Nonparametric state price density estimation using constrained least squares and the bootstrap

The economic theory of option pricing imposes constraints on the structure of call functions and state price densities. Except in a few polar cases, it does not prescribe functional forms. This paper proposes a nonparametric estimator of option pricing models which incorporates various restrictions (such as monotonicity and convexity) within a single least squares procedure. The bootstrap is used to produce confidence intervals for the call function and its first two derivatives and to calibrate a residual regression test of shape constraints. We apply the techniques to option pricing data on the DAX.     

Disentangling Age,Cohort and Time Effects in the Additive Model*

This paper presents a new approach to the old problem of linear dependency of age, cohort and time effects. It is shown that second differences of the effects can be estimated without any normalization restrictions, providing information on the shape of the age‐, cohort‐ and time‐effect profiles, and enabling identification of structural breaks. A Wald test is provided to test the popular linear and quadratic specifications against a very general alternative. The method is illustrated through examples which show its ability to detect structural breaks in time effects as a result of the Mexican peso crisis, and to determine whether the age‐effect profile in the variance of Taiwanese log consumption is concave or convex.     

Lagrange multipliers in incentive-constrained problems

Some recent work on dynamic incentive constraints poses the question of existence and regularity of Lagrange multipliers in optimization problems in infinite dimension with infinitely many-side constraints, which may not be concave. We provide sufficient conditions for Lagrange multipliers to exist and be represented by a sequence.     

Nonparametric Bayes inference for concave distribution functions

Bayesian inference for concave distribution functions is investigated. This is made by transforming a mixture of Dirichlet processes on the space of distribution functions to the space of concave distribution functions. We give a method for sampling from the posterior distribution using a Pólya urn scheme in combination with a Markov chain Monte Carlo algorithm. The methods are extended to estimation of concave distribution functions for incompletely observed data.     

GARCH models without positivity constraints: Exponential or log GARCH?

This paper provides a probabilistic and statistical comparison of the log-GARCH and EGARCH models, which both rely on multiplicative volatility dynamics without positivity constraints. We compare the main probabilistic properties (strict stationarity, existence of moments, tails) of the EGARCH model, which are already known, with those of an asymmetric version of the log-GARCH. The quasi-maximum likelihood estimation of the log-GARCH parameters is shown to be strongly consistent and asymptotically normal. Similar estimation results are only available for the EGARCH (1,1) model, and under much stronger assumptions. The comparison is pursued via simulation experiments and estimation on real data.     

Constrained Kalman Filtering: Additional Results

This paper deals with linear state space modelling subject to general linear constraints on the state vector. The discussion concentrates on four topics: the constrained Kalman filtering versus the recursive restricted least squares estimator; a new proof of the constrained Kalman filtering under a conditional expectation framework; linear constraints under a reduced state space modelling; and state vector prediction under linear constraints. The techniques proposed are illustrated in two real problems. The first problem is related to investment analysis under a dynamic factor model, whereas the second is about making constrained predictions within a GDP benchmarking estimation.     

Optimal risk taking in an uneven tournament game with risk averse players

We analyze the optimal choice of risk in a two-stage tournament game between two players that have different concave utility functions. At the first stage, both players simultaneously choose risk. At the second stage, both observe overall risk and simultaneously decide on effort or investment. The results show that those two effects which mainly determine risk taking – an effort effect and a likelihood effect – are strictly interrelated. This finding sharply contrasts with existing results on risk taking in tournament games with symmetric equilibrium efforts where such linkage can never arise. Conditions are derived under which this linkage leads to a reversed likelihood effect so that the favorite (underdog) can increase his winning probability by increasing (decreasing) risk which is impossible in a completely symmetric setting.     

