Wavelet Analysis

As mentioned before, wavelets analysis is used to obtain the orthogonal decomposition of the

relevant economic variables at different time scales. Due to limited space, we will discuss the

wavelet analysis very briefly here. We refer the interested readers to Ramsey and Lampart

(1998), and Ramsey (2002) for more details.

Wavelet analysis can be seen as an extension of the Fourier analysis. In Fourier analysis, the

sine and cosine functions at various frequencies are used to expand the given function or time

series. Suppose the original function or time series is X(t), wheret∈[0, T]. Based on Fourier

analysis,X(t) can be decomposed into components at different frequencies:

X(t) =a0+ ∞ X ω=1

{aωcos(2πωt/T) +bωsin(2πωt/T)}, t∈[0, T] (2.1)

The orthogonal basis functions cos(2πωt/T) and sin(2πωt/T) capture the variations at dif-

ferent frequency ω = 1,2,3,4, ..., and aω and bω are the amplitudes of corresponding fre-

quencies. Intuitively, the component at frequency ω = 1, a1cos(2πt/T) +b1sin(2πt/T), rep-

resents the long term component or trend in the original time series that fluctuates slowly.

While the high frequency components, for instance the component at the frequency ω = 100,

a100cos(200πt/T) +b100sin(200πt/T), would captures the high frequency variations that os-

cillates 100 time faster. The superposition of components at different frequencies recovers the

original time seriesX(t).

However, the drawback of Fourier analysis is that it assumes that the frequency content of

the time series is invariant across time. After transforming the original time series from time

domain into frequency domain, only the frequency information is preserved, all the temporal

information is lost. Moreover, the sine and cosine basis functions do not die out, hence they

are not adaptive to changes that localized in time. For example, if the time series have high

frequency oscillation in the first ten percent of the time series, and continue with low frequency

fluctuations for the rest, Fourier analysis would correctly identify that there are two frequency

components for the whole time series, but it cannot differentiate the change in frequency content

localized in time, i.e., the temporal information is completely lost. However, the temporal order

relationship and causality among different variables based on their temporal information.

Compared with Fourier analysis, wavelet analysis are more suitable for handling financial/economic

time series. The examples of the Morlet and Haar wavelets are shown in figures 2.1 and 2.2. As

oppose to sine and cosine functions in Fourier analysis, the basis function of the wavelet analysis

are localized in both time and frequency, hence the temporal information would be preserved

as we study the time series variations at different scale/horizon. The decomposed components

based on wavelet analysis at different scale can adaptively capture the local behavior of the

time series in different time periods.

Insert figure 2.1 about here

Insert figure 2.2 about here

To study the variations at different scale/horizon, the scaling or dilation property of wavelets

function is particularly important. Given ψ(t) is the wavelet function at scale 1 and centered

at 0. The Haar wavelet used in this study is in the following form:

ψ(t) =                1 0≤t < 1₂, −1 1₂ ≤t <1, 0 otherwise. (2.2)

which is shown graphically in figure 2.2. The basis function at scale s and centered at time u

is defined as ψu,s(t) = 1 √ sψ t−u s , (2.3)

Intuitively, the wavelet functions at large scales (lower frequency) are dilated or stretched in

the time dimension bystimes, and in order to capture the variation around time u, the whole

wavelet function is shifted to the new location of time u. Therefore, the wavelet ψu,s(t) is

concentrated in a neighborhood of size proportional to s and centered around time u. The

factor √1

s ensure that the energy of wavelet is normalized to one, which means the integration of the squared wavelet function ψ_u,s2 (t) equals to one. Function ψu,s(t) is also referred as the

mother wavelet function, as opposed to the father wavelet or scaling function ψu,s(t) that we

Notice that the integration of the wavelet function ψu,s(t) (eq. 2.3) with respect to t equals

zero, which suggest that at scale s, wavelet function ψu,s(t) captures the fluctuations around

a local average level, but it cannot represent the local average level of the original time series.

Therefore, we need the scaling functions to span the average level of the time series. The scaling

function of Haar wavelet is as follows

φ(t) =        1 0≤t <1, 0 otherwise. (2.4)

The scaling function at scale sand centered at timeu is defined similarly as equation 2.3:

φu,s(t) = √1 sφ t−u s , (2.5)

Therefore, it is easy to see that the scale function φu,s(t) captures the local average of the

original function at scales and locationu.

When wavelet functions are used for multi-resolution analysis, time series Xt is decomposed

into orthogonal components by scale:

Xt= 5 X j=1

Xt[dj] +Xt[a5] =Xt[d1] +Xt[d2] +Xt[d3] +Xt[d4] +Xt[d5] +Xt[a5], (2.6)

whereXt could be one of the variables, such as ∆st,mot, and It, and Xt[dj] is the component

of Xt at the jth scale. Intuitively speaking, the component Xt[di] captures the variations at

2j−1 days horizon, and none of the other scales:

Xt[di] = X

u

cuψu,di(t), (2.7)

where ψu,di(t) is the wavelet function at scale di and location u, and cu is the amplitude of

ψu,di(t) at locationu.

The multi-resolution analysis (equation 2.6) can be seen as the process of peeling the onion.

First of all, we extract the highest frequency variations at the 1-day horizon, which comprise the

variations between 1 day and 2 day horizon; similarly, Xt[d3] captures the variations between

2-day and 4-day horizon, and so on. At last, all the details from scale 1 to 5 have been extracted

from the time series, what is left is the long-term variations that are not captured by Xt[di],

i= 1,2,3,4,5, and they all encapsulated in component Xt[a5]. Xt[ai] represents the long term

variations at the horizon of 25 days or greater:

Xt[ai] = X

u

fuφu,ai(t), (2.8)

wherefu is the coefficient of the scaling function φu,ai(t) at timeu.

Moreover, the components at different scales are completely uncorrelated from each other by

design, since the wavelet basis functions at any two different scales are orthogonal to each other:

Z R

ψu1,s1(t)ψu2,s2(t)dt=δu1,u2δs1,s2, (2.9)

where function δu1,u2 represents the Kronecker delta, which equals to 1 if u1 = u2, and zero

otherwise. Equation 2.9 suggests that the basis functions at different scales or different locations

are orthogonal to each other. Moreover, we also have the detail functionψu,s(t) (such as equation

2.2) and scaling function φu,s(t) (such as equation 2.4) are orthogonal to each other at all

scales. This is why we use wavelet analysis to decompose the relationship between the exchange

rate movement and the intervention by time scale, without worrying about the relationship at

different scales being coupled together.