A nonparametric approach is presented to test whether decisions on a probability simplex could be induced by quasiconcave preferences. Necessary and sufficient conditions are presented. If the answer is affirmative, the methods developed here allow to reconstruct bounds on indifference curves. Furthermore we can construct quasiconcave utility functions in analogy to the utility function constructed in the proof of Afriat's Theorem. The approach is of interest for decisions under risk, stochastic choice, and ex-ante fairness considerations. The method is particularly suitable for data collected in a laboratory experiment.
{"title":"Quasiconcave Preferences and Choices on a Probability Simplex - A Nonparametric Analysis","authors":"Jan Heufer","doi":"10.2139/ssrn.1593473","DOIUrl":"https://doi.org/10.2139/ssrn.1593473","url":null,"abstract":"A nonparametric approach is presented to test whether decisions on a probability simplex could be induced by quasiconcave preferences. Necessary and sufficient conditions are presented. If the answer is affirmative, the methods developed here allow to reconstruct bounds on indifference curves. Furthermore we can construct quasiconcave utility functions in analogy to the utility function constructed in the proof of Afriat's Theorem. The approach is of interest for decisions under risk, stochastic choice, and ex-ante fairness considerations. The method is particularly suitable for data collected in a laboratory experiment.","PeriodicalId":11744,"journal":{"name":"ERN: Nonparametric Methods (Topic)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2010-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90649687","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 article attempts to identify the best model of the short term interest rates that can predict its stochastic process over time. We studied nine different models of the short term interest rates. The choice of these models was the aim of analyzing the relevance of certain specifications of the the short term interest rate stochastic process, the effect of mean reversion and the sensitivity of the volatility to the level of interest rate. The yield on US three months treasury bills is used as a proxy for the short term interest rates. The parameters of the different stochastic process are estimated using the generalized method of moments. The results show that the effect of mean reversion is not statistically significant and that volatility is highly sensitive to the level of interest rates. To further study the performance prediction of the intertemporal behavior of the short term interest rate of the various models; we simulated their stochastic process for different periods. The results show that none of the studied models reproduce the actual path of the short term interest rates. The problem lies in the parametric specification of the mean and volatility of the diffusion process To further study the accurate parametric specification of the interest rate stochastic process we use a nonparametric estimation of the drift and the diffusion functions. The results prove that both should be nonlinear.
{"title":"Comparison of the Short Term Interest Rate Models: Parametric Versus Non Parametric Approach","authors":"Mona Ben Salah","doi":"10.2139/ssrn.2393909","DOIUrl":"https://doi.org/10.2139/ssrn.2393909","url":null,"abstract":"This article attempts to identify the best model of the short term interest rates that can predict its stochastic process over time. We studied nine different models of the short term interest rates. The choice of these models was the aim of analyzing the relevance of certain specifications of the the short term interest rate stochastic process, the effect of mean reversion and the sensitivity of the volatility to the level of interest rate. The yield on US three months treasury bills is used as a proxy for the short term interest rates. The parameters of the different stochastic process are estimated using the generalized method of moments. The results show that the effect of mean reversion is not statistically significant and that volatility is highly sensitive to the level of interest rates. To further study the performance prediction of the intertemporal behavior of the short term interest rate of the various models; we simulated their stochastic process for different periods. The results show that none of the studied models reproduce the actual path of the short term interest rates. The problem lies in the parametric specification of the mean and volatility of the diffusion process To further study the accurate parametric specification of the interest rate stochastic process we use a nonparametric estimation of the drift and the diffusion functions. The results prove that both should be nonlinear.","PeriodicalId":11744,"journal":{"name":"ERN: Nonparametric Methods (Topic)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2010-02-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75034595","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 present new identification results for nonparametric models of differentiated products markets, using only market level observables. We specify a nonparametric random utility discrete choice model of demand allowing rich preference heterogeneity, product/market unobservables, and endogenous prices. Our supply model posits nonparametric cost functions, allows latent cost shocks, and nests a range of standard oligopoly models. We consider identification of demand, identification of changes in aggregate consumer welfare, identification of marginal costs, identification of firms' marginal cost functions, and discrimination between alternative models of firm conduct. We explore two complementary approaches. The first demonstrates identification under the same nonparametric instrumental variables conditions required for identification of regression models. The second treats demand and supply in a system of nonparametric simultaneous equations, leading to constructive proofs exploiting exogenous variation in demand shifters and cost shifters. We also derive testable restrictions that provide the first general formalization of Bresnahan's (1982) intuition for empirically distinguishing between alternative models of oligopoly competition. From a practical perspective, our results clarify the types of instrumental variables needed with market level data, including tradeoffs between functional form and exclusion restrictions.
