Spectra of Stationary Processes
4.2 Definition and Interpretation
4
are given in Proposition3.2(a). We do not assume thatfcjgand hencef.h/gare absolutely summable,1simply because this will not hold under long memory treated in the next chapter. We wish to construct a functionf that allows to express the autocovariances as weighted cosine waves of different periodicity,2
.h/D Z
cos.h/f./d :
The basic ingredient of an analysis of periodicity is the cosine cycle whose properties we want to recall as an introduction.
Periodic Cycles
Byc.t/we denote the cycle based on the cosine,3 c.t/Dcos.t/ ; t2R;
wherewith > 0is calledfrequency. The frequency is inversely related to the periodP,
PD 2 :
For D 1 one obtains the cosine function which is 2 periodic and even (symmetric about the ordinate):
c1.t/Dcos.t/Dcos.tC2/Dc1.tC2/ ; c1. t/Dcos. t/Dcos.t/Dc1.t/:
More generally, it holds withPD2=that:
c.t/Dcos.t/Dcos.tC2/Dcos..tCP//Dc.tCP/ :
Therefore the cosine cyclec.t/with frequencyhas the periodP D 2=: Of course, the symmetry ofc1.t/carries over:
c.t/Dc. t/ :
1The assumption of absolute summability underlies most textbooks when it comes to spectral analysis, see e.g. Hamilton (1994) or Fuller (1996).
2From Brockwell and Davis (1991, Coro. 4.3.1) in connection with Brockwell and Davis (Thm.
5.7.2) one knows that such an expression exists.
3Here, the so-called amplitude is equal to one (jc.t/j 1/, and the phase shift is zero.c.0/D1/.
4.2 Definition and Interpretation 79
−6 −4 −2 0 2 4 6
−1.00.01.0
cosine wave (lambda = 1 and lambda = 2)
[−2pi, 2pi]
−6 −4 −2 0 2 4 6
−1.00.01.0
cosine wave (lambda = 1 and lambda = 0.5)
[−2pi, 2pi]
Fig. 4.1 Cosine cycle with different frequencies
ForD1; D2andD0:5these properties are graphically illustrated in Fig.4.1.
Finally, remember the derivative of the cosine, d c.t/
d t Dc0.t/D sin.t/ ; which we will use repeatedly.
Definition
For convenience, we now rephrase the MA(1) process in terms of the lag polynomialC.L/of infinite order,
xtDCC.L/ "t with C.L/D X1
jD0
cjLj:
Next, we define the so-calledpower transfer function TC./of this polynomial:4 TC./D
X1 jD0
c2j C2 X1 hD1
X1 jD0
cjcjChcos.h/ ; 2Rn fg: (4.2)
Note thatTCmay not exist everywhere, there may be singularities at some frequency such thatTC./goes off to infinity as!; but at least the power transfer function is integrable. The key result in Proposition4.1(e) is from from Brockwell and Davis (1991, Coro. 4.3.1, Thm. 5.7.2); it will be proved explicitly in Problem4.1 under the simplifying assumption of absolute summability. The first four statements in the following proposition are rather straightforward and will be justified below.
Proposition 4.1 (Spectrum) Define forfxtgfrom (4.1) the spectrum f./DTC./2
2 : It has the following properties:
(a) f. /Df./, (b) f./Df.C2/, (c) f./0,
(d) f./is continuous inunder absolute summability,P
jjcjj<1. (e) For all h2Z:
.h/D Z
f./cos.h/dD2 Z
0
f./cos.h/d :
Substituting the autocovariance expression from Proposition 3.2 into (4.2), the following representation of the spectrum exists:
f./D .0/
2 C 2
2 X1 hD1
.h/cos.h/D 1 2
X1
hD 1
.h/cos.h/ : (4.3) The symmetry of the spectrum in Proposition4.1(a) immediately follows from the symmetry of the cosine function. From the periodicity of the cosine, (b) follows as well. Both results jointly explain why the spectrum is normally considered on the restricted domainŒ0; only. Property (c) follows from the definition of the power transfer function, see Footnote6 below. Finally, the continuity off./claimed in
4A more detailed and technical exposition is reserved for the next section. Our expression in (4.2) can be derived from the expression in Brockwell and Davis (1991, eq. 5.7.9), which is given in terms of complex numbers.
4.2 Definition and Interpretation 81 (d) under absolute summability results from uniform convergence, see Fuller (1996, Thm. 3.1.9).
We call the functionf (orfx, if we want to emphasize thatfxtgis the underlying process) thespectrumoffxtg. Frequently, one also talks about spectral density or spectral density function asfis a non-negative function which could be standardized in such a way that the area beneath it would be equal to one.
Interpretation
The usual interpretation of the spectrum is based on Proposition4.1. Result (e) and (4.3) jointly show the spectrum and the autocovariance series to result from each other. In a sense, spectrum and autocovariances are two sides of the same coin.
