Sieve Bootstrap for Fixed-b Phillips–Perron Unit Root Test

This paper extends the fixed-b Phillips–Perron unit root test, namely PP(fb), by using a sieve bootstrap method to deal with serial-correlated errors, especially negative moving average errors. We derive the asymptotic distribution of the proposed \(\mathrm {PP^b(fb)}\) ${\mathrm{PP}}^{\mathrm{b}}(\mathrm{fb})$ test statistics. Our simulation results show that the \(\mathrm {PP^b(fb)}\) ${\mathrm{PP}}^{\mathrm{b}}(\mathrm{fb})$ test substantially improves the size performance without losing power when the error innovations have negative autoregressive or moving average components. Applying the \(\mathrm {PP^b(fb)}\) ${\mathrm{PP}}^{\mathrm{b}}(\mathrm{fb})$ test to both the inflation rates in the U.S. and the Chinese stock market index, we find that the inflation follows a random walk process, just as the Chinese stock market does.