Every essay — page 8
Three series, and a count
A pair is either tied or it is not, so its whole inference is one test with one answer. Three series can carry none, one or two relations at once, and the quantity being estimated stops being a slope and becomes an integer — read off the gap in a spectrum, against a critical value that depends on how many things are left wandering and on nothing else.
Three series and a count
A pair of series is either tied together or it is not, so its whole inference is one test with one answer. Three can carry none, one or two relations at once — and the thing being estimated stops being a slope and becomes an integer, read off the gap in a spectrum whose top eigenvalue holds at 0.25 while the rest fall like 1/n.
Which series goes on the left
The two-step procedure has to pick a series to regress the others on, and nothing in its output records which. With a pair that choice never changes the verdict. With three series and one relation between them, the three choices disagree about whether the system is cointegrated at all 98.0% of the time.
Counting what is still wandering
The statistic that turns a spectrum into an integer has one name and three distributions. Its 5% point is 8.12, 18.64 or 31.74 depending only on how many series are left wandering under the null being tested — and read against the wrong one of those three, it calls unrelated random walks cointegrated most of the time.
Which series does the moving
“y adjusts towards x” and “x adjusts towards y” are different mechanisms with identical long-run relations, and a single-equation model cannot tell them apart because it only writes one equation. Writing all of them recovers a vector — and a gap that closes at 25% a step where one equation alone reports 15%.
The rank is a decision
The sequential procedure's 5% bounds one of its two errors. Over-counting reads between 4.2% and 7.2% at every sample length from fifty observations to three hundred; under-counting reads 69.5% at fifty and 0.0% at three hundred, and nothing in the procedure bounds it.
Which mistake about the rank costs
On a system with two relations, imposing none costs 29.2% of squared forecast error and imposing three costs 2.5%. The expensive mistake is under-counting, which is the error the procedure's 5% does not bound — so the guarantee protects the cheap side.
A space is not a relation
The fitted plane approaches the true one at rate 1/n — 0.1438 at a hundred observations and 0.0075 at sixteen hundred. The angle between the leading fitted relation and the leading generating one reads 29.6° and 29.0° at those same lengths, and never moves.
A restriction the data can check
A rank-two system of three series is identified by exactly two restrictions on its relations, and any two that describe the estimated plane fit it equally well — their forecasts agree with the unrestricted fit's to the last digit, so the data can neither confirm nor refuse them. Supply one relation in full instead, one restriction more than identification needs, and the surplus becomes a test: a relation a tenth wrong is rejected 37.8% of the time at two hundred observations and 98.5% at eight hundred, the error found half the time halving as the sample doubles. A right relation buys the forecast almost nothing; a fifth wrong costs 13% of it at two hundred observations and 25% at eight hundred.
The surface between the corners
A two-level factorial answers which factors matter and is structurally unable to answer what setting is best: every run sits at a corner, every squared term is 1 there, and the column that would estimate curvature is a copy of the intercept. What it takes to see a curve, how far a fitted gradient can be trusted, and why the location of an optimum is a ratio of estimates rather than an estimate.
The design that cannot see a curve
A two-level factorial has every run at a corner, where every squared term equals one — so the column that would estimate curvature is a copy of the intercept, and the design has no information about it at all. A few runs at the centre buy one number back, and only one.
Walking up the gradient
The fitted gradient is wrong by an angle with a closed form, σ/(|β|√N), and what that angle costs is its squared cosine — twelve per cent at twenty degrees. What costs a third of the gain is not the direction at all. It is deciding where to stop.
Three levels, and the ring where the design says the same thing
A central composite design puts its axial runs at ±α, and α is not a matter of taste. At F to the quarter the prediction variance depends only on how far a point is from the centre and not at all on which direction it lies in — a property with no simulation in it, exact or absent.
The optimum is a ratio, and its interval is sometimes the whole line
The best setting is −b₁/2b₂: a ratio of two estimates whose denominator is a curvature the design can often barely see. The delta method reports a finite interval every time and covers 68.8% where the curvature is weak; Fieller's set covers 95% and says so by being unbounded.
