Normal Distribution
nothing is called 95% until it has been counted
Concept

Feasible generalised least squares — where it appears

Generalised least squares with the covariance replaced by an estimate of it, usually read off the residuals of an ordinary fit. The estimate is attenuated because a fit removes dependence along with signal, and iterating the two steps converges somewhere the likelihood does not.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

The dependence, at four removes. Under AR(1) at 0.8, four different sequences all called the dependence. The top line is the law. The middle line is what a sample of 120 errors reports on average — computable exactly, because the expectation of a sample autocovariance is arithmetic once the covariance is known. The lower line is what a candidate's residuals report, which is what every two-step rule in this collection actually reads: a fit removes variance, and it removes more of the persistent part than of the rest. At the first lag the three are 0.800, 0.7773 and 0.7338. The dots are counted from draws and share no arithmetic with the line they sit on; the worst departure is 1.2 standard errors.

A dependence fitted with the line

Every whitening in this collection reads the dependence off a set of residuals, and residuals are not errors. Fitting the two together recovers most of what that costs, and changes almost nothing about the decision it feeds.

together · Dependence One likelihood, three answers. The concentrated Gaussian log-likelihood of one sample of 120 rows under AR(1) at 0.8, as a function of the correlation the errors are whitened at. Three rules put three different numbers on this curve. The two-step rule reads the least-squares residuals and lands at 0.7616, giving up 0.304 of log-likelihood. Iterating moves it to 0.8080 and gives up 0.002. The maximum is at 0.8044. The curve is not flat between them: what a fixed point of the residual update finds is a solution of a different equation, and the difference is the Jacobian term ½log(1 − ρ²), which grows as the correlation does.

Iterating is not maximising

Re-reading a correlation from the generalised residuals and refitting converges in seven steps. What it converges to solves the first-order condition of a sum of squares, and the likelihood has one term more than that.

together · Dependence What iterating the band is worth. How much lower the prediction error is when the Bartlett band is iterated to its fixed point than when it is read once off the least-squares residuals, paired on the draw, with two standard errors, 300 draws at 120 rows for each law and window. First-order autoregression: 4 lags 2.84 ± 0.36, 8 lags 3.43 ± 0.58, 20 lags 4.16 ± 0.80; five-period moving average: 4 lags 2.04 ± 0.28, 8 lags 2.69 ± 0.42, 20 lags 4.32 ± 0.68; long memory: 4 lags 0.83 ± 0.14, 8 lags 1.13 ± 0.25, 20 lags 1.33 ± 0.44 thousandths. Every one is below zero by more than two and a half standard errors; the gain is largest at the widest band under the two short-memory laws.

A fixed point with no likelihood under it

A tapered band of autocovariances has no likelihood to maximise, so the one rule that can fit its dependence together with the regression is to iterate: read the band off the residuals, refit, read it again. It converges on every draw, in seven to eleven fits at the median, gives back about half the memory the first fit took out of the residuals under a short-memory law, and lowers prediction error by two to four thousandths — five to eight paired standard errors. Under the law a first-order likelihood describes, its maximum still wins; under the two it does not, the band's fixed point beats it.

together · Dependence The floor a repair has to choose. Regret of the truncated covariance repaired by flooring its eigenvalues, at a persistence of 0.85 on 120 rows, 120 draws, by floor, on a log scale. Window 4: 1.5649 at 0.001, 0.0945 at 0.01, 0.0385 at 0.05, 0.0285 at 0.1, 0.0226 at 0.2, 0.0235 at 0.4, 0.0292 at 0.7; the taper at the same window 0.0275. Window 8: 1.1919 at 0.001, 0.0980 at 0.01, 0.0334 at 0.05, 0.0258 at 0.1, 0.0197 at 0.2, 0.0209 at 0.4, 0.0249 at 0.7; the taper at the same window 0.0178. Window 12: 0.8692 at 0.001, 0.0621 at 0.01, 0.0239 at 0.05, 0.0209 at 0.1, 0.0212 at 0.2, 0.0239 at 0.4, 0.0282 at 0.7; the taper at the same window 0.0150. Window 20: 1.7110 at 0.001, 0.2495 at 0.01, 0.0399 at 0.05, 0.0238 at 0.1, 0.0202 at 0.2, 0.0219 at 0.4, 0.0259 at 0.7; the taper at the same window 0.0135. Least squares gives up 0.0890 and the whitening told the truth 0.0075.

A covariance repaired

A truncated autocovariance sequence fails to be a covariance matrix on most draws of a persistent series, and this line of essays has always scored it on the draws where it survives. The textbook repair is to floor its eigenvalues. At a floor of 0.001 that moves the matrix by four per cent and gives up 1.56 of regret, seventeen times what least squares gives up, because a whitening reads an inverse and an inverse reads the smallest eigenvalues. At a floor of 0.2 it beats the taper at the shortest windows. So the repair is not a repair: it is a second dial.

banded · Dependence

Named alongside it

The objects these essays reach for when they reach for this one.

Generalised least squaresRegretAutocorrelationWhiteningBiasCovariance estimationFixed pointLong memoryMaximum likelihoodModel selectionNuisance parameterPrais winsten transform

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