---
tags: metrics, core, group
---

$$
% My definitions
\def\ve{{\varepsilon}}
\def\dd{{\text{ d}}}
\newcommand{\dif}[2]{\frac{d #1}{d #2}} % for derivatives
\newcommand{\pd}[2]{\frac{\partial #1}{\partial #2}} % for partial derivatives
\def\R{\text{R}}
$$

# Q1

Consistency: CLT CMT Slutsky

Asymptotic distribution: Delta method

The expectations exist, so we can use WLLN.
WLLN:
$$
\bar{X} \to_p exp^{\alpha \beta},\\
\bar{Y} \to_p exp^{\alpha},
$$

Log is a continuous function, so we can use CMT.
CMT:
$$
\log(\bar{X}) \to_p \alpha \beta\\
\log(\bar{Y}) \to_p \alpha \\
$$

Combine the above conditions, we can use Slutsky:

$$
\frac{\log(\bar{X})}{\log(\bar{Y})} \to_p \beta
$$

Let $Z =[\bar{X}, \bar{Y}]'$ and $\theta = [exp^{\alpha \beta}, exp^{\alpha}]'$

CLT:
$$
\sqrt{n} (z - \theta) \to_d N(0, \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix})
$$

Define $h(x, y) = \log x / \log y$

Delta method:

$$
\sqrt{n} (h(z) - h(\theta)) \to_d N(0, \nabla h(\theta)'\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\nabla h(\theta)),
$$

where

$$
\nabla h(\theta) = [\frac{1}{x \log y}, \frac{-\log x}{y (\log y)^{2}}]'
$$

# Q2
## a
Let $f(X) = m(X; \theta), \theta \neq \theta^*$
$$
E((Y-f(X))^2) = E( (Y-E(Y|X) + E(Y|X) - f(X))^2 ) \\
= E( (Y-E(Y|X)^2) + E( (E(Y|X)- f(X))^2 ) - 2 E( (Y-E(Y|X)) (E(Y|X) - f(X)) )
$$

By LIE,

$$
E((Y-f(X))^2) = E( (Y-E(Y|X)^2) + E( (E(Y|X)-f(X))^2 ) \ge E( (Y-E(Y|X)^2)
$$

WTS
$$
E((Y-f(X))^2) = E( (Y-E(Y|X)^2) + E( (E(Y|X)- f(X))^2 ) > E( (Y-E(Y|X)^2)
$$

Since $\theta^*$ is unique

$$
f(x) \neq E(Y|X)
$$

$$
E( (E(Y|X)- f(X))^2 ) >0
$$
## b
The expectation of mean square error.

## c
Delta method

# Q3

$$
y_i = \begin{cases}
0 &\text{ if } x_i \beta + \epsilon < 0 \\
y_i^* &\text{ if } 0 \le x_i \beta + \epsilon \le 2 \\
-1 &\text{ if } x_i \beta + \epsilon >2
\end{cases}
$$

$$
f(y;x, \beta) = \begin{cases}
F(\frac{-x\beta}{\sigma}) &\text{ if } x_i \beta + \epsilon < 0\\
F(\frac{2-x \beta}{\sigma}) - F(\frac{-x\beta}{\sigma}) \\
1-F(\frac{2-x\beta}{\sigma})
\end{cases}
$$

$$
f(y;x, \beta) = [F(\frac{-x\beta}{\sigma})]^{1(y<0)} \times [F(\frac{2-x \beta}{\sigma}) - F(\frac{-x\beta}{\sigma})]^{1(0 \le y \le 2)} \times [1-F(\frac{2-x\beta}{\sigma})]^{1(y>2)}
$$

log-likelihood : $\log f(y;x,\beta)$

See Peter's DQ9

# Q4

## a

## b

## c

zero subvector
$H_0: \beta_2 = \beta_3 = \beta_4 = \dots = \beta_7 = 0$
$H_a:$ At least one slope coeffiient is not zero.

Test statistic = $\frac{(2000-7)(R^2 - R^{2*})}{1-R^2}$ where $R^2 = 0.308, R{^*}=0$

## d

iii

## e
collinearity