For time series data, certain types of outliers are intrinsically more harmful for parameter estimation and future predictions than others, irrespective of their frequency. In this paper, for the first time, we study the characteristics of such outliers through the lens of the influence functional from robust statistics. In particular, we consider the input time series as a contaminated process, with the recurring outliers generated from an unknown contaminating process. Then we leverage the influence functional to understand the impact of the contaminating process on parameter estimation. The influence functional results in a multi-dimensional vector that measures the sensitivity of the predictive model to the contaminating process, which can be challenging to interpret especially for models with a large number of parameters. To this end, we further propose a comprehensive single-valued metric (the SIF) to measure outlier impacts on future predictions. It provides a quantitative measure regarding the outlier impacts, which can be used in a variety of scenarios, such as the evaluation of outlier detection methods, the creation of more harmful outliers, etc. The empirical results on multiple real data sets demonstrate the effectivenss of the proposed SIF metric.