---
title: 'CSSS/POLS 510 Maximum Likelihood Estimation: Lab 6'
editor_options:
  chunk_output_type: console
date: '2020-11-13'
output:
  beamer_presentation: 
    colortheme: seagull
    fonttheme: structurebold
    keep_tex: yes
    theme: Szeged
  slidy_presentation: default
fontsize: 12pt
subtitle: Nested Logit model GOF
author: Kenya Amano
---
# Agenda
1. Recap lectures

2. Ordered Probit model

3. Homework: HW4


# 1. Recap
Where are we at right now?


# 1. Recap
Where are we at right now?

1. Learn distribution and MLE $\rightarrow$ HW1 & HW2

2. Logit model $\rightarrow$ HW3

3. Ordered Probit model $\rightarrow$ HW4

4. Multinomial logit $\rightarrow$ HW5

5. Count data  $\rightarrow$ HW5

# 2. Ordered Probit model
Review the lecture materials to understand the concept


# 2. Ordered Probit model


Probabilities we want to estimate in four category case
\small
\begin{equation*}
\begin{split}
\text{Pr}(y_i=1|\boldsymbol{x_i})&=\Phi(\tau_1-\alpha-\boldsymbol{x_i\beta})\\
\text{Pr}(y_i=2|\boldsymbol{x_i})&=\Phi(\tau_2-\alpha-\boldsymbol{x_i\beta})-\Phi(\tau_{1}-\alpha-\boldsymbol{x_i\beta})\\
\text{Pr}(y_i=3|\boldsymbol{x_i})&=\Phi(\tau_3-\alpha-\boldsymbol{x_i\beta})-\Phi(\tau_{2}-\alpha-\boldsymbol{x_i\beta})\\
\text{Pr}(y_i=4|\boldsymbol{x_i})&=1-\Phi(\tau_3-\alpha-\boldsymbol{x_i\beta})
\end{split}
\end{equation*}

To identify the model, we commonly make one of two assumptions:

1. Assume that $\tau_{1}=0$. This is also the identifying assumption of logit and probit. `optim()` uses this.

2. Assume that $\alpha=0$. `polr()` uses this.

The likelihood function for ordered probit finds the $\boldsymbol{\beta}$ and $\tau$ that make the observed data most likely.



# 2 Simulating QoI

1. Estimate: MLE $\hat{\beta}, \hat{\tau}$ and its variance $\hat{V}(\hat{\beta}, \hat{\tau})$\
$\textcolor{red}{\rightarrow  \texttt{optim(), polr()}}$


2. Simulate estimation uncertainty from a multivariate normal distribution:\
      Draw $\tilde{\beta} , \tilde{\tau} \sim MVN \big[\hat{(\beta},\hat{\tau}), \hat{V}(\hat{\beta}, \hat{\tau})\big]$\
$\textcolor{red}{\rightarrow  \texttt{MASS::mvrnorm()}}$


3. Create hypothetical scenarios of your substantive interest:\
      Choose valuese of X: $X_c$ 
$\textcolor{red}{\rightarrow  \texttt{simcf::cfmake(), cfchange()} \dots}$


# 2 Simulating QoI

4. Calculate expected values:\
      $\tilde{\pi_c} = g(X_c, \tilde{\beta}, \tilde{\tau})$ \


5. Compute EVs, First Differences or Relative Risks\
      EV: $\mathbb{E}(y = j|X_{c1},\tilde{\beta},\tilde{\tau} )$\
      $\textcolor{red}{\rightarrow  \texttt{simcf::oprobitsimev()} \dots}$\
      FD: $\mathbb{E}(y = j|X_{c2},\tilde{\beta},\tilde{\tau}) - \mathbb{E}(y = j|X_{c1},\tilde{\beta},\tilde{\tau})$\
      $\textcolor{red}{\rightarrow  \texttt{simcf::oprobitsimfd()} \dots}$\
      RR: $\frac{\mathbb{E}(y = j|X_{c2},\tilde{\beta},\tilde{\tau})}{\mathbb{E}(y = j|X_{c1},\tilde{\beta},\tilde{\tau})}$\
      $\textcolor{red}{\rightarrow  \texttt{simcf::oprobitsimrr()} \dots}$\



# 3. Homework: Question HW4
+ Due on Nov 24
+ Email subject: **MLE510HW4**
+ File name: **MLE510HW4KenyaAmano**                                                 

+ One common problem when knitting: the math mode environment doesn't like white space or empty line
        + Try `\begin{alinged}` instead of `\begin{split}`
        + `R Markdown` guide is [\underline{here}](https://bookdown.org/yihui/rmarkdown)