As we saw in the post on vector autoregressive (VAR) models, traditional VAR models suffer from a so-called curse of dimensionality. Each additional variable adds a further equation and a further lag polynomial to every other equation, so the number of coefficients grows with the square of the number of variables. With the few decades of quarterly data that are usually available this limits a VAR to a handful of variables. Policy makers, however, base their decisions on a much larger set of indicators than a VAR can accommodate, and leaving that information out of the model can distort the results.
For some macroeconomic applications it might be interesting to see whether a set of observable variables depends on common drivers. The estimation of such common factors can be done using so-called factor analytical models, which have the form
\[x_t = \lambda f_t + u_t,\]
where \(x_t\) is an \(M\)-dimensional vector of observable variables, \(f_t\) is an \(N \times 1\) vector of unobserved factors, \(\lambda\) is an \(M \times N\) matrix of factor loadings and \(u_t\) is an error term.