Deriving the Left-Censored Tobit Likelihood

Why Censoring Changes the Likelihood

When an outcome is censored, the data do not always reveal its latent value. A recorded value at the censoring point tells us that the latent outcome fell at or below that point; it does not tell us how far below. The likelihood must therefore use the probability of that event rather than the density evaluated at the recorded value. By contrast, an uncensored observation reveals the outcome itself and contributes its density.

The left-censored Tobit model makes this distinction explicit by starting with a latent linear regression and then applying an observation rule. This tutorial derives its likelihood conditional on the regressors. It focuses on how the observation rule determines each contribution, not on interpreting marginal effects for the observed outcome.

Latent Regression and Observation Rule

For individual \(i\), let \(y_i^*\) be the latent outcome and \(x_i\) the vector of observed regressors, including any intercept. The latent regression is

y_i^* = x_i'\beta + u_i,\qquad u_i\mid x_i \sim N(0,\sigma^2),\qquad \sigma>0.

Write \(\mu_i=x_i'\beta\). Thus, conditional on \(x_i\), the latent outcome has a normal distribution with mean \(\mu_i\) and variance \(\sigma^2\). Let \(c\) be a known left-censoring point. The recorded outcome is

y_i = \max(c,y_i^*).

If \(y_i^*\le c\), the recorded value is \(c\). If \(y_i^*>c\), the recorded value equals the latent value. Define \(d_i=1\) when \(y_i>c\), and \(d_i=0\) when \(y_i=c\). Under a continuous normal distribution, the probability that \(y_i^*\) is exactly \(c\) is zero, so the boundary convention does not affect the likelihood.

Model assumptions: Conditional on the observed regressors, the latent outcome is linear with Gaussian errors of mean zero and common variance \(\sigma^2\). Observations are conditionally independent, the censoring point \(c\) is known, and the observation rule is \(y_i=\max(c,y_i^*)\). The likelihood below is conditional on the regressors; it does not require a probability model for how the regressors were sampled.

Let \(\phi(z)=(2\pi)^{-1/2}\exp(-z^2/2)\) denote the standard normal density, and let \(\Phi(z)=P(Z\le z)\) for \(Z\sim N(0,1)\) denote its cumulative distribution function. These functions enter the two types of likelihood contribution.

Deriving the Two Contributions

For an uncensored observation, \(y_i>c\) and \(y_i=y_i^*\). Its conditional density is the latent normal density evaluated at the observed outcome. Standardizing the normal variable gives

f(y_i\mid x_i;\beta,\sigma) = \frac{1}{\sigma} \phi\!\left(\frac{y_i-\mu_i}{\sigma}\right), \qquad y_i>c.

The factor \(1/\sigma\) is required when converting the standard normal density to a density on the outcome scale. This contribution is a density, not a probability at one exact outcome.

For an observation recorded at \(c\), the observed data reveal the event \(y_i^*\le c\). Since \(y_i^*\mid x_i\) is normal with mean \(\mu_i\) and standard deviation \(\sigma\), standardization gives

P(y_i=c\mid x_i;\beta,\sigma) = P(y_i^*\le c\mid x_i) = \Phi\!\left(\frac{c-\mu_i}{\sigma}\right).

This is a probability contribution, representing all latent values at or below the censoring point. Substituting the normal density at \(c\) would incorrectly treat the censored value as if the latent outcome were observed exactly.

Identify what is observed.
If \(y_i>c\), the latent outcome is observed. If \(y_i=c\), only the event \(y_i^*\le c\) is observed.

Translate the observation into a likelihood contribution.
Use the latent density for an uncensored value and the probability of the censoring event for a censored value.

Combine contributions across observations.
Conditional independence makes the sample likelihood the product of the individual contributions.

The Sample Likelihood

Conditional independence means the joint conditional likelihood is the product of the observation-level terms. With \(d_i=1\) for an uncensored observation and \(d_i=0\) for a censored observation, the likelihood for \(n\) observations is

L(\beta,\sigma) = \prod_{i=1}^{n} \left[ \frac{1}{\sigma} \phi\!\left(\frac{y_i-x_i'\beta}{\sigma}\right) \right]^{d_i} \left[ \Phi\!\left(\frac{c-x_i'\beta}{\sigma}\right) \right]^{1-d_i}.

When \(d_i=1\), the density term is raised to the first power and the censoring probability to the zeroth power. When \(d_i=0\), the reverse occurs. This compact expression assumes the observed data follow the stated censoring rule, so recorded outcomes are at \(c\) or above it.

Taking logarithms converts the product to a sum and is usually more convenient for computation:

\ell(\beta,\sigma) = \sum_{i=1}^{n} \left\{ d_i\left[ -\log\sigma+ \log\phi\!\left(\frac{y_i-x_i'\beta}{\sigma}\right) \right] + (1-d_i) \log\Phi\!\left(\frac{c-x_i'\beta}{\sigma}\right) \right\}.

