---
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.
Consider the following two-equation structural equation model:
$$
Y_1 = \alpha_1 Y_2 + \alpha_2 X_1 + \alpha_3 X_2 + \alpha_4 X_3 + U_1\\
Y_2 = \alpha_5 Y_1 + \alpha_6 X_1 + \alpha_7 X_2 + \alpha_8 X_3 + U_2
$$
where the exogenous variables $X_1, X_2, X_3,$ are independent of the structural disturbances $U_1$ and $U_2$. Those disturbances have zero expectations, and may be correlated with each other.
Show whether the following restrictions are sufficient to identify the structal parameters(consider each separately):

a. $\alpha_6 = 0.7$,

b. $\alpha_5 = 0$ and $C(U_1,U_2) = 0$,

c. $\alpha_4 + \alpha_8 = 1$.

## Answer

Combine the structual models, we can obtaint the following equations:

$$
(1-\alpha_1 \alpha_5) Y_1 = (\alpha_1 \alpha_6 + \alpha_2) X_1 + (\alpha_1 \alpha_7 + \alpha_3) X_2 + (\alpha_1 \alpha_8 + \alpha_4) X_3 + \alpha_1 U_2 + U_1\\
(1-\alpha_1 \alpha_5) Y_2 = (\alpha_5 \alpha_2 + \alpha_6) X_1 + (\alpha_5 \alpha_3 + \alpha_7) X_2 + (\alpha_5 \alpha_4 + \alpha_8) X_3 + \alpha_5 U_1 + U_2
$$

By the indepedent assumption, we can consistently estimate the following reduced-form model:

$$
Y_1 = \beta_1 X_1 + \beta_2 X_2 + \beta_3 X_3 + e_1\\
Y_2 = \beta_4 X_1 + \beta_5 X_2 + \beta_6 X_3 + e_2\\
$$

### a
There are four unknows with three exogenous variables, we cannot identify the first equation.

For the second equation, we still cannot identify the parameters. We have

$$
\beta_4 = \frac{\alpha_5 \alpha_2 + \alpha_6}{1-\alpha_1 \alpha_5},
$$

but only $\beta_4$ and $\alpha_6$ are known.

### b

The structual equations become

$$
Y_1 = (\alpha_1 \alpha_6 + \alpha_2) X_1 + (\alpha_1 \alpha_7 + \alpha_3) X_2 + (\alpha_1 \alpha_8 + \alpha_4) X_3 + \alpha_1 U_2 + U_1\\
Y_2 = \alpha_6 X_1 + \alpha_7 X_2 + \alpha_8 X_3 + U_2.
$$

Clearly, the parameters are the same in structual and reduced-form for the second equation. After we identify $\alpha_6, \alpha_7, \alpha_8$, we have
$$
\alpha_1 \alpha_6 + \alpha_2 = \beta_1\\
\alpha_1 \alpha_7 + \alpha_3 = \beta_2\\
\alpha_1 \alpha_8 + \alpha_4 = \beta_3
$$

However, cannot identify additional paramaters. (Because it is not a linear system)

### c

We cannot identify any parameter. Rewrite $\alpha_8 = 1- \alpha_4$, we are still clueless.

# Q2.

There are two mutually exclusive and exhaustive treatments indexed by the binary variable $t = 0,1$. Let $y(t)$ be a binary variable that indicates the potential outcome that would occur under treatment $t$ and let $z$ indicate the actual treatment received. Suppose you are interested in the status quo treatment effect denoted as $E[ y(1) ] - E[ y ]$, where $y$ is the observed outcome, and you draw a random sample size $N$ from the opulation of $y$ and $z$.

a. In the absence of assumptions about the selection process or outcomes, can you learn whether the STE is positive? Would your answer change if the sample size was substantially larger (e.g., 1000*N)?

b. Suppose selection is exogenous: $E[ y(t) ] = E[ y(t) | z ]$. Would this change your answer to (a)? Explain

c. Suppose the outcome under treatment $1$ always (weakly) exceeds the outcome under treatment $0$: $y(1) \ge y(0)$. Would this change your answer to (a)? Explain.

## Answer
:::warning
The question is a little weird.
:::

### a
No. If the individual with negative STE do not receive treatment, we cannot ensure the average STE is positive or not with any number of sample size.

### b
Yes.

### c
:::info
I am not sure.
:::

# Q3.

Suppose the probability a student passes the econometrics Core exam in the first try is $1/3$ and it is $1/2$ to pass it on a re-take. Suppose that infinitely many re-takes are possible.

a. What is the expected number of times a student must take the exam before passing?

b. What is the median number of times a student must take the exam before passing?

c. How many times would a student have to fail the exam on the re-take before you would reject the null hypothesis that his probability of passing is not less than $3/5$ at a $0.05$ level?

# Q4.

Suppose the wage ($W_i$), years of schooling ($S_i$) and ability ($A_i$), for a worker $i\in \{1, 2, \cdots, N\}$ satisfy the following linear relationship:
$$
W_i = \beta_1 \times S_i + \beta_2 \times A_i + \varepsilon \quad \varepsilon \perp (S_i, A_i) (1)
$$
Consider the following estimation steps:
i. Regress wage on ability and determine the residual $\tilde{W}$.
ii. Regress schooling on ability and determine the residual $\tilde{S}$.
iii. Regress $\tilde{W}$ from step (i) on $\tilde{S}$ from step (ii).

a. The residual regression rule implies that the estimate of the coefficient of $\tilde{S}$ in step (iii) is the same as the OLS estimate of $\beta_1$ in Equation (1). Suppose we observe workers' wages and schoolings $(W, S)$ but not their abilities (A). Using the sequential estimation steps (i)-(iii) mentioned above, show that if $S$ and $A$ are independent, then the OLS estimate of $b$ from the following equation
$$
W_i = b \times S_i + \eta_i,
$$
is an unbiased estimate of $\beta_1$.

b. Now, suppose $S$ and $A$ are correlated, and we observe a proxy for $A$, denoted by $A^*$,e.g., IQ score. Furthermore, suppose $A^* = A + \Delta$. Under what additional conditions on $(A^*, \Delta, \varepsilon, W, S)$ can we use $(W, S, A^*)$ in Equation (1) to get an unbiased estimate of $\beta_1$. If there isn't any such condition, explain why.

## Answer
### a

We hvae,
$$
W_i = b \times S_i + \eta_i\\
=\beta_1 \times S_i + \beta_2 \times A_i + \varepsilon
$$

That is, $\eta_i =\beta_2 \times A_i + \varepsilon$. By the independent assumption, $\eta_i \perp S_i$. Hence, $\hat{b}$ is a unbiased estimator of $\beta_1$.

### b

# Q5.
Let g(x) be a density function of a random variable with mean $\mu$ and variance $\sigma^2$. Let $X$ be a random variable with density function
$$
f(x|\theta_0) = g(x)(1 + \theta_0(x-\mu))
$$

Assume $g(x), \mu$ and $\sigma^2$ are known. The unknown parameter is $\theta_0$. Assume that $X$ has bounded support so that $f(x|\theta_0)\ge 0$ for all $x$.

a. Verify that R 1