The sign the curvature has
A fitted surface reports a maximum, a minimum or a saddle, and the report is a comparison of two estimated eigenvalues against zero. At a true second eigenvalue of −0.25 the fit calls a genuine maximum a saddle on 26.4% of studies, and at +0.25 it calls a genuine saddle a maximum on 25.1%.
When the best setting is outside the region
On a flat surface at twice the noise the fitted optimum lands outside the experimental region on 24.9% of studies and more than three coded units out on 11.8%. The answer is a ridge — the best setting at each radius, with a closed form — and the two obvious rules for using it turn out to be within four per cent of each other.
The run that confirms it
The setting a response-surface analysis recommends was chosen because the fitted surface was highest there, so the height the fit predicts at it is a maximum over a random field. At twice the noise the fit predicts 0.858 more than is there — 0.72 of the prediction's own standard error — and the gap is not noise, it is the selection.
The height at the chosen setting
The height a response-surface fit predicts at the setting it recommends reads high, because the setting was chosen where the fit was highest. The obvious repair — choose on some runs, estimate on the rest — is impossible in the usual thirteen-run design, which has nine distinct settings where two separate fits need twelve. A parametric bootstrap of the optimism removes four fifths of it at no cost in runs, 0.884 down to 0.183 at twice the noise. With the design run twice, correcting the full fit beats splitting it: an error of 0.924 against 1.213, at a better setting.
A design chosen rather than looked up
Every design here so far came from a catalogue and was then measured. Turn the arithmetic around and a design is the answer to an optimisation: maximise a functional of X′X and see what comes back. What comes back is the catalogue's own nine settings, re-weighted — and an exact theorem that says when a search is finished without knowing what it was searching for.
A design is a number
A standard design is taken from a catalogue and then measured. Turn the arithmetic round and a design becomes the answer to an optimisation — and over 121 candidate settings the search keeps nine of them, which are exactly the nine a catalogue would have offered, at weights nine equal runs cannot express.
The theorem that says when to stop
A search that maximises the volume of the information has no way of knowing it has finished, because nothing tells it what the maximum is. Kiefer and Wolfowitz's equality does — a design is D-optimal exactly when the worst prediction anywhere in the region equals the number of parameters, which is 6.000000000059 here, gated at machine precision.
Four letters and two camps
D, A, G and I are four ways of turning one matrix into one number, and they do not agree. The design that wins on D is the worst thing here on I. And the same two designs swap places entirely when the region changes from a square to a disc — on all four criteria at once.
The design that has to be integers
The optimal design is a set of real weights and an experiment is a set of runs, so the theory's answer is never available. Thirteen runs reach 99.77% of it and fourteen reach 99.44% — adding a run makes the design worse per run, and the search that finds it does not always find the same one.
How many places a design goes
Carathéodory's bound puts an optimal design's support between six and twenty-one settings, and every design in this field that can fit the model visits exactly nine. The count is not a choice anybody makes, it decides how many degrees of freedom are left to check the model with, and the first spare setting costs six points of efficiency to get back.
Augmenting a design that has already run
The equivalence theorem still certifies when some runs are already spent, and one number in it changes: the bound is no longer p but (p − λ·tr(M⁻¹M_fixed))/(1 − λ). It equals p again exactly when the runs already made can still be absorbed into the design that would have been chosen — so the certificate says whether the experiment is still recoverable.
The criterion with no derivative
E-optimality maximises the smallest eigenvalue of the information matrix, and at its own optimum that eigenvalue is attained twice — which is exactly where the function has a corner. The multiplicative search this field's other three criteria use assumes a derivative that is not there, and stops at 37.2% of the optimum.
A certificate for the worst direction
E-optimality was the one design criterion here whose answer was claimed rather than proved. It has a certificate after all, and on the square it fits on one line — (1⁄5)(1 − x₁² − x₂²)² + (1⁄5)(x₁² − x₂²)², which never exceeds 1⁄5 and touches it at exactly the nine points of the 3² grid. That proves the design with 2⁄5 of its weight at the centre, 1⁄10 at each edge midpoint and 1⁄20 at each corner is E-optimal on the whole square, and it is built from two directions mixed three to two, either of which alone fails.