Autocovariances
Define yh = Ey, yt+h, h = 0, 1, 2,. . . .A sequence (yA) contains important information about the characteristics of a time series {y,}. It is useful to arrange (yA) as an autocovariance matrix
7o 
71 
72 ‘ 
7т – і 
7i 
7o 
7i ‘ 
7t – 2 
72 
7i 
• 
* 
• 
• 
‘ 7i 

7 т – і 
7t – 2 
* 
■ 7i 7o 
This matrix is symmetric, its main diagonal line consists only of y0, the next diagonal lines have only y,, and so on. Such a matrix is called a Toeplitzform.
5.1.1 Spectral Density
Spectral density is the Fourier transform of autocovariances defined by /(<«)= І) 7#’°“°, – яё<ыёя,
A—«
provided the righthand side converges.
Substituting ea = cos A + і sin A, we obtain
as
/(«) = 5) У* [cos (Aft)) — /sin(hft))] (5.1.3)
A——00
00
= 2 yh cos (Act)),
A—00
where the second equality follows from yh = and sin A =—sin (—A). Therefore spectral density is real and symmetric around o) = 0.
Inverting (5.1.2), we obtain
yh = (2tc) 1 J eiAaf((o) da> (5.1.4)
= 7rl J cos (Aft))/(ft)) dw.
An interesting interpretation of (5.1.4) is possible. Suppose y, is a linear combination of cosine and sine waves with random coefficients:
(5.1.5)
where o)k = kn/n and (4) and {(k) are independent of each other and independent across к with E£k = E(k — 0 and V£k = V(k = a. Then we have
7h = 2 cos (w*A), (5.1.6)
fci
which is analogous to (5.1.4). Thus a stationary time series can be interpreted as an infinite sum (actually an integral) of cycles with random coefficients, and a spectral density as a decomposition of the total variance of y, into the variances of the component cycles with various frequencies.
There is a relationship between the characteristic roots of the covariance matrix (5.1.1) and the spectral density (5.1.2). The values of the spectral density/(<u) evaluated at T equidistant points of ft) in [—я, n are approximately the characteristic roots of 2r(see Grenander and Szego, 1958, p. 65, or Amemiya and Fuller, 1967, p. 527).
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