Bootstrap and fast double bootstrap tests of cointegration rank with financial time series

The likelihood ratio test of cointegration rank is the most widely used test for cointegration. Many studies have shown by simulation that the small sample distribution is not well approximated by the limiting distribution. We suggest using the bootstrap to generate small sample critical values instead of correcting the test statistics. The idea of bootstrapping the trace test of cointegration rank is of course not new, but it has mainly been studied in the simple context of first order vector autoregressive models with cointegrating rank equal to zero. For higher order VAR processes with cointegrating rank greater than zero little is known about the size and power properties of different methods. The main contribution of this article is to investigate the behaviour of different bootstrap approaches under the null of cointegration rank zero and fixed alternatives of cointegration rank greater than zero, and to compare their relative performance with small sample corrections based on correcting the test statistics. In particular, it is shown that in most cases the bootstrap is superior to small sample corrections. Two applications to financial time series are considered. The first application is to US interest rates and the second application is to international stock prices.