Field

The observation that has not happened

Two fields here stop at estimation. A forecast is the other question — not what the parameter is but what the next observation will be — and the band round it is computed by substituting estimates into a formula derived for the truth. Counted, that 95% band is not 95%, the shortfall grows with the horizon, and the model has to beat two benchmarks that estimate nothing at all.

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.

What the model says next

The usual account of a time series stops at estimation. A forecast asks the other question — not what the parameter is but what the next observation will be — and the band round it is a closed form that grows with the horizon and then stops growing, at a value the series was going to reach anyway.

What a 95% forecast interval covers, counted. 1200 series of 25 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 95.3% 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: 92.8% at one step and 87.3% at 6. The interval that would cover what it claims is 6.9% wider at one step.

The interval that forgets it estimated

The forecast band is derived for a model whose parameters are known, and then computed by putting estimates into it. Counted, the 95% interval covers 87.3% six steps ahead on twenty-five observations, and the point forecast inside it returns to the mean a third faster than the series does.

What each criterion selects, at 50 observations. 700 series from an AR(2) with coefficients 0.6 and -0.3, every order from 0 to 8 fitted to the same 42 responses so the log-likelihoods are comparable. AIC finds the true order 55.1% of the time and lands above it 25.7%; BIC finds it 54.1% and lands above it 4.0%. 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) = 4.79% for BIC at this size, which falls to zero. Under the true order is the other failure and it is BIC's: 41.9% against 19.1%.

Choosing the order

One criterion is consistent and one is not, which is the whole of what gets said about them. At two hundred observations the consistent one is right 95% of the time and the other 70%; at fifty they are both right 54% of the time and wrong in opposite directions, and consistency has not started to mean anything yet.

What the interval covers once the order is chosen as well. 1200 series of 40 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 95.5% 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: 93.6% at one step and 90.8% at 6. The third line chooses the order by AIC from the same data before computing the interval, which costs a further 0.8 points at h = 6.

The interval after the choice

Estimating the coefficients of a known model costs a 95% forecast interval about two points of coverage. Choosing which coefficients to estimate, from the same forty observations, costs another four and a half — so the step nobody records in the output is the more expensive of the two.

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