Dear Henrique,

Replace one line in your code by

matrix Eigenvalue = mreverse( eigensym(C, &Eigenvector) )

and add the following line

Eigenvector = mreverse( Eigenvector ' ) '

and you get the same results as pca. Unlike ''eigengen'', with ''eigensym'' eigenvalues are returned in ascending order. So you need to reverse the ordering (and the columns of the corresponding eigenvectors).

 

Now, that does not immediately answer your question on signs. This is more tricky (goes beyond my technical capabilitites) but it has to do with probably different libraries (and methodologies?) used to compute eigenvalues and eigenvectors plus the fact that eigenvector signs are non-identified anyway.

In typical econometric factor models you may want the first factor that explains much of the data variability and enters the real gdp growth to be say the business cycle component through which you will built a coincident indicator or do some other "job". However, it turns to be negatively correlated with real gdp (looks upside-down). It's ok. You multiply all produced factors by (-1) and all loadings by (-1) as well (Λ*(-1)*(-1)*F). Nothing changes statistically.

 

Yiannis