“Bond yields and stock returns comparison using wavelet semblance analysis”

Various measures of resemblance are increasingly applied in confrontation of data samples obtained by different sources. Semblance analysis aims at comparison of two sets of data based on their phase and frequency. Conventional semblance analysis following the Fourier transform has several deficiencies resulting from the transform. To overcome these obstacles, another type of semblance analysis was developed applying the wavelet transform. This paper focuses on semblance analysis of stock prices and government bond yields of two major global economies using continuous wavelet transform regarding both scale and time. Nelson-Siegel interest rate term structure model with the crucial yield variables following autore-gressive processes with stochastic volatility. They estimated the model on US treasury yields applying Markov chain Monte Carlo methods and argued that the factor volatilities followed highly persistent processes. Moreover, they showed that yield factors and factor volatilities were closely related to macroeconomic state factors.


INTRODUCTION AND LITERATURE REVIEW
Financial time series have been the subject of academic research for several decades. Because of their specific nature, most of financial data is nonstationary, which means that the statistical properties of the data change over time. Its probabilistic distribution can vary rapidly, and during a short period investors might gain large profits, as well as significant losses. Aiming at government bond yields, Moessner (2015) quantified the impact of policy rate guidance of Federal Open Market Committee at zero on US treasury yields with maturities up to 10 years. His outcomes indicated that open-ended and time-contingent forward guidance announcements caused a reduction in forward yields at a wide range of horizons. On the other hand, Poghosyan (2014) analy zed determinants of government bond yields in 22 European economies between 1980 and 2010 using panel cointegration techniques. His estimations suggested that while in the aftermath of the crisis spreads in some peripheral countries exceeded the level indicated by fundamentals, North European countries have benefited from their safe-haven character which is in line with results of Gonzalez-Rosada and Yeyati (2008). Confirming Diebold and Li (2006), Hautsch and Ou (2012) proposed a Nelson-Siegel interest rate term structure model with the crucial yield variables following autoregressive processes with stochastic volatility. They estimated the model on US treasury yields applying Markov chain Monte Carlo methods and argued that the factor volatilities followed highly persistent processes. Moreover, they showed that yield factors and factor volatilities were closely related to macroeconomic state factors.
Following Lee et al. (2013) and Yang and Shigeyuki (2015), Jammazi et al. (2015) explored the dependence relationship between long-term government bonds and stock returns for a range of countries over the last twenty years using a dynamic DCC-GARCH-copula model which allows obtaining a flexible and comprehensive description of the time variation of various patterns. Their results indicated positive stock-bond association during the 1990s, and negative from the early 2000s. Additionally, authors found no evidence of asymmetric and tail dependence for the majority of countries. Antonakakis and Vergos (2013) examined government bond yield spread spillovers among European countries during the global financial as well as the Euro zone debt crisis. They applied the VAR-based spillover index based on Diebold and Yilmaz (2012) and impulse response analyses. Their outcomes suggested that on average, bond yield spreads shocks tend to increase future bond yield spreads, and that they are connected to policy changes and news announcements. Moreover, bond yield spreads shocks from the peripheral countries have significantly more destabilizing force on other countries than shocks coming from the core countries. Asymmetry in the distribution of government bond returns in developed countries was studied by Fujiwara et al. (2013). Authors provided evidence for asymmetry in government bond returns with short maturities and found that the probability of a large and negative excess return is more likely in a less liquid market.  Kim and In (2007) focused on the relationship between changes in bond yields and stock prices in the G7 economies. They stated that given relationship might be positive, as well as negative. To properly explore the relationship, they proposed wavelet correlation analysis and their results showed that the correlation between changes in bond yields and stock prices can vary in different countries and time scales.
Following previous research, this paper focuses on semblance analysis of government bond yields and stock prices in US and Germany using continuous wavelet transform.

