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# 经济代写|计量经济学代写Introduction to Econometrics代考|ECON771 Continuous Variables

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## 经济代写|计量经济学代写Introduction to Econometrics代考|Continuous Variables

In the previous sections, we implicitly assumed that the conditioning variables are discrete. However, many conditioning variables are continuous. In this section, we take up this case and assume that the variables $(y, \boldsymbol{x})$ are continuously distributed with a joint density function $f(y, \boldsymbol{x})$.

As an example, take $y=\log ($ wage $)$ and $x=$ experience, the number of years of potential labor market experience ${ }^9$. The contours of their joint density are plotted on the left side of Figure $2.4$ for the population of white men with 12 years of education.

(a) Joint Density of $\log$ (wage) and experience
(b) Conditional Density of $\log$ (wage) given experience
Figure 2.4: White Men with High School Degree
Given the joint density $f(y, \boldsymbol{x})$ the variable $\boldsymbol{x}$ has the marginal density
$$f_{\boldsymbol{x}}(\boldsymbol{x})=\int_{-\infty}^{\infty} f(y, \boldsymbol{x}) d y .$$
For any $\boldsymbol{x}$ such that $f_{\boldsymbol{x}}(\boldsymbol{x})>0$ the conditional density of $y$ given $\boldsymbol{x}$ is defined as
$$f_{y \mid x}(y \mid \boldsymbol{x})=\frac{f(y, \boldsymbol{x})}{f_{\boldsymbol{x}}(\boldsymbol{x})}$$

## 经济代写|计量经济学代写Introduction to Econometrics代考|Law of Iterated Expectations

An extremely useful tool from probability theory is the law of iterated expectations. An important special case is the known as the Simple Law.
Theorem 2.1 Simple Law of Iterated Expectations If $\mathbb{E}|y|<\infty$ then for any random vector $\boldsymbol{x}$,
$$\mathbb{E}[\mathbb{E}[y \mid \boldsymbol{x}]]=\mathbb{E}[y] .$$
The simple law states that the expectation of the conditional expectation is the unconditional expectation. In other words the average of the conditional averages is the unconditional average. When $\boldsymbol{x}$ is discrete
$$\mathbb{E}[\mathbb{E}[y \mid \boldsymbol{x}]]=\sum_{j=1}^{\infty} \mathbb{E}\left[y \mid \boldsymbol{x}=\boldsymbol{x}j\right] \mathbb{P}\left[\boldsymbol{x}=\boldsymbol{x}_j\right]$$ and when $\boldsymbol{x}$ is continuous $$\mathbb{E}[\mathbb{E}[y \mid \boldsymbol{x}]]=\int{\mathbb{R}^k} \mathbb{E}[y \mid \boldsymbol{x}] f_{\boldsymbol{x}}(\boldsymbol{x}) d \boldsymbol{x} .$$
Going back to our investigation of average log wages for men and women, the simple law states that
\begin{aligned} & \mathbb{E}[\log (\text { wage }) \mid \text { gender }=\text { man }] \mathbb{P}[\text { gender }=\text { man }] \ & +\mathbb{E}[\log (\text { wage }) \mid \text { gender }=\text { woman }] \mathbb{P}[\text { gender }=\text { woman }] \ & =\mathbb{E}[\log (\text { wage })] \end{aligned}

Or numerically,
$$3.05 \times 0.57+2.81 \times 0.43=2.95 \text {. }$$
The general law of iterated expectations allows two sets of conditioning variables.
Theorem 2.2 Law of Iterated Expectations If $\mathbb{E}|y|<\infty$ then for any random vectors $\boldsymbol{x}_1$ and $\boldsymbol{x}_2$,
$$\mathbb{E}\left[\mathbb{E}\left[y \mid \boldsymbol{x}_1, \boldsymbol{x}_2\right] \mid \boldsymbol{x}_1\right]=\mathbb{E}\left[y \mid \boldsymbol{x}_1\right] .$$

## 经济代写|计量经济学代写Introduction to Econometrics代考|Continuous Variables

(a) 联合密度 $\log$ (工缌) 和经验
(b) 条件密度 $\log$ (工资) 给定经验

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

$$f_{y \mid x}(y \mid \boldsymbol{x})=\frac{f(y, \boldsymbol{x})}{f_{\boldsymbol{x}}(\boldsymbol{x})}$$

## 经济代写|计量经济学代写Introduction to Econometrics代考|Law of Iterated Expectations

$$\mathbb{E}[\mathbb{E}[y \mid \boldsymbol{x}]]=\mathbb{E}[y] .$$

$$\mathbb{E}[\mathbb{E}[y \mid \boldsymbol{x}]]=\sum_{j=1}^{\infty} \mathbb{E}[y \mid \boldsymbol{x}=\boldsymbol{x} j] \mathbb{P}\left[\boldsymbol{x}=\boldsymbol{x}j\right]$$ 什么时候 $\boldsymbol{x}$ 是连续的 $$\mathbb{E}[\mathbb{E}[y \mid \boldsymbol{x}]]=\int \mathbb{R}^k \mathbb{E}[y \mid \boldsymbol{x}] f{\boldsymbol{x}}(\boldsymbol{x}) d \boldsymbol{x}$$

$\mathbb{E}[\log ($ wage $) \mid$ gender $=\operatorname{man}] \mathbb{P}[$ gender $=$ man $] \quad+\mathbb{E}[\log ($ wage $) \mid$ gender $=$ woman $] \mathbb{P}[$ gender $=$ woman $]=\mathbb{E}[\log ($ wage $)]$

$$3.05 \times 0.57+2.81 \times 0.43=2.95 .$$

$$\mathbb{E}\left[\mathbb{E}\left[y \mid \boldsymbol{x}_1, \boldsymbol{x}_2\right] \mid \boldsymbol{x}_1\right]=\mathbb{E}\left[y \mid \boldsymbol{x}_1\right]$$

## MATLAB代写

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