Pub Date : 2025-03-01DOI: 10.1016/j.jeconom.2024.105866
David T. Frazier , Eric Renault , Lina Zhang , Xueyan Zhao
We study the impact of weak identification in discrete choice models, and provide insights into the determinants of identification strength in these models. Using these insights, we propose a novel test that can consistently detect weak identification in commonly applied discrete choice models, such as probit, logit, and many of their extensions. Furthermore, we demonstrate that when the null hypothesis of weak identification is rejected, Wald-based inference can be carried out using standard formulas and critical values. A Monte Carlo study compares our proposed testing approach against commonly applied weak identification tests. The results simultaneously demonstrate the good performance of our approach and the fundamental failure of using conventional weak identification tests for linear models in the discrete choice model context. Lastly, we apply our approach in two empirical examples: married women labor force participation, and US food aid and civil conflicts.
{"title":"Weak identification in discrete choice models","authors":"David T. Frazier , Eric Renault , Lina Zhang , Xueyan Zhao","doi":"10.1016/j.jeconom.2024.105866","DOIUrl":"10.1016/j.jeconom.2024.105866","url":null,"abstract":"<div><div>We study the impact of weak identification in discrete choice models, and provide insights into the determinants of identification strength in these models. Using these insights, we propose a novel test that can consistently detect weak identification in commonly applied discrete choice models, such as probit, logit, and many of their extensions. Furthermore, we demonstrate that when the null hypothesis of weak identification is rejected, Wald-based inference can be carried out using standard formulas and critical values. A Monte Carlo study compares our proposed testing approach against commonly applied weak identification tests. The results simultaneously demonstrate the good performance of our approach and the fundamental failure of using conventional weak identification tests for linear models in the discrete choice model context. Lastly, we apply our approach in two empirical examples: married women labor force participation, and US food aid and civil conflicts.</div></div>","PeriodicalId":15629,"journal":{"name":"Journal of Econometrics","volume":"248 ","pages":"Article 105866"},"PeriodicalIF":9.9,"publicationDate":"2025-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143526832","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-03-01DOI: 10.1016/j.jeconom.2024.105867
Veronika Czellar , René Garcia , François Le Grand
We propose an asset pricing model featuring time-varying limited participation in both bond and stock markets and household heterogeneity. Households participate in financial markets with a certain probability that depends on their individual income and on asset market conditions. We use indirect inference to uncover individual asset market participation from individual consumption data and asset prices. Our model very accurately reproduces the proportions of stockholders in the Survey of Consumer Finances over three-year intervals, provides a reasonable estimate of stock market participation costs, and is able to price characteristic-based stock portfolios with the top decile of households identified as stockholders.
{"title":"Uncovering asset market participation from household consumption and income","authors":"Veronika Czellar , René Garcia , François Le Grand","doi":"10.1016/j.jeconom.2024.105867","DOIUrl":"10.1016/j.jeconom.2024.105867","url":null,"abstract":"<div><div>We propose an asset pricing model featuring time-varying limited participation in both bond and stock markets and household heterogeneity. Households participate in financial markets with a certain probability that depends on their individual income and on asset market conditions. We use indirect inference to uncover individual asset market participation from individual consumption data and asset prices. Our model very accurately reproduces the proportions of stockholders in the Survey of Consumer Finances over three-year intervals, provides a reasonable estimate of stock market participation costs, and is able to price characteristic-based stock portfolios with the top decile of households identified as stockholders.</div></div>","PeriodicalId":15629,"journal":{"name":"Journal of Econometrics","volume":"248 ","pages":"Article 105867"},"PeriodicalIF":9.9,"publicationDate":"2025-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143526837","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-03-01DOI: 10.1016/j.jeconom.2024.105915
Marie-Claude Beaulieu , Jean-Marie Dufour , Lynda Khalaf
This paper proposes exact identification-robust confidence sets for the zero-beta rate and ex-post factor prices in asset pricing models. Exploiting the information from the cross-sectional intercept allows us to impose or formally test model-consistent restrictions, including those resulting from traded factors in excess of the zero beta-rate or from return spreads. Analytical projection-based solutions for confidence set outcomes are developed. The proposed procedures are extended to the case of missing factors. Empirical and simulation results with traded and non-traded factors show that model-consistent restrictions and elusive factors can materially affect model fit, identification, inference and temporal constancy of pricing influence.
