International Macroeconomics and Finance: Theory and Empirical Methods Phần 3 docx

2.7. FILTERING 69

Now let λò ∼ U[0, π]

29. Imagine that we take a draw from this distribu-

-1.5

-1

-0.5

0

0.5

1

1.5

0 0.4 0.8 1.2 1.6 2 2.4 2.8 3.2 3.6 4 4.4 4.8 5.2 5.6 6

Figure 2.2: π/2Phase shift. Solid: cos(t), Dashed: cos(t + π/2).

tion. Let the realization be λ, and form the time-series

qt = a cos(ωt + λ). (2.100)

Once λ is realized, qt is a deterministic function with periodicity 2π

ω and

phase shift λ but qt is a random function ex ante. We will need the

following two basic trigonometric relations.

Two useful trigonometric relations. Let b and c be constants, and i be

the imaginary number where i

2 = −1. Then

cos(b + c) = cos(b) cos(c) − sin(b) sin(c) (2.101)

eib = cos(b) + isin(b) (2.102)

(2.102) is known as de Moivreís theorem. You can rearrange it to get

cos(b) = (eib + e−ib)

2 , and sin(b) = (eib − e−ib)

2i . (2.103)

29You only need to worry about the interval [0, π] because the cosine function is

symmetric about zeroócos(x) = cos(−x) for 0 ≤ x ≤ π

70 CHAPTER 2. SOME USEFUL TIME-SERIES METHODS

Now let b = ωt and c = λ and use (2.101) to represent (2.100) as

qt = a cos(ωt + λ)

= cos(ωt)[a cos(λ)] − sin(ωt)[a sin(λ)].

Next, build the time-series qt = q1t +q2t from the two sub-series q1t and

q2t, where for j = 1, 2

qjt = cos(ωjt)[aj cos(λj )] − sin(ωjt)[aj sin(λj)],

and ω1 < ω2. The result is a periodic function which is displayed on

the left side of Figure 2.3.

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1 6 11 16 21 26 31 36

-30

-20

-10

0

10

20

30

1 6 11 16 21 26 31 36

Figure 2.3: For 0 ≤ ω1 < ··· < ωN ≤ π, qt = PN

j=1 qjt, where qjt =

cos(ωj t)[aj cos(λj)] − sin(ωj t)[aj sin(λj)]. Left panel: N = 2. Right

panel: N = 1000

The composite process with N = 2 is clearly deterministic but if

you build up the analogous series with N = 100 of these components,

as shown in the right panel of Figure 2.3, the series begins to look like

a random process. It turns out that any stationary random process can

be arbitrarily well approximated in this fashion letting N → ∞.

2.7. FILTERING 71

To summarize at this point, for sufficiently large number N of these

underlying periodic components, we can represent a time-series qt as

qt = X

N

j=1

cos(ωj t)uj − sin(ωj t)vj, (2.104)

where uj = aj cos(λj ) and vj = aj sin(λj), E(u2

i) = σ2

i , E(uiuj)=0,

i 6= j, E(v2

i) = σ2

i , E(vivj)=0, i 6= j.

Now suppose that E(uivj) = 0 for all i, j and let N → ∞.

30 You

are carving the interval into successively more subintervals and are

cramming more ωj into the interval [0, π]. Since each uj and vj is

associated with an ωj , in the limit, write u(ω) and v(ω) as functions

of ω. For future reference, notice that because cos(−a) = cos(a), we

have u(−ω) = u(ω) whereas because sin(−a) = − sin(a), you have

v(−ω) = −v(ω). The limit of sums of the areas in these intervals is the

integral

qt =

Z π

0

cos(ωt)du(ω) − sin(ωt)dv(ω). (2.105)

Using (2.103), (2.105) can be represented as

qt =

Z π

0

eiωt + e−iωt

2

du(ω) −

Z π

0

eiωt − e−iωt

2i

dv(ω)

| {z }

(a)

. (2.106)

Let dz(ω) = 1

2 [du(ω) + idv(ω)]. The second integral labeled (a) can be

simplified as ⇐(49)

Z π

0

eiωt − e−iωt

2i

dv(ω) = Z π

0

eiωt − e−iωt

2i

Ã

2dz(ω) − du(ω)

i

!

=

Z π

0

e−iωt − eiωt

2 (2dz(ω) − du(ω))

=

Z π

0

(e−iωt − eiωt

)dz(ω) + Z π

0

eiωt − e−iωt

2

du(ω).

Substitute this last result back into (2.106) and cancel terms to get ⇐(50)

30This is in fact not true because E(uivi) 6= 0, but as we let N → ∞, the

importance of these terms become negligible.