Dynamic estimation of volatility risk premia and investor risk aversion from option-implied and realized volatilities

This paper proposes a method for constructing a volatility risk premium, or investor risk aversion, index. The method is intuitive and simple to implement, relying on the sample moments of the recently popularized model-free realized and option-implied volatility measures. A small-scale Monte Carlo experiment confirms that the procedure works well in practice. Implementing the procedure with actual S&P500 option-implied volatilities and high-frequency five-minute-based realized volatilities indicates significant temporal dependencies in the estimated stochastic volatility risk premium, which we in turn relate to a set of macro-finance state variables. We also find that the extracted volatility risk premium helps predict future stock market returns.     

A note on linked bargaining

A recent result by Jackson and Sonnenschein (2007) describes a general framework for overcoming incentive constraints by linking together independent copies of a Bayesian decision problem. A special case of that work shows that if copies of a standard two-player Bayesian bargaining problem are independently linked (players receive valuations and trade simultaneously on a number of identical copies), then the utility cost associated with incentive constraints tends to 0 as the number of linked problems tends to infinity. We improve upon that result, increasing the rate of convergence from polynomial to exponential and eliminating unwanted trades in the limit, by introducing a mechanism that uses a slightly richer and more refined strategy space. Although very much in the same spirit, our declarations are constrained by a distribution which is skewed away from the expected distribution of player types. When a sufficiently large number of bargaining problems are linked, “truth” is an equilibrium. Moreover, this equilibrium is incentive compatible with the utility cost of incentive constraints almost surely equal to 0.     

Long-term efficiency of public service provision in a context of budget restrictions. An application to the education sector

This paper proposes an extension of the non-parametric long-term evaluation of efficiency, the conditional panel data DEA model, which takes into account the panel structure of the data and, at the same time, incorporates the role of contextual factors in the estimations. Its application to the education sector for the period analyzed (2009–2014) shows the utility of this method, since it obtains more representative efficiency scores for the complete time-period, is more robust to external shocks, and allows improvements to the decision-making process in the allocation of the budget available for the public education sector. The results are clear and present an evolution towards the convergence of the efficiency scores, precisely in a time period when hard budget constraints severely reduced the resources available for public schools.     

Credit market segmentation,essentiality of commodities,and supermodularity

We consider incomplete market economies where agents are subject to price-dependent trading constraints compatible with credit market segmentation. Equilibrium existence is guaranteed when either commodities are essential, i.e, indifference curves through individuals’ endowments do not intersect the boundary of the consumption set, or utility functions are concave and supermodular. The smoothness of mappings representing preferences, financial promises, or trading constraints is not required. Hence, we may include in our framework economies where ambiguity is allowed and agents maximize the minimum expected utility over a set of priors, or where markets include non-recourse collateralized loans.     

Particle filters for continuous likelihood evaluation and maximisation

In this paper, a method is introduced for approximating the likelihood for the unknown parameters of a state space model. The approximation converges to the true likelihood as the simulation size goes to infinity. In addition, the approximating likelihood is continuous as a function of the unknown parameters under rather general conditions. The approach advocated is fast and robust, and it avoids many of the pitfalls associated with current techniques based upon importance sampling. We assess the performance of the method by considering a linear state space model, comparing the results with the Kalman filter, which delivers the true likelihood. We also apply the method to a non-Gaussian state space model, the stochastic volatility model, finding that the approach is efficient and effective. Applications to continuous time finance models and latent panel data models are considered. Two different multivariate approaches are proposed. The neoclassical growth model is considered as an application.     

Exogeneity in Vector Error Correction Models with Purely Exogenous Long-Run Paths

Existing exogeneity conditions of literature are only sufficient and imply 'overly strong' constraints on long-run parameters. This paper presents some new results on exogeneity in vector error correction models. A key concept of the analysis is the 'purely exogenous long-run path', i.e. a cointegrating vector only including 'exogenous' variables. Extending earlier results of Johansen, S. (1992). 'Cointegration in partial systems and the efficiency of single-equation analysis', Journal of Econometrics , Vol. 52, pp. 389–402 and of Toda and Phillips (1991) . Vector Autoregressions and Causality , Cowles Foundation Discussion Paper, No. 977 among others, we propose a framework based on two canonical representations of the long-run matrix, which can constitute a suitable basis to formulate a necessary and sufficient condition for non-causality as well as a condition for strong exogeneity. An interesting property is that the statistics involved in the sequential procedures for testing these conditions are distributed as χ ² variables and can, therefore, easily be calculated with the usual statistical computer packages, which makes our approach fully operational, empirically. Finally, the power and size distortions of the sequential test procedures are analysed using Monte Carlo experiments.     

