1,721,061 research outputs found
Quasi Maximum Likelihood Estimation of High-Dimensional Factor Models
No full textFactor models are some of the most common dimension reduction techniques in time series econometrics. They are based on the idea that each element of a set of N time series is made of a common component driven by few latent factors capturing the main comovements among the series, plus idiosyncratic components oen representing just measurement error or at most being weakly cross-sectionally correlated with the other idiosyncratic components. When N is large the factors can be retrieved by cross-sectional aggregation of the observed time series. This is the so-called blessing of dimensionality, meaning that having N growing to infinity poses no estimation problem but in fact is a necessary condition for consistent estimation of the factors and for identification of the common and idiosyncratic components.
There exist two main ways to estimate a factor model: principal component analysis and maximum likelihood estimation. The former method is more recent and more common in econometrics, but the latter, which is the classical approach, has many appealing features such as allowing one to impose constraints, deal with missing values, and explicitly model the dynamic of the factors. Maximum likelihood estimation of large factor models has been studied in two influential papers: Doz et al.ʼs “A Quasi Maximum Likelihood Approach for Large Approximate Dynamic Factor Models” and Bai and Liʼs “Maximum Likelihood Estimation and Inference for Approximate Factor Models of High Dimension.” The latter considers the static case, which is closer to the classical approach and no model for the factors is assumed, and the former is more general: it considers estimation combined with the use of Kalman filtering techniques, which has grown popular in macroeconomic policy analysis.
Those two papers, together with other recent results, have brought new asymptotic results for which a synthesis is provided. Special attention is paid to the set of assumptions, which is taken to be the minimal set of assumptions required to get the results
The Dynamic, the Static, and the Weak: Factor Models and the Analysis of High‐Dimensional Time Series
No full textSeveral fundamental and closely interconnected issues related to factor models are reviewed and discussed: dynamic versus static
loadings, rate-strong versus rate-weak factors, the concept of weakly common component recently introduced by Gersing, the
irrelevance of cross-sectional ordering and the assumption of cross-sectional exchangeability, the impact of undetected strong
factors, and the problem of combining common and idiosyncratic forecasts. Conclusions all point to the advantages of the General
Dynamic Factor Model approach of Forni, Hallin, Lippi, and Reichlin over the widely used Static Approximate Factor Model
introduced by Chamberlain and Rothschild
Dynamic Factor Models: A Genealogy
No full textDynamic factor models have been developed out of the need of analyzing and forecasting time series in increasingly high dimensions. While mathematical statisticians faced with inference problems in high-dimensional observation spaces were focusing on the so-called spiked-model-asymptotics, econometricians adopted an entirely and considerably more effective asymptotic approach, rooted in the factor models originally considered in psychometrics. The so-called dynamic factor model methods, in two decades, has grown into a wide and successful body of techniques that are widely used in central banks, financial institutions, economic and statistical institutes. The objective of this chapter is not an extensive survey of the topic but a sketch of its historical growth, with emphasis on the various assumptions and interpretations, and a family tree of its main variants
Modelling large dimensional datasets with Markov switching factor models
No full textWe study a novel large dimensional approximate factor model with regime changes in the loadings driven by a latent first order Markov process. By exploiting the equivalent linear representation of the model, we first recover the latent factors by means of Principal Component Analysis. We then cast the model in state–space form, and we estimate loadings and transition probabilities through an EM algorithm based on a modified version of the Baum–Lindgren–Hamilton–Kim filter and smoother that makes use of the factors previously estimated. Our approach is appealing as it provides closed form expressions for all estimators. More importantly, it does not require knowledge of the true number of factors. We derive the theoretical properties of the proposed estimation procedure, and we show their good finite sample performance through a comprehensive set of Monte Carlo experiments. The empirical usefulness of our approach is illustrated through three applications to large U.S. datasets of stock returns, macroeconomic variables, and inflation indexes
Consistent estimation of high-dimensional factor models when the factor number is over-estimated
Full text linkA high-dimensional -factor model for an -dimensional vector time series is characterised by the presence of a large eigengap (increasing with ) between the -th and the -th largest eigenvalues of the covariance matrix. Consequently, Principal Component (PC) analysis is the most popular estimation method for factor models and its consistency, when is correctly estimated, is well-established in the literature. However, popular factor number estimators often suffer from the lack of an obvious eigengap in empirical eigenvalues and tend to over-estimate due, for example, to the existence of non-pervasive factors affecting only a subset of the series. We show that the errors in the PC estimators resulting from the over-estimation of are non-negligible, which in turn lead to the violation of the conditions required for factor-based large covariance estimation. To remedy this, we propose new estimators of the factor model based on scaling the entries of the sample eigenvectors. We show both theoretically and numerically that the proposed estimators successfully control for the over-estimation error, and investigate their performance when applied to risk minimisation of a portfolio of financial time series
Multinetwork of international trade: A commodity-specific analysis
