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 $\mathbf{ \Phi}_n(\tau)$.
Categories: math.ST stat.TH