Dependent Variable Is Positive—Panel OLS or Tobit? What's the Difference? When Should You Switch?

1. Opening: Your Y Is All Positive with Many Zeros—Can You Still Trust the Coefficients from Panel FE?

You are studying the determinants of firm R&D investment. Your dependent variable is firm R&D expenditure—many firms have zero R&D expenditure in a given year (Y = 0), while positive R&D expenditure values span an enormous range (from a few thousand yuan to several billion yuan).

You run a baseline regression with panel fixed effects:

xtreg rd_expenditure size profitability leverage, fe robust

The coefficients are significant and the signs make sense. Then someone asks you: "Your dependent variable is a mixture of zeros and positive values—a large mass piled up at zero, with the positive part right-skewed. Don't you think OLS is inappropriate for this kind of data? Shouldn't you be using Tobit?"

You look it up—the Tobit model is indeed designed for dependent variables that are "censored at zero." But you hesitate:

1. If I only care about whether the coefficient on my key independent variable is significant—is panel OLS enough? Could its significance and sign differ from what Tobit would give?
2. Under what circumstances is it reasonable to keep using OLS? What signs should prompt me to switch to Tobit?
3. Can panel Tobit be estimated in Stata? Does fixed-effects Tobit also suffer from the incidental parameters problem?

Core message: When the dependent variable is bounded below at zero with a non-negligible mass of zeros, the divergence between panel OLS (FE) and Tobit stems from their different modeling of the data-generating process. OLS assumes $Y = X\beta + u_i + \varepsilon$—that is, X affects the conditional mean of Y linearly, regardless of the level of Y. Tobit assumes $Y^* = X\beta + u_i + \varepsilon$ (a latent variable), with $Y = \\max(0, Y^*)$—that is, the effect of X on Y is nonlinearly filtered through the "zero barrier." This filtering mechanism produces a key consequence: the marginal effect of X on the observed Y is not $\beta$, but $\beta \cdot \Pr(Y^* > 0 \mid X)$—that is, the effect of X is a constant $\beta$ at the latent-variable level, but at the level of observed Y, the effect is scaled by the probability of "crossing the zero threshold." If the share of zeros is not large (< 10%), the conditional mean is far above zero, and you only care about the sign and significance of coefficients rather than their absolute magnitudes—panel OLS typically delivers conclusions directionally consistent with Tobit, and is simpler and more transparent. But if the zero mass is substantial (> 15%–20%), predicted values frequently approach zero, or you care (and reviewers care) about how X separately affects "whether to do R&D" (extensive margin) versus "how much R&D expenditure" (intensive margin)—Tobit or its variants (two-part model / Hurdle model) represent a substantively meaningful upgrade. However, in panel settings, FE Tobit also suffers from the incidental parameters problem—just like FE Probit, $\hat{\beta}$ is inconsistent when T is fixed. Therefore, in the practice of panel Tobit, RE Tobit with Mundlak individual means (CRE Tobit) is typically used as a pragmatic compromise.

2. The Modeling Logic of Tobit—What Is Its Most Fundamental Difference from OLS?

2.1 What Is Your Dependent Variable?—Three Fundamentally Different Data-Generating Processes

Before discussing models, distinguish three types of data that are often confused, because they correspond to different models:

The most common case in empirical economics is the corner solution—individuals choose "to participate or not" (Y = 0 or Y > 0), which differs from a censoring mechanism (where values exist but are unobserved). The Tobit model handles exactly the corner solution—you observe $\max(0, Y^*)$, rather than $Y^*$ itself being cut off by an unavoidable external censoring mechanism.

2.2 The Latent Variable Specification of Tobit

The structure of the Tobit model (Type I Tobit) is:

$Y_{it}^* = X_{it} \beta + u_i + \varepsilon_{it}, \quad \varepsilon_{it} \mid X, u_i \sim N(0, \sigma^2_\varepsilon)$

$Y_{it} = \max(0, Y_{it}^*)$

$Y^*$ is a continuous latent variable—which can be understood as the "propensity for R&D investment" or the "intensity of willingness to invest in R&D." When the propensity is negative, the firm chooses not to do R&D (Y = 0); when the propensity is positive, the observed R&D expenditure equals the value of the propensity.

