Common mistake 1 with Truncated regression

Confusing truncation with censoring. Censoring loses the values but keeps the rows; truncation loses the rows. Using the wrong one throws away information or invents it.

Common mistake 2 with Truncated regression

Extrapolating far below the cut. Nothing in the data speaks to that region except the normality assumption.

Common mistake 3 with Truncated regression

Truncating the outcome during data cleaning - dropping the top or bottom few per cent as outliers - and then running OLS. That creates this exact problem by hand, and it is far more common than encountering genuinely truncated data.

Common mistake 4 with Truncated regression

Reading a high R squared on the truncated sample as evidence the bias is small. Truncation bias and goodness of fit are unrelated: the fit is measured on the selected rows, which are precisely the ones that cannot reveal the problem.

EconStatsCraft › Truncated regression

Truncated regression

For a sample that only contains observations on one side of a cut - survey of earners above a threshold, firms above a size. Unlike censoring the missing rows are simply not there, so the errors in what survives no longer average zero and OLS is biased.

What it needs

Outcome
Independent variables

The intuition

Truncation is worse than censoring. Under censoring you at least know how many observations hit the limit; under truncation they are not in your data at all. A survey of people earning above a threshold does not record the people below it, and does not tell you how many there were. Fitting OLS to what survives estimates the relationship among the survivors, which is systematically flatter than the relationship in the population.

When to use it

The sampling frame itself excludes part of the population: a survey of people earning above a threshold, firms above a size, students who scored high enough to enrol.

Where the estimator comes from

Observations are drawn only where y_i>a. The density must be renormalised to integrate to one over the observed region: f(y\mid y>a)=\frac{\phi\!\left(\frac{y-x'\beta}{\sigma}\right)/\sigma}{1-\Phi\!\left(\frac{a-x'\beta}{\sigma}\right)}, and the log-likelihood is the sum of its logs. The bias in OLS follows from the truncated mean, E[y\mid y>a,x]=x'\beta+\sigma\lambda(\alpha) with \alpha=(a-x'\beta)/\sigma and \lambda(\alpha)=\phi(\alpha)/(1-\Phi(\alpha)) the inverse Mills ratio. Because \lambda depends on x, the omitted term is CORRELATED with the regressors, which is precisely the structure of omitted-variable bias.

The model

E[y\mid \mathbf{x},y>c]=\mathbf{x}^{\prime}\boldsymbol{\beta}+\sigma\,\frac{\phi(\alpha)}{1-\Phi(\alpha)},\qquad \alpha=\frac{c-\mathbf{x}^{\prime}\boldsymbol{\beta}}{\sigma}

What it assumes

Normal, homoskedastic errors in the untruncated population.
The truncation point is known and applies to the outcome, not to a regressor.
Truncation, not censoring. If the excluded rows are actually present with the outcome recorded at the limit, use Tobit - it has strictly more information.
The dropped observations are gone entirely and their number is unknown. If you do know how many were removed, that count is information this model cannot use and a censored model can - which makes Tobit the better estimator whenever the rows still exist.

Statistical properties

Truncation discards information that censoring keeps. A Tobit model with the same threshold is strictly more efficient, so use it whenever the limit observations exist at all.
The bias is towards zero and grows with the share truncated. It does not vanish with sample size, because it is not a variance problem.
The estimator relies entirely on normality. Under a misspecified distribution truncated regression is inconsistent, and there is no robust standard error that repairs it.
The truncation point must be known exactly. Estimating it from the minimum of the observed data is not valid, because the minimum is itself random.
Marginal effects on the truncated mean involve the derivative of the inverse Mills ratio and are always smaller in magnitude than beta.
Truncation on the DEPENDENT variable biases the slope; truncation on an independent variable does not, it merely reduces the range and the precision. Confusing the two is common.

How to read the output

The coefficients describe the untruncated population, which is usually what a policy question is about, not the selected sample you can see.
Compare against OLS on the same rows: the gap is the truncation bias.
A large sigma relative to the distance from the cut means heavy truncation and fragile estimates.
The reported sigma describes the untruncated population, so it is always LARGER than the standard deviation of the outcomes you can see. Truncation removes a tail; the estimate puts it back.

A worked reading

A wage equation estimated only on people earning above a reporting threshold. The population slope on schooling is 0.09; OLS on the truncated sample returns roughly 0.06, and truncated regression recovers 0.088. The gap is not sampling noise - it survives any sample size - because the inverse Mills ratio omitted from OLS is a function of the regressors and behaves exactly like an omitted variable correlated with schooling. Note also what cannot be checked: nothing in the surviving data reveals whether the errors below the threshold were normal, and the whole estimate rests on assuming they were.

Symptom → diagnosis → what to do

If you see this	It means	Do this
Estimates much larger than OLS	Expected. OLS was flattened by the truncation	Not evidence of an error
Sigma implausibly large	The model extrapolating far below the truncation point	Weak identification; be sceptical
Truncation share above about half	Very little information survives	Results will be fragile and heavily distribution-dependent
Non-normal residuals	The estimator is inconsistent	No robust standard error fixes this. Consider a semiparametric estimator
You actually know how many were dropped	This is censoring, not truncation	Use Tobit, which is more efficient
Truncation is on a regressor	No bias in the slope	Do not apply this model

Common mistakes

Confusing truncation with censoring. Censoring loses the values but keeps the rows; truncation loses the rows. Using the wrong one throws away information or invents it.
Extrapolating far below the cut. Nothing in the data speaks to that region except the normality assumption.
Truncating the outcome during data cleaning - dropping the top or bottom few per cent as outliers - and then running OLS. That creates this exact problem by hand, and it is far more common than encountering genuinely truncated data.
Reading a high R squared on the truncated sample as evidence the bias is small. Truncation bias and goodness of fit are unrelated: the fit is measured on the selected rows, which are precisely the ones that cannot reveal the problem.

Where to read this properly

Hausman and Wise (1977), "Social Experimentation, Truncated Distributions, and Efficient Estimation", Econometrica 45(4).
Greene, Econometric Analysis (2018), ch. 19.
Wooldridge, Econometric Analysis of Cross Section and Panel Data (2010), ch. 19.

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Truncated regression

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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