Shape-restricted support vector machine (SR-SVM): a SVM classifier taking supplementary shape information of input

Many classification problems contain shape information from input features, such as monotonic, convex, and concave. In this research, we propose a new classifier, called Shape-Restricted Support Vector Machine (SR-SVM), which takes the component-wise shape information to enhance classification accuracy. There exists vast research literature on monotonic classification covering monotonic or ordinal shapes. Our proposed classifier extends to handle convex and concave types of features, and combinations of these types. While standard SVM uses linear separating hyperplanes, our novel SR-SVM essentially constructs non-parametric and nonlinear separating planes subject to component-wise shape restrictions. We formulate SR-SVM classifier as a convex optimization problem and solve it using an active-set algorithm. The approach applies basis function expansions on the input and effectively utilizes the standard SVM solver. We illustrate our methodology using simulation and real world examples, and show that SR-SVM improves the classification performance with additional shape information of input.

A Projection Framework for Testing Shape Restrictions That Form Convex Cones

This paper develops a uniformly valid and asymptotically nonconservative test based on projection for a class of shape restrictions. The key insight we exploit is that these restrictions form convex cones, a simple and yet elegant structure that has been barely harnessed in the literature. Based on a monotonicity property afforded by such a geometric structure, we construct a bootstrap procedure that, unlike many studies in nonstandard settings, dispenses with estimation of local parameter spaces, and the critical values are obtained in a way as simple as computing the test statistic. Moreover, by appealing to strong approximations, our framework accommodates nonparametric regression models as well as distributional/density‐related and structural settings. Since the test entails a tuning parameter (due to the nonstandard nature of the problem), we propose a data‐driven choice and prove its validity. Monte Carlo simulations confirm that our test works well.

The Empirical Content of Binary Choice Models

An important goal of empirical demand analysis is choice and welfare prediction on counterfactual budget sets arising from potential policy interventions. Such predictions are more credible when made without arbitrary functional‐form/distributional assumptions, and instead based solely on economic rationality, that is, that choice is consistent with utility maximization by a heterogeneous population. This paper investigates nonparametric economic rationality in the empirically important context of binary choice. We show that under general unobserved heterogeneity, economic rationality is equivalent to a pair of Slutsky‐like shape restrictions on choice‐probability functions. The forms of these restrictions differ from Slutsky inequalities for continuous goods. Unlike McFadden–Richter's stochastic revealed preference, our shape restrictions (a) are global, that is, their forms do not depend on which and how many budget sets are observed, (b) are closed form, hence easy to impose on parametric/semi/nonparametric models in practical applications, and (c) provide computationally simple, theory‐consistent bounds on demand and welfare predictions on counterfactual budge sets.

Mutually Consistent Revealed Preference Demand Predictions

Revealed preference restrictions are increasingly used to predict demand behavior at new budgets of interest and as shape restrictions in nonparametric estimation exercises. However, the restrictions imposed are not sufficient for rationality when predictions are made at multiple budgets. I highlight the non-convexities in the set of predictions that arise when making multiple predictions. I develop a mixed integer programming characterization of the problem that can be used to impose rationality on multiple predictions. The approach is applied to the UK Family Expenditure Survey to recover rational demand predictions with substantially reduced computational resources compared to known alternatives. (JEL C61, D11, D12)

Inference in nonparametric/semiparametric moment equality models with shape restrictions

This paper studies the inference problem of an infinite‐dimensional parameter with a shape restriction. This parameter is identified by arbitrarily many unconditional moment equalities. The shape restriction leads to a convex restriction set. I propose a test of the shape restriction, which controls size uniformly and applies to both point‐identified and partially identified models. The test can be inverted to construct confidence sets after imposing the shape restriction. Monte Carlo experiments show the finite‐sample properties of this method. In an empirical illustration, I apply the method to ascending auctions held by the US Forest Service and show that imposing shape restrictions can significantly improve inference.