Applied researchers using structural models under rational expectations (RE) often confront empirical evidence of misspecification. In this paper we consider a generic dynamic model that is posed as a vector of unconditional moment restrictions. We suppose that the model is globally misspecified under RE, and thus empirically in a way that is not econometrically subtle. We relax the RE restriction by allowing subjective beliefs to differ from the data-generating probability (DGP) model while still maintaining that the moment conditions are satisfied under the subjective beliefs of economic agents. We use statistical measures of divergence relative to RE to bound the set of subjective probabilities. This form of misspecification alters econometric identification and inferences in a substantial way, leading us to construct robust confidence sets for various set identified functionals.
{"title":"Robust Inference for Moment Condition Models without Rational Expectations","authors":"Xiaohong Chen, L. Hansen, Peter G. Hansen","doi":"10.2139/ssrn.3945856","DOIUrl":"https://doi.org/10.2139/ssrn.3945856","url":null,"abstract":"Applied researchers using structural models under rational expectations (RE) often confront empirical evidence of misspecification. In this paper we consider a generic dynamic model that is posed as a vector of unconditional moment restrictions. We suppose that the model is globally misspecified under RE, and thus empirically in a way that is not econometrically subtle. We relax the RE restriction by allowing subjective beliefs to differ from the data-generating probability (DGP) model while still maintaining that the moment conditions are satisfied under the subjective beliefs of economic agents. We use statistical measures of divergence relative to RE to bound the set of subjective probabilities. This form of misspecification alters econometric identification and inferences in a substantial way, leading us to construct robust confidence sets for various set identified functionals.","PeriodicalId":320844,"journal":{"name":"PSN: Econometrics","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126054481","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Since an economic or financial variable may be affected by both stationary and nonstationary variables, this paper proposes a class of augmented cointegrating linear (ACL) models that accommodate these time series of different types. Moreover, the variables are allowed to be strongly correlated in the sense depicted in the paper. The asymptotic limit theory of estimator proposed is established via jointly convergence of the sample variance and covariance that circumvents the existent drawback in the most nonstationary time series literature; also a self-normalized central limit theorem is given to facilitate statistical inference. Monte Carlo simulations confirm the theoretical results. Finally, ACL regression model is applied to the GDP time series in US, for which we show the proposed model is more accurate and competent than some potential models.
{"title":"Augmented cointegrating linear models with possibly strongly correlated stationary and nonstationary regressors regressors","authors":"Zhen Peng, Chaohua Dong","doi":"10.2139/ssrn.3943779","DOIUrl":"https://doi.org/10.2139/ssrn.3943779","url":null,"abstract":"Since an economic or financial variable may be affected by both stationary and nonstationary variables, this paper proposes a class of augmented cointegrating linear (ACL) models that accommodate these time series of different types. Moreover, the variables are allowed to be strongly correlated in the sense depicted in the paper. The asymptotic limit theory of estimator proposed is established via jointly convergence of the sample variance and covariance that circumvents the existent drawback in the most nonstationary time series literature; also a self-normalized central limit theorem is given to facilitate statistical inference. Monte Carlo simulations confirm the theoretical results. Finally, ACL regression model is applied to the GDP time series in US, for which we show the proposed model is more accurate and competent than some potential models.","PeriodicalId":320844,"journal":{"name":"PSN: Econometrics","volume":"20 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115270490","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Michael Mayer, Steven C. Bourassa, Martin Hoesli, D. Scognamiglio
Structured additive regression (STAR) models are a rich class of regression models that include the generalized linear model (GLM) and the generalized additive model (GAM). STAR models can be fitted by Bayesian approaches, component-wise gradient boosting, penalized least-squares, and deep learning. Using feature interaction constraints, we show that such models can be implemented also by the gradient boosting powerhouses XGBoost and LightGBM, thereby benefiting from their excellent predictive capabilities. Furthermore, we show how STAR models can be used for supervised dimension reduction and explain under what circumstances covariate effects of such models can be described in a transparent way. We illustrate the methodology with case studies pertaining to house price modeling, with very encouraging results regarding both interpretability and predictive performance.
