### 经济代写|计量经济学作业代写Econometrics代考|Conditional Expectation Function

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## 经济代写|计量经济学作业代写Econometrics代考|Conditional Expectation Function

An important determinant of wage levels is education. In many empirical studies economists measure educational attainment by the number of years ${ }^{8}$ of schooling. We will write this variable as education.

The conditional mean of log wages given gender, race, and education is a single number for each

category. For example
$$\mathbb{E}[\log (\text { wage }) \mid \text { gender }=\text { man, } \text { race }=\text { white, education }=12]=2.84 .$$
We display in Figure $2.3$ the conditional means of $\log ($ wage $)$ for white men and white women as a function of education. The plot is quite revealing. We see that the conditional mean is increasing in years of education, but at a different rate for schooling levels above and below nine years. Another striking feature of Figure $2.3$ is that the gap between men and women is roughly constant for all education levels. As the variables are measured in logs this implies a constant average percentage gap between men and women regardless of educational attainment.

In many cases it is convenient to simplify the notation by writing variables using single characters, typically $y, x$ and/or $z$. It is conventional in econometrics to denote the dependent variable (e.g. $\log ($ wage) ) by the letter $y$, a conditioning variable (such as gender) by the letter $x$, and multiple conditioning variables (such as race, education and gender) by the subscripted letters $x_{1}, x_{2}, \ldots, x_{k}$.
Conditional expectations can be written with the generic notation
$$\mathbb{E}\left[y \mid x_{1}, x_{2}, \ldots, x_{k}\right]=m\left(x_{1}, x_{2}, \ldots, x_{k}\right)$$
We call this the conditional expectation function (CEF). The CEF is a function of $\left(x_{1}, x_{2}, \ldots, x_{k}\right)$ as it varies with the variables. For example, the conditional expectation of $y=\log ($ wage $)$ given $\left(x_{1}, x_{2}\right)=($ gender, race) is given by the six entries of Table??. The CEF is a function of (gender, race) as it varies across the entries.
For greater compactness, we will typically write the conditioning variables as a vector in $\mathbb{R}^{k}$ :
$$\boldsymbol{x}=\left(\begin{array}{c} x_{1} \ x_{2} \ \vdots \ x_{k} \end{array}\right)$$

Here we follow the convention of using lower case bold italics $\boldsymbol{x}$ to denote a vector. Given this notation, the CEF can be compactly written as
$$\mathbb{E}[y \mid \boldsymbol{x}]=m(\boldsymbol{x}) \text {. }$$
The CEFE $[y \mid \boldsymbol{x}]$ is a random variable as it is a function of the random variable $\boldsymbol{x}$. It is also sometimes useful to view the CEF as a function of $\boldsymbol{x}$. In this case we can write $m(\boldsymbol{u})=\mathbb{E}[y \mid \boldsymbol{x}=\boldsymbol{u}]$, which is a function of the argument $\boldsymbol{u}$. The expression $\mathbb{E}|y| \boldsymbol{x}=\boldsymbol{u} \mid$ is the conditional expectation of $y$, given that we know that the random variable $\boldsymbol{x}$ equals the specific value $\boldsymbol{u}$. However, sometimes in econometrics we take a notational shortcut and use $\mathbb{E}[y \mid x]$ to refer to this function. Hopefully, the use of $E[y \mid x]$ should be apparent from the context.

