Stochastic Simulation

Definition

Stochastic simulation is a numerical method for propagating uncertainty through a model by repeatedly solving the model over a forecast horizon under random draws of its stochastic components. Given a model $y_t = f(y_{t-1}, x_t, \varepsilon_t, \theta)$, each trial $j = 1, \ldots, J$ draws $\varepsilon_t^{(j)}$ from the estimated error distribution, solves for the implied path $\{y_t^{(j)}\}_{t=1}^H$, and the empirical distribution across $J$ trials characterises forecast uncertainty. Fair (1980) formalises four nested levels of simulation, each adding an additional source of uncertainty.

Key Ideas

• Four levels of stochastic simulation (Fair 1980):
  • (a) Error-term uncertainty only: draw $\varepsilon_t \sim \hat{\mathcal{N}}(0, \hat\Sigma)$, hold $\hat\theta$ fixed and exogenous variables known.
  • (b) + Coefficient uncertainty: also draw $\theta^{(j)} \sim p(\theta \mid \text{data})$ (e.g., from the asymptotic normal distribution of the estimator).
  • (c) + Exogenous-variable uncertainty: also draw exogenous paths $x_t^{(j)}$ from separate forecasting models.
  • (d) + Misspecification: add the gap between the model's in-sample fit and the stochastic-simulation variance, estimated by rolling-window out-of-sample residuals.
• Non-linearity: stochastic simulation is necessary when $f$ is nonlinear — the distribution of $y_t^H$ is not normal even if errors are normal, because $f$ maps normal inputs to non-normal outputs.
• Prospective confidence intervals: unlike classical Root Mean Squared Error (RMSE) evaluation (which requires realized outcomes), stochastic simulation produces intervals before outcomes are observed — directly useful for communicating uncertainty to policymakers.
• Exogenous uncertainty often dominates: in large macroeconometric models, uncertainty about future exogenous variables (oil prices, fiscal policy, trading-partner growth) typically exceeds model-coefficient uncertainty (Fair 1980).

How It Works

1. Estimate the model $\hat\theta$ and residual covariance $\hat\Sigma$.
2. Fix the forecast origin and set $J$ (typically 1,000–5,000 trials).
3. For each trial $j$: draw $\varepsilon_t^{(j)} \sim \hat{\mathcal{N}}(0, \hat\Sigma)$ for $t = 1, \ldots, H$; optionally draw $\theta^{(j)}$ and $x_t^{(j)}$; solve the model dynamically to obtain $y_t^{(j)}$.
4. Compute quantiles of $\{y_t^{(j)}\}_{j=1}^J$ at each horizon $t$ to form prediction intervals.

Rolling-window misspecification estimator: re-estimate the model on each of $T$ rolling samples ending one period before the forecast origin; compute outside-sample squared residual; average over $T$ to estimate the misspecification component $\bar d_i(k)$.

Why It Matters

• Makes probabilistic forecasting operational in large nonlinear models where analytical formulas are unavailable.
• Separates four distinct sources of forecast error — a breakdown unavailable from RMSE alone.
• Underpins fan charts at central banks (e.g., Bank of England Inflation Report, Riksbank Monetary Policy Report).

Open Questions

• Computational cost scales with $J \times H \times \text{model size}$; modern DSGE models with particle-filter estimation have made exact stochastic simulation expensive.
• The constancy assumption (rolling-window misspecification is constant over time) is strong and violated by structural breaks.
• Optimal allocation of simulation effort across the four uncertainty components is not established.

Related

• Forecast Uncertainty Decomposition
• Forecasting
• Macroeconometric Model
• Conditional Forecasting
• Bayesian Multiperiod Prediction
• Ray C. Fair