{"title":"Identification in Differentiated Products Markets Using Market Level Data","authors":"Steven T. Berry, Philip A. Haile","doi":"10.2139/ssrn.2064898","DOIUrl":"https://doi.org/10.2139/ssrn.2064898","url":null,"abstract":"We present new identification results for nonparametric models of differentiated products markets, using only market level observables. We specify a nonparametric random utility discrete choice model of demand allowing rich preference heterogeneity, product/market unobservables, and endogenous prices. Our supply model posits nonparametric cost functions, allows latent cost shocks, and nests a range of standard oligopoly models. We consider identification of demand, identification of changes in aggregate consumer welfare, identification of marginal costs, identification of firms' marginal cost functions, and discrimination between alternative models of firm conduct. We explore two complementary approaches. The first demonstrates identification under the same nonparametric instrumental variables conditions required for identification of regression models. The second treats demand and supply in a system of nonparametric simultaneous equations, leading to constructive proofs exploiting exogenous variation in demand shifters and cost shifters. We also derive testable restrictions that provide the first general formalization of Bresnahan's (1982) intuition for empirically distinguishing between alternative models of oligopoly competition. From a practical perspective, our results clarify the types of instrumental variables needed with market level data, including tradeoffs between functional form and exclusion restrictions.","PeriodicalId":11744,"journal":{"name":"ERN: Nonparametric Methods (Topic)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2010-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76608995","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 considers the identification and estimation of an extension of Roy's model (1951) of occupational choice, which includes a non-pecuniary component in the decision equation and allows for uncertainty on the potential outcomes. This framework is well suited to various economic contexts, including educational and sectoral choices, or migration decisions. We focus in particular on the identification of the non-pecuniary component under the condition that at least one variable affects the selection probability only through potential earnings, that is under the opposite of the usual exclusion restrictions used to identify switching regressions models and treatment effects. Point identification is achieved if such variables are continuous, while bounds are obtained otherwise. As a result, the distribution of the ex ante treatment effects can be point or set identified without any usual instruments. We propose a three-stages semiparametric estimation procedure for this model, which yields root-n consistent and asymptotically normal estimators. We apply our results to the educational context, by providing new evidence from French data that non-pecuniary factors are a key determinant of higher education attendance decisions.
{"title":"Inference on a Generalized Roy Model, with an Application to Schooling Decisions in France","authors":"Xavier D’Haultfœuille, Arnaud Maurel","doi":"10.2139/ssrn.1519242","DOIUrl":"https://doi.org/10.2139/ssrn.1519242","url":null,"abstract":"This paper considers the identification and estimation of an extension of Roy's model (1951) of occupational choice, which includes a non-pecuniary component in the decision equation and allows for uncertainty on the potential outcomes. This framework is well suited to various economic contexts, including educational and sectoral choices, or migration decisions. We focus in particular on the identification of the non-pecuniary component under the condition that at least one variable affects the selection probability only through potential earnings, that is under the opposite of the usual exclusion restrictions used to identify switching regressions models and treatment effects. Point identification is achieved if such variables are continuous, while bounds are obtained otherwise. As a result, the distribution of the ex ante treatment effects can be point or set identified without any usual instruments. We propose a three-stages semiparametric estimation procedure for this model, which yields root-n consistent and asymptotically normal estimators. We apply our results to the educational context, by providing new evidence from French data that non-pecuniary factors are a key determinant of higher education attendance decisions.","PeriodicalId":11744,"journal":{"name":"ERN: Nonparametric Methods (Topic)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2009-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89413649","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}
Recent work by Wang and Phillips (2009b, c) has shown that ill posed inverse problems do not arise in nonstationary nonparametric regression and there is no need for nonparametric instrumental variable estimation. Instead, simple Nadaraya Watson nonparametric estimation of a (possibly nonlinear) cointegrating regression equation is consistent with a limiting (mixed) normal distribution irrespective of the endogeneity in the regressor, near integration as well as integration in the regressor, and serial dependence in the regression equation. The present paper shows that some closely related results apply in the case of structural nonparametric regression with independent data when there are continuous location shifts in the regressor. In such cases, location shifts serve as an instrumental variable in tracing out the regression line similar to the random wandering nature of the regressor in a cointegrating regression. Asymptotic theory is given for local level and local linear nonparametric estimators, links with nonstationary cointegrating regression theory and nonparametric IV regression are explored, and extensions to the stationary strong mixing case are given. In contrast to standard nonparametric limit theory, local level and local linear estimators have identical limit distributions, so the local linear approach has no apparent advantage in the present context. Some interesting cases are discovered, which appear to be new in the literature, where nonparametric estimation is consistent whereas parametric regression is inconsistent even when the true (parametric) regression function is known. The methods are further applied to establish a limit theory for nonparametric estimation of structural panel data models with endogenous regressors and individual effects. Some simulation evidence is reported.