Key point: $Y^*$ can be negative—this corresponds precisely to the state of "I don't want to do R&D." The negative propensity is blocked by the zero barrier—you only see zero.

2.3 What Does OLS Do?—It Does Not Model the "Zero Barrier"

OLS directly fits:

$\mathbb{E}[Y_{it} \mid X_{it}, u_i] = X_{it} \beta + u_i$

But under the Tobit data-generating process, the true $\mathbb{E}[Y \mid X]$ is:

$\mathbb{E}[Y_{it} \mid X_{it}, u_i] = \Phi\left(\frac{X_{it}\beta + u_i}{\sigma_\varepsilon}\right) \cdot (X_{it}\beta + u_i) + \sigma_\varepsilon \cdot \phi\left(\frac{X_{it}\beta + u_i}{\sigma_\varepsilon}\right)$

This is not a linear function of $X\beta + u_i$. It contains two parts:

• $\Phi(\cdot)$: the probability that $Y^* > 0$ (extensive margin).
• $X\beta + \sigma_\varepsilon \cdot \frac{\phi}{\Phi}$: the conditional expectation given $Y^* > 0$ (intensive margin + selection correction term).

OLS ignores the entire nonlinear structure—it fits a straight line to data that has a barrier at zero and curves upward.

2.4 OLS and Tobit Deliver "Effects" on Different Scales

This distinction is crucial:

• OLS's β: the coefficient from the linear projection of X onto observed Y—the "increment" of the effect is assumed to be the same near Y = 0 and where Y is large.
• Tobit's β: the marginal effect of X on the latent variable $Y^*$—constant on the latent-variable scale. But the marginal effect of X on observed Y = $\beta \cdot \Phi\left(\frac{X\beta + u_i}{\sigma_\varepsilon}\right)$—scaled by the "probability of crossing the zero threshold."

The core economic intuition of Tobit: a one-unit increase in X affects Y through two channels—(1) for firms already doing R&D ($Y^* > 0$), Y increases by β; (2) for firms not currently doing R&D but close to the threshold ($Y^*$ negative but near zero), the increase in X may push them across zero—from Y = 0 to Y > 0. The first channel is the intensive margin, the second is the extensive margin. The total effect of X is a weighted average of these two margins—and OLS cannot separate them.

3. "I Only Care Whether the Coefficient Is Significant"—Is Panel OLS Enough?

3.1 Sign and Significance—OLS and Tobit Usually Agree in Direction

If you only care about the sign of your key independent variable and whether it is statistically significant—in the vast majority of empirical settings, panel OLS delivers conclusions that agree in direction with Tobit.

The reason: Tobit's nonlinear scaling factor $\Phi\left(\frac{X\beta + u_i}{\sigma_\varepsilon}\right)$ is always positive—the sign of $\beta$ is the same on the latent-variable scale and on the marginal-effect scale for observed Y. If Tobit says X has a positive effect on $Y^*$, OLS will typically also capture a positive association—although the OLS coefficient differs in magnitude from Tobit's decomposed marginal effects, agreement in sign and significance is very common.

Therefore, if you only need to answer the binary question "is X significantly positive/negative"—panel OLS is usually sufficient and easier for reviewers to understand.

3.2 Situations Where the Two Models May Diverge in Significance

But agreement in sign does not mean significance is always the same. In the following three situations, OLS significance and Tobit significance may diverge substantively:

Situation 1: Severe zero mass and X's effect operates mainly through the extensive margin. If more than 30% of your Y values are zero—X's effect mainly manifests in pushing firms "from zero to positive" (extensive margin), rather than increasing R&D expenditure among firms already doing R&D (intensive margin). OLS is forced to fit a straight line through the pile of zeros—the slope may be pulled down by the "mass" of zeros, causing OLS to fail to detect the true effect. Tobit, by contrast, correctly decomposes the effect into "pushing across zero" and "increasing positive values"—and may detect significance on the key extensive margin that OLS misses.