{"title":"Identification-robust and simultaneous inference in multifactor asset pricing models","authors":"Marie-Claude Beaulieu , Jean-Marie Dufour , Lynda Khalaf","doi":"10.1016/j.jeconom.2024.105915","DOIUrl":"10.1016/j.jeconom.2024.105915","url":null,"abstract":"<div><div>This paper proposes exact identification-robust confidence sets for the zero-beta rate and ex-post factor prices in asset pricing models. Exploiting the information from the cross-sectional intercept allows us to impose or formally test model-consistent restrictions, including those resulting from traded factors in excess of the zero beta-rate or from return spreads. Analytical projection-based solutions for confidence set outcomes are developed. The proposed procedures are extended to the case of missing factors. Empirical and simulation results with traded and non-traded factors show that model-consistent restrictions and elusive factors can materially affect model fit, identification, inference and temporal constancy of pricing influence.</div></div>","PeriodicalId":15629,"journal":{"name":"Journal of Econometrics","volume":"248 ","pages":"Article 105915"},"PeriodicalIF":9.9,"publicationDate":"2025-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143526836","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-03-01DOI: 10.1016/j.jeconom.2024.105821
Firmin Doko Tchatoka , Jean-Marie Dufour
This paper provides new insights on exogeneity tests in linear IV models and their use for estimation, when identification fails or may not be strong. We make two main contributions. First, we show that Durbin–Wu–Hausman (DWH) and Revankar–Hartley (RH) exogeneity tests have correct level asymptotically, even when the first-stage coefficient matrix (which controls identification) is rank-deficient. We provide necessary and sufficient conditions under which these tests are consistent. In particular, we show that test consistency can hold even when identification fails, provided at least one component of the structural parameter vector is identifiable. Second, we study point estimation after estimator (or model) selection, when the outcome of a DWH/RH test determines whether OLS or an IV method is employed in the second-stage. For this purpose, we use (non-local) concepts of asymptotic bias, asymptotic mean squared error (AMSE), and asymptotic relative efficiency (ARE), which remain applicable even when the estimators considered do not have moments (as can happen for 2SLS) or may be inconsistent. We study the asymptotic properties of OLS, 2SLS, and pretest estimators which select OLS or 2SLS based on the outcome of a DWH/RH test. We show that: (i) OLS typically dominates 2SLS estimator asymptotically for MSE across a broad spectrum of cases, including weak identification and moderate endogeneity; (ii) exogeneity-pretest estimators exhibit consistently good performance and asymptotically dominate both OLS and 2SLS. The proposed theoretical findings are documented by Monte Carlo simulations.
{"title":"Exogeneity tests and weak identification in IV regressions: Asymptotic theory and point estimation","authors":"Firmin Doko Tchatoka , Jean-Marie Dufour","doi":"10.1016/j.jeconom.2024.105821","DOIUrl":"10.1016/j.jeconom.2024.105821","url":null,"abstract":"<div><div>This paper provides new insights on exogeneity tests in linear IV models and their use for estimation, when identification fails or may not be strong. We make two main contributions. <em>First</em>, we show that Durbin–Wu–Hausman (DWH) and Revankar–Hartley (RH) exogeneity tests have correct level asymptotically, even when the first-stage coefficient matrix (which controls identification) is rank-deficient. We provide necessary and sufficient conditions under which these tests are consistent. In particular, we show that test consistency can hold even when identification fails, provided <em>at least one</em> component of the structural parameter vector is identifiable. <em>Second</em>, we study point estimation after estimator (or model) selection, when the outcome of a DWH/RH test determines whether OLS or an IV method is employed in the second-stage. For this purpose, we use (<em>non-local</em>) concepts of <em>asymptotic bias</em>, <em>asymptotic mean squared error</em> (AMSE), and <em>asymptotic relative efficiency</em> (ARE), which remain applicable even when the estimators considered do not have moments (as can happen for 2SLS) or may be inconsistent. We study the asymptotic properties of OLS, 2SLS, and pretest estimators which select OLS or 2SLS based on the outcome of a DWH/RH test. We show that: (i) OLS typically dominates 2SLS estimator asymptotically for MSE across a broad spectrum of cases, including weak identification and moderate endogeneity; (ii) exogeneity-pretest estimators exhibit consistently good performance and asymptotically dominate both OLS and 2SLS. The proposed theoretical findings are documented by Monte Carlo simulations.</div></div>","PeriodicalId":15629,"journal":{"name":"Journal of Econometrics","volume":"248 ","pages":"Article 105821"},"PeriodicalIF":9.9,"publicationDate":"2025-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143526833","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-03-01DOI: 10.1016/j.jeconom.2024.105723
Prosper Dovonon , Yves F. Atchadé , Firmin Doko Tchatoka
Moment condition models with mixed identification strength are models that are point identified but with estimating moment functions that are allowed to drift to 0 uniformly over the parameter space. Even though identification fails in the limit, depending on how slow the moment functions vanish, consistent estimation is possible. Existing estimators such as the generalized method of moment (GMM) estimator exhibit a pattern of nonstandard or even heterogeneous rate of convergence that materializes by some parameter directions being estimated at a slower rate than others. This paper derives asymptotic semiparametric efficiency bounds for regular estimators of parameters of these models. We show that GMM estimators are regular and that the so-called two-step GMM estimator – using the inverse of estimating function’s variance as weighting matrix – is semiparametrically efficient as it reaches the minimum variance attainable by regular estimators. This estimator is also asymptotically minimax efficient with respect to a large family of loss functions. Monte Carlo simulations are provided that confirm these results.