## 经济代写|计量经济学作业代写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.

Given the joint density $f(y, \boldsymbol{x})$ the variable $\boldsymbol{x}$ has the marginal density
$$f_{x}(\boldsymbol{x})=\int_{-\infty}^{\infty} f\left(y_{1} \boldsymbol{x}\right) d y$$
For any $\boldsymbol{x}$ such that $f_{x}(\boldsymbol{x})>0$ the conditional density of $y$ given $\boldsymbol{x}$ is defined as
$$f_{y \backslash x}(y \mid x)=\frac{f(y, \boldsymbol{x})}{f_{x}(\boldsymbol{x})} .$$

The conditional density is a (renormalized) slice of the joint density $f(y, x)$ holding $\boldsymbol{x}$ fixed. The slice is renormalized (divided by $f_{x}(\boldsymbol{x})$ so that it integrates to one and is thus a density.) We can visualize this by slicing the joint density function at a specific value of $\boldsymbol{x}$ parallel with the $y$-axis. For example, take the density contours on the left side of Figure $2.4$ and slice through the contour plot at a specific value of experience, and then renormalize the slice so that it is a proper density. This gives us the conditional density of $\log ($ wage $)$ for white men with 12 years of education and this level of experience. We do this for four levels of experience $(5,10,25$, and 40 years), and plot these densities on the right side of Figure 2.4. We can see that the distribution of wages shifts to the right and becomes more diffuse as experience increases from 5 to 10 years, and from 10 to 25 years, but there is little change from 25 to 40 years experience.
The CEF of $y$ given $x$ is the mean of the conditional density (2.4)
$$m(\boldsymbol{x})=\mathbb{E}[y \mid \boldsymbol{x}]=\int_{-\infty}^{\infty} y f_{y \mid x}(y \mid x) d y .$$
Intuitively, $m(\boldsymbol{x})$ is the mean of $y$ for the idealized subpopulation where the conditioning variables are fixed at $\boldsymbol{x}$. This is idealized since $\boldsymbol{x}$ is continuously distributed so this subpopulation is infinitely small.
This definition (2.5) is appropriate when the conditional density (2.4) is well defined. However, the conditional mean $m(\boldsymbol{x})$ exists quite generally. In Theorem $2.13$ in Section $2.31$ we show that $m(\boldsymbol{x})$ exists solong as $\mathbb{E}|y|<\infty$.

In Figure $2.4$ the CEF of log(wage) given experience is plotted as the solid line. We can see that the CEF is a smooth but nonlinear function. The CEF is initially increasing in experience, flattens out around experience $=30$, and then decreases for high levels of experience.

## 经济代写|计量经济学作业代写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 $E|y|<\infty$ then for any random vector $\boldsymbol{x}$,
$$\mathbb{E}[\mathbb{E}[y \mid x]]=\mathbb{E}[y]$$
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
$$\mathbb{E}[\mathbb{E}[y \mid x]]=\sum_{j=1}^{\infty} \mathbb{E}\left[y \mid x=x_{j}\right] \mathbb{P}\left[x=x_{j}\right]$$
and when $\boldsymbol{x}$ is continuous
$$\mathbb{E}[\mathbb{E}[y \mid \boldsymbol{x}]]=\int_{\mathrm{x}^{k}} \mathbb{E}[y \mid \boldsymbol{x}] f_{x}(\boldsymbol{x}) d \boldsymbol{x} \text {. }$$
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,
$$3.05 \times 0.57+2.81 \times 0.43=2.95 .$$
The general law of iterated expectations allows two sets of conditioning variables.

## 经济代写|计量经济学作业代写Econometrics代考|Conditional Expectation Function

X=(X1 X2 ⋮ Xķ)

CEFE[是∣X]是随机变量，因为它是随机变量的函数X. 有时将 CEF 视为X. 在这种情况下，我们可以写米(在)=和[是∣X=在]，它是参数的函数在. 表达方式和|是|X=在∣是条件期望是，假设我们知道随机变量X等于特定值在. 然而，有时在计量经济学中，我们采用符号捷径并使用和[是∣X]参考这个功能。希望使用和[是∣X]从上下文中应该是显而易见的。

## 经济代写|计量经济学作业代写Econometrics代考|Continuous Variables

FX(X)=∫−∞∞F(是1X)d是

F是∖X(是∣X)=F(是,X)FX(X).

## 经济代写|计量经济学作业代写Econometrics代考|Law of Iterated Expectations

3.05×0.57+2.81×0.43=2.95.

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