# First Differenceing

Just started my final review and I am able to solve just about all of the EOCs. However I am just hung up on the overall concept of first differenceing. It makes sense why you do it but I am trying to figure out what you do with it. AR models that are not covariance stationary (i.e. Values which just keep rising over time) need to be first differenced to get a covariance stationary time series. Well rather you check to see if the first differenced time series is covariance stationary or not. Let’s say that the first differenced time series IS covariance stationary. Now what? Do we use that as our model? Does it just prove that the original model was not covariance stationary? I can’t really discern between what we do with it, but I know why it needs to be done. I understand the math behind it but not the use of it. Maybe I am just over thinking this. “Modeling first difference time series in AR(1) model holds no predictive value. It only confirms suspicion that original time series is a random walk. The first differenced R2 shows how well the changes in one period predict the changes in the next” - Wiley If it holds no predictive value then why is there an R2 for it? That is my confusion. Why use it and assign an R2 if it holds no predictive value. What’s the point? According to Wiley it should only confirm suspicion of a random walk… Nothing else.

I only have a basic understanding of this reading so far, so please don’t take my response without doubt. My understanding is that once you’ve first differenced the series and it is covariance stationary, then your assumptions necessary for OLS are satisfied or at least sufficient to proceed with the usual linear regression.

I think the quote from Wiley is just saying that we were better off not using AR to start with since it would hae resulted in incorrect results. Namely because a random walk without drift is equivalent to an AR process with both co-efficients equal to 0.

Anyway, it would be good to know if you’ve thought this one through anymore and could share a better answer/thought process?

I did all of the EOC problems (but not the Topic Tests yet) and had no need to dig into the question further. I would love to see if anyone else has any thoughts.

Il try and answer these in order:

Let’s say that the first differenced time series IS covariance stationary. Now what? Do we use that as our model? Does it just prove that the original model was not covariance stationary?

No, we don’t use it as our model - as you quoted, it serves to show our original AR(1) model is a random walk. Lets go back to the definition of a random walk, right out of the curriculum “A random walk is a time series in which the value of the series in one period is the value in the previous series, plus an unpredictable random error term”

So, if we have a random walk, we CANT model it like are trying to do - its random.

If it holds no predictive value then why is there an R2 for it? That is my confusion. Why use it and assign an R2 if it holds no predictive value. What’s the point? According to Wiley it should only confirm suspicion of a random walk… Nothing else

Any regression output is going to give you an R2 (bounded between 0 and 1) - thats just part of the the process. You’re right, it doesnt hold predictive power - we have a random walk! Take a look in the curriculum at example 10 on page 437 of print version - look at the R2 for the first differenced results. Near zero. This should make sense, since the model is a random walk, there isn’t any predictive power, as we’ve established, its, well, random!

Hope that helps. Edit: Read the two paragraphs above example 10 as well, might help a bit.

Let me know if I can clarify anything else.

Thanks Nereites. It makes sense. I went back through the book (which also had my highlights and underlines from my earlier review) to connect the dots.

I was going to make a flow chart but the curriculum is very succint with its example if you just read it as a flow chart with the example.

I missed the dots of how we got from xt = (xt-1) + Et to Yt = Xt - X(t-1) = Et. It makes sense. It is just like re-arranging the formula. Once I understood that connection it all makes sense. In the end you use the R2 of the first differenced time series to show that the original model is trash. For any R2 to have meaning you need a covariance stationary model (with a finite mean reverting level).

So after restating the time series in a way that makes the stats useful, you can figure out why it is crap.

Interesting.

Thanks