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arXiv2017

A Note on Gale, Kuhn, and Tucker's Reductions of Zero-Sum Games

Gale, Kuhn and Tucker (1950) introduced two ways to reduce a zero-sum game by packaging some strategies with respect to a probability distribution on them. In terms of value, they gave conditions for a desirable reduction. We show that a probability distribution for a desirable reduction relies on optimal strategies in the original game. Also, we correct an improper example given by them to show that the reverse of a theorem does not hold.

econ.EM↗
arXiv2022

When Should You Adjust Standard Errors for Clustering?

In empirical work it is common to estimate parameters of models and report associated standard errors that account for "clustering" of units, where clusters are defined by factors such as geography. Clustering adjustments are typically motivated by the concern that unobserved components of outcomes for units within clusters are correlated. However, this motivation does not provide guidance about questions such as: (i) Why should we adjust standard errors for clustering in some situations but not others? How can we justify the common practice of clustering in observational studies but not randomized experiments, or clustering by state but not by gender? (ii) Why is conventional clustering a potentially conservative "all-or-nothing" adjustment, and are there alternative methods that respond to data and are less conservative? (iii) In what settings does the choice of whether and how to cluster make a difference? We address these questions using a framework of sampling and design inference. We argue that clustering can be needed to address sampling issues if sampling follows a two stage process where in the first stage, a subset of clusters are sampled from a population of clusters, and in the second stage, units are sampled from the sampled clusters. Then, clustered standard errors account for the existence of clusters in the population that we do not see in the sample. Clustering can be needed to account for design issues if treatment assignment is correlated with membership in a cluster. We propose new variance estimators to deal with intermediate settings where conventional cluster standard errors are unnecessarily conservative and robust standard errors are too small.

math.ST↗
arXiv2017

A Unified Approach on the Local Power of Panel Unit Root Tests

In this paper, a unified approach is proposed to derive the exact local asymptotic power for panel unit root tests, which is one of the most important issues in nonstationary panel data literature. Two most widely used panel unit root tests known as Levin-Lin-Chu (LLC, Levin, Lin and Chu (2002)) and Im-Pesaran-Shin (IPS, Im, Pesaran and Shin (2003)) tests are systematically studied for various situations to illustrate our method. Our approach is characteristic function based, and can be used directly in deriving the moments of the asymptotic distributions of these test statistics under the null and the local-to-unity alternatives. For the LLC test, the approach provides an alternative way to obtain the results that can be derived by the existing method. For the IPS test, the new results are obtained, which fills the gap in the literature where few results exist, since the IPS test is non-admissible. Moreover, our approach has the advantage in deriving Edgeworth expansions of these tests, which are also given in the paper. The simulations are presented to illustrate our theoretical findings.

econ.EM↗
arXiv2018

Forecasting Across Time Series Databases using Recurrent Neural Networks on Groups of Similar Series: A Clustering Approach

With the advent of Big Data, nowadays in many applications databases containing large quantities of similar time series are available. Forecasting time series in these domains with traditional univariate forecasting procedures leaves great potentials for producing accurate forecasts untapped. Recurrent neural networks (RNNs), and in particular Long Short-Term Memory (LSTM) networks, have proven recently that they are able to outperform state-of-the-art univariate time series forecasting methods in this context when trained across all available time series. However, if the time series database is heterogeneous, accuracy may degenerate, so that on the way towards fully automatic forecasting methods in this space, a notion of similarity between the time series needs to be built into the methods. To this end, we present a prediction model that can be used with different types of RNN models on subgroups of similar time series, which are identified by time series clustering techniques. We assess our proposed methodology using LSTM networks, a widely popular RNN variant. Our method achieves competitive results on benchmarking datasets under competition evaluation procedures. In particular, in terms of mean sMAPE accuracy, it consistently outperforms the baseline LSTM model and outperforms all other methods on the CIF2016 forecasting competition dataset.

cs.LG↗
arXiv2018

Inference on Auctions with Weak Assumptions on Information

Given a sample of bids from independent auctions, this paper examines the question of inference on auction fundamentals (e.g. valuation distributions, welfare measures) under weak assumptions on information structure. The question is important as it allows us to learn about the valuation distribution in a robust way, i.e., without assuming that a particular information structure holds across observations. We leverage the recent contributions of \cite{Bergemann2013} in the robust mechanism design literature that exploit the link between Bayesian Correlated Equilibria and Bayesian Nash Equilibria in incomplete information games to construct an econometrics framework for learning about auction fundamentals using observed data on bids. We showcase our construction of identified sets in private value and common value auctions. Our approach for constructing these sets inherits the computational simplicity of solving for correlated equilibria: checking whether a particular valuation distribution belongs to the identified set is as simple as determining whether a {\it linear} program is feasible. A similar linear program can be used to construct the identified set on various welfare measures and counterfactual objects. For inference and to summarize statistical uncertainty, we propose novel finite sample methods using tail inequalities that are used to construct confidence regions on sets. We also highlight methods based on Bayesian bootstrap and subsampling. A set of Monte Carlo experiments show adequate finite sample properties of our inference procedures. We illustrate our methods using data from OCS auctions.

