nep-ecm New Economics Papers
on Econometrics
Issue of 2020‒02‒10
twenty papers chosen by
Sune Karlsson
Örebro universitet

  1. Generalized Local IV with Unordered Multiple Treatment Levels: Identification, Efficient Estimation, and Testable Implication By Haitian Xie
  2. Low Frequency Robust Cointegrated Regression in the Presence of a Near-Unity Regressor By Jungbin Hwang; Gonzalo Valdés
  3. Finite-sample Corrected Inference for Two-step GMM in Time Series By Jungbin Hwang; Gonzalo Valdés
  4. Beta observation-driven models with exogenous regressors: a joint analysis of realized correlation and leverage effects By Paolo Gorgi; Siem Jan Koopman
  5. Fundamental Limits of Testing the Independence of Irrelevant Alternatives in Discrete Choice By Arjun Seshadri; Johan Ugander
  6. Estimating Marginal Treatment Effects under Unobserved Group Heterogeneity By Tadao Hoshino; Takahide Yanagi
  7. Bayesian Panel Quantile Regression for Binary Outcomes with Correlated Random Effects: An Application on Crime Recidivism in Canada By Georges Bresson; Guy Lacroix; Mohammad Arshad Rahman
  8. Oracle Efficient Estimation of Structural Breaks in Cointegrating Regressions By Karsten Schweikert
  9. Improving portfolios global performance using a cleaned and robust covariance matrix estimate By Emmanuelle Jay; Thibault Soler; Eugénie Terreaux; Jean-Philippe Ovarlez; Frédéric Pascal; Philippe de Peretti; Christophe Chorro
  10. Extracting Statistical Factors When Betas are Time-Varying By Patrick Gagliardini; Hao Ma
  11. Modeling multivariate operational losses via copula-based distributions with g-and-h marginals By Marco Bee; Julien Hambuckers
  12. Variable-lag Granger Causality and Transfer Entropy for Time Series Analysis By Chainarong Amornbunchornvej; Elena Zheleva; Tanya Berger-Wolf
  13. Robust covariance matrix estimation and portfolio allocation: the case of non-homogeneous assets By Emmanuelle Jay; Thibault Soler; Jean-Philippe Ovarlez; Philippe de Peretti; Christophe Chorro
  14. A Neural-embedded Choice Model: TasteNet-MNL Modeling Taste Heterogeneity with Flexibility and Interpretability By Yafei Han; Christopher Zegras; Francisco Camara Pereira; Moshe Ben-Akiva
  15. Per-Cluster Instrumental Variables Estimation: Uncovering the Price Elasticity of the Demand for Gasoline By Michael Bates; Seolah Kim
  16. Transparency in Structural Research By Isaiah Andrews; Matthew Gentzkow; Jesse M. Shapiro
  17. A Bayesian Long Short-Term Memory Model for Value at Risk and Expected Shortfall Joint Forecasting By Zhengkun Li; Minh-Ngoc Tran; Chao Wang; Richard Gerlach; Junbin Gao
  18. Sharpe Ratio in High Dimensions: Cases of Maximum Out of Sample, Constrained Maximum, and Optimal Portfolio Choice By Mehmet Caner; Marcelo Medeiros; Gabriel Vasconcelos
  19. Do Learning Communities Increase First Year College Retention? Testing Sample Selection and External Validity of Randomized Control Trials By Tarek Azzam; Michael Bates; David Fairris
  20. Refined model of the covariance/correlation matrix between securities By Sebastien Valeyre

  1. By: Haitian Xie
    Abstract: This paper studies the econometric aspects of the generalized local IV framework defined using the unordered monotonicity condition, which accommodates multiple levels of treatment and instrument in program evaluations. The framework is explicitly developed to allow for conditioning covariates. Nonparametric identification results are obtained for a wide range of policy-relevant parameters. Semiparametric efficiency bounds are computed for these identified structural parameters, including the local average structural function and local average structural function on the treated. Two semiparametric estimators are introduced that achieve efficiency. One is the conditional expectation projection estimator defined through the nonparametric identification equation. The other is the double/debiased machine learning estimator defined through the efficient influence function, which is suitable for high-dimensional settings. More generally, for parameters implicitly defined by possibly non-smooth and overidentifying moment conditions, this study provides the calculation for the corresponding semiparametric efficiency bounds and proposes efficient semiparametric GMM estimators again using the efficient influence functions. Then an optimal set of testable implications of the model assumption is proposed. Previous results developed for the binary local IV model and the multivalued treatment model under unconfoundedness are encompassed as special cases in this more general framework. The theoretical results are illustrated by an empirical application investigating the return to schooling across different fields of study, and a Monte Carlo experiment.
