| The traditional approach to time series modeling is based on the Autoregressive Integrated Moving-Average (ARIMA) model (Box and Jenkins, 1970), which is effective in analyzing time series data that exhibit “short-range dependence” (SRD). SRD is characterized by an autocorrelation function (ACF) that goes to zero at an exponential rate, which implies that the observations separated by distant lags are nearly uncorrelated. However, in recent years it has been shown that the ACF tends to zero very slowly in certain types of time-series data, and hence the observations separated by distant lags exhibit strong correlation. This phenomenon is known as long-range dependence (LRD), which has received considerable attention in a wide range of fields including hydrology, biology, computer networks, environmental studies, economics, and finance. Recently, Mandelbrot (1977), Granger and Joyeux (1980), and Hosking (1981) independently introduced the fractional ARIMA (FARIMA) family of models for LRD data. These models have structure similar to the ARIMA(p, d, q ) models, where p and q are the orders of the autoregressive and moving average components respectively, and d is the difference parameter. The key difference between ARIMA and FARIMA models is that in ARIMA models, the parameter d assumes only integer values (i.e., d = 0, 1, 2,…) whereas in FARIMA models, d can be any real number. For a fractional value of d, the difference operator contains an infinite number of terms which introduce long-range dependence into the model. Hence, the identification and estimation of d are critical issues in the FARIMA class of models. In the last few years, many procedures have been proposed for the estimation of d with various degrees of success. Most of these techniques are based on the explosive behavior of the spectral density in the neighborhood of zero, and hence use only frequencies in a carefully chosen neighborhood around zero. This dissertation presents a new semi-parametric method of estimation via a partial linear model. The model is developed based on the semi-parametric structure of the spectral density which separates out the behavior of the spectral density at harmonic frequencies near zero and away from zero. One of the advantage of this technique is that it allows the model to be fitted in the whole range of frequencies, thereby avoiding the issues related to the “choice” of the neighborhood around zero frequencies. Extensive experiments have been conducted to give comprehensive comparisons with other benchmark estimators. These comparisons show that the estimators perform extremely well in the FARIMA(0,d,0) model. However, the proposed estimator, along with the other estimators, has substantial bias in estimating d in general FARIMA(p, d, q) models. This bias is due to the presence of additional short-range dependence structure in the model. To overcome this difficulty, the new estimator is applied to a recursive procedure that accurately estimates all parameters in the FARIMA(p, d, q) model. In particular, this procedure provides a superior estimate of d with enhanced accuracy. |