State space form of the dynamic factor model
Description of the model
The model contains $m$ factors $F_t = \begin{pmatrix}f_{1,t}\ \cdots \ f_{m, t}\end{pmatrix}$. The dynamic of the factors is defined by a VAR with $p$ lags.
\[F_t = A_1 F_{t-1} + \cdots + A_p F_{t-p} + \mu_t\]where $\mu \sim N(0, V)$
State array
The state array is a stack composed of all the lags of each factor. The number of lags introduced in the state array ($q$) must be sufficient to deal with the dynamics of the model ($\ge p$), to deal with the different measurement equations and - eventually - to deal with the needs of a news analysis. More formally, it is defined as:
\[\alpha_t = \begin{pmatrix} f_{1,t} \\ \vdots \\ f_{1,t-q} \\ \vdots \\ f_{m,t} \\ \vdots \\ f_{m,t-q}\end{pmatrix}\]The VAR component is described in the class demetra.var.VarDescriptor.
It is fully identified by the following information:
- nfactors: the number of factors in the VAR ($m$)
- nlags: the number of lags in the VAR ($p$)
- varMatrix: the coefficients of the VAR ($A_1,\dots,A_p$), which is a $m \times (m*p)$ matrix
- the covariance of the innovations ($V$), which is a $m \times m$ matrix