and 33 326 runs for (±1 , ±2 ) = (0.9, ’0.8). The

horizontal axis indicates the run length L.

When we repeated the Monte Carlo experiment

described above for the (0.9, ’0.8) AR(2) process,

we observed 33 355 runs in a 100 000 time unit

simulation. The relative frequency distribution of

Figure 10.9: 240 time step realizations of an AR(2)

process with ±1 = 0.9 and ±2 = ’0.8 and with L that was obtained is shown in Figure 10.10. Note

that the L = 1 category is not the most frequent.

±1 = ±2 = 0.3.

Instead, runs of length L = 3, comprising 44%

of all runs, are most common. This is consistent

L = 1) decreases with increasing ±1 , while the with our perception that this process has a quasi-

frequency of longer runs increases. For instance, periodicity of about six time units in length.

in the white noise case we expect one run in 1000 If the (0.9, ’0.8) AR(2) process is truly quasi-

will be of length 10. In contrast, when ±1 = 0.3, oscillatory with a period of approximately six time

about four runs in 1000 are of length 10, and when steps, we should expect to frequently observe runs

±1 = 0.9, this number increases to 20. of approximately three time units in length. We

therefore counted the number of times that a run

of length, say, L 2 adjoined a run of length L 1 .

10.3.4 AR(2) Processes. AR(2) processes,

The results are given in Table 10.1. Note that

which represent discretized second-order linear

two consecutive runs tend to have joint length

differential equations (see [10.3.1]), have two

L 1 + L 2 = 6 more often than would be expected

degrees of freedom and can oscillate with one

by chance. On the other hand, pairs of intervals

preferred frequency (see also [11.1.8]). Finite

with L 1 + L 2 = 4, 5 or more than 7 are under-

segments of realizations of two AR(2) processes

represented. Any two neighbouring intervals must

are shown in Figure 10.9. The time series with

(±1 , ±2 ) = (0.9, ’0.8) exhibits clear quasi- have different signs, by the de¬nition of L, so that

the (L 1 , L 2 ) = (2, 4) and (3, 3) combinations

periodic behaviour with a period of about six

time steps. The other time series, with (±1 , ±2 ) = represent ˜quasi-oscillatory™ events in the time

(0.3, 0.3), has behaviour comparable to that of an series.

AR(1) process with large memory. The diagram The time series generated with the parameter

combination (±1 , ±2 ) = (0.3, 0.3) exhibits a

hints that there may be a longer quasi-periodicity,

say of the order of 150 or more time steps. strange pattern of extended intervals with con-

However, we will see later that the (0.3, 0.3) tinuous sign reversals and prolonged persistence.

process does not generate periodicities of any The reason for this pattern will become clear in

kind. [10.3.6].

10.3: Auto-regressive Processes 207

±2

L2 L1

1 1 2 3 4 5 (0,1)

1 571 1248 1523 917 334

’875 ’31

351 23 108

2 1428 6418 3304 741 ±

FMA(0.571,0)

1

’282 ’280 650 109 NDJ(1.172,0)

3 7341 5048 1081 MJJ(1.032,-0368) ASO(1.436,-0.471)

783 ’144 ’153

4 846 418 (2,-1)

(-2,-1)

’182 ’70

5 59

0

Figure 10.11: The triangle identi¬es the range

of parameters for which an AR(2) process

is stationary. The four points represent the

Table 10.1: Absolute frequency with which a run

parameters of a seasonal AR(2) process used to

of length L 1 is preceded or followed by a run of

represent the SO index (see [10.3.7]). Processes

length L 2 in a 100 000 time unit simulation of

with parameters below the curve de¬ned by ±1 +

an AR(2) process with (±1 , ±2 ) = (0.9, ’0.8). 2

4±2 = 0 have quasi-oscillatory behaviour (see

The entries in italics display the deviation from

[10.3.6]).

the expected cell frequency computed under the

assumption that consecutive run lengths are

independent.

An AR( p) process with AR coef¬cients ±k , for k =

1, . . . , p, is stationary if and only if all roots of the

characteristic polynomial

10.3.5 Stationarity of AR Processes. The con-

p

ditions under which the AR processes of de¬nition

p(y) = 1 ’ ±k y k (10.11)

[10.3.1] are stationary are not immediately obvi-

k=1

ous. Clearly, AR processes can be non-stationary.

An AR(1) process with ±1 = 2 and µ = 0 lie outside the circle |y| = 1.

initiated from a random variable X0 that has ¬nite Note that (10.11) has p roots y j , some of which are

variance is stationary with respect to the mean but real and others of which may appear in complex

non-stationary with respect to variance. In this case conjugate pairs.

we note that, for t > 0, Thus the stationarity condition for an AR(1)

process is simply

t

|±1 | < 1. (10.12)

Xt = 2 X0 +

t

2t’i Zt’i+1

Stationarity conditions are somewhat more in-

i=1

volved for an AR(2) process, where it is necessary

that

and therefore that

±2 + ±1 < 1

E(Xt ) = 2 E(X0 ) = 0 and

t

±2 ’ ±1 < 1 (10.13)

t

|±2 | < 1.

Var(Xt ) = 4t Var(X0 ) + 4t’i Var(Z )

i=1

The region of admissible process parameters

4t 1 de¬ned by (10.13) consists of points (±1 , ±2 ) in the

= 1’ t .

3 4 two-dimensional plane that also lie in the triangle

depicted in Figure 10.11.

Thus the variance of this process grows at an

exponential rate. 10.3.6 More about the Characteristic

The stationarity of an AR( p) process depends Polynomial. Equation (10.11) has interesting

entirely on the dynamical AR coef¬cients ±k , k = implications. Let y j , for j = 1, . . . , p, be the

0. In fact, roots of the characteristic polynomial p(y). Given

10: Time Series and Stochastic Processes

208

a ¬xed j, set X t’k, j = y k for k = 1, . . . , p.

1.5

j

+

Substitute these values into (10.6), disregard the

+

noise term, and recall that we have assumed

+

that ±0 = 0. Then, using (10.11), we see