<p>Logit and probit models can be derived from an underlying latent variable model. To see this, let <span class="math-container">$y^{\ast}$</span> be an unobserved (i.e. latent) variable, and suppose that
<span class="math-container">\begin{equation}
    y^{\ast} = \beta_0 + x\beta + \epsilon, ~y=1[y^{\ast}&gt;\gamma]
    \label{latent}
\end{equation}</span>
where <span class="math-container">$1[\cdot]$</span> is an indicator function taking value <span class="math-container">$1$</span> if <span class="math-container">$y^{\ast}&gt;\gamma$</span> and zero otherwise. Here, <span class="math-container">$\gamma$</span> is a chosen threshold. If the latent unobserved utility <span class="math-container">$y^\ast$</span> is above the threshold, then <span class="math-container">$y$</span> takes the value one; otherwise, it takes the value zero.</p>
<p>The idea is to see the binary outcome <span class="math-container">$y=\{0,1 \}$</span> as a dichotomization of a latent continuous variable <span class="math-container">$y^{\ast}$</span> modeled with a canonical multiple linear regression model. For example, a bank can issue a loan (<span class="math-container">$y=1$</span>) or choose not to (<span class="math-container">$y=0$</span>). Think of it as having an underlying continuous latent outcome, <span class="math-container">$y^{\ast}$</span> which can be regarded as a measure of the choice utility.</p>
<p>In the model above, we assume that <span class="math-container">$\epsilon$</span> is independent of <span class="math-container">$x$</span> and that <span class="math-container">$\epsilon$</span> either has the standard logistic distribution or the standard normal distribution.</p>
<p>Recall, the standard logistic and standard normal distribution are symmetric about zero. Thus, <span class="math-container">$1-G(-z)=G(z)$</span>.</p>
<p>From the model above, we can derive the response probability for <span class="math-container">$y$</span>. First, we assumed that <span class="math-container">$y_i=1$</span> if and only if <span class="math-container">$y^{\ast}_i &gt; \gamma$</span>, with <span class="math-container">$\gamma$</span> set arbitrarily to zero. Thus,
<span class="math-container">\begin{align*}
    \mathbb{P}(y=1|x) =&amp; \mathbb{P}(y^{\ast} &gt;0|x) = \mathbb{P}(\beta_0 + x\beta + \epsilon &gt;0|x) = \\
    =&amp;\mathbb{P}[ \epsilon &gt; -(\beta_0 + x\beta ) |x] =1 - \mathbb{P}[ \epsilon \le -(\beta_0 + x\beta ) |x] \\
    =&amp; 1 - G[ - (\beta_0 + x\beta)] = G(\beta_0 + x \beta)
\end{align*}</span>
where <span class="math-container">$\mathbb{P}[ \epsilon \le -(\beta_0 + x\beta ) |x]$</span> is the cdf of the error term of the latent outcome.</p>

CC BY-SA 4.0
2023-12-23T14:33:49.930Z
2023-12-23T14:33:49.930Z
Tony
<p>Can you provide me with an intuition about what it means to derive a probit or logit model from an underlying latent variable model?</p>

<econometrics>
Probit and logit Model
Mark
# Probit and logit Model

(Asked by: username_0 on 2023-12-23)

Can you provide me with an intuition about what it means to derive a probit or logit model from an underlying latent variable model?

**Tags:** econometrics


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## 1 answers

### Answer by username_1 on 2023-12-23. Score: 5 | Accepted answer

Logit and probit models can be derived from an underlying latent variable model. To see this, let $y^{\ast}$ be an unobserved (i.e. latent) variable, and suppose that
\begin{equation}
y^{\ast} = \beta\_0 + x\beta + \epsilon, ~y=1[y^{\ast}>\gamma]
\label{latent}
\end{equation}
where $1[\cdot]$ is an indicator function taking value $1$ if $y^{\ast}>\gamma$ and zero otherwise. Here, $\gamma$ is a chosen threshold. If the latent unobserved utility $y^\ast$ is above the threshold, then $y$ takes the value one; otherwise, it takes the value zero.

The idea is to see the binary outcome $y=\{0,1 \}$ as a dichotomization of a latent continuous variable $y^{\ast}$ modeled with a canonical multiple linear regression model. For example, a bank can issue a loan ($y=1$) or choose not to ($y=0$). Think of it as having an underlying continuous latent outcome, $y^{\ast}$ which can be regarded as a measure of the choice utility.

In the model above, we assume that $\epsilon$ is independent of $x$ and that $\epsilon$ either has the standard logistic distribution or the standard normal distribution.

Recall, the standard logistic and standard normal distribution are symmetric about zero. Thus, $1-G(-z)=G(z)$.

From the model above, we can derive the response probability for $y$. First, we assumed that $y\_i=1$ if and only if $y^{\ast}\_i > \gamma$, with $\gamma$ set arbitrarily to zero. Thus,
\begin{align\*}
\mathbb{P}(y=1|x) =& \mathbb{P}(y^{\ast} >0|x) = \mathbb{P}(\beta\_0 + x\beta + \epsilon >0|x) = \\
=&\mathbb{P}[ \epsilon > -(\beta\_0 + x\beta ) |x] =1 - \mathbb{P}[ \epsilon \le -(\beta\_0 + x\beta ) |x] \\
=& 1 - G[ - (\beta\_0 + x\beta)] = G(\beta\_0 + x \beta)
\end{align\*}
where $\mathbb{P}[ \epsilon \le -(\beta\_0 + x\beta ) |x]$ is the cdf of the error term of the latent outcome.


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