    Date: 2020–01
  2. By: Jungbin Hwang (University of Connecticut); Gonzalo Valdés (Universidad de Tarapacá)
    Abstract: This paper develops a robust t and F inferences on a triangular cointegrated system when one may not be sure the economic variables are exact unit root processes. We show that the low frequency transformed augmented (TA) OLS method possesses an asymptotic bias term in the limiting distribution, and corresponding t and F inferences in Hwang and Sun (2017) are asymptotically invalid. As a result, the size of the cointegration vector can be extremely large for even very small deviations from the unit root regressors. We develop a method to correct the asymptotic bias of the TA-OLS test statistics for the cointegration vector. Our modi ed statistics not only adjusts the locational bias but also reects the estimation uncertainty of the long-run endogenity parameter in the bias correction term and has asymptotic t and F limits. Based on the modi ed TA-OLS test statistics, the paper provides a simple Bonferroni method to test for the cointegration parameter.
    Keywords: Cointegration, Local to Unity, t and F tests, Alternative Asymptotics, Low Fre-quency Econometrics, Transformed and Augmented OLS
    JEL: C12 C13 C32
    Date: 2020–01
  3. By: Jungbin Hwang (University of Connecticut); Gonzalo Valdés (Universidad de Tarapacá)
    Abstract: This paper develops a nite-sample corrected and robust inference for e¢ cient two-step generalized method of moments (GMM). One of the main challenges in e¢ cient GMM is that we do not observe the moment process and have to use the estimated moment process to construct a GMM weighting matrix. We use a non-parametric long run variance estimator as the optimal GMM weighting matrix. To capture the estimation uncertainty embodied in the weight matrix, we extend the nite-sample corrected formula of Windmeijer (2005) to a heteroskedasticity autocorrelated robust (HAR) inference in time series setting. Using xed-smoothing asymptotics, we show that our new test statistics lead to standard asymptotic F or t critical values and improve the nite sample performance of existing HAR robust GMM tests.
    Keywords: Generalized Method of Moments, Heteroskedasticity Autocorrelated Robust, Finite-sample Correction, Fixed-smoothing Asymptotics, t and F tests.
    JEL: C12 C13 C32
    Date: 2020–01
  4. By: Paolo Gorgi (Vrije Universiteit Amsterdam); Siem Jan Koopman (Vrije Universiteit Amsterdam)
    Abstract: We consider a general class of observation-driven models with exogenous regressors for double bounded data that are based on the beta distribution. We obtain a stationary and ergodic beta observation-driven process subject to a contraction condition on the stochastic dynamic model equation. We derive conditions for strong consistency and asymptotic normality of the maximum likelihood estimator. The general results are used to study the properties of a beta autoregressive process with threshold effects and to establish the asymptotic properties of the maximum likelihood estimator. We employ the threshold autoregressive model with leverage effects to analyze realized correlations for several sets of stock returns. We find that the impact of past values of realized correlation on future values is at least 10% higher when stock returns are negative rather than positive. This finding supports the conjecture that correlation between stock returns tends to be higher when stock prices are falling, and lower when there is a surge in stock prices. Finally, we conduct an out-of-sample study that shows that our model with leverage effects can enhance the accuracy of point and density forecasts of realized correlations.
