This paper studies Quasi Maximum Likelihood estimation of dynamic factor models for large panels of time series. Specifically, we consider the case in which the autocorrelation of the factors is explicitly accounted for and therefore the factor model has a state-space form. Estimation of the factors and  their loadings is implemented by means of the Expectation Maximization  algorithm, jointly with the Kalman smoother.  We prove that, as both the dimension of the panel n and the sample size T diverge to infinity , the estimated loadings , factors , and common components are \min(\sqrt n,\sqrt T\to \to  )-consistent and asymptotically normal  . Although the model is estimated under the unrealistic constraint of independent idiosyncratic errors, this mis-specification does not affect consistency. Moreover, we give conditions under which the derived asymptotic distribution can still be used for inference even in case of mis-specifications. Our results are confirmed by a MonteCarlo simulation exercise where we compare the performance of our estimators with Principal Components .
This paper studies Quasi Maximum Likelihood estimation of dynamic factor models for large panels of time series. Specifically, we consider the case in which the autocorrelation of the factors is explicitly accounted for and therefore the model has a state-space form. Estimation of the factors and of their loadings is implemented by means of the Expectation Maximization (EM) algorithm, jointly with the Kalman smoother. ~ We prove that, as both the dimension of the panel n and the sample size T diverge to infinity : (i) the estimated loadings are \sqrt T-consistent and asymptotically normal if \sqrt T/n\to 0; (ii) the estimated factors are \sqrt n-consistent and asymptotically normal if \sqrt n/T\to 0; (iii) the estimated common component is \min(\sqrt T,\sqrt n )-consistent and asymptotically normal regardless of the relative rate of divergence of n and T . Although the model is estimated as if the idiosyncratic terms were cross-sectionally and serially uncorrelated, we show that these mis-specifications do not affect consistency. Moreover, the estimated loadings are asymptotically as efficient as those obtained with the Principal Components estimator, whereas numerical results show that the loss in efficiency of the estimated factors becomes negligible as n and T increase.~We then propose robust estimators of the asymptotic covariances, which can be used to conduct inference on the loadings and to compute confidence intervals for the factors and common components. In a MonteCarlo simulation exercise and an analysis of US macroeconomic data, we study the performance of our estimators and we compare them with the traditional Principal Components approach .
D
factor
A
of
A
(EM)
A
~
R
,
: (i)
R
, factors , and common components are \min(\sqrt n,\sqrt T
are \sqrt T-consistent and asymptotically normal if \sqrt T/n
A
0; (ii) the estimated factors are \sqrt n-consistent and asymptotically normal if \sqrt n/T
A
0; (iii) the estimated common component is \min(\sqrt T,\sqrt n
A
regardless of the relative rate of divergence of n and T
R
under the unrealistic constraint of independent idiosyncratic errors, this mis-specification does
as if the idiosyncratic terms were cross-sectionally and serially uncorrelated, we show that these mis-specifications do
R
we give conditions under which the derived asymptotic distribution can still be used for inference even in case of mis-specifications. Our results are confirmed by
the estimated loadings are asymptotically as efficient as those obtained with the Principal Components estimator, whereas numerical results show that the loss in efficiency of the estimated factors becomes negligible as n and T increase.~We then propose robust estimators of the asymptotic covariances, which can be used to conduct inference on the loadings and to compute confidence intervals for the factors and common components. In
R
where we compare
and an analysis of US macroeconomic data, we study
R
with Principal Components
and we compare them with the traditional Principal Components approach