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 2 essays across one field — each of them below, with the objects they name alongside it.

Also named here as prais winsten transform — the same set of essays touches all of them, so they are one junction rather than several.

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

Named alongside it

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

AutocorrelationBiasGeneralised least squaresMaximum likelihoodNuisance parameterPrais winsten transformRegretWhiteningAttenuationClosed formFixed pointHat matrix

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