Andreas Noack:Some Mathematical and Computational Results for Vector Error Correction Models

The first chapter analyses the nesting structure of the I(2) cointegrated vector autoregressive models. I(2) models with different number of cointegrating relations are shown to be nested and the implications for rank determination are discussed. A surprising result is that even though the I(2) models are formulated as submodels of I(1) models of same rank, some I(1) models are in fact submodels of I(2) models of higher rank.

Recently, there has been some interest in so called near I(2) processes, but it has not yet been investigated how inference on cointegration vectors is affected when the process is close to the I(2) boundary. In the second chapter, definitions of near I(2) processes are discussed which proves to be a more complicated problem than defining near I(1) processes and inference on cointegration parameters is analysed by simulations.

In the third chapter, my coauthor and I provide a fast algorithm for calculating the fractional difference of a time series. In standard implementations, the calculation speed (number of arithmetic operations) is of order $T^2$, where $T$ is the length of the time series. Our algorithm allows calculation speed of order $T \log T$. For moderate and large sample sizes, the difference in computation time is substantial.In the last chapter a similar method is used for fast simulation of fractionally cointegrated processes.