The past decade witnessed the rapid development of high dimensional statistics in deterministic design. High dimensional time series analysis, due to the time dependency, still faces several theoretical challenges. Among the time series models, the Vector Error Correction Model (VECM) is especially complicated because of the non-stationary components. The classical estimation strategies (e.g. Johansen's approach) fail to provide consistent estimates for dimensions larger than three. Moreover, it is impossible to apply existing statistical methods to determine VECM in high dimensions, i.e. when the dimension is allowed to increase with the number of observations or even larger than that.
This dissertation aims at providing feasible regularized methods, which can determine and estimate high dimensional VECM with robust statistical properties. The detailed analysis is divided into three parts. First I develop new tailored Lasso-type methods to estimate VECM and prove their statistical properties, in a setting where the cointegration rank is fixed but unknown and the dimension is large but not increasing with sample size. Then this m ... mehrethodology is extended to cover also the high dimensional case with moving rank and dimension. For this the estimation strategy must be changed and the statistical analysis requires completely different high-dimensional techniques. Under specific assumptions, I propose the estimation strategy in the ultra-high dimensional case. From the application side, these techniques are highly valuable for appropriately treating complex potentially non-stationary systems not only in economics and finance but also in weather and climate systems. I also illustrate this for a portfolio of Credit Default Swaps in a banking-sovereign network and identify different risk clusters by measuring the interconnectedness over time, which is beyond the scope of previous methodologies. Moreover, I provide a detailed empirical study on a high-frequency portfolio where new high-dimensional time series techniques allow to account for liquidity effects through the Limit Order Book in a very detailed way. With this the new spillover channels in the system can be identified.