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CPS2176 Chiraz KARAMTI et al.
of abrupt changes and volatility. Recently, this methodology has received
great interest in the financial literature (Rua and Nunes, 2009; He et al., 2009;
Masih et al., 2010; Jammazi, 2012). Researchers, have consistently endorsed
that the wavelet tool is superior to the conventional statistical ones that have
been used to decompose, filter and denoise signals. Wavelet transformations
has the capacity to breakdown macroeconomic variables into their scale
parcels. According to the Mallat’s (2001) theory, the original discrete time
series () can be decomposed into a series of linearity independent
approximation and detail signals by using wavelet transform. For that, the
wavelet technique uses multiresolution analysis by which different frequencies
are analyzed with different resolutions. There is a scaling function () (also
called father wavelet) such that:

⁄

() = 2 2 (2 − ) (1)
,
where the level j controls the degree of stretching of the function (the larger
the j, the more stretched is the basis function); the smaller the scale, the higher
the frequency of the decomposed series, and k is the parameter that controls
the translation of the basis function. Assuming that the detail space, { }, are

orthogonal to each other, we can define a sequence { ()} (called mother
,

wavelet) of orthonormal basis function that spans ²(ℝ):
⁄

() = 2 2 (2 − ) (2)
,
where wavelets are generated by shifts and stretches of the mother
wavelet () . Let () the original time series, we represent the
,
multiresolution representation of () by:
() = ∑ () + ∑ ∑ ()
, ,
, ,
(3)
= () + () + −1 () + ⋯ + ()

1
The series () provides a smooth of original time series () (called also

approximation) at levels J and captures the long term properties (i.e. the low-
frequency dynamics), and the series () for = 1, … , denote wavelet details

and capture small variations (i.e. the higher-frequency characteristics) in the
data over the entire period at each scale. The last expression (3) denotes the
decomposition of () into orthogonal components at different resolutions
and represents the so-called wavelet multiresolution analysis (MRA).

2.2 Wavelet-based EGARCH model
To model the time-varying dynamics of exchange rates, the wavelet details
(d1-d7) are used as input in the ARMA-EGARCH model of Nelson (1991). Recall
that the EGARCH model was developed to allow for asymmetric effects

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