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CHAPTER 2 GENERAL REGRESSION ANALYSIS 一般回归分析

2.1 Conditional Probability Distribution

假设$Z=\left(Y, X^{\prime}\right)^{\prime}$是一个随机向量，$E\left(Y^{2}\right)<\infty$, $Y$是一个标量，$X$是一个$(k+1)\times1$的向量。 $f(x,y)$为联合概率密度分布（probability density function, PDF）,则对于$X$的边缘概率分布为：

$$f_{X}(x)=\int_{-\infty}^{\infty} f(x, y) d y$$

给定$X$,对于$Y$的条件概率密度分布为：

$$f_{Y | X}(y | x)=\frac{f(x, y)}{f_{X}(x)}$$

则条件概率分布能够完整表示出$Y$依赖$X$的程度。根据$f_{Y | X}(y | x)$可以计算以下：

• The conditional mean

$$\begin{aligned} E(Y | x) & \equiv E(Y | X=x) \\ &=\int_{-\infty}^{\infty} y f_{Y | X}(y | x) d y \end{aligned}$$

• the conditional variance

$$\begin{aligned} \operatorname{var}(Y | x) & \equiv \operatorname{var}(Y | X=x) \\ &=\int_{-\infty}^{\infty}[y-E(Y | x)]^{2} f_{Y | X}(y | x) d y \\ &=E\left(Y^{2} | x\right)-[E(Y | x)]^{2} \end{aligned}$$

• the conditional skewness （刻画Y分布是否对称）

$$S(Y | x)=\frac{E\left[(Y-E(Y | x))^{3} | x\right]}{[\operatorname{var}(Y | x)]^{3 / 2}}$$

• the conditional kurtosis (刻画有没有heavy tail)

$$K(Y | x)=\frac{E\left[(Y-E(Y | x))^{4} | x\right]}{[\operatorname{var}(Y | x)]^{2}}$$

• the $\alpha$-conditional quantile $Q(x,\alpha)$ (分位数):

$$P[Y \leq Q(X, \alpha) | X=x]=\alpha \in(0,1)$$

Note that when $\alpha=0.5, Q(x, 0.5)$is the conditional median, which is the cutoff point or threshold that divides the population into two equal halves, conditional on $X = x$.

A mathematical model (i.e., an assumed functional form with a finite number of unknown parameters) for a conditional moment is called an econometric model for that conditional moment.

2.2 Regression Analysis

Deffnition 2.1 [Regression Function]: The conditional mean $E(Y|X)$ is called a regression function of $Y$ on $X$:

$$E(y|X=x) = \int_{- \infty}^{\infty}yf_{Y|X}(y|x)dy = g(x)\text{(只是关于$x$的函数)}$$

Lemma 2.1 :$E[E(Y|X)=E(Y)]$

Definition 2.2[MSE]: Suppose function $g(X)$ is used to predict $Y$. Then the mean squared error of function $g(X)$ is defined as:

$$MSE(g)=E[Y-g(X)]^2$$