conducted under ideal conditions: The process is normal and its

8 Even when the correct model has been chosen, z behaves

order is known. The performance in practice will not be quite t

as white noise only in the limit as T ’ ∞.

as good.

12.2: Identifying and Fitting Auto-regressive Models 261

by the null hypothesis.9 Thus the critical value for Plot of Standardized Residuals

testing the null hypothesis that the spectrum of zt

3

is white at the p — 100% level is K p / T ’1 ,

2

2

where K p is given in the table below and where the

1

notation x refers to the largest integer contained

0

in x.

-1

Signi¬cance level K p

-2

0.01 1.63

0 50 100 150 200

0.05 1.36

0.10 1.22

ACF Plot of Residuals

0.25 1.02

-0.2 0.0 0.2 0.4 0.6 0.8 1.0

12.2.9 Example: Diagnostics. The ¬rst ex-

ample we consider is the time series of length 240

ACF

generated from an AR(2) process with parameters

(±1 , ±2 ) = (0.9, ’0.8) (see also [12.2.1,3,5,7].

The full and partial auto-correlation function esti-

mates computed from this time series are shown 0 5 10 15 20

Lag

in Figure 12.3. Both functions show behaviour

characteristic of an AR(2) model, so p = 2 is a p-value for Portmanteau Statistic

•

good tentative choice. •

• •

•

0.8

•

•

Diagnostic plots of the residuals, the estimated

•

0.4 0.6

auto-correlation function of the residuals, and

p-value

•

the p-values of the portmanteau goodness-of-¬t

statistic (12.15) are shown in Figure 12.5. These •

0.0 0.2

plots con¬rm our tentative choice of model. The

auto-correlation function of the residuals (middle

2 4 6 8 10

panel) is essentially zero for nonzero lags, and all Lag

p-values of the portmanteau statistic (lower panel)

are greater than the 5% critical value, which is

shown as a dashed line. The upper panel hints at Figure 12.5: Plots diagnosing the goodness of a

behaviour that might bear investigation if we had maximum likelihood ¬t of an AR(2) model to a

¬tted this model to real data; the variability of time series of length 240 generated from and AR(2)

the residuals for t = 2 to t ≈ 50 seems to be process with (±1 , ±2 ) = (0.9, ’0.8).

somewhat less than that of subsequent residuals. Top: The residuals zt .

Middle: The auto-correlation function rzz („ ) of the

The cumulative periodogram of the residuals (not ˆˆ

shown) supports the hypothesis that the correct residuals.

model has been selected. Bottom: p-values of the portmanteau statistic

q(k).

In our second example we deliberately ¬t

an AR(1) model to the AR(2) time series

to produce an extreme example of a set of

there is quasi-periodic variation at frequencies

diagnostic plots (Figure 12.6) that show lack-

roughly in the interval (0.1, 0.2).

of-¬t. The plot of the residuals reveals quasi-

periodic behaviour that has not been captured

by the ¬tted model. This is also revealed in the

12.2.10 Objective Order Determination: AIC.

auto-correlation function of the residuals. The

The Box“Jenkins method of model identi¬cation

p-values of the portmanteau statistics (not shown)

and ¬tting is labour intensive: the investigator must

are uniformly less than 0.05. In addition, the

be actively (and skilfully) involved. This is a very

cumulative periodogram (Figure 12.7) shows that

strong advantage, because such close interaction

9 We will see in [12.3.5“7] that Q is the cumulative sum with the data will help to identify problems with

of independent identically distributed random variables when lack-of-¬t, but also a disadvantage because the

zt is white. The Kolmogorov-Smirnov statistics is written in

method can not be practically applied to the large

the same way as (12.17) except that Q is replaced with the

¬elds of time series often encountered in climate

empirical distribution function. The latter is also a cumulative

research. Objective order determining criteria are

sum of independent identically distributed random variables.

12: Estimating Covariance Functions and Spectra

262

[6][7]) determines the order by minimizing