Talk:Unit root

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Normally, "a" would be the intercept, and "b" would be the slope. Is it different in this equation for some reason?

There is not intercept in this equation. Note that this article is still a stub and does not provide full coverage of the subject. Karina.l.k 17:17, 28 April 2007 (UTC)[reply]

The link to characteristic equation isn't making a lot of sense to me so far. That is an article about characteristics equations of matrices, whose roots are eignvalues. Is that what is intended here? It doesn't look like it. Michael Hardy (talk) 22:13, 2 August 2008 (UTC)[reply]

Agree the links to characteristic equation don't make sense. In one place the link is to the charactaristic equation of a differential equation, and in another to charactaristic polynomial of a matrix. Can an autoregressive process be expressed as a differential equation with identical coefficients, or as a matrix with the same charactaristic polynomial? I think this needs to be clarified. (April 22 2016). — Preceding unsigned comment added by 205.254.147.8 (talk) 16:49, 22 April 2016 (UTC)[reply]

I believe we should split this page into two. The section at the bottom about the Unit Root Hypothesis is well written, interesting, and important for/to economists. The Unit Root, however, is a more general topic that applies to all time-series analysis in all fields. Therefore, I think the Unit Root Hypothesis should be only mentioned as an example with the bulk of its content on a separate page.

This is a big enough change that I am reluctant to do this unless there is evidence of a consensus - and more importantly, since the information is useful for econ people (like me), I would hate to make the change in such a way that it is unavailable while the new page is waiting on approval. — Preceding unsigned comment added by Balaamsgrayass (talk • contribs) 23:40, 3 December 2017 (UTC)[reply]

The audience for this article is severely limited. I suggest an introduction that targets a broader audience with a more limited understanding of statistics. The article is heavy on statistical jargon and does not make the subject more accessible. I would be interested in others thoughts on this. Perhaps accessibility is not a concern. 206.193.225.70 (talk) 19:01, 26 September 2008 (UTC)[reply]

You could then maybe first introduce the implications of non stationarity (section properties at the bottom), discuss the random walk and then come to problems of characteristic roots and integration order to make it more comprehensible? EtudiantEco (talk) 15:50, 20 October 2008 (UTC)[reply]

Just to drive this point home, I'm a bioengineering/neuroscience student with a signals processing background and this page makes no sense to me whatsoever. I have taken courses on time series analysis and I'm trying to work with Granger causality, so this means that someone who is somewhat/moderately familiar with the field cannot read it. Any help rewording this page would be appreciated. -- eykanal talk 21:34, 8 December 2008 (UTC)[reply]

I agree strongly with the comment that started this thread and with eykanal. (Aside: This is really about math article policy, but despite several minutes searching I haven't found the right page for this policy matter, sorry. I'll be grateful if someone merciful will link to the right page, and of course you may copy this there if you like.) The wording of this article (and of many, many math articles) assumes that what every reader wants is a detailed, exact, exhaustive explanation about the kind of mathematical intricacies that will allow you to do mathematical work based on the information given. This is only one of the roles of an encyclopedia. There are many readers who have no such interest at all, and instead look for a general notion, an outline, a rough idea, a view from afar. For example, only on this talk page do I learn that the term that the article defines has something to do with statistics. The article doesn't even mention this simple fact. To illustrate, if a doctor asks me, a programmer, to help him with his computer, I don't answer by showing him program code, I answer in terms that he finds relevant for his computer usage, even though this is radically different from my own needs and interests. I wish every math article started with a brief introductory explanation that followed the example of this Nobel prize winner, who undoubtedly could write far more intricate math than this article does, but as you can see, despite the great complexity of his subject, he's also able to discuss in beautifully readable English.--QuickFox (talk) 21:14, 4 March 2009 (UTC)[reply]

Like the above commentors, I have a background in maths and statistics, but I am having enough trouble with this article that I'm not sure if I've found an error, or simply misunderstood it.