{"title":"Structured Additive Regression and Tree Boosting","authors":"Michael Mayer, Steven C. Bourassa, Martin Hoesli, D. Scognamiglio","doi":"10.2139/ssrn.3924412","DOIUrl":"https://doi.org/10.2139/ssrn.3924412","url":null,"abstract":"Structured additive regression (STAR) models are a rich class of regression models that include the generalized linear model (GLM) and the generalized additive model (GAM). STAR models can be fitted by Bayesian approaches, component-wise gradient boosting, penalized least-squares, and deep learning. Using feature interaction constraints, we show that such models can be implemented also by the gradient boosting powerhouses XGBoost and LightGBM, thereby benefiting from their excellent predictive capabilities. Furthermore, we show how STAR models can be used for supervised dimension reduction and explain under what circumstances covariate effects of such models can be described in a transparent way. We illustrate the methodology with case studies pertaining to house price modeling, with very encouraging results regarding both interpretability and predictive performance.","PeriodicalId":320844,"journal":{"name":"PSN: Econometrics","volume":"40 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121439649","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Aryan Eftekhari, Lisa Gaedke-Merzhaeuser, D. Pasadakis, M. Bollhoefer, S. Scheidegger, O. Schenk
High-dimensional sparse precision matrix estimation is a ubiquitous task in multivariate analysis with applications that cross many disciplines. In this paper, we introduce the SQUIC package, which benefits from superior runtime performance and scalability, significantly exceeding the available state-of-the-art packages. This package is a second-order method that solves the L1--regularized maximum likelihood problem using highly optimized linear algebra subroutines, which leverage the underlying sparsity and the intrinsic parallelism in the computation. We provide two sets of numerical tests; the first one consists of didactic examples using synthetic datasets highlighting the performance and accuracy of the package, and the second one is a real-world classification problem of high dimensional medical datasets. The base algorithm is implemented in C++ with interfaces for R and Python.
{"title":"Large-Scale Precision Matrix Estimation With SQUIC","authors":"Aryan Eftekhari, Lisa Gaedke-Merzhaeuser, D. Pasadakis, M. Bollhoefer, S. Scheidegger, O. Schenk","doi":"10.2139/ssrn.3904001","DOIUrl":"https://doi.org/10.2139/ssrn.3904001","url":null,"abstract":"High-dimensional sparse precision matrix estimation is a ubiquitous task in multivariate analysis with applications that cross many disciplines. In this paper, we introduce the SQUIC package, which benefits from superior runtime performance and scalability, significantly exceeding the available state-of-the-art packages. This package is a second-order method that solves the L1--regularized maximum likelihood problem using highly optimized linear algebra subroutines, which leverage the underlying sparsity and the intrinsic parallelism in the computation. We provide two sets of numerical tests; the first one consists of didactic examples using synthetic datasets highlighting the performance and accuracy of the package, and the second one is a real-world classification problem of high dimensional medical datasets. The base algorithm is implemented in C++ with interfaces for R and Python.","PeriodicalId":320844,"journal":{"name":"PSN: Econometrics","volume":"22 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121200695","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We introduce valid regression models and valid error correction models for the non-cointegrated variables. These models are also valid for the cointegrated variables. Consequently, cointegration tests and analysis become needless. Furthermore, our approach overcomes the lag selection problem.
{"title":"Error Correction Models and Regressions for Non-Cointegrated Variables","authors":"Moawia Alghalith","doi":"10.2139/ssrn.3902889","DOIUrl":"https://doi.org/10.2139/ssrn.3902889","url":null,"abstract":"We introduce valid regression models and valid error correction models for the non-cointegrated variables. These models are also valid for the cointegrated variables. Consequently, cointegration tests and analysis become needless. Furthermore, our approach overcomes the lag selection problem.","PeriodicalId":320844,"journal":{"name":"PSN: Econometrics","volume":"73 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124448062","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A simple yet easy to implement method is proposed to further improve the finite sample approximation by central limit theorems for aggregate efficiency. By adopt- ing the correction method in Simar and Zelenyuk (2020, EJOR), we further propose plugging the bias-corrected mean efficiency estimate rather than just mean efficiency estimate, into the variance estimator of aggregate efficiency. In extensive Monte-Carlo experiments, although our newly proposed method is found to have smaller coverages than the method using the true variance, it is found to have larger coverages across virtually all finite sample sizes and across dimensions than the original method in Simar and Zelenyuk (2018,OR) and the correction method in Simar and Zelenyuk (2020, EJOR). A real data set is employed to show the differences between these three methods in the estimated variance and the estimated confidence intervals.
