CONSISTENCY OF THE DRIFT PARAMETERS ESTIMATES IN THE FRACTIONAL BROWNIAN DIFFUSION MODEL AND ESTIMATION OF THE HURST PARAMETER BY MAXIMUM LIKELIHOOD METHOD

In this paper, we study the problem of estimating the unknow parameters in a long memory process based on the maximum likelihood method. We consider again a diffusion model involving fractional Brownian motion. Our goal is to study the consistency of the drift parameter estimates depending on the form of the model.

A Random walk down the Dhaka Stock Exchange lane

Market efficiency has always been a matter of keen interest to the researchers of finance. Since the advancement of this concept, researchers are consistently investigating the market efficiency of different financial markets. Bangladesh, being one of the emerging economies, has also attracted the attention of many researchers. The researchers have investigated the realities regarding the market efficiency of both the stock exchanges of the country. Most of their investigations reveal that the Dhaka Stock Exchange (DSE) and the Chittagong Stock Exchange (CSE) are inefficient. This research, however, did not stop at revisiting market efficiency alone. Whether the return series follows a long-memory process, has also been tested. Besides, non-parametric tests have also been conducted to confirm the results of the parametric tests and vice versa. It generated a more reliable estimate of market efficiency for the period under study. Results of the Autoregressive Fractionally Integrated Moving Average (ARFIMA) model confirm that the return series does not follow a long memory process, and any shock in the system will eventually vanish. The findings of other tests (the run test, the Augmented Dickey-Fuller (ADF) test, the Kwiatkowski–Phillips–Schmidt–Shin (KPSS) test, and the Kolmogorov-Smirnov (K-S) test) suggest that the return series of the DSE are time-series stationary, non-normal, and do not follow a random walk. Given these results, we must echo the prior researchers to conclude that the stock market of Bangladesh is not efficient for the period of 2015 to 2020. These findings add new knowledge to the existing knowledge pool about market efficiency and long memory of the stock market of Bangladesh.

Long-memory modeling and forecasting: evidence from the U.S. historical series of inflation

We report the results of applying several long-memory models to the historical monthly U.S. inflation rate series and analyze their out-of-sample forecasting performance over different horizons. We find that the time-varying approach to estimating inflation persistence outperforms the models that assume a constant long-memory process. In addition, we examine the link between inflation persistence and exchange rate regimes. Our results support the hypothesis that floating exchange rates associate with increased inflation persistence. This finding, however, is less pronounced during the era of the Great Moderation and the Federal Reserve System’s commitment to inflation targeting.

Cointegrated Dynamics for a Generalized Long Memory Process: Application to Interest Rates

Recent developments in econometric methods enable estimation and testing of general long memory processes, which include the general Gegenbauer process. This paper considers the error correction model for a vector general long memory process, which encompasses the vector autoregressive fractionally integrated moving average and general Gegenbauer processes. We modify the tests for unit roots and cointegration, based on the concept of heterogeneous autoregression. The Monte Carlo simulations show that the finite sample properties of the modified tests for unit roots are satisfactory, while the conventional tests suffer from size distortion. The experiments also indicate that the modified tests for cointegration improve the problem of finding too many cointegration relationships which arises for fractionally integrated series. Empirical results for interest rates series for the USA and Australia indicate that: (1) the modified unit root test detected unit roots for all series; (2) after differencing, all series favour the general Gegenbauer (GG) process; (3) the modified test for cointegration found only two cointegrating vectors; and (4) the zero interest rate policy in the USA had no effect on the cointegrating vectors for the two countries.

Long Memory and Parity Reversion in Real Exchange Rate

This paper examines the post Bretton Woods experience of the Malaysian Ringgit. In this period, Malaysia moved from a managed to a floating exchange rate environment.We examine persistence in real exchange rates by estimating fractionally integrated ARIMA models and find evidence of long memory, which induces persistence though this long memory need not be associated with a unit root. The results show that three out of four exchange rates being examined display mean reversion. The long memory process re-establishes the Purchasing Power Parity as a meaningful concept of long-run equilibrium relation between the exchange rate and relative prices.

Fractional relaxation noises, motions and the fractional energy balance equation

We consider the statistical properties of solutions of the stochastic fractional relaxation equation that has been proposed as a model for the earth's energy balance. In this equation, the (scaling) fractional derivative term models energy storage processes that occur over a wide range of space and time scales. Up until now, stochastic fractional relaxation processes have only been considered with Riemann-Liouville fractional derivatives in the context of random walk processes where it yields highly nonstationary behaviour. For our purposes we require the stationary processes that are the solutions of the Weyl fractional relaxation equations whose domain is −∞ to t rather than 0 to t. We develop a framework for handling fractional equations driven by white noise forcings. To avoid divergences, we follow the approach used in fractional Brownian motion (fBm). The resulting fractional relaxation motions (fRm) and fractional relaxation noises (fRn) generalize the more familiar fBm and fGn (fractional Gaussian noise). We analytically determine both the small and large scale limits and show extensive analytic and numerical results on the autocorrelation functions, Haar fluctuations and spectra. We display sample realizations. Finally, we discuss the prediction of fRn, fRm which – due to long memories is a past value problem, not an initial value problem. We develop an analytic formula for the fRn forecast skill and compare it to fGn. Although the large scale limit is an (unpredictable) white noise that is attained in a slow power law manner, when the temporal resolution of the series is small compared to the relaxation time, fRn can mimick a long memory process with a wide range of exponents ranging from fGn to fBm and beyond. We discuss the implications for monthly, seasonal, annual forecasts of the earth's temperature.

Fractional Calculus as a Simple Tool for Modeling and Analysis of Long Memory Process in Industry

This paper deals with the application of the fractional calculus as a tool for mathematical modeling and analysis of real processes, so called fractional-order processes. It is well-known that most real industrial processes are fractional-order ones. The main purpose of the article is to demonstrate a simple and effective method for the treatment of the output of fractional processes in the form of time series. The proposed method is based on fractional-order differentiation/integration using the Grünwald–Letnikov definition of the fractional-order operators. With this simple approach, we observe important properties in the time series and make decisions in real process control. Finally, an illustrative example for a real data set from a steelmaking process is presented.