distinctive
  • 1.8.2. Declaring the planner’s problem
  • 1.8.3. Switching parameters on leads and lags
  • 1.8.4. Switching parameters in an objective
  • 1.8.5. A lead that a lag multiplies
  • 1.8.6. Equations outside the optimization
  • 1.8.7. Robust policy: a policymaker that distrusts its model
  • 1.8.8. Solve-time choices
  • 1.8.9. Solver selection
  • 1.8.10. Perturbation types
  • 1.8.11. The planner-discount knife-edge
  • 1.8.12. Loose commitment and stochastic replanning
  • 1.8.13. Loose commitment and stochastic replanning are different problems
  • 1.8.14. A linear-quadratic solver for stochastic replanning
  • 1.8.15. When the structural matrices also switch
  • 1.8.16. Solving a non-cooperative game without the policy-derivative machinery
  • 1.8.17. A worked example: Tatiana’s monetary-fiscal game
  • 1.8.18. Nonstationary optimal policy: geometric multipliers
  • 1.8.19. Where to look next
• 1.9. Optimal (optimized) simple rules
  • 1.9.1. The loss function
  • 1.9.2. (Approximate) analytical loss
  • 1.9.3. Simulated loss
  • 1.9.4. Switching models: how regimes are aggregated
  • 1.9.5. OSR and indirect inference
  • 1.9.6. OSR with a user-defined objective
  • 1.9.7. Policy frontiers
• 1.10. Occasionally-binding constraints
  • 1.10.1. Form 1. Kinky functions
  • 1.10.2. Form 2. Regime-switching approximation
  • 1.10.3. Constraints held by a named shock
  • 1.10.4. Mixing the two forms in one model
  • 1.10.5. Side constraints inside an optimization problem
  • 1.10.6. Simulation and IRFs under OBC
  • 1.10.7. Where to look next
• 1.11. Deterministic and quasi-deterministic solutions
  • 1.11.1. The perfect-foresight problem
  • 1.11.2. Setting the conditions
  • 1.11.3. Free terminal: commitment and optimal-policy models
  • 1.11.4. Choosing the horizon
  • 1.11.5. Solvers and performance
  • 1.11.6. Quasi-deterministic solutions: the extended path
  • 1.11.7. Regime switching
  • 1.11.8. See also
• 1.12. Heterogeneous agents (HANK)
  • 1.12.1. Heterogeneity axes
  • 1.12.2. Deterministic axes
  • 1.12.3. Individual variables
  • 1.12.4. The individual block
  • 1.12.5. Aggregation: @Agg
  • 1.12.6. The aggregate block
  • 1.12.7. Setting grid and transition values
  • 1.12.8. Bin indicators: addressing a cell by position
  • 1.12.9. The bracket window: span
  • 1.12.10. Solving the steady state: the het engine
  • 1.12.11. Pruning the constants: solve_prune_hank_constants
  • 1.12.12. Composing with regime switching
  • 1.12.13. Worked example
• 1.13. Very large models
  • 1.13.1. Why RISE requires MATLAB R2023b
  • 1.13.2. Writing a large model
  • 1.13.3. Parsing, differentiation and memory
  • 1.13.4. Solving a large model
• 1.14. Model reduction
  • 1.14.1. What the reduction does
  • 1.14.2. The basis
  • 1.14.3. Regime switching
  • 1.14.4. Occasionally-binding constraints
  • 1.14.5. Higher orders
  • 1.14.6. Supplied bases and projections
  • 1.14.7. Contracted derivatives
  • 1.14.8. Validating a reduction
  • 1.14.9. Options of build
  • 1.14.10. Limits
• 1.15. Template differentiation
  • 1.15.1. It is on by default
  • 1.15.2. What it recognizes
  • 1.15.3. What breaks a template
  • 1.15.4. What it saves
  • 1.15.5. Guarantee
  • 1.15.6. Limits
  • 1.15.7. How it works
• 1.16. Automatic translation of files
  • 1.16.1. Dynare files
  • 1.16.2. Dynare steady state files
  • 1.16.3. AIM/YADA files