Conditional Mean and Regression Analysis where the operator E(X)is the expectation with respect to fyx(X), and the operator E()is the expectation with respect to the marginal density fx()of X.The law of iterated expectation is also called the law of total expectation inin probability theory.This completes the proof. ADVANCED ECONOMETRICS General Regression Analysis May12,2021 26
ADVANCED ECONOMETRICS General Regression Analysis May 12, 2021 26 Conditional Probability Distribution Conditional Mean and Regression Analysis
Conditional Mean and Regression Analysis Questions: How to interpret the law of iterated expectations from an economic per- spective? ADVANCED ECONOMETRICS General Regression Analysis May12,2021 27
ADVANCED ECONOMETRICS General Regression Analysis May 12, 2021 27 Conditional Mean and Regression Analysis Questions:
Conditional Mean and Regression Analysis Example 4 Suppose Y is wage,and X is a gender dummy variable,taking value 1 if an employee is female and value 0 if an employee is male.Then E(YX =1)=average wage of a female worker, E(YX -0)=average wage of a male worker, and the overall average wage E(Y)）=E[E(YIX)] =P(X=1)E(YX=1)+P(X=0)E(YX=0), where P(X =1)is the proportion of female employees in the labor force, and P(X =0)is the proportion of the male employees in the labor force. The use of LIE here thus provides some insight into the income distribution between genders. ADVANCED ECONOMETRICS General Regression Analysis May12,2021 28
ADVANCED ECONOMETRICS General Regression Analysis May 12, 2021 28 Conditional Mean and Regression Analysis Example 4
Conditional Mean and Regression Analysis Example 5 Suppose Y is an asset return and we have two information sets:I and J, where ICJ so that all information in I is contained in J but J contains some extra information.Then we have a conditional version of the law of iterated expectations says that E(YII)=E[E(YIJ)川I or equivalently E{[Y-E(YIJ)川I}=0. ADVANCED ECONOMETRICS General Regression Analysis May12,2021 29
ADVANCED ECONOMETRICS General Regression Analysis May 12, 2021 29 Conditional Probability Distribution Conditional Mean and Regression Analysis Example 5
Conditional Mean and Regression Analysis where Y-E(YJ)is the prediction error using the superior information set J.The conditional LIE says that one cannot use a limited informa- tion I to predict the prediction error one would make if one had superior information J.In other words,the larger information set J has supe- rior forecasting power than the information set I.See Campbell,Lo and MacKinlay (1997,p.23)for more discussion. ADVANCED ECONOMETRICS General Regression Analysis May12,2021 30
ADVANCED ECONOMETRICS General Regression Analysis May 12, 2021 30 Conditional Probability Distribution Conditional Mean and Regression Analysis