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# 经济代写|计量经济学代写Introduction to Econometrics代考|ECON471 Linear CEF with Nonlinear Effects

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## 经济代写|计量经济学代写Introduction to Econometrics代考|Linear CEF with Nonlinear Effects

The linear CEF model of the previous section is less restrictive than it might appear, as we can include as regressors nonlinear transformations of the original variables. In this sense, the linear CEF framework is flexible and can capture many nonlinear effects.

For example, suppose we have two scalar variables $x_1$ and $x_2$. The CEF could take the quadratic form
$$m\left(x_1, x_2\right)=x_1 \beta_1+x_2 \beta_2+x_1^2 \beta_3+x_2^2 \beta_4+x_1 x_2 \beta_5+\beta_6 .$$
This equation is quadratic in the regressors $\left(x_1, x_2\right)$ yet linear in the coefficients $\boldsymbol{\beta}=\left(\beta_1, \ldots, \beta_6\right)^{\prime}$. We will descriptively call (2.13) a quadratic CEF, and yet (2.13) is also a linear CEF in the sense of being linear in the coefficients. The key is to understand that (2.13) is quadratic in the variables $\left(x_1, x_2\right)$ yet linear in the coefficients $\boldsymbol{\beta}$.

To simplify the expression we define the transformations $x_3=x_1^2, x_4=x_2^2, x_5=x_1 x_2$, and $x_6=1$, and redefine the regressor vector as $\boldsymbol{x}=\left(x_1, \ldots, x_6\right)^{\prime}$. With this redefinition,
$$m\left(x_1, x_2\right)=\boldsymbol{x}^{\prime} \boldsymbol{\beta}$$
which is linear in $\boldsymbol{\beta}$. For most econometric purposes (estimation and inference on $\boldsymbol{\beta}$ ) the linearity in $\boldsymbol{\beta}$ is all that is important.

An exception is in the analysis of regression derivatives. In nonlinear equations such as (2.13) the regression derivative should be defined with respect to the original variables not with respect to the transformed variables. Thus
\begin{aligned} & \frac{\partial}{\partial x_1} m\left(x_1, x_2\right)=\beta_1+2 x_1 \beta_3+x_2 \beta_5 \ & \frac{\partial}{\partial x_2} m\left(x_1, x_2\right)=\beta_2+2 x_2 \beta_4+x_1 \beta_5 \end{aligned}

## 经济代写|计量经济学代写Introduction to Econometrics代考|Law of Iterated Expectations

An extremely useful tool from probability theory is the law of iterated expectations. An important special case is the known as the Simple Law.
Theorem 2.1 Simple Law of Iterated Expectations If $\mathbb{E}|y|<\infty$ then for any random vector $\boldsymbol{x}$,
$$\mathbb{E}[\mathbb{E}[y \mid \boldsymbol{x}]]=\mathbb{E}[y] .$$
The simple law states that the expectation of the conditional expectation is the unconditional expectation. In other words the average of the conditional averages is the unconditional average. When $\boldsymbol{x}$ is discrete
$$\mathbb{E}[\mathbb{E}[y \mid \boldsymbol{x}]]=\sum_{j=1}^{\infty} \mathbb{E}\left[y \mid \boldsymbol{x}=\boldsymbol{x}j\right] \mathbb{P}\left[\boldsymbol{x}=\boldsymbol{x}_j\right]$$ and when $\boldsymbol{x}$ is continuous $$\mathbb{E}[\mathbb{E}[y \mid \boldsymbol{x}]]=\int{\mathbb{R}^k} \mathbb{E}[y \mid \boldsymbol{x}] f_{\boldsymbol{x}}(\boldsymbol{x}) d \boldsymbol{x} .$$
Going back to our investigation of average log wages for men and women, the simple law states that
\begin{aligned} & \mathbb{E}[\log (\text { wage }) \mid \text { gender }=\text { man }] \mathbb{P}[\text { gender }=\text { man }] \ & +\mathbb{E}[\log (\text { wage }) \mid \text { gender }=\text { woman }] \mathbb{P}[\text { gender }=\text { woman }] \ & =\mathbb{E}[\log (\text { wage })] \end{aligned}

Or numerically,
$$3.05 \times 0.57+2.81 \times 0.43=2.95 \text {. }$$
The general law of iterated expectations allows two sets of conditioning variables.
Theorem 2.2 Law of Iterated Expectations If $\mathbb{E}|y|<\infty$ then for any random vectors $\boldsymbol{x}_1$ and $\boldsymbol{x}_2$,
$$\mathbb{E}\left[\mathbb{E}\left[y \mid \boldsymbol{x}_1, \boldsymbol{x}_2\right] \mid \boldsymbol{x}_1\right]=\mathbb{E}\left[y \mid \boldsymbol{x}_1\right] .$$

## 经济代写|计量经济学代写Introduction to Econometrics代考|Linear CEF with Nonlinear Effects

$$m\left(x_1, x_2\right)=x_1 \beta_1+x_2 \beta_2+x_1^2 \beta_3+x_2^2 \beta_4+x_1 x_2 \beta_5+\beta_6 .$$

$$m\left(x_1, x_2\right)=\boldsymbol{x}^{\prime} \boldsymbol{\beta}$$

$$\frac{\partial}{\partial x_1} m\left(x_1, x_2\right)=\beta_1+2 x_1 \beta_3+x_2 \beta_5 \quad \frac{\partial}{\partial x_2} m\left(x_1, x_2\right)=\beta_2+2 x_2 \beta_4+x_1 \beta_5$$

## 经济代苛|计是组济学代 Expectations

$$\mathbb{E}[\mathbb{E}[y \mid \boldsymbol{x}]]=\mathbb{E}[y]$$

$$\mathbb{E}[\mathbb{E}[y \mid \boldsymbol{x}]]=\sum_{j=1}^{\infty} \mathbb{E}[y \mid \boldsymbol{x}=\boldsymbol{x} j] \mathbb{P}\left[\boldsymbol{x}=\boldsymbol{x}j\right]$$ 什么时候 $\boldsymbol{x}$ 是连续的 $$\mathbb{E}[\mathbb{E}[y \mid \boldsymbol{x}]]=\int \mathbb{R}^k \mathbb{E}[y \mid \boldsymbol{x}] f{\boldsymbol{x}}(\boldsymbol{x}) d \boldsymbol{x} .$$

$\mathbb{E}[\log$ (wage ) $\mid$ gender $=\operatorname{man}] \mathbb{P}[$ gender $=\operatorname{man}] \quad+\mathbb{E}[\log ($ wage $) \mid$ gender $=$ woman $] \mathbb{P}[$ gender $=$ woman $]=\mathbb{E}[\log ($ wage $)]$

$$3.05 \times 0.57+2.81 \times 0.43=2.95 \text {. }$$

$$\mathbb{E}\left[\mathbb{E}\left[y \mid \boldsymbol{x}_1, \boldsymbol{x}_2\right] \mid \boldsymbol{x}_1\right]=\mathbb{E}\left[y \mid \boldsymbol{x}_1\right]$$

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