{"title":"Efficiency bounds for moment condition models with mixed identification strength","authors":"Prosper Dovonon , Yves F. Atchadé , Firmin Doko Tchatoka","doi":"10.1016/j.jeconom.2024.105723","DOIUrl":"10.1016/j.jeconom.2024.105723","url":null,"abstract":"<div><div>Moment condition models with mixed identification strength are models that are point identified but with estimating moment functions that are allowed to drift to 0 uniformly over the parameter space. Even though identification fails in the limit, depending on how slow the moment functions vanish, consistent estimation is possible. Existing estimators such as the generalized method of moment (GMM) estimator exhibit a pattern of nonstandard or even heterogeneous rate of convergence that materializes by some parameter directions being estimated at a slower rate than others. This paper derives asymptotic semiparametric efficiency bounds for regular estimators of parameters of these models. We show that GMM estimators are regular and that the so-called two-step GMM estimator – using the inverse of estimating function’s variance as weighting matrix – is semiparametrically efficient as it reaches the minimum variance attainable by regular estimators. This estimator is also asymptotically minimax efficient with respect to a large family of loss functions. Monte Carlo simulations are provided that confirm these results.</div></div>","PeriodicalId":15629,"journal":{"name":"Journal of Econometrics","volume":"248 ","pages":"Article 105723"},"PeriodicalIF":9.9,"publicationDate":"2025-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140154621","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-03-01DOI: 10.1016/j.jeconom.2024.105743
E. Andreou , P. Gagliardini , E. Ghysels , M. Rubin
Factor analysis is a widely used tool to summarize high dimensional panel data via a small dimensional set of latent factors. Many applications in finance and macroeconomics, are often focused on observable factors with an economic interpretation. The objective of this paper is to provide a test to answer a question which naturally comes up in discussions regarding latent versus observable factors: do latent and observable factors span the same space? We derive asymptotic properties of a formal test and propose a bootstrap version with improved small sample properties. We find empirical evidence for a small number of factors common between a small number of traditional Fama–French risk factors – or returns on a few stocks (i.e. “magnificent” 5 or 7) – and large panels of US, North American and international portfolio returns.
{"title":"Spanning latent and observable factors","authors":"E. Andreou , P. Gagliardini , E. Ghysels , M. Rubin","doi":"10.1016/j.jeconom.2024.105743","DOIUrl":"10.1016/j.jeconom.2024.105743","url":null,"abstract":"<div><div><span>Factor analysis is a widely used tool to summarize high dimensional panel data via a small dimensional set of latent factors<span>. Many applications in finance<span><span> and macroeconomics, are often focused on observable factors with an economic interpretation. The objective of this paper is to provide a test to answer a question which naturally comes up in discussions regarding latent versus observable factors: do latent and observable factors span the same space? We derive </span>asymptotic properties of a formal test and propose a bootstrap version with improved small sample properties. We find empirical evidence for a small number of factors common between a small number of traditional Fama–French </span></span></span>risk factors – or returns on a few stocks (i.e. “magnificent” 5 or 7) – and large panels of US, North American and international portfolio returns.</div></div>","PeriodicalId":15629,"journal":{"name":"Journal of Econometrics","volume":"248 ","pages":"Article 105743"},"PeriodicalIF":9.9,"publicationDate":"2025-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141144485","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-03-01DOI: 10.1016/j.jeconom.2024.105905
Christian Gourieroux , Joann Jasiak
This paper introduces a local-to-unity/small sigma model for stationary processes with long-range persistence and non-negligible long-run prediction and estimation risks. The model represents a process containing unobserved short and long-run components measured on different time scales. The short-run component is defined in calendar time, while the long-run component evolves in rescaled time with ultra-long units. We develop estimation and long-run prediction methods for time series with multivariate Vector Autoregressive (VAR) short-run components and reveal the impossibility of estimating consistently some of the long-run parameters, which causes significant estimation and prediction risks in the long run. A simulation study and an application to macroeconomic data illustrate the approach.
