Factor model is an appealing and effective analytic tool for high-dimensional time series, with a wide range of applications in economics, finance and statistics. One of the fundamental issues in using factor model for time series in practice is the determination of the number of factors to use. This paper develops two criteria for such a task for tensor factor models where the signal part of an observed time series in tensor form assumes a Tucker decomposition with the core tensor as the factor tensor. The task is to determine the dimensions of the core tensor. One of the proposed criteria is similar to information based criteria of model selection, and the other is an extension of the approaches based on the ratios of consecutive eigenvalues often used in factor analysis for panel time series. The new criteria are designed to locate the gap between the true smallest non-zero eigenvalue and the zero eigenvalues of a functional of the population version of the auto-cross-covariances of the tensor time series using their sample versions. As sample size and tensor dimension increase, such a gap increases under regularity conditions, resulting in consistency of the rank estimator. The criteria are built upon the existing non-iterative and iterative estimation procedures of tensor factor model, yielding different performances. We provide sufficient conditions and convergence rate for the consistency of the criteria as the sample size $T$ and the dimensions of the observed tensor time series go to infinity. The results include the vector factor models as special cases, with an additional convergence rates. The results also include the cases when there exist factors with different signal strength. In addition, the convergence rates of the eigenvalue estimators are established. Simulation studies provide promising finite sample performance for the two criteria.
{"title":"Rank Determination in Tensor Factor Model","authors":"Yuefeng Han, Cun-Hui Zhang, Rong Chen","doi":"10.1214/22-EJS1991","DOIUrl":"https://doi.org/10.1214/22-EJS1991","url":null,"abstract":"Factor model is an appealing and effective analytic tool for high-dimensional time series, with a wide range of applications in economics, finance and statistics. One of the fundamental issues in using factor model for time series in practice is the determination of the number of factors to use. This paper develops two criteria for such a task for tensor factor models where the signal part of an observed time series in tensor form assumes a Tucker decomposition with the core tensor as the factor tensor. The task is to determine the dimensions of the core tensor. One of the proposed criteria is similar to information based criteria of model selection, and the other is an extension of the approaches based on the ratios of consecutive eigenvalues often used in factor analysis for panel time series. The new criteria are designed to locate the gap between the true smallest non-zero eigenvalue and the zero eigenvalues of a functional of the population version of the auto-cross-covariances of the tensor time series using their sample versions. As sample size and tensor dimension increase, such a gap increases under regularity conditions, resulting in consistency of the rank estimator. The criteria are built upon the existing non-iterative and iterative estimation procedures of tensor factor model, yielding different performances. We provide sufficient conditions and convergence rate for the consistency of the criteria as the sample size $T$ and the dimensions of the observed tensor time series go to infinity. The results include the vector factor models as special cases, with an additional convergence rates. The results also include the cases when there exist factors with different signal strength. In addition, the convergence rates of the eigenvalue estimators are established. Simulation studies provide promising finite sample performance for the two criteria.","PeriodicalId":106740,"journal":{"name":"ERN: Other Econometrics: Econometric Model Construction","volume":"2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130392650","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper assembles a toolkit for the assessment of model risk when model uncertainty sets are defined in terms of an F-divergence ball around a reference model. We propose a new family of F-divergences that are easy to implement and flexible enough to imply convincing uncertainty sets for broad classes of reference models. We use our theoretical results to construct concrete examples of divergences that allow for significant amounts of uncertainty about lognormal or heavy-tailed Weibull reference models without implying that the worst case is necessarily infinitely bad. We implement our tools in an open-source software package and apply them to three risk management problems from operations management, insurance, and finance. This paper was accepted by Baris Ata, stochastic models and simulation.
{"title":"A Toolkit for Robust Risk Assessment Using F-Divergences","authors":"T. Kruse, Judith C. Schneider, Nikolaus Schweizer","doi":"10.2139/ssrn.3680475","DOIUrl":"https://doi.org/10.2139/ssrn.3680475","url":null,"abstract":"This paper assembles a toolkit for the assessment of model risk when model uncertainty sets are defined in terms of an F-divergence ball around a reference model. We propose a new family of F-divergences that are easy to implement and flexible enough to imply convincing uncertainty sets for broad classes of reference models. We use our theoretical results to construct concrete examples of divergences that allow for significant amounts of uncertainty about lognormal or heavy-tailed Weibull reference models without implying that the worst case is necessarily infinitely bad. We implement our tools in an open-source software package and apply them to three risk management problems from operations management, insurance, and finance. This paper was accepted by Baris Ata, stochastic models and simulation.","PeriodicalId":106740,"journal":{"name":"ERN: Other Econometrics: Econometric Model Construction","volume":"226 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-08-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124497358","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper is the first of a multi-part series on the calibration of the one-factor Hull-White short rate model for the purpose of computing CVAs (and xVAs) with an xVA system. It introduces an atypical bootstrapping scheme for the calibration of the short rate volatility. The second part focuses on the selection of the mean reversion parameter. In both expositions we present long-term time series results for EUR, JPY, and USD, covering the period from the beginning of 2009 (at the earliest) to spring 2020.
