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.

Regret under AR(1) at 0.8 as the tapered estimate is given more lags, over 120 draws at n = 120. The best window is L = 12; the automatic bandwidth is 4 and the error model's own likelihood chooses 5.2 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.

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.

Regret on a sample whose persistence changes from 0.95 to 0.65 at row 60, over 200 draws. The three stationary rules — told one number, told a window, told an order — are within 0.4 standard errors of each other, and all three stop in the same place: they are general in the lag direction, and the departure is in the other one. Letting the model change once, at a point estimated from the same residuals, is worth 0.05021 more at 4.5 paired standard errors — about as much again as the whole of the first repair. Being told where the break is adds 0.01926, and being told the entire covariance adds 0.02465.