In this paper, we present a testing procedure for fractional orders of integration in the context of non-linear terms approximated by Fourier functions. The test statistic has an asymptotic standard normal distribution and several Monte Carlo experiments conducted in the paper show that it performs well in finite samples. Various applications using real life time series, such as US unemployment rates, US GNP and Purchasing Power Parity (PPP) of G7 countries are presented at the end of the paper.

This paper examines the coherence of extreme returns between green bonds and a unique set of green stocks. We use the novel quantile cross-spectral coherence methodology of quantile spectral coherency model, cross-quantilogram correlation approach, windowed time-lagged cross-correlation, and windowed scalogram difference models as estimation techniques. The study period spans from 28 November 2008 to 23 September 2020. Our measure of green stocks comprises the constituents of the MSCI Global Environment Price Index: Alternative Energy, Green Building, Pollution Prevention or Clean Technology while our green bond market is proxied by S&P Green Bond Index. We find the dependency between Green Bonds and green stocks to be weak, and this is high during market downturn periods in the short- to medium-term dynamics. This suggests that Green Bonds do act as a hedge, diversifier, or safe-haven instrument for environment portfolio in the short-term, medium-term and long-term dynamics during bearish market conditions. We conclude that green bonds and green stocks are two distinct asset classes with a distinct risk-return profile despite their common climate-friendly nature.