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