Functional lagged regression with sparse noisy observations

Abstract

A functional (lagged) time series regression model involves the regression of scalar response time series on a time series of regressors that consists of a sequence of random functions. In practice, the underlying regressor curve time series are not always directly accessible, but are latent processes observed (sampled) only at discrete measurement locations. In this article, we consider the so-called sparse observation scenario where only a relatively small number of measurement locations have been observed, possibly different for each curve. The measurements can be further contaminated by additive measurement error. A spectral approach to the estimation of the model dynamics is considered. The spectral density of the regressor time series and the cross-spectral density between the regressors and response time series are estimated by kernel smoothing methods from the sparse observations. The impulse response regression coefficients of the lagged regression model are then estimated by means of ridge regression (Tikhonov regularization) or principal component analysis (PCA) regression (spectral truncation). The latent functional time series are then recovered by means of prediction, conditioning on all the observed data. The performance and implementation of our methods are illustrated by means of a simulation study and the analysis of meteorological data.

Yield curve and macroeconomy interaction: evidence from the non-parametric functional lagged regression approach

“…t ) = X t are the elements of the vector of macroeconomic variables, we propose to model the link between the yield curve functional time series {Y t (τ )} and the multivariate time series of macroeconomic variables {X } by the lagged regression model [Brillinger, 1981, Hörmann et al, 2015b, Rubín and Panaretos, 2019 manifested by the equation…”

“…Under the model (2.2) and the assumptions (2.3), (2.4), the lagged cross-covariance functions satisfies also the weak dependence [Rubín and Panaretos, 2019] h∈Z…”

“…we have now estimated all components in order to estimate the filter coefficients by the empirical version of the formula (2.7). It is noteworthy to stress that in contrast to the functional regression models with functional regressor time series [Hörmann et al, 2015b, Pham and Panaretos, 2018, Rubín and Panaretos, 2019, our setting does not require regularisation of the spectral density matrix inversion FX ω , assuming the population spectral density operator F X ω is invertible for each ω ∈ [−π, π], see Brillinger [1981]. Hence…”

“…We consider the US Treasury yield curve as sparsely observed functional time series and estimate the crossdependence between this data set and the US macroeconomic variables using the local-polynomial smoother techniques Panaretos, 2019, 2020]. We model the dependence between the yield curve and the macroeconomic variables by the means of the functional lagged regression [Hörmann et al, 2015b, Pham and Panaretos, 2018, Rubín and Panaretos, 2019 and estimated the filter-based regression coefficients. The results of our analysis confirm the findings of Diebold and Li [2006] attained under parametric assumptions.…”

Bayesian surface regression versus spatial spectral nonparametric curve regression

“…The asymptotic normality of the functional discrete Fourier transform of the curve data is previously proved, under suitable functional cumulant mixing conditions, and the summability in time of the trace norm of the elements of the covariance operator family (see also Tavakoli [71]). In Panaretos and Tavakoli [62], a Karhunen-Loéve-like decomposition in the temporal functional spectral domain is derived, the so-called Cramér-Karhunen-Loéve representation, providing a harmonic principal component analysis of functional time series (see also some recent applications in the context of functional regression in Pham and Panaretos [64], and Rubin and Panaretos [66]). In addition, Rubin and Panaretos [67] propose simulation techniques based on the Cramér-Karhunen-Loéve representation.…”