This thesis contains four essays on fractional cointegration and spurious long memory following the introduction and related literature in the first chapter. Chapter 2 provides an analysis of fractional integration and cointegration using the high, the low, and the range stock prices instead of closing prices. We analyze the long memory trends across six Asian stock markets including Korea, Indonesia, Malaysia, Sri Lanka, India, and Pakistan. The empirical analysis provide a univariate analysis which includes the unit root tests and estimation of fractional integration in the highs, the lows, and the ranges. Range, being a linear combination of the nonstationary highs and lows, is a stationary long memory process which specifies the need to model these two extreme values and the range simultaneously in a multivariate fractional cointegration context. The fractional vector error correction model fulfills this specification while considering the short-run and long-run relationships. We also perform a forecast comparison of FVECM with alternative models. The autoregressive fractionally integrated and the Heterogeneous autoregressive models are considered to model the long memory in the ranges. Our results support the use of daily ranges as volatility estimator and FVECM to model the long-run convergence and short-run divergence in the highs and lows at the same time.In chapter 3, we analyze the true long memory or spurious long memory in the range based volatilities of spot exchange rates across 30 currencies against the USD including the developed, the developing and the emerging exchange rates. The persistence of exchange rates is of much interest for the central bank, for policymakers, and to understand the inflation dynamics in an economy. The frequency domain analysis exhibits the spurious long memory in most of the currencies due to some low-frequency contaminations, level shifts, or structural changes. We proceed with the estimation of structural points with an unknown number of breaks. Our results provide a different number of breaks across currencies which may relate to some shocks, economic crisis, and financial policies.Chapter 4 contains a detailed analysis of persistent trends in all share index and ten sectoral indices in an emerging stock market of Pakistan. There is a general consideration regarding the inefficiency of emerging markets compared to the developed markets. Our results show the existence of predictable trends across KSE100 and ten sectors. Moreover, we investigate that the existing trends are true or a result of some level shifts with a semiparametric test. According to the adaptive market hypothesis the long-range dependence is not a constant phenomenon and it varies over time corresponding to the market conditions. We analyze this time-varying long memory with a rolling window technique and observe the fluctuating trends such as persistence, antipersistence, efficiency, and inefficiency at different times.Finally, we analyze the fractional cointegration between the volatilities of the conventional index and Islamic index in chapter 5. Islamic finance attracts the attention of investors and traders regarding its different features such as zero interest rates and profit loss sharing strategies. Some researchers believe that Islamic financial markets can work as a good diversification candidate due to different performance levels during the phases of economic and political shocks. This analysis considers the conventional and Islamic indices across nine Islamic countries: Bahrain, UAE, Oman, Qatar, Malaysia, Indonesia, Egypt, Turkey, and Pakistan. Our results suggest the existence of fractional cointegration and absence of diversification opportunities between the indices in seven out of nine countries in the long-run. This implies that both types of indices follow same trends while there may exist the diversification alternatives in the other two cases.