Concept

Fixed point — where it appears

A value that an update rule returns unchanged: read the quantity off a fit, refit with it, and get the same quantity back. An iterated estimator stops there, which says it is self-consistent and says nothing, by itself, about whether it maximises anything.

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

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

Named alongside it

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

Autocorrelation Feasible generalised least squares Generalised least squares Regret Whitening Bias Closed form Covariance estimation Jacobian Long memory Maximum likelihood Moving average

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