Situation 2: The positive part of Y is highly skewed. Data like R&D expenditure and charitable donations—the positive part is severely right-skewed (a few firms spend enormous amounts). OLS is sensitive to extreme values—a firm with extremely high R&D expenditure can substantially pull the OLS slope. Tobit's MLE is more robust to extreme values than OLS (because the likelihood function assigns lower weight to observations far from the mean). If OLS's significance is propped up by a few extreme observations—Tobit may deliver an insignificant or marginally significant result.

Situation 3: The model includes interaction terms or quadratic terms. If you include nonlinear terms in your model ($X^2$, interaction term $X_1 \times X_2$), the inconsistency between OLS and Tobit may be amplified—because nonlinear terms have completely different coefficient meanings on the latent-variable scale versus the observed-Y scale. In OLS, the coefficient on $X^2$ captures the partial linear association between Y and $X^2$—whereas in Tobit, the coefficient on $X^2$ captures the "curvature of the latent variable $Y^*$." Both the economic interpretation and the statistical significance may differ.

3.3 A Practical Truth

In the vast majority of academic papers—if the author's core message is "X significantly affects Y (where Y is a mixture of zeros and positive values)," and the share of zeros is not too high (< 15%), panel FE is the standard and reviewer-accepted baseline specification. Reviewers typically will not reject your paper because "you didn't use Tobit"—they care more about your identification strategy (how endogeneity is handled) than about the nonlinearity of the dependent variable. But if you use OLS and find reviewers repeatedly asking "your Y has so many zeros—is OLS appropriate?"—that is a signal: in your specific context, Tobit may be necessary.

4. What Signs Suggest You Should Use Tobit Rather Than Panel OLS?

4.1 Sign 1: Severe Zero Mass—Share Exceeds 15%–20%

Calculate the proportion of zeros in your Y:

count if y == 0
di r(N) / _N

If it exceeds 15%–20%—the zeros are not a marginal phenomenon but a structural feature. At this proportion, the "mass" of zeros is large enough that OLS's linear approximation may produce meaningful bias. The larger the zero share → the more important the extensive margin → the greater the advantage of Tobit.

4.2 Sign 2: Your Research Question Itself Contains Two Layers—"Whether to Participate" and "How Much to Participate"

If your paper's title or core question is "Do firms conduct R&D? How much do they spend?"—then these two questions are logically separate, and your model should be able to distinguish the extensive margin from the intensive margin.

OLS compresses both margins into a single coefficient—you cannot answer "Does X cause more firms to start doing R&D, or cause firms already doing it to do more?"—which is precisely the core question of your paper. Tobit's decomposition (or better, the two-part model / Hurdle model—see Section 5 below) allows you to answer this question head-on.

4.3 Sign 3: The Residuals-versus-Fitted Plot Shows a "Funnel-Shaped" Truncation Pattern

After running OLS, plot residuals against fitted values:

xtreg y x, fe
predict yhat, xb
predict e, e
twoway (scatter e yhat), yline(0)

If the residuals show a "truncation" pattern on the side with low fitted values—that is, residuals are not uniformly scattered around zero but are squeezed into a diagonal line at the low-fitted-value end—this is a signal that OLS is not appropriate for the current data structure. Tobit residual analysis is typically conducted on the latent-variable scale, so Tobit model diagnostics use different tools—but the "truncation pattern" in OLS residuals is an intuitive reminder that "the zero barrier is distorting the OLS fit."

4.4 Sign 4: Your Key X Produces Predicted Values of Y That Frequently Approach Zero

If, at certain values of your key independent variable, the OLS predicted values $\hat{Y}$ frequently approach or fall below zero (even though actual Y cannot be negative)—this indicates that the linear model does not logically respect the nonnegativity constraint on Y. Tobit, by construction, ensures predicted values ≥ 0.

Check in Stata:

predict yhat_ols
count if yhat_ols < 0

If the proportion of yhat_ols < 0 is not negligible—Tobit may be more appropriate.

4.5 Sign 5: Reviewers Explicitly Ask About Your Dependent Variable's Structure

If a reviewer writes in their comments "the dependent variable is censored at zero" or "a non-trivial fraction of the DV is zero"—this is an unambiguous signal. Adding a Tobit / Hurdle model as a robustness check in your next revision is typically the path to resolution. Even if your main conclusions are unchanged under OLS, demonstrating that the conclusions are robust under Tobit is the most effective way to respond to reviewers.

