Change points in high dimensional settings∗

While there is considerable work on change point analysis in univariate time series, more and more data being collected comes from high dimensional multi-variate settings. This paper investigates change point detection procedures using projections and develops asymptotic theory for how full panel (multivariate) tests compare with both oracle and random projections. This is done by considering an analogous concept to asymptotic relative efficiency termed high dimensional efficiency. This provides the rate at which the change can get smaller with dimen-sion while still being detectable. The effect of misspecification of the covariance on the power of the tests is investigated, because in many high dimensional sit-uations estimation of the full dependency (covariance) between the multivariate observations in the panel is often either computationally or even theoretically infeasible. It is shown that if information concerning the direction of change is available, then projecting in this direction is always advantageous over the use of a panel statistic, in terms of size and power, particularly when the covariance is misspecified. Even if the change is not known, the projection method achieves a better power as long as the difference between the true change and the direction of the projection is small. The features of the tests are quantified in theory and simulations indicate that these results are indicative of small sample behaviour