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Vugar Rahimov: Relationship between PPI and CPI in Azerbaijan: A Wavelet Approach

with parameter . The parameter governs the number of oscillations within
0
0
Gaussian envelope. As shown in Çepni et al. (2020), in practice, the preferred value
of the parameter is 6, as it creates balance between time and frequency localization.
Interesting quantities, such as variance and covariance, can be captured wavelet
domain. Since, in this study, I focus on two series case, say ( ) and ( ), I can
construct the cross-wavelet power spectrum in the following way:

( , ) = ( , ) ( , ) (5)
∗

As discussed in the previous studies, such as Rua (2012), Jiang et al. (2015), Çepni et
al. (2020), wavelet coherency can be expressed as:

2
| ( −1 ( , ))|

2
( , ) = 2 (6)
2
( −1 | ( , )| ) ( −1 | ( , )| )

2
where (. ) represents smoothing parameter both in time and scale. ( , ) measures
the strength of relationship between the series and varies between 0 and 1 in a time
frequency space, where the higher the value, the stronger the relationship between the
variables.

Wavelet coherence phase differences between the two series can also be described as
follows:

Γ ( ( , ))

( , ) = −1 ( ) (7)
ℵ ( ( , ))

where Γ and ℵ are imaginary and real parts, respectively, of the smoother cross-
wavelet transform. Phase differences are shown by the arrows, where arrows pointing
to the right indicate that the two series are positively correlated (in phase) and arrows
pointing to the left represent negatively correlated (out of phase) variables. In addition,
the lead-lag relationship can also be shown with the arrows, where arrows positioned
upward mean the first variable leads the second one, while arrows positioned
downward suggest that the second variable leads the first one.

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