Null hypothesis on intercept in unit root test

[karsten_kohler]

Hello there!

I would like to know what the null hypothesis on the intercept in an Augmented Dickey Fuller (ADF) unit root test in EViews is. When running an ADF test in EViews, there are three options

a) no intercept and no trend
b) intercept
c) intercept and trend.

If I'm not mistaken, Hamilton (1994, Time Series Analysis, chap. 17), however, distinguishes four cases:

1) y(t) = p.y(t-1) + e(t)
HO: p = 1
HA: |p| < 1

2) y(t) = a + p.y(t-1) + e(t)
HO: p = 1, a = 0
HA: |p| < 1, a ≠ 0

3) y(t) = a + p.y(t-1) + e(t)
HO: p = 1, a ≠ 0
HA: |p| < 1, a = 0

4) y(t) = a + p.y(t-1) + d.t + e(t)
HO: p = 1, d = 0
HA: p < 1,d ≠ 0

It is thus not clear to me, what the null hypothesis on the intercept (a in my notation) is when I select option b) in EViews. Does it correspond to Hamilton's case 2) a=0, or 3) a ≠ 0 ?

I couldn't find the answer in the Eviews Guide. Any help is highly appreciated.

[KrilleJ]

when perorming the ADF tests.In the EViews test procedures there are no restrictions placed on the deterministic components of the test regression.

Under a) a pure AR model (without deterministic trend components) is specified: y_t=\rho*y_{t-1}+\eps_t. The test performed is on \rho. Under b) and c), deterministic components are added to the test regression (only intercept or intercept and time trend), but the test is still on \rho. Hence, the null hypothesis involves only \rho. You can of course construct joint tests on \rho and the deterministic components (by setting up an Equation object), but the critical values will not be the (A)DF critical values.

[karsten_kohler]

Thank you for your answer.

The problem is that according to Hamilton, the distribution of the test statistic for H0: p = 1 depends on the assumption made about the intercept.

[KrilleJ]

That is absolutely true. The critical values depends both on what deterministic components you include and whether you test a simple or joint hypothesis. Tabulations exist for all of them. As long as you know what hypothesis you test, you can find the right critical values. The built-in EViews procedures only test the simple hypotheses on the AR-term.