5. Not Just Tobit—A Comparison of the Related Model Family

5.1 Standard Tobit (Type I)—The Most Classic Specification

$Y = \max(0, X\beta + \varepsilon)$

Assumption: The same $\beta$ drives the extensive margin (whether Y > 0) and the intensive margin (the magnitude of Y when Y > 0). Moreover, the coefficients on the extensive margin and the intensive margin are constrained to be equal ($\beta_{\text{ext}} = \beta_{\text{int}}$), up to a scale factor $\sigma_\varepsilon$.

This constraint is strong but usually tolerated—if theory and data suggest that the same X has effects of the same sign and comparable magnitude on both margins, Tobit is appropriate. If the effects on the two margins are opposite in sign (e.g., an increase in X causes more people to participate but the average expenditure of participants falls)—Tobit's constraint will produce a biased β—in this case, a more flexible model (Hurdle / two-part) should be used.

5.2 Two-Part Model—Separating the Extensive and Intensive Margins

First part: Probit / Logit—models $\Pr(Y > 0 \mid X)$. Second part: OLS (on the subsample with Y > 0)—models $\mathbb{E}(\ln Y \mid Y > 0, X)$ or $\mathbb{E}(Y \mid Y > 0, X)$.

The two-part model imposes no constraint on the relationship between the extensive and intensive margins—the β's in the two parts can have different signs, different magnitudes, and be driven by different variables.

* First part: Probit
gen pos = (y > 0)
probit pos x1 x2
 
* Second part: OLS on the subsample with Y > 0
reg ln_y x1 x2 if y > 0

Choosing between the two-part model and Tobit:

• Tobit assumes the same latent variable drives both margins → if this assumption holds, Tobit is more efficient.
• The two-part model does not impose this constraint → more flexible, but may sacrifice efficiency.
• If you have reason to believe the two margins are driven by different processes (e.g., "whether to export" depends on a firm's productivity threshold, while "how much to export" depends on destination-market size) → use the two-part model.

5.3 Hurdle Model—Restarting After "Crossing Zero"

The Hurdle model is similar to the two-part model—the first part models "whether the threshold is crossed" (Probit/Logit), and the second part models "Y after crossing the threshold" (truncated normal or truncated Poisson, etc.).

The difference between Hurdle and two-part models is subtle but important: the second part of the two-part model is "OLS of ln Y given Y > 0"—it does not assume a distribution for the residuals. The second part of the Hurdle model is typically a truncated distribution (e.g., truncated normal)—meaning the second part is also a parametric model. In practice, OLS in the second part of the two-part model is more common (simpler, more robust), while the Hurdle model's stronger parametric assumptions in the second part make it more efficient.

6. Tobit in Panel Data—The Incidental Parameters Problem Returns

6.1 FE Tobit Is Inconsistent with Small T

Just like FE Probit—FE Tobit also suffers from the incidental parameters problem. You need to estimate $u_i$ for each individual—when T is fixed and small, these $\hat{u}_i$ are inconsistent, and their inconsistency contaminates $\hat{\beta}$. Greene's (2004) Monte Carlo simulations show that the bias of FE Tobit is substantial at T = 5.

This fact creates a gap in empirical practice: many researchers do not know that FE Tobit is inconsistent with small T—Stata's xttobit, fe command exists and runs, but its output is systematically biased when T is small. "Stata can run it" ≠ "the model is valid."

6.2 Empirical Strategies for Panel Tobit

Strategy A: RE Tobit + Mundlak individual means (CRE Tobit)—recommended

This is currently the most pragmatic approach for micro panels with small T. Add the individual mean $\bar{X}_i$ of each time-varying X to the RE Tobit:

$Y_{it}^* = X_{it} \beta + \bar{X}_i \gamma + v_i + \varepsilon_{it}$

$v_i \sim N(0, \sigma^2_v)$, $\varepsilon_{it} \sim N(0, \sigma^2_\varepsilon)$. $\bar{X}_i$ absorbs the part of $u_i$ that is linearly correlated with X—the remaining $v_i$ is assumed uncorrelated with X (under the Mundlak assumption).

