Publication: More is not always better : back to the Kalman filter in dynamic factor models
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2012-10
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Abstract
In the context of dynamic factor models (DFM), it is known that, if the cross-sectional
and time dimensions tend to infinity, the Kalman filter yields consistent smoothed
estimates of the underlying factors. When looking at asymptotic properties, the cross-
sectional dimension needs to increase for the filter or stochastic error uncertainty to
decrease while the time dimension needs to increase for the parameter uncertainty to
decrease. ln this paper, assuming that the model specification is known, we separate the
finite sample contribution of each of both uncertainties to the total uncertainty
associated with the estimation of the underlying factors. Assuming that the parameters
are known, we show that, as far as the serial dependence of the idiosyncratic noises is
not very persistent and regardless of whether their contemporaneous correlations are
weak or strong, the filter un-certainty is a non-increasing function of the cross-sectional
dimension. Furthermore, in situations of empirical interest, if the cross-sectional
dimension is beyond a relatively small number, the filter uncertainty only decreases
marginally. Assuming weak contemporaneous correlations among the serially
uncorrelated idiosyncratic noises, we prove the consistency not only of smooth but also
of real time filtered estimates of the underlying factors in a simple case, extending the
results to non-stationary DFM. In practice, the model parameters are un-known and
have to be estimated, adding further uncertainty to the estimated factors. We use
simulations to measure this uncertainty in finite samples and show that, for the sample
sizes usually encountered in practice when DFM are fitted to macroeconomic variables,
the contribution of the parameter uncertainty can represent a large percentage of the
total uncertainty involved in factor extraction. All results are illustrated estimating
common factors of simulated time series
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Common factors, Cross-sectional dimension, Filter uncertainty, Parameter uncertainty, Steady-state