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# 经济代写|计量经济学代写Introduction to Econometrics代考|ECON345 Conditional Expectation

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## 经济代写|计量经济学代写Introduction to Econometrics代考|Conditional Expectation

We saw in Figure 2.1(b) the density of log wages. Is this distribution the same for all workers, or does the wage distribution vary across subpopulations? To answer this question, we can compare wage distributions for different groups – for example, men and women. To investigate, we plot in Figure $2.2$ (a) the densities of log wages for U.S. men and women. We can see that the two wage densities take similar shapes but the density for men is somewhat shifted to the right.

The values $3.05$ and $2.81$ are the mean log wages in the subpopulations of men and women workers. They are called the conditional expectation (or conditional mean) of log wages given gender. We can write their specific values as
$$\mathbb{E}[\log (\text { wage }) \mid \text { gender }=\text { man }]=3.05$$
$$\mathbb{E}[\log (\text { wage }) \mid \text { gender }=\text { woman }]=2.81 \text {. }$$
We call these expectations “conditional” as they are conditioning on a fixed value of the variable gender. While you might not think of a person’s gender as a random variable, it is random from the viewpoint of econometric analysis. If you randomly select an individual, the gender of the individual is unknown and thus random. (In the population of U.S. workers, the probability that a worker is a woman happens to be $43 \%$.) In observational data, it is most appropriate to view all measurements as random variables, and the means of subpopulations are then conditional means.

It is important to mention at this point that we in no way attribute causality or interpretation to the difference in the conditional expectation of log wages between men and women. There are multiple potential explanations.

## 经济代写|计量经济学代写Introduction to Econometrics代考|Logs and Percentages

In this section we want to motivate and clarify the use of the logarithm in regression analysis by making two observations. First, when applied to numbers the difference of logarithms approximately equals the percentage difference. Second, when applied to averages the difference in logarithms approximately equals the percentage difference in the geometric mean. We now explain these ideas and the nature of the approximations involved.
Take two positive numbers $a$ and $b$. The percentage difference between $a$ and $b$ is
$$p=100\left(\frac{a-b}{b}\right) .$$
Rewriting,
$$\frac{a}{b}=1+\frac{p}{100} .$$
Taking natural logarithms,
$$\log a-\log b=\log \left(1+\frac{p}{100}\right) .$$
A useful approximation for small $x$ is
$$\log (1+x) \simeq x .$$
This can be derived from the infinite series expansion of $\log (1+x)$ :
$$\log (1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\cdots=x+O\left(x^2\right)$$
The symbol $O\left(x^2\right)$ means that the remainder is bounded by $A x^2$ as $x \rightarrow 0$ for some $A<\infty$. Numerically, the approximation $\log (1+x) \simeq x$ is within $0.001$ for $|x| \leq 0.1$, and the approximation error increases with $|x|$.
Applying (2.3) to (2.2) and multiplying by 100 we find
$$p \simeq 100(\log a)-\log b) .$$

## 经济代写|计量经济学代写计量经济学导论代考|条件期望

.

$3.05$和$2.81$的值是男性和女性工人亚群体的平均对数工资。它们被称为给定性别的对数工资的条件期望(或条件均值)。我们可以将它们的具体值写成
$$\mathbb{E}[\log (\text { wage }) \mid \text { gender }=\text { man }]=3.05$$
$$\mathbb{E}[\log (\text { wage }) \mid \text { gender }=\text { woman }]=2.81 \text {. }$$

## 经济代写|计量经济学代写计量经济学导论代考|日志和百分比

. 在本节中，我们想通过两个观察来激发和阐明对数在回归分析中的使用。首先，当应用于数字时，对数的差值近似等于百分比差值。第二，当应用于平均值时，对数的差值近似等于几何平均值的差值百分比。我们现在解释这些概念和所涉及的近似的性质。取两个正数 $a$ 和 $b$。两者之间的百分比差异 $a$ 和 $b$
$$p=100\left(\frac{a-b}{b}\right) .$$

$$\frac{a}{b}=1+\frac{p}{100} .$$取自然对数，
$$\log a-\log b=\log \left(1+\frac{p}{100}\right) .$$
small的一个有用的近似 $x$
$$\log (1+x) \simeq x .$$这可以由的无穷级数展开得到 $\log (1+x)$ :
$$\log (1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\cdots=x+O\left(x^2\right)$$

$$p \simeq 100(\log a)-\log b) .$$

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