Derivation of Imputation Estimators for ARMA Models with GARCH Innovations

The paper investigated the problem of restoring missing values in time series data analysis. The aim of this study was to advance the imputation of missing values for some autoregressive moving average models (ARMA) with generalized autoregressive conditioned heteroscedastic (GARCH) models. In this work, the novel imputation estimators for ARMA (1, 1) ＋ GARCH (1, 1) and ARMA (2, 2) ＋ GARCH (2, 2) were derived. The study utilized the method of optimal interpolation, whereby the innovation terms of the involved processes were minimized in the sense of dispersion. The study tested the consistency of the estimators using simulated data. A sample of a thousand (1000) observations was generated using R software following the proposed models. A hundred (100) positions of missing values were created at random within the data generated. Besides, the study carried out a comparison between the derived estimators and the celebrated machine learning Artificial Neural Networks (ANN), K-Nearness Neighbors (KNN) and the Kalman filters techniques. The imputation performance was carried out using the following metrics; Mean Error (ME), Root Mean Squared Error (RMSE), Mean Absolute Error (MAE), Mean Percentage Error (MPE), Mean Absolute Percentage Error (MAPE). The study found that the derived novel imputation estimator for ARMA (2, 2) ＋GARCH (2, 2) process was superior to the imputation estimator of ARMA (1, 1) ＋ GARCH (1, 1) process. The derived estimators competed well compared to modern missing values imputation techniques. This paper gave a clear comparison that the ANN technique was the best followed by optimal interpolation technique while the Kalman technique was the last according to imputation performance. The study recommends that the derived estimators be utilized to input missing values for time series with GARCH innovations. The rational for this study is to contribute to the field of missing values imputation for non-linear time series modeling.