* CRE Tobit (Mundlak approach)
bysort id: egen x1_mean = mean(x1)
bysort id: egen x2_mean = mean(x2)
xttobit y x1 x2 x1_mean x2_mean, re ll(0)

Limitation of CRE Tobit: it assumes the correlation between $u_i$ and X is linear (captured through $\bar{X}_i$)—if the true correlation is nonlinear, CRE Tobit provides only partial protection.

Strategy B: Pooled Tobit + clustered standard errors—pragmatic but not ideal

The simplest approach is to ignore the panel structure, use cross-sectional Tobit, and cluster standard errors at the individual level. This strategy is partially accepted when N is large, T is small, and within-individual variation in Y comes from observable changes.

But it is not ideal: it does not control for $u_i$—if your core endogeneity concern is "time-invariant omitted variables," Pooled Tobit faces the same bias problem as Pooled OLS.

Strategy C: Honoré (1992) semiparametric panel censored estimator—theoretically elegant, rarely used in practice

Honoré proposed an alternative to panel Tobit that does not rely on distributional assumptions or parametric estimation of $u_i$—based on rank conditions from pairwise comparisons. This approach does not require assuming a distribution for $\varepsilon$ and bypasses the incidental parameters problem. But it is rarely used in practice because (a) it only estimates the sign and relative magnitude of β rather than its level, (b) implementation is complex, and (c) standard error estimation is not straightforward.

6.3 How to Compute AMEs After Panel Tobit?

As with FE Probit—after RE Tobit + CRE, you have an estimate of $\hat{\sigma}_v^2$. Using this estimate along with the contribution of $\bar{X}_i$, you can compute marginal effects on both the latent-variable and observed-Y scales:

xttobit y x1 x2 x1_mean x2_mean, re ll(0)
margins, dydx(*) predict(ystar(0,.))   // on the latent-variable scale
margins, dydx(*) predict(e(0,.))       // on the observed-Y scale (conditional mean after censoring)

Note that predict(e(0,.)) is the conditional expectation of Tobit after censoring—it is the most central economic quantity under the Tobit model. We recommend reporting AMEs on both scales in your paper—or at least clearly stating which scale you are interpreting.

7. Common Misconceptions

7.1 Misconception 1: "Y Is All Positive → I Should Use Tobit"

Y being all positive does not automatically mean you need Tobit. Tobit is designed for corner solutions with "a mass at zero + continuous positive values." If your Y only takes positive values (e.g., wages, prices, height—never equal to zero), and there is no distributional truncation—OLS is perfectly reasonable. "Y > 0" and "Y is censored at zero" are two different concepts. The former only means Y takes values on the positive domain—if Y is far from zero and there is no mass at any boundary, OLS's linear approximation typically works well. The latter means there is a barrier—zero—that squeezes a portion of observations onto zero.

7.2 Misconception 2: "Tobit Is More 'Advanced' Than OLS—I Should Switch to Tobit"

While Tobit relaxes OLS's linearity assumption, it also introduces new, stronger assumptions: normality and homoskedasticity of $\varepsilon$. If these assumptions do not hold in your data—Tobit's MLE is inconsistent, whereas OLS (with robust standard errors) remains a consistent estimator of $\mathbb{E}[Y \mid X]$ (under the linearity assumption). Tobit is not an "upgraded version" of OLS—it is a "different-assumptions version" of OLS. If your data support Tobit's assumptions, it can provide richer information (decomposition of extensive vs. intensive margins). But if the assumptions fail—OLS is more robust.

7.3 Misconception 3: "Tobit's β Is the Marginal Effect of X on Y"

Tobit's $\hat{\beta}$ is the marginal effect of X on the latent variable $Y^*$—not on the observed Y. The marginal effect on observed Y = $\hat{\beta} \cdot \Phi\left(\frac{X\hat{\beta}}{\hat{\sigma}_\varepsilon}\right)$—scaled by a factor. Reporting Tobit coefficients without computing marginal effects is the same error as reporting Logit coefficients without computing AMEs—your readers cannot directly understand what β = 0.38 means for Y (on the censored, observed scale).

