Approximate Factor Model Research Articles

Efficient Estimation of Approximate Factor Models via Regularized Maximum Likelihood

We study the estimation of a high dimensional approximate factor model in the presence of both cross sectional dependence and heteroskedasticity. The classical method of principal components analysis (PCA) does not efficiently estimate the factor loadings or common factors because it essentially treats the idiosyncratic error to be homoskedastic and cross sectionally uncorrelated. For the efficient estimation, it is essential to estimate a large error covariance matrix. We assume the model to be conditionally sparse, and propose two approaches to estimating the common factors and factor loadings; both are based on maximizing a Gaussian quasi-likelihood and involve regularizing a large covariance sparse matrix. In the first approach the factor loadings and the error covariance are estimated separately while in the second approach they are estimated jointly. Extensive asymptotic analysis has been carried out. In particular, we develop the inferential theory for the two-step estimation. Because the proposed approaches take into account the large error covariance matrix, they produce more efficient estimators than the classical PCA methods or methods based on a strict factor model.

Estimating the number of common factors in serially dependent approximate factor models

A simple data-dependent filtering method is proposed before applying the Bai–Ng method to estimate the number of common factors in the conventional approximate factor model. The asymptotic justification is provided and the finite-sample performance is examined.

Investigating finite sample properties of estimators for approximate factor models when [formula omitted] is small

We examine finite sample properties of estimators for approximate factor models when N is small. Contrary to the “rule-of-thumb”, we find that the principal component analysis estimator and the quasi-maximum likelihood estimator perform well even when N is small.

Identifying the Independent Sources of Consumption Variation

By representing a system of budget shares as an approximate factor model we determine its rank, i.e. the number of common functional forms, or factors and we estimate a base of the factor space by means of approximate principal components. We assume that the extracted factors span the same space of basic Engel curves representing the fundamental forces driving consumers’ behaviour. We identify these curves by imposing statistical independence and by studying their dependence on total expenditure using local linear regressions. We prove consistency of the estimates. Using data from the U.K. Family Expenditure Survey from 1977 to 2006, we find strong evidence of two common factors and mixed evidence of a third factor. These are identified as decreasing, increasing, and almost constant Engel curves. The household consumption behaviour is therefore driven by two factors respectively related to necessities (e.g. food), luxuries (e.g. vehicles), and in some cases by a third factor related to goods to which is allocated the same percentage of total budget both by rich and poor households (e.g. housing).

A Model for Vast Panels of Volatilities

We introduce an approximate dynamic factor model for modeling and forecasting large panels of realized volatilities. Since the model is estimated by means of principal components and low dimensional maximum likelihood, it does not suffer from the curse of dimensionality. We apply the model to a panel of 90 daily realized volatilities pertaining to S&P100 from January 2001 to December 2008. Results show that our model is able to capture the stylized facts of panels of volatilities (comovements, clustering, long memory, dynamic volatility, skewness and heavy tails), and that it performs fairly well in forecasting, in particular in period of turmoil in which it outperforms standard univariate benchmarks.

Large Covariance Estimation by Thresholding Principal Orthogonal Complements

This paper deals with the estimation of a high-dimensional covariance with a conditional sparsity structure and fast-diverging eigenvalues. By assuming sparse error covariance matrix in an approximate factor model, we allow for the presence of some cross-sectional correlation even after taking out common but unobservable factors. We introduce the Principal Orthogonal complEment Thresholding (POET) method to explore such an approximate factor structure with sparsity. The POET estimator includes the sample covariance matrix, the factor-based covariance matrix (Fan, Fan, and Lv, 2008), the thresholding estimator (Bickel and Levina, 2008) and the adaptive thresholding estimator (Cai and Liu, 2011) as specific examples. We provide mathematical insights when the factor analysis is approximately the same as the principal component analysis for high-dimensional data. The rates of convergence of the sparse residual covariance matrix and the conditional sparse covariance matrix are studied under various norms. It is shown that the impact of estimating the unknown factors vanishes as the dimensionality increases. The uniform rates of convergence for the unobserved factors and their factor loadings are derived. The asymptotic results are also verified by extensive simulation studies. Finally, a real data application on portfolio allocation is presented.

Forecasting US state-level employment growth: An amalgamation approach

Abstract We forecast US state-level employment growth using several distinct econometric approaches: combinations of individual autoregressive distributed lag models, general-to-specific modeling with bootstrap aggregation (GETS-bagging), and approximate factor (or “beta”) models. Our results show that these forecasting approaches consistently deliver sizable reductions in mean squared forecast error (MSFE) relative to an autoregressive (AR) benchmark model across the 50 US states. On the basis of forecast encompassing test results, we also consider amalgamating these approaches and find that this strategy yields additional forecasting improvements. These improvements are particularly evident during national business-cycle recessions, where the amalgamation approach outperforms the AR benchmark for nearly all states and leads to a 40% reduction in MSFE on average across states relative to the AR benchmark.

