On Nesting Monte Carlo Estimators

Many problems in machine learning and statistics involve nested expectations and thus do not permit conventional Monte Carlo (MC) estimation. For such problems, one must nest estimators, such that terms in an outer estimator themselves involve calculation of a separate, nested, estimation. We investigate the statistical implications of nesting MC estimators, including cases of multiple levels of nesting, and establish the conditions under which they converge. We derive corresponding rates of convergence and provide empirical evidence that these rates are observed in practice. We further establish a number of pitfalls that can arise from naive nesting of MC estimators, provide guidelines about how these can be avoided, and lay out novel methods for reformulating certain classes of nested expectation problems into single expectations, leading to improved convergence rates. We demonstrate the applicability of our work by using our results to develop a new estimator for discrete Bayesian experimental design problems and derive error bounds for a class of variational objectives.Comment: To appear at International Conference on Machine Learning 201

Improving the Multi-Dimensional Comparison of Simulation Results: A Spatial Visualization Approach

Results from simulation experiments are important in applied spatial econometrics to, for instance, assess the performance of spatial estimators and tests for finite samples. However, the traditional tabular and graphi- cal formats for displaying simulation results in the literature have several disadvantages. These include loss of results, lack of intuitive synthesis, and difficulty in comparing results across multiple dimensions. We pro- pose to address these challenges through a spatial visualization approach. This approach visualizes model precision and bias as well as the size and power of tests in map format. The advantage of this spatial approach is that these maps can display all results succinctly, enable an intuitive interpretation, and compare results efficiently across multiple dimensions of a simulation experiment. Due to the respective strengths of tables, graphs and maps, we propose this spatial approach as a supplement to traditional tabular and graphical display formats. To allow readers to generate maps such as the ones presented in this article, a package (written in Python) has been made available by the authors as free/libre software. The package includes an example as well as a short tutorial for researchers without programming experience and can be downloaded at: https://github.com/darribas/simVizMap.

Decentralization Estimators for Instrumental Variable Quantile Regression Models

The instrumental variable quantile regression (IVQR) model (Chernozhukov and Hansen, 2005) is a popular tool for estimating causal quantile effects with endogenous covariates. However, estimation is complicated by the non-smoothness and non-convexity of the IVQR GMM objective function. This paper shows that the IVQR estimation problem can be decomposed into a set of conventional quantile regression sub-problems which are convex and can be solved efficiently. This reformulation leads to new identification results and to fast, easy to implement, and tuning-free estimators that do not require the availability of high-level "black box" optimization routines

Estimation and Welfare Calculations in a Generalized Corner Solution Model with an Application to Recreation Demand

The Kuhn-Tucker model of Wales and Woodland (1983) provides a utility theoretic framework for estimating preferences over commodities for which individuals choose not to consume one or more of the goods. Due to the complexity of the model, however, there have been few applications in the literature and little attention has been paid to the problems of welfare analysis within the Kuhn-Tucker framework. This paper provides an application of the model to the problem of recreation demand. In addition, we develop and apply a methodology for estimating compensating variation, relying on Monte Carlo integration to derive expected welfare changes.

Ferromagnetism in the Two-Dimensional Periodic Anderson Model

Using the constrained-path Monte Carlo method, we studied the magnetic properties of the two-dimensional periodic Anderson model for electron fillings between 1/4 and 1/2. We also derived two effective low energy theories to assist in interpreting the numerical results. For 1/4 filling we found that the system can be a Mott or a charge transfer insulator, depending on the relative values of the Coulomb interaction and the charge transfer gap between the two non-interacting bands. The insulator may be a paramagnet or antiferromagnet. We concentrated on the effect of electron doping on these insulating phases. Upon doping we obtained a partially saturated ferromagnetic phase for low concentrations of conduction electrons. If the system were a charge transfer insulator, we would find that the ferromagnetism is induced by the well-known RKKY interaction. However, we found a novel correlated hopping mechanism inducing the ferromagnetism in the region where the non-doped system is a Mott insulator. Our regions of ferromagnetism spanned a much smaller doping range than suggested by recent slave boson and dynamical mean field theory calculations, but they were consistent with that obtained by density matrix renormalization group calculations of the one-dimensional periodic Anderson model

A numerical method to compute derivatives of functions of large complex matrices and its application to the overlap Dirac operator at finite chemical potential

We present a method for the numerical calculation of derivatives of functions of general complex matrices. The method can be used in combination with any algorithm that evaluates or approximates the desired matrix function, in particular with implicit Krylov-Ritz-type approximations. An important use case for the method is the evaluation of the overlap Dirac operator in lattice Quantum Chromodynamics (QCD) at finite chemical potential, which requires the application of the sign function of a non-Hermitian matrix to some source vector. While the sign function of non-Hermitian matrices in practice cannot be efficiently approximated with source-independent polynomials or rational functions, sufficiently good approximating polynomials can still be constructed for each particular source vector. Our method allows for an efficient calculation of the derivatives of such implicit approximations with respect to the gauge field or other external parameters, which is necessary for the calculation of conserved lattice currents or the fermionic force in Hybrid Monte-Carlo or Langevin simulations. We also give an explicit deflation prescription for the case when one knows several eigenvalues and eigenvectors of the matrix being the argument of the differentiated function. We test the method for the two-sided Lanczos approximation of the finite-density overlap Dirac operator on realistic $SU(3)$ gauge field configurations on lattices with sizes as large as $14\times14^3$ and $6\times18^3$.Comment: 26 pages elsarticle style, 5 figures minor text changes, journal versio

Unbalanced Nested Component Error Model and the Value of Soil Insecticide and Bt Corn for Controlling Western Corn Rootworm

We describe four recently developed panel data estimators for unbalanced and nested data, a common problem for economic and experimental data. We estimate a western corn rootworm damage function with each estimator, including separate parameters for random effects from year, location, and experimental errors. We then use each estimator to assess the cost of the western corn rootworm soybean variant and the net benefit of soil insecticide and Bt corn for controlling this pest. At current prices, we find that soil insecticide generates a net loss ranging about $0.50-$3.25/ac, while Bt corn generates a net benefit ranging $2.50-$7.00/ac.

Advances in forecast evaluation

This paper surveys recent developments in the evaluation of point forecasts. Taking West's (2006) survey as a starting point, we briefly cover the state of the literature as of the time of West's writing. We then focus on recent developments, including advancements in the evaluation of forecasts at the population level (based on true, unknown model coefficients), the evaluation of forecasts in the finite sample (based on estimated model coefficients), and the evaluation of conditional versus unconditional forecasts. We present original results in a few subject areas: the optimization of power in determining the split of a sample into in-sample and out-of-sample portions; whether the accuracy of inference in evaluation of multi-step forecasts can be improved with judicious choice of HAC estimator (it can); and the extension of West's (1996) theory results for population-level, unconditional forecast evaluation to the case of conditional forecast evaluation.Forecasting