{"title":"Long-run risk in stationary vector autoregressive models","authors":"Christian Gourieroux , Joann Jasiak","doi":"10.1016/j.jeconom.2024.105905","DOIUrl":"10.1016/j.jeconom.2024.105905","url":null,"abstract":"<div><div>This paper introduces a local-to-unity/small sigma model for stationary processes with long-range persistence and non-negligible long-run prediction and estimation risks. The model represents a process containing unobserved short and long-run components measured on different time scales. The short-run component is defined in calendar time, while the long-run component evolves in rescaled time with ultra-long units. We develop estimation and long-run prediction methods for time series with multivariate Vector Autoregressive (VAR) short-run components and reveal the impossibility of estimating consistently some of the long-run parameters, which causes significant estimation and prediction risks in the long run. A simulation study and an application to macroeconomic data illustrate the approach.</div></div>","PeriodicalId":15629,"journal":{"name":"Journal of Econometrics","volume":"248 ","pages":"Article 105905"},"PeriodicalIF":9.9,"publicationDate":"2025-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143526828","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-03-01DOI: 10.1016/j.jeconom.2024.105728
Frank Kleibergen , Lingwei Kong
We propose identification robust statistics for testing hypotheses on the risk premia in dynamic affine term structure models. We do so using the moment equation specification proposed in Adrian et al. (2013). Statistical inference based on their three-stage estimator requires knowledge of the risk factors’ quality and can be misleading when the ’s are weak, which results when sampling errors are of comparable order of magnitude as the risk factor loadings. We extend the subset (factor) Anderson–Rubin test from Guggenberger et al. (2012) to models with multiple dynamic factors and time-varying risk prices. It provides a computationally tractable manner to conduct identification robust tests on a few risk premia when a larger number is present. We use it to analyze potential identification issues arising in the data from Adrian et al. (2013) for which we show that some factors, though potentially weak, may drive the time variation of risk prices, and weak identification issues are more prominent in multi-factor models.
{"title":"Identification robust inference for the risk premium in term structure models","authors":"Frank Kleibergen , Lingwei Kong","doi":"10.1016/j.jeconom.2024.105728","DOIUrl":"10.1016/j.jeconom.2024.105728","url":null,"abstract":"<div><div>We propose identification robust statistics for testing hypotheses on the risk premia in dynamic affine term structure models. We do so using the moment equation specification proposed in <span><span>Adrian et al. (2013)</span></span>. Statistical inference based on their three-stage estimator requires knowledge of the risk factors’ quality and can be misleading when the <span><math><mi>β</mi></math></span>’s are weak, which results when sampling errors are of comparable order of magnitude as the risk factor loadings. We extend the subset (factor) Anderson–Rubin test from <span><span>Guggenberger et al. (2012)</span></span> to models with multiple dynamic factors and time-varying risk prices. It provides a computationally tractable manner to conduct identification robust tests on a few risk premia when a larger number is present. We use it to analyze potential identification issues arising in the data from <span><span>Adrian et al. (2013)</span></span> for which we show that some factors, though potentially weak, may drive the time variation of risk prices, and weak identification issues are more prominent in multi-factor models.</div></div>","PeriodicalId":15629,"journal":{"name":"Journal of Econometrics","volume":"248 ","pages":"Article 105728"},"PeriodicalIF":9.9,"publicationDate":"2025-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143526835","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-03-01DOI: 10.1016/j.jeconom.2024.105863
Federico M. Bandi , Yinan Su
We model predictive scale-specific cycles. By employing suitable matrix representations, we express the forecast errors of covariance-stationary multivariate time series in terms of conditionally orthonormal scale-specific bases. The representations yield conditionally orthogonal decompositions of these forecast errors. They also provide decompositions of their variances and betas in terms of scale-specific variances and betas capturing predictive variability and co-variability over cycles of alternative lengths without spillovers across cycles. Making use of the proposed representations within the classical family of time-varying conditional volatility models, we document the role of time-varying volatility forecasts in generating orthogonal predictive scale-specific cycles in returns. We conclude by providing suggestive evidence that the conditional variances of the predictive return cycles () may be priced over short-to-medium horizons and () may offer economically-relevant trading signals over these same horizons.
