The limiting density of unit root test statistics: A unifying technique (Q2703259)

The authors consider the AR(1) process with a constant term NEWLINE\[NEWLINE X_k-\mu=\rho(X_{k-1}-\mu)+\varepsilon_k,\;k=1,\dots,n, NEWLINE\]NEWLINE where \(\varepsilon_k\) are i.i.d. \(N(0,\sigma^2)\), \(\rho\) and \(\sigma\) are unknown parameters, \(\mu\) can be known or unknown. A least squares estimator \(\hat\rho_n\) for \(\rho\) is considered. The limit PDF of \(n(\hat\rho_n-1)\) as \(n\to\infty\) under the null hypothesis \(\rho=1\) and under local alternatives \(\rho=1-\vartheta/n\) is evaluated in terms of the inverse Fourier transform of an infinite series in \(R^2\). The case of dependent errors is also discussed briefly.