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17
Quantile Autoregression
 Convergence of Stochastic Processes
, 2006
"... Abstract. We consider quantile autoregression (QAR) models in which the autoregressive coefficients can be expressed as monotone functions of a single, scalar random variable. The models can capture systematic influences of conditioning variables on the location, scale and shape of the conditional d ..."
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Cited by 45 (5 self)
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Abstract. We consider quantile autoregression (QAR) models in which the autoregressive coefficients can be expressed as monotone functions of a single, scalar random variable. The models can capture systematic influences of conditioning variables on the location, scale and shape of the conditional distribution of the response, and therefore constitute a significant extension of classical constant coefficient linear time series models in which the effect of conditioning is confined to a location shift. The models may be interpreted as a special case of the general random coefficient autoregression model with strongly dependent coefficients. Statistical properties of the proposed model and associated estimators are studied. The limiting distributions of the autoregression quantile process are derived. Quantile autoregression inference methods are also investigated. Empirical applications of the model to the U.S. unemployment rate and U.S. gasoline prices highlight the potential of the model. 1.
Behavioral portfolio selection in continuous time
 Mathematical Finance
, 2008
"... This paper formulates and studies a general continuoustime behavioral portfolio selection model under Kahneman and Tversky’s (cumulative) prospect theory, featuring Sshaped utility (value) functions and probability distortions. Unlike the conventional expected utility maximization model, such a be ..."
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Cited by 19 (2 self)
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This paper formulates and studies a general continuoustime behavioral portfolio selection model under Kahneman and Tversky’s (cumulative) prospect theory, featuring Sshaped utility (value) functions and probability distortions. Unlike the conventional expected utility maximization model, such a behavioral model could be easily misformulated (a.k.a. illposed) if its different components do not coordinate well with each other. Certain classes of an illposed model are identified. A systematic approach, which is fundamentally different from the ones employed for the utility model, is developed to solve a wellposed model, assuming a complete market and general Itô processes for asset prices. The optimal terminal wealth positions, derived in fairly explicit forms, possess surprisingly simple structure reminiscent of a gambling policy betting on a good state of the world while accepting a fixed, known loss in case of a bad one. An example with a twopiece CRRA utility is presented to illustrate the general results obtained, and is solved completely for all admissible parameters. The effect of the behavioral criterion on the risky allocations is finally discussed.
Multivariate quantiles and multipleoutput regression quantiles: from L1 optimization to halfspace depth (with discussion
 Annals of Statistics
, 2010
"... A new multivariate concept of quantile, based on a directional version of Koenker and Bassett’s traditional regression quantiles, is introduced for multivariate location and multipleoutput regression problems. In their empirical version, those quantiles can be computed efficiently via linear progra ..."
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Cited by 17 (5 self)
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A new multivariate concept of quantile, based on a directional version of Koenker and Bassett’s traditional regression quantiles, is introduced for multivariate location and multipleoutput regression problems. In their empirical version, those quantiles can be computed efficiently via linear programming techniques. Consistency, Bahadur representation and asymptotic normality results are established. Most importantly, the contours generated by those quantiles are shown to coincide with the classical halfspace depth contours associated with the name of Tukey. This relation does not only allow for efficient depth contour computations by means of parametric linear programming, but also for transferring from the quantile to the depth universe such asymptotic results as Bahadur representations. Finally, linear programming duality opens the way to promising developments in depthrelated multivariate rankbased inference.
SPECTRAL RISK MEASURES AND PORTFOLIO SELECTION
, 2007
"... This paper deals with risk measurement and portfolio optimization under risk constraints. Firstly we give an overview of risk assessment from the viewpoint of risk theory, focusing on momentbased, distortion and spectral risk measures. We subsequently apply these ideas to an asset management framew ..."
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Cited by 8 (0 self)
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This paper deals with risk measurement and portfolio optimization under risk constraints. Firstly we give an overview of risk assessment from the viewpoint of risk theory, focusing on momentbased, distortion and spectral risk measures. We subsequently apply these ideas to an asset management framework using a database of hedge funds returns chosen for their nonGaussian features. We deal with the problem of portfolio optimization under risk constraints and lead a comparative analysis of efficient portfolios. We show some robustness of optimal portfolios with respect to the choice of risk measure. Unsurprisingly, risk measures that emphasize large losses lead to slightly more diversified portfolios. However, risk measures that account primarily for worst case scenarios overweight funds with smaller tails which mitigates the relevance of diversification.
