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# 经济代写|计量经济学代写Introduction to Econometrics代考|Joint Normality and Linear Regression

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## 经济代写|计量经济学代写Introduction to Econometrics代考|Joint Normality and Linear Regression

Suppose the variables $(y, \boldsymbol{x})$ are jointly normally distributed. Consider the best linear predictor of $y$ given $\boldsymbol{x}$
$$y=\boldsymbol{x}^{\prime} \boldsymbol{\beta}+\alpha+e .$$
By the properties of the best linear predictor, $\mathbb{E}[x e]=0$ and $\mathbb{E}[e]=0$, so $x$ and $e$ are uncorrelated. Since $(e, \boldsymbol{x})$ is an affine transformation of the normal vector $(y, x)$ it follows that $(e, \boldsymbol{x})$ is jointly normal (Theorem 5.2). Since $(e, x)$ is jointly normal and uncorrelated they are independent (Theorem 5.3). Independence implies that
$$\mathbb{E}[e \mid x]=\mathbb{E}[e]=0$$
and
$$\mathbb{E}\left[e^2 \mid \boldsymbol{x}\right]=\mathbb{E}\left[e^2\right]=\sigma^2$$
which are properties of a homoskedastic linear CEF.
We have shown that when $(y, \boldsymbol{x})$ are jointly normally distributed they satisfy a normal linear CEF
$$y=\boldsymbol{x}^{\prime} \boldsymbol{\beta}+\alpha+e$$
where
$$e \sim \mathrm{N}\left(0, \sigma^2\right)$$
is independent of $\boldsymbol{x}$. This result can also be deduced from Theorem 5.3.7.
This is a classical motivation for the linear regression model.

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

The normal regression model is the linear regression model with an independent normal error
\begin{aligned} & y=\boldsymbol{x}^{\prime} \boldsymbol{\beta}+e \ & e \sim \mathrm{N}\left(0, \sigma^2\right) . \end{aligned}
As we learned in Section 5.4 the normal regression model holds when $(y, \boldsymbol{x})$ are jointly normally distributed. Normal regression, however, does not require joint normality. All that is required is that the conditional distribution of $y$ given $\boldsymbol{x}$ is normal (the marginal distribution of $\boldsymbol{x}$ is unrestricted). In this sense the normal regression model is broader than joint normality. Notice that for notational convenience we have written (5.1) so that $\boldsymbol{x}$ contains the intercept.

Normal regression is a parametric model where likelihood methods can be used for estimation, testing, and distribution theory. The likelihood is the name for the joint probability density of the data, evaluated at the observed sample, and viewed as a function of the parameters. The maximum likelihood estimator is the value which maximizes this likelihood function. Let us now derive the likelihood of the normal regression model.

First, observe that model (5.1) is equivalent to the statement that the conditional density of $y$ given $\boldsymbol{x}$ takes the form
$$f(y \mid \boldsymbol{x})=\frac{1}{\left(2 \pi \sigma^2\right)^{1 / 2}} \exp \left(-\frac{1}{2 \sigma^2}\left(y-\boldsymbol{x}^{\prime} \boldsymbol{\beta}\right)^2\right) .$$

Under the assumption that the observations are mutually independent this implies that the conditional density of $\left(y_1, \ldots, y_n\right)$ given $\left(\boldsymbol{x}1, \ldots, \boldsymbol{x}_n\right)$ is \begin{aligned} f\left(y_1, \ldots, y_n \mid \boldsymbol{x}_1, \ldots, \boldsymbol{x}_n\right) & =\prod{i=1}^n f\left(y_i \mid \boldsymbol{x}i\right) \ & =\prod{i=1}^n \frac{1}{\left(2 \pi \sigma^2\right)^{1 / 2}} \exp \left(-\frac{1}{2 \sigma^2}\left(y_i-\boldsymbol{x}i^{\prime} \boldsymbol{\beta}\right)^2\right) \ & =\frac{1}{\left(2 \pi \sigma^2\right)^{n / 2}} \exp \left(-\frac{1}{2 \sigma^2} \sum{i=1}^n\left(y_i-\boldsymbol{x}_i^{\prime} \boldsymbol{\beta}\right)^2\right) \ & \stackrel{\text { def }}{=} L_n\left(\boldsymbol{\beta}, \sigma^2\right) \end{aligned}
and is called the likelihood function.

## 经济代写|计量经济学代写Introduction to Econometrics代考|Joint Normality and Linear Regression

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

$$\mathbb{E}[e \mid x]=\mathbb{E}[e]=0$$

$$\mathbb{E}\left[e^2 \mid \boldsymbol{x}\right]=\mathbb{E}\left[e^2\right]=\sigma^2$$

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

$$e \sim \mathrm{N}\left(0, \sigma^2\right)$$

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

$$y=\boldsymbol{x}^{\prime} \boldsymbol{\beta}+e \quad e \sim \mathrm{N}\left(0, \sigma^2\right)$$

$$f(y \mid x)=\frac{1}{\left(2 \pi \sigma^2\right)^{1 / 2}} \exp \left(-\frac{1}{2 \sigma^2}\left(y-\boldsymbol{x}^{\prime} \boldsymbol{\beta}\right)^2\right)$$

$$f\left(y_1, \ldots, y_n \mid \boldsymbol{x}_1, \ldots, \boldsymbol{x}_n\right)=\prod i=1^n f\left(y_i \mid \boldsymbol{x} i\right) \quad=\prod i=1^n \frac{1}{\left(2 \pi \sigma^2\right)^{1 / 2}} \exp \left(-\frac{1}{2 \sigma^2}\left(y_i-\boldsymbol{x} i^{\prime} \boldsymbol{\beta}\right)^2\right)$$

## MATLAB代写

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