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Tobit model

A latent Gaussian regression with censoring: why zeros contribute probability mass, how recorded and latent means differ, and when Tobit assumptions fail.

A Tobit model is a regression model in which a continuous latent outcome follows a linear relationship and the recorded outcome is censored at a known limit. In the classical version, the latent errors are normally distributed with constant variance. For a quant analysing a floor-coded financial measure, the model can distinguish changes in the underlying outcome from changes in what the reporting system records. A pile of zeros alone does not establish that this is the right model. Stata's Tobit manual distinguishes genuine measurement censoring from a corner solution, where zero is an actual economic choice.

One regression, two kinds of observation

For left censoring at zero, write

yi∗=μi+εi,μi=xi⊤β,εi∣xi∼N(0,σ2),σ>0,y_i^*=\mu_i+\varepsilon_i,\qquad \mu_i=\mathbf{x}_i^\top\boldsymbol\beta,\qquad \varepsilon_i\mid\mathbf{x}_i\sim\mathcal{N}(0,\sigma^2),\quad \sigma>0, yi=max⁡(0,yi∗).y_i=\max(0,y_i^*).

Here yi∗y_i^* is latent and yiy_i is recorded. The vector xi\mathbf{x}_i can include an intercept. The classical likelihood also assumes independent observations conditional on the covariates and a correctly specified censoring rule.

Imagine a fictional reporting system that records a signed adjustment exactly when positive but replaces every nonpositive adjustment with zero. Latent values −2-2, −0.2-0.2 and 0.40.4 become recorded values 00, 00 and 0.40.4. Both censored rows remain in the data, with their covariates. A zero says “at or below the floor”; it does not reveal the latent value.

Let ϕ\phi and Φ\Phi denote the standard normal density and CDF. The recorded distribution has an atom at zero plus a continuous density above zero:
P(yi=0∣xi)=Φ(−μi/σ),fy(y∣xi)=1σϕ ⁣(y−μiσ),y>0.\begin{aligned} \mathbb{P}(y_i=0\mid\mathbf{x}_i)&=\Phi(-\mu_i/\sigma),\\ f_y(y\mid\mathbf{x}_i)&=\frac{1}{\sigma}\phi\!\left(\frac{y-\mu_i}{\sigma}\right),\qquad y>0. \end{aligned}
The second line integrates to Φ(μi/σ)\Phi(\mu_i/\sigma), not one. Its missing mass is at zero. Dividing it by Φ(μi/σ)\Phi(\mu_i/\sigma) instead gives the density conditional on being positive, a different object.

This distinction drives the censored likelihood. For independent rows, each contribution is

Li(β,σ)={Φ(−μi/σ),yi=0,σ−1ϕ((yi−μi)/σ),yi>0.L_i(\boldsymbol\beta,\sigma)= \begin{cases} \Phi(-\mu_i/\sigma),&y_i=0,\\ \sigma^{-1}\phi((y_i-\mu_i)/\sigma),&y_i>0. \end{cases}
Maximum likelihood estimation combines these contributions to estimate the latent coefficients and scale. This is the zero-floor specialization of the probability and density contributions in Stata's interval-regression formulas.

Three means from the same model

At one fixed covariate pattern, suppose μ=−0.5\mu=-0.5 and σ=1\sigma=1, in arbitrary outcome units. Then

P(y=0)=Φ(0.5)≈0.69146246.\mathbb{P}(y=0)=\Phi(0.5)\approx0.69146246.
About 69.15% of the population is recorded at zero, and 30.85% is positive. There are three distinct averages:
TargetFormulaValue
Latent outcomeE[y∗]=μ\mathbb{E}[y^*]=\mu−0.50000000-0.50000000
Recorded outcome, including zerosE[y]=μΦ(μ/σ)+σϕ(μ/σ)\mathbb{E}[y]=\mu\Phi(\mu/\sigma)+\sigma\phi(\mu/\sigma)0.197796560.19779656
Positive observations onlyE[y∣y>0]=μ+σϕ(μ/σ)/Φ(μ/σ)\mathbb{E}[y\mid y>0]=\mu+\sigma\phi(\mu/\sigma)/\Phi(\mu/\sigma)0.641077770.64107777

The recorded mean equals the positive probability times the positive-only mean: 0.30853754×0.64107777≈0.197796560.30853754\times0.64107777\approx0.19779656. The other observations contribute zero. Flooring the latent mean would give max⁡(0,−0.5)=0\max(0,-0.5)=0, which misses the positive tail's contribution.

Synthetic Tobit model with latent mean minus 0.5 and standard deviation 1: 69.15% of probability becomes an atom at zero and 30.85% remains above zero; latent, recorded and positive-only means are minus 0.50000, 0.19780 and 0.64108.
Synthetic Tobit model with latent mean minus 0.5 and standard deviation 1: 69.15% of probability becomes an atom at zero and 30.85% remains above zero; latent, recorded and positive-only means are minus 0.50000, 0.19780 and 0.64108.
Original synthetic illustration. Probability mass and density use separate vertical scales; the atom's height must not be compared with the density curve's height.

Which effect does a coefficient describe?

A Tobit coefficient βk\beta_k is the slope of the latent conditional mean. With a continuous regressor entering μ\mu linearly, a fixed zero floor and constant σ\sigma, differentiating the recorded mean gives
∂E[y∣x]∂xk=βkΦ(μ/σ).\frac{\partial\mathbb{E}[y\mid\mathbf{x}]}{\partial x_k} =\beta_k\Phi(\mu/\sigma).
In this example, a latent slope of 11 gives a recorded-mean marginal effect of about 0.30854 at this covariate pattern. It is a local derivative, not the exact effect of a finite one-unit change. An average marginal effect averages such derivatives over chosen covariate patterns. Binary regressors require finite differences. Stata's postestimation manual separates latent predictions, censored means and positive-conditional means.
Ordinary linear regression on recorded values generally does not recover β\boldsymbol\beta, because the recorded conditional mean is nonlinear in x\mathbf{x}. Dropping the zeros does not recover it either: the positive-only conditional mean contains an extra selection term.

Diagnose the zeros before choosing the model

  • Censoring: the row remains; a known reporting rule hides the outcome beyond a threshold. The likelihood uses the probability of that hidden interval.
  • Truncation: rows outside the observation range are excluded. For a sample restricted to positive latent outcomes, the normal density is divided by the probability of being positive. Arbitrary missing outcomes or participation-based selection require their own observation model.
  • Structural zeros: inactivity, nonparticipation or no demand may generate real zeros. A Tobit corner-solution interpretation can be appropriate, but it imposes one latent index on both the probability of a positive outcome and its magnitude. A separate participation process can call for a two-part or selection model.
Normality and constant conditional variance are substantive restrictions. Heteroskedasticity or nonnormal latent errors can make classical Tobit coefficient estimates inconsistent; changing standard errors alone does not repair the likelihood. These limits are discussed in Croissant's Tobit chapter, §11.2. Also check whether the normal latent support makes sense for the quantity being modelled.
For the likelihood foundations, Estimation and Maximum Likelihood derives probability-mass and density likelihoods and explains their regularity conditions. Its public preview is open; the full lesson requires membership.

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