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

Moving average — where it appears

A series formed by averaging a fixed number of independent shocks, so that its autocorrelation is exactly zero past that many lags. It is the natural opposite of an autoregression, whose autocorrelation never reaches zero, and no autoregression of finite order reproduces one exactly.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

Four dependences a single parameter cannot tell apart. Every law here is standardised to a lag-one autocorrelation of 0.8, so a rule told the errors are a first-order autoregression finds the same number in all four and has no way of seeing what separates them. The geometric decay is the world in which estimating a covariance rather than naming it was priced, and found to cost. The five-period moving average has 0.200 at the fourth lag and exactly nothing past it, where the geometric law says 0.328 at the fifth. Long memory at d = 4/9 is still at 0.576 by the twentieth lag, where the geometric law has reached 0.012. The break has no autocorrelation function at all: what is drawn for it is the average over the pairs at each gap, which is what a stationary estimate converges to.

A dependence with a shape

Four ways for errors to repeat, all with the same first lag and nothing else in common. A rule told the errors are a first-order autoregression finds the same number in all four, and is right about one of them.

general · Dependence The window a whitening wants is not the memory of the errors. Regret under a five-period moving average as the tapered estimate is given more lags, over 120 draws at n = 120. The best window is L = 30; the automatic bandwidth is 4 and the error model's own likelihood chooses 9.7 on average. Both land in the same place and both are short, and the reason is the taper: a Bartlett weight at lag k is 1 − k/(L + 1), so a window of 8 keeps 0.556 of whatever the fourth lag carries and a window of 30 keeps 0.871. A window has to be several times the memory before it stops removing the memory. The dashed line is the rule told the errors are a first-order autoregression, which needs no window at all.

The window a whitening wants

Every law here is best whitened by a window several times longer than its own memory, including the one whose memory ends at the fourth lag. The three ways of choosing it from the sample all land in the same place, and it is the wrong one.

general · Order-selection Four sequences, and the rule only ever sees the last one. Under long memory at d = 4/9, four things that are all called the dependence. The law itself is the top line. What a sample of 120 rows reports on average is the second, computed exactly: subtracting a sample mean takes the first lag from 0.800 to 0.538. What a candidate's residuals report is the third, lower again at 0.472, because a fit removes dependence along with signal. The autoregressions are fitted to that third sequence and reproduce it exactly out to their own order — the Yule–Walker equations are solved to make it so — so everything they say past that is extrapolation. At the twentieth lag the law has 0.576, the residuals report 0.006, and an AR(8) extrapolates 0.028.

The order the tail is drawn at

A fitted autoregression reproduces the sample exactly at the lags it was fitted on, so everything it says past them is extrapolation — and the order is the dial that decides how much of it there is.

general · Order-selection

Named alongside it

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

AutocorrelationGeneralised least squaresInformation criterionLong memoryModel selectionNuisance parameterRegretSample autocovarianceWhiteningAutoregressionCovariance matrixBandwidth selection

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