    Keywords: Double bounded time series, financial econometrics, leverage effects, observation- driven models, realized correlation
    JEL: C32 C52 C58
    Date: 2020–01–27
  5. By: Arjun Seshadri; Johan Ugander
    Abstract: The Multinomial Logit (MNL) model and the axiom it satisfies, the Independence of Irrelevant Alternatives (IIA), are together the most widely used tools of discrete choice. The MNL model serves as the workhorse model for a variety of fields, but is also widely criticized, with a large body of experimental literature claiming to document real-world settings where IIA fails to hold. Statistical tests of IIA as a modelling assumption have been the subject of many practical tests focusing on specific deviations from IIA over the past several decades, but the formal size properties of hypothesis testing IIA are still not well understood. In this work we replace some of the ambiguity in this literature with rigorous pessimism, demonstrating that any general test for IIA with low worst-case error would require a number of samples exponential in the number of alternatives of the choice problem. A major benefit of our analysis over previous work is that it lies entirely in the finite-sample domain, a feature crucial to understanding the behavior of tests in the common data-poor settings of discrete choice. Our lower bounds are structure-dependent, and as a potential cause for optimism, we find that if one restricts the test of IIA to violations that can occur in a specific collection of choice sets (e.g., pairs), one obtains structure-dependent lower bounds that are much less pessimistic. Our analysis of this testing problem is unorthodox in being highly combinatorial, counting Eulerian orientations of cycle decompositions of a particular bipartite graph constructed from a data set of choices. By identifying fundamental relationships between the comparison structure of a given testing problem and its sample efficiency, we hope these relationships will help lay the groundwork for a rigorous rethinking of the IIA testing problem as well as other testing problems in discrete choice.
    Date: 2020–01
  6. By: Tadao Hoshino; Takahide Yanagi
    Abstract: This paper studies endogenous treatment effect models in which individuals are classified into unobserved groups based on heterogeneous treatment choice rules. Such heterogeneity may arise, for example, when multiple treatment eligibility criteria and different preference patterns exist. Using a finite mixture approach, we propose a marginal treatment effect (MTE) framework in which the treatment choice and outcome equations can be heterogeneous across groups. Under the availability of valid instrumental variables specific to each group, we show that the MTE for each group can be separately identified using the local instrumental variable method. Based on our identification result, we propose a two-step semiparametric procedure for estimating the group-wise MTE parameters. We first estimate the finite-mixture treatment choice model by a maximum likelihood method and then estimate the MTEs using a series approximation method. We prove that the proposed MTE estimator is consistent and asymptotically normally distributed. We illustrate the usefulness of the proposed method with an application to economic returns to college education.
    Date: 2020–01
  7. By: Georges Bresson; Guy Lacroix; Mohammad Arshad Rahman
    Abstract: This article develops a Bayesian approach for estimating panel quantile regression with binary outcomes in the presence of correlated random effects. We construct a working likelihood using an asymmetric Laplace (AL) error distribution and combine it with suitable prior distributions to obtain the complete joint posterior distribution. For posterior inference, we propose two Markov chain Monte Carlo (MCMC) algorithms but prefer the algorithm that exploits the blocking procedure to produce lower autocorrelation in the MCMC draws. We also explain how to use the MCMC draws to calculate the marginal effects, relative risk and odds ratio. The performance of our preferred algorithm is demonstrated in multiple simulation studies and shown to perform extremely well. Furthermore, we implement the proposed framework to study crime recidivism in Quebec, a Canadian Province, using a novel data from the administrative correctional files. Our results suggest that the recently implemented "tough-on-crime" policy of the Canadian government has been largely successful in reducing the probability of repeat offenses in the post-policy period. Besides, our results support existing findings on crime recidivism and offer new insights at various quantiles.
    Date: 2020–01
  8. By: Karsten Schweikert
    Abstract: In this paper, we propose an adaptive group lasso procedure to efficiently estimate structural breaks in cointegrating regressions. It is well-known that the group lasso estimator is not simultaneously estimation consistent and model selection consistent in structural break settings. Hence, we use a first step group lasso estimation of a diverging number of breakpoint candidates to produce weights for a second adaptive group lasso estimation. We prove that parameter changes are estimated consistently by group lasso if it is tuned correctly and show that the number of estimated breaks is greater than the true number but still sufficiently close to it. Then, we use these results and prove that the adaptive group lasso has oracle properties if weights are obtained from our first step estimation and the tuning parameter satisfies some further restrictions. Simulation results show that the proposed estimator delivers the expected results. An economic application to the long-run US money demand function demonstrates the practical importance of this methodology.
