Can You Run OLS Directly with a Dummy Variable as the Dependent Variable? — The 'Three Sins' of the Linear Probability Model and Why People Still Use It

1. Opening: Y Is Only 0 and 1—You Casually Run reg y x, and Then What?

You are studying "what kind of students choose to take the postgraduate entrance exam." Your dependent variable is kaoyan—taking the exam = 1, not taking it = 0. You have a set of explanatory variables: GPA, school tier, family income, gender……

You casually type in Stata:

reg kaoyan gpa school income female

The coefficients come out, and the R² comes out too. Everything looks normal.

Then you glance at the fitted values from predict—some people's "predicted probability" of taking the exam is −0.15, and some people's is 1.38.

−0.15? A negative probability? 1.38? A probability exceeding 100%?

Congratulations—you have just run into the first sin of the Linear Probability Model (LPM). But don't rush to delete reg and switch to logit—because even with these problems, LPM remains the choice of many economists in numerous settings, and for quite good reasons.

Core message: Running OLS directly with a 0/1 dependent variable—this is called the Linear Probability Model (LPM). It has three well-known problems: predicted values can fall outside [0, 1], the error term is inherently heteroskedastic, and the linear form assumes constant marginal effects. But LPM has three equally important advantages: coefficients are directly interpreted as probability changes (no need to convert marginal effects), it works seamlessly with instrumental variables and fixed effects, and when the sample is large enough and the values of X are not too extreme, the problem of predicted values falling outside [0, 1] is not severe. Logit/Probit solves the first three sins, but also falls short of LPM on the latter three points. There is no absolute right or wrong in method choice—it depends on your research context and data characteristics.

2. What Is LPM? — Running OLS with 0/1 Treated as an Ordinary Number

2.1 Model Specification

$Y_i = \beta_0 + \beta_1 X_i + \varepsilon_i, \quad Y_i \in \{0, 1\}$

You pretend Y is a continuous variable and use OLS to estimate β₀ and β₁. Under the orthodox assumptions of OLS, Y should be a continuous variable that can take any value on (−∞, +∞)—clearly, a 0/1 variable does not satisfy this condition.

But because the algebra of OLS does not care whether Y is 0/1 or continuous, you can still obtain coefficients. The question is: are those coefficients still usable? Can they still be interpreted?

2.2 Interpreting the Coefficients—This Is Where LPM's Greatest Strength Lies

In LPM, the regression line $\hat{Y} = \hat{\beta}_0 + \hat{\beta}_1 X$ has a very intuitive meaning:

$\hat{\beta}_1 = \frac{\Delta \hat{Y}}{\Delta X} = \frac{\Delta \hat{P}(Y=1)}{\Delta X}$

β̂₁ = for each one-unit increase in X, the predicted probability that Y = 1 changes by this amount.

For example, β̂₁ = 0.05 means "for each 1-point increase in GPA, the probability of taking the postgraduate exam increases by 5 percentage points." This statement requires no translation—it is "the change in probability," and you can write it directly into the policy recommendations of your paper.

This is LPM's greatest advantage over Logit/Probit. The raw coefficients of a Logit model are changes in "log odds ratios"—you cannot directly say "a coefficient of 0.3 means the probability changes by XX." You need post-estimation marginal effects (margins, dydx), and the marginal effects differ at different values of X—unlike LPM, which gives you a single, "globally constant" number.

3. The "Three Sins" of LPM—Why Your Instructor Tells You to Be Careful

3.1 The First Sin: Predicted Probabilities Can Fall Outside [0, 1]

This is LPM's most obvious problem. An OLS regression line is a straight line—it extends to any position on (−∞, +∞) when X is sufficiently small or large. But the domain of a probability is [0, 1].

When is this a real problem? When the variation in X in your data is large, or when the individuals you care about are at extreme values of X. For example, if you use "college entrance exam score" to predict "whether admitted to Tsinghua"—scores range from 200 to 750. At 200, LPM's predicted probability might be −0.3; at 750, it might be 1.5.

When is this not a big problem? When the range of X variation you care about is small, and most observations' predicted probabilities fall between [0.1, 0.9]—within this interval, a straight line is not very different from a more reasonable S-shaped curve, few observations fall outside [0, 1], and OLS's main estimation results are not seriously affected.

A practical tip: after running LPM, predict p_hat, then sum p_hat. If min < 0 or max > 1, note it down—report it honestly in your paper, and explain that the proportion of such extreme predictions is small (if it indeed is).

3.2 The Second Sin: The Error Term Is Inherently Heteroskedastic—Not Because the Data Are Bad, but Because of the Mathematics

In LPM, the dependent variable Y can only take values 0 or 1. Given Xᵢ, Yᵢ follows a Bernoulli distribution with probability pᵢ = P(Yᵢ = 1 | Xᵢ) = β₀ + β₁Xᵢ. The variance of a Bernoulli distribution is pᵢ(1 − pᵢ)—this variance inherently depends on the value of Xᵢ.

When pᵢ ≈ 0.5, the variance ≈ 0.25 (maximum). When pᵢ ≈ 0 or 1, the variance → 0. This means the dispersion of the errors differs systematically at different values of X—this is "structural heteroskedasticity," not a problem with your data; it is built into the mathematical structure of a 0/1 dependent variable.

