Truncation and Truncated Regression—What Problems Do They Address? Consequences of Misusing OLS? Minimal Necessary Knowledge and a Decision Framework

I. Opening: Your Data Is Not "Complete"—But Your OLS Pretends It Is

You study "the determinants of wages." You use a micro-survey dataset—but this dataset only contains people who are currently employed. You have no wages for the unemployed (they have no wages), nor information on those who have exited the labor force.

You run OLS:

reg wage education experience age

The coefficients are significant, and the return to education is approximately 8%. You are about to conclude—"each additional year of education raises wages by about 8%."

But someone reminds you: your sample is a "post-selection sample"—only those who choose to work and found a job have entered your regression. If education affects not only wages but also whether a person chooses to work (more educated people are more likely to participate in the labor force)—then your OLS estimate captures not just "the effect of education on wages" but a mixture of two effects: the true effect of education on wages + the selection effect of education on "who enters the wage equation."

The problem you face—that in your sample, Y (wages) is observable only for a subset of people, and "who is observed" is not random—is the core of Truncation and Sample Selection problems.

This differs from Censoring and Tobit, which you have seen elsewhere in this series—but they share the same family: Limited Dependent Variable Models. Understanding the "scope of application" of each member of this family—and the cost of misusing OLS—is the minimal necessary knowledge this article aims to give you.

Core message: Limited dependent variables have three different faces, but they all say the same thing—your Y is not "fully visible" for all people and all values. The first face is Censoring**—you know the true value of Y is cut off at a threshold, but you know X for all observations (including censored ones). For example, income is coded in a survey as "500,000 and above"—you know this person's income is ≥ 500,000, and you see all their X (education, age, occupation). Censored regression exploits this "partial information"—censored observations contribute the probability that "Y is at least some value," rather than being discarded. The second face is** Truncation**—when Y falls below (or above) a threshold, the entire observation disappears from your sample. You know nothing about truncated individuals—no Y, no X. For example, you only have wage data for the employed—the wages and characteristics of the unemployed are both invisible. Truncated regression can only estimate using visible observations—but it must correct for the fact that "I only saw those who crossed the threshold." The third face is** Sample Selection**—whether Y is observed depends on another process. For example, "whether employed" is the first step (selection equation), and "what the wage is" is the second step (outcome equation). If the errors of the two equations are correlated (the same individual shares unobservable factors across the two steps—such as ability)—then running OLS on only the second step produces selection bias. The Heckman two-step method is the standard tool for handling this type of selection. The common consequence of these three data structures is: OLS is inconsistent in truncated or selected data (not just inefficient—the coefficients are also biased). Truncation discards information below the threshold—OLS uses only observations above the threshold, yielding a β that is not the true marginal effect in the population. Censoring retains all observations, but OLS's linear fit on censored values distorts the shape of the true relationship. The key to choosing a model wisely is not 'which model is more advanced'—but rather 'at which stage does your data lose information: is Y partially blocked (censoring), is the entire observation discarded (truncation), or is the very visibility of Y itself an economic choice (sample selection)?'**

II. Three Faces—The Essential Differences Among Censoring, Truncation, and Sample Selection

2.1 Using the Same Data-Generating Process to Illustrate the Differences Among the Three Concepts

Suppose the true wage-generating process is:

$Y_i^* = \beta_0 + \beta_1 \cdot \text{Education}_i + \varepsilon_i, \quad \varepsilon_i \sim N(0, \sigma^2)$

$Y_i^*$ is each person's "potential wage" in the labor market—regardless of whether they are actually working.

Scenario One: Censoring—Censoring from Above

Your survey codes wages as "500,000 and above" (top-coding). What you observe is:

$Y_i = \begin{cases} Y_i^* & \text{if } Y_i^* < 500,000 \\ 500,000 & \text{if } Y_i^* \geq 500,000 \end{cases}$

You know all X (education, age, occupation) for everyone—regardless of whether their wages are censored. For those with wages ≥ 500,000—you know their Y is at least 500,000; your information is "Y ≥ 500,000," not the exact value of Y. Censored regression does not discard these observations—instead, it incorporates the probability that "Y is at least some value" into the likelihood function.

Scenario Two: Truncation—Truncation from Below

Your data contains only people whose wages exceed a certain threshold—for example, you only have complete data for people with wages above the minimum wage of 2,000 yuan/month. For those below this threshold—both wages and X are invisible.

$Y_i = Y_i^* \text{ is observed only when } Y_i^* > 2,000$

You only see those who crossed the threshold. Those below the threshold—complete darkness. Truncated regression can only estimate on visible observations—but it adds a correction term because you only see high-wage individuals, and high-wage individuals happen to be those whose "combination of education and ability exceeds the threshold."

