Page 57 - Azerbaijan State University of Economics

THE JOURNAL OF ECONOMIC SCIENCES: THEORY AND PRACTICE, V.81, # 1, 2024, pp. 51-64

Methodology
To analyse relationship between producer prices and consumer inflation, I apply
wavelet analysis. Wavelet analysis has been popularized in scientific fields since
1980s as an alternative to Fourier transform. The wavelet approach allows us to study
the relationship between the variables both over time and frequency domain.

Basically, the Fourier transform can be represented in the following way:
+∞

( ) = ∫ ( ) − (1)

−∞
where is the angular formula and − = cos( ) − sin( ) according to
Euler’s formula.

Despite Fourier transform is widely used in frequency analysis in economics, it does
not reveal the relationship between the variables over time. Furthermore, Fourier
analysis requires stationarity conditions which is easy to be violated in economic
series. In this regard, wavelet analysis can be considered a useful tool which does not
suffer these drawbacks.

The continuous wavelet transform can be expressed as follows:

+∞
∗
( , ) = ∫ ( ) ( ) (2)
,

−∞

where * denotes the complex conjugate. In this context, ( ) represent the
∗
,
conjugate functions of the daughter wavelet functions of ( ) (Jiang, et al., 2015).
,
As a starting point, so-called mother wavelet can be derived as:

1 −
( ) = ( )
,
√ (3)

where denotes scaling factor that determines the length of the wavelet and indicates
the position of wavelet, i.e., time. (Aguiar-Conraria et al. 2008). There are different
types of mother wavelets that can be employed for different analysis. Jiang et al.
(2015) state that the most common mother wavelet used for feature extraction
purposes is Morlet wavelet which is defined in simplified form as

2
( ) = −1/4 0 − /2 (4)

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