A NEW ROBUST M-ESTIMATOR WITH AN APPLICATION TO NON-STATIONARY TIME SERIES FORECASTING

Abstract

M-estimators are used as a robust replacement of the general classical estimators used in the field of statistics. Redescending M-estimators are those estimators which reject the extreme values completely. In the first project, a new redescending M-estimator “Uk’s redescending M-estimator” for robust regression and outlier detection has been presented which provide protection against outliers. Moreover, the  -function of the Uk’s estimator is closer to being linear in the central segment before it redescends. Simulation studies show that Uk’s Redescending M-estimator is more efficient than the other estimators. We also have applied the estimator to the real world data taken from the literature. The newly developed Uk’s Redescending M-estimator provides a general idea to interconnect all the Redescending M-estimators with that of the idea used in Andrews sine function. A couple of which has been solved and the rest are under study for the mathematical solution. Bootstrap distribution of the new redescending estimator has been derived for real data sets. The second project explains the application of the newly developed redescending M- estimator to time series forecasting in the presence of outliers. A comparison is made for the forecasted values using the robust estimation procedure and without using robust approach in the presence of outliers. In case when the response variable (dependent variable) is the function of its own lagged values as an explanatory variable (independent variable) or when the response variable is the function of its own lagged value in addition to other different explanatory variables, then such a model is called time series model. The third project is about exponential smooth transition autoregressive models in time series. Exponential Smooth Transition Autoregressive (ESTAR) models can capture non-linear correction of the deviations from equilibrium conditions which may explain the economic behavior of many variables that appear non-stationary from a linear viewpoint. Many researchers employ the test of Kapetanios et al. (2003) that has a unit root as the null and a stationary nonlinear model as the alternative. However, this test statistics is based on the assumption of normally distributed errors in the DGP. Cook (2008) has analyzed the size of the nonlinear unit root of this test in the presence of heavy-tailed innovation process and obtained the critical values for both finite variance and infinite variance cases. However, the test statistics of Cook are oversized. Pavlidis et al. (2010) find that using conventional tests is dangerous though the best performance among these is a HCCME used by Mackinnon and white (1985). The over sizing for LM tests can be reduced by employing fixed design wild bootstrap remedies which provide a valuable alternative to the conventional tests. The size of the Kapetanios et al. test statistic employing hetroscedastic consistent covariance matrices has been derived and the results are reported for various sample sizes in which size distortion is reduced. The properties for estimates of ESTAR models have been investigated when errors are assumed non-normal. We compare the results obtained through the fitting of nonlinear least square with that of the quantile regression fitting in the presence of outliers and the error distribution was considered to be from t-distribution for various sample sizes.