A household survey asks five thousand families how much they gave to charity last year, and two thousand of them answer zero. Regress giving on income with ordinary least squares and the fitted line has to pass through that wall of zeros, so it comes out too flat, and it predicts negative giving for the poorest households, which nobody gave. Drop the zeros and regress only the givers, and the line is wrong in a different way, because the households that gave a little are exactly the ones whose unobserved generosity pushed them over the line. The Tobit model was built for this shape of data. It treats a zero not as a measurement but as a boundary the outcome could not cross, and it recovers the relationship that the wall of zeros hides. The price is one strong assumption about what a zero means, and most of the mistakes made with the model come from not asking whether that assumption holds.
Why a Pile of Zeros Breaks Least Squares
The problem has a name and a shape. A dependent variable is censored when it is observed only above or below some limit and takes the limit value otherwise: spending cannot be negative, hours worked cannot be negative, a top-coded income is recorded as the ceiling however far above it the true figure lies. The regressors are observed for everyone; it is the outcome that is squashed against the limit for part of the sample. Censoring is different from truncation, where observations beyond the limit are missing altogether, regressors and all, as when a dataset contains only households that gave something. Both produce a distorted picture, and the standard multiple regression fails on each for a related reason: the error term is no longer independent of the regressors once the sample has been shaped by the outcome.
James Tobin saw the shape in 1958 in household spending on durable goods, where a large fraction of families bought nothing in a given year. He noticed that the conditional mean of such a variable is not a straight line. At low income the mean is held near zero by the households at the boundary; at high income almost everyone spends, and the mean rises with income at the true rate. Between the two the relationship bends. A least squares line fitted to the whole sample averages the flat region and the rising region into a slope that is too small everywhere, and a line fitted only to the positive observations is biased as well, because the sample it uses was selected by the very error term the regression assumes it can ignore.
The figure makes the two failures visible. The grey points on the axis are households whose desired giving, if it could be measured, would have been negative; the survey records them as zero. The dashed line is the relationship that generated the data. The red line is what least squares returns when it treats the zeros as ordinary observations, and it is too flat because the zeros drag its left end up. The green curve is the conditional mean of the observed outcome, which is what the data can actually show, and it bends. Any straight line through it is wrong somewhere.
The Model and the Likelihood Behind It
The Tobit resolves the problem by writing down the process in two lines. There is a latent variable, the outcome as it would be without the boundary, which follows an ordinary linear model. And there is an observation rule that maps the latent variable to what the data records.
The estimation then uses each observation for exactly what it reveals. A positive observation reveals the latent value itself, and contributes the density of the error at that point. A zero reveals only that the latent value was at or below zero, and contributes the probability of that event, which is the normal distribution function evaluated at the negative of the index divided by the standard deviation. The log-likelihood adds the two kinds of contribution across the sample.
The first sum is the probit part of the model, and the second is the regression part, which is why the Tobit is sometimes described as a probit and a linear regression sharing one set of coefficients. The parameters are found by maximum likelihood, and the error standard deviation is estimated alongside the coefficients rather than recovered afterwards from residuals, because it appears inside the probability that an observation is censored. The same structure extends to a limit other than zero, to an upper limit such as a top-coded income, and to a variable censored at both ends, with the probability terms adjusted to the limits in question.
One Coefficient, Three Effects
The most common error in reading Tobit output is treating the coefficient as a marginal effect on the outcome, the way a least squares coefficient would be read. It is not. The coefficient is the effect of the regressor on the latent variable, the outcome as it would be without the boundary. Three different questions can be asked of the model and each has a different answer.
The effect on the latent variable is the coefficient itself. It answers a question about desired giving including the negative values nobody can act on, which is rarely the question a policy analyst has. The effect on the expected observed outcome across everyone, givers and non-givers together, is the coefficient scaled down by the probability of being above the limit at that point. And the effect on the expected outcome among those who are above the limit is smaller again, because it strips out the households who would be pulled across the boundary. The second is usually what an applied question means by the effect of income on giving.
