Likelihood for Two-Limit Tobit Models

Why Two Limits Require Three Kinds of Contributions

A recorded outcome may be constrained by both a lower and an upper reporting limit. For example, a measurement below a detection threshold may be recorded at that threshold, while a measurement above the instrument’s range may be recorded at its maximum. The observed value at either limit does not reveal the exact latent value; it tells us that the latent value lies in a tail of its distribution.

That distinction determines the likelihood. An observation recorded at the lower limit contributes the probability of being at or below that limit. An observation recorded at the upper limit contributes the probability of being at or above that limit. An observation strictly between the limits is observed at its actual value and contributes a probability density. The model below combines these three contributions.

Model and Likelihood

Definition: In a two-limit Tobit model, the latent outcome for observation \(i\) is \(y_i^*=x_i'\beta+\varepsilon_i\), where \(x_i\) is a vector of observed covariates, \(\beta\) is a vector of regression coefficients, and \(\varepsilon_i\mid x_i\sim N(0,\sigma^2)\) with \(\sigma>0\). Write \(\mu_i=x_i'\beta\). For known limits \(L<U\), the observed outcome is \(y_i=L\) if \(y_i^*\leq L\), \(y_i=y_i^*\) if \(L<y_i^*<U\), and \(y_i=U\) if \(y_i^*\geq U\).

The normality assumption implies that, conditional on \(x_i\), the latent outcome has density \(f_i(t)=\frac{1}{\sigma}\phi\!\left(\frac{t-\mu_i}{\sigma}\right)\), where \(\phi\) is the standard normal density. Let \(\Phi\) denote the standard normal cumulative distribution function. The model assumes that the censoring rule is known and applied as stated, and that the observations are independent conditional on their covariates. These assumptions allow the sample likelihood to be written as a product of observation-specific contributions.

Theorem: For observed data \(y_1,\ldots,y_n\), the likelihood contribution \(L_i(\beta,\sigma)\) is \[ L_i(\beta,\sigma)= \begin{cases} \Phi\!\left(\dfrac{L-\mu_i}{\sigma}\right), & y_i=L,\\[6pt] \dfrac{1}{\sigma}\phi\!\left(\dfrac{y_i-\mu_i}{\sigma}\right), & L<y_i<U,\\[6pt] 1-\Phi\!\left(\dfrac{U-\mu_i}{\sigma}\right), & y_i=U. \end{cases} \] Under conditional independence, \(L(\beta,\sigma)=\prod_{i=1}^n L_i(\beta,\sigma)\), and the log-likelihood is \(\ell(\beta,\sigma)=\sum_{i=1}^n\log L_i(\beta,\sigma)\).

A value exactly equal to a limit is treated as censored only when the recording rule says it represents a latent value at or beyond that limit. If the data system can also record an exact uncensored value equal to a threshold, the observation mechanism must distinguish that case; the recorded number alone would not identify its likelihood contribution.

Why the Contributions Have Different Forms

At the lower limit, observing \(y_i=L\) means the event \(y_i^*\leq L\). Standardizing the latent normal outcome gives \(P(y_i^*\leq L\mid x_i)=P((y_i^*-\mu_i)/\sigma\leq(L-\mu_i)/\sigma)=\Phi((L-\mu_i)/\sigma)\). At the upper limit, the recorded value means \(y_i^*\geq U\), so its contribution is \(P(y_i^*\geq U\mid x_i)=1-\Phi((U-\mu_i)/\sigma)\).

For an interior observation, the latent outcome is observed rather than known only to fall in a tail. Its likelihood contribution is therefore the density at the observed value, \(f_i(y_i)\). More formally, the probability of observing an interior value in a small interval from \(y_i\) to \(y_i+\Delta\) is approximately \(f_i(y_i)\Delta\) for small \(\Delta\); the density is the likelihood contribution after accounting for the interval width. The density is not itself a probability, and it may have units of inverse outcome units. This is standard for likelihoods with continuous observations.

The log form follows directly from the product rule and the identity that the logarithm of a product is the sum of the logarithms. It is often easier to work with because it avoids multiplying many small numbers. For an interior observation, its log contribution can be written \(\log f_i(y_i)=-\log\sigma-\frac{1}{2}\log(2\pi)-\frac{(y_i-\mu_i)^2}{2\sigma^2}\). The lower- and upper-limit log contributions are the logarithms of their respective tail probabilities.

Worked Example: Writing the Likelihood for a Mixed Sample

Suppose a hypothetical instrument reports outcomes only between \(L=0\) and \(U=10\), inclusive, censoring latent values outside that range to the nearest limit. For three observations, suppose the observed values are \(0,4,10\), and their model means are \(\mu_1=2,\mu_2=3,\mu_3=8\). Let \(\sigma=2\). Construct and evaluate the likelihood at these parameter values.

The first observation is at the lower limit, so it contributes \(\Phi((0-2)/2)=\Phi(-1)=0.1586553\). The second is interior, so its contribution is the density at 4: \(\frac{1}{2}\phi((4-3)/2)=\frac{1}{2}\phi(0.5)=0.1760327\). The third is at the upper limit, so its contribution is \(1-\Phi((10-8)/2)=1-\Phi(1)=0.1586553\).

The sample likelihood is the product of these contributions:

L\approx (0.1586553)(0.1760327)(0.1586553)\approx 0.0044310.

Using the unrounded contributions when summing their logarithms gives \(\ell\approx -1.8410-1.7371-1.8410=-5.4191\). This is the log-likelihood at the specified parameter values, not a probability that the parameter values are correct.

Constructing the Likelihood in Practice

For a data set, the classification is observation-specific. The means \(\mu_i=x_i'\beta\) may differ across observations, but the contribution is selected by whether the recorded outcome is at the lower limit, strictly inside the limits, or at the upper limit. The following sequence keeps the observation rule and the probability calculation aligned.