econ.EM↗
arXiv2017

Revenue-based Attribution Modeling for Online Advertising

This paper examines and proposes several attribution modeling methods that quantify how revenue should be attributed to online advertising inputs. We adopt and further develop relative importance method, which is based on regression models that have been extensively studied and utilized to investigate the relationship between advertising efforts and market reaction (revenue). Relative importance method aims at decomposing and allocating marginal contributions to the coefficient of determination (R^2) of regression models as attribution values. In particular, we adopt two alternative submethods to perform this decomposition: dominance analysis and relative weight analysis. Moreover, we demonstrate an extension of the decomposition methods from standard linear model to additive model. We claim that our new approaches are more flexible and accurate in modeling the underlying relationship and calculating the attribution values. We use simulation examples to demonstrate the superior performance of our new approaches over traditional methods. We further illustrate the value of our proposed approaches using a real advertising campaign dataset.

econ.EM↗
arXiv2017

Minimax Linear Estimation at a Boundary Point

This paper characterizes the minimax linear estimator of the value of an unknown function at a boundary point of its domain in a Gaussian white noise model under the restriction that the first-order derivative of the unknown function is Lipschitz continuous (the second-order Hölder class). The result is then applied to construct the minimax optimal estimator for the regression discontinuity design model, where the parameter of interest involves function values at boundary points.

econ.EM↗
arXiv2017

Modal Regression using Kernel Density Estimation: a Review

We review recent advances in modal regression studies using kernel density estimation. Modal regression is an alternative approach for investigating relationship between a response variable and its covariates. Specifically, modal regression summarizes the interactions between the response variable and covariates using the conditional mode or local modes. We first describe the underlying model of modal regression and its estimators based on kernel density estimation. We then review the asymptotic properties of the estimators and strategies for choosing the smoothing bandwidth. We also discuss useful algorithms and similar alternative approaches for modal regression, and propose future direction in this field.

stat.ME↗
arXiv2017

Electricity Market Theory Based on Continuous Time Commodity Model

The recent research report of U.S. Department of Energy prompts us to re-examine the pricing theories applied in electricity market design. The theory of spot pricing is the basis of electricity market design in many countries, but it has two major drawbacks: one is that it is still based on the traditional hourly scheduling/dispatch model, ignores the crucial time continuity in electric power production and consumption and does not treat the inter-temporal constraints seriously; the second is that it assumes that the electricity products are homogeneous in the same dispatch period and cannot distinguish the base, intermediate and peak power with obviously different technical and economic characteristics. To overcome the shortcomings, this paper presents a continuous time commodity model of electricity, including spot pricing model and load duration model. The market optimization models under the two pricing mechanisms are established with the Riemann and Lebesgue integrals respectively and the functional optimization problem are solved by the Euler-Lagrange equation to obtain the market equilibria. The feasibility of pricing according to load duration is proved by strict mathematical derivation. Simulation results show that load duration pricing can correctly identify and value different attributes of generators, reduce the total electricity purchasing cost, and distribute profits among the power plants more equitably. The theory and methods proposed in this paper will provide new ideas and theoretical foundation for the development of electric power markets.

econ.EM↗
arXiv2018

Existence in Multidimensional Screening with General Nonlinear Preferences

We generalize the approach of Carlier (2001) and provide an existence proof for the multidimensional screening problem with general nonlinear preferences. We first formulate the principal's problem as a maximization problem with $G$-convexity constraints and then use $G$-convex analysis to prove existence.

econ.EM↗
arXiv2017

Calibration of Machine Learning Classifiers for Probability of Default Modelling

Binary classification is highly used in credit scoring in the estimation of probability of default. The validation of such predictive models is based both on rank ability, and also on calibration (i.e. how accurately the probabilities output by the model map to the observed probabilities). In this study we cover the current best practices regarding calibration for binary classification, and explore how different approaches yield different results on real world credit scoring data. The limitations of evaluating credit scoring models using only rank ability metrics are explored. A benchmark is run on 18 real world datasets, and results compared. The calibration techniques used are Platt Scaling and Isotonic Regression. Also, different machine learning models are used: Logistic Regression, Random Forest Classifiers, and Gradient Boosting Classifiers. Results show that when the dataset is treated as a time series, the use of re-calibration with Isotonic Regression is able to improve the long term calibration better than the alternative methods. Using re-calibration, the non-parametric models are able to outperform the Logistic Regression on Brier Score Loss.