Scenario Three: Sample Selection—Whether Y Is Visible Is Itself a Choice

In your data, X (education, age, etc.) is observable for everyone—regardless of whether they are employed. But Y (wages) is visible only for those who choose to enter employment.

$Z_i^* = \gamma_0 + \gamma_1 \cdot \text{Education}_i + \gamma_2 \cdot \text{Age}_i + u_i \quad \text{(selection equation—whether employed)}$

$Y_i^* = \beta_0 + \beta_1 \cdot \text{Education}_i + \varepsilon_i \quad \text{(outcome equation—wages)}$

When $\text{Corr}(u_i, \varepsilon_i) \neq 0$—that is, the unobservable factors affecting whether a person is employed (such as motivation, ability) are correlated with the unobservable factors affecting wages—running OLS only on the employed produces selection bias.

2.2 The Fundamental Differences Among the Three—A Table for Clarity

III. If OLS Is Misused—Specific Consequences Under the Three Data Structures

3.1 OLS on Censored Data—Coefficients Attenuated Toward Zero

Scenario: Y is censored at 500,000 (top-coding). OLS treats the Y values of censored observations as exactly 500,000—but in reality, these individuals' Y is ≥ 500,000, and the true Y may be far higher than 500,000.

Consequence: OLS's slope is pulled toward zero. Because individuals with high X are more likely to be censored—and their true Y is underestimated (capped at 500,000). OLS sees "high X → Y capped at 500,000"—which underestimates the true effect of X on Y.

Intuition: The true wages of highly educated individuals far exceed 500,000, but censoring makes OLS think their wages are exactly 500,000—OLS is forced to underestimate the return to education.

Magnitude: The higher the censoring proportion → the more severe the attenuation. If 5% of observations are censored, attenuation is in the 5%–10% range. If 20%–30% are censored → attenuation can be substantial.

3.2 OLS on Truncated Data—Coefficients Attenuated Toward Zero (Typically)

Scenario: You only have complete data for people with Y > 2,000. Those below the truncation threshold—completely invisible in the truncated data.

Consequence: OLS uses only observations with Y > 2,000—these are "those who crossed the threshold." In this subsample, the true relationship between X and Y is compressed—because the threshold cuts off the left tail of the distribution, and the variation in the sample is artificially reduced. OLS's β is typically attenuated toward zero—because you are running a regression on a truncated sample with reduced variation.

Intuition: If your threshold is 2,000—those with wages just above 2,000 and those with wages far above 2,000 are both in the sample. But the effect of X near the threshold is distorted by the "threshold effect"—because the threshold artificially truncates the distribution of Y at the cutoff (only those who crossed are seen), which interferes with the true covariance between X and Y.

3.3 OLS Using Only Visible Observations on Sample Selection Data—Bias Direction Depends on Selection

Scenario: You run a wage regression only on the employed. The probability of entering employment and wages share unobservable factors (ability, motivation).

Consequence: The direction of bias in $\hat{\beta}$ depends on the role of ability in both the employment and wage equations.

• If high-ability individuals are more likely to be employed and ability raises wages among the employed → $\text{Corr}(u, \varepsilon) > 0$ → OLS overestimates the return to education (because the employed happen to be the highest-ability group).
• If high-ability individuals are more likely to be employed and the wage distribution of ability among the employed differs from that of highly educated individuals → the direction and magnitude of bias depend on the specific covariance structure—there is no uniform "upward" or "downward" bias.

This is the most dangerous of the three—because not only is the magnitude biased, but the direction may also be biased, and the bias cannot be corrected within the OLS framework (a selection equation is needed).

IV. Minimal Necessary Knowledge—A Decision Framework for Choosing Models Wisely

4.1 Decision Tree—Starting from Your Data Structure

What is wrong with your Y?
│
├── Y is "blocked" at some value (upper or lower limit)
│   │
│   ├── You know X for everyone (including those blocked)
│   │   └── → Censoring
│   │       ├── Censored only from below (Y ≥ 0, with many Y = 0) → Tobit (corner solution)
│   │       ├── Censored only from above (top-coding) → Censored Regression / `cnreg`
│   │       └── Censored from both sides → interval censoring model
│   │
│   └── For those blocked—you have no information about them at all
│       └── → Truncation
│           └── → Truncated Regression / `truncreg, ll()`
│
├── Y is visible only for certain individuals—and "who is seen" is not random
│   ├── You have X for everyone (including those for whom Y is invisible)
│   │   └── → Sample Selection
│   │       ├── A selection equation + an outcome equation → Heckman two-step / `heckman`
│   │       └── The correlation between the selection equation error and the outcome equation error is the key parameter to focus on
│   │
│   └── You do not have X for those for whom Y is invisible
│       └── → Truncation—back to truncated regression
│
└── Y has mass at zero + continuous positive values (corner solution)
    └── → Tobit (standard left-censoring)—see the Tobit article in this series

4.2 Three Concrete Cases—Applying the Decision Framework

Case One: Top-coding in a wage survey

"In your wage data, all individuals with monthly income > 50,000 yuan are coded as 50,000."

