# 第 11 章 Multinomial choice model

## 11.1 Ordered choice（可排序選擇）

• 0-10分自我衡量健康滿意度

• 1-5分課程評量

There are J+1 options from 0 to J. An individual’s choice depends on the latent variable Y_i^* such that $Y_{i}^*=X_{i}^{'}\beta+\epsilon_{i},$ where $$\epsilon$$ has a pdf $$f(.)$$ and a CDF $$F(.)$$.

The larger the $$Y_i^*$$ the higher option number that he will choose. There must be J thresholds $\mu_{0}<...<\mu_{J-1}$ such that the option of person $$i$$: $Y_{i}=\left\{ \begin{array}{ccc} 0 & if & Y_i^*<\mu_{0}\\ k & if & \mu_{k-1}\leq Y_i^*<\mu_{k}\\ J & if & \mu_{J-1}\leq Y_i^* \end{array}.\right.$ It follows that \begin{align} Pr(Y_{i} =y_{i})=\begin{cases} Pr(X_{i}^{'}\beta+\epsilon_{i}<\mu_{0})=F(\mu_{0}-X_{i}^{'}\beta), & y_{i}=0\\ Pr(\mu_{k-1}\leq X_{i}^{'}\beta+\epsilon_{i}<\mu_{k})=F(\mu_{k}-X_{i}^{'}\beta)-F(\mu_{k-1}-X_{i}^{'}\beta), & 1\leq y_{i}=k\leq J-1\\ Pr(X_{i}^{'}\beta+\epsilon_{i}\geq\mu_{J-1})=1-F(\mu_{J-1}-X_{i}^{'}\beta), & y_{i}=J \end{cases} \end{align}

### Goodness-of-Fit

1. Pseudo-$$R^{2}$$.

2. Prediction: Predicted choice $\hat{y}_{i}=\arg\max_{\{y_{i}=0,1,...,J\}}\Pr(Y_{i}=y_{i}|\hat{\beta},\hat{\mu}).$ And compute the percentage of correct prediction.

### 概似函數

The likelihood function and MLE

Since $$Y_{i}$$ is discrete, $$L(Y_{i})=\Pr(Y_{i}=y_{i})$$,where $$\Pr(Y_{i}=y_{i})$$ is defined by ([eq:orderPr]). The sample log-likelihood function

$\ln L=\sum\ln L(Y_{i})=\sum\ln\Pr(Y_{i}=y_{i}).$

The MLEs is to solve for $$\{\beta,\mu_{0},\cdots,\mu_{j-1}\}$$.

### Marginal Effect

If $$X_{i}$$ is continuous :

• Before (two options) :

$\Pr(Y_{i}=0)+\Pr(Y_{i}=1)=1.$

$\frac{\partial Pr(Y_{i}=1)}{\partial X_{i}}+\frac{\partial Pr(Y_{i}=0)}{\partial X_{i}}=0.$

• Now (multiple ordered options):

$\Pr(Y_{i}=0)+\dots+\Pr(Y_{i}=J)=1.$

## 11.2 Unordered choice（不可排序選擇）

### 11.2.1 Random Utility

$U_{ij}=V_{ij}+\epsilon_{ij},$ 其中$$V_{ij}$$為可被解釋的部份, $$\epsilon_{ij}$$為殘差項。

### 11.2.2 Multinomial Logit Model

1. $$\epsilon\sim\text{Gumbel distribution}$$

\begin{align} f(\epsilon) & = e^{-\epsilon}e^{-e^{-\epsilon}},\\ F(\epsilon) & = e^{-e^{-\epsilon}} \end{align}

2. 不同選項間的殘差互相獨立，即$$\epsilon_{ij}\perp\epsilon_{ij'}$$$$j\neq j'$$

### 11.2.3 Identification

For $$U=V+\epsilon$$, $$V$$ consists of all the regressors $${\bf X}$$ so that $$V={\bf X}\beta$$. However, not all $$\beta$$ can be estimated (or identified more specifically) since we can only infer the difference of $$V$$ between options. Consider the following $$V$$ setup :

$V_{ij}=\alpha_j+\beta x_{ij}+\gamma_j z_i+\delta_j w_{ij}+\tau q_{i}.$

To what extend can we estimate those parameters?

1. only the $$\alpha_j-\alpha_{k}$$ can be estimated, but their not separate levels.
2. $$\beta$$ can be estimated.
3. only the $$\gamma_j-\gamma_k$$ can be estimated, but their not separate levels.
4. all $$\delta_j$$s can be estimated.
5. $$\tau$$ can not be estimated.
$$V_{ij}-V_{i1}$$為例，最後我們只會有 $V_{ij}-V_{i1}=(\alpha_j-\alpha_1)+\beta (x_{ij}-x_{i1})+(\gamma_j-\gamma_1) z_i+\delta_j w_{ij}-\delta_j w_{i1}.$ 可以分成三大區塊：
1. $$x_{ij}$$ with constant coefficient: $$\beta (x_{ij}-x_{i1})$$
2. $$z_i$$ with alternative varying coefficient: $$(\alpha_j-\alpha_1)+(\gamma_j-\gamma_1) z_i$$
3. $$w_{ij}$$ with alternative varying coefficient: $$\delta_j w_{ij}-\delta_j w_{i1}$$.

### 11.2.4 Multinomial Probit

Multinomial Probit比起Multinomial Logit還多了選項間的variance-covariance matrix得估算。

\begin{align} U_B-U_A &= V_B-V_A+(\epsilon_B-\epsilon_A)\\ U_C-U_A &= V_C-V_A+(\epsilon_C-\epsilon_A) \end{align}

$\left[\begin{array}{c} \epsilon_{A}-\epsilon_B\\ \epsilon_{A}-\epsilon_C\\ \end{array}\right]\sim N(0,\tilde{\Sigma})$

$$\Theta$$代表$$V_A-V_B,V_A-V_C$$裡的參數，故模型的概似函數可寫成$$L(\Theta,\tilde{\Sigma})$$，請說明$$(\Theta,\tilde{\Sigma})=(\Theta_0,\tilde{\Sigma}_0)$$$$(\Theta,\tilde{\Sigma})=(\alpha\Theta_0,\alpha^2\tilde{\Sigma}_0)$$會有相同的概似函數值。

Multinomial Probit在估算時，除了和Multinomial Logit一樣要有一個選項為比較選項外，必需選擇一個$$\alpha$$值來滿足認定條件。一般是選$$\alpha=1/\sigma(\epsilon_B-\epsilon_A)$$使得$$\tilde{\Sigma}$$對角線第一個variance值： $\tilde{\Sigma}_{11}=1$