The log likelihood retains the same distinction: a log density for each uncensored value and a log probability for each censored observation. Maximum likelihood estimation chooses parameter values that maximize this joint likelihood, or equivalently its log. The likelihood itself is not a probability that the parameters are true.

Worked Example: One Censored and One Uncensored Observation

Suppose the censoring point is \(c=0\), both observations have latent conditional mean \(\mu_i=1\), and \(\sigma=1\). One outcome is recorded as \(0\); the other is observed as \(1.1\). Assume the observations are conditionally independent. Calculate their joint likelihood contribution.

For the censored observation, the latent outcome must be at or below zero. Its contribution is

\Phi\!\left(\frac{0-1}{1}\right)=\Phi(-1)\approx 0.1586553.

For the uncensored observation, the residual is \(1.1-1=0.1\), so its density contribution is

\frac{1}{1}\phi\!\left(\frac{1.1-1}{1}\right) =\phi(0.1)\approx 0.3969525.

Conditional independence gives the joint contribution. Using the unrounded values,

\Phi(-1)\phi(0.1) \approx 0.1586553(0.3969525) \approx 0.0629786 \approx 0.06298.

The joint contribution is approximately \(0.06298\). Multiplying the displayed values rounded to \(0.1587\) and \(0.3970\) instead gives approximately \(0.06300\); that small difference is due to rounding the factors first.

Worked Example: A Nonzero Censoring Point and Nonunit Scale

Let \(c=2\), \(\mu_i=3\), and \(\sigma=2\). Find the contribution for an observation recorded at the censoring point, then for an uncensored observation \(y_i=4\).

For the censored observation, standardize the threshold relative to the latent mean:

P(y_i=c\mid x_i) = \Phi\!\left(\frac{2-3}{2}\right) = \Phi(-0.5) \approx 0.3085375.

For the uncensored value \(4\), the standardized residual is \((4-3)/2=0.5\). The density on the original outcome scale includes the factor \(1/\sigma=1/2\):

f(4\mid x_i) = \frac{1}{2}\phi(0.5) \approx \frac{1}{2}(0.3520653) \approx 0.1760327.

Thus the two contributions are approximately \(0.3085\) and \(0.1760\), respectively. They have different roles: the first is a dimensionless probability, while the second is a density per unit of the outcome.

Worked Example: Log Likelihood for a Small Sample

Consider three conditionally independent observations with censoring point \(c=0\), common latent conditional mean \(\mu_i=0\), and \(\sigma=1\). The first is recorded at zero, the second is observed at \(1\), and the third is observed at \(0.5\). Calculate the likelihood and log likelihood at these parameter values.

The censored observation contributes \(\Phi(0)=0.5\). The uncensored observations contribute \(\phi(1)\) and \(\phi(0.5)\). Therefore,

L = \Phi(0)\phi(1)\phi(0.5) = (0.5)(0.2419707)(0.3520653) \approx 0.0425948.

The log likelihood can be calculated as the sum of the log contributions:

\ell = \log(0.5)+\log\phi(1)+\log\phi(0.5) = -0.6931472-1.4189385-1.0439385 \approx -3.1560242.

The product and log-sum describe the same sample likelihood because \(\log(ab)=\log(a)+\log(b)\) for positive contributions. The log likelihood is negative here; that is not an error, since a density can be less than one and its logarithm can be negative.

Why the Mixed Contributions Matter

The observed outcome has a point mass at \(c\) and a continuous density above \(c\). The point mass comes from all latent outcomes at or below \(c\); the continuous part comes from latent values that are observed without censoring. Consequently, this distribution cannot be represented by evaluating one ordinary continuous density at every recorded value. The product likelihood combines probabilities for the censored observations with densities for the uncensored observations, as required by what the observation rule reveals.

A common mistake is to plug \(y_i=c\) into the latent normal density for a censored observation. That uses the likelihood for observing the latent outcome exactly at \(c\), whereas the data only say it lies at or below \(c\). The correct contribution is \(\Phi((c-x_i'\beta)/\sigma)\). Another frequent error is to omit \(1/\sigma\) from the uncensored density; doing so gives the wrong density scale and therefore the wrong likelihood when \(\sigma\) varies.

Key takeaway: Under left censoring at \(c\), an observation at \(c\) contributes \(\Phi((c-x_i'\beta)/\sigma)\), while an observation above \(c\) contributes \(\sigma^{-1}\phi((y_i-x_i'\beta)/\sigma)\). Multiply these terms across conditionally independent observations to obtain the Tobit likelihood.

Check Your Understanding

For each question, identify the observation event before writing its likelihood contribution.

With \(c=0\), \(x_i'\beta=0.8\), and \(\sigma=1.5\), what expression gives the contribution when \(y_i=0\)?
With \(c=1\), \(x_i'\beta=2\), and \(\sigma=2\), write the contribution for an uncensored observation \(y_i=3\).
Why does a censored observation contribute a probability rather than the latent density evaluated at \(c\)?
State the role of conditional independence in constructing the sample likelihood.
What factor must multiply the standard normal density for an uncensored observation when \(\sigma\ne1\), and why?