2.11. Stochastic simulations

Once a model is solved, RISE exposes a small set of stochastic-analysis routines on the model object. All of them share the same convention: the return type is a struct of ts (RISE time-series) objects keyed by endogenous-variable name, with multiple pages when there are multiple parameterizations or solution branches.

2.11.1. Artificial data

simulate returns simulated time series from the solved model:

sim = simulate(m, 'simul_periods', 200);

plot(sim.PAI, sim.Y)

The most relevant options:

Option	Effect
`simul_periods`	Number of periods to simulate (default `100`).
`simul_burn`	Burn-in periods discarded from the head of the returned series (default `0`).
`simul_seed`	Seed forwarded to MATLAB’s RNG before drawing. `[]` keeps the current state.
`simul_shock_uncertainty`	When `true` (default), draws structural shocks from their declared distributions; when `false`, sets every shock to its mean and returns the deterministic continuation.
`simul_historical_data`	Initial conditions and / or a conditioning path. Accepts a struct of `ts` or a simulation plan.

The output struct is suitable for fan charts, regime-conditioning diagnostics, and as data for an estimation step. Pages of the returned ts correspond to the model’s parameterizations.

2.11.2. Pruned simulations

RISE implements 5-th order approximation of the perturbation solution of regime-switching DSGE models. At order \(\geq 2\) the straight-perturbation simulation is not guaranteed to be bounded: an unlucky shock realisation can send the state off to inf. The pruning algorithm of [Andreasen et al., 2013], generalised to regime-switching models, removes the unbounded higher-order terms at simulation time without changing the policy function. Pruning is on by default at order \(\geq 2\) and can be controlled via the standard solve / simulate options.

2.11.3. Theoretical and Simulated moments

For a linear solved model the unconditional first and second moments are available in closed form via the Lyapunov equation. RISE exposes them through theoretical_autocovariances (and the theoretical_autocorrelations convenience wrapper):

[Acov, info, retcode] = theoretical_autocovariances(m, 10);
Acorr                 = theoretical_autocorrelations(m, 'autocorr_ar', 10);

For higher-order or regime-switching models, where closed-form moments are not available, the standard practice is to simulate a long path with simulate and compute sample moments from the returned ts (cov, corrcoef, var overloaded on ts). The two should agree at order 1 up to Monte Carlo error.

2.11.4. Variance decomposition

variance_decomposition returns the share of forecast-error variance attributable to each structural shock, at each horizon, for each endogenous variable:

vd = variance_decomposition(m);

The output is a struct of ts (one entry per endogenous variable) where each ts carries one column per shock and one row per horizon (conditional) or a single row for the unconditional decomposition (unconditional). It is available for the linear (order-1) solution of constant-parameter and regime-switching models.

2.11.5. Impulse responses

irf returns impulse responses on the solved model:

myirfs = irf(m);

The output is a struct of ts keyed by shock name; each entry is itself a struct of ts keyed by endogenous variable. quick_irfs (see Plotting tools) lays the result out in a publication grid.

Simple Impulse Responses

“Simple” IRFs trace the model’s response to a one-time, one-shock deviation from a fixed initial state (typically the unconditional mean or the deterministic steady state). For a linear model the simple IRF is the same regardless of initial state and shock sign, so it is the natural default. The main controls are:

irf_periods (default 40): horizon.
irf_shock_sign and irf_shock_scale: sign and size of the impulse. Use -1 for a contractionary shock and 2 to feed a two-standard-deviation impulse instead of the default one.
solve_shock_horizon: anticipation horizon. With a strictly positive value the IRF traces the response to a shock that is announced the corresponding number of periods ahead (the “anticipation” pattern); with 0 the shock is unanticipated.
Delayed reaction of the response is obtained by appending zero realisations to the shock path through a simulation plan. The same plan, with simplan.anticipate(true) vs anticipate(false), produces the anticipated and unanticipated variants.

Generalized Impulse Responses

For nonlinear or regime-switching models, the simple IRF depends on the initial state and on whether the shock is positive or negative, small or large. The generalized impulse response (GIRF), defined as the difference between the conditional expectation of the path with a given shock and without, averaged over the model’s stationary distribution of states, is the right object to look at. Switch irf into GIRF mode via:

girf = irf(m, 'irf_type', 'girf', 'irf_draws', 1000);

irf_draws controls the number of state-and-shock realisations averaged. The cost scales linearly with irf_draws; values in the low thousands are usually enough for smooth output.