{"title":"One-factor Hull-White Model Calibration for CVA - Part I: Instrument Selection With a Kink","authors":"Christoph M. Puetter, Stefano Renzitti","doi":"10.2139/ssrn.3659430","DOIUrl":"https://doi.org/10.2139/ssrn.3659430","url":null,"abstract":"This paper is the first of a multi-part series on the calibration of the one-factor Hull-White short rate model for the purpose of computing CVAs (and xVAs) with an xVA system. It introduces an atypical bootstrapping scheme for the calibration of the short rate volatility. The second part focuses on the selection of the mean reversion parameter. In both expositions we present long-term time series results for EUR, JPY, and USD, covering the period from the beginning of 2009 (at the earliest) to spring 2020.","PeriodicalId":106740,"journal":{"name":"ERN: Other Econometrics: Econometric Model Construction","volume":"52 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123667732","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study Granger causality testing for high-dimensional time series using regularized regressions. To perform proper inference, we rely on heteroskedasticity and autocorrelation consistent (HAC) estimation of the asymptotic variance and develop the inferential theory in the high-dimensional setting. To recognize the time-series data structures, we focus on the sparse-group LASSO (sg-LASSO) estimator, which includes the LASSO and the group LASSO as special cases. We establish the debiased central limit theorem for low-dimensional groups of regression coefficients and study the HAC estimator of the long-run variance based on the sg-LASSO residuals. This leads to valid time-series inference for individual regression coefficients as well as groups, including Granger causality tests. The treatment relies on a new Fuk–Nagaev inequality for a class of τ-mixing processes with heavier than Gaussian tails, which is of independent interest. In an empirical application, we study the Granger causal relationship between the VIX and financial news.
{"title":"High-Dimensional Granger Causality Tests with an Application to VIX and News","authors":"Andrii Babii, Eric Ghysels, Jonas Striaukas","doi":"10.2139/ssrn.3615718","DOIUrl":"https://doi.org/10.2139/ssrn.3615718","url":null,"abstract":"\u0000 We study Granger causality testing for high-dimensional time series using regularized regressions. To perform proper inference, we rely on heteroskedasticity and autocorrelation consistent (HAC) estimation of the asymptotic variance and develop the inferential theory in the high-dimensional setting. To recognize the time-series data structures, we focus on the sparse-group LASSO (sg-LASSO) estimator, which includes the LASSO and the group LASSO as special cases. We establish the debiased central limit theorem for low-dimensional groups of regression coefficients and study the HAC estimator of the long-run variance based on the sg-LASSO residuals. This leads to valid time-series inference for individual regression coefficients as well as groups, including Granger causality tests. The treatment relies on a new Fuk–Nagaev inequality for a class of τ-mixing processes with heavier than Gaussian tails, which is of independent interest. In an empirical application, we study the Granger causal relationship between the VIX and financial news.","PeriodicalId":106740,"journal":{"name":"ERN: Other Econometrics: Econometric Model Construction","volume":"91 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115757879","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A. Dufays, Elysée Aristide Houndetoungan, Alain Coen
Change-point processes are one flexible approach to model long time series. We propose a method to uncover which model parameter truly vary when a change-point is detected. Given a set of breakpoints, we use a penalized likelihood approach to select the best set of parameters that changes over time and we prove that the penalty function leads to a consistent selection of the true model. Estimation is carried out via the deterministic annealing expectation-maximization algorithm. Our method accounts for model selection uncertainty and associates a probability to all the possible time-varying parameter specifications. Monte Carlo simulations highlight that the method works well for many time series models including heteroskedastic processes. For a sample of 14 Hedge funds (HF) strategies, using an asset based style pricing model, we shed light on the promising ability of our method to detect the time-varying dynamics of risk exposures as well as to forecast HF returns.
