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# 经济代写|计量经济学代写ECONOMETRICS代考|ECO562A Best Linear Approximation

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## 经济代写|计量经济学代写ECONOMETRICS代考|Best Linear Approximation

There are alternative ways we could construct a linear approximation $X^{\prime} \beta$ to the conditional mean $m(X)$. In this section we show that one alternative approach turns out to yield the same answer as the best linear predictor.

We start by defining the mean-square approximation error of $X^{\prime} \beta$ to $m(X)$ as the expected squared difference between $X^{\prime} \beta$ and the conditional mean $m(X)$
$$d(\beta)=\mathbb{E}\left[\left(m(X)-X^{\prime} \beta\right)^{2}\right] .$$
The function $d(\beta)$ is a measure of the deviation of $X^{\prime} \beta$ from $m(X)$. If the two functions are identical then $d(\beta)=0$, otherwise $d(\beta)>0$. We can also view the mean-square difference $d(\beta)$ as a density-weighted average of the function $\left(m(X)-X^{\prime} \beta\right)^{2}$ since
$$d(\beta)=\int_{\mathbb{R}^{k}}\left(m(x)-x^{\prime} \beta\right)^{2} f_{X}(x) d x$$
where $f_{X}(x)$ is the marginal density of $X$.
We can then define the best linear approximation to the conditional $m(X)$ as the function $X^{\prime} \beta$ obtained by selecting $\beta$ to minimize $d(\beta)$ :
$$\beta=\underset{b \in \mathbb{R}^{k}}{\operatorname{argmin}} d(b) .$$

## 经济代写|计量经济学代写ECONOMETRICS代考|Regression to the Mean

The term regression originated in an influential paper by Francis Galton (1886) where he examined the joint distribution of the stature (height) of parents and children. Effectively, he was estimating the conditional mean of children’s height given their parent’s height. Galton discovered that this conditional mean was approximately linear with a slope of $2 / 3$. This implies that on average a child’s height is more mediocre (average) than his or her parent’s height. Galton called this phenomenon regression to the mean, and the label regression has stuck to this day to describe most conditional relationships.

One of Galton’s fundamental insights was to recognize that if the marginal distributions of $Y$ and $X$ are the same (e.g. the heights of children and parents in a stable environment) then the regression slope in a linear projection is always less than one.
To be more precise, take the simple linear projection
$$Y=X \beta+\alpha+e$$
where $Y$ equals the height of the child and $X$ equals the height of the parent. Assume that $Y$ and $X$ have the same mean so that $\mu_{Y}=\mu_{X}=\mu$. Then from (2.39) $\alpha=(1-\beta) \mu$ so we can write the linear projection (2.49) as
$$\mathscr{P}(Y \mid X)=(1-\beta) \mu+X \beta .$$
This shows that the projected height of the child is a weighted average of the population average height $\mu$ and the parent’s height $X$ with the weight equal to $\beta$. When the height distribution is stable across generations so that $\operatorname{var}[Y]=\operatorname{var}[X]$, then this slope is the simple correlation of $Y$ and $X$. Using (2.40)
$$\beta=\frac{\operatorname{cov}(X, Y)}{\operatorname{var}[X]}=\operatorname{corr}(X, Y)$$

## 经济代写|计量经济学代写ECONOMETRICS代考|Best Linear Approximation

$$d(\beta)=\mathbb{E}\left[\left(m(X)-X^{\prime} \beta\right)^{2}\right] .$$

$$d(\beta)=\int_{\mathbb{R}^{k}}\left(m(x)-x^{\prime} \beta\right)^{2} f_{X}(x) d x$$

$$\beta=\underset{b \in \mathbb{R}^{k}}{\operatorname{argmin}} d(b) .$$

## 经济代写|计量经济学代写ECONOMETRICS代考|Regression to the Mean

$$Y=X \beta+\alpha+e$$

$$\mathscr{P}(Y \mid X)=(1-\beta) \mu+X \beta$$

$\operatorname{var}[Y]=\operatorname{var}[X]$, 那么这个斜率就是 $Y$ 和 $X$. 使用 $(2.40)$
$$\beta=\frac{\operatorname{cov}(X, Y)}{\operatorname{var}[X]}=\operatorname{corr}(X, Y)$$

## MATLAB代写

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