^{1}

^{2}

^{3}

The recent bouts of crises in the Europe and the US have spurred increasing interest on the co-movement of output and stock market across the European countries. The evidence of such co-movements is rooted mainly from the spillover of uncertainty among the European countries. Hence, using a new uncertainty index from Baker et al . [1] , we investigate both the time-varying as well as frequency based co-movement of uncertainty in the selected European countries. Our results suggest both time and frequency varying co-mov ement of the uncertainty indices. I n particular, these co-movements are found to be more pronounced during the crises periods.

The vast body of literature in economics and finance focuses the co-movement of output, commodity prices and stock prices [^{1} Since our study period is characterized by the periods of crisis and tranquility, the time varying co-movement is of particular interest, because periods of uncertainty and tranquility could indicate different nature of co-movements. The motivation for studying the uncertainty co-movement at different frequencies for this study arises out of the fact that the key drivers of uncertainty like fiscal, monetary and real factors operate at different frequencies.^{2} For example, output uncertainty may transmit to other countries at business, Juglar or Kuznets cycles. Similarly, inflation uncertainty could flow to other countries because shocks to the international prices of the imported goods are not adjusted by exchange rates. These shocks flow to domestic consumer prices over the short to medium frequencies. Yet, the stock market uncertainty may be transmitted at different trading frequencies to other countries by the heterogeneous behavior of investors. Since output, inflation and stock market uncertainty form important components of overall uncertainty, it is important to know how uncertainty is synchronized between countries both over time as well as over frequencies. Thus uncertainty co-movement in our case may be defined in terms of co-movement of uncertainty pertaining to the socio-economic variables. We investigate both of these issues using the wavelet based measure of cohesion for uncertainty indices of selected European countries.

The innovation of this paper has been done in two ways. First, we test the time varying co-movement of uncertainty in the major pair wise European countries. Second, we investigate this co-movement at different frequencies with defined time dynamics. Last but not least major contribution of this study lies in analyzing, for the first time, the co-movement of uncertainty in the major European countries using new index of uncertainty. Our results are interesting and unravel both time varying as well as frequency varying co-movement of uncertainty which could not have been obtained by following the time domain or frequency domain approach only.

The organization of the paper follows as. Section 2, the methodology describing the wavelet based measure of co-movement is described. In Section 3, results are described and discussed. Section 4 draws the conclusion.

The technique of wavelet transformation enables us to understand the evolution of a signal or time series across time as well as over frequency. It adjusts the time resolution to the frequency; narrows down the window width on high frequencies and widens while dealing with low frequencies. It makes use of local base functions that can be translated and stretched into both time and frequency. Moreover, the Wavelets are characterized by finite energy such that they grow and die out within a period. Mathematically, the Wavelets are defined as

where,

are elementary functions obtained by decomposition of a time series through wavelet transform and are derived from a time-localized mother wavelet

The convolution of continuous wavelet transform (CWT) of a time

where, * denotes the complex conjugate.

Moreover, several interesting quantities can be captured within wavelet domain. The measure of wavelet power spectrum that captures the relative contribution at each time and at each scale to the time series’ variance is defined as

Following Croux et al. [

where,

The striking feature of cross-wavelet spectrum lies with its ability in providing information about the co- movement both at the frequency as well as over time. Moreover, assessing the contour plot of the wavelet cross spectrum it can be identified the time-frequency regions over which the two series commove as well can be assessed the features of the time and frequency variation of the co-movement. The suggested wavelet-based measure, hence, enriches the analysis of co-movement between a set of variables.

We assess the co-movement of uncertainty among the major European countries by using the uncertainty index originally constructed by Baker [^{3} The vertical axis measures the frequency and horizontal axis measures frequency interpreted in terms of time (years). The colour scale measures cohesion and the ranges of from −1 to +1. The deep blue colour indicates the perfect negative cohesion and deep red colour indicates the perfect positive cohesion. The identification of time and frequency varying coherence is done by inspecting the contour plot. One can therefore identify both frequency bands (in the vertical axis) and time intervals (in the horizontal axis) where the uncertainty indices are synchronized.

The results based on his analysis show several interesting findings. It is observed that there are higher co- movements at intermediate and lower frequencies for all the studied pair of countries (France and Germany being exceptional to it). The co-movement of uncertainty is very high at business cycle frequencies (2 - 8 years) for all the country pairs. A note worthy finding is that the uncertainty indices are highly synchronized during the subprime mortgage crisis of 2007-08 for all the country pairs. This synchronization extends from mid 2007 (the time of origin of subprime crisis) till 2012 at lowest frequencies for all the country pairs. For some country pairs like France-Germany, France-Italy France-UK and Italy-UK, the synchronization of uncertainty extends to lower higher frequencies also. Another interesting point is that co-movement is also very high during 2001-02 for almost all the country pairs. This co-movement ranges from higher frequencies to intermediate frequencies and could be probably because of the dotcom bubble burst in 2001-02. These findings indicate that uncertainty could be synchronized at different frequencies. Moreover, it unveils the fact that during the periods of crises there are more chances of uncertainty spillovers.

We have analyzed the uncertainty spillovers over time and across frequencies in selected European countries. Using the new uncertainty index from Baker et al. [