A Tobit model fits a regression to an outcome that piles up at a limit — donations at zero, hours at zero, scores at a ceiling — by treating a value at the limit as evidence that the underlying quantity was at or beyond it, rather than as a measurement. Least squares reads those values literally, which flattens the fitted line toward zero, and the bias grows with how much of the sample is stacked at the limit.
Why dropping the censored rows fails too
The naive mistake is obvious once stated: a household recorded as donating £0 did not necessarily want to donate £0. The zero records that its latent inclination was at or below zero, and treating it as the number zero puts those households higher on the outcome scale than they belong, which flattens the line. On the preset that costs 44.3%.
The second response looks much more careful. If those zeros are not real measurements, remove them and fit the line to households whose donations are genuine amounts. That gives 0.5725 — 44.1% short. Within 0.002 of the naive answer. All the care bought nothing.
The reason is that the rows removed are not a random subset. Households at zero are systematically the ones with lower income, so dropping them removes the whole bottom of the predictor range along with the bottom of the outcome range. What is left is a truncated sample where the surviving low- income households are the unusually generous ones, and fitting a line through that compresses the slope just as effectively.
Both mistakes come from the same place: treating a boundary as data. The censored fit does neither. An uncensored row contributes its value, and a censored row contributes only the statement that the latent value was at or below the limit — which is exactly what was observed and no more.
How big the bias is
The size is predictable, and worth knowing because it tells you when to care. Two hundred datasets were simulated at each of four censoring levels with a true slope of 1.5.
At 13.2% censored, least squares averaged 1.2971. At 38.9%, 0.9142. At 61.0%, 0.5885. At 75.7%, 0.3689. The censored regression averaged 1.4956, 1.4973, 1.5064 and 1.5187 across the same four — essentially exact at every level.
Line those up against the share of rows surviving uncensored and the pattern is hard to miss: 86.8%, 61.1%, 39.0% and 24.3% of the truth would be 1.302, 0.917, 0.585 and 0.365. The measured values are 1.2971, 0.9142, 0.5885 and 0.3689. The attenuation factor is, near enough, the uncensored share.
That gives a usable rule. With a tenth of the sample at the limit the bias is around a tenth and probably not worth a different model. With half the sample at the limit the slope is roughly halved, and no amount of careful interpretation recovers it. The lightly censored preset is on the page for exactly this reason: at 8.2% censored the shortfall is 6.8%, and reporting the ordinary regression would be defensible.
Two different effects
The coefficient describes the latent variable — the underlying inclination to donate, which can be negative even though donations cannot. The effect on the outcome you actually observe is smaller, because a household already at zero may not move off zero when its income rises a little.
Both are reported. The coefficient answers “how does the underlying propensity respond?” and the marginal effect answers “how does observed spending respond?”. The second is the one most audiences want and the first is the one most papers print, which is a reliable source of overstatement by the ratio between them.
The gap depends on how much of the sample sits at the limit: the marginal effect is roughly the coefficient times the probability of being above it. When almost nothing is censored the two nearly coincide; when half the sample is at the floor the observed effect is about half the coefficient.
What the model assumes
Tobit assumes a single process decides both whether a unit sits at the limit and, if not, how far above it goes. The same income coefficient governs whether a household donates at all and how much it gives.
That is often too strong. Deciding to donate and deciding how much may be different decisions driven by different things — one by whether a charity asked, the other by disposable income. When that is the case a two-part model fits them separately: a binary model for crossing the limit, then a regression on the units that did. It is more flexible and more work, and the right choice when the two decisions are substantively distinct.
The model also leans on normality harder than least squares does. Ordinary regression is fairly forgiving of non-normal errors; here the normal assumption enters the likelihood through the censored rows’ probability term, so getting the distribution wrong biases the coefficients rather than just the standard errors. On skewed outcomes that is a real risk, and quantile regression makes no distributional assumption at all.
One distinction worth keeping straight: censoring is not truncation. Censored data records that a unit was beyond the limit; truncated data excludes those units from the sample entirely, so you do not know how many there were. This model handles the first. Fitting it to truncated data would assume information you do not have.
Reporting it
Report the censored share first. It is what tells a reader whether any of this mattered: at 8% the ordinary regression would have been nearly right, at 46% it is off by nearly half, and the number makes that judgement available without trusting anyone’s model choice.
Report both the coefficient and the marginal effect, and label which is which. A reader given only the coefficient will read it as the effect on observed spending and overstate it by the ratio between them.
Say whether you considered a two-part model and why you did not. The single-process assumption is the substantive one here and it is rarely mentioned at all. And state the limit and which side is censored — a ceiling behaves identically to a floor with the signs reversed, and the two are easy to conflate when only the coefficient is shown.
Sources and methodology
References for the estimator and the alternatives to it.
Method. The censored-normal likelihood is maximised by Newton steps with a numerically differentiated Hessian and a backtracking line search; standard errors come from the observed information, and both least-squares comparisons are computed from the same rows. The suite checks that shifting the outcome and the limit together leaves the slope and sigma unchanged, that the reported least-squares slope matches a direct covariance computation, that the marginal effect never exceeds the coefficient in size and keeps its sign, and that both least-squares fits are attenuated relative to the censored one on every dataset. The attenuation figures quoted come from 200 replications at each of four censoring levels with a known true slope. That engine is verified on every change against 490 assertions. The count and the per-case breakdown are published on the formula verification page.