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

Stochastic Expansions and Moment Approximations for Three Indirect Estimators

This paper deals with properties of three indirect estimators that are known to be (first order) asymptotically equivalent. Specifically, we examine a) the issue of validity of the formal Edgeworth expansion of an arbitrary order. b) Given a), we are concerned with valid moment approximations and employ them to characterize the second order bias structure of the estimators. Our motivation resides on the fact that one of the three is reported by the relevant literature to be second order unbiased. However, this result was derived without any establishment of validity. We provide this establishment, but we are also able to massively generalize the conditions under which this second order property remains true. In this way, we essentially prove their higher order inequivalence. We generalize indirect estimators by introducing recursive ones, emerging from multistep optimization procedures. We are able to establish higher order unbiaseness for estimators of this sort.Asymptotic Approximation, Second Order Bias Structure, Binding Function, Local Canonical Representation, Convex Variational Distance, Recursive Indirect Estimators, Higher order Bias.

Stochastic spanning

This study develops and implements a theory and method for analyzing whether introducing new securities or relaxing investment constraints improves the investment opportunity set for risk averse investors. We develop a test procedure for 'stochastic spanning' for two nested polyhedral portfolio sets based on subsampling and Linear Programming. The procedure is statistically consistent and asymptotically exact for a class of weakly dependent processes. Using the stochastic spanning tests, we accept market portfolio efficiency but reject two-fund separation in standard data sets of historical stock market returns. The divergence between the results of the two tests illustrates the role for higher-order moment risk in portfolio choice and challenges representative-investor models of capital market equilibrium

Stochastic Expansions and Moment Approximations for Three Indirect Estimators

This paper deals with properties of three indirect estimators that are known to be (first order) asymptotically equivalent. Specifically, we examine a) the issue of validity of the formal Edgeworth expansion of an arbitrary order. b) Given a), we are concerned with valid moment approximations and employ them to characterize the second order bias structure of the estimators. Our motivation resides on the fact that one of the three is reported by the relevant literature to be second order unbiased. However, this result was derived without any establishment of validity. We provide this establishment, but we are also able to massively generalize the conditions under which this second order property remains true. In this way, we essentially prove their higher order inequivalence. We generalize indirect estimators by introducing recursive ones, emerging from multistep optimization procedures. We are able to establish higher order unbiaseness for estimators of this sort

Stochastic Spanning

https://www.tandfonline.com/doi/full/10.1080/07350015.2017.1391099This study develops and implements methods for determining whether introducing new securities or relaxing investment constraints improves the investment opportunity set for all risk averse investors. We develop a test procedure for “stochastic spanning” for two nested portfolio sets based on subsampling and linear programming. The test is statistically consistent and asymptotically exact for a class of weakly dependent processes. A Monte Carlo simulation experiment shows good statistical size and power properties in finite samples of realistic dimensions. In an application to standard datasets of historical stock market returns, we accept market portfolio efficiency but reject two-fund separation, which suggests an important role for higher-order moment risk in portfolio theory and asset pricing. Supplementary materials for this article are available online

On the Existence of Strongly Consistent Indirect Estimators When the Binding Function Is Compact Valued

We provide sufficient conditions for the definition and the existence of strongly consistent indirect estimators when the binding function is a compact valued correspondence. We use conditions that concern the asymptotic behavior of the epigraphs of the criteria involved, a relevant notion of continuity for the binding correspondence as well as an indirect identification condition that restricts the behavior of the aforementioned correspondence. These are generalizations of the analogous results in the relevant literature and hence permit a broader scope of statistical models. We examine simple examples involving Levy and ergodic conditionally heteroskedastic processes

Sparse spanning portfolios and under-diversification with second-order stochastic dominance

We develop and implement methods for determining whether relaxing sparsity constraints on portfolios improves the investment opportunity set for risk-averse investors. We formulate a new estimation procedure for sparse second-order stochastic spanning based on a greedy algorithm and Linear Programming. We show the optimal recovery of the sparse solution asymptotically whether spanning holds or not. From large equity datasets, we estimate the expected utility loss due to possible under-diversification, and find that there is no benefit from expanding a sparse opportunity set beyond 45 assets. The optimal sparse portfolio invests in 10 industry sectors and cuts tail risk when compared to a sparse mean-variance portfolio. On a rolling-window basis, the number of assets shrinks to 25 assets in crisis periods, while standard factor models cannot explain the performance of the sparse portfolios

Norm constrained empirical portfolio optimization with stochastic dominance: Robust optimization non-asymptotics

The present note provides an initial theoretical explanation of the way norm regularizations may provide a means of controlling the non-asymptotic probability of False Dominance classification for empirically optimal portfolios satisfying empirical Stochastic Dominance restrictions in an iid setting. It does so via a dual characterization of the norm-constrained problem, as a problem of Distributional Robust Optimization. This enables the use of concentration inequalities involving the Wasserstein distance from the empirical distribution, to obtain an upper bound for the non-asymptotic probability of False Dominance classification. This leads to information about the minimal sample size required for this probability to be dominated by a predetermined significance level