Simultaneous Bootstrap Inference for Autocovariances and Autocorrelations under strict stationarity

This thesis analyses simultaneous bootstrap inference for autocovariances and autocorrelations and examines structural identifiability of vector autoregressive models with singular innovation covariance matrix. The first part discusses the theory of two bootstrap methods that allow to conduct simultaneous inference for autocovariances and autocorrelations. Natural statistics in this case are maximum deviations of sample autocovariances and autocorrelations from their population counterparts. In contrast to classical multivariate asymptotic theory, we allow for the number of lags to increase with sample size. We obtain results for a general class of physically dependent strictly stationary time series. Such a class includes many well-known linear and non-linear time series. We show that the autoregressive (AR) sieve bootstrap, which is a classical time domain bootstrap method that is simple to apply to data, is asymptotically valid under the important condition that the number of lags grows with the number of observations to infinity. We also show that the AR-sieve bootstrap speed of convergence is of polynomial order if the underlying process is a linear time series. The convergence rate to the asymptotic distribution, which is of Gumbel-type, is at best of logarithmic order. In a next step we show that a dependent wild-type bootstrap is asymptotically valid for our maximum deviation statistics at a polynomial convergence rate. This result holds for physically dependent strictly stationary processes and it does not need the condition that the number of lags grows polynomially with the sample size. The number of lags is still allowed to grow polynomially. This makes it a valuable result for simultaneous statistical inference. In order to obtain such a result we apply modern tools of Gaussian approximation for sums of high-dimensional vectors. Moreover we give potential applications including spectral density estimation and investigate by simulations the finite sample properties of both bootstrap proposals. In our finite sample study we observe that the AR-sieve and the dependent wild-type bootstrap work better than the asymptotic approximation. However, especially for non-linear time series undercoverage of simultaneous confidence bands can be observed for the AR-sieve. The dependent wild-type bootstrap which also captures higher order properties of the underlying time series performs better in such cases. In the second part of the thesis, we generalize well-known results on structural identifiability of vector autoregressive (VAR) models to the case where the innovation covariance matrix is of reduced rank. Singular structural VAR models appear, for example, as solutions of rational expectation models where the number of shocks is usually smaller than the number of endogenous variables, and as an essential building block in dynamic factor models. We show that order conditions for identifiability are misleading in the singular case and we provide a rank condition for identifiability of the noise parameters. Since the Yule-Walker equations may have multiple solutions, we analyse the effect of restricting system parameters on over- and underidentification in detail and provide easily verifiable conditions.