This paper studies variable selection problem in structural equation of a two-stage least squares （2SLS） model in presence of endogeneity which is commonly encountered in empirical economic studies. Model uncertaint...This paper studies variable selection problem in structural equation of a two-stage least squares （2SLS） model in presence of endogeneity which is commonly encountered in empirical economic studies. Model uncertainty and variable selection in the structural equation is an important issue as described in Andrews and Lu （2001） and Caner （2009）. The authors propose an adaptive Lasso 2SLS estimator for linear structural equation with endogeneity and show that it enjoys the oracle properties, i.e., the consistency in both estimation and model selection. In Monte Carlo simulations, the authors demonstrate that the proposed estimator has smaller bias and MSE compared with the bridge-type GMM estimator （Caner, 2009）. In a case study, the authors revisit the classic returns to education problem （Angrist and Krueger, 1991） using the China Population census data. The authors find that the education level not only has strong effects on income but also shows heterogeneity in different age cohorts.展开更多
The notion of broken k-diamond partitions was introduced by Andrews and Paule.Let△k(n)denote the number of broken k-diamond partitions of n.Andrews and Paule also posed three conjectures on the congruences of△2(n)mo...The notion of broken k-diamond partitions was introduced by Andrews and Paule.Let△k(n)denote the number of broken k-diamond partitions of n.Andrews and Paule also posed three conjectures on the congruences of△2(n)modulo 2,5 and 25.Hirschhorn and Sellers proved the conjectures for modulo 2,and Chan proved the two cases of modulo 5.For the case of modulo 3,Radu and Sellers obtained an infinite family of congruences for△2(n).In this paper,we obtain two infinite families of congruences for△2(n)modulo 3 based on a formula of Radu and Sellers,a 3-dissection formula of the generating function of triangular number due to Berndt,and the properties of the U-operator,the V-operator,the Hecke operator and the Hecke eigenform.For example,we find that△2(243n+142)≡△2(243n+223)≡0(mod 3).The infinite family of Radu and Sellers and the two infinite families derived in this paper have two congruences in common,namely,△2(27n+16)≡△2(27n+25)≡0(mod 3).展开更多
基金Fan’s research was supported by the National Natural Science Foundation of China under Grant No.71671149the Fundamental Research Funds for the Central Universities under Grant No.20720171042+1 种基金the Natural Science Foundation of Fujian Province of China under Grant No.2016J01340Zhong’s research was supported by the National Natural Science Foundation of China under Grant Nos.11671334,11301435,and 11401497
文摘This paper studies variable selection problem in structural equation of a two-stage least squares （2SLS） model in presence of endogeneity which is commonly encountered in empirical economic studies. Model uncertainty and variable selection in the structural equation is an important issue as described in Andrews and Lu （2001） and Caner （2009）. The authors propose an adaptive Lasso 2SLS estimator for linear structural equation with endogeneity and show that it enjoys the oracle properties, i.e., the consistency in both estimation and model selection. In Monte Carlo simulations, the authors demonstrate that the proposed estimator has smaller bias and MSE compared with the bridge-type GMM estimator （Caner, 2009）. In a case study, the authors revisit the classic returns to education problem （Angrist and Krueger, 1991） using the China Population census data. The authors find that the education level not only has strong effects on income but also shows heterogeneity in different age cohorts.
基金supported by National Basic Research Program of China (973 Project) (Grant No. 2011CB808003)the PCSIRT Project of the Ministry of EducationNational Natural Science Foundation of China (Grant No. 11231004)
文摘The notion of broken k-diamond partitions was introduced by Andrews and Paule.Let△k(n)denote the number of broken k-diamond partitions of n.Andrews and Paule also posed three conjectures on the congruences of△2(n)modulo 2,5 and 25.Hirschhorn and Sellers proved the conjectures for modulo 2,and Chan proved the two cases of modulo 5.For the case of modulo 3,Radu and Sellers obtained an infinite family of congruences for△2(n).In this paper,we obtain two infinite families of congruences for△2(n)modulo 3 based on a formula of Radu and Sellers,a 3-dissection formula of the generating function of triangular number due to Berndt,and the properties of the U-operator,the V-operator,the Hecke operator and the Hecke eigenform.For example,we find that△2(243n+142)≡△2(243n+223)≡0(mod 3).The infinite family of Radu and Sellers and the two infinite families derived in this paper have two congruences in common,namely,△2(27n+16)≡△2(27n+25)≡0(mod 3).
