Abstract
Factor model methods recently have become extremely popular in the theory and practice of large panels of time series data. Those methods rely on various factor models which all are particular cases of the Generalized Dynamic Factor Model (GDFM) introduced in Forniet al. (2000). That paper, however, rests on Brillinger’s dynamic principal components. The corresponding estimators are two-sided filters whose performance at the end of the observation period or for forecasting purposes is rather poor. No such problem arises with estimators based on standard principal components, which have been dominant in this literature. On the other hand, those estimators require the assumption that the space spanned by the factors has finite dimension. In the present paper, we argue that such an assumption is extremely restrictive and potentially quite harmful. Elaborating upon recent results by Anderson and Deistler (2008a, b) on singular stationary processes with rational spectrum, we obtain one-sided representations for the GDFM without assuming finite dimension of the factor space. Construction of the corresponding estimators is also briefly outlined. In a companion paper, we establish consistency and rates for such estimators, and provide Monte Carlo results further motivating our approach.
Highlights
1.1 Dynamic factor modelsHigh-dimensional factor model methods can be traced back to two seminal papers by Chamberlain (1983) and Chamberlain and Rothschild (1983)
Apart for some minor features, most factor models considered in the literature are particular cases of the so-called Generalized Dynamic Factor Model (GDFM) introduced in Forni et al (2000)
We are interested in the case n > q. Such “tall systems” have been studied recently by Anderson, Deistler and their coauthors. One of their results is that when n > q, there exists a nowhere dense set N ⊂ Πn, i.e. a set whose closure has no interior points, such that if the parameter vector lies in Πn − N, yt has an autoregressive representation of the form
Summary
High-dimensional factor model methods can be traced back to two seminal papers by Chamberlain (1983) and Chamberlain and Rothschild (1983). Using the frequency-domain principal components (Brillinger 1981), and without any finite-dimensional assumption of the form (1.2), Forni et al (2000) obtain an estimator of the spectral density of the common components χit and show how to consistently recover the common components themselves. Such examples provide a strong theoretical motivation for solving the one-sidedness problem in model (1.1) without turning to the finite-dimension restriction and the related assumptions and methods This is done in the present paper under assumptions that include rational spectral density for the common components χit.[5]. We must point out that, even when the finite-dimension assumption does not hold, model (1.2), or (1.3), can provide a good approximation to model (1.1), or, in empirical situations, with n and T given, a good fit or a good performance in forecasting These problems are not studied in the present paper, in which we only deal with representation issues and make use of population covariances.
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