Talk:Unit root

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)

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)

Audience?

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)

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)

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)

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)

Much clarification required

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, $y_{t}=a_{1}y_{t-1}+\varepsilon _{t}$,

The example goes on to show that $Var(y_{t})$ 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:

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

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

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

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

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

Which then (generally) simplifies to:

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

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

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

So, did I miss some rule that says we are only interested in $a_{1}^{2}\leq 1$? 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)