{"title":"Nonparametric Structural Estimation Via Continuous Location Shifts in an Endogenous Regressor","authors":"P. Phillips, Liangjun Su","doi":"10.2139/ssrn.1414338","DOIUrl":"https://doi.org/10.2139/ssrn.1414338","url":null,"abstract":"Recent work by Wang and Phillips (2009b, c) has shown that ill posed inverse problems do not arise in nonstationary nonparametric regression and there is no need for nonparametric instrumental variable estimation. Instead, simple Nadaraya Watson nonparametric estimation of a (possibly nonlinear) cointegrating regression equation is consistent with a limiting (mixed) normal distribution irrespective of the endogeneity in the regressor, near integration as well as integration in the regressor, and serial dependence in the regression equation. The present paper shows that some closely related results apply in the case of structural nonparametric regression with independent data when there are continuous location shifts in the regressor. In such cases, location shifts serve as an instrumental variable in tracing out the regression line similar to the random wandering nature of the regressor in a cointegrating regression. Asymptotic theory is given for local level and local linear nonparametric estimators, links with nonstationary cointegrating regression theory and nonparametric IV regression are explored, and extensions to the stationary strong mixing case are given. In contrast to standard nonparametric limit theory, local level and local linear estimators have identical limit distributions, so the local linear approach has no apparent advantage in the present context. Some interesting cases are discovered, which appear to be new in the literature, where nonparametric estimation is consistent whereas parametric regression is inconsistent even when the true (parametric) regression function is known. The methods are further applied to establish a limit theory for nonparametric estimation of structural panel data models with endogenous regressors and individual effects. Some simulation evidence is reported.","PeriodicalId":11744,"journal":{"name":"ERN: Nonparametric Methods (Topic)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2009-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77512718","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 develop nonparametric tests for the null hypothesis that a function has a prescribed form, to apply to data sets with missing observations. Omnibus nonparametric tests do not need to specify a particular alternative parametric form, and have power against a large range of alternatives, the order selection tests that we study are one example. We extend such order selection tests to be applicable in the context of missing data. In particular, we consider likelihood-based order selection tests for multiply-imputed data. A simulation study and data analysis illustrate the performance of the tests. A model selection method in the style of Akaike's information criterion for multiply imputed datasets results along the same lines.
{"title":"Order Selection Tests with Multiply-Imputed Data","authors":"Fabrizio Consentino, G. Claeskens","doi":"10.2139/ssrn.1430275","DOIUrl":"https://doi.org/10.2139/ssrn.1430275","url":null,"abstract":"We develop nonparametric tests for the null hypothesis that a function has a prescribed form, to apply to data sets with missing observations. Omnibus nonparametric tests do not need to specify a particular alternative parametric form, and have power against a large range of alternatives, the order selection tests that we study are one example. We extend such order selection tests to be applicable in the context of missing data. In particular, we consider likelihood-based order selection tests for multiply-imputed data. A simulation study and data analysis illustrate the performance of the tests. A model selection method in the style of Akaike's information criterion for multiply imputed datasets results along the same lines.","PeriodicalId":11744,"journal":{"name":"ERN: Nonparametric Methods (Topic)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2009-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79263450","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 studies two-step extremum estimation that involves the first step estimation of nonparametric functions of single-indices. First, this paper finds that under certain regularity conditions for conditional measures, linear functionals of conditional expectations are insensitive to the first order perturbation of the parameters in the conditioning variable. Applying this result to symmetrized nearest neighborhood estimation of the nonparametric functions, this paper shows that the influence of the estimated single-indices on the estimator of main interest is asymptotically negligible even when the estimated single-indices follow cube root asymptotics. As a practical use of this finding, this paper proposes a bootstrap method for conditional moment restrictions that are asymptotically valid in the presence of cube root-converging single-index estimators. Some results from Monte Carlo simulations are presented and discussed.
