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Quasi Maximum Likelihood Estimation and Inference of Large Approximate Dynamic Factor Models via the EM algorithm

Matteo Barigozzi, Matteo Luciani

Abstract

We study estimation of large Dynamic Factor models implemented through the Expectation Maximization (EM) algorithm, jointly with the Kalman smoother. We prove that as both the cross-sectional dimension, $n$, and the sample size, $T$, diverge to infinity: (i) the estimated loadings are $\sqrt T$-consistent, asymptotically normal and equivalent to their Quasi Maximum Likelihood estimates; (ii) the estimated factors are $\sqrt n$-consistent, asymptotically normal and equivalent to their Weighted Least Squares estimates. Moreover, the estimated loadings are asymptotically as efficient as those obtained by Principal Components analysis, while the estimated factors are more efficient if the idiosyncratic covariance is sparse enough.

Quasi Maximum Likelihood Estimation and Inference of Large Approximate Dynamic Factor Models via the EM algorithm

Abstract

We study estimation of large Dynamic Factor models implemented through the Expectation Maximization (EM) algorithm, jointly with the Kalman smoother. We prove that as both the cross-sectional dimension, , and the sample size, , diverge to infinity: (i) the estimated loadings are -consistent, asymptotically normal and equivalent to their Quasi Maximum Likelihood estimates; (ii) the estimated factors are -consistent, asymptotically normal and equivalent to their Weighted Least Squares estimates. Moreover, the estimated loadings are asymptotically as efficient as those obtained by Principal Components analysis, while the estimated factors are more efficient if the idiosyncratic covariance is sparse enough.

Paper Structure

This paper contains 42 sections, 78 theorems, 614 equations, 8 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Related literature
  3. Estimation via the EM algorithm
  4. E-step and Kalman smoother
  5. M-step
  6. Convergence of the EM algorithm and final estimators
  7. The Large Approximate Dynamic Factor Model
  8. Main assumptions
  9. Additional assumptions
  10. Identification conditions
  11. Identification of the number of factors and of the common component
  12. Identification of the loadings, the factors, and the VAR parameters
  13. Identification of the linear system
  14. Existence of the EM and QML estimators
  15. Asymptotic properties
  16. ...and 27 more sections

Key Result

Proposition 1

Consider the EM estimators of the parameters $\widehat{\bm\Lambda}_n=(\widehat{\bm \lambda}_1\cdots \widehat{\bm \lambda}_n)^\prime$ with $\widehat{\bm \lambda}_i \equiv \widehat{\bm \lambda}_i^{(k^*+1)}$, $\widehat{\sigma}_i^2\equiv \widehat{\sigma}_i^{2(k^*+1)}$ , $i=1,\ldots, n$, $\widehat{\mathb

Figures (8)

  • Figure 1: Simulation results - Convergence of the EM algorithm
  • Figure 2: Simulation results - Common components
  • Figure 3: Simulation results - Factors and loadings
  • Figure 4: Simulation results - Factors and loadings
  • Figure 5: Simulation results - Histograms of $Z_{it}^{(b)}$
  • ...and 3 more figures

Theorems & Definitions (93)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Proposition 1
  • Remark 6
  • Remark 7
  • Proposition 2: Loadings
  • Proposition 3: Factors
  • ...and 83 more