{"title":"Selective Linear Segmentation For Detecting Relevant Parameter Changes","authors":"A. Dufays, Elysée Aristide Houndetoungan, Alain Coen","doi":"10.2139/ssrn.3461554","DOIUrl":"https://doi.org/10.2139/ssrn.3461554","url":null,"abstract":"Change-point processes are one flexible approach to model long time series. We propose a method to uncover which model parameter truly vary when a change-point is detected. Given a set of breakpoints, we use a penalized likelihood approach to select the best set of parameters that changes over time and we prove that the penalty function leads to a consistent selection of the true model. Estimation is carried out via the deterministic annealing expectation-maximization algorithm. Our method accounts for model selection uncertainty and associates a probability to all the possible time-varying parameter specifications. Monte Carlo simulations highlight that the method works well for many time series models including heteroskedastic processes. For a sample of 14 Hedge funds (HF) strategies, using an asset based style pricing model, we shed light on the promising ability of our method to detect the time-varying dynamics of risk exposures as well as to forecast HF returns.","PeriodicalId":106740,"journal":{"name":"ERN: Other Econometrics: Econometric Model Construction","volume":"39 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114794489","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Simple, multi-step estimators are developed for the popular GARCH(1,1) model, where these estimators are either available entirely in closed form or dependent upon a preliminary estimate from, for example, quasi-maximum likelihood. Identification sources to asymmetry in the model's innovations, casting skewness as an instrument in a linear, two-stage least squares estimator. Properties of regular variation coupled with point process theory establish the distributional limits of these estimators as stable, though highly non-Gaussian, with slow convergence rates relative to the ??n-case. Moment existence criteria necessary for these results are consistent with the heavy-tailed features of many financial returns. In light-tailed cases that support asymptotic normality for these simple estimators, conditions are discovered where the simple estimators can enhance the asymptotic efficiency of quasi-maximum likelihood estimation. In small samples, extensive Monte Carlo experime nts reveal these efficiency enhancements to be available for (very) heavy tailed cases. Consequently, the proposed simple estimators are members of the class of multi-step estimators aimed at improving the efficiency of the quasi-maximum likelihood estimator.
{"title":"When Simplicity Offers a Benefit, Not a Cost: Closed-Form Estimation of the GARCH(1,1) Model that Enhances the Efficiency of Quasi-Maximum Likelihood","authors":"Todd Prono","doi":"10.17016/FEDS.2019.030","DOIUrl":"https://doi.org/10.17016/FEDS.2019.030","url":null,"abstract":"Simple, multi-step estimators are developed for the popular GARCH(1,1) model, where these estimators are either available entirely in closed form or dependent upon a preliminary estimate from, for example, quasi-maximum likelihood. Identification sources to asymmetry in the model's innovations, casting skewness as an instrument in a linear, two-stage least squares estimator. Properties of regular variation coupled with point process theory establish the distributional limits of these estimators as stable, though highly non-Gaussian, with slow convergence rates relative to the ??n-case. Moment existence criteria necessary for these results are consistent with the heavy-tailed features of many financial returns. In light-tailed cases that support asymptotic normality for these simple estimators, conditions are discovered where the simple estimators can enhance the asymptotic efficiency of quasi-maximum likelihood estimation. In small samples, extensive Monte Carlo experime nts reveal these efficiency enhancements to be available for (very) heavy tailed cases. Consequently, the proposed simple estimators are members of the class of multi-step estimators aimed at improving the efficiency of the quasi-maximum likelihood estimator.","PeriodicalId":106740,"journal":{"name":"ERN: Other Econometrics: Econometric Model Construction","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130486982","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Mario Larch, Joschka Wanner, Y. Yotov, Thomas Zylkin
Recent work on the effects of currency unions (CUs) on trade stresses the importance of using many countries and years in order to obtain reliable estimates. However, for large samples, computational issues associated with the three‐way (exporter‐time, importer‐time, and country pair) fixed effects currently recommended in the gravity literature have heretofore limited the choice of estimator, leaving an important methodological gap. To address this gap, we introduce an iterative poisson pseudo‐maximum likelihood (PPML) estimation procedure that facilitates the inclusion of these fixed effects for large data sets and also allows for correlated errors across countries and time. When applied to a comprehensive sample with more than 200 countries trading over 65 years, these innovations flip the conclusions of an otherwise rigorously specified linear model. Most importantly, our estimates for both the overall CU effect and the Euro effect specifically are economically small and statistically insignificant. We also document that linear and PPML estimates of the Euro effect increasingly diverge as the sample size grows.
