Missing data is a ubiquitous problem in biomedical and social science research and almost all areas involving data collection. Data imputation is commonly recommended to make the best use of valuable data at hand and gain more efficient estimates of quantities of interest. Mobile technology (e.g., mobile phones and wearable devices) allows an unprecedented way to closely monitor individuals' behavior, social interaction, symptoms, and other health conditions in real-time, holding great potential for scientific discoveries in psychiatric research and advancements in the treatment of severe mental illness. Continuous data collection using mobile technology gives rise to a new type of data -- entangled multivariate time series of outcome, exposure, and covariates, and poses new challenges in missing data imputation for valid causal inference. Most existing imputation methods are either designed for longitudinal data with limited follow-up times or for stationary time series, which may not be suitable in the field of psychiatry when mental health symptoms display dramatic changes over time or patients experience shifts in treatment regime over their course of recovery. We propose a novel multiple imputation method based on the state space model (SSMmp) to address missing data in multivariate time series that are potentially non-stationary. We evaluate its theoretical properties and performance in extensive simulations of both stationary and non-stationary time series under different missing mechanisms, showing its advantages over other commonly used strategies for missing data. We apply the SSMmp method in the analysis of a multi-year observational smartphone study of bipolar patients to evaluate the association between social network size and psychiatric symptoms adjusting for confounding.