What a band of lags actually costs, against what it is charged

Three quantities against the width of a tapered band, on 2000 pairs of independent samples of 120 rows under AR(1) at 0.8. The upper line is what a criterion charges — one log-likelihood unit a lag, which is Akaike's penalty applied to the band as though every lag were a free coefficient. The middle line is what the window's own weights predict, Σ(1 − k/(L+1)), which is exactly half the width. The lower line with its error bars is the measurement: the objective at the fitted covariance, minus the same objective on a second sample drawn independently from the same law, differenced from the narrowest band so that the coefficients' own optimism drops out. At 30 lags it is 10.855 ± 0.281 against a charge of 29 — 2.67 times too large — and against 14.50 predicted by the weights. The measurement is nearer the weights than the convention and is under both, at every width in the sweep.

The measured optimism as a share of the summed weights, for four windows at a band of 30 lags, over 2000 pairs of independent samples of 120 rows. One would mean the weights are the arithmetic; the three Bartlett powers read 0.749, 0.771, 0.781, which averages 0.767 and spans 0.033. Their weight sums differ by a factor of two and their shares do not differ at all, so the weights carry the whole of the difference between those three windows and about three quarters of the level. Parzen reads 0.871 ± 0.033, which is 0.104 above the family at 3.1 standard errors: the same amount of weight, put at the lags where the sample knows most, costs more. So a derived charge needs the window's shape and not only its area, and no charge that reads L alone can have either.

The band width each charge picks, averaged over 400 draws of 120 rows under AR(1) at 0.8, with the standard deviation across draws beside it. Schwarz's charge — half a log n a lag, which is 2.39 here — picks 3.67. Akaike's picks 6.02. Charging the numbers the window actually leaves free, which is half the width, picks 10.12; charging what the optimism measures, 0.767 of that, picks 14.15. A charge and the width it buys are very nearly reciprocal, which is what a likelihood rising at a fixed rate a lag implies and is why the four answers span a factor of 3.86. The width that was actually best on the draw averages 13.90 and moves by 10.30 from draw to draw — three times as much as any rule's answer does.

The optimism one more lag of a Bartlett band adds, measured by fitting on one sample and scoring on an independent one, 2,400 pairs at each size, with two standard errors: 0.3745 ± 0.0176 at 120 rows, 0.3842 ± 0.0266 at 240 rows, 0.4523 ± 0.0392 at 480 rows, 0.4022 ± 0.0563 at 960 rows. The window's summed weights give one half a lag and the convention charges one.

The prediction error at every band width from four lags to eighty, above the lowest on the extended list, under long memory at d = 4/9 at three sample sizes, 600 draws each. At 120 rows the floor is at 12 lags and eighty is 0.0039 above it. At 480 the floor is at 30 lags and every width from twenty-four to eighty is within 0.0003 of it. At 960 it is at 48 and thirty to eighty lags are within 0.0000. The dashed line is the list's cap: the surface does not rise behind it, it levels off.