# The Dickey Fuller 1 or 0 problem

Okay. I want to put this to bed because everybody seems confused here (myself included) and this directly relates to LOS 13.j. Schweser p. 240 Background: Everybody agrees that if b1=1 than you have a unit root. Where we don’t agree is how you figure out if your coefficient is statically equal to 1 or not. Argument (straight from Schweser): For statistical reasons, you cannot directly test whether the coefficent on the indpendent variable in an AR time series is equal to 1.0. To compensate, Dickey and Fuller created a rather ingenious test for a unit root. Remember, if and AR(1) model has a coefficent of 1.0, it has a unit root and no finite mean reverting level (i.e it is not covariance stationary). Dickey and Fuller (DF) transform the AR(1) model to run a simple regression. To transfore the model, they (1) start with the basic form of the AR(1) model and (2) subtract xt-1 from both sides: (1) xt=bo+b1xt-1+e (2) xt-xt-1=bo+b1xt-1-xt-1+e xt-xt-1=bo+(b1-1)xt-1+e Then, rather than directly testing wherter the coefficent is different from 1.0, then they test whether the new, transformed coefficent (b1-1) is different from ZERO using a modified t-test. If (b1-1) is NOT significantly different from zero, they say b1 must be equal to 1.0 and, therefore, the series must have a unit root. ********************************************************** Notice that the null hypothesis is a unit root. So if you can’t reject the null than you have a unit root. This is the opposite of most tests that we run where the Ha is what we are tying to prove/provide evidence for… I am looking for some feedback and after that I am NEVER looking at this stuff again. Molodovsky.

I’m not sure if this clarifies but the cfai text described it this way: DF tranformed the AR(1) model by subtracting x(t-1) from both sides x(t) - x(t-1) = bo + g1 * x(t-1) + et, where g1 = (b1 - 1) then Ho: g = 0 (the time series has a unit root and is nonstationary) Ha: g /= 0 (the time series does not have a unit root and is stationary) I think I get your question… what we want to do is reject the null that g=0 and provide evid that g /= 0 I thought an interesting test question would include some tricky format asking why? … and the answer being that if you try to test Ho: b1 = 1 and b1 = 1 then xt is not covariance stationary and the t-values do not follow the t-dist and so a t-test is invalid… and thats where DF come in. The other tidbit that sticks in my head is that even after all that the DF test still uses a “revised set of critical values” that are larger in absolute value than the conventional t-table. Oh, and those revised values are computed for you by none other than Mr. Dickey and Ms Fuller. **Edit: On a side note…the cfai text "demonstrates how a unit root can be transformed so that it can be analyzed " aka the DF test, yeah that is described in exactly 5 sentences out of 2500 pages give or take. This test is a lot of fun. I may decide to take it again next year, cause I like it so much.

slouiscar Wrote: ------------------------------------------------------- > I’m not sure if this clarifies but the cfai text > described it this way: > > DF tranformed the AR(1) model by subtracting > x(t-1) from both sides > > x(t) - x(t-1) = bo + g1 * x(t-1) + et, where g1 = > (b1 - 1) > > then > Ho: g = 0 (the time series has a unit root and is > nonstationary) > Ha: g /= 0 (the time series does not have a unit > root and is stationary) > > I think I get your question… what we want to do > is reject the null that g=0 and provide evid that > g /= 0 > > I thought an interesting test question would > include some tricky format asking why? … and > the answer being that if you try to test Ho: b1 = > 1 and b1 = 1 then xt is not covariance stationary > and the t-values do not follow the t-dist and so a > t-test is invalid… and thats where DF come in. > The other tidbit that sticks in my head is that > even after all that the DF test still uses a > “revised set of critical values” that are larger > in absolute value than the conventional t-table. > Oh, and those revised values are computed for you > by none other than Mr. Dickey and Ms Fuller. > Yeah, I could see something like that… This test is a lot of > fun. I may decide to take it again next year, > cause I like it so much. Not a chance.

mvt9, dude, you are 100% with respect to your description. g = b1 - 1 H0 : g == 0 (unit root exists) HA : g <> 0 (unit root does not exist) Reject H0 if DF t statistic is significant which means we don’t have the unit root problem. I believe this is also connected to DF-EG if there’s cointegration… Good stuff, MVT9!

I thought the DF-EG test was for cointegration only.

Who is this Dickey Fuller guy? Does he post here often?

Dickey Fuller is patient of Dr. Dustbin … err… Durbin Watson!

Remember that if you are trying to do any predictions or menaningful modelling from a random walk, you are wasting your time. When you test x[t]-x[t-1] = b0 + (b1 - 1)*x[t-1] + e for (b1 - 1) = 0, you are rejecting the idea that the increment is b0 + e. That’s just a normal step with mean b0 which is pretty much the normal state of the universe (the process is Markov, martingale, Brownian motion like the drift of planets or the diffusion of heat or the movement of a molecule of gas). To show that you have some job to do, you need to show that there is not just nothingness in the data. It seems very reasonable to insist that someone prove beyond a 95% certainty that the data has some structure before proceeding with the analysis.

I got into some brownian motion stuff with some personal readings on chaos theory. Pretty cool ideas.

Think they ought to give Einstein a Nobel prize for his groundbreaking research in it? Wait, …

This is the only thread I could find that discussed cointegration, so I would like to continue this discussion, if possible. So…if I want to run a regression using two time series, but I find that each series has a unit root, it turns out I can use this regression if I find that the two time series are cointegrated. To test for cointegration, I regress one time series on the other and test the RESIDUALS for a unit root using DF test and critical t-values from EG (so this is a DF-EG test). If the Ho: is rejected, than the error terms generated by the two times series do not have a unit root and they are cointegrated, so it turns out I can use this model. Q1: How do you construct Ho: for a test of the RESIDUALS for a unit root? Q2: Do we ASSUME that two series are cointegrated if each separately has a unit root and a regression of one against the other produces error terms that are covariance stationary?

Bump