econ.EM↗
arXiv2017

Asymptotic Distribution and Simultaneous Confidence Bands for Ratios of Quantile Functions

Ratio of medians or other suitable quantiles of two distributions is widely used in medical research to compare treatment and control groups or in economics to compare various economic variables when repeated cross-sectional data are available. Inspired by the so-called growth incidence curves introduced in poverty research, we argue that the ratio of quantile functions is a more appropriate and informative tool to compare two distributions. We present an estimator for the ratio of quantile functions and develop corresponding simultaneous confidence bands, which allow to assess significance of certain features of the quantile functions ratio. Derived simultaneous confidence bands rely on the asymptotic distribution of the quantile functions ratio and do not require re-sampling techniques. The performance of the simultaneous confidence bands is demonstrated in simulations. Analysis of the expenditure data from Uganda in years 1999, 2002 and 2005 illustrates the relevance of our approach.

stat.ME↗
arXiv2018

Shape-Constrained Density Estimation via Optimal Transport

Constraining the maximum likelihood density estimator to satisfy a sufficiently strong constraint, $\log-$concavity being a common example, has the effect of restoring consistency without requiring additional parameters. Since many results in economics require densities to satisfy a regularity condition, these estimators are also attractive for the structural estimation of economic models. In all of the examples of regularity conditions provided by Bagnoli and Bergstrom (2005) and Ewerhart (2013), $\log-$concavity is sufficient to ensure that the density satisfies the required conditions. However, in many cases $\log-$concavity is far from necessary, and it has the unfortunate side effect of ruling out sub-exponential tail behavior. In this paper, we use optimal transport to formulate a shape constrained density estimator. We initially describe the estimator using a $ρ-$concavity constraint. In this setting we provide results on consistency, asymptotic distribution, convexity of the optimization problem defining the estimator, and formulate a test for the null hypothesis that the population density satisfies a shape constraint. Afterward, we provide sufficient conditions for these results to hold using an arbitrary shape constraint. This generalization is used to explore whether the California Department of Transportation's decision to award construction contracts with the use of a first price auction is cost minimizing. We estimate the marginal costs of construction firms subject to Myerson's (1981) regularity condition, which is a requirement for the first price reverse auction to be cost minimizing. The proposed test fails to reject that the regularity condition is satisfied.

econ.EM↗
arXiv2017

Calibrated Projection in MATLAB: Users' Manual

We present the calibrated-projection MATLAB package implementing the method to construct confidence intervals proposed by Kaido, Molinari and Stoye (2017). This manual provides details on how to use the package for inference on projections of partially identified parameters. It also explains how to use the MATLAB functions we developed to compute confidence intervals on solutions of nonlinear optimization problems with estimated constraints.

econ.EM↗
arXiv2018

Nonparametric Identification in Index Models of Link Formation

We consider an index model of dyadic link formation with a homophily effect index and a degree heterogeneity index. We provide nonparametric identification results in a single large network setting for the potentially nonparametric homophily effect function, the realizations of unobserved individual fixed effects and the unknown distribution of idiosyncratic pairwise shocks, up to normalization, for each possible true value of the unknown parameters. We propose a novel form of scale normalization on an arbitrary interquantile range, which is not only theoretically robust but also proves particularly convenient for the identification analysis, as quantiles provide direct linkages between the observable conditional probabilities and the unknown index values. We then use an inductive "in-fill and out-expansion" algorithm to establish our main results, and consider extensions to more general settings that allow nonseparable dependence between homophily and degree heterogeneity, as well as certain extents of network sparsity and weaker assumptions on the support of unobserved heterogeneity. As a byproduct, we also propose a concept called "modeling equivalence" as a refinement of "observational equivalence", and use it to provide a formal discussion about normalization, identification and their interplay with counterfactuals.

econ.EM↗
arXiv2017

Macroeconomics and FinTech: Uncovering Latent Macroeconomic Effects on Peer-to-Peer Lending

Peer-to-peer (P2P) lending is a fast growing financial technology (FinTech) trend that is displacing traditional retail banking. Studies on P2P lending have focused on predicting individual interest rates or default probabilities. However, the relationship between aggregated P2P interest rates and the general economy will be of interest to investors and borrowers as the P2P credit market matures. We show that the variation in P2P interest rates across grade types are determined by three macroeconomic latent factors formed by Canonical Correlation Analysis (CCA) - macro default, investor uncertainty, and the fundamental value of the market. However, the variation in P2P interest rates across term types cannot be explained by the general economy.

econ.EM↗
arXiv2017

On some further properties and application of Weibull-R family of distributions

In this paper, we provide some new results for the Weibull-R family of distributions (Alzaghal, Ghosh and Alzaatreh (2016)). We derive some new structural properties of the Weibull-R family of distributions. We provide various characterizations of the family via conditional moments, some functions of order statistics and via record values.

math.ST↗