Timeline for Can Dickey-Fuller be used if the residuals are non-normal?
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| Aug 29, 2020 at 0:02 | comment | added | mlofton | Thaks Michael. I'll have to look at what you said more carefully when I have more time. At the moment, it's not clear but it has some potential for becoming clearer with effort on my part. Thanks. | |
| Aug 28, 2020 at 8:05 | history | edited | Michael | CC BY-SA 4.0 |
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| Aug 27, 2020 at 23:31 | comment | added | Michael | Similarly, the relationship between a MA process and strong mixing-type condition such as Phillips and Perron's Condition (vi) is not automatic. E.g. if $\epsilon_t$ has Bernoulli distribution, then $u_t$ would not be strong mixing. Usually, to show an ARMA process satisfy a mixing condition, one needs assumptions on the marginal distribution, in addition to other properties. (C.f. stats.stackexchange.com/questions/484767/…) | |
| Aug 27, 2020 at 23:29 | comment | added | Michael | @mlofton Incidentally, I believe that the sufficient condition for FCLT stated in Hamilton is that the innovations $u_t$ is of the MA form $u_t = \sum \theta_h \epsilon_{t-h}$ where $\epsilon_t$ is i.i.d. is finite fourth moments and $\theta_h \in l^2$. No proof is given and I don't believe it actually suffices. The i.i.d. with finite fourth moment assumption would imply $\sup_t E[u_t^2] < \infty$---which does not satisfy McLeish's UI condition, much less Phillips and Perron's more restrictive Condition (ii). | |
| Aug 27, 2020 at 13:32 | comment | added | mlofton | Michael: That was a beautiful analogy. I get it now. Thanks so much and apologies. Also, my apologies to Peter Phillips also since what he did was a bigger deal than I ever realized. For those reading this, Hamilton's chapter on this topic is also quite good, although he doesn't explain the relation between Dickey Fuller, 1979 and Phillips results like Michael did. | |
| Aug 26, 2020 at 17:55 | comment | added | Michael | @mlofton (Cont'd) So they give a characterization of the limit distribution---in this case N(0,1). That's Phillips et al. The difference in the FCLT case is that even after your characterize it, the explicit density is not known and you still have to simulate. But the point is that the i.i.d. normal assumption is dropped completely. | |
| Aug 26, 2020 at 17:53 | comment | added | Michael | @mlofton The difference between those tables would be numerical, the distribution being simulated is the same. Consider the following analogous situation. You proved that, when data is i.i.d. normal, the t-stat has a limit distribution---just existence, no characterization of any kind. This existence then implies it makes sense to simulate, which you do and provide a table. This is DF. Now someone else comes along and points out the connection to CLT and drops the i.i.d. normality assumption completely---CLT holds under very general conditions. | |
| Aug 25, 2020 at 18:29 | comment | added | mlofton | Michael: Gotcha. Thanks for clarification. So, when one looks up the DF tables in say the Hamilton text, are they looking up the ones that DF 1979 constructed ? On the other hand, when one refers to "DF tables" in such a text, maybe they are updated ones that are no longer based on DF 1979 ? Either way, I understand the issue a lot more clearly now. Thanks a lot and I'm sorry for confusing the issue and causing so much of your time to be spent explaining it. Maybe you can tell that I'm old since I'm stuck on DF 1979. | |
| Aug 24, 2020 at 17:13 | comment | added | Michael | @mlofton "...the [Dickey-Fuller JASA] 1979 paper was assuming normality of the error term and then the later papers made improvements...and led to the less restrictive assumptions?" Yes, significant improvements. Phillips et al noted that the asymptotic distribution is given by certain continuous functionals of the Brownian sample paths, whereas DF 1979 assumed i.i.d. normal error and even under this restrictive assumption only showed the existence of a limit distribution without any closed form characterization, and did the rest by simulation. | |
| Aug 24, 2020 at 12:15 | comment | added | mlofton | HI MIchael: Thanks for your edited answer and again my apologies. I'm definitely confused now ( given what Professor Hamilton said ) but, based on what you said in your reply, it sounds like the 1979 paper was assuming normality of the error term and then the later papers made improvements to what they did and led to the less restrictive assumptions ? Thanks for clarifying and all of the references. When I get some time, I'll check them out. | |
