We develop a pseudo maximum likelihood method for latent factor analysis in short panels without imposing sphericity nor Gaussianity. We derive an asymptotically uniformly most powerful invariant test for the number of factors. On a large panel of monthly U.S. stock returns, we separate month after month systematic and idiosyncratic risks in short subperiods of bear vs. bull markets. We observe an uptrend in the paths of total and idiosyncratic volatilities. The systematic risk explains a large part of the cross-sectional total variance in bear markets but is not driven by a single factor and not spanned by observed factors.

Factor analysis is a widely used tool to summarize high dimensional panel data via a small dimensional set of latent factors. Many applications in finance and macroeconomics, are often focused on observable factors with an economic interpretation. The objective of this paper is to provide a test to answer a question which naturally comes up in discussions regarding latent versus observable factors: do latent and observable factors span the same space? We derive asymptotic properties of a formal test and propose a bootstrap version with improved small sample properties. We find empirical evidence for a small number of factors common between a small number of traditional Fama-French risk factors - or returns on a few stocks (i.e. "magnificent'' 5 or 7) - and large panels of US, North American and international portfolio returns.

Hint: these are not the Fama-French 3 factors and they are not even spanned by the Fama-French 5 factors. More importantly, they feature superior out-of-sample pricing performance compared to standard asset pricing models. What is "common" about these factors? We identify the factor space common between individual stocks and sorted portfolios - neither affected by time-varying betas nor by the sorting characteristics.

We develop a novel approach to separate alpha and beta under model misspecification. It comes with formal tests to identify less misspecified models and sharpen the return decomposition of individual funds. Our hedge fund analysis reveals that: (i) prominent models are as misspecified as the CAPM, (ii) several factors (time-series momentum, variance, carry) capture alternative strategies and lower performance in all investment categories, (iii) fund heterogeneity in alpha and beta is large—an important result for fund selection and models of active management, (iv) performance is increasingly similar to mutual funds, (v) fund valuation is sensitive to investor sophistication.

This paper studies new tests for the number of latent factors in a large cross-sectional factor model with small time dimension. These tests are based on the eigenvalues of variance-covariance matrices of (possibly weighted) asset returns, and rely on either the assumption of spherical errors, or instrumental variables for factor betas. We establish the asymptotic distributional results using expansion theorems based on perturbation theory for symmetric matrices. Our framework accommodates semi-strong factors in the systematic components. We propose a novel statistical test for weak factors against strong or semi-strong factors. We provide an empirical application to US equity data. Evidence for a different number of latent factors according to market downturns and market upturns, is statistically ambiguous in the considered subperiods. In particular, our results contradicts the common wisdom of a single factor model in bear markets.

Regression learning is one of the long-standing problems in statistics, machine learning, and deep learning (DL). We show that writing this problem as a probabilistic expectation over (unknown) feature probabilities – thus increasing the number of unknown parameters and seemingly making the problem more complex—actually leads to its simplification, and allows incorporating the physical principle of entropy maximization. It helps decompose a very general setting of this learning problem (including discretization, feature selection, and learning multiple piece-wise linear regressions) into an iterative sequence of simple substeps, which are either analytically solvable or cheaply computable through an efficient second-order numerical solver with a sublinear cost scaling. This leads to the computationally cheap and robust non-DL second-order Sparse Probabilistic Approximation for Regression Task Analysis (SPARTAn) algorithm, that can be efficiently applied to problems with millions of feature dimensions on a commodity laptop, when the state-of-the-art learning tools would require supercomputers. SPARTAn is compared to a range of commonly used regression learning tools on synthetic problems and on the prediction of the El Niño Southern Oscillation, the dominant interannual mode of tropical climate variability. The obtained SPARTAn learners provide more predictive, sparse, and physically explainable data descriptions, clearly discerning the important role of ocean temperature variability at the thermocline in the equatorial Pacific. SPARTAn provides an easily interpretable description of the timescales by which these thermocline temperature features evolve and eventually express at the surface, thereby enabling enhanced predictability of the key drivers of the interannual climate.

