分层分位模拟——理论、方法及应用(英文版)

Part I QUANTILE REGRESSION MODELLING Chapter 1 LINEAR QUANTILE REGRESSION 1.1 Education: Mathematical Achievements Family background factor can be a very important part of a person’s life. One ofthe main interests of this chapter is to investigate whether the family backgroundfactors alter performance on mathematical achievement of the stronger studentsthe same way that weaker students are affected. Using large sample of 2000， 2001and 2002 mathematics participation in Alberta， Canada， such questions have beeninvestigated by means of quantile regression approach. The findings suggest thatthere may be differential family-background-factor effects at different points in theconditional distribution of mathematical achievements. 1.1.1 Introduction Children’s mathematical achievements have long been a concern of society. Mastering mathematics has become more important than ever. Previous research indicates thata senior high school student with a strong grasp of mathematics has an advantage inacademics and in the job market， i.e.， mathematical achievement is a key to collegeentrance and success in the labor force. For several decades a number of studies have been focus on gathering andinvestigating information from many variables with effects on the mathematicalachievement. The social， economic， and cultural factors that are either in favorof or not conducive to children are not well understood. Among our main researchinterests here are the variables including family background factors， such as numberof parents， number of siblings， mother’s socioeconomic status， father’s socioeconomicstatus， gender， immigrants， language problem， native and minority， etc. As for gender difference in learning mathematics， evidence shows that femalesare not likely to believe that mathematics has utility in their lives (Fennema andSherman， 1978). They see mathematics as unconnected to a relationship model ofthinking. Even if females continue to take mathematical courses， they are apt tofind that they themselves do not like these courses. However， liking a subject is keyto succeeding at it (Lockhead et al.， 1985). Some researches on immigrants’ school performance suggest that their performanceis above averages (e.g.， Rumbaut， 1996; Viadero， 1997; Lapin， 1998). Whilethere is also evidence that immigrant children， especially Hispanics and others withimpoverished background， suffer poor academic achievement and lower educationalattainment (e.g.， McPartland， 1998; Vernez and Abrahamse， 1996). Also， morerecent studies of immigrant children’s academic achievement provide some insightsfor understanding the variation among immigrant children’s academic achievement.For example， Hao and Portes(1998) used the concept social capital to explainimmigrant children’s academic performance. It’s well known that language problem limits immigrant children’s learningon key subject areas such as mathematics and science. Living in socially andlinguistically isolated communities， poor immigrant children can hardly improvetheir new language skills and the language barriers persist over the school years.On the other hand， bilingual proficiency， defined as the mastery of both the mothertongue and a new language， is found to be a strength for immigrant children’scognitive growth (e.g.， Hao and Portes， 1998). Several authors have recognized that minorities may see mathematics as a Whitedomain， are less likely than Whites to understand its future value， and are negativelyinfluenced by the school staff’s attitudes toward them and their work (Mathews，1983). Educational reform advocated by the politicians and policy makers hasbeen performed in enhancing minority mathematical achievement including gooddiscipline and attendance， small class size， placement in advanced tracks， andmaterials that affirm the important role of minorities in mathematics (Mathews， 1983; Taylor， 1983). Generally speaking， studies mentioned above have primarily dependent onclassical mean regression methods such as ordinary least squares(OLS) or instrumentalvariables (IV). These methodology may miss crucial points such as howfamily background factors affect mathematical achievement differently at differentquantiles of the conditional test score distribution. What is worse， these approachescannot be used to characterize the entire conditional distribution of mathematicalachievement given high-dimensional covariates (family background factors) and theestimated coefficient vector (marginal effects) is not robust to outlier observation onmathematical achievement. Fortunately， the drawbacks mentioned above can be overcome by combinganother statistical method called quantile regression(QR)， which was proposed byKoenker and Bassett (1978) and has become a comprehensive approach to the useof linear and nonlinear response models for conditional quantile functions. Roughlyspeaking， QR， which is based on minimizing “check function” r