| Aug 24, 2020 at 0:13 | comment | added | user54285 | I was reminded by the software that there is not supposed to be long discussions in comments. If you ever want to talk about time series in chat, assuming I am allowed there I would be delighted to continue this conversation. | |
| Aug 24, 2020 at 0:07 | comment | added | Michael | @user54285 Empirical knowledge of your particular context helps a lot and should probably enter into your prior as much as purely statistical considerations. In Nelson-Plosser's case, Perron pointed out that the 1970 oil shock is a possible structural break and used his test to provide statistical evidence. Zivot-Andrews doesn't assume a known break date---a possible trade-off there is statistical vs. empirical evidence (if you're lucky, they point in the same direction). | |
| Aug 23, 2020 at 23:37 | comment | added | user54285 | I found 4 methods in my research, but am going to stick with Zivot Andrews assuming R does this. Thanks again Michael. This is one of my greatest concerns about statistics. The importance of structural breaks in unit roots is obviously important, but I never heard about until a week ago despite a lot of reading on unit root test. Given that there are limits on time to do reading I sometimes wonder if its possible to ever run it correctly except for a handful of experts maybe. | |
| Aug 23, 2020 at 23:13 | comment | added | Michael | @user54285 I believe this particular test is just called the Perron test but you should check documentation for a specific software. It's also pretty easy to compute the Perron statistic yourself---mechanically it's a slight modification of the ADF regression. So you can always compute the statistic yourself and use Perron's tables for critical value. For structural breaks, there's also the more recent Zivot-Andrews test (and others). | |
| Aug 23, 2020 at 23:08 | comment | added | user54285 | Is this the Phillip Pheron unit root test (sometimes people have more than one test associated with them). The test for this I know of is Lee and Strazicich - well I have heard of it I am still not sure if R runs it :) | |
| Aug 23, 2020 at 22:57 | comment | added | Michael | @user54285 "...structural breaks in my data..."---structural break is an issue. A one-time structural break in otherwise stationary data may lead to a false non-rejection of the unit root null. This was first noted and addressed in Perron (1989 Econometrica) where he proposed a unit root test that accounts for possible structural break under the alternative and applied his test to the Nelson-Plosser dataset. Any statistical software that implements the ADF test should also include the Perron test. | |
| Aug 23, 2020 at 22:46 | comment | added | user54285 | thanks very much Michael. Now all I have to worry if there are structural breaks in my data which as I understand it creates problems for the unit root tests [although few authors bring this up so maybe its not true] | |
| Aug 23, 2020 at 21:39 | comment | added | Michael | @user54285 "I commonly have sixty..."---60 is pushing the lower end of the range where you could be fine (but I, for one, cannot say for sure without knowing more). However, if you look at Nelson and Plosser (1982 JME)---a (very) important early application of unit root testing in the empirical economics literature, they applied the DF test with asymptotic critical values and sample size ranging from 60 to 100. This precedent is in your favor. | |
| Aug 23, 2020 at 20:52 | comment | added | user54285 | thanks Michael. I am not a statistician [I just run statistical models relying on the experts I read such as this forum] and I have yet to learn how to simulate data the way you reference. I have never heard before of a FCLT. You mention a hundred points. I commonly have sixty, is that too few to rely on the FCLT. And if you don't have more than 60 how do you test for non-stationarity? If it matters I use both the ADF and KPPS and look to see if they agree (as I have found suggested given power issues). | |
| Aug 23, 2020 at 20:14 | history | edited | Michael | CC BY-SA 4.0 |
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| Aug 23, 2020 at 20:06 | comment | added | Michael | @user54285 "...lag structure...does not matter"---that's correct, asymptotically. For example, an ARMA error term with innovations which has continuous distributions, that you can simulate, would fall under the Phillips-Perron assumption (more precisely, the corrected formulation of their assumption). | |
| Aug 23, 2020 at 20:01 | comment | added | Michael | In this respect, unit root asymptotics is not unlike, say, that of the t-test. Also, the FCLT is a much stronger result than CLT. So, informally speaking, you would expect the convergence to be slower. If you really believe the error is i.i.d. normal, then the DF tables for finite sample can be used as @mlofton says (or just do a simulation for your given sample size). | |
