Conditional Mean and Regression Analysis Definition [Regression Function]: The conditional mean E(YX)is called a re-gression function of Y on X. ● Many economic theories can be characterized by the conditional mean E(YX)of Y given X.Most,though not all,of dynamic economic theories and or dynamic optimization models,such as rational expectations,effi- cient markets hypothesis,expectations hypothesis,and optimal dynamic asset pricing,have important implications on (and only on)the conditional mean of underlying economic variables given the information available to economic agents(e.g.,Cochrane 2001,Sargent and Ljungqvist 2002) ADVANCED ECONOMETRICS General Regression Analysis May12,2021 16
ADVANCED ECONOMETRICS General Regression Analysis May 12, 2021 16 Conditional Mean and Regression Analysis Definition
Conditional Mean and Regression Analysis For example,the classical efficient market hypothesis states that the ex- pected asset return given the information available,is zero,or at most,is constant over time;the optimal dynamic asset pricing theory implies that the expectation of the pricing error given the information available is zero for each asset (Cochrane 2001). The term "regression"was coined by Francis Galton (1877,1885)in the nineteenth century to describe a biological phenomenon.The phenomenon was that the heights of descendants of tall ancestors tend to regress down towards a normal average,a phenomenon also known as regression toward the mean. ADVANCED ECONOMETRICS General Regression Analysis May12,2021 17
ADVANCED ECONOMETRICS General Regression Analysis May 12, 2021 17 Conditional Mean and Regression Analysis
Conditional Mean and Regression Analysis For Galton,regression had only this biological meaning,but his work was later extended by Udny Yule (1897)and Karl Pearson (1903)to a more general statistical context.In the work of Yule and Pearson,the joint distribution of the response and explanatory variables is assumed to be Gaussian.This assumption was weakened by R.A.Fisher (1922,1925) who assumed that the conditional distribution of the response variable is Gaussian,but the joint distribution need not be.In this respect,Fisher's assumption is closer to Gauss'(1821)formulation. ADVANCED ECONOMETRICS General Regression Analysis May12,2021 18
ADVANCED ECONOMETRICS General Regression Analysis May 12, 2021 18 Conditional Mean and Regression Analysis
Conditional Mean and Regression Analysis Most commonly,regression analysis is concerned with the conditional ex- pectation of the dependent variable given the explanatory variables,that is,the average value of the dependent variable when the explanatory vari- ables are fixed. ADVANCED ECONOMETRICS General Regression Analysis May12,2021 19
ADVANCED ECONOMETRICS General Regression Analysis May 12, 2021 19 Conditional Probability Distribution Conditional Mean and Regression Analysis
Conditional Mean and Regression Analysis Less commonly,the focus can also be on a quantile,or other location parameter of the conditional distribution of the dependent variable given the explanatory variables.For the case of a conditional quantitle,one usually calls it the quantitle regression function.In all cases,a function of the explanatory variables called the regression function. Conditional mean analysis or regression analysis is one of the most pop- ular statistical methods in econometrics,and has wide applications in economics. ADVANCED ECONOMETRICS General Regression Analysis May12,2021 20
ADVANCED ECONOMETRICS General Regression Analysis May 12, 2021 20 Conditional Mean and Regression Analysis