{"title":"Two-Step Extremum Estimation with Estimated Single-Indices","authors":"Kyungchul Song","doi":"10.2139/ssrn.1360975","DOIUrl":"https://doi.org/10.2139/ssrn.1360975","url":null,"abstract":"This paper studies two-step extremum estimation that involves the first step estimation of nonparametric functions of single-indices. First, this paper finds that under certain regularity conditions for conditional measures, linear functionals of conditional expectations are insensitive to the first order perturbation of the parameters in the conditioning variable. Applying this result to symmetrized nearest neighborhood estimation of the nonparametric functions, this paper shows that the influence of the estimated single-indices on the estimator of main interest is asymptotically negligible even when the estimated single-indices follow cube root asymptotics. As a practical use of this finding, this paper proposes a bootstrap method for conditional moment restrictions that are asymptotically valid in the presence of cube root-converging single-index estimators. Some results from Monte Carlo simulations are presented and discussed.","PeriodicalId":11744,"journal":{"name":"ERN: Nonparametric Methods (Topic)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2009-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84221064","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}
Existing indices of residential segregation are based on a partition of the city in neighborhoods: given a spatial distribution of racial groups, the index measures different segregation levels for different partitions. I propose a spatial approach, which estimates segregation at the individual level and produces the entire spatial distribution of segregation. This method provides different rankings of cities in terms of segregation and new insights on the effect of segregation on socioeconomic outcomes. Using Census data and controlling for endogeneity using instrumental variables, I show that reduced form estimates of the impact of segregation on socioeconomic outcomes are not robust to the spatial approach.
{"title":"Poisson Indices of Segregation","authors":"A. Mele","doi":"10.2139/ssrn.1413246","DOIUrl":"https://doi.org/10.2139/ssrn.1413246","url":null,"abstract":"Existing indices of residential segregation are based on a partition of the city in neighborhoods: given a spatial distribution of racial groups, the index measures different segregation levels for different partitions. I propose a spatial approach, which estimates segregation at the individual level and produces the entire spatial distribution of segregation. This method provides different rankings of cities in terms of segregation and new insights on the effect of segregation on socioeconomic outcomes. Using Census data and controlling for endogeneity using instrumental variables, I show that reduced form estimates of the impact of segregation on socioeconomic outcomes are not robust to the spatial approach.","PeriodicalId":11744,"journal":{"name":"ERN: Nonparametric Methods (Topic)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2009-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84444723","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}
Following Bali and Weinbaum (2005) and Maillet et al. (2008), we present several estimates of volatilities computed with high- and low-frequency data and complement their results using additional measures of risk and several alternative methods for tail-index estimation. The aim is here to confirm previous results regarding the slope of the tail of various risk measure distributions, in order to define the high watermarks of market risks. We also produce synthetic general results concerning the method of estimation of the tail- indexes related to expressions of the L-moments. Based on estimates of tail indexes, backed-out from the high frequency 30' sampled CAC40 French stock Index series on the period 1997-2006, using Non-parametric Generalized Hill, Maximum Likelihood and various kinds of L-moment Methods for the estimation of both a Generalized Extreme Value density and a Generalized Pareto Distribution, we confirm that a heavy-tail density specification of the Log-volatility is not necessary.
{"title":"Extreme Volatilities, Financial Crises and L-Moment Estimations of Tail Indexes","authors":"Bertrand B. Maillet, Jean-Philippe Médecin","doi":"10.2139/ssrn.1288661","DOIUrl":"https://doi.org/10.2139/ssrn.1288661","url":null,"abstract":"Following Bali and Weinbaum (2005) and Maillet et al. (2008), we present several estimates of volatilities computed with high- and low-frequency data and complement their results using additional measures of risk and several alternative methods for tail-index estimation. The aim is here to confirm previous results regarding the slope of the tail of various risk measure distributions, in order to define the high watermarks of market risks. We also produce synthetic general results concerning the method of estimation of the tail- indexes related to expressions of the L-moments. Based on estimates of tail indexes, backed-out from the high frequency 30' sampled CAC40 French stock Index series on the period 1997-2006, using Non-parametric Generalized Hill, Maximum Likelihood and various kinds of L-moment Methods for the estimation of both a Generalized Extreme Value density and a Generalized Pareto Distribution, we confirm that a heavy-tail density specification of the Log-volatility is not necessary.","PeriodicalId":11744,"journal":{"name":"ERN: Nonparametric Methods (Topic)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2009-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82544854","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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