We introduce a novel covariance estimator that exploits the heteroscedastic nature of financial time series by employing exponential weighted moving averages and shrinking the in-sample eigenvalues through cross-validation. Our estimator is model-agnostic in that we make no assumptions on the distribution of the random entries of the matrix or structure of the covariance matrix. Additionally, we show how Random Matrix Theory can provide guidance for automatic tuning of the hyperparameter which characterizes the time scale for the dynamics of the estimator. By attenuating the noise from both the cross-sectional and time-series dimensions, we empirically demonstrate the superiority of our estimator over competing estimators that are based on exponentially-weighted and uniformly-weighted covariance matrices.
{"title":"Large Non-Stationary Noisy Covariance Matrices: A Cross-Validation Approach","authors":"Vincent W. C. Tan, S. Zohren","doi":"10.2139/ssrn.3745692","DOIUrl":"https://doi.org/10.2139/ssrn.3745692","url":null,"abstract":"We introduce a novel covariance estimator that exploits the heteroscedastic nature of financial time series by employing exponential weighted moving averages and shrinking the in-sample eigenvalues through cross-validation. Our estimator is model-agnostic in that we make no assumptions on the distribution of the random entries of the matrix or structure of the covariance matrix. Additionally, we show how Random Matrix Theory can provide guidance for automatic tuning of the hyperparameter which characterizes the time scale for the dynamics of the estimator. By attenuating the noise from both the cross-sectional and time-series dimensions, we empirically demonstrate the superiority of our estimator over competing estimators that are based on exponentially-weighted and uniformly-weighted covariance matrices.","PeriodicalId":299310,"journal":{"name":"Econometrics: Mathematical Methods & Programming eJournal","volume":"68 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124020313","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}
Research on quantum technology spans multiple disciplines: physics, computer science, engineering, and mathematics. The objective of this manuscript is to provide an accessible introduction to this emerging field for economists that is centered around quantum computing and quantum money. We proceed in three steps. First, we discuss basic concepts in quantum computing and quantum communication, assuming knowledge of linear algebra and statistics, but not of computer science or physics. This covers fundamental topics, such as qubits, superposition, entanglement, quantum circuits, oracles, and the no-cloning theorem. Second, we provide an overview of quantum money, an early invention of the quantum communication literature that has recently been partially implemented in an experimental setting. One form of quantum money offers the privacy and anonymity of physical cash, the option to transact without the involvement of a third party, and the efficiency and convenience of a debit card payment. Such features cannot be achieved in combination with any other form of money. Finally, we review all existing quantum speedups that have been identified for algorithms used to solve and estimate economic models. This includes function approximation, linear systems analysis, Monte Carlo simulation, matrix inversion, principal component analysis, linear regression, interpolation, numerical differentiation, and true random number generation. We also discuss the difficulty of achieving quantum speedups and comment on common misconceptions about what is achievable with quantum computing.
{"title":"Quantum Technology for Economists","authors":"Isaiah Hull, Or Sattath, E. Diamanti, G. Wendin","doi":"10.2139/ssrn.3745608","DOIUrl":"https://doi.org/10.2139/ssrn.3745608","url":null,"abstract":"Research on quantum technology spans multiple disciplines: physics, computer science, engineering, and mathematics. The objective of this manuscript is to provide an accessible introduction to this emerging field for economists that is centered around quantum computing and quantum money. We proceed in three steps. First, we discuss basic concepts in quantum computing and quantum communication, assuming knowledge of linear algebra and statistics, but not of computer science or physics. This covers fundamental topics, such as qubits, superposition, entanglement, quantum circuits, oracles, and the no-cloning theorem. Second, we provide an overview of quantum money, an early invention of the quantum communication literature that has recently been partially implemented in an experimental setting. One form of quantum money offers the privacy and anonymity of physical cash, the option to transact without the involvement of a third party, and the efficiency and convenience of a debit card payment. Such features cannot be achieved in combination with any other form of money. Finally, we review all existing quantum speedups that have been identified for algorithms used to solve and estimate economic models. This includes function approximation, linear systems analysis, Monte Carlo simulation, matrix inversion, principal component analysis, linear regression, interpolation, numerical differentiation, and true random number generation. We also discuss the difficulty of achieving quantum speedups and comment on common misconceptions about what is achievable with quantum computing.","PeriodicalId":299310,"journal":{"name":"Econometrics: Mathematical Methods & Programming eJournal","volume":"39 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121571394","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}
Abstract We study utility indifference pricing of untradable assets in incomplete markets using a symmetric asymptotic hyperbolic absolute risk aversion (SAHARA) utility function, both from the buyer’s and seller’s perspective. The use of the SAHARA utility function allows us to tackle the “short call” problem, which power and exponential utility functions are unable to solve. While no closed-form solutions are available for the indifference prices, we are able to derive some pricing bounds. Furthermore, we rely on the dynamic programming approach to solve the associated utility maximization problem, which leads to a two-dimension HJB equation. A complex algorithm discussed in Ma and Forsyth (2016) is consequently adopted to numerically solve the HJB equation. We determine utility indifference prices for options written on the untradable underlying assets and some insurance contracts.
