# Can a linear combination of an I(2) and I(1) variable be I(1)?

I have two variables; log household credit and log real GDP. I use the Augmented Dickey Fuller test following Dolado 1990's testing procedure (https://ideas.repec.org/a/bla/jecsur/v4y1990i3p249-73.html). I find that log household credit is an I(2) process, while log real GDP is I(1). However, when I run the Augmented Dickey Fuller test on the difference; log household credit - log real GDP, I find that this is an I(1) process.

My understanding was that a linear combination of I(2) and I(1) process was always I(2). How can this be possible? What can I do to reconcile this.

I've checked that I am including enough lag terms in the ADF test such that the residuals aren't statistically autocorrelated. Sample size = 170.

Edit: lcrhh = log household credit, lyr = log real gdp, lcryhh = lcrhh - lyr, d prefix is first-differenced, d2 prefix is second-differenced
ur.df(dlcrhh, lags = 3, type = "none") %>% summary()
The value of the test statistic is: -1.5906
Critical values for test statistics at 5pct: tau1: -1.95

ur.df(dlyr, lags = 3, type = "none") %>% summary()
The value of the test statistic is: -3.4411
Critical values for test statistics at 5pct: tau1: -1.95

ur.df(d2lcrhh, lags = 3, type = "none") %>% summary()
The value of the test statistic is: -9.1321
Critical values for test statistics at 5pct: tau1: -1.95

ur.df(dlcryhh, lags = 3, type = "none") %>% summary()
The value of the test statistic is: -2.6539
Critical values for test statistics at 5pct: tau1: -1.95

I've tested including "drift" as well - conclusions do not change.

Using Phillips-Peron test:
ur.pp(dlcrhh, type = "Z-tau", model = "trend") %>% summary()
Value of test-statistic, type: Z-tau is: -5.405
Critical values for Z statistic at 5pct: -3.435589

However, I know that Phillips-Peron test has low power for small sample sizes. Furthermore, I do not believe that log household credit is I(1) from observing its graph. I use the Elliott-Rothenberg-Stock Test to test whether dlcryhh is actually I(1).
ur.ers(lcryhh, type = "DF-GLS", model = "trend" Value of test-statistic is: -1.2599 critical values of DF-GLS at 5pct is -2.93

• Bear in mind that no result of a hypothesis test is definitive and that there is no assurance multiple tests applied to a dataset will yield consistent results. To interpret your results we would need--at a minimum--to be told the p-values. Other details would be welcome, too.
– whuber
Jun 23, 2021 at 14:19
• Thanks, I have added test statistic values which show that log household credit is I(2), log real gdp is I(1) and log household credit - log real gdp is I(1). Jun 23, 2021 at 14:30
• Theoretically, this is impossible. What you found in the data tells you that either your assumptions are wrong or you got a very unlikely result that is due to chance. Jun 23, 2021 at 16:05
• I used Phillips-Perron test and found that log household credit is I(1). How does I decide which test to trust Jun 23, 2021 at 19:18
• Why are you using type='none'? In ADF the none type has the alternative hypothesis of stationary series with mean zero. Your differenced series are approximately growth rates which are not supposed to be zero. So you should use the second type with a constant. Jun 24, 2021 at 0:28