Discussion Peter Guttorp Norwegian Computing Center University of Washington
Discussion
Peter GuttorpNorwegian Computing Center
University of Washington
Some points
The Hurst effect
Effect of long term memory on standard analyses
Trend and error
Hurst effect
H. E. Hurst
Storage capacity
of dams
is input at time k,
Hurst found that often RN~NH
where H > ½ is the Hurst coefficient.
Feller showed H = ½ for iid
k ZN kk1
NZt,N Zt
t
NZN RN maxZt,N minZt,N
k
Hurst coefficient is not fractal dimension
For self-scaling (fractal) processes on the line D+H = 2
D is fractal dimension of path. For stationary Gaussian process
.
Long memory has
with Hurst coefficient
In the spectral domain the corresponding features are
1 c(h) : h
as h 0 D 2
2
c(h) : h
as h H 1
2
f() : 1as and f() : 1
as 0.
The spectrum
1.95
0.1
D=2.75 D=2.5 D=2
H=.9875
H=.95
H=.55
Hurst coefficient is not only long memory
Bhattacharya, Gupta & Waymire:
A short term memory process with trend has Hurst coefficient
Statistical problems: Estimating H
Removing trend
(Demetris does not remove trend; Peter and Armin do)
(c n) 2
12
, 0 1
But does it matter?
Smith and Chen (1996) applied to Hadley global temperature series
Linear trend estimate 0.43°C per decade, se 1.2x10-3
OLS same trend, se 2.6x10-4
H .92
Effect of trend removal
Residuals from either AR(4)
H .8
Residual plots