中国科学院数学与系统科学研究院：《高级计量经济学》课程教学资源（课件讲稿）第二章 General Regression Analysis

中国科学院数学与系统科学研究院 Academy of Mathematics and Systems Science Chinese Academy of Sciences General Regression Analysis Professor Yongmiao Hong March 23,2021

General Regression Analysis Professor Yongmiao Hong March 23, 2021

CONTENTS 2.1 Conditional Probability Distribution 2.2 Conditional Mean and Regression Analysis 2.3 Linear Regression Modelling 2.4 Correct Model Specification for Conditional Mean 2.5 Conclusion ADVANCED ECONOMETRICS General Regression Analysis May12,2021 2

ADVANCED ECONOMETRICS General Regression Analysis May 12, 2021 2 2.1 Conditional Probability Distribution 2.2 Conditional Mean and Regression Analysis 2.3 Linear Regression Modelling 2.4 Correct Model Specification for Conditional Mean 2.5 Conclusion CONTENTS

Conditional Probability Distribution Some Notations: 。 Throughout this book,we use the following notational conventions:capital letters (e.g.,Y)denote random variables or random vectors,lower case letters (e.g.,y)denote realizations of random variables. We assume that 2=(Y,X')'is a random vector with E(Y2)<oo,where Y is a scalar,X is a (k+1)x 1 vector of variables with its first component being a constant,and X'denotes the transpose of X. ADVANCED ECONOMETRICS General Regression Analysis May12,2021 3

ADVANCED ECONOMETRICS General Regression Analysis May 12, 2021 3 Some Notations: Conditional Probability Distribution

Conditional Probability Distribution .Statistically speaking,the relationship between two random variables or vectors X (e.g.,oil price change)and Y (e.g.,economic growth)can be characterized by their joint distribution function. 忆50 22 139 0.83 0.56 A2066 0.28 20.00 10 0.2 0.0 0.6 0204060.8 0.4Y 0.2 100.0 ADVANCED ECONOMETRICS General Regression Analysis May12,2021 4

ADVANCED ECONOMETRICS General Regression Analysis May 12, 2021 4 Conditional Probability Distribution

Conditional Probability Distribution Suppose (X,Y)'are continuous random vectors,and the joint probability density function (pdf)of (X',Y)'is f(x,y).Then the marginal pdf of X is Jx(z)-f(z,u)dv, and the conditional pdf of Y given X =x is rx()= f(x,y) fx(x) provided fx()>0.The conditional pdf fyx(y)completely describes how Y depends on X.In other words,it characterizes a predictive rela- tionship of Y using X. ADVANCED ECONOMETRICS General Regression Analysis May12,2021

ADVANCED ECONOMETRICS General Regression Analysis May 12, 2021 5 Conditional Probability Distribution

Conditional Probability Distribution With this conditional pdf fyx(y),we can compute the following quan- tities: 。The conditional mean E(Yx)三E(YX=x) yfy x(ylx)dy ADVANCED ECONOMETRICS General Regression Analysis May12,2021 6

ADVANCED ECONOMETRICS General Regression Analysis May 12, 2021 6 Conditional Probability Distribution

Conditional Probability Distribution The conditional variance var(Y|x)≡var(YX=c) =[-E(Yk)"fyix(ulkz)dy ●X● =E(Y21x)-[E(Yx)]2; The conditional skewness S(Yx)= E(Y-E(Y x)3x [var(Yx)3/2 ADVANCED ECONOMETRICS General Regression Analysis May12,2021 7

ADVANCED ECONOMETRICS General Regression Analysis May 12, 2021 7 Conditional Probability Distribution

Conditional Probability Distribution The conditional kurtosis K(Yx)= E[(Y-E(Y )) [var(Y)]2 The a-conditional quantile Q(x,a),which is determined by the following equation: P[Y≤Q(X,a)川X=]=a∈(0,1). Note that when a =,Q(,)is the conditional median,which is the cutoff point or threshold that divides the population into two equal halves, conditional on X =x. ADVANCED ECONOMETRICS General Regression Analysis May12,2021 8

ADVANCED ECONOMETRICS General Regression Analysis May 12, 2021 8 Conditional Probability Distribution Some Notations:

Conditional Probability Distribution The class of conditional moments is a summary characterization of the con- ditional distribution fyx(y).A mathematical model for a conditional moment is called an econometric model for that conditional moment. Questions: Which moment to model and use in practice? ADVANCED ECONOMETRICS General Regression Analysis May12,2021 9

ADVANCED ECONOMETRICS General Regression Analysis May 12, 2021 9 Conditional Probability Distribution  Questions:

Conditional Probability Distribution It depends on economic applications. For some applications,we only need to model the first conditional moment,namely the conditional mean.For example,asset pricing aims at explaining excess asset returns by systematic risk factors.An asset pricing model is essentially a model for the conditional mean of asset returns on risk factors. To be continued ADVANCED ECONOMETRICS General Regression Analysis May12,2021 10

ADVANCED ECONOMETRICS General Regression Analysis May 12, 2021 10 Conditional Probability Distribution To be continued