Large vector autoregressive exogenous factor (VARX) model with network regularization

Estimating the common break date in large factor models

Summary This paper provides a review of the recent literature concerned with large factor models as forecast devices.We focus on factor models that account for mixed-frequency data and missing observations at the end of the sample. These are data irregularities applied forecasters have to cope with in real time. To extract the factors from the irregular data, special factor estimation techniques are necessary, expanding on the standard approaches for balanced data such as principal components (PC). The estimation methods include variants of the Expectation-Maximisation (EM) algorithm together with PC and factor estimation using state-space models. Given the estimated factors, forecasts can be obtained from bridge equations, mixed-data sampling (MIDAS) regressions and the Kalman smoother applied to fully-fledged factor models in state-space form. Empirical applications for German GDP growth often find that forecasts based on factor models are informative only a few months ahead compared to naive benchmarks. Thus, these models can be regarded as short-term forecast tools only. However, the factor models estimated on mixed-frequency data with missing observations tend to outperform factor models based on balanced data time-aggregated from high-frequency data.

Factor 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 often 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.’s “A Quasi Maximum Likelihood Approach for Large Approximate Dynamic Factor Models” and Bai and Li’s “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.

A large factor model for forecasting macroeconomic variables in South Africa

Real-time forecasting of German GDP based on a large factor model with monthly and quarterly data

Large factor models use a few latent factors to characterize the co-movement of economic variables in a high-dimensional data set. High dimensionality brings challenges as well as new insights into the advancement of econometric theory. Because of their ability to effectively summarize information in large data sets, factor models have been increasingly used in economics and finance. The factors, estimated from the high-dimensional data, can, for example, help improve forecasting, provide efficient instruments, control for nonlinear unobserved heterogeneity, and capture cross-sectional dependence. This article reviews the theory on estimation and statistical inference of large factor models. It also discusses important applications and highlights future directions.

This paper proposes a moving sum methodology for detecting multiple change points in high‐dimensional time series under a factor model, where changes are attributed to those in loadings as well as emergence or disappearance of factors. We establish the asymptotic null distribution of the proposed test for family‐wise error control and show the consistency of the procedure for multiple change point estimation. Simulation studies and an application to a large dataset of volatilities demonstrate the competitive performance of the proposed method.

This paper discusses a factor model for estimating monthly GDP using a large number of monthly and quarterly time series in real-time. To take into account the different periodicities of the data and missing observations at the end of the sample, the factors are estimated by applying an EM algorithm combined with a principal components estimator. We discuss the in-sample properties of the estimator in real-time environments and methods for out-of-sample forecasting. As an empirical application, we estimate monthly German GDP in real-time, discuss the nowcast and forecast accuracy of the model and the role of revisions. Furthermore, we assess the contribution of timely monthly data to the forecast performance.

The importance of units with pervasive impacts on a large number of other units in a network has become increasingly recognized in the literature. In this paper we propose a new method to detect such pervasive units by basing our analysis on unit-speci c residual error variances in the context of a standard factor model, subject to suitable adjustments due to multiple testing. Our proposed method allows us to estimate and identify pervasive units having neither a priori knowledge of the interconnections amongst cross-section units nor a short list of candidate units. It is applicable even if the cross section dimension exceeds the time dimension, and most importantly it could end up with none of the units selected as pervasive when this is in fact the case. The sequential multiple testing procedure proposed exhibits satisfactory small-sample performance in Monte Carlo simulations and compares well relative to existing approaches. We apply the proposed detection method to sectoral indices of US industrial production, US house price changes by states, and the rates of change of real GDP and real equity prices across the world's largest economies.

This paper discusses a factor model for estimating monthly GDP using a large number of monthly and quarterly time series in real-time. To take into account the different periodicities of the data and missing observations at the end of the sample, the factors are estimated by applying an EM algorithm combined with a principal components estimator. We discuss the in-sample properties of the estimator in real-time environments and methods for out-of-sample forecasting. As an empirical application, we estimate monthly German GDP in real-time, discuss the nowcast and forecast accuracy of the model and the role of revisions. Furthermore, we assess the contribution of timely monthly data to the forecast performance.

The importance of units with pervasive impacts on a large number of other units in a network has become increasingly recognized in the literature. In this paper we propose a new method to detect such influential or dominant units by basing our analysis on unit-specific residual error variances in the context of a standard factor model, subject to suitable adjustments due to multiple testing. Our proposed method allows us to estimate and identify the dominant units without the a priori knowledge of the interconnections amongst the units, or using a short list of potential dominant units. It is applicable even if the cross section dimension exceeds the time dimension, and most importantly it could end up with none of the units selected as dominant when this is in fact the case. The sequential multiple testing procedure proposed exhibits satisfactory small-sample performance in Monte Carlo simulations and compares well relative to existing approaches. We apply the proposed detection method to sectoral indices of US industrial production, US house price changes by states, and the rates of change of real GDP and real equity prices across the world’s largest economies.

We propose a new class of large approximate factor models which enable us to study the full spectrum of quarterly Industrial Production (IP) sector data combined with annual non-IP sectors of the economy. We derive the large sample properties of the estimators and test statistics for the new class of unobservable factor models involving mixed frequency data and common as well as frequency-specific factors. Despite the growth of service sectors, we find that a single common factor explaining 90% of the variability in IP output growth index also explains 60% of total GDP output growth fluctuations. A single low frequency factor unrelated to manufacturing explains 14% of GDP growth. The picture with a structural factor model featuring technological innovations is quite different. Last but not least, our identification and inference methodology rely on novel results on group factor models that are of general interest beyond the mixed frequency framework and the application of the paper.

We consider large factor models with unobserved factors. We formalize the notion of common factors between different groups of variables and propose to use it as a general approach to study the structure of factors, i.e., which factors drive which variables. The spanning hypothesis, which states that factors driving one group are spanned by those driving another group, can be studied as a special case under our framework. We develop a statistical procedure for testing the number of common factors. Our inference procedure is built upon recent results on high-dimensional bootstrap and is shown to be valid under the asymptotic framework of large n and large T. In Monte Carlo simulations, our procedure performs well in finite samples. As an empirical application, we construct confidence sets for the number of common factors between the macroeconomy and the financial markets.

Monetary, fiscal and oil shocks: Evidence based on mixed frequency structural FAVARs

Asymptotic analysis of the squared estimation error in misspecified factor models