Financial decision-making problems based on relatively few observations and several explanatory variables can be problematic for the common machine learning (ML) tools, since they cannot efficiently discriminate the relevant information. To investigate the challenges of this “small data” regime, we employ several state-of-the-art ML methods for predicting whether three selected stocks from the Swiss Market Index will outperform the market, by using, as classification features, a set of commonly used technical indicators. We show that the recently introduced entropic Scalable Probabilistic Approximation (eSPA) algorithm significantly surpasses its competitors in both prediction accuracy and computational cost. We then discuss the interpretability of the employed ML methods and suggest some statistically derived heuristics to select the most appropriate and parsimonious financial decision-making candidate model.

We study semiparametric inference in a small-dimensional vector autoregressive (VAR) model of order p augmented by unobservable common factors with a dynamic described by a VAR process of order q . This state-space specification is useful to measure separately the direct causality effects and the responses to dynamic common factors. We show that the state-space parameters are identifiable from the autocovariance function of the observed process. We estimate the model by means of a multistep procedure in closed-form, which combines an eigenvalue–eigenvector matrix decomposition and a linear instrumental variable estimation allowing for Hansen–Sargan specification tests. We study the asymptotic and finite-sample properties of the parameter estimators and of rank tests for selecting the number of unobservable factors and VAR orders. In an empirical illustration, we investigate the dynamic common factors and the spillover effects that explain the co-movements among the log daily realized volatilities of four European stock market indices.

We study positional portfolio management strategies in which the manager maximizes an expected utility function written on the cross-sectional rank (position) of the portfolio return. The objective function reflects the manager’s goal to be well-ranked among competitors. To implement positional allocation strategies, we specify a nonlinear unobservable factor model for the asset returns which disentangles the dynamics of the cross-sectional distribution and the dynamics of the ranks of the individual assets. Using a large dataset of stocks returns we find that positional strategies outperform standard momentum, reversal and mean-variance allocation strategies, as well as equally weighted portfolio for criteria based on position.

We develop a new approach for evaluating performance across hedge funds. Our approach allows for performance comparisons between models that are misspecified – a common feature given the numerous factors that drive hedge fund returns. The empirical results show that the standard models used in previous work omit similar factors because they (i) perform exactly like the CAPM, and (ii) produce large and positive alphas. In contrast, we observe a large and statistically significant decrease in performance with a new model formed with alternative factors that capture variance, correlation, liquidity, betting-against-beta, carry, and time-series momentum strategies. Overall, the results suggest that the average returns of hedge funds are largely explained by mechanical trading strategies.

The paper by Antoine, Proulx, and Renault (2018) (APR) deals with the econometric definition, economic interpretation, and statistical estimation of the pseudo-true stochastic discount factor (SDF) in misspecified conditional asset pricing models. The paper revolves around fundamental issues like the role of conditioning information and omitted risk factors, and has non-trivial interactions with the current debate in the literature on the impact of weak factors (weak identification) for assessing asset pricing models. Building on, and substantially extending, previous contributions in the literature, the approach of the authors to define a pseudo-true SDF relies on the minimizers of econometric criteria based on a conditional version of the Hansen–Jagannathan (HJ) distance, that is, an average across states of squared conditional pricing errors. The authors provide an insightful discussion of the economic interpretation of pseudo-true SDFs. APR advocate the use of a fixed bandwidth (i.e., independent of the sample size) when estimating the conditional pricing errors by kernel regression methods to facilitate statistical analysis. This route leads to bandwidth-dependent pseudo-true SDF parameters and estimators thereof.

We compare non-nested parametric specifications of the Stochastic Discount Factor (SDF) using the conditional Hansen-Jagannathan (HJ-) distance. This distance measures the discrepancy between a parametric model-implied SDF and the admissible SDF's satisfying all the conditional (dynamic) no-arbitrage restrictions, instead of just few unconditional no-arbitrage restrictions for managed portfolios chosen through the instrument selection. We estimate the conditional HJ-distance by a Generalized Method of Moments estimator and establish its large sample properties for model selection purposes. We compare empirically several SDF models including multifactor beta pricing specifications and some recently proposed SDF models that are conditionally linear in consumption growth.