    Date: 2020–01
  9. By: Emmanuelle Jay (Quanted & Europlace Institute of Finance, Fideas Capital); Thibault Soler (Fideas Capital, CES - Centre d'économie de la Sorbonne - UP1 - Université Panthéon-Sorbonne - CNRS - Centre National de la Recherche Scientifique); Eugénie Terreaux (DEMR, ONERA, Université Paris Saclay [Palaiseau] - ONERA - Université Paris-Saclay); Jean-Philippe Ovarlez (DEMR, ONERA, Université Paris Saclay [Palaiseau] - ONERA - Université Paris-Saclay); Frédéric Pascal (L2S - Laboratoire des signaux et systèmes - CNRS - Centre National de la Recherche Scientifique - CentraleSupélec - UP11 - Université Paris-Sud - Paris 11, CentraleSupélec); Philippe de Peretti (CES - Centre d'économie de la Sorbonne - UP1 - Université Panthéon-Sorbonne - CNRS - Centre National de la Recherche Scientifique, UP1 - Université Panthéon-Sorbonne); Christophe Chorro (CES - Centre d'économie de la Sorbonne - UP1 - Université Panthéon-Sorbonne - CNRS - Centre National de la Recherche Scientifique, UP1 - Université Panthéon-Sorbonne)
    Abstract: This paper presents how the most recent improvements made on covariance matrix estimation and model order selection can be applied to the portfolio optimization problem. The particular case of the Maximum Variety Portfolio is treated but the same improvements apply also in the other optimization problems such as the Minimum Variance Portfolio. We assume that the most important information (or the latent factors) are embedded in correlated Elliptical Symmetric noise extending classical Gaussian assumptions. We propose here to focus on a recent method of model order selection allowing to efficiently estimate the subspace of main factors describing the market. This non-standard model order selection problem is solved through Random Matrix Theory and robust covariance matrix estimation. Moreover we extend the method to non-homogeneous assets returns. The proposed procedure will be explained through synthetic data and be applied and compared with standard techniques on real market data showing promising improvements.
    Keywords: Maximum Variety Portfolio,Elliptical Symmetric Noise,Robust Covariance Matrix Estimation,Model Order Selection,Random Matrix Theory,Portfolio Optimisation,Financial Time Series,Multi-Factor Model
    Date: 2019–10
  10. By: Patrick Gagliardini (USI Università della Svizzera italiana; Swiss Finance Institute); Hao Ma (USI Università della Svizzera italiana; Swiss Finance Institute, Students)
    Abstract: 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.
    Keywords: Large Panel, Unobservable Factors, Conditioning Information, Instrumental Variables, Machine Learning, Post-Lasso, Artificial Neural Networks
    JEL: G12
    Date: 2019–07
  11. By: Marco Bee; Julien Hambuckers
    Abstract: We propose a family of copula-based multivariate distributions with g-and- h marginals. After studying the properties of the distribution, we develop a two-step estimation strategy and analyze via simulation the sampling distribution of the estimators. The methodology is used for the analysis of a 7-dimensional dataset containing 40,871 operational losses. The empirical evidence suggests that a distribution based on a single copula is not flexible enough, thus we model the dependence structure by means of vine copulas. We show that the approach based on regular vines improves the fit. Moreover, even though losses corresponding to different event types are found to be dependent, the assumption of perfect positive dependence is not supported by our analysis. As a result, the Value-at-Risk of the total operational loss distribution obtained from the copula- based technique is substantially smaller at high confidence levels, with respect to the one obtained using the common practice of summing the univariate Value-at-Risks.
    Keywords: loss model; dependence structure; vine copula; Value-at-Risk
    JEL: C46 C63
    Date: 2020
  12. By: Chainarong Amornbunchornvej; Elena Zheleva; Tanya Berger-Wolf
    Abstract: Granger causality is a fundamental technique for causal inference in time series data, commonly used in the social and biological sciences. Typical operationalizations of Granger causality make a strong assumption that every time point of the effect time series is influenced by a combination of other time series with a fixed time delay. The assumption of fixed time delay also exists in Transfer Entropy, which is considered to be a non-linear version of Granger causality. However, the assumption of the fixed time delay does not hold in many applications, such as collective behavior, financial markets, and many natural phenomena. To address this issue, we develop variable-lag Granger causality and Transfer Entropy, generalizations of both Granger causality and Transfer Entropy that relax the assumption of the fixed time delay and allows causes to influence effects with arbitrary time delays. In addition, we propose a method for inferring both variable-lag Granger causality and Transfer Entropy relations. We demonstrate our approach on an application for studying coordinated collective behavior and other real-world casual-inference datasets and show that our proposed approaches perform better than several existing methods in both simulated and real-world datasets. Our approach can be applied in any domain of time series analysis. The software of this work is available in the R package: VLTimeSeriesCausality.
