e use of different methods and approaches to the analysis of the stability of the banking system can expand limits of proper analysis and get new results. Due to such a generalization we propose to focus on discovering the stability of the banking system by the methods of wavelet analysis. Disclosure of available content of stability of the banking system will be produced according to the generalized cross-analysis between time series data that determine or are individual components of such stability. This is what defines the basis for the purpose of this study.
3. Wavelet Analysis Methodology as a Tool to Study Time Series
In general, wavelet analysis methodology allows for the selected schedule time series system of individual wavelets. These wavelets may be obtained by shifting and scaling so-called function-generating wavelet  . Therefore, in this case, wavelet is a function which rapidly decreases at infinity and whose average value is zero.
The basis of formal generalization of continuous wavelet transformation on the time interval t is converting the input time series using wavelet parent  -  :
where denotes a normalization,
u is a location parameter,
s is a scale parameter,
In this case, feasibility of wavelet analysis in time series study is determined by the fact that the method of wavelet analysis allows to discover the local features of the studied time series due to the decomposition of the input data   . That is, the wavelet transformation defines the hierarchical structure of the input of the studied time series. Underlying this statement is the fact that information flows, generated by fractal time series have properties detected by wavelet transform, making it even more informative  . Then wavelet transform allows adding input characteristics of time series that are investigated.
In addition, the empowerment of study time series using wavelet analysis methodology promotes the use of various procedures of wavelet transformation: even-scaled analysis, cross wavelet transformation, wavelet coherence. Then wavelet analysis methodology has been widely used in the disclosure of dynamics of time series that define various economic data   .
One of the main wavelet transformation methods used for generalized cross-reference analysis between different time series is wavelet coherence. It allows calculating local correlation of two time series in a region of time-frequency. It uses the following formalized model: wavelet coherence as the squared absolute value of the smoothed cross wavelet spectra (time (x) and frequency (y) of the time series), normalized by the product of the smoothed individual wavelet power spectra of each series   :
where Q is a smoothing operator.
We use Morlet wavelet that is a complex wavelet with a good time-frequency localization, as a parent one   .
The squared wavelet coherency coefficient is in the range, values close to zero indicate weak correlation, while values close to one are evidences of strong correlation.
Thus, wavelet coherency analysis enables interconnection between the studied time series and analyzes the frequency of such communications.
4. Data for the Analysis
For the analysis we will use relevant data for the banking system of Ukraine. This choice is made due to the need for a synthesis of the analysis in terms of the banking system of Ukraine, accompanied by significant and sudden changes in the political system of Ukraine for many years, due to the global changes taking place in the banking system of Ukraine in recent years.
The main objective of the analysis of the stability of the banking system in Ukraine is, above all, the comparison of certain indicators of such stability and determining the dynamics of their consistency with each other over time. That is, we try to answer the question: how fully and adequately some indicators reflect the current state of the stable functioning of the banking system in Ukraine.
As some pairs of indicators for appropriate analysis we consider:
the ratio of non-performing loans excluding capital reserves (KK) and the ratio of non-performing loans to total gross loans (KVK); the rate of return on assets (PA) and the rate of return on capital (PK); the ratio of interest margin to gross income (MD) and the ratio of non-interest expenses to gross income (ND); the ratio of liquid assets to total assets (LA) and the ratio of liquid assets to current liabilities (LZ); the ratio of trading income to gross income (TD) and the ratio of staff costs to non-interest expenses (VP); the spread between rates on loans and deposits (PKD) and the spread between the highest and lowest interbank rates (KD); the spread between rates on loans and deposits (PKD) and the ratio of customer deposits to total gross loans (excluding interbank) (VKD).
The above selection of individual indicators was made in accordance with their importance and corresponds with mobility of inward and outward flows as well as with available methodological foundations of stability disclosure. Grouping selected indicators of stability of the banking system is determined by the ability of conducting comparative and crossed correlation of effects between financial flows of the banking system and bringing coherence between these flows or installation on problematic aspects of banking activity.
In particular, the ratio of non-performing loans excluding capital reserves and the ratio of non-performing loans to total gross loans identifies and maps a dynamic display of the current stability of the banking system in Ukraine under the existing conditions for banking, comparison of the rate of return on assets and rate of return on capital can reveal the conditions of the stability of the banking system in Ukraine to the efficiency of banking activities according to indicators such as assets and capital, the ratio of trading income to gross income and the ratio of staff costs non-interest expenses to determine expression stability of the banking system in Ukraine by the performance of the personnel of banks, the spread between rates on loans and deposits (basis points) and the spread between the highest and lowest interbank rates (basis points), determine the conditions of stability of the banking system in Ukraine in accordance with the balance set interest rates according to the circumstances of their formation on the market, the spread between rates on loans and deposits (basis points) and the ratio of customer deposits to total gross loans (excluding interbank), reveals signs of stability of the banking system in Ukraine by the volume and terms of attracting and allocation of resources of the banking system.
As the time interval for the study of indicators on stability of the banking system in Ukraine we selected data for the period Q4 2005-Q4 2015. So we have 41 values for each data series. Each of the values is appropriate for data specific indicator of stability at quarter end from the selected time period. This allows you to take into account the different periods of the banking system in Ukraine in terms of ensuring its stability.
All data for analysis was taken from the official website of the regulator of banking activities in Ukraine-the National Bank of Ukraine (http://www.bank.gov.ua―statistics of indicators of financial stability). Table 1 shows generalized statistical data that characterizes indicators chosen for the study.