• Censoring or truncation? → Censoring—you know the X of these individuals; only Y is blocked.
• How many observations are affected? → Suppose 5% of people have wages ≥ 50,000. The censoring proportion is not high, but high earners happen to be the most educated group—OLS's attenuation is concentrated at high X values, pulling down the overall slope.
• Recommended model: Censored Regression / cnreg. Alternative: Tobit (in the case of upper censoring—i.e., $Y = \min(500,000, Y^*)$). OLS + a note that "income top-coding may cause coefficients to attenuate toward zero" as a baseline.

Case Two: Observing wages only for the employed

"Your labor force survey data includes education, age, and gender for everyone. But the wage variable is observable only for those currently employed."

• Censoring, truncation, or selection? → Sample selection—X is visible for everyone; Y is visible only for those who choose to enter employment.
• Do you have variables that affect employment but not wages directly (exclusion restriction)? → This is the key to Heckman model identification: at least one variable appears in the selection equation but not in the outcome equation (e.g., "having young children" affects whether one is employed but does not directly affect wages). If not—Heckman relies only on nonlinear functional form for identification, which is fragile.
• Recommended model: Heckman two-step / heckman, with explicit discussion of the exclusion restriction. Alternative: OLS (baseline) + discussion of the direction and possible magnitude of selection bias + Oster δ sensitivity analysis.

Case Three: Observing only firms with "advertising expenditure > 1 million"

"Your firm-level marketing data contains only firms with annual advertising expenditure exceeding 1 million yuan—firms spending below 1 million on advertising are not in the database."

• Censoring or truncation? → Truncation—both X and Y of truncated individuals are invisible.
• Is the truncation endogenous? → Firms with advertising expenditure > 1 million may be "large firms"—the observable relationship between size, industry, and advertising expenditure is distorted in the truncated sample.
• Recommended model: Truncated Regression / truncreg, ll(100). Alternative: OLS (which typically attenuates toward zero under truncation) as a conservative baseline—with a note that "the OLS estimates in this paper may underestimate the true effects because the sample is limited to firms with advertising expenditure > 1 million."

V. Core Members of the Model Family and Command Quick Reference

5.1 Model Overview

5.2 Key Diagnostics and Tests

Tobit / Censored / Truncated Regression:

• Testing the normality assumption: Tobit and truncated regression are both sensitive to the normality of $\varepsilon$. After Tobit, conditional moment tests can be used to assess normality. In practice, however, these tests are rarely strictly enforced—a more common approach is to report OLS alongside as a robustness reference (since OLS does not require normality).
• Heteroskedasticity: Tobit is estimated under the homoskedasticity assumption—if $\sigma^2_\varepsilon$ varies with X, the Tobit MLE is inconsistent. Robust Tobit estimators are complicated; a two-part model or OLS with clustered standard errors can be used as alternatives.

Heckman model:

• Is the coefficient on the inverse Mills ratio (λ) significant? If λ is not significant → we cannot reject the null hypothesis of zero selection bias (but we cannot say "selection bias does not exist"—because the test may lack power).
• Is ρ (error correlation) significant? ρ = 0 → the errors of the two equations are uncorrelated → selection bias may not exist or is at least not severe.
• Validity of the exclusion restriction: Is there at least one variable that appears in the selection equation but theoretically not in the outcome equation? This is the strongest argument for Heckman model identification—and the aspect reviewers focus on most. Without an exclusion restriction, Heckman identification relies only on functional form (normality), which is empirically fragile.

VI. Common Misconceptions

6.1 Misconception One: "Censoring and truncation are the same thing"

Censoring = you know X for everyone; Y is only partially visible. Truncation = both Y and X are invisible for those below the threshold. The structure of available data determines which model you can use: censoring retains partial information (X and the "at least some value" for censored individuals), while truncation completely discards information below the threshold. Using truncated regression on censored data (or vice versa) will not give you the correct answer—although the two models have similar names, their likelihood functions have different structures.

