Adding extra variables to the manifold

It can be advantageous to combine data from multiple sources into a single EDM analysis.

The extra command will incorporate additional pieces of data into the manifold. As an example, consider the Stata command

edm explore a, extra(b)

Choose the number of observations

Choose a value for \(E\)

Choose a value for \(\tau\)

The time-delayed embedding of the \(a\) time series with the given \(E\) and \(\tau\) is the manifold:

However, with the extra(b) option, we use the time-delayed embedding with the extra time series included like:

After extra variables are added, the manifold \(M_{a,b}\) no longer has \(E\) columns. In these cases, we make a distinction between \(E\) which selects the number of lags for each time series, and the actual \(E\) which is size of each point (i.e. the number of columns).

By default just one \(b\) observation is added to each point in the manifold. However, this will only allow us to capture to the contemporaneous effect of \(b\) on the system.

If instead we want to add the last \(E\) lags of \(b\) (just as we do with the \(a\) time series series), then the command should be altered slightly to

edm explore a, extra(b(e))

and then the manifold will be:

More than one extra variable can be added, and any combinations of \(E\)-varying and non-\(E\)-varying extras are permitted.

Note

If some extras are lagged extra variables (i.e. \(E\)-varying extras) and they are specified after some unlagged extras, then the package will reorder them so that all the lagged extras are first.