### 经济代写|计量经济学作业代写Econometrics代考|Regression Variance

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• Foundations of Data Science 数据科学基础

## 经济代写|计量经济学作业代写Econometrics代考|Intercept-Only Model

An important measure of the dispersion about the CEF function is the unconditional variance of the CEF error $e$. We write this as
$$\sigma^{2}=\operatorname{var}[e]=\mathbb{E}\left[(e-\mathbb{E}[e])^{2}\right]=\mathbb{E}\left[e^{2}\right]$$
Theorem 2.4.3 implies the following simple but useful result.
Theorem $2.5$ If $E\left[y^{2}\right]<\infty$ then $\sigma^{2}<\infty$.
We can call $\sigma^{2}$ the regression variance or the variance of the regression error. The magnitude of $\sigma^{2}$ measures the amount of variation in $y$ which is not “explained” or accounted for in the conditional mean $\mathbb{E}[y \mid \boldsymbol{x}]$.

The regression variance depends on the regressors $\boldsymbol{x}$. Consider two regressions
\begin{aligned} &y=\mathbb{E}\left[y \mid x_{1}\right]+e_{1} \ &y=\mathbb{E}\left[y \mid x_{1}, x_{2}\right]+e_{2} \end{aligned}
We write the two errors distinctly as $e_{1}$ and $e_{2}$ as they are different – changing the conditioning information changes the conditional mean and therefore the regression crror as well.

In our discussion of iterated expectations we have seen that by increasing the conditioning set the conditional expectation reveals greater detail about the distribution of $y$. What is the implication for the regression error?

It turns out that there is a simple relationship. We can think of the conditional mean $\mathbb{E}[y \mid \boldsymbol{x}]$ as the “explained portion” of $y$. The remainder $e=y-\mathbb{E}[y \mid x]$ is the “unexplained portion”. The simple relationship we now derive shows that the variance of this unexplained portion decreases when we condition on more variables. This relationship is monotonic in the sense that increasing the amont of information always decreases the variance of the unexplained portion.
Theorem 2.6 If $E\left[y^{2}\right]<\infty$ then
$$\operatorname{var}[y] \geq \operatorname{var}\left[y-\mathbb{E}\left[y \mid \boldsymbol{x}{1}\right]\right] \geq \operatorname{var}\left[y-\mathbb{E}\left[y \mid \boldsymbol{x}{1}, \boldsymbol{x}_{2}\right]\right]$$
Theorem $2.6$ says that the variance of the difference between $y$ and its conditional mean (weakly) decreases whenever an additional variable is added to the conditioning information.
The proof of Theorem $2.6$ is given in Section 2.33.

## 经济代写|计量经济学作业代写Econometrics代考|Best Predictor

Suppose that given a realized value of $x$ we want to create a prediction or forecast of $y$. We can write any predictor as a function $g(\boldsymbol{x})$ of $\boldsymbol{x}$. The prediction error is the realized difference $y-g(\boldsymbol{x})$. A nonstochastic measure of the magnitude of the prediction error is the expectation of its square
$$\mathbb{E}\left[(y-g(x))^{2}\right]$$
We can define the best predictor as the function $g(x)$ which minimizes (2.9). What function is the best predictor? It turns out that the answer is the CEF $m(\boldsymbol{x})$. This holds regardless of the joint distribution of $(y, \boldsymbol{x})$.
To see this, note that the mean squared error of a predictor $g(\boldsymbol{x})$ is
\begin{aligned} \mathbb{E}\left[(y-g(\boldsymbol{x}))^{2}\right] &=\mathbb{E}\left[(e+m(\boldsymbol{x})-g(\boldsymbol{x}))^{2}\right] \ &=\mathbb{E}\left[e^{2}\right]+2 \mathbb{E}[e(m(\boldsymbol{x})-g(\boldsymbol{x}))]+\mathbb{E}\left[(m(\boldsymbol{x})-g(\boldsymbol{x}))^{2}\right] \ &=\mathbb{E}\left[e^{2}\right] \text { । }\left[(m(\boldsymbol{x}) \quad g(\boldsymbol{x}))^{2}\right] \ & \geq \mathbb{E}\left[e^{2}\right] \ &=\mathbb{E}\left[(y-m(\boldsymbol{x}))^{2}\right] \end{aligned}

where the first equality makes the substitution $y=m(\boldsymbol{x})+e$ and the third equality uses Theorem $2.4 .4$. The right-hand-side after the third equality is minimized by setting $g(\boldsymbol{x})=m(\boldsymbol{x})$, yielding the inequality in the fourth line. The minimum is finite under the assumption $\mathbb{E}\left[y^{2}\right]<\infty$ as shown by Theorem $2.5$.
We state this formally in the following result.
Theorem 2.7 Conditional Mean as Best Predictor If $E\left[y^{2}\right]<\infty$, then for any predictor $g(x)$,
It may be helpful to consider this result in the context of the intercept-only model
\begin{aligned} y &=\mu+e \ \mathbb{E}[e] &=0 \end{aligned}
Theorem $2.7$ shows that the best predictor for $y$ (in the class of constants) is the unconditional mean $\mu=\mathbb{E}[y]$, in the sense that the mean minimizes the mean squared prediction error.

