Large Dimensional Factor Models Research Articles

Ridge Regression Under Dense Factor Augmented Models

This article establishes a comprehensive theory of the optimality, robustness, and cross-validation selection consistency for the ridge regression under factor-augmented models with possibly dense idiosyncratic information. Using spectral analysis for random matrices, we show that the ridge regression is asymptotically efficient in capturing both factor and idiosyncratic information by minimizing the limiting predictive loss among the entire class of spectral regularized estimators under large-dimensional factor models and mixed-effects hypothesis. We derive an asymptotically optimal ridge penalty in closed form and prove that a bias-corrected k-fold cross-validation procedure can adaptively select the best ridge penalty in large samples. We extend the theory to the autoregressive models with many exogenous variables and establish a consistent cross-validation procedure using the what-we-called double ridge regression method. Our results allow for nonparametric distributions for, possibly heavy-tailed, martingale difference errors and idiosyncratic random coefficients and adapt to the cross-sectional and temporal dependence structures of the large-dimensional predictors. We demonstrate the performance of our ridge estimators in simulated examples as well as an economic dataset. All the proofs are available in the supplementary materials, which also includes more technical discussions and remarks, extra simulation results, and useful lemmas that may be of independent interest.

Identifying Common and Idiosyncratic Explosive Behaviors in the Large Dimensional Factor Model with an Application to U.S. State-Level House Prices

Abstract This study applies the date-stamping methodologies for explosive behaviors proposed in the seminal work of Phillips, P. C. B., and J. Yu. (2011. “Dating the Timeline of Financial Bubbles during the Subprime Crisis.” Quantitative Economics 2 (3): 455–91), Phillips, P. C. B., S. Shi, and J. Yu. (2015a. “Testing for Multiple Bubbles: Historical Episodes of Exuberance and Collapse in the S&P 500.” International Economic Review 56 (4): 1043–78), and Phillips, P. C. B., S. Shi, and J. Yu. (2015b. “Testing for Multiple Bubbles: Limit Theory of Real Time Detectors.” International Economic Review 56 (4): 1079–134) to a large dimensional factor model. To this end, we compare two methods of identifying common and idiosyncratic components: the Panel Analysis of Nonstationarity in Idiosyncratic and Common Components (PANIC) method by Bai, J., and S. Ng. (2004. “A Panic Attack on Unit Roots and Cointegration.” Econometrica 72 (4): 1127–77) and the Cross-Sectional regression (CS) method by Yamamoto, Y., and T. Horie. (2022. “A Cross-Sectional Method for Right-Tailed PANIC Tests under a Moderately Local to Unity Framework.” Econometric Theory (forthcoming)). We show that, when the explosive behavior lies only in the common component, the origination and termination dates are precisely estimated by either method. However, when the explosive behaviors exist in idiosyncratic components, only the CS method can detect them. We apply our method to the U.S. state-level real house price indices. We find that the 2000s boom was driven by not only the national bubble factors but also local components, while the 2010s onward expansion is dominated by the effect of national components.

Testing for structural changes in large dimensional factor models via discrete Fourier transform

We propose a new test for structural changes in large dimensional factor models using a discrete Fourier transform (DFT) approach. When structural changes occur, the conventional principal component analysis may fail to estimate the common factors and factor loadings consistently, and the estimated residuals contain information about the structural changes. This allows us to compare the DFT of the estimated residuals weighted by the estimated common factors with the null (zero) spectrum implied by no structural change. The proposed test is powerful against both smooth structural changes and abrupt structural breaks with an unknown number of breaks and unknown break dates in factor loadings. It can detect a class of local alternatives at the rate N−1/2T−1/2, where N and T are the numbers of cross-sectional units and time periods, respectively. Monte Carlo studies demonstrate that the proposed test has reasonable size and excellent power in detecting various structural changes in factor loadings. When applied to the U.S. macroeconomic data, the test reveals significant and robust evidence of time-varying factor loadings for the post-Great Moderation sample and the pre-Great Recession subsample, which the existing literature may fail to address.

Quasi-maximum likelihood estimation of break point in high-dimensional factor models

This paper proposes a quasi-maximum likelihood (QML) estimator of the break point for large-dimensional factor models with a single structural break in the factor loading matrix. We show that the QML estimator is consistent for the true break point when the covariance matrix of the pre- or post-break factor loading (or both) is singular. Consistency here means that the deviation of the estimated break date from the true break date k0 converges to zero as the sample size grows. This is a much stronger result than the break fraction kˆ/T being T-consistent (super-consistent) for k0/T. Also, singularity occurs for most types of structural changes, except for a rotational change. Even for a rotational change, the QML estimator is still T-consistent in terms of the break fraction. Simulation results confirm the theoretical properties of our estimator, and it significantly outperforms existing estimators for change points in factor models.

