The empirical distribution of sequential LS factors in Multi-level Dynamic Factor Models

Abstract

The research question we answer in this paper is whether the asymptotic distribution derived by Bai (2003) for Principal Components (PC) factors in dynamic factor models (DFMs) can approximate the empirical distribution of the sequential Least Squares (SLS) estimator of global and group-specific factors in multi-level dynamic factor models (ML-DFMs). Monte Carlo experiments confirm that under general forms of the idiosyncratic covariance matrix, the finite-sample distribution of SLS global and group-specific factors can be well approximated using the asymptotic distribution of PC factors. We also analyse the performance of alternative estimators of the asymptotic mean squared error (MSE) of the SLS factors and show that the MSE estimator that allows for idiosyncratic cross-sectional correlation and accounts for estimation uncertainty of factor loadings is best.

Figures (6)

• Figure 1: Loadings of the global factor (blue), the first group-specific factor (orange), and the second group-specific factor (green) in the ML-DFM with $N_1 = N_2 = 25$.
• Figure 2: Loadings of each of the first (blue), second (orange) and third (green) factors in the DFM with $N=50$.
• Figure 3: Histograms of the estimated factors in a DFM with $r = 3$ at three time points: $t = 1$ (first column), $t = T/2$ (second column), and $t = T$ (third column), for the first factor (top row), second factor (middle row), and third factor (bottom row). The red vertical line indicates the true factor value. The orange curve represents the asymptotic density, computed using the MSE with the true parameter values. The idiosyncratic errors are cross-sectionally homoscedastic and uncorrelated. Each panel corresponds to a different $(N,T)$ configuration.
• Figure 4: Histograms of the estimated global factor, $G_t$ (first column), specific factor of the first block $L_{1t}$ (second column), and specific factor of the second block, $L_{2t}$ (third column), at three particular moments of time: i) $t=1$ (first column); ii) $t=T/2$ (second column); and iii) $t=T$ (third column). The red vertical line represents the true factor's value. The idiosyncratic errors are cross-sectionally homoscedastic and uncorrelated.
• Figure 5: Histograms of the estimated global factor, $G_t$ (first column), specific factor of the first block $L_{1t}$ (second column), and specific factor of the second block, $L_{2t}$ (third column), at three particular moments of time: i) $t=1$ (first column); ii) $t=T/2$ (second column); and iii) $t=T$ (third column). The red vertical line represents the true factor's value. The idiosyncratic errors are cross-sectionally uncorrelated and heteroscedastic.