Which econometric test applied here?
As they clearly say on pp 2267:

Alternative specifications were compared using the
specification test by Davidson and MacKinnon and
the Wald test of coefficient restrictions.

So they two tests Davidson and MacKinnon specification test and Wald test.
Wald test is a classic test for coefficient restrictions you can learn can learn more about it in any econometrics textbook, for example Verbeek (2008). Guide to modern econometrics. 4ed.
For Davidson and MacKinnon specification test you can have a look at the original Davidson and MacKinnon (1990) paper.