Ambiguity and climate policy
, 2010
"... Economic evaluation of climate policy traditionally treats uncertainty by appealing to expected utility theory. Yet our knowledge of the impacts of climate policy may not be of sufficient quality to justify probabilistic beliefs. In such circumstances, it has been argued that the axioms of expected ..."
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Cited by 3 (1 self)
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Economic evaluation of climate policy traditionally treats uncertainty by appealing to expected utility theory. Yet our knowledge of the impacts of climate policy may not be of sufficient quality to justify probabilistic beliefs. In such circumstances, it has been argued that the axioms of expected utility theory may not be the correct standard of rationality. By contrast, several axiomatic frameworks have recently been proposed that account for ambiguous beliefs. In this paper, we apply static and dynamic versions of a smooth ambiguity model to climate mitigation policy. We obtain a general result on the comparative statics of optimal abatement and ambiguity aversion and illustrate this sufficient condition in some simple examples. We then extend our analysis to a more realistic, dynamic setting, and adapt a wellknown empirical model of the climateeconomy system to show that the value of emissions abatement increases as ambiguity aversion increases, and that this ‘ambiguity premium ’ can in some plausible cases be very large. 1
Regression Quantiles: From L1 Optimization to Halfspace Depth
 Annals of Statistics
, 2010
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CONDITIONAL QUANTILE ESTIMATION FOR GARCH MODELS
"... Abstract. Conditional quantile estimation is an essential ingredient in modern risk management. Although GARCH processes have proven highly successful in modeling financial data it is generally recognized that it would be useful to consider a broader class of processes capable of representing more f ..."
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Abstract. Conditional quantile estimation is an essential ingredient in modern risk management. Although GARCH processes have proven highly successful in modeling financial data it is generally recognized that it would be useful to consider a broader class of processes capable of representing more flexibly both asymmetry and tail behavior of conditional returns distributions. In this paper, we study estimation of conditional quantiles for GARCH models using quantile regression. Quantile regression estimation of GARCH models is highly nonlinear; we propose a simple and effective twostep approach of quantile regression estimation for linear GARCH time series. In the first step, we employ a quantile autoregression sieve approximation for the GARCH model by combining information over different quantiles; second stage estimation for the GARCH model is then carried out based on the first stage minimum distance estimation of the scale process of the time series. Asymptotic properties of the sieve approximation, the minimum distance estimators, and the final quantile regression estimators employing generated regressors are studied. These results are of independent interest and have applications in other quantile regression settings. Monte Carlo and empirical application results indicate that the proposed estimation methods outperform some existing conditional quantile estimation methods. 1.
SOME EXERCISES ON QUANTILE REGRESSION
"... These exercises are intended to provide an introduction to quantile regression computing and illustrate some econometric applications of quantile regression methods. For purposes of the course my intention would be to encourage all students to do the first exercise, which gives an overview of the q ..."
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These exercises are intended to provide an introduction to quantile regression computing and illustrate some econometric applications of quantile regression methods. For purposes of the course my intention would be to encourage all students to do the first exercise, which gives an overview of the quantile regression software
View Bias towards Ambiguity, Expectile CAPM and the Anomalies ✩,✩✩
"... Information ambiguity introduces view bias. By defining this view bias, we develop a novel rewardrisk measurement framework, an extended CAPM a sequence of empirical test procedures to explain asset pricing anomalies. U.S. stock market data (19261999) implies a pessimistic view on average for peop ..."
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Information ambiguity introduces view bias. By defining this view bias, we develop a novel rewardrisk measurement framework, an extended CAPM a sequence of empirical test procedures to explain asset pricing anomalies. U.S. stock market data (19261999) implies a pessimistic view on average for people with rational risk preference; that explains the equity premium puzzle. The extended CAPM still admits a single beta representation. The amount of risk becomes the weighted average of systematic risk and latent risk. The price of risk, or the expected market excess return, is adjusted by view bias. The momentum effect has two alternative explanations within this framework. Either the winner has a low systematic risk but a high latent risk, and the adjusted price of risk is positive; or the winner has a low systematic risk and a low amount of risk (a weighted average of systematic risk and latent risk), but the adjusted price of risk is negative. Postwar U.S. data supports the latter explanation. Keywords: