The same dial under a second law

A step that is not a ratio ran this field’s separation sweep on two tuning parameters at once. Each candidate in a table of fifteen regressions is whitened at its own value of a tuning parameter, chosen from a list of eight, and the question is how often that per-candidate choice changes which candidate the table selects. Pull the candidates apart by scaling the coefficients they omit, and two factors move in opposite directions: the candidates disagree about the tuning parameter more often, and a disagreement changes the winner less often. That held on both the whitening window, whose list is a geometric grid, and the sieve order, whose list is an interval of integers.

What did not hold was the size of the movement. Across the sweep the probability that the list changes the winner spanned a factor of 11.75 on the window and 5.00 on the order, 2.35 times as much on one as on the other. The essay explained it through the field’s own measure of indifference: a change of order was worth 0.666 of criterion where a change of window was worth 1.857, so an integer step was worth about a third of a geometric one, and fewer candidates sat within one step’s reach of the winner on the order — 1.90 near the null against the window’s 3.63.

It ended with the caution that applied: every cell was a first-order autoregression, and a dial that reverses which of two parameters is larger is exactly the kind of finding that should be re-run somewhere else before it is carried anywhere. This essay re-runs it, with errors from a five-period moving average — the field’s law with nothing at all past the fourth lag, chosen because it is the law on which a geometric shape is most wrong — and everything else held: the fifteen candidates, the eight values on each list, the five separations, the sample size and the seeds.

The probability, under the second law #

On the window the probability is 26.3% with every candidate true, 24.0% at half the standing coefficients, 14.3% at the standing coefficients, 5.3% at twice them and 4.8% at four times — a span of 5.53. On the order it is 29.5%, 25.9%, 13.6%, 6.0% and 5.8%, a span of 5.13. The ratio between the two spans is 1.08.

Under the autoregression the same five cells ran 16.4%, 17.6%, 13.1%, 2.8% and 1.5% on the window and 13.1%, 11.8%, 9.3%, 3.6% and 2.6% on the order. So the moving average does two things at once. It raises the probability near the null on both lists, by about half on the window and more than double on the order. And it removes the difference between the lists: at every separation the two probabilities are now within a few points of each other, where under the autoregression the window’s ran up to half as large again as the order’s near the null and fell to little more than half of it at the far end.

The two factors still point opposite ways #

The field’s central finding survives the change of law intact. On the order the candidates disagree 58.9% of the time with every candidate true and 99.6% at four times the standing coefficients; the share of disagreements that change the winner falls from 50.1% to 5.8%. On the window, 56.3% to 98.3% and 46.7% to 4.8%. The rate rises by a factor of about 1.7 on both lists and the turnover falls by a factor of nine.

So the direction is a property of the separation, as two factors pointing opposite ways argued, and it holds under a law whose dependence has a completely different shape. What moved is the starting point: under the moving average the candidates disagree about the tuning parameter far more often even when every candidate is true — 58.9% on the order against 31.0% under the autoregression — and half of those disagreements change the winner.

What the indifference measure says #

The measure that was the same on both lists is the count of candidates within one tuning change’s worth of criterion of the winner. Under the autoregression it separated the lists cleanly: 3.63 on the window and 1.90 on the order with every candidate true. Under the moving average it is 5.51 on the window and 5.69 on the order, and it falls on both to under two at four times the standing coefficients — 1.85 and 1.96.

Between the ends the two lists track each other closely. At half the standing coefficients 4.22 candidates are within reach on the window and 4.42 on the order; at the standing coefficients 2.49 and 2.31; at twice them 1.67 and 1.75. Under the autoregression the same three cells were 3.30 and 1.87, 2.37 and 1.43, and 1.56 and 1.24 — the window ahead at every separation, by as much as one and three quarter candidates near the null and by a third of a candidate even at twice the standing coefficients, where both lists are close to deciding. Under the moving average the gap is under a quarter of a candidate everywhere and changes sign twice, which is what two lists measuring the same indifference look like.

So under the second law the order’s step is no longer small in the criterion’s units. Near the null more than a third of the table is within one change of order of the winner, as it is within one change of window; a tuning change can reach almost as many candidates on the integer list as on the geometric one. The earlier essay’s account was that a step’s size is what sets how many candidates a list can tip, and that the two lists’ steps differ by a factor of three. That account still describes the autoregression. It does not describe the moving average, where the two steps reach the same candidates.

Why the moving average equalises the lists #

The likeliest reason is in what each list is being asked to approximate. Under a first-order autoregression a sieve of order one is exactly right, and the criterion is sharp about it: a change from order one to two costs a definite amount of fit, so an integer step moves the criterion decisively and few candidates are within its reach. A window, by contrast, is a family of truncations of a geometric sequence that never ends, so neighbouring windows differ by little and the criterion is flat along the grid — the cheap place to lose resolution that the earlier essay pointed to.

Under the five-period moving average both of those reverse. A window holding the fourth lag holds every autocovariance the law has, so the long end of the grid is flat for the same reason as before, and the short end is decisive. And no finite autoregression is right, because the moving average’s polynomial has its roots on the unit circle: a model against a window found a sieve whose order is chosen by the criterion asking for eleven or twelve lags on a law this short-memoried, with the bias falling slowly as the order grows. A list of orders that is never right is a list along which the criterion is flat, and a flat list has many candidates within reach of one step — which is what the window’s list was under the autoregression.

So the property the earlier essay attributed to the shape of the step — integer against ratio — is better described as a property of whether the parameter on the list can be right. Under the autoregression the order can and the window cannot; under the moving average neither can, and the two lists behave alike. That is a reading of the measurements rather than a measurement of the mechanism; the sweep says the difference disappears, and the account of why is what is consistent with it and with the sieve’s behaviour under this law elsewhere.