The spectrum can be determined from the autocovariances by definition and having the spectrum, Proposition4.1provides the autocovariances. The casehD0with
Var.xt/D.0/D Z
f./dD2 Z
0
f./d
is particularly interesting. This equation implies: The spectrum at0measures how strongly the cycle with frequency0and therefore of periodP0 D 2=0 adds to the variance of the process. Iff has a maximum at0, then the dynamics offxtg is dominated by the corresponding cycle or period; inversely, if the spectrum has a minimum at0, then the corresponding cycle is of less relevance for the behavior offxtgthan all other cycles. For!0, periodPconverges to infinity. A cycle with an infinitely long period is interpreted as a trend or a long-run component. Hence, f.0/indicates how strongly the process is dominated by atrend component.
Frequently, the analysis of the autocovariance structure or the autocorrelation structure of a process is called “analysis in the time domain” as.h/measures the direct temporary dependence betweenxt and xtCh. Correspondingly, the spectral analysis is often referred to as “analysis in the frequency domain”. Proposition4.1 and the definition in (4.3) show how to move back and forth between time and frequency domain.
Examples
Example 4.1 (White Noise) Let us consider the white noise processxt D "tbeing free from serial correlation. By definition it immediately follows that the spectrum is constant:
f"./D2=2 ; 2Œ0; :
According to Proposition 4.1 all frequencies account equally strongly for the variance of the process. Analogously to the perspective in optics that the “color”
white results if all frequencies are present equally strongly, serially uncorrelated processes are also often called “white noise”.
Example 4.2 (Season) Let us consider the ordinary seasonal MA process from Example3.1,
xtD"tCb"t S
with
.0/D2 1Cb2
; .S/D2b
and .h/D0else. By definition we obtain for the spectrum from (4.3) 2f./D .0/C2 .S/cos.S/
or
f./D
1Cb2C2bcos.S/
2=2:
In Problem4.2we determine that there are extrema at
0;
S; 2
S ; : : : ; .S 1/
S ; : The corresponding values are
f.0/Df 2
S
D: : :D.1Cb/22=2 ; f
S
Df 3
S
D: : :D.1 b/22=2:
Depending on the sign of b, maxima and minima are followed by each other, respectively. In Fig.4.2we find two typical shapes of the spectrum of the seasonal MA process for5 S D 4 (quarterly data) withb D 0:7 andb D 0:5. First, let us interpret the caseb > 0. There are maxima at the frequencies0; =2 and. Corresponding cycles are of the period
P0D 2
0 D 1; P1D 2
=2 D4; P2D 2
D2:
5The variance of the white noise is set to one ,2 D1. This is also true for all spectra of this chapter depicted in the following.
4.2 Definition and Interpretation 83
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.01.02.03.0
MA(4) with b=0.7 and b=−0.5
[0, pi]
b>0 b<0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.01.02.03.0
MA(1) with b=0.7 and b=−0.5
[0, pi]
b>0 b<0
Fig. 4.2 Spectra (2f./) of the MA(S) process from Example4.2
The trend is the first infinitely long “period”. The second cycle has the periodP1D 4, i.e. four quarters which is why this is the annual cycle. The third cycle with P2D2is the semi-annual cycle with only two quarters. These three cycles dominate the process for b > 0. Inversely, for b < 0 it holds that these very cycles add particularly little to the variance of the process.
Example 4.3 (MA(1)) Specifically forS D1the seasonal MA process passes into the MA(1) process. Accordingly, one obtains two extrema at zero and:
f.0/D.1Cb/22=2 ; f./D.1 b/22=2:
In between the spectrum reads f./D
1Cb2C2bcos./
2=2:
Forb D 0:7andb D 0:5, respectively, the spectra were calculated, see Fig.4.2.
Forb < 0one spots the relative absence of a trend (frequency zero matters least) while forb> 0precisely the long-run component as a trend dominates the process.
0.0 0.5 1.0 1.5 2.0 2.5 3.0
02468
maximum at frequency lambda = 0.73
[0, pi]
AR(2) spectrum
Fig. 4.3 Spectrum (2f./) of business cycle with a period of 8.6 years
Example 4.4 (Business Cycle) The spectrum is not only used for modelling sea- sonal patterns but as well for determining the length of a typical business cycle.
Let us assume a process with annual observations having the spectrum depicted in Fig.4.3. The maximum is at D 0:73. How do we interpret this fact with regard to contents? The dominating frequencyD0:73corresponds to a period of about 8.6 (years). A frequency of this magnitude is often called “business cycle frequency”
being interpreted as the frequency which corresponds to the business cycle. In fact, Fig.4.3does not comprise an empirical spectrum. Rather, one detects the theoretical spectrum of the AR(2) model whose autocorrelogram is depicted in Fig.3.4down to the right. The cycle, which can be seen in the autocorrelogram there, translates into the spectral maximum from Fig.4.3.