Weighted Least Squares Estimation for AR(1) Model With Incomplete Data

Abstract

Time series forecasting is the main objective in many life applications such as weather prediction, natural phenomena analysis, financial or economic analysis, etc. In real-life data analysis, missing data can be considered as a feature that the researcher faces because of human error, technical damage, or catastrophic natural phenomena, etc. When one or more observations are missing, it might be urgent to estimate the model as well as to estimate the missing values which lead to a better understanding of the data, and more accurate prediction. Different time series require different effective techniques to have better estimates for those missing values. Traditionally, the missing values are simply replaced by mean and mode imputation, deleted or handled using other methods, which are not convenient enough to address missing values, as those methods can cause bias. One of the most popular models used in estimating time-series data is autoregressive models. Autoregressive models forecast the future values in terms of the previous ones. The first-order autoregressive AR (1) model is the one which the current value is based on the immediately preceding value, then estimating parameters of AR (1) with missing observations is an urgent topic in time series analysis. Many approaches have been developed to address the estimation problems in time series such as ordinary least square (OLS), Yule Walker (YW). Therefore, a suggested method will be introduced to estimate the parameter of the model by using weighted least squares. The properties of the (WLS) estimator are investigated. Moreover, a comparison between those methods using AR (1) model with missing observations is conducted through a Monte Carlo simulation at various sample sizes and different proportions of missing observations, this comparison is conducted in terms of mean square error (MSE) and mean absolute error (MAE). The results of the simulation study state that (WLS) estimator can be considered as the preferable method of estimation. Also, time series real data with missing observations were estimated.