On the nature of certainty equivalent functionals

We explore connections between the certainty equivalent return (CER) functional and the underlying utility function. Curvature properties of the functional depend upon how utility function attributes relate to hyperbolic absolute risk aversion (HARA) type utility functions. If the CER functional is concave, i.e., if risk tolerance is concave in wealth, then preferences are standard. The CER functional is linear in lotteries if utility is HARA and lottery payoffs are on a line in state space. Implications for the optimality of portfolio diversification are given. When utility is concave and non-increasing relative risk averse, then the CER functional is superadditive in lotteries. Depending upon the nature of association among lottery payoffs, CERs for constant absolute risk averse utility functions may be subadditive or superadditive in lotteries. Our approach lends itself to straightforward experiments to elicit higher order attributes on risk preferences.     

On the nature of certainty equivalent functionals

We explore connections between the certainty equivalent return (CER) functional and the underlying utility function. Curvature properties of the functional depend upon how utility function attributes relate to hyperbolic absolute risk aversion (HARA) type utility functions. If the CER functional is concave, i.e., if risk tolerance is concave in wealth, then preferences are standard. The CER functional is linear in lotteries if utility is HARA and lottery payoffs are on a line in state space. Implications for the optimality of portfolio diversification are given. When utility is concave and non-increasing relative risk averse, then the CER functional is superadditive in lotteries. Depending upon the nature of association among lottery payoffs, CERs for constant absolute risk averse utility functions may be subadditive or superadditive in lotteries. Our approach lends itself to straightforward experiments to elicit higher order attributes on risk preferences.     

Distribution-free tests of stochastic monotonicity

This article proposes a nonparametric test of monotonicity for conditional distributions and its moments. Unlike previous proposals, our method does not require smooth estimation of the derivatives of nonparametric curves. Distinguishing features of our approach are that critical values are pivotal under the null in finite samples and that the test is invariant to any monotonic continuous transformation of the explanatory variable. The test statistic is the sup-norm of the difference between the empirical copula function and its least concave majorant with respect to the explanatory variable coordinate. The resulting test is able to detect local alternatives converging to the null at the parametric rate n^−1/2$n^{-1/2}$, with n$n$ the sample size. The finite sample performance of the test is examined by means of a Monte Carlo experiment and an application to testing intergenerational income mobility.     

Nonparametric estimation and inference on conditional quantile processes

This paper presents estimation methods and asymptotic theory for the analysis of a nonparametrically specified conditional quantile process. Two estimators based on local linear regressions are proposed. The first estimator applies simple inequality constraints while the second uses rearrangement to maintain quantile monotonicity. The bandwidth parameter is allowed to vary across quantiles to adapt to data sparsity. For inference, the paper first establishes a uniform Bahadur representation and then shows that the two estimators converge weakly to the same limiting Gaussian process. As an empirical illustration, the paper considers a dataset from Project STAR and delivers two new findings.     

Hölder continuity of the policy function approximation in the value function approximation

We show that given a value function approximation V of a strongly concave stochastic dynamic programming problem (SDDP), the associated policy function approximation is Hölder continuous in V.     

Exact computation of max weighted score estimators

We show that exact computation of a family of ‘max weighted score’ estimators, including Manski’s max score estimator, can be achieved efficiently by reformulating them as mixed integer programs (MIP) with disjunctive constraints. The advantage of our MIP formulation is that estimates are exact and can be computed using widely available solvers in reasonable time. In a classic work-trip mode choice application, our method delivers exact estimates that lead to a different economic interpretation of the data than previous heuristic estimates. In a small Monte Carlo study we find that our approach is computationally efficient for usual estimation problem sizes.