# Discrete Choices

# Models for Binary Outcomes

Let $𝑈_{𝑎}$ and $𝑈_{𝑏}$ represents an individual’s utility of two choices:

𝑈_{𝑎} = 𝑊 β_{𝑎} + 𝑍_{𝑎} γ_{𝑎} + ε_{𝑎},

𝑈_{𝑏} = 𝑊 β_{𝑏} + 𝑍_{𝑏} γ_{𝑏} + ε_{𝑏} .

If we denote by $Y = 1$ the consumer’s choice of alternative $a$ , we infer from $Y = 1$ that $𝑈_{𝑎} > 𝑈_{𝑏}$ .

\begin{aligned} P r o b [𝑌 = 1 | 𝑊, 𝑍_{𝑎}, 𝑍_{𝑏}] & = P r o b [𝑈_{𝑎} > 𝑈_{𝑏}] \\ = P r o b [𝑋 β + ε > 0 | 𝑋] . \end{aligned}

W (β_{a} - β_{b}) + Z_{a} γ_{a} - Z_{b} γ_{b} + ε_{a} - ε_{b} > 0.

We model the net benefit of a choice as an variable $y^{*}$ such that

y^{*} = 𝑋 β + ε,

where $ε$ has mean zero and has either a standardized logistic or normal distribution.

We do not observe $y^{*}$ , instead, our observation is

y = 1 if y^{*} > 0,

y = 0 if y^{*} \leq 0.

Then we have

P r o b [y^{*} > 0 | 𝑋] = P 𝑟 𝑜 𝑏 [ε < 𝑋 β | 𝑋] = 𝐹 (𝑋 β) .

Note that the assumptions of known variance and zero cutoff are innocent normalization.

Linear Probability Model:

y^{*} = 𝑋 β + ε .

Shortcomings:

We want to construct a model produce predictions consistent with the underlying theory

P r o b [y^{*} > 0 | 𝑋] = P 𝑟 𝑜 𝑏 [ε < 𝑋 β | 𝑋] = 𝐹 (𝑋 β),

and we expect that

lim_{𝑋 β \to + \infty} P r o b [𝑌 = 1 | 𝑋] = 1,

lim_{X β \to - \infty} P r o b [𝑌 = 1 | 𝑋] = 0.

The normal distribution has been commonly used, denoted as the probit model,

P r o b (𝑌 = 1 | 𝑋) = \int_{- \infty}^{X β} ϕ_{(} t) d t = Φ (X β) .

Another commonly used model is the logit model, assuming logistic distribution,

P r o b (𝑌 = 1 | 𝑋) = \frac{e x p (𝑋 β)}{1 + e x p (𝑋 β)} = Λ (X β) .

The probability model is

𝐸 [𝑦 | 𝑋] = 𝐹 (𝑋 β) .

The parameters of the model are not necessarily the marginal effects

\frac{\partial 𝐸 [𝑦 | 𝑋]}{\partial 𝑋} = [\frac{𝑑 𝐹 (𝑋 β)}{𝑑 (𝑋 β)}] \times β = 𝑓 (𝑋 β) \times β .

Likelihood equations:

\frac{\partial \ln 𝐿}{\partial β} = \sum_{𝑖 = 1}^{𝑛} [\frac{𝑦_{𝑖} 𝑓_{𝑖}}{𝐹_{𝑖}} + (1 - 𝑦_{𝑖}) \frac{- 𝑓 𝑖}{(1 - 𝐹 𝑖)}] 𝑥_{𝑖} = 0.

Logit model

\frac{\partial \ln 𝐿}{\partial β} = \sum_{𝑖 = 1}^{𝑛} (y_{i} - Λ_{𝑖}) 𝑥_{𝑖} = 0.

Probit model

\frac{\partial \ln 𝐿}{\partial β} = \sum_{𝑖 = 1}^{𝑛} (\frac{q_{𝑖} ϕ (𝑞_{𝑖} 𝑥_{𝑖} β)}{Φ (𝑞_{𝑖} 𝑥_{𝑖} β)}) 𝑥_{𝑖} = 0,

where $𝑞_{𝑖} = 2 𝑦_{𝑖} - 1$ .

Goodness of Fit

Likelihood Ratio Index:

𝐿 𝑅 𝐼 = 1 - \frac{\ln 𝐿}{\ln 𝐿_{0}}

For a single estimator, use the $t$ test.

For more involved restrictions