Consider the example, concerning:

The first order autoregressive model, ${\displaystyle y_{t}=a_{1}y_{t-1}+\varepsilon _{t}}$,

The example goes on to show that ${\displaystyle Var(y_{t})}$ is a function of t if the characteristic equation has a unit root. However, it is also a function of t for non-unit roots! Proceding just as in the example, but not restricting outselves to m = 1:

${\displaystyle y_{t}=a_{1}{\frac {1-a_{1}^{t}}{1-a_{1}}}y_{0}+\varepsilon _{t}+a_{1}\varepsilon _{t-1}+...+a_{1}^{t-1}\varepsilon _{0}}$.

Then the variance of ${\displaystyle y_{t}}$ is given by:

${\displaystyle Var(y_{t})=Var(a_{1}{\frac {1-a_{1}^{t}}{1-a_{1}}}y_{0})+Var(\varepsilon _{t}+a_{1}\varepsilon _{t-1}+...+a_{1}^{t-1}\varepsilon _{0})}$

The first term is zero, since the ${\displaystyle a_{1},y_{0}}$ are not random variates and so have variance 0. The second term expands to:

${\displaystyle Var(y_{t})=Var(\varepsilon _{t})+a_{1}^{2}Var(\varepsilon _{t-1})+...+a_{1}^{2(t-1)}Var(\varepsilon _{0}))}$

Which then (generally) simplifies to:

${\displaystyle Var(y_{t})={\frac {1-a_{1}^{2t}}{1-a_{1}^{2}}}\sigma ^{2}}$

which is clearly just as non-stationary as ${\displaystyle t\sigma ^{2}}$, if not more so!! - 202.63.39.58 (talk) 20:45, 25 March 2013 (UTC)[reply]

I might just exapnd on my own remark slightly. For that term, ${\displaystyle {\frac {1-a_{1}^{2t}}{1-a_{1}^{2}}}\sigma ^{2}}$, we can distinguish three cases. For ${\displaystyle a_{1}^{2}<1}$, then in the limit it converges to ${\displaystyle {\frac {1}{1-a_{1}^{2}}}\sigma ^{2}}$ which is stationary. For ${\displaystyle a_{1}^{2}=1}$ then we get the special case mentioned in the article, where ${\displaystyle Var(y_{t})=t\sigma ^{2}}$, which is (slowly) divergent and non-stationary. Note that this works for ${\displaystyle a_{1}=\pm 1}$, not just +1. Finally, for ${\displaystyle a_{1}^{2}>1}$1}"/>, in the limit we get ${\displaystyle Var(y_{t})=a_{1}^{2t-2}\sigma ^{2}}$ which is rapidly divergent and also non-stationary.

So, did I miss some rule that says we are only interested in ${\displaystyle a_{1}^{2}\leq 1}$? Certainly there is no physical reason presented why this should be so, and we can easily think of practical examples where it is not (e.g. Fibonacci's rabbits!) -- 202.63.39.58 (talk) 21:13, 25 March 2013 (UTC)[reply]

• Article says: "To estimate the slope coefficients, one should first conduct a unit root test, whose null hypothesis is that a unit root is present." Stop me if I'm wrong, but the null hypothesis of ADF is that a unit root is present, but that of KPSS is that none is present. So the article is not quite correct..? • Serviceable†Villain 07:40, 12 April 2014 (UTC)[reply]

In the Definition section, it almost seems as if the exponents in the polynomial in m are in reverse order. Most of what I could find online deals with AR(1), but in what little I can find for unit roots of AR(p), the characteristic polynomial has the *lowest* power in lag operator (presumably m in this case) bound to the constant coefficient associated with the least lag (a_1 in this case). See, for example, http://www.bauer.uh.edu/rsusmel/phd/ec2-5.pdf. This is, of course, assuming that the operator m is a stand-in for a lag operation. Nowhere is this stated, but if it isn't the case, it sure would be unorthodox and deserving of an explanation. — Preceding unsigned comment added by 131.136.242.1 (talk) 23:28, 21 April 2015 (UTC)[reply]

It seems to be pretty fundamental to time series. I don't know how to bump up the importance.

