FRB,FEDS Paper: Quasi Maximum Likelihood Estimation and Inference of Large Approximate Dynamic Factor Models via the EM algorithm

FEDS Paper: Quasi Maximum Likelihood Estimation and Inference of Large Approximate Dynamic Factor Models via the EM Algorithm

Abstract

This paper presents a new estimation and inference method for large approximate dynamic factor models (ADFMs) using the Expectation-Maximization (EM) algorithm. The proposed method, called quasi maximum likelihood (QML) estimation, is based on a reformulation of the ADFMs as a conditional latent variable model. This reformulation allows us to apply the EM algorithm to obtain QML estimates of the model parameters. The paper shows that the QML estimates are consistent and asymptotically normal, and provides a detailed discussion of the implementation of the EM algorithm for ADFMs. The paper also presents a simulation study that compares the performance of the QML estimates to the performance of alternative estimation methods. The simulation results show that the QML estimates are more accurate and efficient than the alternative estimates, especially when the sample size is small and the number of factors is large.

Introduction

Dynamic factor models (DFMs) are a popular class of models used to represent the dynamics of a large number of time series. ADFMs are a special case of DFMs that are used to represent the dynamics of a large number of time series that are approximately cointegrated. ADFMs have been used in a variety of applications, including forecasting, risk management, and monetary policy analysis.

The traditional approach to estimating ADFMs is based on maximum likelihood (ML) estimation. However, ML estimation of ADFMs can be computationally intensive, especially when the sample size is large and the number of factors is large. This has motivated the development of alternative estimation methods, such as principal components (PC) analysis and partial least squares (PLS) regression. However, these alternative estimation methods are not as efficient as ML estimation.

Quasi Maximum Likelihood Estimation

The QML estimation method proposed in this paper is a computationally efficient alternative to ML estimation. The QML method is based on a reformulation of the ADFMs as a conditional latent variable model. This reformulation allows us to apply the EM algorithm to obtain QML estimates of the model parameters.

The EM algorithm is an iterative algorithm that alternates between two steps: the expectation step (E-step) and the maximization step (M-step). In the E-step, the conditional expectation of the unobserved latent variables is computed given the observed data and the current estimates of the model parameters. In the M-step, the model parameters are updated to maximize the expected log-likelihood function. The EM algorithm continues to iterate between the E-step and the M-step until it converges.

Implementation of the EM Algorithm

The implementation of the EM algorithm for ADFMs is straightforward. The E-step involves computing the conditional expectation of the unobserved latent variables, which is given by

E[F_t | X_1, ..., X_T] = Sigma_F Sigma_X^{-1} X_t

where F_t is the vector of latent factors at time t, X_t is the vector of observed time series at time t, Sigma_F is the covariance matrix of the latent factors, and Sigma_X is the covariance matrix of the observed time series.

The M-step involves updating the model parameters to maximize the expected log-likelihood function. The update equations for the model parameters are given by

Sigma_F = E[(F_t - Sigma_F Sigma_X^{-1} X_t)(F_t - Sigma_F Sigma_X^{-1} X_t)'] Sigma_X = E[(X_t - Sigma_X Sigma_F^{-1} F_t)(X_t - Sigma_X Sigma_F^{-1} F_t)'] Lambda = E[X_t Sigma_F^{-1} F_t']

where Lambda is the matrix of factor loadings.

Simulation Study

A simulation study was conducted to compare the performance of the QML estimates to the performance of the alternative estimates. The simulation results show that the QML estimates are more accurate and efficient than the alternative estimates, especially when the sample size is small and the number of factors is large.

Conclusion

The QML estimation method proposed in this paper is a computationally efficient alternative to ML estimation of ADFMs. The QML method is based on a reformulation of the ADFMs as a conditional latent variable model, which allows us to apply the EM algorithm to obtain QML estimates of the model parameters. The simulation results show that the QML estimates are more accurate and efficient than the alternative estimates, especially when the sample size is small and the number of factors is large.


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

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