Testing Predictive Ability and Power Robustification

One of the approaches to compare forecasts is to test whether the loss from a benchmark prediction is smaller than the others. The test can be embedded into the general problem of testing functional inequalities using a one-sided Kolmogorov-Smirnov functional. This paper shows that such a test generally suffers from unstable power properties, meaning that the asymptotic power against certain local alternatives can be much smaller than the size. This paper proposes a general method to robustify the power properties. This method can also be applied to testing inequalities such as stochastic dominance and moment inequalities. Simulation studies demonstrate that tests based on this paper’s approach perform quite well relative to the existing methods.Inequality Restrictions, Testing Predictive Ability, One-sided Nonparametric Tests, Power Robustification

Inference Based on Many Conditional Moment Inequalities

In this paper, we construct confidence sets for models defined by many conditional moment inequalities/equalities. The conditional moment restrictions in the models can be finite, countably infinite, or uncountably infinite. To deal with the complication brought about by the vast number of moment restrictions, we exploit the manageability (Pollard (1990)) of the class of moment functions. We verify the manageability condition in five examples from the recent partial identification literature. The proposed confidence sets are shown to have correct asymptotic size in a uniform sense and to exclude parameter values outside the identified set with probability approaching one. Monte Carlo experiments for a conditional stochastic dominance example and a random-coefficients binary-outcome example support the theoretical results

Inference Based on Many Conditional Moment Inequalities

In this paper, we construct confidence sets for models defined by many conditional moment inequalities/equalities. The conditional moment restrictions in the models can be finite, countably in finite, or uncountably in finite. To deal with the complication brought about by the vast number of moment restrictions, we exploit the manageability (Pollard (1990)) of the class of moment functions. We verify the manageability condition in five examples from the recent partial identification literature. The proposed confidence sets are shown to have correct asymptotic size in a uniform sense and to exclude parameter values outside the identified set with probability approaching one. Monte Carlo experiments for a conditional stochastic dominance example and a random-coefficients binary-outcome example support the theoretical results

Spanning Tests for Markowitz Stochastic Dominance

We derive properties of the cdf of random variables defined as saddle-type points of real valued continuous stochastic processes. This facilitates the derivation of the first-order asymptotic properties of tests for stochastic spanning given some stochastic dominance relation. We define the concept of Markowitz stochastic dominance spanning, and develop an analytical representation of the spanning property. We construct a non-parametric test for spanning based on subsampling, and derive its asymptotic exactness and consistency. The spanning methodology determines whether introducing new securities or relaxing investment constraints improves the investment opportunity set of investors driven by Markowitz stochastic dominance. In an application to standard data sets of historical stock market returns, we reject market portfolio Markowitz efficiency as well as two-fund separation. Hence, we find evidence that equity management through base assets can outperform the market, for investors with Markowitz type preferences

Conditional stochastic dominance testing

This article proposes bootstrap-based stochastic dominance tests for nonparametric conditional distributions and their moments. We exploit the fact that a conditional distribution dominates the other if and only if the difference between the marginal joint distributions is monotonic in the explanatory variable for each value of the dependent variable. The proposed test statistic compares restricted and unrestricted estimators of the difference between the joint distributions, and can be implemented under minimal smoothness requirements on the underlying nonparametric curves and without resorting to smooth estimation. The finite sample properties of the proposed tests are examined by means of a Monte Carlo study. We report an application to studying the impact on post-intervention earnings of the National Supported Work Demonstration, a randomized labor training program carried out in the 1970s.Nonparametric testing, Conditional stochastic dominance, Conditional inequality restrictions, Least concave majorant, Treatment effects

Hypothesis testing of multiple inequalities: the method of constraint chaining

Econometric inequality hypotheses arise in diverse ways. Examples include concavity restrictions on technological and behavioural functions, monotonicity and dominance relations, one-sided constraints on conditional moments in GMM estimation, bounds on parameters which are only partially identified, and orderings of predictive performance measures for competing models. In this paper we set forth four key properties which tests of multiple inequality constraints should ideally satisfy. These are (1) (asymptotic) exactness, (2) (asymptotic)similarity on the boundary, (3) absence of nuisance parameters from the asymptotic null distribution of the test statistic, (4) low computational complexity and boostrapping cost. We observe that the predominant tests currently used in econometrics do not appear to enjoy all these properties simultaneously. We therefore ask the question : Does there exist any nontrivial test which, as a mathematical fact, satisfies the first three properties and, by any reasonable measure, satisfies the fourth ? Remarkably the answer is affirmative. The paper demonstrates this constructively. We introduce a method of test construction called chaining which begins by writing multiple inequalities as a single equality using zero-one indicator functions. We then smooth the indicator functions. The approximate equality thus obtained is the basis of a well-behaved test. This test may be considered as the baseline of a wider class of tests. A full asymptotic theory is provided for the baseline. Simulation results show that the finite-sample performance of the test matches the theory quite well

Bounds on Counterfactual Distributions Under Semi-Monotonicity Constraints

This paper explores semi-monotonicity constraints in the distribution of potential outcomes, first, conditional on an instrument, and second, in terms of the response function. The imposed assumptions are strictly weaker than traditional instrumental variables assumptions and can be gainfully employed to bound the counterfactual distributions, even though point identification is only achieved in special cases. The bounds have a simple analytical form and thus have much practical relevance in all instances when strong exogeneity assumptions cannot be credibly invoked. The bounding strategy is illustrated in a simulated data example and applied to the effect of education on smoking.nonparametric bounds, treatment effects, causality, endogeneity, instrumental variables, policy evaluation