A Quasi–Maximum Likelihood Approach for Large, Approximate Dynamic Factor Models

Is maximum likelihood suitable for factor models in large cross-sections of time series? We answer this question from both an asymptotic and an empirical perspective. We show that estimates of the common factors based on maximum likelihood are consistent for the size of the cross-section (n) and the sample size (T) going to infinity along any path of n and T and that therefore maximum likelihood is viable for n large. The estimator is robust to misspecification of the cross-sectional and time series correlation of the the idiosyncratic components. In practice, the estimator can be easily implemented using the Kalman smoother and the EM algorithm as in traditional factor analysis.

Estimation of high-dimensional linear factor models with grouped variables

We introduce a generalization of the approximate factor model that divides the observable variables into groups, allows for arbitrarily strong cross-correlation between the disturbance terms of variables that belong to the same group, and for weak correlation between the disturbances of variables that belong to different groups. We call this model the Grouped Variable Approximate Factor Model. We establish identification, propose an estimation approach based on instrumental variable conditions that hold in the limit, and prove consistency in a dual limit framework. Monte Carlo simulations are used to investigate the performance of the estimator, and the techniques are applied to an analysis of industrial output in the US.

More than just one labor market cycle in Germany? An analysis of regional unemployment data

We analyze unemployment dynamics for Germany on a regional basis by means of an approximate factor model. We first estimate the number of factors corresponding to the number of cycles. At least for the pre-“Hartz” reform data we find strong evidence for more than just one dynamic labor market cycle present in German regions. Thus, labor market dynamics are driven by more than a single nationwide business cycle. Next, we look for regional partitions reflecting the different cycles best. Our results indicate pronounced differences between East and West Germany for 1997 to 2004 and ongoing but reduced differences between 2005 and 2010. A convergence process is found to have taken place up until late 2001. There is evidence for the differences observed before 2004 to be driven by active labor market policy, which thus had a volatility-increasing effect on the labor market.

LOCALLY STATIONARY FACTOR MODELS: IDENTIFICATION AND NONPARAMETRIC ESTIMATION

In this paper we propose a new approximate factor model for large cross-section and time dimensions. Factor loadings are assumed to be smooth functions of time, which allows considering the model aslocally stationarywhile permitting empirically observed time-varying second moments. Factor loadings are estimated by the eigenvectors of a nonparametrically estimated covariance matrix. As is well known in the stationary case, this principal components estimator is consistent in approximate factor models if the eigenvalues of the noise covariance matrix are bounded. To show that this carries over to our locally stationary factor model is the main objective of our paper. Under simultaneous asymptotics (cross-section and time dimension go to infinity simultaneously), we give conditions for consistency of our estimators. A simulation study illustrates the performance of these estimators.

High-dimensional covariance matrix estimation in approximate factor models

The variance covariance matrix plays a central role in the inferential theories of high dimensional factor models in finance and economics. Popular regularization methods of directly exploiting sparsity are not directly applicable to many financial problems. Classical methods of estimating the covariance matrices are based on the strict factor models, assuming independent idiosyncratic components. This assumption, however, is restrictive in practical applications. By assuming sparse error covariance matrix, we allow the presence of the cross-sectional correlation even after taking out common factors, and it enables us to combine the merits of both methods. We estimate the sparse covariance using the adaptive thresholding technique as in Cai and Liu (2011), taking into account the fact that direct observations of the idiosyncratic components are unavailable. The impact of high dimensionality on the covariance matrix estimation based on the factor structure is then studied.

A two-step estimator for large approximate dynamic factor models based on Kalman filtering

This paper shows consistency of a two-step estimation of the factors in a dynamic approximate factor model when the panel of time series is large ( n large). In the first step, the parameters of the model are estimated from an OLS on principal components. In the second step, the factors are estimated via the Kalman smoother. The analysis develops the theory for the estimator considered in Giannone et al. (2004) and Giannone et al. (2008) and for the many empirical papers using this framework for nowcasting.

Panel Cointegration Rank Testing with Cross-Section Dependence

In this paper, we propose a test statistic to determine the cointegration rank of VAR processes in panel data allowing for cross-section dependence among the time series in the panel data. The cross-section dependence is accounted for through the specification of an approximate common factor model, which covers situations where there is cointegration among the cross-section dimension. Finite sample performance is investigated via a Monte Carlo experiment. We show that in some cases not accounting for common factors when they are present can lead to overestimating the cointegrating rank. We apply our proposed tests to two empirical applications and find strong evidence for panel cointegration once common factors are accounted for.

