Time Series Modelling With Semiparametric Factor Dynamics

Abstract

High-dimensional regression problems which reveal dynamic behavior are typically analyzed by time propagation of a few number of factors. The inference on the whole system is then based on the low-dimensional time series analysis. Such highdimensional problems occur frequently in many different fields of science. In this paper we address the problem of inference when the factors and factor loadings are estimated by semiparametric methods. This more flexible modelling approach poses an important question: Is it justified, from inferential point of view, to base statistical inference on the estimated times series factors? We show that the difference of the inference based on the estimated time series and 'true' unobserved time series is asymptotically negligible. Our results justify fitting vector autoregressive processes to the estimated factors, which allows one to study the dynamics of the whole high-dimensional system with a low-dimensional representation. We illustrate the theory with a simulation study.Also, we apply the method to a study of the dynamic behavior of implied volatilities and discuss other possible applications in finance and economics.

Modelling spatio-temporal variability of temperature

“…Fengler et al (2007) and Park et al (2009) generalize a dynamic semiparametric factor model for high-dimensional time series, in which linear combinations of factors are utilized for a low-dimensional representation of the data. The estimated factors and factor loadings are driven by historical observations and reflect the dynamics of time series and the spatial, time-invariant component, respectively.…”

“…and are also known as common factors and factor loadings, respectively. The basis functions are estimated via B-spline series (Park et al 2009):…”

“…As an alternative to standard Kriging procedures, we consider a dynamic semiparametric factor model (DSFM) which has recently been suggested by Song et al (2010Song et al ( , 2013 for analyzing high dimensional data. A DSFM is able to smooth high dimensional data via dimension reduction and to forecast in both time and space (see Park et al (2009)). The nature of a DSFM is based on parametric estimation in time and nonparametric estimation in space.…”

Yield curve modeling and forecasting using semiparametric factor dynamics

“…Some weak conditions on the initial choice of vec(A (0) ), Z (0) t ensure the convergence to the true unknown parameters matrix A and factor loadings Z t . It was proved by Park et al (2009), that the differences between the estimates Z t and the true, unobserved loadings Z t can be asymptotically neglected. This fact allows us to model the dynamics of factor loadings based on estimated time series and therefore study the dynamics of the main, high-dimensional object of interest.…”

“…The evolution in time is driven by time-varying factor loadings (in FPCA defined as scores), which are modeled parametrically employing a multivariate autoregressive approach. The factor decomposition is obtained by the Dynamic Semiparametric Factor Model (DSFM) also analyzed in Fengler et al (2007), Brüggemann et al (2008) and Park et al (2009). Accordingly, the term structure of interests rates is modeled in terms of underlying latent factors, which are defined on the time to maturity grid space and may depend on additional explanatory variables.…”