# Dickey Fuller Test on ARCH model

Hi, does anyone else find the DF test on ARCH model section in Schweser notes painfully confusing? So far I’ve spotted three places where the notes makes comment regarding how to interpret the DF test result. 1. If Ho cannot be rejected, the time series has unit root, if Ho is rejected, no unit root. 2. If Ho cannot be rejected, no cointergration. If Ho rejected, two time series is cointergrated and we can use regression to model relationship. 3. If Ho cannot be rejected, we can use model. If Ho rejected, ARCH is present and have to correct using GLS. It looks like to me that 2 and 3 are conflicting. Also…does having a unit root mean that its covariance is stationary or nonstationary? Thanks!

Series with Unit Root (b1=1) is NOT covariance Stationary i.e. non-stationary.

This is what I could concur from the Schweser notes. Unit root test ------------------- if(Ho = rejected) { Time Series does not have a unit root } else { Time Series has a unit root } Covariance Stationary test ---------------------------------- if(Ho = rejected) { Time Series is Covariance Stationary } else { Time Series is not Covariance Stationary } CoIntegration test ---------------------------------- if(Ho = rejected) { Time Series are CoIntegrated } else { Time Series are not CoIntegrated } To test if time series is non-stationary (unit root test), we have to prove that the coefficient b1 = 1.0 and therefore the Mean Reverting Level (MRL) goes to an indeterminate state. But for DF test, instead of doing this tradition test, he modified the base AR(1) model question to subtract the prior-period-lag and instead test if (b1 - 1) = g = 0 and we have to use a some kind of a revised DF-t-table to find the t(critical) value for the decision rule. - Dinesh S

Rainy, I was reading through the schweser notes, and I don’t see any part saying that the DF test is used to test for ARCH.

HydrogenRainbow Wrote: ------------------------------------------------------- > Rainy, I was reading through the schweser notes, > and I don’t see any part saying that the DF test > is used to test for ARCH. It’s not…it’s used for testing Coveriance Nonstationarity…and Cointergration. The last few pages of Book I.

As far as I understand…they are testing the coefficient from the ARCH model in the test.

You can use at least some forms of Dickey-Fuller in Arch modelling. Dinesh has this right except "we have to use a some kind of a revised DF-t-table to find the t(critical) value for the decision rule. " where the table is a Dickey-Fuller table not a t-table (I know - just semantics).

And does everyone else chuckle when they say Dickey-Fuller? Honey, how about we integrate by parts and do some Dickey Fuller remediation?

Thanks Joey for clearing the fog. I surely need to read this TS a couple of times to be able to do something on the exam. And yes, the name sounds more like a porn-metaphor

JoeyDVivre Wrote: ------------------------------------------------------- > And does everyone else chuckle when they say > Dickey-Fuller? Honey, how about we integrate by > parts and do some Dickey Fuller remediation? Good to hear I’m not alone on that. Every time I hear Dickey-Fuller I get flashbacks of Austin Powers jokes.

Don’t worry about where the Dickey-Fuller distribution (and its corresponding tables) come from. They are basically a modified t-table and will look exactly the same. Just know that they are not the same 1.96 or whatever numbers that are on the t tables, but slightly biased in one direction (can’t remember which). It will likely be given on the test if it is on there at all. It comes from a Monte Carlo simulation in the original DF paper.

Joey, I was reading through the stuff in the Schweser notes, and I found something which I found particularly weird and it made me wonder if there is something wrong with the material. Honestly this is the first time I have come across the discussion on ARCH in the context of it being a problem “If a time-series model has been found to contain ARCH errors, regression procedures that correct for heteroskedasticity, e.g. GLS must be applied for the predictive model. Otherwise, the standard errors of the model’s coefficients will be wrong, leading to incorrect predictions” The question is, why should an ARCH model be corrected for heteroskedasticity using GLS? My understanding is (Et|Et-1)= a + b(Et-1)^2 if we have y=B’Xt+Et and Et=Ut*(a+b(Et-1)^2)^0.5. I.e. Et is CONDITIONALLY HETEROSKEDASTIC. However, the UNCONDITIONAL variance is HOMOSKEDASTIC, i.e. var(Et)=a/(1-b) if the process generating the error term is weakly covariance stationary, and therefore the model satisfies the classical assumptions and therefore OLS will work fine, although there are other estimators that are more efficient. You need only the unconditional variance because you are doing the prediction on y. Thanks for your help Joey!