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Working Papers
A Test for Sparsity (Job Market Paper)
Many properties of sparse estimators rely on the assumption that the underlying data generating process (DGP) is sparse. When this assumption does not hold, a sparse estimator can perform less well than non-sparse estimators such as the ridge estimator. We propose a test of sparsity for linear regression models. Our null hypothesis is that the number of non-zero parameters does not exceed a small preset fraction of the total number of parameters. The null can be interpreted as a family of Bayesian prior distributions where each parameter equals zero with large probability. As the alternative, we consider the case where all parameters are nonzero and of root-p-order for all p number of parameters. Formally the alternative is a normal prior distribution, which is the maximum entropy prior with zero mean and whose variance determined by the ANOVA identity. We derive a test statistic using the theory of robust statistics and show its asymptotic consistency under some regularity assumptions. This statistic is minmax-optimal when the design matrix is orthogonal, and can be used for general design matrices as a conservative test.
Prior Free Bayesian Estimation through the AIC
Joint work with Werner Ploberger
In this paper we show that, in linear models with an increasing number of parameters, the estimator resulting from the maximization of Akaike's Information Criterion is asymptotically equivalent to some Bayesian estimators. The family of prior distributions which generates our estimators consists of normal distributions, defined on the space of all sequence, and is characterized by an exponential decay of the variance for the higher order components of the parameter.
Rational Contextual Choices under Imperfect Perception of Attributes
Revision requested at Management Science
Classical rational choice theory proposes that preferences do not depend on context, i.e. are independent of irrelevant alternatives. Empirical choice data, however, display several contextual choice effects that seem inconsistent with rational preferences. This paper studies a choice model in the classical rational paradigm with a novel information friction: the agent's perception of the options is affected by an attribute-specific noise. Under this friction, the agent obtains useful information when additional options are introduced. Therefore, the agent chooses contextually, exhibiting intransitivity, joint-separate evaluation reversal, attraction effect, compromise effect, similarity effect, and the phantom decoy effect. Our model also provides a connection between random utility models and reference-dependent models. It approximates the conditional probit model and a reference-dependent model at different parameter values.
Moderate Expected Utility
Joint work with Paulo Natenzon
Individual choice data often violate strong stochastic transitivity (SST) while conforming to moderate stochastic transitivity (MST). We propose a slightly stronger version of the MST postulate, which we call MST+, and show that MST and MST+ retain significantly more predictive power than weak stochastic transitivity (WST). Our first theorem shows that a binary choice rule satisfies MST+ if and only if it can be represented by a moderate utility model with two parameters: a utility function describing the value of each option, and a distance metric of comparability. When choice is over lotteries, we show that it can be represented by a moderate expected utility model if and only if it is continuous, linear, convex, symmetric and satisfies MST. We also show how the utility and distance can be identified from choice data over lotteries.
Optimal Estimation when the Parameter Space is of Infinite Dimension
Joint work with Werner Ploberger. Revision requested at Econometrics
Many classical non-parametric estimation problems can be reduced to the estimation of an infinite dimensional parameter vector. When maximum likelihood estimators do not exist, we construct simple prior distributions for the parameters that force the parameters to obey some higher order smoothness conditions. We show that under such prior distributions, certain shrunken sieve estimators are asymptotically optimal for a family of loss functions.
A Test for Sparsity (Job Market Paper)
Many properties of sparse estimators rely on the assumption that the underlying data generating process (DGP) is sparse. When this assumption does not hold, a sparse estimator can perform less well than non-sparse estimators such as the ridge estimator. We propose a test of sparsity for linear regression models. Our null hypothesis is that the number of non-zero parameters does not exceed a small preset fraction of the total number of parameters. The null can be interpreted as a family of Bayesian prior distributions where each parameter equals zero with large probability. As the alternative, we consider the case where all parameters are nonzero and of root-p-order for all p number of parameters. Formally the alternative is a normal prior distribution, which is the maximum entropy prior with zero mean and whose variance determined by the ANOVA identity. We derive a test statistic using the theory of robust statistics and show its asymptotic consistency under some regularity assumptions. This statistic is minmax-optimal when the design matrix is orthogonal, and can be used for general design matrices as a conservative test.
