In this study we propose a fractional frequency flexible Fourier form fractionally integrated ADF unit-root test, which combines the fractional integration and nonlinear trend as a form of the Fourier function. We provide the asymptotics of the newly proposed test and investigate its small-sample properties. Moreover, we show the best estimators for both fractional frequency and fractional difference operator for our newly proposed test. Finally, an empirical study demonstrates that not considering the structural break and fractional integration simultaneously in the testing process may lead to misleading results about the stochastic behavior of the Covid-19 pandemic.

Recent developments in time series analysis allow proper modelling of nonlinearities in economic and financial variables. A growing body of research was dedicated to investigation of potential nonlinearities in conditional mean of many economic and financial variables, mainly concentrating in developed economies. However, nonlinearities in financial variables in developing economies have not been fully examined yet. In this article we investigate potential nonlinearity and cyclical behaviour of stock returns in Europe's two largest emerging stock markets, mainly in the Greek and Turkish stock markets. Specifically, we use STAR family models, which allow to model nonlinearities in the conditional mean, for modelling monthly returns on stock exchange indices of the Athens Stock Exchange and Istanbul Stock Exchange. Although we find no nonlinearity in conditional variance, we do find strong evidence in favour of nonlinear adjustment of stock returns. It is found that allowing for nonlinearity in conditional mean results in a superior model and provides good out-of-sample forecasts, which contradicts to efficient market hypothesis.

We test the empirical validity of the PPP proposition under temporary structural breaks and dynamic nonlinear adjustments. Although several testing procedures have recently been proposed in the existing literature to investigate stochastic properties of the series under gradual breaks and nonlinear adjustments, none of these tests are compatible with the PPP proposition. Therefore, we propose new testing procedures that restrict the break to be temporary while simultaneously allowing for asymmetric dynamic nonlinear adjustment towards equilibrium. Using these newly proposed tests, we test stationarity of real exchange rate of 24 OECD countries vis-à-vis USA, and find support in favour of PPP proposition in majority of the countries.