7.4 Misconception 4: "ln(Y+1) + OLS Is Roughly the Same as Tobit"

Adding a constant to Y and taking the logarithm (e.g., $\ln(Y+1)$)—then running OLS—is a common "ad hoc" approach. But the expectation of $\ln(Y+1)$ is not a linear function of $X\beta$ (even if $Y^*$ truly follows a Tobit), and the arbitrary choice of the constant 1 affects the results. Tobit avoids this problem by directly modeling the censoring mechanism—$\ln(Y+1)$ + OLS has no such theoretical protection, and marginal effects are more difficult to interpret back on the original scale.

7.5 Misconception 5: "FE Tobit Runs in Stata → It Can Be Used"

xttobit, fe exists in Stata—but it uses the same approach as FE Probit of directly estimating N individual $u_i$'s, and suffers from the incidental parameters problem when T is fixed. In panels with small T (e.g., T ≤ 10), the β estimates from FE Tobit are biased. Use CRE Tobit (RE + Mundlak) instead—or accept LPM + FE as a more robust baseline.

8. Summary

Dependent variable is positive—six core takeaways on OLS vs. Tobit:

1. The fundamental difference between OLS and Tobit is whether the "zero barrier" is modeled. OLS assumes $Y = X\beta + \varepsilon$; Tobit assumes $Y = \max(0, X\beta + \varepsilon)$. If the data-generating process truly features a "zero barrier" (many individuals choose not to participate, Y = 0), Tobit is closer to the true structure; if Y merely happens to be greater than zero with no mass at any boundary, OLS is entirely sufficient.

2. If you only care about the sign and significance of coefficients—OLS and Tobit usually agree in direction. Divergence mainly arises in three scenarios: severe zero mass, extreme skewness in the positive part of Y, or models with nonlinear terms.

3. Signals for switching to Tobit: zero share > 15%–20%, your research question naturally involves both "whether to participate" and "how much to participate," reviewers are questioning the structure of the dependent variable, or OLS predicted values frequently approach or fall below zero.

4. Panel FE Tobit is inconsistent with small T—the incidental parameters problem. The same logic as FE Probit—you cannot consistently estimate N individual $u_i$'s when T is fixed. Practical strategy: CRE Tobit (RE + Mundlak)—add the individual mean of each X to the RE Tobit.

5. Tobit's β is the marginal effect on the latent variable, not on observed Y. Compute AMEs (on the observed-Y scale) in your paper—use margins, predict(e(0,.)).

6. Tobit is not the only option—the two-part model / Hurdle model is more flexible when the extensive and intensive margins are driven by different processes. If you have reason to believe X affects "whether to participate" and "how much to participate" in fundamentally different ways—the two-part model is more honest than Tobit (it does not impose the equality constraint).

One-sentence closing:

"OLS draws a straight line through your Y—it passes through zero, extends toward negative infinity, and treats the pile of zeros and the positive values as the same thing. Tobit places a glass barrier at zero—those with a negative propensity are blocked by the glass, and you only see zero. The effect of X is refracted by this glass—part of it pushes people across the glass, part of it makes those already above the glass stand higher. If enough people in your data are standing on the glass—the curve is closer to where they truly are than the straight line. But before you switch from the line to the curve, ask yourself two questions: how many people in your data are standing on zero (> 15%?), and do you really need to distinguish between 'having it or not' and 'how much'? If the answers are yes—Tobit's curve is more honest than OLS's line. But in panels—remember that FE Tobit's curve is biased in small T (the incidental parameters problem); cross over using the bridge of CRE Tobit."