Nowcasting norwegian GDP: the role of asset prices in a small open economy

This article finds that asset prices on Oslo Stock Exchange is the single most important block of data to improve estimates of current quarter GDP in Norway. We use an approximate dynamic factor model that is able to handle new information as it is released, thus the marginal impact on mean square nowcasting error can be studied for a large number of variables. We use a panel of 148 non-synchronous variables. The high informational content in asset prices is explained by reference to the small size of companies on Oslo Stock Exchange and the small and open nature of the Norwegian economy.

Constrained Factor Models

This article considers estimation and applications of constrained and partially constrained factor models when the dimension of explanatory variables is high. Both the classical and approximate factor models are investigated. For estimation, we employ both the maximum likelihood and least squares methods. We show that the least squares estimation is based on constrained principal component analysis and provides consistent estimates for the model under certain conditions. The normality condition is not used in the derivation. We then propose likelihood ratio statistics to test the adequacy of factor constraints. The test statistic is developed under the normality assumption, but simulation results show that it continues to perform well even if the underlying distribution is Student-t. The constraints are useful tools to incorporate prior information or substantive theory in applications of factor models. In addition, the constraints also serve as a statistical tool to obtain parsimonious econometric models for forecasting, to simplify the interpretations of common factors, and to reduce the dimension. We use simulation and real examples to investigate the performance of constrained estimation in finite samples and to highlight the importance of noise-to-signal ratio in factor analysis. Finally, we compare the constrained model with its unconstrained counterpart both in estimation and in forecasting. This article has supplementary material online.

Fitting dynamic factor models to non-stationary time series

Factor modelling of a large time series panel has widely proven useful to reduce its cross-sectional dimensionality. This is done by explaining common co-movements in the panel through the existence of a small number of common components, up to some idiosyncratic behaviour of each individual series. To capture serial correlation in the common components, a dynamic structure is used as in traditional (uni- or multivariate) time series analysis of second order structure, i.e. allowing for infinite-length filtering of the factors via dynamic loadings. In this paper, motivated from economic data observed over long time periods which show smooth transitions over time in their covariance structure, we allow the dynamic structure of the factor model to be non-stationary over time by proposing a deterministic time variation of its loadings. In this respect we generalize the existing recent work on static factor models with time-varying loadings as well as the classical, i.e. stationary, dynamic approximate factor model. Motivated from the stationary case, we estimate the common components of our dynamic factor model by the eigenvectors of a consistent estimator of the now time-varying spectral density matrix of the underlying data-generating process. This can be seen as a time-varying principal components approach in the frequency domain. We derive consistency of this estimator in a “double-asymptotic” framework of both cross-section and time dimension tending to infinity. The performance of the estimators is illustrated by a simulation study and an application to a macroeconomic data set.

Determining the Number of Factors from Empirical Distribution of Eigenvalues

We develop a new estimator of the number of factors in the approximate factor models. The estimator works well even when the idiosyncratic terms are substantially correlated. It is based on the fact, established in the paper, that any finite number of the largest “idiosyncratic” eigenvalues of the sample covariance matrix cluster around a single point. In contrast, all the “systematic” eigenvalues, the number of which equals the number of factors, diverge to infinity. The estimator consistently separates the diverging eigenvalues from the cluster and counts the number of the separated eigenvalues. We consider a macroeconomic and a financial application.

Improved penalization for determining the number of factors in approximate factor models

The procedure proposed by Bai and Ng (2002) for identifying the number of factors in static factor models is revisited. In order to improve its performance, we introduce a tuning multiplicative constant in the penalty, an idea that was proposed by Hallin and Liška (2007) in the context of dynamic factor models. Simulations show that our method in general delivers more reliable estimates, in particular in the case of large idiosyncratic disturbances.

A Testing Procedure for Determining the Number of Factors in Approximate Factor Models With Large Datasets

The paradigm of a factor model is very appealing and has been used extensively in economic analyses. Underlying the factor model is the idea that a large number of economic variables can be adequately modeled by a small number of indicator variables. Throughout this extensive research activity on large dimensional factor models a major preoccupation has been the development of tools for determining the number of factors needed for modeling. This article provides an alternative method to information criteria as a tool for estimating the number of factors in large dimensional factor models. The new method is robust to considerable cross-sectional and temporal dependence. The theoretical properties of the method are explored and an extensive Monte Carlo study is undertaken. Results are favorable for the new method and suggest that it is a reasonable alternative to existing methods.