经济代写|EC309 Econometrics

Exercise 2.1 Find $\mathbb{E}\left[\mathbb{E}\left[\mathbb{E}\left[y \mid \boldsymbol{x}_1, \boldsymbol{x}_2, \boldsymbol{x}_3\right] \mid \boldsymbol{x}_1, \boldsymbol{x}_2\right] \mid \boldsymbol{x}_1\right]$.
Exercise 2.2 If $\mathbb{E}[y \mid x]=a+b x$, find $\mathbb{E}[y x]$ as a function of moments of $x$.
Exercise 2.3 Prove Theorem 2.4.4 using the law of iterated expectations.
Exercise 2.4 Suppose that the random variables $y$ and $x$ only take the values 0 and 1 , and have the following joint probability distribution
\begin{tabular}{|c|cc|}
\hline & $x=0$ & $x=1$ \
\hline$y=0$ & .1 & .2 \
$y=1$ & .4 & .3 \
\hline
\end{tabular}
Find $\mathbb{E}[y \mid x], \mathbb{E}\left[y^2 \mid x\right]$ and var $[y \mid x]$ for $x=0$ and $x=1$.

Exercise 2.5 Show that $\sigma^2(\boldsymbol{x})$ is the best predictor of $e^2$ given $\boldsymbol{x}$ :
(a) Write down the mean-squared error of a predictor $h(\boldsymbol{x})$ for $e^2$.
(b) What does it mean to be predicting $e^2$ ?
(c) Show that $\sigma^2(\boldsymbol{x})$ minimizes the mean-squared error and is thus the best predictor.
Exercise 2.6 Use $y=m(\boldsymbol{x})+e$ to show that
$$
\operatorname{var}[y]=\operatorname{var}[m(\boldsymbol{x})]+\sigma^2
$$
Exercise 2.7 Show that the conditional variance can be written as
$$
\sigma^2(\boldsymbol{x})=\mathbb{E}\left[y^2 \mid \boldsymbol{x}\right]-(\mathbb{E}[y \mid \boldsymbol{x}])^2
$$
Exercise 2.8 Suppose that $y$ is discrete-valued, taking values only on the non-negative integers, and the conditional distribution of $y$ given $\boldsymbol{x}$ is Poisson:
$$
\mathbb{P}[y=j \mid \boldsymbol{x}]=\frac{\exp \left(-\boldsymbol{x}^{\prime} \boldsymbol{\beta}\right)\left(\boldsymbol{x}^{\prime} \boldsymbol{\beta}\right)^j}{j !}, \quad j=0,1,2, \ldots
$$
Compute $\mathbb{E}[y \mid \boldsymbol{x}]$ and var $[y \mid \boldsymbol{x}]$. Does this justify a linear regression model of the form $y=\boldsymbol{x}^{\prime} \boldsymbol{\beta}+e$ ?
Hint: If $\mathbb{P}[y=j]=\frac{\exp (-\lambda) \lambda^j}{j !}$ then $\mathbb{E}[y]=\lambda$ and $\operatorname{var}(y)=\lambda$.

Exercise 2.9 Suppose you have two regressors: $x_1$ is binary (takes values 0 and 1) and $x_2$ is categorical with 3 categories $(A, B, C)$. Write $\mathbb{E}\left[y \mid x_1, x_2\right]$ as a linear regression.
Exercise 2.10 True or False. If $y=x \beta+e, x \in \mathbb{R}$, and $\mathbb{E}[e \mid x]=0$, then $\mathbb{E}\left[x^2 e\right]=0$.
Exercise 2.11 True or False. If $y=x \beta+e, x \in \mathbb{R}$, and $\mathbb{E}[x e]=0$, then $\mathbb{E}\left[x^2 e\right]=0$.
Exercise 2.12 True or False. If $y=\boldsymbol{x}^{\prime} \boldsymbol{\beta}+e$ and $\mathbb{E}[e \mid \boldsymbol{x}]=0$, then $e$ is independent of $\boldsymbol{x}$.

Exercise 2.13 True or False. If $y=\boldsymbol{x}^{\prime} \boldsymbol{\beta}+e$ and $\mathbb{E}[\boldsymbol{x} e]=\mathbf{0}$, then $\mathbb{E}[e \mid \boldsymbol{x}]=0$.
Exercise 2.14 True or False. If $y=\boldsymbol{x}^{\prime} \boldsymbol{\beta}+e, \mathbb{E}[e \mid \boldsymbol{x}]=0$, and $\mathbb{E}\left[e^2 \mid \boldsymbol{x}\right]=\sigma^2$, a constant, then $e$ is independent of $\boldsymbol{x}$.

Exercise 2.15 Consider the intercept-only model $y=\alpha+e$ defined as the best linear predictor. Show that $\alpha=\mathbb{E}[y]$.

Exercise 2.16 Let $x$ and $y$ have the joint density $f(x, y)=\frac{3}{2}\left(x^2+y^2\right)$ on $0 \leq x \leq 1,0 \leq y \leq 1$. Compute the coefficients of the best linear predictor $y=\alpha+\beta x+e$. Compute the conditional mean $m(x)=\mathbb{E}[y \mid x]$. Are the best linear predictor and conditional mean different?
Exercise 2.17 Let $x$ be a random variable with $\mu=\mathbb{E}[x]$ and $\sigma^2=\operatorname{var}[x]$. Define
$$
g\left(x \mid \mu, \sigma^2\right)=\left(\begin{array}{c}
x-\mu \
(x-\mu)^2-\sigma^2
\end{array}\right) .
$$
Show that $\mathbb{E}[g(x \mid m, s)]=0$ if and only if $m=\mu$ and $s=\sigma^2$.
Exercise 2.18 Suppose that
$$
\boldsymbol{x}=\left(\begin{array}{c}
1 \
x_2 \
x_3
\end{array}\right)
$$
and $x_3=\alpha_1+\alpha_2 x_2$ is a linear function of $x_2$.
(a) Show that $\boldsymbol{Q}_{\boldsymbol{x} \boldsymbol{x}}=\mathbb{E}\left[\boldsymbol{x}^{\prime}\right]$ is not invertible.
(b) Use a linear transformation of $\boldsymbol{x}$ to find an expression for the best linear predictor of $y$ given $\boldsymbol{x}$. (Be explicit, do not just use the generalized inverse formula.)