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Time-Series-Based EconometricsUnit Roots and Co-integrations$

Michio Hatanaka

Print publication date: 1996

Print ISBN-13: 9780198773535

Published to British Academy Scholarship Online: November 2003

DOI: 10.1093/0198773536.001.0001

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(p.247) Appendix 1 Spectral Analysis

(p.247) Appendix 1 Spectral Analysis

Source:
Time-Series-Based Econometrics
Publisher:
Oxford University Press

This appendix supplies proofs for several well‐known statements on the spectral analysis that appear in the text of Chapters 3 and 7. Mathematical rigour is sacrificed for emphasis on the basic ideas. The readers who want more about the spectral analysis are advised to study Fuller (1976, ch. 7), Granger and Newbold (1986, ch. 2), and Hamilton (1994, ch. 6). More advanced readings are Anderson (1971, chs. 7, 8, 9, and 10) and Brillinger (1981).

(a) In a sequence of real numbers that extends in both directions, . . . , γ−1, γ0, γ1, . . . suppose that γj = γj, j = 1, 2, . . . .

f ( λ ) 1 2 π j = γ j exp ( i λ j ) , π λ π
(A1.1)
is the Fourier transform of the sequence. In (A1.1) i = √ − 1. Noting that exp (−iλ j) = cos λ ji sin λ j, cos λ j = cos λ (−j), sin λ j = −sin λ (−j), (A1.1) is written as
1 2 π ( γ 0 + 2 j = 1 γ j cos λ j )
(A1.1′)
and also as
1 2 π j = γ j exp ( i λ j )
(A1.1″)

When f(λ) is defined by (A1.1) we have for j = . . . , −1, 0, 1, . . .

π π f ( λ ) exp ( i λ j ) d λ , = π π 1 2 π h = γ h exp ( i λ h ) exp ( i λ j ) d λ = π π 1 2 π γ j d λ = γ j ,
(A1.2)
where the second equality follows from
π π exp ( i λ ( j h ) ) d λ = 0 if  j h .
The expression (A1.2) is called the Fourier inverse transform of f(λ).

Given a stationary stochastic process, {x t}, such that E(x t) = 0, the autocovariance for lag j is E(x t x tj) = E(x t + j x t), and thus γj = γj. The Fourier transform of the sequence, . . . , γ−1, γ0, γ1, . . . is the (power) spectral density (p.248) function, f(λ), where the argument λ is called the frequency. The Fourier inverse transform of f(λ) is the autocovariance sequence.

(b) Given two sequences, . . . , α−1, α0, α1, . . . and . . . , γ−1, γ0, γ1, . . . we are interested in the sequence, . . . , α−1γ−1, α0γ0, α1γ1, . . . . Define

f α ( λ ) = 1 2 π j α j exp ( i λ j ) f γ ( λ ) = 1 2 π j γ j exp ( i λ j ) f α γ ( λ ) = 1 2 π j α j γ j exp ( i λ j ) .
Then
π π f α ( ζ ) f γ ( λ ζ ) d ζ = π π 1 2 π j α j exp ( i ζ j ) 1 2 π h γ h exp ( i ( λ ζ ) h ) d ζ = π π ( 1 2 π ) 2 j h α j γ h exp ( i λ h ) exp ( i ζ ( j h ) ) d ζ = 1 2 π h α h γ h exp ( i λ h ) = f α γ ( λ ) ,
(A1.3)
where we have used that π π exp ( i ζ ( j h ) ) d ζ = 0 if jh, and = 2π if j = h.

(c) If {x t} is a scalar stationary process with E(x t) = 0 and the spectral density function f(λ), there is the Cramer representation

x t = π π exp ( it λ ) dZ ( λ ) ,
where Z(·) is a complex valued, random function such that
E ( dZ ( λ ) dZ ( ζ ) ̲ ) = { 0 if  λ ζ f ( λ ) d λ if  λ = ζ .
Consider y t j = a j x t j . Then
y t = j a j π π exp ( i ( t j ) λ ) dZ ( λ ) = π π exp ( it λ ) ( j a j exp ( ij λ ) ) dZ ( λ ) .
The spectral density function of {y t} is
| j a j exp ( ij λ ) | 2 f ( λ ) .
(p.249) Setting a 0 = 1, a 1 = −1, all other as to zero, the spectral density function of Δ x t is
| 1 exp ( i λ ) | 2 f ( λ ) = 2 ( 1 cos λ ) f ( λ ) .