Common mistake 1 with Tobit (censored)

Using Tobit on a variable that merely has many zeros for a real reason. If zero is a genuine value rather than a censored one, Tobit is the wrong model - a count model or a two-part model may be right.

Common mistake 2 with Tobit (censored)

Reading the coefficient as the effect on the observed outcome. It is not; it is roughly 1/Φ times too big.

Common mistake 3 with Tobit (censored)

Forgetting that Tobit forces one mechanism onto two decisions. If the sign of an effect on participation differs from its effect on the amount, Tobit cannot represent that and will average the two.

Common mistake 4 with Tobit (censored)

Comparing a Tobit coefficient with an OLS coefficient from another study. They are on different scales - one describes the latent variable, the other the observed one - and the ratio between them depends on how much of each sample is censored.

EconStatsCraft › Tobit (censored)

Tobit (censored)

For an outcome that cannot go past a limit, so values bunch up at it - hours worked at zero, spending at zero, a top-coded wage. OLS treats the pile-up as genuine low values and is biased towards zero; Tobit models the censoring and reports both the latent and the observed effect.

What it needs

Censored outcome - The variable with a pile-up at a limit.
Independent variables

The intuition

Some outcomes pile up at a limit. Hours worked cannot go below zero, and a third of your sample sits exactly at zero - not because those people wanted exactly zero hours, but because the value they wanted was negative and reality clipped it. OLS treats those zeros as genuine observations of the outcome, which flattens the fitted line towards the floor. Tobit says the zeros are censored reports of a latent variable and estimates the line that variable actually follows.

When to use it

A histogram of the outcome shows a spike at one end and a smooth distribution above it. That spike is a corner solution, not a cluster of genuinely tiny values.

Where the estimator comes from

Posit a latent y_i^=x_i'\beta+u_i with u_i\sim N(0,\sigma^2), observed as y_i=\max(0,y_i^). Each observation contributes to the likelihood differently: an uncensored one contributes the density, a censored one contributes the whole probability mass below the limit. \ln L=\sum_{y_i>0}\left[-\ln\sigma+\ln\phi\!\left(\tfrac{y_i-x_i'\beta}{\sigma}\right)\right]+\sum_{y_i=0}\ln\Phi\!\left(-\tfrac{x_i'\beta}{\sigma}\right). Note that \sigma appears in the censored term, so unlike OLS the scale is not a nuisance parameter that separates out - \beta and \sigma must be estimated jointly, which is why a two-step fix does not work.

The model

y^{*}=\mathbf{x}^{\prime}\boldsymbol{\beta}+\varepsilon,\qquad y=\max(y^{*},L),\qquad \frac{\partial E[y\mid \mathbf{x}]}{\partial x_{j}}=\beta_{j}\,\Phi\!\left(\frac{\mathbf{x}^{\prime}\boldsymbol{\beta}-L}{\sigma}\right)

What it assumes

The errors are normal and homoskedastic. Tobit is far less forgiving about this than OLS - non-normality biases the coefficients, it does not just spoil the standard errors.
One single process drives both whether you are at the limit and how far above it you are. If different forces decide participation and quantity, you want Heckman instead.
The limit is a genuine censoring point, not a measurement convention you could undo.
The censoring threshold is known for every observation. A cap that changes over time - a top-coded income recorded against a different ceiling each year - has to be supplied per row rather than assumed constant, or the model attributes to behaviour what is really a change in the recording rule.

Statistical properties

Beta is the effect on the LATENT variable, not on what you observe. The effect on the observed mean is beta times Phi(xb/sigma), which is always smaller, and the effect on the mean among the uncensored is smaller still.
OLS on the censored data is biased towards zero, and dropping the zeros does not help - it makes it slightly worse, because selecting on y > 0 selects on the error.
Tobit imposes one parameter vector for both decisions: the same beta governs whether y is positive and how large it is. If different forces drive participation and intensity, that restriction is false and a hurdle or Cragg model is the right specification.
The estimator is far more sensitive to non-normality and heteroskedasticity than OLS is. Under either, Tobit is INCONSISTENT, not merely inefficient, because the likelihood for the censored mass depends on the whole distributional assumption.
Censoring is a property of the recording, not of the population. Do not use Tobit for a variable that is genuinely zero for many people - a corner solution is a real choice, not a clipped one.
A censoring point other than zero is handled by shifting; a two-sided limit needs both terms in the likelihood.

How to read the output

There are TWO effects. The coefficient is the effect on the latent y*; multiply by Φ(·) to get the effect on what you actually observe. Quote the second one unless you are explicitly talking about the latent variable.
Compare the Tobit and OLS columns. OLS on a censored outcome is attenuated towards zero, and the gap shows you how much the censoring was costing.
sigma is the standard deviation of the latent error, so it is not comparable with an OLS root MSE from the same data.
The marginal effect on the observed outcome is not constant across the sample: it is scaled by the probability of being above the limit, which differs for every unit. The figure reported is an average over the sample in front of you and does not transfer to a population censored at a different rate.

A worked reading

Annual hours worked for 900 women, of whom 338 - 37.6% - report zero. Tobit puts the effect of a year of schooling on the LATENT outcome at 82.37 hours (SE 10.59), while OLS on the same data returns 53.27, an error of -35.3%. Experience is understated by 40.1%, children by 48.1% and husband's income by 41.6%: censoring flattens every slope, and it flattens them all in the same direction. The resolution of the apparent paradox is in the third table. The Tobit effect on the OBSERVED outcome - what actually changes in the data, averaged over women who may stay at zero - is 52.60 for schooling, almost exactly what OLS reported. OLS was not estimating the latent slope badly; it was estimating a different quantity correctly. Which of the two you want depends on the question: 82 hours is the answer for a woman already in work, 53 is the answer for the population including those who will not enter at all.

Symptom → diagnosis → what to do

If you see this	It means	Do this
Almost nothing censored	Tobit and OLS will agree	Use OLS; the extra assumptions buy nothing
Most of the sample at the limit	Very little information about the slope	Estimates will be imprecise and fragile
Tobit coefficient far above OLS	Expected. They measure different things	Report the marginal effect, not the raw coefficient
Residuals heteroskedastic	Tobit is inconsistent, not just inefficient	Model the variance or use a semiparametric estimator
Participation and amount driven by different variables	The single-index restriction is false	Use a hurdle model and test the restriction
Predicted values below the limit	Those are latent predictions	Truncate them before comparing with data

Common mistakes

Using Tobit on a variable that merely has many zeros for a real reason. If zero is a genuine value rather than a censored one, Tobit is the wrong model - a count model or a two-part model may be right.
Reading the coefficient as the effect on the observed outcome. It is not; it is roughly 1/Φ times too big.
Forgetting that Tobit forces one mechanism onto two decisions. If the sign of an effect on participation differs from its effect on the amount, Tobit cannot represent that and will average the two.
Comparing a Tobit coefficient with an OLS coefficient from another study. They are on different scales - one describes the latent variable, the other the observed one - and the ratio between them depends on how much of each sample is censored.

Where to read this properly

Tobin (1958), "Estimation of Relationships for Limited Dependent Variables", Econometrica 26(1).
Amemiya (1984), "Tobit Models: A Survey", Journal of Econometrics 24(1).
Wooldridge, Econometric Analysis of Cross Section and Panel Data (2010), ch. 17.
Cragg (1971), Econometrica 39(5), for the hurdle alternative.

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Tobit (censored)

What it needs

The intuition

When to use it

Where the estimator comes from

The model

What it assumes

Statistical properties

How to read the output

A worked reading

Symptom → diagnosis → what to do

Common mistakes

Where to read this properly

See also

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