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 $\mathbf{\Lambda}_i$ is an $n\times k$ loading matrix of full rank, $\{\mathbf{f}_t\}$ are i.i.d. $k\times1$-factors, and $\mathbf{e}_t$ 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 $$\mathbf{\Phi}_n(\tau)=\frac{1}{2T}\sum_{j=1}^T (\mathbf{R}_j \mathbf{R}_{j+\tau}^* + \mathbf{R}_{j+\tau} \mathbf{R}_j^*),$$ where $\tau\ge 0$ is an integer. When there are no factors, the matrix $\Phi_{n}(\tau)$ reduces to $$\mathbf{M}_n(\tau) = \frac{1}{2T} \sum_{j=1}^T (\mathbf{e}_j \mathbf{e}_{j+\tau}^* + \mathbf{e}_{j+\tau} \mathbf{e}_j^*).$$ 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 $\mathbf{\Phi}_n(\tau)$.