One forecast, and the band the arithmetic puts round it

An AR(1) with φ = 0.75, 60 observations, fitted by least squares and forecast 14 steps ahead. The point forecast decays towards the fitted mean at φ̂^h; the band is ±1.96 standard errors from σ̂²Σψ̂², which grows with the horizon and stops at the unconditional spread 1.72. The dashed pair is the same band computed at the true parameters, which nobody has. The marks past zero are what actually arrived: 12 of 14 inside the band this once, which is one draw and settles nothing.

1200 series of 50 observations from an AR(1) with φ = 0.7, at each horizon, on one set of seeds. The upper line is the interval computed at the true parameters — it covers 94.5% on average, which is the check that σ²Σψ² is the right formula rather than a claim about anything a forecaster can do. The lower line is the same formula fed σ̂² and φ̂: 92.7% at one step and 90.6% at 6. The interval that would cover what it claims is 5.5% wider at one step.

700 series from an AR(2) with coefficients 0.6 and -0.3, every order from 0 to 8 fitted to the same 192 responses so the log-likelihoods are comparable. AIC finds the true order 70.3% of the time and lands above it 29.7%; BIC finds it 95.0% and lands above it 2.3%. The closed form for one extra lag is P(χ²₁ > 2) = 15.73% for AIC, which does not depend on n at all, and P(χ²₁ > ln n) = 2.13% for BIC at this size, which falls to zero. Under the true order is the other failure and it is BIC's: 2.7% against 0.0%.

One-step squared error for three forecasts of the same next observation, on the same series and the same seeds, over 1500 series of 50 observations at each φ. The sample mean estimates no dynamics and beats the fitted model below φ = 0.119; the last value carried forward estimates nothing at all and beats it above φ = 0.923. Both crossings are solved from closed forms — σ²(1 + k/n) for the fitted model against 2γ₀(1 − φ) and γ₀(1 + (1+φ)/(n(1−φ))) — and both are functions of the length of the series alone. The band widens at both ends as n grows and never reaches either edge.