    Date: 2020–02
  13. By: Emmanuelle Jay (Fideas Capital, Quanted & Europlace Institute of Finance); Thibault Soler (Fideas Capital, CES - Centre d'économie de la Sorbonne - UP1 - Université Panthéon-Sorbonne - CNRS - Centre National de la Recherche Scientifique); Jean-Philippe Ovarlez (DEMR, ONERA, Université Paris Saclay [Palaiseau] - ONERA - Université Paris-Saclay); Philippe de Peretti (CES - Centre d'économie de la Sorbonne - UP1 - Université Panthéon-Sorbonne - CNRS - Centre National de la Recherche Scientifique); Christophe Chorro (CES - Centre d'économie de la Sorbonne - UP1 - Université Panthéon-Sorbonne - CNRS - Centre National de la Recherche Scientifique)
    Abstract: This paper presents how the most recent improvements made on covariance matrix estimation and model order selection can be applied to the portfolio optimization problem. Our study is based on the case of the Maximum Variety Portfolio and may be obviously extended to other classical frameworks with analogous results. We focus on the fact that the assets should preferably be classified in homogeneous groups before applying the proposed methodology which is to whiten the data before estimating the covariance matrix using the robust Tyler M-estimator and the Random Matrix Theory (RMT). The proposed procedure is applied and compared to standard techniques on real market data showing promising improvements.
    Keywords: Elliptical Symmetric Noise,Robust Covariance Matrix Estimation,Model Order Selection,Random Matrix Theory,Portfolio Optimization
    Date: 2019–10
  14. By: Yafei Han; Christopher Zegras; Francisco Camara Pereira; Moshe Ben-Akiva
    Abstract: Discrete choice models (DCMs) and neural networks (NNs) can complement each other. We propose a neural network embedded choice model - TasteNet-MNL, to improve the flexibility in modeling taste heterogeneity while keeping model interpretability. The hybrid model consists of a TasteNet module: a feed-forward neural network that learns taste parameters as flexible functions of individual characteristics; and a choice module: a multinomial logit model (MNL) with manually specified utility. TasteNet and MNL are fully integrated and jointly estimated. By embedding a neural network into a DCM, we exploit a neural network's function approximation capacity to reduce specification bias. Through special structure and parameter constraints, we incorporate expert knowledge to regularize the neural network and maintain interpretability. On synthetic data, we show that TasteNet-MNL can recover the underlying non-linear utility function, and provide predictions and interpretations as accurate as the true model; while examples of logit or random coefficient logit models with misspecified utility functions result in large parameter bias and low predictability. In the case study of Swissmetro mode choice, TasteNet-MNL outperforms benchmarking MNLs' predictability; and discovers a wider spectrum of taste variations within the population, and higher values of time on average. This study takes an initial step towards developing a framework to combine theory-based and data-driven approaches for discrete choice modeling.
    Date: 2020–02
  15. By: Michael Bates (Department of Economics, University of California Riverside); Seolah Kim (UCR)
    Abstract: We propose a per-cluster instrumental variables estimator (PCIV) for estimating population average effects under correlated random coefficient models in the presence of endogeneity. We demonstrate consistency, showing robustness over standard estimators, and provide analytic standard errors for robust inference. We compare PCIV, fixed-effects instrumental variables, and pooled 2-stage least squares estimators using Monte Carlo simulation verifying that PCIV performs relatively well. We also apply the approaches, examining the monthly responsiveness of gasoline consumption to prices as instrumented by state fuel taxes. We find that US consumers are on average more elastic in their demand for gasoline than previous estimates imply.
    Keywords: population average effects, climate policy, gasoline taxation
    JEL: C33 C36 Q41 Q54 Q58
    Date: 2019–08
  16. By: Isaiah Andrews; Matthew Gentzkow; Jesse M. Shapiro
    Abstract: We propose a formal definition of transparency in empirical research and apply it to structural estimation in economics. We discuss how some existing practices can be understood as attempts to improve transparency, and we suggest ways to improve current practice, emphasizing approaches that impose a minimal computational burden on the researcher. We illustrate with examples.