METHODOLOGY
There exist various methods of how to compare two time series, such as cross-correlation, however standard techniques can often result into outcomes that are difficult to exploit or do not give sufficient perspective about the position of the datasets in time. Reboredo et al. (2017) therefore explored causality and co-movement between oil and renewable energy stock prices using continuous and discrete wavelets, Akoum et al. (2012) examined the short and long term dependencies between stock market returns and OPEC basket oil utilizing the wavelet coherency methodology, and Samadder (2015) investigated the periodicity of the two prime Indian and American stock market indices applying Ferraz-Mello method of date-compensated discrete Fourier transform.
Semblance analysis has been commonly utilized in seismology, however, its construction seems ap-propriate for financial time series as well (Cooper & Cowan, 2008). It confronts two samples regarding the correlations between their phase angles, as a function of frequency. Blackedge (2006) and Ahmed and Rao (2012) describe the Fourier trans- Hf of a sample ( ) ht as:  between -1 and 1, where 1 means absolute phase correlation, -1 implies perfect negative correlation and 0 represents no correlation. Semblance procedure divides each input sample into two output samples comprising of the set of the input sample that is correlated, and the set that is not, which is done in the frequency domain.
After the selection of the threshold correlation, the Fourier transform of each sample is divided into two groups. First one consists of the Fourier coefficients with a semblance above the threshold and the second group consists of the remainder. In each case are the missing coefficients substituted by zeros. Consequently, the inverse Fourier transform is performed to each group, so that there are two output samples for each input sample.
In contrast to the Fourier transform, the waveletbased techniques provide the possibility to cover temporal variability in spectral character. Even though the wavelets are relatively young method, their applications have expanded to many different areas. In the field of finance, Liu et al. (2017) investigated the evolution of mean and volatility spillovers between oil and stock markets in the time and frequency dimensions using a wavelet-based GARCH-BEKK method and showed that spillover effects vary across wavelet scales in terms of strength and direction. Similar studies focusing on financial issues were also presented by Khalfaoui where t is the time, u describes the displacement, s represents the scale, ψ enotes the mother wavelet, and * stands for complex conjugate. Same as in (Cooper & Cowan, 2008), the complex Morlet wavelet was applied in this work, which Teolis (2012) defines as:

RESULTS
As it has already been stated, continuous wavelet transform provides better temporal resolution than semblance models based on Fourier transform. To explore the abilities of continuous wavelet transform based semblance analysis in financial time series research, we examined two major global economies, i.e., US and Germany. In order to measure the semblance between German stocks and government bond yields we focused on daily returns of stock index DAX, as well as daily 10 year Bund yield returns. Both time series are benchmark values defining the Euro zone performance. Since stock and bonds are usually considered as investment substitutes, we might expect positive relationship between stock indices returns and changes in government bond yields as it is explained below.  where dark grey corresponds to a semblance of +1 and black corresponds to a semblance of −1. Light grey color corresponds to a semblance of 50%. We might conclude that besides the period between 2013 and 2014, daily changes of 10 year treasury yield and daily S&P500 returns were negatively correlated primarily on scale of 10-40 days, in period of 2013 to 2014 we found negative correlation between US stock prices and treasury yields, i.e., returns of both type of usual substitutes in US moved in same direction (black color on the fifth segment of Figure 2).
In order to extend our analysis, we compared also the relationship between government bond yields and stock market returns in Germany and US. Figure 3 depicts the semblance between

CONCLUSION
The relationship between stock returns and bond yield changes has been extensively explored in academic research. Even though different methods of resemblance analysis were applied in comparison of financial data samples, only small number of studies focused on wavelets as a perspective approach to enhance conventional semblance analysis. Conventional semblance analysis is based on the Fourier transform, however, it has several shortcomings resulting from its nature. This work aims at semblance analysis of stock prices and government bond yields of US and Germany using continuous wavelet transform. To examine the semblance between German government bond yields and stocks, we focused on the sample of daily returns of stock index DAX and 10 year Bond yield returns consisting of 2238 daily observations between 28th February 2008 and 16th November 2016. Our results indicate that daily DAX returns and daily changes of 10 year German government bond yields are usually correlated on scale of 20-40 days, while scale of 10-20 days suggests significantly lower correlation on the explored data sample. However, the rise of German government 10 year bond yields was usually accompanied by the rise of stock index DAX. Focusing on the relationship between Standard & Poor's stock index and 10 year US treasury yields we might conclude that besides the period between 2013 and 2014, daily changes of 10 year treasury yield and daily S&P500 returns were negatively correlated primarily on scale of 10-40 days, in period of 2013 to 2014 we found negative correlation between US stock prices and treasury yields.
Moreover, we also compared the relationship between government bond yields and stock market returns in both countries. Until the European debt crisis, bond yields and stock returns were significantly correlated on almost all scales, however, after the expansion of the crisis in 2012, development of financial assets in both countries started to significantly differ with semblance close to -1 on almost all scales in case of stock indices and with semblance close to -1 on scale of 15-35 days in case of bond yields.