I’m a little confused about these tests. In the last section of time series analysis of schweser notes, the unit root test does not show up in those steps. This makes sense for me since it already check for serial correlation, seasonality and ARCH stuff, why would I need to check unit root? But if we can check unit root directly, then why would we need to run all the t-stat stuff for the lags? I have some gaps in these concepts, but not sure where they are. Unit root, random walk and covariance non-stationary are the same things? It seems that some non-stationary (linear trend) is not a random walk. Thanks, Tao
If there is a unit root (i.e. b_{1} = 1), the time series is not time reverting and so it cannot be covariance stationary (not a good thing). If you are regressing two time series and exactly one series is non-covariance stationary, then the resultant series is also not covariance stationary. If both series are non-covariance stationary, you need to check for co-integration. Basically, use unit root as a quick way to check for non-covariance stationary. I believe the converse is not true: if a series does not have a unit root, it does not mean it is covariance stationary. That’s when you need to run the t-test on all the lags.
Thanks, now I have a better idea on this stationary thing. It seems Dockey-Fuller test (even with available software) can tell if a time series has a unit root or not, while the traditional serial correlation method(t-stat for lags) are still used. A random walk has a unit root in it, and is unpredictable, no matter how to transform it. First differantial can transform a random walk into a stationary time series, but how to use it if it is not predictable?
I am not too clear what your question is. First differencing transforms a non-covariance stationary series into a covariance stationary one. Dickey-Fuller test is used to test whether or not two time series are cointegrated. You only do the DF test when you are regressing two time series into a new time series and you discovered that both input time series are non-covariance stationary.
My confusion is about unit root and what to do if a unit root is found. Dickey-Fuller test can also be used to determine if a time series is covariance stationary or not(Schweser notes B1-P241) though I can’t find enough details there. What drives me crazy is: it seems that linear trend model has a unit root it. For example, y(t)=0.5+3.5t can be transformed into y(t)-y(t-1)=3.5 or y(t)=y(t-1)+3.5 or b1=1. Can I still use it for forecasting even if it has a unit root in it? Is it equivalent to a random walk?
If the unit root hypothesis is not rejected, then we have a nonstationary process (not necessarily a random walk). We can still use it for forecasting, but not in its levels. First-differencing will often make the process stationary, and some sort of forecasting can be done with the differences. If our regressors are nonstationary as well, then cointegration analysis may have some potential.
For the purpose of the exam, the D-F test is meant for testing cointegration when you regress two time series into one. I believe to be sure of a single time series is covariance stationary or not, you need to run the t-test on each lag terms. If you are testing for covariance stationary of a time series that is the result of regressing over two time series, you need to test for covariance stationary of the input series first, and then (if applicable) test for cointegration using D-F test. Note: even though the linear trend model looks a lot like the resultant model when you regress two time series, they are not the same because of the input. In the latter, we assume we know the underlying input are two time series. Whereas the former, there is no such assumption (we assume there is a linear relationship between time and the dependent variable only). This is why for linear trend model, it’s sufficient to treat it like a linear regression and apply DW test. And for the two time series regression, we need to go through both covariance stationary test as well as cointegration test (if applicable).
What’s the confusion here? We use DF to test for unit root. If there is unit root, then most probably, first differencing will make the regression model covariance stationary and then we can use that regression against the regressor (if the regressor does not exhibit unit walk and/or walk with drift) for co-integration. To test for cointegration we use DG-EF test and if the 2 regressions are cointegrated - then it’s safe to use the regression.
DF test is for unit root test. DG-EF test is for testing cointegration. (Sorry about the mix up earlier.) I believe OP’s confusion is that, the form of a linear trend model looks very similar to an AR(1) or a two time series regression model. So he is probably puzzled why we don’t need to check for unit root in linear trend model, when we need to check in the latter two cases.
Yeps actually this equation [y(t)=y(t-1)+3.5] looks like a unit root with drift. I am confused too now.
The equation [y(t)=y(t-1)+3.5] corresponds to an AR(1) model (assuming brackets mean subscripts) that has a unit root (since b(1) = 1). I believe the question is, does unit root apply to linear trend model as well, such as y(t) = t + 3.5. I am more inclined to say “no” since it’s only an ordinary linear regression using time as the independent variable. Seasonality could still exist in a linear trend model, but the resulting model would be inaccurate because the underlying relationship is not linear–a violation of the model’s assumption. I’ll ask on Thursday during Schweser’s office hours to be sure.
I asked Dr. Lekvin today. He said unit root, random walk, covariance stationary, cointegration, etc do not apply because the independent variable in trend models is not a lagged value of the dependent variable.
eltia, thank you for the follow-up. It makes the life a lot easier.
Sorry eltia, to follow up … should I just replace D-F test with DG-EF test in your posts above and the rest of the explanation is fine? Many thanks for all your help …
Yes, I mixed up D-F w/ DG-EF test in a previous post. For unit root test, use D-F test. For cointegration test, use DG-EF test. The rest (as far as I know) is accurate.