Hi, I am taking Dr. Chan's Python for Mean Reversion Strategies course. I am on the Johansen test section. It seems to me that there isn't a very clear explanation of Trace Statistic and Eigen Statistic, so I need to ask for some clarification of these. Is there a way to get a better intuitive understanding of these? Is it correct that we only look at the first line (R<=0) for both Trace and Eigen Statistic to ascertain significance? Also, what about the case when one of them is significant while the other isn't? How aer Trace and Eigen Statistic different, and do we need both of them to be significant for the stationarity to hold?
Thank you very much for your help
Hi Abhinav,
The Eigenvalues of the Johansen test are the autoregressive coefficients of a system of Vector Error Correction (VEC) equations transformed to a set of variables that are independent of each other. So you have a separate scalar autoregressive equation for each transformed variable, and whether that autoregressive coefficient is significantly different from zero determines whether we have a stationary variable.
Recall that for a scalar autoregressive equation, Y(t)-Y(t-1)=a*(Y(t-1)-mu)+ …+noise
where mu is a constant (the "mean"). If a is zero, this is a random walk, and we don't have mean reversion. a must be significantly different from zero (and negative) in order for Y to be mean reverting. The Johansen test will find out whether, after linear transformation of the original variables, there are any a's that are significantly different from zero. If there is one (or more) such non-zero a's, then we have found one (or more) cointegrating relationship, because a linear combination of the original variables form a stationary, mean-reverting, time series.
The Eigen statistic is a statistical signficance test on whether there is a non-zero a. The first line of the Johansen test indicates whether we can reject the null hypothesis that the number of such non-zero a's is zero. If we can reject that, that means there is at least one non-zero a.
Recall that both eigenvalues and trace of a matrix are invariant under linear transformation of variables. So to ask whether there is a non-zero a's is the same as asking whether the trace of the matrix is different from zero. Both of them usually provide consistent results. But if not, that means we have borderline cointegration.
Hope this helps!
Ernie
Thank you for that detailed explanation Dr. Chan.