Advanced Econometrics Takeshi Amemiya

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

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 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:

(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).