Full text linkWe study the topological properties of the multinetwork of commodity-specific trade relations among world countries over the 1992-2003 period, comparing them with those of the aggregate-trade network, known in the literature as the international-trade network (ITN). We show that link-weight distributions of commodity-specific networks are extremely heterogeneous and (quasi) log normality of aggregate linkweight distribution is generated as a sheer outcome of aggregation. Commodity-specific networks also display average connectivity, clustering, and centrality levels very different from their aggregate counterpart. We also find that ITN complete connectivity is mainly achieved through the presence of many weak links that keep commodity-specific networks together and that the correlation structure existing between topological statistics within each single network is fairly robust and mimics that of the aggregate network. Finally, we employ cross-commodity correlations between link weights to build hierarchies of commodities. Our results suggest that on the top of a relatively time-invariant intrinsic taxonomy (based on inherent between-commodity similarities), the roles played by different commodities in the ITN have become more and more dissimilar, possibly as the result of an increased trade specialization. Our approach is general and can be used to characterize any multinetwork emerging as a nontrivial aggregation of several interdependent layers
Inference in Heavy-Tailed Nonstationary Multivariate Time Series
Full text linkWe study inference on the common stochastic trends in a nonstationary, N-variate time series yt, in the possible presence of heavy tails. We propose a novel methodology which does not require any knowledge or estimation of the tail index, or even knowledge as to whether certain moments (such as the variance) exist or not, and develop an estimator of the number of stochastic trends m based on the eigenvalues of the sample second moment matrix of yt. We study the rates of such eigenvalues, showing that the first m ones diverge, as the sample size T passes to infinity, at a rate faster by (Formula presented.) than the remaining N–m ones, irrespective of the tail index. We thus exploit this eigen-gap by constructing, for each eigenvalue, a test statistic which diverges to positive infinity or drifts to zero according to whether the relevant eigenvalue belongs to the set of the first m eigenvalues or not. We then construct a randomized statistic based on this, using it as part of a sequential testing procedure, ensuring consistency of the resulting estimator of m. We also discuss an estimator of the common trends based on principal components and show that, up to a an invertible linear transformation, such estimator is consistent in the sense that the estimation error is of smaller order than the trend itself. Importantly, we present the case in which we relax the standard assumption of iid innovations, by allowing for heterogeneity of a very general form in the scale of the innovations. Finally, we develop an extension to the large dimensional case. A Monte Carlo study shows that the proposed estimator for m performs particularly well, even in samples of small size. We complete the article by presenting two illustrative applications covering commodity prices and interest rates data. Supplementary materials for this article are available online
General spatio-temporal factor models for high-dimensional random fields on a lattice
No full textMotivated by the need for analysing large spatio-temporal panel data,
we introduce a novel nonparametric methodology for n-dimensional random
fields observed across S spatial locations and T time periods. We call it general spatio-temporal factor model (GSTFM). First, we provide the probabilistic and mathematical underpinning needed for the representation of a random field as the sum of two components: the common component (driven
by a small number q of latent factors) and the idiosyncratic component
(mildly cross-correlated). We show that the two components are identified
as n → ∞. Second, we propose an estimator of the common component
and derive its statistical guarantees (consistency and rate of convergence)
as min(n,S,T ) → ∞. Third, we propose an information criterion to determine the number of factors. Estimation makes use of Fourier analysis in the
frequency domain and thus it fully exploits the information on the spatiotemporal covariance structure of the whole panel. Synthetic data examples
illustrate the applicability of GSTFM and its advantages over the extant generalized dynamic factor model that ignores the spatial correlations
fnets: An R Package for Network Estimation and Forecasting via Factor-Adjusted VAR Modelling
No full textVector autoregressive (VAR) models are useful for modelling high-dimensional time series data. This paper introduces the package fnets, which implements the suite of methodologies proposed by (Barigozzi et al. 2023) for the network estimation and forecasting of high-dimensional time series under a factor-adjusted vector autoregressive model, which permits strong spatial and temporal correlations in the data. Additionally, we provide tools for visualising the networks underlying the time series data after adjusting for the presence of factors. The package also offers data-driven methods for selecting tuning parameters including the number of factors, the order of autoregression, and thresholds for estimating the edge sets of the networks of interest in time series analysis. We demonstrate various features of fnets on simulated datasets as well as real data on electricity prices
Factor Network Autoregressions
No full textWe propose a factor network autoregressive (FNAR) model for time series with complex network structures. The coefficients of the model reflect many different types of connections between economic agents ("multilayer network"), which are summarized into a smaller number of network matrices ("network factors") through a novel tensor-based principal component approach. We provide consistency and asymptotic normality results for the estimation of the factors, their loadings, and the coefficients of the FNAR, as the number of layers, nodes and time points diverges to infinity. Our approach combines two different dimension-reduction techniques and can be applied to high-dimensional datasets. Simulation results show the goodness of our estimators in finite samples. In an empirical application, we use the FNAR to investigate the cross-country interdependence of GDP growth rates based on a variety of international trade and financial linkages. The model provides a rich characterization of macroeconomic network effects as well as good forecasts of GDP growth rates
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