Prior Free Bayesian Estimation through the AIC
Joint work with Werner Ploberger
In this paper we show that, in linear models with an increasing number of parameters, the estimator resulting from the maximization of Akaike's Information Criterion is asymptotically equivalent to some Bayesian estimators. The family of prior distributions which generates our estimators consists of normal distributions, defined on the space of all sequence, and is characterized by an exponential decay of the variance for the higher order components of the parameter.
Rational Contextual Choices under Imperfect Perception of Attributes
Revision requested at Management Science
Classical rational choice theory proposes that preferences do not depend on context, i.e. are independent of irrelevant alternatives. Empirical choice data, however, display several contextual choice effects that seem inconsistent with rational preferences. This paper studies a choice model in the classical rational paradigm with a novel information friction: the agent's perception of the options is affected by an attribute-specific noise. Under this friction, the agent obtains useful information when additional options are introduced. Therefore, the agent chooses contextually, exhibiting intransitivity, joint-separate evaluation reversal, attraction effect, compromise effect, similarity effect, and the phantom decoy effect. Our model also provides a connection between random utility models and reference-dependent models. It approximates the conditional probit model and a reference-dependent model at different parameter values.
Moderate Expected Utility
Joint work with Paulo Natenzon
Individual choice data often violate strong stochastic transitivity (SST) while conforming to moderate stochastic transitivity (MST). We propose a slightly stronger version of the MST postulate, which we call MST+, and show that MST and MST+ retain significantly more predictive power than weak stochastic transitivity (WST). Our first theorem shows that a binary choice rule satisfies MST+ if and only if it can be represented by a moderate utility model with two parameters: a utility function describing the value of each option, and a distance metric of comparability. When choice is over lotteries, we show that it can be represented by a moderate expected utility model if and only if it is continuous, linear, convex, symmetric and satisfies MST. We also show how the utility and distance can be identified from choice data over lotteries.
Optimal Estimation when the Parameter Space is of Infinite Dimension
Joint work with Werner Ploberger. Revision requested at Econometrics
Many classical non-parametric estimation problems can be reduced to the estimation of an infinite dimensional parameter vector. When maximum likelihood estimators do not exist, we construct simple prior distributions for the parameters that force the parameters to obey some higher order smoothness conditions. We show that under such prior distributions, certain shrunken sieve estimators are asymptotically optimal for a family of loss functions.
Work in Progress
Rational Choice of the Alpha-level
Abstract. Conventionally, the alpha-level in hypothesis testing is chosen as 5% or 1% as a rule of thumb. We study hypothesis testing from in a decision theoretic framework and analyze the properties of the optimal alpha level. As opposed to the rule of thumb, the rational alpha-level decreases as sample size increases. We also analyze the properties of the rational rejection region for a family of the rejection regions.
On the Loss of Efficiency from Inequality
Joint work with Inkee Jang.
Abstract. The First Welfare Theorem states that when the market is complete, the competitive market is efficient. It is also implicit in the assumption that private property right is enforced. When there is no institution enforcing private property right, people have to spend resources to defend against appropriation. We show that in a competitive equilibrium without property right, the fraction of asset spent on protection converges to 1 as inequality increases, resulting in a loss in efficiency.
Rational Choice of the Alpha-level
Abstract. Conventionally, the alpha-level in hypothesis testing is chosen as 5% or 1% as a rule of thumb. We study hypothesis testing from in a decision theoretic framework and analyze the properties of the optimal alpha level. As opposed to the rule of thumb, the rational alpha-level decreases as sample size increases. We also analyze the properties of the rational rejection region for a family of the rejection regions.
On the Loss of Efficiency from Inequality
Joint work with Inkee Jang.
Abstract. The First Welfare Theorem states that when the market is complete, the competitive market is efficient. It is also implicit in the assumption that private property right is enforced. When there is no institution enforcing private property right, people have to spend resources to defend against appropriation. We show that in a competitive equilibrium without property right, the fraction of asset spent on protection converges to 1 as inequality increases, resulting in a loss in efficiency.
Publications
Risk Reducers in Convex Order. Joint work with Qihe Tang and Huan Zhang.
Insurance: Mathematics and Economics. Sep. 2016
On the Necessity of Pairs and Triplets for the Equivalence between Rationality Axioms.
Economics E-Journal. Aug. 2012
Risk Reducers in Convex Order. Joint work with Qihe Tang and Huan Zhang.
Insurance: Mathematics and Economics. Sep. 2016
On the Necessity of Pairs and Triplets for the Equivalence between Rationality Axioms.
Economics E-Journal. Aug. 2012
