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.

若樣本數有500個，$y\in\{1,2,3\}$, 其中$y=1$的有30個, $y=2$有300個，$y=3$有170個，請問在Probit和Logit模型下,$\ln L_0$為多少？

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 $\{\beta,\mu_{0},\cdots,\mu_{j-1}\}$.

寫下明確概似函數定義。

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）

假設有A,B,C三個選擇，同一層次3選1。

11.2.1 Random Utility

個人$i$選擇$j$的效用：

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

11.2.2 Multinomial Logit Model

假設

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$\epsilon_{ij}\perp\epsilon_{ij'}$當$j\neq j'$。

以三個選項{A,B,C}為例，我們可以證明 $$\begin{eqnarray*} \Pr(\text{choice =}A) & = & \Pr(U_A>U_B,U_A>U_C)\\ & = & \frac{1}{1+\exp(V_{B}-V_A)+\exp(V_{C}-V_A)}. \end{eqnarray*}$$

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.

以$V_{ij}-V_{i1}$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

假設 $$\left[\begin{array}{c} \epsilon_{i0}\\ \epsilon_{i1}\\ \vdots\\ \epsilon_{iJ} \end{array}\right]\sim N(0,\Sigma)$$

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

以A,B,C三選項為例，任何選擇只會透露兩兩效用比較結果，我們可以只看$U_B-U_A,U_C-U_A$

說明不論選擇結果為何，$U_B-U_A$及$U_C-U_A$兩個隨機變數即足夠表示所有對應的訊息。

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

因為$\epsilon$的常態假設，故

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

也是常態分配。

效用函數同乘$\alpha>0$倍不會改變選擇結果，也就是說$U_{j}$與$U'_{j}$若有此倍數關係，它們顯示的選擇會相同。

令$\Theta$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$(\Theta,\tilde{\Sigma})=(\Theta_0,\tilde{\Sigma}_0)$與$(\Theta,\tilde{\Sigma})=(\alpha\Theta_0,\alpha^2\tilde{\Sigma}_0)$會有相同的概似函數值。

上面論述表示，若不進一步限制模型參數空間，最大概似估計會有無窮多組解，也就是認定不足（under-identifying）的現象。

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： $$\tilde{\Sigma}_{11}=1$$