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

Подпись:Spectral density is the Fourier transform of autocovariances defined by /(<«)= І) 7#’°“°, – яё<ыёя,

A—«

provided the right-hand 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:

Подпись: П (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)

fc-i

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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