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# 经济代写|计量经济学代写Introduction to Econometrics代考|BEA242 Regression to the Mean

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## 经济代写|计量经济学代写Introduction to 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 (2.38)
$$\alpha=(1-\beta) \mu$$
so we can write the linear projection (2.48) as
$$\mathscr{P}(y \mid x)=(1-\beta) \mu+x \beta .$$

## 经济代写|计量经济学代写Introduction to Econometrics代考|Reverse Regression

Galton noticed another interesting feature of the bivariate distribution. There is nothing special about a regression of $y$ on $x$. We can also regress $x$ on $y$. (In his heredity example this is the best linear predictor of the height of parents given the height of their children.) This regression takes the form
$$x=y \beta^+\alpha^+e^*$$
This is sometimes called the reverse regression. In this equation, the coefficients $\alpha^, \beta^$ and error $e^$ are defined by linear projection. In a stable population we find that \begin{aligned} & \beta^=\operatorname{corr}(x, y)=\beta \ & \alpha^*=(1-\beta) \mu=\alpha \end{aligned}
which are exactly the same as in the projection of $y$ on $x$ ! The intercept and slope have exactly the same values in the forward and reverse proiections!

## 经济代写|计量经济学代写Introduction to Econometrics代 考|Regression to the Mean

$$y=x \beta+\alpha+e$$

$$\alpha=(1-\beta) \mu$$

$$\mathscr{P}(y \mid x)=(1-\beta) \mu+x \beta$$

## 经济代写|计量经济学代写Introduction to Econometrics代 考|Reverse Regression

$$x=y \beta^{+} \alpha^{+} e^*$$

$$\beta^{=} \operatorname{corr}(x, y)=\beta \quad \alpha^*=(1-\beta) \mu=\alpha$$

## MATLAB代写

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