Forecasting stock returns with large dimensional factor models

We study equity premium out-of-sample predictability by extracting the information contained in a high number of macroeconomic predictors via large dimensional factor models. We compare the well-known factor model with a static representation of the common components with the Generalized Dynamic Factor Model, which accounts for time series dependence in the common components. Using statistical and economic evaluation criteria, we empirically show that the Generalized Dynamic Factor Model helps predicting the equity premium. Exploiting the link between business cycle and return predictability, we find accurate predictions also by combining rolling and recursive forecasts in real-time.

Interpretable Sparse Proximate Factors for Large Dimensions

This article proposes sparse and easy-to-interpret proximate factors to approximate statistical latent factors. Latent factors in a large-dimensional factor model can be estimated by principal component analysis (PCA), but are usually hard to interpret. We obtain proximate factors that are easier to interpret by shrinking the PCA factor weights and setting them to zero except for the largest absolute ones. We show that proximate factors constructed with only 5%–10% of the data are usually sufficient to almost perfectly replicate the population and PCA factors without actually assuming a sparse structure in the weights or loadings. Using extreme value theory we explain why sparse proximate factors can be substitutes for non-sparse PCA factors. We derive analytical asymptotic bounds for the correlation of appropriately rotated proximate factors with the population factors. These bounds provide guidance on how to construct the proximate factors. In simulations and empirical analyses of financial portfolio and macroeconomic data, we illustrate that sparse proximate factors are close substitutes for PCA factors with average correlations of around 97.5%, while being interpretable.

Determining the number of breaks in large dimensional factor models with structural changes

This paper considers a large dimensional factor model with multiple structural changes in factor loadings. A novel two-step procedure is proposed to estimate the number of breaks. First, a sequential estimation method is applied to estimate break points one by one for a given number of breaks. Then the number of breaks can be consistently determined by some information criteria. Simulation results confirm the desired performance of the estimation procedure in finite samples.

Large-Dimensional Factor Analysis Without Moment Constraints

Large-dimensional factor model has drawn much attention in the big-data era, to reduce the dimensionality and extract underlying features using a few latent common factors. Conventional methods for estimating the factor model typically requires finite fourth moment of the data, which ignores the effect of heavy-tailedness and thus may result in unrobust or even inconsistent estimation of the factor space and common components. In this article, we propose to recover the factor space by performing principal component analysis to the spatial Kendall’s tau matrix instead of the sample covariance matrix. In a second step, we estimate the factor scores by the ordinary least square regression. Theoretically, we show that under the elliptical distribution framework the factor loadings and scores as well as the common components can be estimated consistently without any moment constraint. The convergence rates of the estimated factor loadings, scores, and common components are provided. The finite sample performance of the proposed procedure is assessed through thorough simulations. An analysis of a financial dataset of asset returns shows the superiority of the proposed method over the classical principle component analysis method.

Cointegration and Error Correction Mechanisms for Singular Stochastic Vectors

Large-dimensional dynamic factor models and dynamic stochastic general equilibrium models, both widely used in empirical macroeconomics, deal with singular stochastic vectors, i.e., vectors of dimension r which are driven by a q-dimensional white noise, with q < r . The present paper studies cointegration and error correction representations for an I ( 1 ) singular stochastic vector y t . It is easily seen that y t is necessarily cointegrated with cointegrating rank c ≥ r − q . Our contributions are: (i) we generalize Johansen’s proof of the Granger representation theorem to I ( 1 ) singular vectors under the assumption that y t has rational spectral density; (ii) using recent results on singular vectors by Anderson and Deistler, we prove that for generic values of the parameters the autoregressive representation of y t has a finite-degree polynomial. The relationship between the cointegration of the factors and the cointegration of the observable variables in a large-dimensional factor model is also discussed.

Testing for Structural Changes in Large Dimensional Factor Models via Discrete Fourier Transform

We propose a new test for structural changes in large dimensional factor models via a discrete Fourier transform (DFT) approach. If structural changes occur, the conventional principal component analysis fails to estimate common factors and factor loadings consistently. The estimated residuals will contain information about structural changes. Therefore, we can compare the DFT of the estimated residuals with the null (zero) spectrum implied by no structural change. The proposed test is powerful against both smooth structural changes and abrupt structural breaks with a possibly unknown number of breaks and unknown break dates in factor loadings. It can detect a class of local alternatives at the parametric rate. As a result, the test is asymptotically more efficient than the existing tests in the factor model literature. And our test is also robust to serial and cross-sectional dependence of unknown form without having to estimate any long-run variance-covariance matrix. Moreover, it is easy to implement and tuning parameter-free. Monte Carlo studies demonstrate its reasonable size and excellent power in detecting various forms of structural changes in factor loadings. In an application to the U.S. macroeconomic data, we find significant and robust evidence of time-varying factor loadings.

