UC San Diego

Time series data often exhibit high dimensionality, temporal dependence, distributional heterogeneity, and limited labeled observations. These features pose substantial challenges for statistical estimation, prediction, and inference. This dissertation develops statistical methods for dependent data, with particular emphasis on high-dimensional regression, discrete-valued time series, predictive inference, and transfer learning. First, a factor-augmented regression framework is developed for high-dimensional time series. By separating pervasive cross-sectional and temporal dependence from sparse predictive signals, the proposed method improves estimation and variable selection when the observed covariates exhibit strong dependence. Second, statistical inference for high-dimensional binary time series is challenging because the underlying transition structure becomes difficult to estimate as the dimension grows. A generalized binary vector autoregressive model is developed to characterize dynamic dependence among binary-valued processes. Post-selection inference procedures are further established using Gaussian approximation and wild bootstrap methods, providing valid inference in high-dimensional time series settings. Moreover, conformal prediction provides model-free predictive inference under exchangeability or independence assumptions. We develop distributional conformal prediction for dependent observations and establish theoretical coverage guarantees for its application to Markov processes. The proposed framework exploits the underlying temporal dependence while retaining rigorous uncertainty quantification. Finally, estimation can be difficult when only a limited number of observations are available in the target time series, while directly pooling data from different sources may introduce bias due to distributional heterogeneity. We extend a transfer learning framework to high-dimensional time series, allowing information to be integrated from similar and related time series. The resulting methods improve estimation and prediction when target-domain observations are limited, with theoretical guarantees.