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

While the conditional mean is a good measure of the location of a conditional distribution it does not provide information ahout the spread of the distrihution. A common measure of the dispersion is the conditional variance. We first give the general definition of the conditional variance of a random variable $w$.
Definition 2.1 If $\mathbb{E}\left[w^{2}\right]<\infty$, the conditional variance of $w$ given $\boldsymbol{x}$ is
$$\operatorname{var}[w \mid \boldsymbol{x}]=\mathbb{E}\left[(w-\mathbb{E}[w \mid \boldsymbol{x}])^{2} \mid \boldsymbol{x}\right]$$
The conditional variance is distinct from the unconditional variance var $[w]$. The difference is that the conditional variance is a function of the conditioning variables. Notice that the conditional variance is the conditional second moment, centered around the conditional first moment.
Given this definition we define the conditional variance of the regression error.
Definition 2.2 If $\mathbb{E}\left[e^{2}\right]<\infty$, the conditional variance of the regression error $e$ is
$$\sigma^{2}(\boldsymbol{x})=\operatorname{var}[e \mid \boldsymbol{x}]=\mathbb{E}\left[e^{2} \mid \boldsymbol{x}\right] .$$

Again, the conditional variance $\sigma^{2}(\boldsymbol{x})$ is distinct from the unconditional variance $\sigma^{2}$. The conditional variance is a function of the regressors, the unconditional variance is not. Generally, $\sigma^{2}(\boldsymbol{x})$ is a non-trivial function of $\boldsymbol{x}$ and can take any form subject to the restriction that it is non-negative. One way to think about $\sigma^{2}(\boldsymbol{x})$ is that it is the conditional mean of $e^{2}$ given $\boldsymbol{x}$. Notice as well that $\sigma^{2}(\boldsymbol{x})=\operatorname{var}[y \mid \boldsymbol{x}]$ so it is equivalently the conditional variance of the dependent variable.

The variance is in a different unit of measurement than the original variable. To convert the variance back to the same unit of measure we define the conditional standard deviation as its square root $\sigma(\boldsymbol{x})=$ $\sqrt{\sigma^{2}(x)}$.

As an example of how the conditional variance depends on observables, compare the conditional log wage densities for men and women displayed in Figure 2.2. The difference between the densities is not purely a location shift but is also a difference in spread. Specifically, we can see that the density for men’s log wages is somewhat more spread out than that for women, while the density for women’s wages is somewhat more peaked. Indeed, the conditional standard deviation for men’s wages is $3.05$ and that for women is $2.81$. So while men have higher average wages they are also somewhat more dispersed.
In general the unconditional variance is related to the conditional variance by the following relationship.
Theorem $2.8$ If $E\left[w^{2}\right]<\infty$ then
$$\operatorname{var}[w]=\mathbb{E}[\operatorname{var}[w \mid \boldsymbol{x}]]+\operatorname{var}[\mathbb{E}[w \mid \boldsymbol{x}]]$$
See Theorem $4.14$ of Introduction to Econometrics. Theorem $2.8$ decomposes the unconditional variance into what are sometimes called the “within group variance” and the “across group variance”. For example, if $x$ is education level, then the first term is the expected variance of the conditional means by education level. The second term is the variance after controlling for education.

The regression error has a conditional mean of zero, so its unconditional error variance equals the expected conditional variance, or equivalently can be found by the law of iterated expectations
$$a^{2}-\mathbb{L}\left[e^{2}\right]-\mathbb{L}\left[\mathbb{L}\left[e^{2} \mid x\right]\right]-\mathbb{L}\left[\sigma^{2}(x)\right]$$
That is, the unconditional error variance is the average conditional variance.
Given the conditional variance we can define a rescaled error
$$u=\frac{e}{\sigma(x)}$$
We can calculate that since $\sigma(x)$ is a function of $\boldsymbol{x}$
$$\mathbb{E}[u \mid \boldsymbol{x}]=\mathbb{E}\left[\frac{e}{\sigma(\boldsymbol{x})} \mid \boldsymbol{x}\right]=\frac{1}{\sigma(\boldsymbol{x})} \mathbb{E}[e \mid \boldsymbol{x}]=0$$
and
$$\operatorname{var}[u \mid \boldsymbol{x}]=\mathbb{E}\left[u^{2} \mid \boldsymbol{x}\right]=\mathbb{E}\left[\frac{e^{2}}{\sigma^{2}(\boldsymbol{x})} \mid \boldsymbol{x}\right]=\frac{1}{\sigma^{2}(\boldsymbol{x})} \mathbb{E}\left[e^{2} \mid \boldsymbol{x}\right]=\frac{\sigma^{2}(\boldsymbol{x})}{\sigma^{2}(\boldsymbol{x})}=1$$
Thus $u$ has a conditional mean of zero and a conditional variance of 1 .
Notice that (2.10) can be rewritten as
$$e=\sigma(\boldsymbol{x}) u$$

## 经济代写|计量经济学作业代写Econometrics代考|Intercept-Only Model

CEF 函数离散度的一个重要度量是 CEF 误差的无条件方差和. 我们把它写成
σ2=曾是⁡[和]=和[(和−和[和])2]=和[和2]

## 经济代写|计量经济学作业代写Econometrics代考|Best Predictor

।和[(是−G(X))2]=和[(和+米(X)−G(X))2] =和[和2]+2和[和(米(X)−G(X))]+和[(米(X)−G(X))2] =和[和2] । [(米(X)G(X))2] ≥和[和2] =和[(是−米(X))2]

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

σ2(X)=曾是⁡[和∣X]=和[和2∣X].

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