Factor-Adjusted Ridge Prediction Using Large-Dimensional Mixed-Effects Models

This paper establishes a comprehensive theory of the optimality, robustness, and cross-validation selection-consistency for the ridge regression under factor-augmented models with possibly dense idiosyncratic information. Using spectral analysis for random matrices, we show that the ridge regression is asymptotically efficient in capturing both factor and idiosyncratic information, by minimizing the limiting predictive loss among the entire class of spectral regularized estimators under large-dimensional factor models and mixed-effects hypothesis. We derive an asymptotically optimal ridge penalty in closed form, and prove that a bias-corrected k-fold cross-validation procedure can select the best ridge penalty adaptively in large samples. We extend the theory to the autoregressive models with many exogenous variables and establish a consistent cross-validation procedure using the what-we-called double ridge regression method. Our results allow for non-parametric distributions for martingale difference errors and idiosyncratic random coefficients, and adapt to both the cross-sectional and temporal dependence structures of the large-dimensional predictors. A simulation study shows the efficiency of the ridge regression estimators under dense models and its robustness against sparsity. We apply our methods to forecast the growth rate of US industrial production and S\&P 500 index using the FRED-MD database. Our double ridge regression improves the autoregressive prediction and outperforms the principal component regression and the factor-adjusted LASSO regression consistently.

A rank test for the number of factors with high-frequency data

In the literature, consistency of the estimates of the number of factors for large-dimensional factor models had been extensively studied recently. But the second-order property of the estimator has long been unsolved due to lack of limiting distribution of the estimators. In this paper, we propose a rank test of the number of factors using large panel high-frequency data contaminated with microstructure noise. The rank test is realized by forming a fixed number of portfolios which reduce the dimension to a finite number. In the process of constructing portfolios, the number of factors is equal to the rank of the volatility matrix of the diversified portfolios asymptotically. Via estimating the volatility rank of a low-dimensional price dynamics of the portfolios, we establish a central limit theorem of the estimated factor number. We then apply the asymptotic normality to testing on the number of factors. Numerical experiments including the Monte-Carlo simulations and real data analysis justify our theory.

Estimation of Large Dimensional Conditional Factor Models in Finance

This chapter provides an econometric methodology for inference in large-dimensional conditional factor models in finance. Changes in the business cycle and asset characteristics induce time variation in factor loadings and risk premia to be accounted for. The growing trend in the use of disaggregated data for individual securities motivates our focus on methodologies for a large number of assets. The beginning of the chapter outlines the concept of approximate factor structure in the presence of conditional information, and develops an arbitrage pricing theory for large-dimensional factor models in this framework. Then we distinguish between two different cases for inference depending on whether factors are observable or not. We focus on diagnosing model specification, estimating conditional risk premia, and testing asset pricing restrictions under increasing cross-sectional and time series dimensions. At the end of the chapter, we review some of the empirical findings and contrast analysis based on individual stocks and standard sets of portfolios. We also discuss the impact on computing time-varying cost of equity for a firm, and summarize differences between results for developed and emerging markets in an international setting.

Estimation of large dimensional factor models with an unknown number of breaks

In this paper we study the estimation of a large dimensional factor model when the factor loadings exhibit an unknown number of changes over time. We propose a novel three-step procedure to detect the breaks if any and then identify their locations. In the first step, we divide the whole time span into subintervals and fit a conventional factor model on each interval. In the second step, we apply the adaptive fused group Lasso to identify intervals containing a break. In the third step, we devise a grid search method to estimate the location of the break on each identified interval. We show that with probability approaching one our method can identify the correct number of changes and estimate the break locations. Simulation studies indicate superb finite sample performance of our method. We apply our method to investigate Stock and Watson’s (2009) U.S. monthly macroeconomic dataset and identify five breaks in the factor loadings, spanning 1959–2006.

