Least squares estimation for the high-order uncertain moving average model with application to carbon dioxide emissions

Abstract

Uncertain time series analysis aims to explore how the current observation is affected by the disturbance terms and past imprecise observations characterized as uncertain variables. For the case that the current observation is affected by a single past disturbance term, the 1-order uncertain moving average (UMA) model has been tentatively explored. While for the situation that the current observation is affected by multiple past disturbance terms, this paper initiates a high-order UMA model to more accurately describe this relationship. By transforming the high-order UMA model into an uncertain autoregressive model via a backward shift operator, the unknown parameters are calculated through the least squares method. Then, a tth residual is defined to describe the properties of disturbance terms. Furthermore, the forecast value and the confidence interval are derived from the fitted model. Finally, two examples are presented to demonstrate the effectiveness of this method.