State the latent model and limits.
Write \(\mu_i=x_i'\beta\), specify \(\sigma>0\), and identify known \(L<U\) and the censoring rule.

Classify each recorded outcome.
Use the recording mechanism to determine whether it indicates \(y_i^*\leq L\), an observed interior value, or \(y_i^*\geq U\).

Write each contribution.
Use the lower-tail probability, interior density, or upper-tail probability for that observation.

Combine contributions.
Multiply them for the likelihood or sum their logarithms for the log-likelihood, using conditional independence.

This procedure constructs the likelihood; it does not by itself establish that the Gaussian model is appropriate or that the parameters can be estimated precisely. The likelihood is a function of \(\beta\) and \(\sigma\) for the observed data. Estimation would select parameter values that maximize it, but the central issue here is correctly representing what each recorded observation tells us about its latent value.

Worked Example: Calculating a Log-Likelihood With Different Tail Events

Consider a second hypothetical sample with \(L=0\), \(U=10\), and \(\sigma=1\). Three observations have \(y=(0,2,10)\), with model means \(\mu=(1,2,11)\). Find the log-likelihood at these parameter values.

For \(y_1=0\), the lower-limit contribution is \(\Phi((0-1)/1)=\Phi(-1)=0.1586553\), with log contribution approximately \(-1.8410\). For the interior value \(y_2=2\), the density contribution is \(\phi((2-2)/1)=\phi(0)=0.3989423\), with log contribution approximately \(-0.9189\).

For \(y_3=10\), the upper-limit contribution is \(1-\Phi((10-11)/1)=1-\Phi(-1)=\Phi(1)=0.8413447\), with log contribution approximately \(-0.1728\). Therefore,

\ell\approx -1.8410-0.9189-0.1728=-2.9327.

The upper-limit contribution is relatively large here because the model mean for that observation is 11, above the reporting limit. It is still a tail probability: the recorded value 10 indicates that the latent outcome is at least 10, not that it equals 10 exactly.

A Further Worked Likelihood

Worked Example: Four Observations With Both Limits

For a hypothetical set of measurements, let \(L=0\), \(U=10\), and \(\sigma=2\). The observed outcomes are \(0,3,10,6\), and the corresponding model means are \(1,4,8,5\). Calculate the log-likelihood at these values.

The first observation is lower-censored and contributes \(\Phi((0-1)/2)=\Phi(-0.5)\approx0.3085375\), whose logarithm is approximately \(-1.1759\). The second is interior and contributes \(\frac{1}{2}\phi((3-4)/2)=\frac{1}{2}\phi(-0.5)\approx0.1760327\), with log approximately \(-1.7371\).

The third observation is upper-censored and contributes \(1-\Phi((10-8)/2)=1-\Phi(1)\approx0.1586553\), with log approximately \(-1.8410\). The fourth is interior and contributes \(\frac{1}{2}\phi((6-5)/2)=\frac{1}{2}\phi(0.5)\approx0.1760327\), again with log approximately \(-1.7371\).

Adding the four log contributions gives \(\ell\approx-1.1759-1.7371-1.8410-1.7371=-6.4911\). The two interior observations have the same density contribution because their standardized residuals have the same absolute value under the normal density. The two censored observations contribute different probabilities because their distances from the respective limits, relative to their model means, differ.

Assumptions and Common Pitfalls

The likelihood’s interpretation depends on the full observation process, not just the shape of the formula. The limits must be known, and a value at a limit must represent censoring as specified. The Gaussian latent regression and common \(\sigma\) are maintained throughout this model. Conditional independence is what justifies multiplying observation-level contributions; if observations are dependent, that product need not be their joint likelihood. Also, an interior observation must lie strictly between \(L\) and \(U\) under the stated rule.

Using a density at a censoring limit. A recorded lower-limit value does not identify an exact latent outcome of \(L\); it identifies the event \(y_i^*\leq L\). Use the lower-tail probability. Apply the corresponding upper-tail probability at \(U\).
Using a tail probability for an interior value. An interior value is observed as a continuous outcome, so its contribution is a density evaluated at that value, not the probability of being below or above it.
Reversing the upper tail. The event at the upper limit is \(y_i^*\geq U\), so the contribution is \(1-\Phi((U-\mu_i)/\sigma)\), not \(\Phi((U-\mu_i)/\sigma)\).
Forgetting the scale factor in the density. The interior density includes \(1/\sigma\). Leaving it out does not give the Gaussian density of an outcome measured in its original units.
Treating a likelihood as a probability about parameters. The likelihood evaluates how compatible the observed data are with parameter values. It is not the probability that those parameter values are true.

Key takeaway: In a two-limit Tobit model, the observation rule maps latent outcomes to recorded values. That mapping gives a lower-tail probability at the lower limit, a Gaussian density in the interior, and an upper-tail probability at the upper limit. The sample likelihood multiplies the contributions, or equivalently sums their logarithms.

Check Your Understanding

Answer each question using the two-limit Gaussian model and its stated censoring rule.

For \(L=0\) and \(U=10\), what event about the latent outcome is indicated by an observation recorded as 0? Write its likelihood contribution.
Why does an interior observation contribute a density rather than a tail probability?
If an observation is recorded as \(U\), should its contribution be \(\Phi((U-\mu_i)/\sigma)\) or \(1-\Phi((U-\mu_i)/\sigma)\)? Explain using the event represented.
State the assumption that permits the sample likelihood to be written as a product of observation-specific contributions.
For \(L=0\), \(U=10\), \(\mu_i=2\), and \(\sigma=2\), write the contribution for an observation at the lower limit and simplify its standardized argument.