Order Determination of Large Dimensional Dynamic Factor Model

Consider the following dynamic factor model: $\bbR_{t}=\sum_{i=0}^{q}\bgL_{i}\bbf_{t-i}+\bbe_{t},t=1,...,T$, where $\bgL_{i}$ is an $n\times k$ loading matrix of full rank, $\{\bbf_t\}$ are i.i.d. $k\times1$-factors, and $\bbe_t$ are independent $n\times1$ white noises. Now, assuming that $n/T\to c>0$, we want to estimate the orders $k$ and $q$ respectively. Define a random matrix $\bbPhi_{n}(\tau)=\frac{1}{2T}\sum_{j=1}^{T}(\bbR_{j}\bbR_{j+\tau}^{*}+\bbR_{j+\tau}\bbR_{j}^{*}),$ where $\tau\ge 0$ is an integer. When there are no factors, the matrix $\Phi_{n}(\tau)$ reduces to $\bbM_n(\tau)=\frac{1}{2T}\sum_{j=1}^T(\bbe_{j}\bbe_{j+\tau}^{*}+\bbe_{j+\tau}\bbe_{j}^{*}).$ When $\tau=0$, $\bbM_n(\tau)$ reduces to the usual sample covariance matrix whose ESD tends to the well known MP law and $\bbPhi_n(0)$ reduces to the standard spike model. Hence the number $k(q+1)$ can be estimated by the number of spiked eigenvalues of $\bbPhi_n(0)$. To obtain separate estimates of $k$ and $q$ , we have employed the spectral analysis of $\bbM_n(\tau)$ and established the spiked model analysis for $\bbPhi_n(\tau)$.