Dear Prof. Ibarra,

Thank you very much for your answer!

However, I have some questions regarding PPUROOT that I would appreciate if you would answer.

**1. **I cannot see how the critical values (for the unit root coefficient) can be simulated for e.g. 100 observations, WHEN WE ALLOW FOR A BREAK UNDER H0. With no break under H0 and only a break under H1, then I understand - but not if there is a break under both H0 and H1. If we, under H0, have a structural break of magnitude 1, this will give a different critical value for the unit root coefficient, compared to the critical value in the presence of a break of magnitude 0.5. This would lead to different critical values for every structural break magnitude – break magnitudes which are unknown.

E.g. if one under the null hypothesis (H0) would simulate the following DGP with structural breaks... I do not get it (I use EViews syntax so that we do now need to deal with font problems in any formulas, and this is based on the formulas in the attachment):

SERIES Y=mu_sc+DVTB_ser+DVU_ser+Y(-1)+NRND

where the scalar mu_sc is equal to 1 this gives a drift, the series DVTB_ser is always zero except for at e.g. observation 51 where it is 1, the series DVU_ser is always zero except for observations after observation 50 which follow a @trend. However, if the single 1 is replaced by 0.5 in DVTB_ser, this will lead to a different statistical size when running y.ppuroot(lag=4, model=B)

However, if one would just run the following DGP (under H0): SERIES Y=Y(-1)+NRND then I would understand that one could get only one critical value. The problem is that it says in the instructions that we allow structural breaks under H0 (I understand this under H1, but not also under H0. I cannot see what one puts in a break under the H0, the magnitude of the break under H0 will give decide the value of the critical value.

(See the attachment over the formulas and notations used. It is a photo over this Perron's 1997 method (from Pattersson, 2000). It should not infringe any copyright rules since I am posting my own photo of only a part of a page).

*Notations and formulas used* - Notations and formulas.jpg (1.56 MiB) Viewed 4715 times

In summary, if I change the magnitude of the impulse dummy from e.g. 1 to 0.5 this should change the critical value (the critical value that should give a statistical size of exactly 5%). Therefore, the empirical statistical size would be different from 5% when the value of the impulse dummy in DVTB_ser is changed.

I can see how critical values can be simulated when we do NOT have a structural break under H0 (or a possibly a structural break of known size), but not when having a structural break under H0.

- Or with what data generating process (DGP), and with which parameters, is the critical value simulated??? Please alter the code above so that I can see exactly how the DGP really looks like under the H0.

**2. **Another thing, when checking the critical values at page 362-363 in Perron (1997) is it the “k(t-sig)” row that table one should use for getting the critical values? I get this impression by reading your pdf-documentation. Is “(a) Model 1” min t(alpha-hat) in Perron (1997) related to your Model A in y.ppuroot(lag=4, model=A)? Is “(d) Model 2” min t(alpha-hat) in Perron (1997) related to your Model B in y.ppuroot(lag=4, model=B)? Is “(g) Model 3” min t(alpha-hat) in Perron (1997) related to your Model C in y.ppuroot(lag=4, model=C)?

Here are the critical values from Perron (1997):

*Table1(2)* - Table1.jpg (1.61 MiB) Viewed 4722 times

*Table2(2)* - Table2.jpg (1.64 MiB) Viewed 4722 times

**3. ** Also it seems like your y.ppuroot(lag=4, model=

**B**) corresponds to Patterson's (2000) Model

**C**, and when using your y.ppuroot(lag=4, model=

**C**) this corresponds to Patterson's (2000) Model

**B**. Patterson (2000) is based on Perron (1997). Am I right? See the first attachment in this post with Patterson's formulas.

**4. ** What is the maximum lag that may be chosen by PPUROOT?

Thank you very much for your generous assistance!!!