| Aug 23, 2020 at 20:01 | comment | added | Michael | @user54285 "...since this uses the CLT..."---it's much more than a CLT. "Is this impacted by by how many data points you have..."---yes, indeed, unit root tests all make use of FCLT for asymptotics and in general are known for small sample distortions regarding their size, and power. This is a caveat I didn't mention. In the theory literature, when a Monte Carlo exercise is performed for a newly proposed unit root test, one rarely see the sample size chosen to be <100. If "you...have as few as 20 points", my null hypothesis would be that any unit root test results are meaningless. | |
| Aug 23, 2020 at 19:49 | comment | added | user54285 | since this uses the CLT Is this impacted by by how many data points you have (I often have 50 or so)? If I understood the comment above you could have as few as 20 points and it would be ok. I assume, since I did not see it mentioned, that the lag structure (the number of lags in the ARDL) does not matter. | |
| Aug 23, 2020 at 19:35 | history | edited | Michael | CC BY-SA 4.0 |
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| Aug 23, 2020 at 19:32 | comment | added | Michael | @mlofton No problem. The Elliott, Rothenberg, and Stock (1996) citation has been added to the answer. | |
| Aug 23, 2020 at 19:30 | history | edited | Michael | CC BY-SA 4.0 |
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| Aug 23, 2020 at 19:01 | comment | added | mlofton | Hi Micheal and Richard: I just emailed Jamies Hamilton an he said below so my apologies tp Michael and Richard Hardy. I'm more confused but atleast I know for sure.. Again, my sincere apologies and causing confusion. "DF does not assume normality. It is based on the asymptotic distribution that holds under a variety of settings. An example of conditions under which it is valid are included in my text. Alternative versions of sufficient conditions exist in a number of sources." | |
| Aug 23, 2020 at 18:29 | comment | added | Michael | @mlofton For something more "recent", see Elliott, Rothenberg, and Stock Efficient Tests for an Autoregressive Unit Root (Econometrica 1996), where they apply a Neyman-Pearson-like approach to benchmark the asymptotic power envelope of unit root tests. The normality assumption is long gone by then. | |
| Aug 23, 2020 at 14:15 | comment | added | Michael | @mlofton The statements in Phillips-Perron clearly says normality is not required at all, that's a basic fact (those results are plainly true, if not exactly under the conditions they claim). "...DF tables do assume normality..."---yes, in Dickey and Fuller (JASA 1979 and Econometrica 1981). That's why unit root tests were not widely considered by empirical folks prior to Phillips---it's not reasonable to have to always assume i.i.d. N(0,sigma^2) errors for a time series. The assumptions and derivations in those papers are very primitive by today's standards. | |
| Aug 23, 2020 at 14:05 | history | edited | Michael | CC BY-SA 4.0 |
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| Aug 23, 2020 at 13:55 | comment | added | mlofton | I think it is but we can agree to disagree. And I'm sorry, the precvious link , I pointed to was DW test rather than DF. When I have time, I'll try to ask someone ( who can point me to somewhere that says yes or no ). whether DF assumes normality of error term. I'm pretty certain it does but the only place where it says so is Greene. As I said, best way to know would be through simulation. | |
| Aug 23, 2020 at 13:45 | comment | added | Michael | @mlofton "...to use Phillips' result to claim that the DF tables don't assume normality, is not the correct thing to do"---well, that is simply not true or correct. | |
| Aug 23, 2020 at 13:40 | comment | added | mlofton | Hi Michael: I've already had a long discussion with Richard on this. My point is that, to use Phillips' result to claim that the DF tables don't assume normality, is not the correct thing to do. I'm pretty certain that the DF tables do assume normality ( see the link in Greene that I point to) but, the best thing to do would be to do EXACTLY what DF do, except with non-normal errors, and see if the same critical values are obtained. Note that the limitation of the early unit-root literature was that , under the unit root null, the t-statistic is not t-distributed when the errors are normal. | |
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| Aug 23, 2020 at 5:25 | history | answered | Michael | CC BY-SA 4.0 |