Field

Comparing two forecasters

The forecast field ranks forecasts by mean squared error and stops. Whether one forecaster is really better is a test, its terms are dependent because neighbouring forecasts overlap, and where one model contains the other it fails completely — declaring the smaller model significantly better while the null it claims to test is true, and more certainly the more data it is given.

One comparison, and the two error bars it can be given. 60 rolling origins, a window of 60 observations, forecasts 4 steps ahead, at the persistence φ = 0.8256 where the two benchmarks have exactly equal population mean squared error. Each mark is one origin's difference in squared error; the horizontal line is their mean, 0.6522. The two vertical bars at the right are ±1.96 standard errors round that mean computed two ways — 0.5337 treating the differences as independent, 0.6880 allowing for the overlap between neighbouring forecasts. The null is true here by construction, so an interval that excludes zero is a mistake, and the narrow one does it far more often than the wide one.

Which forecast is better

Two forecasters, one series, and a difference in mean squared error. Whether that difference is real is a hypothesis test, its terms are not independent, and the standard error it needs is not the one a t-test computes.

A test between nested models, under a null that is true. 1000 comparisons: an AR(1) truth, forecast by a fitted AR(1) and by a fitted AR4 whose extra coefficients are zero. In population the two forecasts are identical, so every rejection is false. The larger model's mean squared error is 1.1663 against 1.0583 — worse, by exactly the noise in estimating coefficients that are not there — and the ordinary test therefore declares the smaller model significantly better 67.2% of the time. Read one-sided in the direction anybody asks about, it finds the larger model better 0.0% of the time. Adding the squared difference between the two forecasts back into the loss differential puts the level at 4.9%.

When one model contains the other

The comparison a forecaster most often wants is between a model and the same model with one more term. That is exactly the comparison the standard test cannot make — and it fails by declaring the smaller model significantly better, more confidently the more data it is given.

Least squares estimates persistence low, by an amount with a formula. 3000 series of 50 observations at each persistence. The lower curve is the counted bias of the least-squares estimate of φ, and the open marks on it are −(1 + 3φ)/n, computed rather than fitted. The upper curve is the bias left after adding that quantity back, evaluated at the estimate rather than at the truth nobody has: -0.0020 at φ = 0.3, -0.0023 at φ = 0.5, -0.0039 at φ = 0.7, -0.0059 at φ = 0.8, -0.0108 at φ = 0.9, -0.0165 at φ = 0.95. The formula is a leading-order expression and it understates the bias where the persistence is nearest one — -0.0882 counted against -0.0770 predicted at φ = 0.95, which is the corner of the parameter space every one of these approximations is worst in.

Correcting the persistence

Least squares estimates how much a series remembers of itself as smaller than it is, at every value it can take, by an amount with a closed form. Subtracting that amount back is one line of arithmetic, and what the line costs is variance.

The correction does not arrive at the truth, it passes it. The average decay factor a forecast applies to the last observation, at φ = 0.85 and 50 observations, 3000 series per horizon. The middle curve is φʰ, what the model actually does. Below it is the uncorrected forecast, which uses φ̂ʰ and reverts too fast — 24.8% short at h = 4, 30.0% short at h = 6, 32.7% short at h = 8. Above it is the forecast built on the corrected estimate, which overshoots, and the reason is arithmetic rather than a bad correction: raising an unbiased estimate to a power does not give an unbiased estimate of the power, and the higher the power the more the spread of φ̂ is converted into overshoot.

The repair that moves the wrong number

Correcting the bias in a persistence parameter is one line of arithmetic that works. Feeding the corrected estimate into a forecast repairs the number everybody looks at, makes the forecast worse by squared error at moderate persistence, and improves the interval for a reason that has nothing to do with bias.

The average decay factor each route produces, φ = 0.85, 6 steps ahead. The truth is φ^6 = 0.3771. no correction averages 0.2616 with a spread of 0.1646 and a squared forecast error of 3.2516; the formula, on the persistence averages 0.4213 with a spread of 0.2528 and a squared forecast error of 3.4827; the bootstrap, on the persistence averages 0.4355 with a spread of 0.2655 and a squared forecast error of 3.5120; the bootstrap, on the decay factor averages 0.3375 with a spread of 0.2278 and a squared forecast error of 3.4132. 800 series, 100 bootstrap refits each.

Correcting the forecast instead

The complaint against the usual repair is that a correction aimed at the persistence lands on the wrong quantity. Aiming it at the decay factor the forecast actually uses fixes exactly that — the error stops compounding with the horizon, 69.7% becomes 9.5% at twelve steps — and the forecast still gets worse.

Five treatments of an estimate above one, φ = 0.95, n = 25. The correction exceeds one on 31.1% of series at this setting. left where it lands: squared forecast error 12.828, average decay factor 0.7974 against a true 0.7351; capped at 0.995: squared forecast error 5.680, average decay factor 0.5950 against a true 0.7351; capped at 1 − 1/n: squared forecast error 5.535, average decay factor 0.5256 against a true 0.7351; correction scaled to fit: squared forecast error 5.535, average decay factor 0.5256 against a true 0.7351; correction refused where it leaves: squared forecast error 5.868, average decay factor 0.4423 against a true 0.7351.

The correction that leaves the region

The bias correction adds (1 + 3φ̂)/n whatever φ̂ is, so it pushes the estimate above one whenever φ̂ exceeds (n − 1)/(n + 3) — on 31.1% of series at φ = 0.95 and twenty-five observations. Five obvious things to do about it differ by a factor of 2.3 in squared forecast error, and none of them is documented as a choice.

What the forecast interval is short by, φ = 0.85, 6 steps ahead. The plug-in interval covers 88.42% against a claimed 95%. Correcting the variance recovers 0.56 points, propagating the persistence's own standard error recovers 0.40, correcting the persistence recovers 2.66, and all three together recover 4.20 — leaving 2.38 points unaccounted for.

What the interval is short by

The forecast interval covers 88.42% where it claims 95%. Correcting the persistence recovers 2.66 points, correcting the innovation variance 0.56, propagating the persistence's own standard error 0.40 — and all three together recover 4.20 of the 6.58, leaving a residual none of the standard repairs reaches.

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