Locally Stationary Factor Models: Identification and Nonparametric Estimation

Abstract

In this paper we propose a new approximate factor model for large cross-section and time dimensions. Factor loadings are assumed to be smooth functions of time, which allows considering the model aslocally stationarywhile permitting empirically observed time-varying second moments. Factor loadings are estimated by the eigenvectors of a nonparametrically estimated covariance matrix. As is well known 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 the main objective of our paper. Under simultaneous asymptotics (cross-section and time dimension go to infinity simultaneously), we give conditions for consistency of our estimators. A simulation study illustrates the performance of these estimators.

Shrinkage estimation in the frequency domain of multivariate time series

“…In [8], it is used to distinguish between idiosyncratic and global common components in the analysis of economic panel data. The authors of [22] use it to identify time variant factors driving a nonstationary time series, where time is rescaled according to the Dahlhaus model of locally stationary time series [23]. In our work, we use it to derive an enhanced estimator of the spectrum that asymptotically has minimal risk in a class of linear estimators that is chosen to approximately compensate for the bias of the eigenvalues of the averaged periodogram.…”

Fitting dynamic factor models to non-stationary time series

“…Furthermore we do not content ourselves to work with a static factor model in order to avoid taking a potentially large number of factors into account. In this respect we generalize both the work by Motta et al (2011) on time-varying static factors and by Forni et al (2000) on stationary dynamic factor modelling. With our approach we contribute to the yet recent literature on dimensionreduction of multivariate time series with a possibly time-varying correlation, where the latter one will allow to work with a considerably smaller number of factors (common components) to explain the co-movements in a large panel of observed time series: it is obvious that allowing the dynamics of a few common components to slowly change over time will allow for very sparse modelling.…”

“…and it is estimated non-parametrically according to Motta et al (2011) (the computation of confidence intervals is based on the asymptotic normality of the estimator). Fig.…”

“…With this approach, we provide for a true generalization of two classes of existing factor models in the literature: static factors with time-varying loadings recently developed by Motta et al (2011), where no serial correlation has been allowed in the common components; and stationary, i.e. time-constant, dynamic factor models with a dynamic structure using (possibly infinite-length) filters of the factors via dynamic loadings, such as in Forni et al (2000).…”

“…As such, the development of a rigorous asymptotic theory for consistently estimating the time-varying loadings has been possible by embedding this model into the framework of locally stationary processes as derived by a series of papers by Dahlhaus (1996Dahlhaus ( , 1997Dahlhaus ( , 2000. We note that our approach is also related to the time-varying static approach of Motta et al (2011), which generalizes the static factor model of Bai (2003).…”