Interpretable Proximate Factors for Large Dimensions

This paper proposes sparse and easy-to-interpret proximate factors to approximate statistical latent factors. Latent factors in a large-dimensional factor model can be estimated by principal component analysis (PCA), but are usually hard to interpret. We obtain proximate factors that are easier to interpret by shrinking the PCA factor weights and setting them to zero except for the largest absolute ones. We show that proximate factors constructed with only 5-10% of the data are usually sufficient to almost perfectly replicate the population and PCA factors without actually assuming a sparse structure in the weights or loadings. Using extreme value theory we explain why sparse proximate factors can be substitutes for non-sparse PCA factors. We derive analytical asymptotic bounds for the correlation of appropriately rotated proximate factors with the population factors. These bounds provide guidance on how to construct the proximate factors. In simulations and empirical analyses of financial portfolio and macroeconomic data we illustrate that sparse proximate factors are close substitutes for PCA factors with average correlations of around 97.5% while being interpretable.

Testing for Regime Changes in Large Dimensional Factor Models

We develop a new test for threshold-type regime changes in the risk exposures in portfolios with a large number of financial assets whose returns exhibit an approximate factor structure. Unlike existing procedures to detect discrete shifts in factor models, our test is robust to regime-specific second moment of the common factors. We rely on an auxiliary threshold regression: we take a weighted cross-sectional average of the cross-sectional units; we estimate the factors from the original model under the null hypothesis of no regime changes; we construct a Lagrange multiplier statistic to test for threshold effect in the auxiliary regression. Numerical results show the good finite sample properties of our procedure. The empirical analysis uncovers the dynamics of portfolio weights and diversification benefits in factor mimicking portfolios across different regimes.

Forecasting Stock Returns with Large Dimensional Factor Models

Assuming that economic variables can be decomposed into common and idiosyncratic components, we study equity premium out-of-sample predictability when the information contained in a high number of predictors is extracted using large dimensional factor models. We consider factor models with a static representation of the common components, and models that allow for dynamic and infinite-dimensional representations. Using statistical and economic evaluation criteria, we show that large dimensional factor models with a dynamic representation of the common components do help predicting the equity premium. Furthermore, exploiting the well-known link between the business cycle and return predictability, we find more accurate predictions by combining rolling and recursive forecasts.

Constrained Principal Components Estimation of Large Approximate Factor Models

This paper proposes a constrained principal components (CnPC) estimator for efficient estimation of large-dimensional factor models when errors are crosssectionally correlated and the number of cross-sections (N) may be larger than the number of observations (T). Although principal components (PC) method is consistent for any path of the panel dimensions, it is inefficient as the errors are treated to be homoskedastic and uncorrelated. The new CnPC exploits the assumption of bounded cross-sectional dependence, which defines Chamberlain and Rothschild’s (1983) approximate factor structure, as an explicit constraint and solves a constrained PC problem. The CnPC method is computationally equivalent to the PC method applied to a regularized form of the data covariance matrix. Unlike maximum likelihood type methods, the CnPC method does not require inverting a large covariance matrix and thus is valid for panels with N ≥ T. The paper derives a convergence rate and an asymptotic normality result for the CnPC estimators of the common factors. We provide feasible estimators and show in a simulation study that they are more accurate than the PC estimator, especially for panels with N larger than T, and the generalized PC type estimators (Choi [2012]), especially for panels with N almost as large as T.

Diffusion Indexes With Sparse Loadings

The use of large-dimensional factor models in forecasting has received much attention in the literature with the consensus being that improvements on forecasts can be achieved when comparing with standard models. However, recent contributions in the literature have demonstrated that care needs to be taken when choosing which variables to include in the model. A number of different approaches to determining these variables have been put forward. These are, however, often based on ad hoc procedures or abandon the underlying theoretical factor model. In this article, we will take a different approach to the problem by using the least absolute shrinkage and selection operator (LASSO) as a variable selection method to choose between the possible variables and thus obtain sparse loadings from which factors or diffusion indexes can be formed. This allows us to build a more parsimonious factor model that is better suited for forecasting compared to the traditional principal components (PC) approach. We provide an asymptotic analysis of the estimator and illustrate its merits empirically in a forecasting experiment based on U.S. macroeconomic data. Overall we find that compared to PC we obtain improvements in forecasting accuracy and thus find it to be an important alternative to PC. Supplementary materials for this article are available online.

Least squares estimation of large dimensional threshold factor models

This paper studies large dimensional factor models with threshold-type regime shifts in the loadings. We estimate the threshold by concentrated least squares, and factors and loadings by principal components. The estimator for the threshold is superconsistent, with convergence rate that depends on the time and cross-sectional dimensions of the panel, and it does not affect the estimator for factors and loadings: this has the same convergence rate as in linear factor models. We propose model selection criteria and a linearity test. Empirical application of the model shows that connectedness in financial variables increases during periods of high economic policy uncertainty.