{"title":"Conditional spectral methods","authors":"Federico M. Bandi , Yinan Su","doi":"10.1016/j.jeconom.2024.105863","DOIUrl":"10.1016/j.jeconom.2024.105863","url":null,"abstract":"<div><div>We model predictive scale-specific cycles. By employing suitable matrix representations, we express the forecast errors of covariance-stationary multivariate time series in terms of conditionally orthonormal scale-specific bases. The representations yield conditionally orthogonal decompositions of these forecast errors. They also provide decompositions of their variances and betas in terms of scale-specific variances and betas capturing predictive variability and co-variability over cycles of alternative lengths without spillovers across cycles. Making use of the proposed representations within the classical family of time-varying conditional volatility models, we document the role of time-varying volatility forecasts in generating orthogonal predictive scale-specific cycles in returns. We conclude by providing suggestive evidence that the conditional variances of the predictive return cycles (<span><math><mi>i</mi></math></span>) may be priced over short-to-medium horizons and (<span><math><mrow><mi>i</mi><mi>i</mi></mrow></math></span>) may offer economically-relevant trading signals over these same horizons.</div></div>","PeriodicalId":15629,"journal":{"name":"Journal of Econometrics","volume":"248 ","pages":"Article 105863"},"PeriodicalIF":9.9,"publicationDate":"2025-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143526916","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-03-01DOI: 10.1016/j.jeconom.2024.105918
Christian Bontemps , Jean-Pierre Florens , Nour Meddahi
In this paper, we consider the problem of ecological inference when one observes the conditional distributions of and from aggregate data and attempts to infer the conditional distribution of without observing and in the same sample. First, we show that this problem can be transformed into a linear equation involving operators for which, under suitable regularity assumptions, least squares solutions are available. We then propose the use of the least squares solution with the minimum Hilbert–Schmidt norm, which, in our context, can be structurally interpreted as the solution with minimum dependence between and . Interestingly, in the case where the conditioning variable is discrete and belongs to a finite set, such as the labels of units/groups/cities, the solution of this minimal dependence has a closed form. In the more general case, we use a regularization scheme and show the convergence of our proposed estimator. A numerical evaluation of our procedure is proposed.
{"title":"Functional ecological inference","authors":"Christian Bontemps , Jean-Pierre Florens , Nour Meddahi","doi":"10.1016/j.jeconom.2024.105918","DOIUrl":"10.1016/j.jeconom.2024.105918","url":null,"abstract":"<div><div>In this paper, we consider the problem of ecological inference when one observes the conditional distributions of <span><math><mrow><mi>Y</mi><mo>|</mo><mi>W</mi></mrow></math></span> and <span><math><mrow><mi>Z</mi><mo>|</mo><mi>W</mi></mrow></math></span> from aggregate data and attempts to infer the conditional distribution of <span><math><mrow><mi>Y</mi><mo>|</mo><mi>Z</mi></mrow></math></span> without observing <span><math><mi>Y</mi></math></span> and <span><math><mi>Z</mi></math></span> in the same sample. First, we show that this problem can be transformed into a linear equation involving operators for which, under suitable regularity assumptions, least squares solutions are available. We then propose the use of the least squares solution with the minimum Hilbert–Schmidt norm, which, in our context, can be structurally interpreted as the solution with minimum dependence between <span><math><mi>Y</mi></math></span> and <span><math><mi>Z</mi></math></span>. Interestingly, in the case where the conditioning variable <span><math><mi>W</mi></math></span> is discrete and belongs to a finite set, such as the labels of units/groups/cities, the solution of this minimal dependence has a closed form. In the more general case, we use a regularization scheme and show the convergence of our proposed estimator. A numerical evaluation of our procedure is proposed.</div></div>","PeriodicalId":15629,"journal":{"name":"Journal of Econometrics","volume":"248 ","pages":"Article 105918"},"PeriodicalIF":9.9,"publicationDate":"2025-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143526834","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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