    JEL: C10
    Date: 2020–01
  17. By: Zhengkun Li; Minh-Ngoc Tran; Chao Wang; Richard Gerlach; Junbin Gao
    Abstract: Value-at-Risk (VaR) and Expected Shortfall (ES) are widely used in the financial sector to measure the market risk and manage the extreme market movement. The recent link between the quantile score function and the Asymmetric Laplace density has led to a flexible likelihood-based framework for joint modelling of VaR and ES. It is of high interest in financial applications to be able to capture the underlying joint dynamics of these two quantities. We address this problem by developing a hybrid model that is based on the Asymmetric Laplace quasi-likelihood and employs the Long Short-Term Memory (LSTM) time series modelling technique from Machine Learning to capture efficiently the underlying dynamics of VaR and ES. We refer to this model as LSTM-AL. We adopt the adaptive Markov chain Monte Carlo (MCMC) algorithm for Bayesian inference in the LSTM-AL model. Empirical results show that the proposed LSTM-AL model can improve the VaR and ES forecasting accuracy over a range of well-established competing models.
    Date: 2020–01
  18. By: Mehmet Caner; Marcelo Medeiros; Gabriel Vasconcelos
    Abstract: In this paper, we analyze maximum Sharpe ratio when the number of assets in a portfolio is larger than its time span. One obstacle in this large dimensional setup is the singularity of the sample covariance matrix of the excess asset returns. To solve this issue, we benefit from a technique called nodewise regression, which was developed by Meinshausen and Buhlmann (2006). It provides a sparse/weakly sparse and consistent estimate of the precision matrix, using the Lasso method. We analyze three issues. One of the key results in our paper is that mean-variance efficiency for the portfolios in large dimensions is established. Then tied to that result, we also show that the maximum out-of-sample Sharpe ratio can be consistently estimated in this large portfolio of assets. Furthermore, we provide convergence rates and see that the number of assets slow down the convergence up to a logarithmic factor. Then, we provide consistency of maximum Sharpe Ratio when the portfolio weights add up to one, and also provide a new formula and an estimate for constrained maximum Sharpe ratio. Finally, we provide consistent estimates of the Sharpe ratios of global minimum variance portfolio and Markowitz's (1952) mean variance portfolio. In terms of assumptions, we allow for time series data. Simulation and out-of-sample forecasting exercise shows that our new method performs well compared to factor and shrinkage based techniques.
    Date: 2020–02
  19. By: Tarek Azzam (UCSB); Michael Bates (Department of Economics, University of California Riverside); David Fairris (UCR)
    Abstract: Voluntary selection into experimental samples is ubiquitous and leads researchers to question the external validity of experimental findings. We introduce tests for sample selection on unobserved variables to discern the generalizability of randomized control trials. We estimate the impact of a learning community on first-year college retention using an RCT, and employ our tests in this setting. We compare observational and experimental estimates, considering the internal and external validity of both approaches. Intent-to-treat and local-average-treatment-effect estimates reveal no discernable programmatic effects, whereas observational estimates are significantly positive. The experimental sample is positively selected on unobserved characteristics suggesting limited external validity.
    Keywords: External validity, college retention, selection on unobserved variables
    JEL: C93 I23
    Date: 2019–09
  20. By: Sebastien Valeyre
    Abstract: A new methodology has been introduced to clean the correlation matrix of single stocks returns based on a constrained principal component analysis using financial data. Portfolios were introduced, namely "Fundamental Maximum Variance Portfolios", to capture in an optimal way the risks defined by financial criteria ("Book", "Capitalization", etc.). The constrained eigenvectors of the correlation matrix, which are the linear combination of these portfolios, are then analyzed. Thanks to this methodology, several stylized patterns of the matrix were identified: i) the increase of the first eigenvalue with a time scale from 1 minute to several months seems to follow the same law for all the significant eigenvalues with 2 regimes; ii) a universal law seems to govern the weights of all the "Maximum variance" portfolios, so according to that law, the optimal weights should be proportional to the ranking based on the financial studied criteria; iii) the volatility of the volatility of the "Maximum Variance" portfolios, which are not orthogonal, could be enough to explain a large part of the diffusion of the correlation matrix; iv) the leverage effect (increase of the first eigenvalue with the decline of the stock market) occurs only for the first mode and cannot be generalized for other factors of risk. The leverage effect on the beta, which is the sensitivity of stocks with the market mode, makes variable the weights of the first eigenvector.
    Date: 2020–01

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