{"title":"Indifference Pricing Under Sahara Utility","authors":"A. Chen, Thai Q. Nguyen, Nils Sørensen","doi":"10.2139/ssrn.3730609","DOIUrl":"https://doi.org/10.2139/ssrn.3730609","url":null,"abstract":"Abstract We study utility indifference pricing of untradable assets in incomplete markets using a symmetric asymptotic hyperbolic absolute risk aversion (SAHARA) utility function, both from the buyer’s and seller’s perspective. The use of the SAHARA utility function allows us to tackle the “short call” problem, which power and exponential utility functions are unable to solve. While no closed-form solutions are available for the indifference prices, we are able to derive some pricing bounds. Furthermore, we rely on the dynamic programming approach to solve the associated utility maximization problem, which leads to a two-dimension HJB equation. A complex algorithm discussed in Ma and Forsyth (2016) is consequently adopted to numerically solve the HJB equation. We determine utility indifference prices for options written on the untradable underlying assets and some insurance contracts.","PeriodicalId":299310,"journal":{"name":"Econometrics: Mathematical Methods & Programming eJournal","volume":"51 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":"129787664","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}
T. Athanasopoulos, Burak Dindaroğlu, G. Petropoulos
How do incentives to collude depend on how asymmetric firms are? In digital and technology markets product quality is an important parameter that determines firms' market strategies. We study collusion in a quality differentiated duopoly and we adopt a Nash bargaining approach to compute the collusive equilibrium and assess its stability. We derive collusive and deviation strategies as continuous functions of quality asymmetry. We obtain novel and surprising results. Stability of collusion is associated with quality differentiation in a non-monotonic way. For low levels of differentiation, an increase in quality difference makes collusion less stable. The opposite holds for high levels of differentiation. Also, while the low quality firm is more likely to leave the cartel for small quality differences, the high quality firm determines cartel stability when the quality difference is sufficiently high. Our results have implications for empirical research, strategy theory, and antitrust enforcement.
{"title":"Stability of Collusion and Quality Differentiation: A Nash Bargaining Approach","authors":"T. Athanasopoulos, Burak Dindaroğlu, G. Petropoulos","doi":"10.2139/ssrn.3728502","DOIUrl":"https://doi.org/10.2139/ssrn.3728502","url":null,"abstract":"How do incentives to collude depend on how asymmetric firms are? In digital and technology markets product quality is an important parameter that determines firms' market strategies. We study collusion in a quality differentiated duopoly and we adopt a Nash bargaining approach to compute the collusive equilibrium and assess its stability. We derive collusive and deviation strategies as continuous functions of quality asymmetry. We obtain novel and surprising results. Stability of collusion is associated with quality differentiation in a non-monotonic way. For low levels of differentiation, an increase in quality difference makes collusion less stable. The opposite holds for high levels of differentiation. Also, while the low quality firm is more likely to leave the cartel for small quality differences, the high quality firm determines cartel stability when the quality difference is sufficiently high. Our results have implications for empirical research, strategy theory, and antitrust enforcement.","PeriodicalId":299310,"journal":{"name":"Econometrics: Mathematical Methods & Programming eJournal","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128976337","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 an equilibrium-based continuous asset pricing problem for the securities market. In the previous work [16], we have shown that a certain price process, which is given by the solution to a forward backward stochastic differential equation of conditional McKean-Vlasov type, asymptotically clears the market in the large population limit. In the current work, under suitable conditions, we show the existence of a finite agent equilibrium and its strong convergence to the corresponding mean-field limit given in [16]. As an important byproduct, we get the direct estimate on the difference of the equilibrium price between the two markets; the one consisting of heterogeneous agents of finite population size and the other of homogeneous agents of infinite population size.