9. Presentation Suggestions for Bilibili / WeChat Official Account

• Bilibili video: Use "the zero glass ceiling" as the core visual metaphor. Opening: a group of firms (data points) scattered along the Y-axis (R&D expenditure). Some firms stand on the floor (Y = 0); others are scattered in the air (Y > 0, widely dispersed positive values). A transparent glass ceiling (actually a floor—the "zero barrier") presses down on the floor—below the glass are negative values ("negative R&D propensity"), blocked underneath by the glass. Narration: "In Tobit's world—R&D expenditure is not naturally nonnegative. There is a latent variable—'R&D propensity'—which can be negative. When the propensity is negative, the firm does no R&D—you only see a zero. When the propensity is positive, R&D expenditure equals the size of the propensity." A straight OLS line cuts through the data—it struggles at the pile of zeros (unable to bend downward), forcibly penetrating into the negative region below the glass. Tobit's curve is "blocked" at the glass—no negative predictions, bending at zero. Narration: "OLS can only draw straight lines—it is forced to pierce the glass and predict negative R&D expenditure. Tobit knows where the glass is—its line breaks at zero, respecting the barrier of the glass." Act 1 "extensive margin vs. intensive margin": a firm bumps against the glass—crosses zero—going from Y = 0 to Y > 0. Another firm is already above the glass—the increment in X merely makes it stand higher. Two arrows emanate from X—one pointing to "crossing the glass" (extensive margin), the other to "standing higher" (intensive margin). Narration: "The total effect of X on Y is a weighted average of the two margins. Tobit allows you to separate them. OLS compresses them into one number—it does not tell you whether your X is pushing people across the threshold or making those who crossed walk further. If your research question is precisely one of these two questions—you need Tobit." Act 2 "FE Tobit is biased in small T": a vortex appears (the incidental parameters problem). FE Tobit is sucked in—the β estimate line deviates from the true value. Annotation: "At T = 5, FE Tobit's β can be biased by 30%–50%." The CRE Tobit bridge bypasses the vortex—Mundlak's individual means (X̄_i) form a path. Narration: "CRE Tobit—adding individual means within the RE framework to absorb endogeneity. It is not perfect—it assumes linear correlation between u_i and X. But in small-T panels, it is the safest way to do Tobit."

• WeChat Official Account: The comparison table of the three dependent-variable types (corner solution, truncation, censoring) should be made into a diagnostic card—this is the most commonly confused concept in empirical work. The diagram of Tobit's latent variable specification (Y* → max(0, Y*) = Y) should be the core infographic. The comparison of effect scales between OLS and Tobit—the explanation of "β_OLS ≠ β_Tobit"—should be made into an intuitive comparison card. The five warning signs for switching to Tobit should be made into a checklist card. The comparison table of the Tobit family (standard Tobit, two-part model, Hurdle model) should be made into a model-selection reference card. The three strategies for panel Tobit (CRE Tobit, Pooled Tobit, Honoré) with usability judgments should be made into a decision reference card. The five common misconceptions should be made into a correction card.

• Recommended titles:

  • Main title: 《Dependent Variable Is Positive—Panel OLS or Tobit? What's the Difference?》
  • Alternative title: 《Y Is Censored at Zero—What Signs Suggest You Should Switch from OLS to Tobit?》
  • New-media title: 《Your Dependent Variable Has a Huge Pile of Zeros—Is Panel FE Still Enough? When Should You Switch to Tobit?》

• Key quotes:

  "OLS draws a straight line through your Y—passing through zero, extending into negative values, pretending the pile of zeros and the positive values are the same thing. Tobit places a piece of glass at zero—those with propensity below zero are blocked by the glass, and you only see zero. The effect of X is refracted by this glass—part pushes people across the glass, part makes those above the glass stand higher. If enough people in your data are standing on the glass—Tobit is more honest than OLS."

  "Tobit's β is not the marginal effect of X on Y—it is the marginal effect of X on the latent variable. The effect on observed Y is scaled by the 'probability of crossing zero.' In your paper, do not just report Tobit's coefficient table—use margins to compute the AME on the Y scale. Reviewers and readers need the latter."

  "FE Tobit has a one-line command in Stata—but its output is unreliable in small T. The incidental parameters problem—you must estimate N individual u_i's, each with only T observations, and the bias at T = 5 can reach 30%–50%. CRE Tobit is the way around it—using individual means within the RE framework to absorb endogeneity. Not perfect—but among the limited options, it is the safest path."

  "The difference between the two-part model and Tobit is not that one is the 'correct model' and the other is not—it is that one assumes the β's on the extensive and intensive margins are the same (Tobit), while the other does not (two-part model). If your X has the same sign on both margins—Tobit is more efficient than the two-part model. If the signs may differ—the two-part model is the more flexible choice. This is economic thinking, not statistical algorithm—your theory should determine this choice."