0.5

+

+ ++

that X t = y 0 = 1.8 That is, each root y j + +

j

+

+

identi¬es a set of ˜typical initial conditions™

++

+

-0.5

+

IC j = (X t’1, j , . . . , X t’ p, j ) that lead to X t = 1 +

+

when the noise Zt is disregarded. Since these

+

˜initial conditions™ are linearly independent, any

-1.5

set of states (X t’1 , . . . , X t’ p ) can be represented

p

as a linear combination j=1 β j IC j of the initial

0 5 10 15

states. In the absence of noise, the future evolution

Lag

of these states will be

p

β j y ’„ .

X t+„ = (10.14)

j

Figure 10.12: Initial conditions at times X t’2

j=1

and X t’1 which lead an AR(2) process, with

Note that some of the X t’k, j s may be complex and parameters 0.9 and ’0.8, to X t = 1, and their

therefore will appear in conjugate complex pairs. future development X t+„ in the absence of noise.

When this is true, the corresponding coef¬cients

β j will also appear as complex conjugate pairs.

The roots of the characteristic polynomial of an

When X t is an AR(1) process and the noise is

absent, X t’1 = 1/±1 is the only initial condition AR(2) process are complex when ±1 < ’4±2 , and

2

that leads to X t = 1 in one time step. can therefore be written in the form

In the case of an AR(2) process, the roots of the y = r · exp ’(’1) j iφ , j = 1, 2. (10.15)

j

characteristic polynomial (10.11) are

It is easily shown that r = 1.11 and φ = π 3

’±1 ’ (’1) j ± 2 + 4±

when (±1 , ±2 ) = (0.9, ’0.8). Since the process

2

1

yj = , j = 1, 2. parameters are real, (10.11) may be rewritten as

2± 2

0 = 1 ’ (±1 Re(y) + ±2 Re(y 2 ))

The roots are either both real or they are complex

= 1 ’ (±1r cos(φ) + ±2r 2 cos(2φ))

conjugates.

Both roots are real when ±1 > ’4±2 . The

2

AR(2) process with (±1 , ±2 ) = (0.3, 0.3) belongs and

to this category. Its characteristic polynomial has 0 = ±1 Im(y) + ±2 Im(y 2 )

roots y1 = 1.39 and y2 = ’2.39, and ˜typical

= ±1r sin(φ) + ±2r 2 sin(2φ)

initial conditions,™ which lead to X t = 1, are

IC 1 = (X t’2,1 , X t’1,1 ) = (1.93, 1.39) and so that the two sets of ˜typical initial conditions™

IC 2 = (X t’2,2 , X t’1,2 ) = (5.71, ’2.39). that evolve into X t = 1 are

The ¬rst ˜mode,™ which is initiated by IC 1 , has a

IC j = (X t’2, j , X t’1, j )

damping rate of X t’1,1 / X t’2,1 = X t,1 / X t’1,1 =

1/y1 = 0.72. The time development initiated by with

such an initial state is that of an exponential decay

X t’2, j = r 2 (cos(2φ) ’ (’1) j sin(2φ))

with constant sign.

The second mode has a damping rate 1/|y2 | =

and

0.42 and a clear tendency for perpetual sign

X t’1, j = r (cos(φ) ’ (’1) j sin(φ)).

reversals.

These two modes underlie the ˜strange pattern™

Thus (10.14) determines the future states as

of variation seen in Figure 10.9. There are some

X t+„,1 = r ’„ (cos(„ φ) + sin(„ φ))

periods when the process undergoes continual sign

reversals, and others when the system retains the

and

same sign. Change between the two regimes is

instigated by the noise Zt .

X t+„,2 = r ’„ (cos(„ φ) ’ sin(„ φ)).

8 Note that now we are neither dealing with the stochastic

The two sets of initial conditions (labelled ˜1™ and

process Xt nor with a random realization xt . We therefore use

˜2™) and the future evolution of the process without

the notation X t .

10.3: Auto-regressive Processes 209

2 there is a sequence of independent, zero mean

the noise Zt are plotted in Figure 10.12. We

random variables {Zt„ : t ∈ Z, „ =

see that the process generates damped oscillations

π

with a period of φ = 6 time steps for arbitrary 1, . . . , N } that have variance σZ „ which

2

depends only on „ and such that the sequence

nonzero initial conditions. The initial conditions

{Zt„ /σ Z „ : t ∈ Z, „ = 1, . . . , N } behaves as

serve only to determine the phase and amplitude

of the oscillation. white noise, and

The region of admissible process parameters

3 Xt„ satis¬es the difference equation

(10.13) for a stationary AR(2) process (see

Figure 10.11) can be split into two sub-regions. An

upper area, delimited by ±1 + 4±2 > 0, indexes

2 p

Xt,„ = ±0,„ + ±k„ Xt,„ ’k + Zt„ (10.17)

AR(2) processes whose characteristic polynomials

k=1

have two real solutions and thus consist of two

non-oscillatory damped modes. The rest of the

for all (t, „ ).

parameter space, delimited by (10.13) and the

constraint ±1 + 4±2 < 0, indexes processes

2

with characteristic polynomials that have a pair Such processes are able to exhibit cycles of

of conjugate roots, and thus one quasi-oscillatory length N of the mean, the variance, and the

auto-covariance function.

mode.

Suppose, now, that a process satisfying (10.17)

is weakly cyclo-stationary. This means that the

10.3.7 Characteristic Time. What is the process parameters are constrained in such a way

characteristic time (10.1) of an AR( p) process? that all means, variances, and covariances exist.

According to (10.1), we must ¬nd a lag „ M such This constraint, together with (10.17), is suf¬cient

that auto-correlations ρ X t ,X t+„ vanish for lags „ ≥ to ensure that the mean and variance are only a

„ M . In the case of an AR(1) process with µ = 0 function of „ and that the auto-covariance function

we ¬nd is only a function of the absolute time difference

E(Xt Xt+„ ) and the location in the external cycle. With these

ρ X t ,X t+„ = assumptions it is possible to derive the ˜seasonal

Var(Xt )

„ cycle™ of mean, variance and auto-covariance.

±1 E(Xt Xt )