I've been quite confused by the various unit root testing strategies recommended in the literature, so I was hoping others may have some advice on the best way to proceed using ADF and KPSS tests.
Pfaff (2008) describes the following procedure using the ur.df()
function to determine whether a series (i) is stationary around a zero mean, (ii) is stationary around a non-zero mean, (iii) is stationary around a linear trend, (iv) has a unit root with a zero drift, (v) has a unit root with a non-zero drift.
Estimate $\Delta y_{t}=\beta_{1}+\beta_{2}t+\pi y_{t-1}+\sum_{j=1}^{k}\gamma_{j}\Delta y_{t-j}+u_{1t}$ (option
type = "trend"
)1.1. Check tau3: $H_0: \pi = 0$ (t-test for presence of a unit root), reject if tau3 < critical value at 5pct and conclude there is no unit root, otherwise go to step 1.2.
1.2. Check phi3: $H_0: (\beta_{1}, \beta_{2}, \pi) = (\beta_{1}, 0, 0)$ (F-test for absence of trend), reject $H_0$ if phi3 > critical value at 5pct and conclude that there is a unit root, otherwise go to step 2.
Estimate $\Delta y_{t}=\beta_{1}+\pi y_{t-1}+\sum_{j=1}^{k}\gamma_{j}\Delta y_{t-j}+u_{2t}$ (option
type = "drift"
)2.1. Check tau2: $H_0: \pi = 0$ (t-test for presence of a unit root), reject if tau2 < critical value at 5pct level and conclude that there is no unit root, otherwise proceed to step 2.2.
2.2. Check phi1: $H_0 (\beta_{1}, \pi) = (0, 0)$ (F-test for absence of constant), reject $H_0$ if phi1 > critical value at 5pct and conclude that there is a unit root, otherwise proceed to step 3.
Estimate $\Delta y_{t}=\pi y_{t-1}+\sum_{j=1}^{k}\gamma_{j}\Delta y_{t-j}+u_{3t}$ (option
type = "none"
)3.1. Check tau1: $H_0: \pi = 0$ (t-test for presence of a unit root), reject $H_0$ if tau1 < critical value at 5pct level and conclude that there is no unit root, otherwise conclude that there is a unit root
If a unit root is found, take first differences and repeat procedure to find order of integration.
a. While I understand the outlined procedure, I am not quite certain how to relate the five different types of series (i)-(v) above to the test results. I guess failure to reject $H_0$ in step 3.1. is equivalent to (iv), whereas rejection of $H_0$ in step 2.2. would be (v), but what about others?
b. It is often recommended to test again whether $\pi = 0$ under a normal distribution if $H_0$ is rejected at steps 1.2. or 2.2. How is this done and why would this be necessary?
c. Could the KPSS test, which has an opposite $H_0$ (series is stationary), be used to distinguish between the five different types of series (i)-(v)? How should contradictions with the ADF be handled?