6.2 Misconception Two: "Truncated Regression is just running OLS on the subsample with Y > 0"

Running OLS on the subsample with Y > 0 is precisely the operation of "ignoring truncation"—which is exactly what Truncated Regression aims to correct. OLS typically attenuates toward zero in truncated samples—because truncation compresses the variation in Y. Truncated Regression recovers the β of the untruncated population by dividing by the probability of being observed in the likelihood.

6.3 Misconception Three: "The Heckman two-step method can solve all selection bias problems"

The Heckman model relies on two strong assumptions: (a) normality of the errors; (b) the existence of a variable satisfying the exclusion restriction—appearing in the selection equation but not in the outcome equation. Without an exclusion restriction, Heckman identification relies solely on the nonlinearity of the normal distribution—which is extremely fragile empirically. A good Heckman application should clearly discuss the exclusion restriction: why does this variable affect "whether one is observed," but not directly affect Y? The most common reason reviewers reject Heckman is "the exclusion restriction is not credible."

6.4 Misconception Four: "Tobit can handle truncated data—because both are MLE"

Tobit assumes that the X of censored observations is known—this holds for censored data but not for truncated data (the X of truncated observations is also lost). In the Tobit likelihood, censored observations contribute the probability that "Y is below the threshold," and this probability contains information from X. The truncated regression likelihood does not have this component—because X itself is invisible. Tobit ≠ truncated regression—using Tobit on truncated data or vice versa will yield inconsistent estimates.

6.5 Misconception Five: "My data has a selection problem → Heckman is the only way out"

Heckman is not the only tool for handling selection problems—nor is it always the best. Alternatives include:

• DID / fixed effects: If selection is based on time-invariant unobservable factors ($u_i$) → panel FE can directly eliminate it without needing a selection model.
• Instrumental variables: If selection can be instrumented (there is a Z that affects "whether one is observed" but not Y).
• Partial identification / bounds analysis: Instead of assuming the full structure of the selection model, provide a plausible range for β. This is more honest when the assumptions of the selection model are difficult to verify.

VII. Summary

Truncation, censoring, and sample selection—six core takeaways:

1. Censoring ≠ truncation ≠ sample selection. Censoring: Y is blocked, X is fully visible. Truncation: both Y and X are invisible for those below the threshold. Sample selection: whether Y is visible is an economic decision, and X is visible for everyone.

2. The consequences of OLS differ across the three: Censoring → OLS coefficients attenuate toward zero (censored Y values are incorrectly treated as exact values). Truncation → OLS coefficients typically attenuate toward zero (sample variation is artificially compressed). Sample selection → OLS is inconsistent—the direction of bias depends on the correlation between the errors of the selection and outcome equations (may be upward or downward).

3. The choice of model depends on the missing-data structure—not on "which model is more advanced." The distinction among censoring/truncation/selection comes from your data collection and data-generating process—not from statistical tests.

4. Censored and truncated regression rely on normality and homoskedasticity assumptions. If these assumptions are severely violated, OLS (despite its bias) may be the more robust choice in some cases—because OLS retains consistency (in the sense of linear projection) when these assumptions are violated. A pragmatic strategy: report Tobit / Truncated Regression as the main model, with OLS as a robustness reference.

5. The validity of the Heckman model depends on the exclusion restriction. Without a variable that appears in the selection equation but not in the outcome equation → Heckman identification relies only on functional form (fragile). This is the primary reason reviewers reject Heckman—you need to explicitly argue for your exclusion restriction in the paper.

6. There is no perfect limited dependent variable model. All these models trade off "being closer to the true data-generating process" against "requiring stronger distributional assumptions." Good empirical work is not about finding a perfect model—it is about choosing the most reasonable model given your data constraints and candidly discussing its limitations.

One-sentence closing:

"Some information in your data is missing—this is not your fault. But pretending the missing information does not exist—continuing to use OLS—is a mistake. Censoring tells you 'some people are blocked, but I know who they are'—censored regression exploits this partial information. Truncation tells you 'those below the threshold have disappeared—I have no trace of them'—truncated regression corrects for this disappearance in the visible part. Sample selection tells you 'some people chose not to be seen—but I know why they made this choice'—the Heckman model exploits the information in this selection process. Using the correct model is not because you want more sophisticated statistics—but because your data has already lost information at some stage. What you need to do is not pretend the missing information is complete—but place the correct mathematical structure at the point of loss, so that the remaining information is used correctly."