This article presents tests for the existence of common factors spanning two large panels/groups of macroeconomic and financial variables, and the estimation of common and group-specific factors. New analytical results are derived regarding (i) the difference in the asymptotic distribution of the test statistics when aggregating the data first and then extracting the principal components (PCs), or vice versa, as well as (ii) the estimation of the common factor and its asymptotic distribution, extending the work of Andreou et al. (2019). We find that although there is no empirical evidence for one common factor, with constant loadings, in the United States during the period 1963-2017, there is evidence of one common macro-finance factor during the pre- and post-Great Moderation regimes. The aforementioned approaches of estimating PCs yield almost identical common and group-specific (financial and macro) factors which turn out to be significant in predicting key economic indicators, such as real Gross Domestic Product (GDP) growth and the CBOE Volatility Index, among others.

Data-driven modelling and computational predictions based on maximum entropy principle (MaxEnt-principle) aim at finding as-simple-as-possible - but not simpler then necessary - models that allow to avoid the data overfitting problem. We derive a multivariate non-parametric and non-stationary formulation of the MaxEnt-principle and show that its solution can be approximated through a numerical maximisation of the sparse constrained optimization problem with regularization. Application of the resulting algorithm to popular financial benchmarks reveals memoryless models allowing for simple and qualitative descriptions of the major stock market indexes data. We compare the obtained MaxEnt-models to the heteroschedastic models from the computational econometrics (GARCH, GARCH-GJR, MS-GARCH, GARCH-PML4) in terms of the model fit, complexity and prediction quality. We compare the resulting model log-likelihoods, the values of the Bayesian Information Criterion, posterior model probabilities, the quality of the data autocorrelation function fits as well as the Value-at-Risk prediction quality. We show that all of the considered seven major financial benchmark time series (DJI, SPX, FTSE, STOXX, SMI, HSI and N225) are better described by conditionally memoryless MaxEnt-models with nonstationary regime-switching than by the common econometric models with finite memory. This analysis also reveals a sparse network of statistically-significant temporal relations for the positive and negative latent variance changes among different markets. The code is provided for open access.

We derive asymptotic properties of estimators and test statistics to determine - in a grouped data setting - common versus group-specific factors. Despite the fact that our test statistic for the number of common factors, under the null, involves a parameter at the boundary (related to unit canonical correlations) we derive a parameter-free asymptotic Gaussian distribution. We show how the group factor setting applies to mixed frequency data. As an empirical illustration we address the question whether Industrial Production (IP) is still the dominant factor driving the U.S. economy using a mixed-frequency data panel of IP and non-IP sectors. We find that a single common factor explains 89% of IP output growth and 61% of total GDP growth despite the diminishing role of manufacturing.

This paper deals with identification and inference on the unobservable conditional factor space and its dimension in large unbalanced panels of asset returns. The model specification is nonparametric regarding the way the loadings vary in time as functions of common shocks and individual characteristics. The number of active factors can also be time-varying as an effect of the changing macroeconomic environment. The method deploys Instrumental Variables (IV) which have full-rank covariation with the factor betas in the cross-section. It allows for a large dimension of the vector generating the conditioning information by machine learning techniques. In an empirical application, we infer the conditional factor space in the panel of monthly returns of individual stocks in the CRSP dataset between January 1971 and December 2017.

This chapter surveys recent econometric methodologies 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 chapter starts with an historical perspective on conditional factor models with a small number of assets for comparison purpose. Then, it outlines the concept of approximate factor structure in the presence of conditional information, and reviews an arbitrage pricing theory for large dimensional factor models in this framework. For inference, we distinguish between two different cases 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 provide new empirical findings based on a broad set of factor models 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.

We build a simple diagnostic criterion for approximate factor structure in large cross-sectional equity datasets. Given a model for asset returns with observable factors, the criterion checks whether the error terms are weakly cross-sectionally correlated or share at least one unobservable common factor. It only requires computing the largest eigenvalue of the empirical cross-sectional covariance matrix of the residuals of a large unbalanced panel. A general version of this criterion allows us to determine the number of omitted common factors. The panel data model accommodates both time-invariant and time-varying factor structures. The theory applies to random coefficient panel models with interactive fixed effects under large cross-section and time-series dimensions. The empirical analysis runs on monthly and quarterly returns for about ten thousand US stocks from January 1968 to December 2011 for several time-invariant and time-varying specifications. For monthly returns, we can choose either among time-invariant specifications with at least four financial factors, or a scaled three-factor specification. For quarterly returns, we cannot select macroeconomic models without the market factor.