McDonald and Moffitt showed in 1980 that this unconditional effect splits into two parts that are worth reporting separately: the change in the amount given by those already giving, weighted by the share who give, plus the change in the share who give, weighted by how much givers give. A programme that mostly moves people across the boundary and one that mostly raises the amounts of existing participants can have the same total effect and call for different designs. Because the scaling factor is a probability evaluated at particular values of the regressors, the marginal effect differs across the sample; the convention is to report it at the sample means or as an average across observations, and to say which.
| Quantity | Formula | Question it answers | Typical size |
|---|---|---|---|
| Effect on the latent variable | \( \beta_k \) | How desired giving would change, including values below zero that cannot be observed | Largest of the three |
| Effect on the expected observed outcome, everyone | \( \beta_k\,\Phi(z) \), with \( z = \mathbf{x}’\boldsymbol{\beta}/\sigma \) | How average giving in the population changes, counting non-givers as zero | Coefficient scaled by the share above the limit |
| Effect on the expected outcome, participants only | \( \beta_k\,[1 – \lambda(z)(z + \lambda(z))] \), with \( \lambda = \phi/\Phi \) | How much more existing givers give | Smallest of the three |
| Effect on the probability of a positive outcome | \( \phi(z)\,\beta_k/\sigma \) | How the share of givers changes | Read like a probit marginal effect |
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A rough check follows from the same arithmetic. If least squares is run on the full sample, its coefficients should come out at roughly the Tobit coefficients multiplied by the share of uncensored observations. A Tobit coefficient of ten on income with sixty percent of households giving should sit beside a least squares coefficient of about six. When the two are far apart from that relationship, something other than censoring is going on in the data, and the next two sections describe what it usually is.
When the Zeros Are Not Corner Solutions
The Tobit makes one assumption that does all the work: the same process decides whether the outcome is positive and how large it is. A household gives nothing because its desired giving, on the same scale as everyone else’s, fell below zero. That is a corner solution, the optimising story where a boundary is hit rather than a separate choice being made. For durable purchases, charitable gifts, and hours worked at the margin, it is often reasonable. For a great many other zeros it is not.
Spending on cigarettes is the standard example. A non-smoker’s zero is not the record of a person who would buy a small quantity if the price were lower; it is the record of a person outside the market. Two decisions are being made, whether to participate and how much to consume, and the variables that drive them can differ, as can the signs. Forcing both through one coefficient produces an estimate that describes neither decision. The remedy is a two-part or hurdle model, proposed by Cragg in 1971: a probit for the participation decision and a regression, truncated at zero, for the amount among participants. The Tobit is the special case of the hurdle model in which the two equations share the same coefficients up to scale, and that restriction can be tested with a likelihood ratio. If it is rejected, the Tobit was the wrong tool.
A third situation looks similar and needs a third model. When the zero is not a genuine outcome but a missing one, as with the wages of people who do not work, the data are not censored; they are selected. A non-worker does not have a wage of zero. She has a wage the market would pay that is unobserved because she declined it, and the decision to decline is correlated with what that wage would have been. That is the setting for the Heckman selection model, which estimates the participation equation first and corrects the wage equation for the selection it implies. The three models are easy to confuse because all three begin with a probit-shaped question, and the way to tell them apart is to ask what a zero in the data actually records.
| Model | What a zero means | Key assumption | Standard example |
|---|---|---|---|
| Tobit (censored regression) | A corner solution: desired value at or below the limit | One process decides participation and amount | Household spending on durables; charitable giving |
| Hurdle (two-part) model | A separate decision not to participate | Participation and amount can have different drivers | Cigarette or alcohol spending; visits to a doctor |
| Heckman selection model | An outcome that exists but is unobserved | Selection is driven by unobservables correlated with the outcome | Wages of non-workers; prices of unsold houses |
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What Breaks the Estimator
Least squares is forgiving in a way the Tobit is not. Heteroskedastic errors leave least squares coefficients consistent and only spoil the standard errors. In the Tobit, the probability that an observation is censored depends on the error variance, so if the variance changes with the regressors, the model has the probabilities wrong and the coefficients themselves are inconsistent. Non-normal errors do the same damage for the same reason: the probability of a zero is computed from a normal distribution function, and if the errors are skewed or heavy-tailed that probability is misstated for every observation. These are not refinements. Simulation evidence going back to the 1980s shows the biases can be large, and a Tobit estimated on survey expenditure data, where variances typically grow with income, is exposed to exactly this.
