Order Determination of Large Dimensional Dynamic Factor Model

Consider the following dynamic factor model: $$\mathbf{R}$t=\sum{i=0}^q $\mathbf{\Lambda }$i $\mathbf{f}${t-i}+$\mathbf{e}$_t,t=1,…,T$,where$$\mathbf{\Lambda }$_i$isan$n\times k$loadingmatrixoffullrank,${$\mathbf{f}$t}$arei\mathrm{.}i\mathrm{.}d\mathrm{.}$k\times1$-factors,and$$\mathbf{e}$t$ are independent $n\times 1$ white noises. Now, assuming that $n\mathrm{/}T\to c>0$, we want to estimate the orders $k$ and $q$ respectively. Define a random matrix $${\mathbf{\Phi }}_n(\tau )=\frac{1}{2T}\sum _{j=1}^T({\mathbf{R}}_j{\mathbf{R}}_{j+\tau }^{\ast }+{\mathbf{R}}_{j+\tau }{\mathbf{R}}_j^{\ast }),$$ where $\tau\ge 0$isaninteger\mathrm{.}Whentherearenofactors,thematrix$\Phi{n}(\tau)$reducesto$$$\mathbf{M}$n(\tau) = \frac{1}{2T} \sum{j=1}^T ($\mathbf{e}$j $\mathbf{e}${j+\tau}^* + $\mathbf{e}${j+\tau} $\mathbf{e}$_j^*).$$When$\tau=0$,$$\mathbf{M}$_n(\tau)$ reduces to the usual sample covariance matrix whose ESD tends to the well known MP law and ${\mathbf{\Phi }}_n(0)$ reduces to the standard spike model. Hence the number $k(q+1)$ can be estimated by the number of spiked eigenvalues of ${\mathbf{\Phi }}_n(0)$. To obtain separate estimates of $k$ and $q$ , we have employed the spectral analysis of ${\mathbf{M}}_n(\tau )$ and established the spiked model analysis for ${\mathbf{\Phi }}_n(\tau )$.