{"title":"Strong Convergence to the Mean-Field Limit of A Finite Agent Equilibrium","authors":"M. Fujii, Akihiko Takahashi","doi":"10.2139/ssrn.3905899","DOIUrl":"https://doi.org/10.2139/ssrn.3905899","url":null,"abstract":"We study an equilibrium-based continuous asset pricing problem for the securities market. In the previous work [16], we have shown that a certain price process, which is given by the solution to a forward backward stochastic differential equation of conditional McKean-Vlasov type, asymptotically clears the market in the large population limit. In the current work, under suitable conditions, we show the existence of a finite agent equilibrium and its strong convergence to the corresponding mean-field limit given in [16]. As an important byproduct, we get the direct estimate on the difference of the equilibrium price between the two markets; the one consisting of heterogeneous agents of finite population size and the other of homogeneous agents of infinite population size.","PeriodicalId":299310,"journal":{"name":"Econometrics: Mathematical Methods & Programming eJournal","volume":"36 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126741571","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 introduce the new price probability measure, which entirely depends on the probability measures of the value and the volume of the market trades. We define the nth statistical moment of the price as the ratio of the nth statistical moment of the value to the nth statistical moment of the volume of all trades performed during an averaging time interval Δ. The set of the price statistical moments determines the price characteristic function and its Fourier transform defines the price probability measure. The price volatility depends on the 1st and the 2nd statistical moments of the value and the volume of the trades. The prediction of the price volatility requires a description of the sums of squares of the value and the volume of the market trades during the interval Δ and we call it the second-order economic theory. To develop that theory, we introduce numerical continuous risk ratings and distribute the agents by the risk ratings as coordinates. Based on distributions of the agents by the risk coordinates, we introduce a continuous economic media approximation that describes the collective trades. The agents perform the trades under the action of their expectations. We model the mutual impact of the expectations and the trades and derive equations that describe their evolution. To illustrate the benefits of our approach, in a linear approximation we describe perturbations of the mean price, the mean square price and the price volatility as functions of the first and the second-degree trades’ disturbances.
{"title":"Price, Volatility and the Second-Order Economic Theory","authors":"Victor Olkhov","doi":"10.2139/ssrn.3688109","DOIUrl":"https://doi.org/10.2139/ssrn.3688109","url":null,"abstract":"We introduce the new price probability measure, which entirely depends on the probability measures of the value and the volume of the market trades. We define the nth statistical moment of the price as the ratio of the nth statistical moment of the value to the nth statistical moment of the volume of all trades performed during an averaging time interval Δ. The set of the price statistical moments determines the price characteristic function and its Fourier transform defines the price probability measure. The price volatility depends on the 1st and the 2nd statistical moments of the value and the volume of the trades. The prediction of the price volatility requires a description of the sums of squares of the value and the volume of the market trades during the interval Δ and we call it the second-order economic theory. To develop that theory, we introduce numerical continuous risk ratings and distribute the agents by the risk ratings as coordinates. Based on distributions of the agents by the risk coordinates, we introduce a continuous economic media approximation that describes the collective trades. The agents perform the trades under the action of their expectations. We model the mutual impact of the expectations and the trades and derive equations that describe their evolution. To illustrate the benefits of our approach, in a linear approximation we describe perturbations of the mean price, the mean square price and the price volatility as functions of the first and the second-degree trades’ disturbances.","PeriodicalId":299310,"journal":{"name":"Econometrics: Mathematical Methods & Programming eJournal","volume":"42 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132128683","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 : 2020-08-31DOI: 10.15587/1729-4061.2020.210771
Sergii Patkovskyi, L. Kharsun
This paper deals with intermodal operations optimization methods to be implemented by the Block Train Operator upon cargo flows asymmetries at the hinterland. The algorithm of containerized cargo flows analysis and mathematical model were developed based on the relevant intermodal operation system. Inland leg of inbound containers from seaport to customer door arranged by truck is dominating within the emerging markets environment. Also imbalance in in- and outbound container flows as far as volumes, container size and payload is the case for largest inland destinations. Hence, the issue of rail-road transport prioritization and operational manageability is becoming of utmost importance. Centralizing those operations under a holistic service company – block train operator has been proven feasible. Last mile deliveries prioritization approach is offered to achieve the highest number of inbound containers processing with their further utilization for outbound export shipments. Mathematical modeling was conducted for distinct sets of operational scenarios that might take place. The scenario that allows the block train operator to achieve the highest revenue numbers and emptied inbound containers utilization for exports was selected. The number of truck heads, chassis and truck driver mitigation has become a secondary objective. The optimal scenario selected helps to reduce the overheads risk at the time of weekly cargo volumes fluctuations. The optimization approach represented can be applied to intermodal operations within markets where volume imbalance is rather possible
