A quasi maximum likelihood approach for large approximate dynamic factor models

TLDR: In this article, the authors show that the common factors based on maximum likelihood are consistent for the size of the cross-section (n) and the sample size (T) going to infinity along any path of n and T and therefore maximum likelihood is viable for n large.

Abstract: Is maximum likelihood suitable for factor models in large cross-sections of time series? We answer this question from both an asymptotic and an empirical perspective. We show that estimates of the common factors based on maximum likelihood are consistent for the size of the cross-section (n) and the sample size (T) going to infinity along any path of n and T and that therefore maximum likelihood is viable for n large. The estimator is robust to misspecification of the cross-sectional and time series correlation of the the idiosyncratic components. In practice, the estimator can be easily implemented using the Kalman smoother and the EM algorithm as in traditional factor analysis.

Figures

Table 3: Simulation results for the model: ρ = .9, d = 0, τ = 0, u = .1, r = 3

Full text

ISSN 1561081-0
9 771561 081005
WORKING PAPER SERIES
NO 674 / SEPTEMBER 2006
A QUASI MAXIMUM
LIKELIHOOD APPROACH
FOR LARGE APPROXIMATE
DYNAMIC FACTOR
MODELS
by Catherine Doz,
Domenico Giannone
and Lucrezia Reichlin

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WORKING PAPER SERIES
NO 674 / SEPTEMBER 2006
This paper can be downloaded without charge from
http://www.ecb.int or from the Social Science Research Network
electronic library at http://ssrn.com/abstract_id=927425
A QUASI MAXIMUM
LIKELIHOOD APPROACH
FOR LARGE APPROXIMATE
DYNAMIC FACTOR
MODELS
1
by Catherine Doz
2
,
Domenico Giannone
3
and Lucrezia Reichlin
4
1 We would like to thank Ursula Gather and Marco Lippi for helpful suggestions and seminar participants at the International
Statistical Institute in Berlin 2003, the European Central Bank, 2003, the Statistical Institute at the Catholic University
of Louvain la Neuve, 2004, the Institute for Advanced Studies in Vienna, 2004, the Department of Statistics
in Madrid, 2004. The opinions in this paper are those of the authors and do not necessarily
reflect the views of the European Central Bank.
2 Directrice de l'UFR Economie Gestion, University of Cergy-Pontoise - Department of Economics, 33 Boulevard du port,
F-95011 Cergy-Pontoise Cedex, France; e-mail: catherine.doz@eco.u-cergy.fr
3 Free University of Brussels (VUB/ULB) - European Center for Advanced Research in Economics and Statistics (ECARES),
Ave. Franklin D Roosevelt, 50 - C.P. 114, B-1050 Brussels, Belgium; e-mail: dgiannon@ulb.ac.be
4 European Central Bank, ECARES and CEPR; Kaiserstrasse 29, 60311 Frankfurt am Main, Germany;
e-mail: lucrezia.reichlin@ecb.int

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ISSN 1561-0810 (print)
ISSN 1725-2806 (online)

3
ECB
Working Paper Series No 674
September 2006
CONTENTS
Abstract 4
Non-technical summary 5
1 Introduction 6
2 Models 8
2.1 Notation 8
2.2 The approximate dynamic factor model 8
2.3 The approximating factor models 10
3 The asymptotic properties of the QML
estimator of the common factors 12
4 Monte Carlo study 14
5 Summary and conclusions 19
References 20
6 Appendix 23
European Central Bank Working Paper Series 33

Abstract
This paper considers quasi-maximum likelihood estimations of a dynamic ap-
proximate factor model when the panel of time series is large. Maximum likelihood
is analyzed under different sources of misspecification: omitted serial correlation of
the observations and cross-sectional correlation of the idiosyncratic components. It
is shown that the effects of misspecification on the estimation of the common factors
is negligible for large sample size (T) and the cross-sectional dimension (n). The
estimator is feasible when n is large and eas ily implementable using the Kalman
smoother and the EM algorithm as in traditional factor analysis. Simulation results
illustrate what are the empirical conditions in which we can expect improvement
with respec t to simple principle components considered by Bai (2003), Bai and
Ng (2002), Forni, Hallin, Lippi, and Reichlin (2000, 2005b), Stock and Watson
(2002a,b).
JEL Classification: C51, C32, C33.
Keywords: Factor Model, large cross-sections, Quasi Maximum Likelihood.
4
ECB
Working Paper Series No 674
September 2006

Frequently asked questions

Q1. What are the contributions in "A quasi maximum likelihood approach for large approximate dynamic factor models" ?

In this paper, the authors proposed an approximate factor model with orthogonal idiosyncratic elements, which is called an exact factor model. 

Q2. What have the authors stated for future works in "A quasi maximum likelihood approach for large approximate dynamic factor models" ?

These features are not studied in this paper but they are natural extensions to explore in further work. 

Q3. What is the computational complexity of the Kalman smoother?

The computational complexity of the Kalman smoother depends mainly on the number of states which in their approximating model corresponds to the number of factors, r, and hence is independent of the size of the cross-section n. 

Q4. How many times do the authors apply the algorithm to standardized data?

The authors simulate the model 500 times and, at each repetition, the authors apply the algorithm to standardized data since the principal components used for initialization are not scale invariant. 

Q5. Why is it necessary to impose a parametrization while retaining parsimony?

The reason is that, in order to estimate the model by maximum likelihood, it is necessary to impose a parametrization while retaining parsimony. 

Q6. How can the authors compute the expected value of the common factors?

Given the quasi maximum likelihood estimates of the parameters θ, the common factors can be approximated by their expected value, which can be computed using the Kalman smoother: 

Q7. Why do the ML estimates dominate the principal components?

Because of the explicit modelling of the dynamics and the cross-sectional heteroscedasticity, the maximum likelihood estimates dominate the principal components and, to a less extent, the two two-step procedure. 

Q8. What is the effect of the misspecification error?

This result tells us that the misspecification error due to the approximate structure of the idiosyncratic component vanishes asymptotically for n and T large, provided that the cross-correlation of the idiosyncratic processes is limited and that of the common components is pervasive throughout the cross section as n increases. 

Q9. What is the bias arising from the approximating model?

The bias arising from this misspecification of the approximating model is negligible if the cross-sectional dimension is large enough.