{"title":"Currency Unions and Trade: A PPML Re‐Assessment with High‐Dimensional Fixed Effects","authors":"Mario Larch, Joschka Wanner, Y. Yotov, Thomas Zylkin","doi":"10.1111/obes.12283","DOIUrl":"https://doi.org/10.1111/obes.12283","url":null,"abstract":"Recent work on the effects of currency unions (CUs) on trade stresses the importance of using many countries and years in order to obtain reliable estimates. However, for large samples, computational issues associated with the three‐way (exporter‐time, importer‐time, and country pair) fixed effects currently recommended in the gravity literature have heretofore limited the choice of estimator, leaving an important methodological gap. To address this gap, we introduce an iterative poisson pseudo‐maximum likelihood (PPML) estimation procedure that facilitates the inclusion of these fixed effects for large data sets and also allows for correlated errors across countries and time. When applied to a comprehensive sample with more than 200 countries trading over 65 years, these innovations flip the conclusions of an otherwise rigorously specified linear model. Most importantly, our estimates for both the overall CU effect and the Euro effect specifically are economically small and statistically insignificant. We also document that linear and PPML estimates of the Euro effect increasingly diverge as the sample size grows.","PeriodicalId":106740,"journal":{"name":"ERN: Other Econometrics: Econometric Model Construction","volume":"16 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123123195","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-11-12DOI: 10.1016/BS.HESCOM.2018.02.003
T. Lux, Remco C. J. Zwinkels
{"title":"Empirical Validation of Agent-Based Models","authors":"T. Lux, Remco C. J. Zwinkels","doi":"10.1016/BS.HESCOM.2018.02.003","DOIUrl":"https://doi.org/10.1016/BS.HESCOM.2018.02.003","url":null,"abstract":"","PeriodicalId":106740,"journal":{"name":"ERN: Other Econometrics: Econometric Model Construction","volume":"22 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122837427","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper discusses the specifics of forecasting using factor-augmented predictive regressions under general loss functions. In line with the literature, we employ principal component analysis to extract factors from the set of predictors. In addition, we also extract information on the volatility of the series to be predicted, since the volatility is forecast-relevant under non-quadratic loss functions. We ensure asymptotic unbiasedness of the forecasts under the relevant loss by estimating the predictive regression through the minimization of the in-sample average loss. Finally, we select the most promising predictors for the series to be forecast by employing an information criterion that is tailored to the relevant loss. Using a large monthly data set for the US economy, we assess the proposed adjustments in a pseudo out-of-sample forecasting exercise for various variables. As expected, the use of estimation under the relevant loss is found to be effective. Using an additional volatility proxy as the predictor and conducting model selection that is tailored to the relevant loss function enhances the forecast performance significantly.
{"title":"Predictive Regressions Under Asymmetric Loss: Factor Augmentation and Model Selection","authors":"M. Demetrescu, Sinem Hacioglu Hoke","doi":"10.2139/ssrn.3180690","DOIUrl":"https://doi.org/10.2139/ssrn.3180690","url":null,"abstract":"This paper discusses the specifics of forecasting using factor-augmented predictive regressions under general loss functions. In line with the literature, we employ principal component analysis to extract factors from the set of predictors. In addition, we also extract information on the volatility of the series to be predicted, since the volatility is forecast-relevant under non-quadratic loss functions. We ensure asymptotic unbiasedness of the forecasts under the relevant loss by estimating the predictive regression through the minimization of the in-sample average loss. Finally, we select the most promising predictors for the series to be forecast by employing an information criterion that is tailored to the relevant loss. Using a large monthly data set for the US economy, we assess the proposed adjustments in a pseudo out-of-sample forecasting exercise for various variables. As expected, the use of estimation under the relevant loss is found to be effective. Using an additional volatility proxy as the predictor and conducting model selection that is tailored to the relevant loss function enhances the forecast performance significantly.","PeriodicalId":106740,"journal":{"name":"ERN: Other Econometrics: Econometric Model Construction","volume":"33 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126586669","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Ignoring the model selection step in inference after selection is harmful. This paper studies the asymptotic distribution of estimators after model selection using the Akaike information criterion. First, we consider the classical setting in which a true model exists and is included in the candidate set of models. We exploit the overselection property of this criterion in the construction of a selection region, and obtain the asymptotic distribution of estimators and linear combinations thereof conditional on the selected model. The limiting distribution depends on the set of competitive models and on the smallest overparameterized model. Second, we relax the assumption about the existence of a true model, and obtain uniform asymptotic results. We use simulation to study the resulting postselection distributions and to calculate confidence regions for the model parameters. We apply the method to data.
{"title":"Asymptotic Post-Selection Inference for Akaike's Information Criterion","authors":"Ali Charkhi, G. Claeskens","doi":"10.2139/ssrn.3167253","DOIUrl":"https://doi.org/10.2139/ssrn.3167253","url":null,"abstract":"Ignoring the model selection step in inference after selection is harmful. This paper studies the asymptotic distribution of estimators after model selection using the Akaike information criterion. First, we consider the classical setting in which a true model exists and is included in the candidate set of models. We exploit the overselection property of this criterion in the construction of a selection region, and obtain the asymptotic distribution of estimators and linear combinations thereof conditional on the selected model. The limiting distribution depends on the set of competitive models and on the smallest overparameterized model. Second, we relax the assumption about the existence of a true model, and obtain uniform asymptotic results. We use simulation to study the resulting postselection distributions and to calculate confidence regions for the model parameters. We apply the method to data.","PeriodicalId":106740,"journal":{"name":"ERN: Other Econometrics: Econometric Model Construction","volume":"54 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131385927","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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