The defences are practical. Model the variance as a function of the regressors and estimate the heteroskedastic Tobit, which most software allows. Compare the results with Powell’s censored least absolute deviations estimator, which is consistent under heteroskedasticity and non-normality because it works with the median rather than the mean; a wide gap between the two is evidence that the standard Tobit is misspecified. Check the least squares ratio described above. And resist the transformation that seems to make the problem go away, regressing the logarithm of the outcome plus one, which does nothing about the boundary and adds an arbitrary constant whose choice moves the coefficients.
Two boundaries deserve a word. Top-coding, where incomes above a threshold are recorded as the threshold, is censoring from above, and the same model handles it with the limit reversed; our article on survey and tax records of top incomes shows how much a top code can hide. And a bounded outcome, a share between zero and one with piles at both ends, can be treated with a two-limit Tobit, though a fractional response model often fits its shape better, a choice that belongs to the family our article on nonlinear econometric models describes. Survey instruments themselves create censoring when they cap an answer or offer zero as the first option, which is one reason the design choices in our article on survey design reach the estimator later.
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Conclusion
The Tobit model answers a specific question well: when an outcome is squashed against a limit for part of the sample and the same process decides whether and how much, what is the relationship the boundary hides? It answers by using each observation for exactly what it reveals, the value for the uncensored and the probability for the censored, and it returns a coefficient that describes the latent outcome rather than the recorded one. That last point is the one most often lost. The coefficient has to be scaled by the probability of being above the limit before it says anything about average giving, hours or spending, and the split between more participants and larger amounts is usually the finding that matters.
The discipline is in asking what a zero means before choosing the model. A corner solution calls for the Tobit. A separate decision not to take part calls for a hurdle model, and the Tobit’s restriction that both decisions share one set of coefficients can be tested rather than assumed. An outcome that exists but was never recorded calls for a selection model. And whichever is chosen, the estimator’s dependence on normal, constant-variance errors is a real weakness rather than a footnote, so the comparison with a median-based estimator and the least squares ratio check belong in the results, not the drawer.
Frequently Asked Questions
What is the Tobit model used for?
For a dependent variable that is observed only above or below a limit and piles up at that limit for part of the sample, such as household spending that cannot fall below zero, hours worked, or an income that is top-coded. It estimates the relationship between the regressors and the underlying outcome that the limit conceals, using maximum likelihood rather than least squares.
What is the difference between censoring and truncation?
Under censoring, observations beyond the limit are recorded at the limit and their regressors are still observed, so the sample is complete but the outcome is squashed. Under truncation, those observations are missing altogether. The Tobit is a censored regression. A sample containing only households that gave something is truncated and needs a truncated regression instead.
Are Tobit coefficients marginal effects?
No. A coefficient measures the effect on the latent variable, the outcome as it would be without the boundary. The effect on the expected observed outcome is the coefficient multiplied by the probability of being above the limit, and the effect among participants only is smaller still. Reporting the coefficient as if it were the effect on average spending overstates it, often by a large margin.
When should a hurdle model be used instead of a Tobit?
When the zeros represent a decision not to participate rather than a corner solution, as with people who do not smoke at all. A hurdle model estimates participation and amount separately and allows them different drivers. The Tobit is the special case where both share one set of coefficients, and that restriction can be tested with a likelihood ratio test; rejection means the hurdle model is the right one.
Can the Tobit model handle top-coded or two-sided limits?
Yes. Top-coding is censoring from above, and the likelihood is adjusted so that observations at the ceiling contribute the probability of exceeding it. A variable with limits at both ends, such as a share between zero and one, can be estimated with a two-limit Tobit, though a fractional response model is often a better description of that shape.
What happens to the Tobit if the errors are heteroskedastic?
The coefficients become inconsistent, not merely the standard errors, because the probability of censoring depends on the error variance and is computed wrongly when the variance changes across observations. Non-normal errors cause the same problem. A heteroskedastic Tobit, or a comparison with Powell’s censored least absolute deviations estimator, is the usual response.
Thanks for reading! A zero in the data is a question before it is a number, and the model has to match the answer. Happy learning with MASEconomics
