DIMENSION REDUCTION IN TIME SERIES UNDER THE PRESENCE OF CONDITIONAL HETEROSCEDASTICITY

A fundamental task of time series analysis is inference about the conditional distribution or moments of the current value given its past realizations. This not only helps unveil the underlying dynamics of a series, but also enables meaningful forecast of future values.In this dissertation, we consider a univariate discrete-time series {xt;t≥1}, where the conditional mean of xt is assumed to be an unknown function of linear combinations of past observations and the conditional variance of xt is also assumed to be an unknown function of linear combinations of past squared residuals. These two linear combinations are such that they contain all the necessary information about xt that is available from the conditional mean and conditional variance, respectively.We have developed an iterative estimation approach for dimension reduction in time series, which, in the first step, uses the Nadaraya-Watson estimator of the unknown (con-ditional) mean function and minimizes a sum of squared error to estimate the parameter associated with the linear combinations of past observations. This initial estimator is then used to form the observed residuals. In the second-step, our estimation approach once again uses a Nadaraya-Watson estimator of the unknown (conditional) variance function and minimizes a sum of squares to estimate the parameter associated with the linear combination of past observed residuals. In the third step, a revised estimate of the parameter associated

with the linear combinations of past observations is obtained by minimizing an appropriately weighted sum of squares. This iterative process of estimation is then repeated until convergence of the estimates.We have theoretically shown that the iterative estimators obtained in the above manner are consistent as sample size tends to infinity. Our theoretical results are validated through comprehensive simulation studies. Furthermore, we have applied the iterative estimation procedure to the task of forecasting the BRL/USD Exchange Rate. For this time series data, we have demonstrated that the estimated linear combinations can be used to generate competitive forecasts of the series as compared to those generated using an AR-ARCH model.Finally, to overcome some of the computational challenges, we have developed a new parametrization technique in order to guarantee that the numerical optimization is more efficient. The advantages of this new parametrization are that: 1) it ensures that the constraints imposed are fully met, 2) it makes convergence more frequent, 3) it reduces the computational time, and 4) it makes the optimization feasible to a wider range of algorithms and software.