Under both laws, side by side #

Read the two figures together and three statements survive the change of law and one does not.

The separation is the dial that matters. Under both laws the probability that a tuning list changes the winner falls by a factor of five or more as the candidates are pulled apart, on both parameters, while the list’s length is held at eight.

The two factors point opposite ways. Under both laws the rate of disagreement rises and the share that decides falls, and the second dominates.

Indifference is where the list decides. Under both laws the count of candidates within reach of one tuning change falls with the separation, and the probability follows it.

Which parameter moves more is not a property of the parameter. Under the autoregression the window moved 2.35 times as much as the order; under the moving average 1.08 times. A practitioner choosing between whitening by a window and by an order cannot carry the earlier ranking to a series whose dependence is not geometric.

The eighth, under a second law #

The number this line of essays started from was an eighth. A rate times a size found the cost of letting every candidate choose its own tuning parameter flat across the length of the list, and the eighth that was not a constant traced it to a probability of about twelve per cent that the list changes the winner, which turned out to depend on how far apart the candidates are. Under the autoregression, at the standing coefficients, that probability was 13.1% on the window and 9.3% on the order. Under the moving average it is 14.3% and 13.6%.

At the standing coefficients, then, the second law barely moves the window’s figure and lifts the order’s by a half. Near the null it lifts both by a half or more. So the eighth was the autoregression’s number at one separation, and on a law that no list can describe well the same procedure changes the winner on more than a quarter of the draws where the candidates are close.

That has a direct reading for practice, and it strengthens the advice the field already gave. A window for every candidate found that choosing the tuning parameter once, from the fullest candidate’s residuals, and sharing it across the table is better than letting each candidate choose its own, because the per-candidate choice adds a selection the criterion then reads as fit. Under the moving average that selection decides the winner more often, so sharing the choice is worth more, not less. The law that makes every tuning value wrong is the law under which the freedom to choose a different wrong value for each candidate costs most.

What a practitioner should carry #

Two of the field’s findings travel and one does not.

The separation travels: a per-candidate tuning list matters most when the candidates are nearly indifferent, under both laws, and a practitioner can read how indifferent their candidates are from the spread of the criterion across the table before choosing whether to share the tuning. The direction of the two factors travels with it.

The ranking of the two parameters does not. The earlier essay found the window’s choice consequential and the order’s less so, and offered the step’s size as the reason. On a series whose dependence stops — a moving average, a seasonal pattern that dies out, any process with a finite memory — that ranking is gone, and a sieve order is as consequential a choice as a window. The table and the list already found the ranking of tables reversing against prediction; the ranking of parameters is a second ranking in this line that depends on something the earlier sweep held fixed.

What the earlier essay’s open items become #

The earlier essay named three things it did not measure. The second law was the third of them, and it is answered: the reversal did not survive it. The first was a finer spacing of the window’s grid, on the prediction that a grid with less room to lose resolution would make the window’s probability start moving along a row as the order’s does. The moving average gives that prediction an indirect test, since it changes where on the grid the criterion is flat without changing the grid; the window’s probability under the moving average is higher near the null and the two lists agree, which is what the prediction would expect if flatness rather than spacing were doing the work. The second open item — whether the criterion’s spans agreeing to two per cent was an accident — is not touched, because the spans under the moving average were not recorded here.

How often it matters first turned the field’s question from a disagreement rate into a probability that the winner changes. That probability is now measured on two parameters, two laws and five separations, and the one constant across all twenty cells is its direction along the separation. Its level depends on the law, and its comparison between parameters depends on the law too.

The autoregression’s seeds, under the moving average #

Eight hundred draws at each separation and parameter under the moving average, on the seeds the autoregression’s sweep used, with the same fifteen candidates, the same eight-value lists and the same hundred and twenty rows; the indifference measure is computed on four hundred of them, as in the earlier essay. The moving average is the field’s five-period law, standardised to unit variance, with nothing at all past the fourth lag. The autoregression’s rows are the earlier essay’s, recomputed on its own seeds, and agree with it to the digit: spans of 11.75 and 5.00. Each probability has a standard error of about a point and a half near the null and under one at the far end, so differences between the two lists of two or three points are read as ties.

The window’s span collapses towards the order’s, and the mechanism is a reading #

The claims are that under a five-period moving average the probability that a per-candidate tuning list changes the winner spans a factor of 5.53 on the whitening window and 5.13 on the sieve order across the separation sweep, against 11.75 and 5.00 under the first-order autoregression; that the rate of disagreement still rises and the share that decides still falls on both; and that near the null both lists leave about five and a half candidates within reach of one tuning change, where the autoregression left 3.63 on the window and 1.90 on the order.

Not claimed: that the mechanism offered for the equalisation is established, since it is a reading of these measurements and the sieve’s behaviour elsewhere rather than a measurement of its own; anything about long memory or a break, the field’s other two laws; and anything about the spans of the criterion across the table under the moving average, which were not recorded.

Still open: a law the order can be right about and the window cannot reach #

The account above predicts a third case. A law whose dependence is a finite autoregression of order two or three — exactly right for some order on the list, and geometric enough that no window holds all of it — should restore the autoregression’s ordering, with the order’s step decisive and the window’s grid flat. A law that is a moving average of order one, held by a window of two and approximated badly by every autoregression, should push it the other way, with the window decisive and the order flat. Running the sweep under those two laws is the measurement that would say whether “can the parameter be right” is the variable the earlier essay’s step size was standing in for, or whether the moving average equalised the lists for a reason of its own.