VIII. Bilibili/WeChat Presentation Suggestions

• Bilibili video: It is recommended to use "three different doors" as the core visual metaphor. Three scenarios run in parallel—Door A (censoring): an opaque barrier blocks the upper part of the Y-axis. You can see everyone's X, but high Y values are blurred by the barrier. Door B (truncation): a threshold lies across the lower part of the Y-axis. Those below the threshold—their entire figure (X and Y) disappears beyond the frame—complete darkness. Door C (sample selection): X is visible for everyone (all people stand in the frame), but Y emerges only for a subset—the Y labels light up for those who "choose to enter." Narration: "Limited dependent variables are not one type—they are three. Censoring: you know everyone's characteristics, but Y is blocked by a threshold. Truncation: those below the threshold—characteristics and outcomes alike—completely disappear from your data. Sample selection: you see everyone's characteristics, but outcomes are observable only for a subset—and 'who is observed' is itself an economic choice." Act One, "Consequences of misusing OLS": each of the three scenarios demonstrates the failure of OLS. Scenario one—the OLS line is pinned at the censoring threshold (attenuation—bias toward zero). Scenario two—the OLS line tends toward horizontal in the truncated sample (variation compressed—bias toward zero). Scenario three—the OLS line is tilted in the selected sample (selection bias—direction depends on selection). Narration: "OLS is biased in all three scenarios—but the direction and cause of bias differ in each. Censoring compresses Y downward, truncation compresses the variation in Y, and selection restricts the sample to a non-random subset. Three different mechanisms—but OLS is powerless against all of them." Act Two, "Decision tree": a tree grows—three main branches extend: censoring → Tobit / cnreg, truncation → truncreg, selection → heckman. On each branch hang miniature depictions of the corresponding scenario. At the core node of the decision tree is a question: "Is X visible for everyone?" → Yes → censoring or selection. → No → truncation. Second question: "Is Y invisible because of an economic choice?" → Yes → sample selection. → No → censoring. Narration: "You don't need to memorize all the models—you only need to answer two key questions: (1) Is X visible for everyone? (2) Why is Y invisible—because it is blocked, or because it has been 'selected' away?"

• WeChat article: The comparison table of censoring vs. truncation vs. sample selection (conceptual definition, whether X is fully visible, consequences of misusing OLS, standard model, Stata command) should be made into an infographic as the core—this is the most important visual output of the entire article. The decision tree should be made into a vertical flowchart. The four models (Tobit, Censored, Truncated, Heckman) should be made into a quick-reference card with "usage conditions + one-line Stata command + key risks." The direction of OLS bias across the three (censoring→toward zero, truncation→toward zero, selection→direction depends on selection) should be made into three comparison cards. The complete walkthroughs of Cases 1–3 (data structure → model choice → Stata) should be made into practical reference cards. The five common misconceptions should be made into correction cards.

• Recommended titles:

  • Main title: 《Censoring, Truncation, and Sample Selection—At Which Stage Does Your Y Lose Information?》
  • Alternative title: 《Tobit, Truncated, Heckman—Minimal Necessary Knowledge and a Decision Framework for Limited Dependent Variable Models》
  • New media title: 《Your Data Is Incomplete—But OLS Is Pretending It Is Complete. The Differences and Choices Among Censoring, Truncation, and Sample Selection》

• Key quotes:

  "Censoring, truncation, and sample selection all say the same thing—your data is incomplete. But different types of 'incompleteness' require different models. Censoring: information is blocked by a threshold—but you at least know who is blocked. Truncation: information disappears along with the observation—leaving only darkness. Sample selection: the information is still there, but it has been 'selected'—you need to understand the logic of this selection to correct the bias in your results."

  "OLS pretends everything is normal where your information is missing. Censoring → OLS treats 'blocked' as 'this is exactly the blocked value' (underestimating the true value). Truncation → OLS ignores that 'those below the threshold have disappeared' (regressing on a sample with compressed variation). Sample selection → OLS pretends that 'who appears in the Y equation' is random (ignoring selection). Three different pretenses—three different biases."

  "The Heckman model places 'who is seen' and 'how much the seen individuals have' in the same framework. But its Achilles' heel is the exclusion restriction—at least one variable in the selection equation that does not affect Y. If you cannot find such a variable—your Heckman identifies only through the shape of the normal distribution—as thin as a sheet of normal curve paper."

  "Limited dependent variable models are not statistical showing-off—they are your mathematical honesty when your data is incomplete. You acknowledge that your information was lost at some stage, and you place the correct structure at that stage. You do this not to make your coefficients more significant—but to make your coefficients reflect the true story of your data on the most honest scale possible."