{"title":"Development of Agile Management Approaches Towards Intermodal Operations Upon Cargo Flows Imbalance","authors":"Sergii Patkovskyi, L. Kharsun","doi":"10.15587/1729-4061.2020.210771","DOIUrl":"https://doi.org/10.15587/1729-4061.2020.210771","url":null,"abstract":"This paper deals with intermodal operations optimization methods to be implemented by the Block Train Operator upon cargo flows asymmetries at the hinterland. The algorithm of containerized cargo flows analysis and mathematical model were developed based on the relevant intermodal operation system. Inland leg of inbound containers from seaport to customer door arranged by truck is dominating within the emerging markets environment. Also imbalance in in- and outbound container flows as far as volumes, container size and payload is the case for largest inland destinations. Hence, the issue of rail-road transport prioritization and operational manageability is becoming of utmost importance. Centralizing those operations under a holistic service company – block train operator has been proven feasible. Last mile deliveries prioritization approach is offered to achieve the highest number of inbound containers processing with their further utilization for outbound export shipments. Mathematical modeling was conducted for distinct sets of operational scenarios that might take place. The scenario that allows the block train operator to achieve the highest revenue numbers and emptied inbound containers utilization for exports was selected. The number of truck heads, chassis and truck driver mitigation has become a secondary objective. The optimal scenario selected helps to reduce the overheads risk at the time of weekly cargo volumes fluctuations. The optimization approach represented can be applied to intermodal operations within markets where volume imbalance is rather possible","PeriodicalId":299310,"journal":{"name":"Econometrics: Mathematical Methods & Programming eJournal","volume":"80 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129233500","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 linear structure is a family of matrices that satisfy a given set of linear restrictions, such as symmetry or diagonality. We add to the literature on linear structures by studying the family of matrices where all diagonal elements are zero, and discuss econometric examples where these results can be fruitfully applied.
{"title":"Zero-Diagonality as a Linear Structure","authors":"Jan R. Magnus, Enrique Sentana","doi":"10.2139/ssrn.3637942","DOIUrl":"https://doi.org/10.2139/ssrn.3637942","url":null,"abstract":"A linear structure is a family of matrices that satisfy a given set of linear restrictions, such as symmetry or diagonality. We add to the literature on linear structures by studying the family of matrices where all diagonal elements are zero, and discuss econometric examples where these results can be fruitfully applied.","PeriodicalId":299310,"journal":{"name":"Econometrics: Mathematical Methods & Programming eJournal","volume":"74 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122609910","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}
Abstract We derive closed-form expressions for the Jacobian of the matrix exponential function for both diagonalizable and defective matrices. The results are applied to two cases of interest in macroeconometrics: a continuous-time macro model and the parameterization of rotation matrices governing impulse response functions in structural vector autoregressions.
{"title":"The Jacobian of the Exponential Function","authors":"J. Magnus, H. Pijls, Enrique Sentana","doi":"10.2139/ssrn.3631767","DOIUrl":"https://doi.org/10.2139/ssrn.3631767","url":null,"abstract":"Abstract We derive closed-form expressions for the Jacobian of the matrix exponential function for both diagonalizable and defective matrices. The results are applied to two cases of interest in macroeconometrics: a continuous-time macro model and the parameterization of rotation matrices governing impulse response functions in structural vector autoregressions.","PeriodicalId":299310,"journal":{"name":"Econometrics: Mathematical Methods & Programming eJournal","volume":"46 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115654493","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}
Investors recently are really concerned about the risk aspects associated with the investment in securities. Volatility calculation, therefore, has become an important aspect in the financial markets. For these reasons time series models are greatly used to forecast volatility. One such model is the different variants of the Econometric model. Along with it the use of Econophysics methods is also helpful for the same. Understanding a better model of the forecast is what investors are looking forward to as it helps reduce the risk associated with investing. So a comparison of the models will be of important which will give us an insight on the same. For this purpose, the German DAX is considered.
{"title":"Estimating Volatility of German Dax From Econometric and Econophysics Perspectives","authors":"Jose Paul Pulickal","doi":"10.2139/ssrn.3639209","DOIUrl":"https://doi.org/10.2139/ssrn.3639209","url":null,"abstract":"Investors recently are really concerned about the risk aspects associated with the investment in securities. Volatility calculation, therefore, has become an important aspect in the financial markets. For these reasons time series models are greatly used to forecast volatility. One such model is the different variants of the Econometric model. Along with it the use of Econophysics methods is also helpful for the same. Understanding a better model of the forecast is what investors are looking forward to as it helps reduce the risk associated with investing. So a comparison of the models will be of important which will give us an insight on the same. For this purpose, the German DAX is considered.","PeriodicalId":299310,"journal":{"name":"Econometrics: Mathematical Methods & Programming eJournal","volume":"119 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131705159","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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