Dr. Dogru has reviewed this Wikipedia page, and provided us with the following comments to improve its quality:

1. Unit root hypothesis should be taken into account with and withoud draft. Besides that it is okay.

2. You could add this reference to the further readings:

Dogru, B. (2014). Analysis of Long-and Short-run Balance of Money Demand In Turkey Using ARDL and VEC Approaches” The International Journal of Economic and Social Research, Vol. 10, No. 2, 19-32,

We hope Wikipedians on this talk page can take advantage of these comments and improve the quality of the article accordingly.

Dr. Dogru has published scholarly research which seems to be relevant to this Wikipedia article:

• Reference 1: dogru, bulent, 2013. "Dynamic Analysis of Money Demand Function: Case of Turkey," MPRA Paper 48402, University Library of Munich, Germany.

• Reference 2: dogru, bulent, 2013. "Are Output Fluctuations Transitory in the MENA Region?," MPRA Paper 49080, University Library of Munich, Germany.

ExpertIdeas (talk) 23:46, 23 August 2015 (UTC)[reply]

Dr. Platen has reviewed this Wikipedia page, and provided us with the following comments to improve its quality:

The page is well written and allows the reader to get a reasonable overview about the question raised.

We hope Wikipedians on this talk page can take advantage of these comments and improve the quality of the article accordingly.

Dr. Platen has published scholarly research which seems to be relevant to this Wikipedia article:

• Reference : Wolfgang Breymann & Leah Kelly & Eckhard Platen, 2004. "Intraday Empirical Analysis and Modeling of Diversified World Stock Indices," Research Paper Series 125, Quantitative Finance Research Centre, University of Technology, Sydney.

ExpertIdeasBot (talk) 14:58, 24 June 2016 (UTC)[reply]

Dr. Reed has reviewed this Wikipedia page, and provided us with the following comments to improve its quality:

"time serie" should be "time series"

"transitory, the time serie will converge" could be better stated as "over time, the time series will converge"

"while unit-root processes" could be better stated as "while shocks to unit-root processes"

We hope Wikipedians on this talk page can take advantage of these comments and improve the quality of the article accordingly.

We believe Dr. Reed has expertise on the topic of this article, since he has published relevant scholarly research:

• Reference : W. Robert Reed, 2015. "Testing For Unit Roots With Cointegrated Data," Working Papers in Economics 15/11, University of Canterbury, Department of Economics and Finance.

ExpertIdeasBot (talk) 20:40, 1 July 2016 (UTC)[reply]

Dr. Shi has reviewed this Wikipedia page, and provided us with the following comments to improve its quality:

1. "that can cause problems in statistical inference involving time series models. "

Not if one uses correct model for estimation.

2. "If the other roots of the characteristic equation lie inside the unit circle" The unit root process may have more than two roots. The introduction, in general, is not accurate.

3. "Granger and Newbold called such estimates 'spurious regression' results:" Spurious regression refers to the regression of one I(1) variable on the other I(1) variable when they are not cointegrated. Not the case of regressing y_{t} on y_{t-1} -- the slope of the autoregressive model. The statement is wrong.

4. Estimation Section: when two variables are I(1) and cointegrated, the estimated coefficient is super-consistent. Information along this line is not mentioned.

We hope Wikipedians on this talk page can take advantage of these comments and improve the quality of the article accordingly.

We believe Dr. Shi has expertise on the topic of this article, since he has published relevant scholarly research:

• Reference : Peter C.B. Phillips & Shu-Ping Shi & Jun Yu, 2012. "Specification Sensitivity in Right-Tailed Unit Root Testing for Explosive Behavior," Cowles Foundation Discussion Papers 1842, Cowles Foundation for Research in Economics, Yale University.

ExpertIdeasBot (talk) 16:06, 12 July 2016 (UTC)[reply]