Timeline for Unit root near unit circle or near 1?
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| Jul 30, 2015 at 4:36 | comment | added | mlofton | Hi: Note that differencing and looking at acf and pacf is the "box-jenkins" philosophy for how to build arma models ( which need require stationarity so possibly differencing ). Unit root testing is somewhat more theoretical and used by an econometrician who wants to say A) test for trend stationarity versus difference stationariry or B) test for whether two series are cointegrated. So, the whole concept of stationarity and how it applies, really depends on what the particular question-project is. All the best. | |
| Jul 23, 2015 at 18:36 | comment | added | mlofton | oh, note that the question of "needing to difference" can also be inferred from the acf. if the acf doesn't die down after a reasonable # of lags, this is a strong indication that you need to difference. using hypothesis testing ( i.e: testing for a unit root ) to determine whether to difference may be a little too "formal" of an approach and could lead to problems. box and jenkins recommend using the acf. the acf of an I(1) series will be pretty obvious. | |
| Jul 23, 2015 at 18:32 | comment | added | mlofton | yes, there'a big controversy in econometrics about trend stationary versus difference stationary so the unit root testing is done often to "prove" difference stationarity. stationarity testing ( using the ADF ) is also a big deal in cointegration because, if the residuals are stationary after regressing two I(1) series on each other, then this shows that the two series are cointegrated. hamilton's text is really great for all of this. another good text is hayashi's text.. mckinnon and davidson also. I don't have greene or judge et al but they are supposedly good also. good luck. | |
| Jul 22, 2015 at 18:56 | comment | added | StatSmartWannaB | OK, thanks for carifying. At this point, I just wanted to clear up a conceptual confusion on my part as to what the concern about unit roots really was -- indeed, what unit roots referred to -- as I read up on "best practice" for model development. So far, it seems that you should test for unit root to determine whether to difference, but even after you resolve that, you still need stationarity testing. CAVEAT on that last speculation: I'm still have much more reading ahead. | |
| Jul 21, 2015 at 15:42 | comment | added | mlofton | my mistake. there wasn't an earlier comment about this. That was a different thread. My point is that the ADF assumes a unit root as the null hypothesis and rejects if the estimate is away from 1.0 in either direction. There are other tests ( that I'm much less familar with ) that assume stationarity ( estimate is less than 1.0 ) for the null. then they only reject the null if there's evidence of a unit root. I'm not clear on how they go about doing this but you may want to check out KPSS and some of the other tests that assume stationarity as the null. The ADF may not really fit your case. | |
| Jul 21, 2015 at 15:35 | comment | added | mlofton | @StatSmartWannaB. yes, unit root theory focuses on the value of 1. but note that when you do the augmented dickey fuller test for a unit root, rejection ( of the null that there's a unit root ) is two-sided in the sense that it's a two sided test. So, the unit root is a specific kind of non-stationarity but the region of non-stationarity is larger. This may actually be why other unit root tests where the null is that there is not a unit root are preferred as the earlier comment noted. I'm glad were on the same page and good luck. | |
| Jul 21, 2015 at 14:07 | comment | added | StatSmartWannaB | @mlofton: I think we're in agreement there. I modified the question yesterday to reflect exactly that, though referencing Cochrane's electronic textbook instead of Hamilton. I also perused Hamilton's Chapter 1 & 2, which is good background. However, my motivation for posting this question was to determine whether the body of unit root theory, methods, and tests focus only on real roots of value 1.0. As per the expansion of my question, I strongly suspect that it does. | |
| Jul 21, 2015 at 3:44 | comment | added | mlofton | @StatSmartWannaB I looked at the chapter that whuber was referring to and that is covering the non-stationary case where the series is a random walk. But that's not the only way that a series can be non-stationary. The root of the characteristic equation being on or outside the unit circle ( it depends on which characteristic equation you're referring to ) is the general case . I would read chapters 1 and 2 of Hamilton for the general case. There are a lot of gory details in those chapters but my point is that a random walk-unit root is a specific case of non-stationarity. | |
| Jul 20, 2015 at 18:27 | comment | added | StatSmartWannaB | @mlofton: I looked at Hamilton just now (and I might have done so in my surfing before posting). His treatment is consistent the prevailing that a unit root is a root of 1.0 (real number). It is not inconsistent with Barlett's polynomial characteristics of stationarity and causality above. | |
| Jul 20, 2015 at 18:26 | comment | added | StatSmartWannaB | @whuber: On slide 6, Barlett distinguishes between simply avoiding the unit circle versus being outside the circle: Stationarity = Roots not on the unit circle, while causality = roots outside of the unit circle. So a stationary process might not be causal. For such processes the roots of the AR polynomial are inside the unit circle. I have to remind myself that this means: (i) the statistical parameters of the noise source do not have a time dependence, and (ii) sample values of the time series are not expressible in terms of past values of the noise source. | |
| Jul 20, 2015 at 18:25 | comment | added | StatSmartWannaB | @whuber: Agreed that clarity is needed in the form of the polynomial. Being an electrical engineer by training, I'm use to z-transforms. In contrast, time series is often used for econometrics (and environmental/health sciences), where the backshift operator B (or L in some texts) is the inverse of Z. I just clarified my post by highlighting slide 6 in Bartlett's lecture, which adopts the convention $\phi(z)=1-\phi_1z-\ldots-\phi_pz^p$. I find this to be consistent throughout the many textbooks and lecture material I've perused (not necessarily read in detail or understood!). | |
| Jul 20, 2015 at 0:11 | comment | added | mlofton | @whuber: Thanks for explaining where you saw that. I don't have the book in front of me and won't till Tuesday but I'll check it out and let you know what I think. It's not an easy book by any stretch but I'd say its my favorite and I have many, many time series books. | |
| Jul 19, 2015 at 13:10 | comment | added | whuber♦ | Good point. I looked up "Unit Root" in the index, which refers to chapter 15, Models of Nonstationary Time Series. The book has many merits, but providing clear definitions is not one of them. | |
| Jul 19, 2015 at 3:38 | comment | added | mlofton | @whuber: It's an amazing book and full of various info in different chapters. Chapter 1 is consistent with what I wrote in my answer. But maybe there's a different chapter that talks about it in a different way and more along the lines of wikipedia. | |
| Jul 18, 2015 at 18:31 | comment | added | whuber♦ | My version of Hamilton's text (1994) is consistent with Wikipedia. | |
| Jul 18, 2015 at 18:17 | history | edited | mlofton | CC BY-SA 3.0 |
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| Jul 18, 2015 at 18:12 | history | answered | mlofton | CC BY-SA 3.0 |