5. Findings and Discussion
On Figure 2, Figure 3, Figures 5-9 you can check the results of wavelet coherence between selected indicators for the study on stability of the banking system in Ukraine. Each of the following figures indicated a separate group of indicators that match each other. This comparison of selected groups of indicators is displayed in the time-frequency plane. Horizontal axis shows time scale, which corresponds to the sequence analyzed data since the 4th quarter of 2005 and ending with Q4 2015. On the vertical axis are the weighted characteristics of the
Figure 2. Wavelet coherence between KK and KVK.
Figure 3. Wavelet coherence between PA and PK.
Table 1. Statistical indicators of data series that define the selected set of indicators on stability of the banking system in Ukraine.
analyzed data series in frequency space measuring their results are grouped by time periods (quarters) according to the total time interval. Along each of the figures importance scale is presented as separate columns for reflections. Defined lines are a manifestation of localization for individual irregularities within studied time series according to importance of irregularities. In general, each point of wavelet reflects shown in Figure 2, Figure 3, Figures 5-9 is their value in the time-frequency space, which are calculated through wavelet transformation. The phase difference, indicated by arrows, gives us details about delays of oscillation of the two examined time series. Arrows pointing to the right (left) when the time series are in-phase (anti-phase) or are positively (negatively) correlated. Arrow pointing up means that the first time series leads the second one, arrow pointing down indicates that the second time series leads the first one.
Therefore, Figure 2 shows that over a longer period of time investigated there is a steady mutual tendency between ratio of non-performing loans excluding capital reserves (KK) and the ratio of non-performing loans to total gross loans (KVK). The manifestation of breach of reciprocity between the ratio of non-performing loans excluding capital reserves (KK) and the ratio of non-performing loans to total gross loans (KVK) begins to be determined on the basis of the banking system in Ukraine in the 1st quarter of 2012. That is 1 quarter of 2012 should be considered the point of departure, when the stability of the banking system is impaired due to lack of capital as a factor of stability and respect for this increase in non-performing loans.
From the data in Figure 3 above all we can see quite identical consistency and reciprocity between the rate of return on assets (PA) and the rate of return on capital (PK). However, given the downward trend as the of values of return on assets (PA) (in Figure 4) and also of values the rate of return on capital (PK) it is worth noting that in terms of wavelet coherence manifestation of instability of banking system is evident.
According to Figure 5 you can talk about lack of complete consistency between the ratio of interest margin to
Figure 4. Dynamics of values for rate of return on assets (PA) and the rate of return on capital (PK) for the banking system in Ukraine in Q4 2005-Q4 2015.
Figure 5. Wavelet coherence between MD and ND.
gross income (MD) and the ratio of non-interest expenses to gross income (ND). Therefore, it is worth noting the fact that in terms of the banking system in Ukraine there may be a lack of balance between the margin and non-interest expenses. This may be an evidence of lack of manifestation of prudential policy for setting remuneration of banking activities in the form of interest or in the form of commission income and expenses. So the dynamics of the analyzed data series and analysis of such dynamics based on wavelet coherence can be seen as a negative factor to determine the stability of the banking system in Ukraine.
Lack of consistency in analyzed data series is observed in terms of the ratio of liquid assets to total assets (LA) and the ratio of liquid assets to current liabilities (LZ) (Figure 6). This fact can be considered in terms of lack of liquid assets in case of urgent necessity to repay short-term obligations. Therefore it is also necessary to take disclosing weight stability of the banking system into account.
In addition, the relevant data from the analysis of wavelet coherence between the ratio of trading income to gross income (TD) and the ratio of staff costs to non-interest expenses (VP) (Figure 7) can be seen as a manifestation of a lack of balance in staff costs to a result which such expenses determined in terms of one of the areas forming gross income. So there definitely is a lack of conformity of productivity of its workers to costs that are aimed at keeping staff. Therefore, it is also necessary to identify this issue as one of the factors affecting the stability of the banking system.
According to the data in Figure 8 we should also note that in Ukraine spread between rates on loans and deposits in general is different from the general trend of the corresponding spread that sets the interbank market.
Figure 6. Wavelet coherence between LA and LZ.
Figure 7. Wavelet coherence between TD and VP.
Figure 8. Wavelet coherence between PRD and KD.
Figure 9. Wavelet coherence between PRD and VKD.
So we can say that setting rates on loans and deposits is, above all, given the current conditions of each institutional unit within the banking system. At the same time the impact of market factors evident only in the long term. However, for next to Figure 5 of ignoring the current market leverage on the formation of interest rates may affect the current stability of the banking system, which in turn has a greater chance of becoming a more significant banking crisis. This is observed in Ukraine in recent years.
There is a similar discrepancy in the formation of interest rates on loans and deposits according to market and leverage affects formation of scale deposits and loans of the banking system (Figure 9). As a result we can identify both periods of significant outflow of deposits and long periods of moderate inflows of resources on deposit accounts, declines in lending in real economy and the growth of bad loans    .
The results of the study can determine the appropriate application of wavelet analysis methodology as disclosure of wavelet coherency between the studied data series as a tool for the study on stability of the banking system.
Application of wavelet coherence analysis for the stability of the banking system can not only explore the manifestations of such stability or instability of the banking system, but also emphasize and reveal existing conditions to ensure the efficiency of the banking system.
In particular the work states that among the factors influencing the stability of the banking system in Ukraine great role is played by:
lack of capital;
increase in non-performing loans;
lack of prudence rooted policy for setting remuneration of banking activities in the form of interest or in the form of commission income and expenditure;
lack of liquid assets in case of immediate need short-term repayment obligations;
lack of conformity productivity of its workers to expenses aimed at staff retention;
incomplete compliance of market leverage on the formation of the current values of interest rates on loans and deposits.
At the same time we observed that the use of wavelet coherence analysis data series which allowed summarizing stability of the banking system allowed drawing conclusions about reasonable adjustments of dynamics of the components of the studied data series and forming a balanced policy impact on the banking system.