Download Locally Stationary Factor Models

Locally Stationary Factor Models
Giovanni Motta
Supervisors: Professors Rainer von Sachs & Christian M. Hafner
Abstract
Linear factor models have attracted considerable interest over recent years especially in the econometrics literature. The intuitively appealing idea to explain a panel of economic variables by a few common
factors is one of the reasons for their popularity. From a statistical viewpoint, the need to reduce the
cross-section dimension to a much smaller factor space dimension is obvious considering the large data
sets available in economics and finance.
The traditional approach of fixing either the time dimension T or the cross-section dimension N and
letting the other dimension go to infinity is likely to be inappropriate in situations where both dimensions
are large. The large factor model literature, including Stock and Watson (2002a, 2002b), Forni et al.
(2000, 2005), Forni and Lippi (2001), Bai and Ng (2002) and Bai(2003), use the concept introduced by
Chamberlain and Rothschild (1983) of simultaneous asymptotics, where both N and T go to infinity
with rates that are rather flexible.
One of the characteristics of the traditional factor model is that the process is stationary in the time
dimension. This appears restrictive, given the fact that over long time periods it is unlikely that e.g.
factor loadings remain constant. For example, in the capital asset pricing model (CAPM) of Sharpe
(1964) and Lintner (1965), typical empirical results show that factor loadings are time-varying, which in
the CAPM is caused by time-varying second moments.
In this thesis we propose two new approximate factor models for large cross-section and time dimensions.
In the first model factor loadings are assumed to be smooth functions of time, which allows to consider
the model as locally stationary while permitting empirically observed time-varying second moments.
Factor loadings are estimated by the eigenvectors of a nonparametrically covariance matrix. As is wellknown in the stationary case, this principal components estimator is consistent in approximate factor
models if the eigenvalues of the noise covariance matrix are bounded. To show that this carries over
to our locally stationary factor model is one of the main objective of this thesis. Under simultaneous
asymptotics (cross-section and time dimension go to infinity simultaneously), we give conditions for
consistency of our estimators of the time varying covariance matrix, the loadings and the factors. We
generalize to the locally stationary case the results given by Bai (2003) in the stationary framework. A
simulation study illustrates the performance of these estimators.
This approach can be generalized even further. In the second approach we propose a factor model with
infinite dynamics characterized by a locally stationary behavior à la Dahlhaus (1997) and nonorthogonal
idiosyncratic components as in the static approximate factor model of Chamberlain and Rothschild
(1983). This model generalizes the dynamic (but stationary) factor model of Forni et al. (2000), as well
as the nonstationary (but static) factor model of Motta et al. (2006). In the stationary (dynamic) case,
Forni et al. (2000) show that the common components are estimated by the eigenvectors of a consistent
estimator of the spectral density matrix, which is a matrix depending only on the frequency. In the locally
stationary framework the dynamics of the model is explained by a time-varying spectral density matrix.
This operator is a function of time as well as of the frequency. We show that the common components of a
locally stationary dynamic factor model can be estimated consistently by the eigenvectors of a consistent
estimator of the time-varying spectral density matrix.
References
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