S2000 Think I need you help on this one --- Quant Conflict

CFA Level II
#1

S2000 Magician. If you can help with this — I will love it. Or anybody else but I got a feeling S2000 might be the goto guy here.

How does the Unit root test not conflict with the test for Serial Correlation for AR models. test On Page 430 it states “For a stationary time series either AUTOCORRELATIONS at ALL lags are indistinguishable from 0 OR it rapidly drops off to zero as lags become large”

For the second part focusing on as the LAGS beome large it drops off to 0. That would mean the autocorrelations are not 0 (Say for example for the first 3 and then drops right to 0). In this case we would Reject that the autocorrelcation = 0 at the first 3 lags and then Fail to reject afterwards). Then we could say there is No Unit root and it’s covariance stationary and the AR Model fits by that logic.

But with that same idea — Using the Test for serial correlation using the autocorrelations — Using the same numbers we would Reject the null hypothesis the autocorrelations = 0 -----> Therefor they are serial correlated and the AR model can NOT be used. The two ideas conflict.

Question 2.

Another Conflict. Comparing MA models with AR models using the autocorrelations. There’s an example given on page 437 where the autocorrelations are ALL near 0 and you fail to to reject and conclude it comes from an MA(0) model. It states later the for an AR(1) model the first autocorrelation would differ significantly from 0 and the autocorrelations would have gradually declined. 1. If the first autocorrelation is signficantly different then 0 then it would be serially correlated and it couldn’t come from an AR 1 model. 2. There’s two examples in the text (Eg. 9 and 10). All the autocorrelations are near 0 and you fail to reject —> it then concludes they’re not serially correlated and the AR model fits. At the first lag it is NOT signficantly different then 0 as example 14 states at the end on page 438.

So that’s my confusion.

#2

So I think this is what’s happening. Looked into a little bit. For my first question. For the statement “For a stationary time series either AUTOCORRELATIONS at ALL lags are indistinguishable from 0 OR it rapidly drops off to zero as lags become large”. This refers to AR with Drift and is not linked to the Serial correlation test for a simple AR(1) model and that’s why I was confused and thought there was a conflict. Correct?

  1. So for Simple AR(1) model WITHOUT DRIFT all autocorrelations have to be 0 then you can do the serial correlation test and if you fail to reject the model fits.

  2. AR with drift. The first few autocorrleation can differ from 0 but then they must drop to 0 or all the aurocorrelations have to be 0.

  3. MA model - Depending on if its MA(0), MA(1), MA(2) - In order all autocorrelations will be 0(MA(0) , 1 autocorrelation will differ significantly from 0 then all will be 0 (MA(1)). and so forth.

I think thats where the confusion was for the first part of my question. Can you confirm that? BUT still on page 438 it states If a model had come from an AR(1) model, the first autocorrelation would have differed significantly then 0 and the autocorrelation would have declined GRADUALLY.

I dont agree with that. How is that possible —> that would mean there is serial correlation and then the AR(1) model is NOT correct. That goes against what I stated in statement 1. above. I dont think it should be gradual but all AR(1) models should have all autocorrelations at 0 and not gradually declining to 0.

And having said that – Looking at model that gives us autocorrelations and t-tests, we should NOT be able to distinguish between and AR(1) model WITHOUT drift and a MA(0) Model. They both should have all autocorrelations at 0 and we fail to reject. Example 14 doesnt make sense to me or the logic cause it seems to contradict everything.