{"title":"Further Improving Finite Sample Approximation by Central Limit Theorems for Aggregate Efficiency","authors":"Shirong Zhao","doi":"10.2139/ssrn.3901240","DOIUrl":"https://doi.org/10.2139/ssrn.3901240","url":null,"abstract":"A simple yet easy to implement method is proposed to further improve the finite sample approximation by central limit theorems for aggregate efficiency. By adopt- ing the correction method in Simar and Zelenyuk (2020, EJOR), we further propose plugging the bias-corrected mean efficiency estimate rather than just mean efficiency estimate, into the variance estimator of aggregate efficiency. In extensive Monte-Carlo experiments, although our newly proposed method is found to have smaller coverages than the method using the true variance, it is found to have larger coverages across virtually all finite sample sizes and across dimensions than the original method in Simar and Zelenyuk (2018,OR) and the correction method in Simar and Zelenyuk (2020, EJOR). A real data set is employed to show the differences between these three methods in the estimated variance and the estimated confidence intervals.","PeriodicalId":320844,"journal":{"name":"PSN: Econometrics","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125244539","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper proposes a test for the signal-to-noise ratio applicable to a range of significance tests and model diagnostics in a linear regression. It is particularly useful under a large or massive sample size, where a conventional test frequently rejects an economically negligible deviation from the null hypothesis. The test is conducted in the context of the traditional $F$-test, with its critical values increasing with sample size. It maintains desirable size properties under a large or massive sample size, when the null hypothesis is violated by a practically negligible margin.
{"title":"Testing for Signal-to-Noise Ratio in Linear Regression: A Test for Big Data Era","authors":"Jae H. Kim","doi":"10.2139/ssrn.3884683","DOIUrl":"https://doi.org/10.2139/ssrn.3884683","url":null,"abstract":"This paper proposes a test for the signal-to-noise ratio applicable to a range of significance tests and model diagnostics in a linear regression. It is particularly useful under a large or massive sample size, where a conventional test frequently rejects an economically negligible deviation from the null hypothesis. The test is conducted in the context of the traditional $F$-test, with its critical values increasing with sample size. It maintains desirable size properties under a large or massive sample size, when the null hypothesis is violated by a practically negligible margin.","PeriodicalId":320844,"journal":{"name":"PSN: Econometrics","volume":"359 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121642166","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we consider an assortment optimization problem in which a platform must choose pairwise disjoint sets of assortments to offer across a series of T stages. Arriving customers begin their search process in the first stage and progress sequentially through the stages until their patience expires, at which point they make a multinomial-logit-based purchasing decision from among all products they have viewed throughout their search process. The goal is to choose the sequential displays of product offerings to maximize expected revenue. Additionally, we impose stage-specific constraints that ensure that as each customer progresses farther and farther through the T stages, there is a minimum level of “desirability” met by the collections of displayed products. We consider two related measures of desirability: purchase likelihood and expected utility derived from the offered assortments. In this way, the offered sequence of assortment must be both high earning and well-liked, which breaks from the traditional assortment setting, where customer considerations are generally not explicitly accounted for. We show that our assortment problem of interest is strongly NP-Hard, thus ruling out the existence of a fully polynomial-time approximation scheme (FPTAS). From an algorithmic standpoint, as a warm-up, we develop a simple constant factor approximation scheme in which we carefully stitch together myopically selected assortments for each stage. Our main algorithmic result consists of a polynomial-time approximation scheme (PTAS), which combines a handful of structural results related to the make-up of the optimal assortment sequence within an approximate dynamic programming framework.
{"title":"Display Optimization Under the Multinomial Logit Choice Model: Balancing Revenue and Customer Satisfaction","authors":"Jacob B. Feldman, Puping (Phil) Jiang","doi":"10.2139/ssrn.3909033","DOIUrl":"https://doi.org/10.2139/ssrn.3909033","url":null,"abstract":"In this paper, we consider an assortment optimization problem in which a platform must choose pairwise disjoint sets of assortments to offer across a series of T stages. Arriving customers begin their search process in the first stage and progress sequentially through the stages until their patience expires, at which point they make a multinomial-logit-based purchasing decision from among all products they have viewed throughout their search process. The goal is to choose the sequential displays of product offerings to maximize expected revenue. Additionally, we impose stage-specific constraints that ensure that as each customer progresses farther and farther through the T stages, there is a minimum level of “desirability” met by the collections of displayed products. We consider two related measures of desirability: purchase likelihood and expected utility derived from the offered assortments. In this way, the offered sequence of assortment must be both high earning and well-liked, which breaks from the traditional assortment setting, where customer considerations are generally not explicitly accounted for. We show that our assortment problem of interest is strongly NP-Hard, thus ruling out the existence of a fully polynomial-time approximation scheme (FPTAS). From an algorithmic standpoint, as a warm-up, we develop a simple constant factor approximation scheme in which we carefully stitch together myopically selected assortments for each stage. Our main algorithmic result consists of a polynomial-time approximation scheme (PTAS), which combines a handful of structural results related to the make-up of the optimal assortment sequence within an approximate dynamic programming framework.","PeriodicalId":320844,"journal":{"name":"PSN: Econometrics","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122252954","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A rich variety of threshold regression models has been in use starting with Tobin (1958). However, several applications indicate the necessity for a class of threshold regression models that have not been considered so far. This note presents a specification of such models and offers a novel method of estimation.
