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