Order Determination of Large Dimensional Dynamic Factor Model

TL;DR

This paper proposes a method to determine the order of large-dimensional dynamic factor models using spectral analysis of certain random matrices, extending spike model analysis to estimate factors and lags.

Contribution

It introduces a novel spectral analysis approach for estimating the number of factors and lags in large dynamic factor models, leveraging spiked eigenvalues of specific random matrices.

Findings

Consistent estimation of factor number and lag order in high-dimensional settings.

Extension of spike model analysis to dynamic factor models with lag structure.

Theoretical validation of eigenvalue-based estimation methods.

Abstract

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 $\mathbf{\Phi}_n(\tau)$.