{"title":"On the Estimation of a Class of Threshold Regression Models","authors":"Ramamohan Rao","doi":"10.2139/ssrn.3850678","DOIUrl":"https://doi.org/10.2139/ssrn.3850678","url":null,"abstract":"A rich variety of threshold regression models has been in use starting with Tobin (1958). However, several applications indicate the necessity for a class of threshold regression models that have not been considered so far. This note presents a specification of such models and offers a novel method of estimation.","PeriodicalId":320844,"journal":{"name":"PSN: Econometrics","volume":"29 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125898325","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Understanding individual customers’ sensitivities to prices, promotions, brand, and other aspects of the marketing mix is fundamental to a wide swath of marketing problems, including targeting and pricing. Companies that operate across many product categories have a unique opportunity, insofar as they can use purchasing data from one category to augment their insights in another. Such cross-category insights are especially crucial in situations where purchasing data may be rich in one category, and scarce in another. An important aspect of how consumers behave across categories is dynamics: preferences are not stable over time, and changes in individual-level preference parameters in one category may be indicative of changes in other categories, especially if those changes are driven by external factors. Yet, despite the rich history of modeling cross-category preferences, the marketing literature lacks a framework that flexibly accounts for correlated dynamics, or the cross-category interlinkages of individual-level sensitivity dynamics. In this work, we propose such a framework, leveraging individual-level, latent, multi-output Gaussian processes to build a non-parametric Bayesian choice model that allows information sharing of preference parameters across customers, time, and categories. We apply our model to grocery purchase data, and show that our model detects interesting dynamics of customers’ price sensitivities across multiple categories. Managerially, we show that capturing correlated dynamics yields substantial predictive gains, relative to benchmarks. Moreover, we find that capturing correlated dynamics can have implications for understanding changes in consumers preferences over time, and developing targeted marketing strategies based on those dynamics.
{"title":"A Gaussian Process Model of Cross-Category Dynamics in Brand Choice","authors":"Ryan Dew, Yuhao Fan","doi":"10.2139/ssrn.3832290","DOIUrl":"https://doi.org/10.2139/ssrn.3832290","url":null,"abstract":"Understanding individual customers’ sensitivities to prices, promotions, brand, and other aspects of the marketing mix is fundamental to a wide swath of marketing problems, including targeting and pricing. Companies that operate across many product categories have a unique opportunity, insofar as they can use purchasing data from one category to augment their insights in another. Such cross-category insights are especially crucial in situations where purchasing data may be rich in one category, and scarce in another. An important aspect of how consumers behave across categories is dynamics: preferences are not stable over time, and changes in individual-level preference parameters in one category may be indicative of changes in other categories, especially if those changes are driven by external factors. Yet, despite the rich history of modeling cross-category preferences, the marketing literature lacks a framework that flexibly accounts for correlated dynamics, or the cross-category interlinkages of individual-level sensitivity dynamics. In this work, we propose such a framework, leveraging individual-level, latent, multi-output Gaussian processes to build a non-parametric Bayesian choice model that allows information sharing of preference parameters across customers, time, and categories. We apply our model to grocery purchase data, and show that our model detects interesting dynamics of customers’ price sensitivities across multiple categories. Managerially, we show that capturing correlated dynamics yields substantial predictive gains, relative to benchmarks. Moreover, we find that capturing correlated dynamics can have implications for understanding changes in consumers preferences over time, and developing targeted marketing strategies based on those dynamics.","PeriodicalId":320844,"journal":{"name":"PSN: Econometrics","volume":"42 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123164457","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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