Forecasting and the Wold decomposition

Forecasting is the deliverable of most time-series modelling. The forecast itself, the uncertainty around it, and the diagnostics that justify both are the three things every forecasting framework must produce. Two facts dominate: forecast errors grow with horizon, and the unconditional mean dominates long-horizon point forecasts.

Optimal point forecast

Under squared-error loss, the optimal h-step-ahead forecast is the conditional expectation: X̂_{t+h|t} = E[X_{t+h} | F_t]. Under absolute-error loss, it's the conditional median. Under more exotic loss functions (asymmetric, quantile), other functionals.

AR(1) forecasts

For X_t = c + φ X_{t-1} + ε_t with stationary AR(1): X̂_{t+h|t} = c(1 + φ + ... + φ^(h-1)) + φ^h X_t. As h → ∞, X̂_{t+h|t} → c / (1 - φ) — the unconditional mean. Forecasts mean-revert geometrically.

Forecast errors

The h-step forecast error is e_{t+h|t} = X_{t+h} - X̂_{t+h|t}. For ARMA(p, q) with Wold MA(∞) coefficients ψ_j, the h-step forecast error variance is:

Var(e_{t+h|t}) = σ²(ψ_0² + ψ_1² + ... + ψ_{h-1}²)

Variance grows with h, plateauing at the unconditional variance as h → ∞ for stationary processes. For random walks, variance grows linearly — never converging.

Prediction intervals

Assuming Gaussian errors: PI_α = X̂_{t+h|t} ± z_α · √Var(e_{t+h|t}). For non-Gaussian errors, bootstrap or simulation-based intervals are more honest. Empirical financial data is heavy-tailed; Gaussian intervals are often too narrow.

Forecast evaluation

• MAE — mean absolute error.
• RMSE — root mean squared error.
• MAPE — mean absolute percentage error; problematic when actual is near zero.
• Directional accuracy — fraction of correct sign predictions; relevant for trading.
• Diebold-Mariano test — formal comparison of two forecasts' loss differentials.

Wold decomposition revisited

Every stationary, purely-nondeterministic process is an MA(∞) of its own innovations. The Wold theorem guarantees the existence; it doesn't guarantee identifiability of the MA coefficients from finite data. The practical content: ARMA models are flexible enough to approximate any stationary linear process, but with finite samples we can only get the first few ψ_j precisely.

Combining forecasts

Equal-weighted combinations of multiple forecasts often beat individual forecasts, even sophisticated ones. The Bates-Granger (1969) finding has held up across decades and domains. Bayesian model averaging is the principled generalisation; simple averaging is the practitioner's default.