Consequences: OLS coefficients remain unbiased (under the assumption that Cov(X, ε) = 0), but the standard error estimates are wrong (typically underestimated → inflated t-values → you are more likely to incorrectly reject the null hypothesis that "the coefficient is zero").

Solution: simple—use robust standard errors. reg y x, robust. In LPM, using robust standard errors is not an option; it is a must. Any LPM that does not report robust standard errors should be questioned.

Angrist & Pischke (2009) offer a classic piece of advice in Mostly Harmless Econometrics: LPM + robust standard errors is often good enough in many settings—you do not necessarily need to use Logit/Probit.

3.3 The Third Sin: The Linear Form Assumes "Constant Marginal Effects"

LPM assumes that for each one-unit increase in X, the probability that Y = 1 changes by a fixed β₁ percentage points—regardless of the starting value of X.

This is unreasonable in many cases. Consider "the effect of family income on whether one buys a car": going from an annual income of 10,000 to 20,000 might increase the probability of buying a car by 5 percentage points. Going from 990,000 to 1,000,000—would the probability also increase by 5 percentage points? Most likely not—when you are already wealthy, earning a bit more has almost zero effect on the "whether to buy a car" decision (the probability is already close to 1, with little room to rise).

This is why Logit/Probit use an S-shaped (Sigmoid) curve—the marginal effect is largest in the middle of X and approaches zero at both ends. LPM's straight line cannot capture this "diminishing marginal effect" pattern.

But conversely—if in your research context the effect of X is indeed approximately constant over the sample range, LPM is not only unproblematic but also easier to interpret than Logit. For example, the effect of "whether one owns a mobile phone" on "whether one uses mobile payment"—the jump from no phone to having a phone is discrete; there is no issue of "the marginal effect differing at the 0.3rd phone versus the 0.7th phone."

4. Given the Three Sins, Why Do Economists Still Use LPM?

4.1 Reason One: Interaction Terms and Instrumental Variables—LPM's Natural Advantage

This is the most robust justification for LPM in empirical economics.

In Logit/Probit, interpreting interaction terms is very complicated—the "interaction effect" of two variables in a nonlinear model is not equal to the sign or magnitude of the interaction term coefficient (Ai & Norton, 2003). You need additional computations to correctly determine the magnitude and significance of the interaction effect. In LPM, an interaction term is just an interaction term—exactly the same as in ordinary linear regression, and margins handles it in one line.

More importantly—instrumental variables (IV) are standard practice in LPM, and 2SLS is directly applicable. If your endogenous variable is 0/1, you need an IV strategy—LPM + 2SLS is straightforward. In contrast, incorporating IV into Logit/Probit requires special estimation methods (such as IV Probit or control function approaches), which are more complex to implement and interpret.

Conclusion: If your research involves interaction terms or instrumental variables, and your primary interest lies in the direction and significance of coefficients (rather than precise probability predictions), LPM is often simpler and more transparent than Logit/Probit.

4.2 Reason Two: Fixed Effects—Logit's "Incidental Parameters Problem"

In panel data, if you want to control for unit fixed effects—that is, time-invariant individual characteristics—the Logit model suffers from the well-known "Incidental Parameters Problem": when T (number of time periods) is fixed and N (number of individuals) → ∞, the coefficient estimates from fixed-effects Logit are inconsistent. The only exception is Conditional Logit, but it can only use observations where Y varies within groups.

In LPM, fixed effects simply mean adding a set of individual dummy variables (or within-group demeaning)—exactly the same as a panel fixed-effects model with a continuous dependent variable. xtreg y x, fe with robust standard errors, and the problem is solved.

This is especially important in policy evaluation: if you have panel data and want to control for "time-invariant individual heterogeneity" (such as ability, personality, geographic location), LPM + unit FE is your best choice—Logit/Probit do not have an equally simple and reliable alternative in this scenario.

4.3 Reason Three: Coefficients Are Directly Readable—Lowest Communication Cost

Your paper is not written only for econometricians. Policymakers, corporate decision-makers, and interdisciplinary collaborators need to hear "when this variable increases by one unit, the probability changes by X percentage points," not "the log odds change by 0.3, and the marginal effect at the mean is 0.05."

LPM coefficients require no translation. They are marginal effects by construction. This reduces the communication barrier between you and your readers—in policy reports, industry analyses, and interdisciplinary collaborations, this is a tangible advantage.

5. LPM vs. Logit vs. Probit—How to Choose?

A common practice in empirical work: use Logit/Probit as the baseline model in the main text, but report LPM results in a footnote or appendix—if the direction and significance of the core coefficients are consistent across both models, reviewers will find it hard to question the sensitivity of your conclusions to model choice.

6. Common Misconceptions

6.1 Misconception One: R² Is Too Low, So LPM Cannot Be Used

LPM's R² is typically very low (0.05–0.20 is the norm). Some people see R² = 0.08 and conclude the model is hopeless.

Wrong. The R² for a 0/1 dependent variable is naturally low—because at the individual level, Y is 0 or 1, while the model predicts a probability between 0 and 1. Even if the model is completely correct (you know the true DGP), the actual values of Y are 0 or 1, while the regression line is a continuous probability line—you can never perfectly fit a bunch of 0s and 1s with a line. In this case, R² is no longer an accurate measure of "how good the model is." In LPM, the significance and economic meaning of coefficients matter far more than R².

6.2 Misconception Two: Because the Errors Are Not Normally Distributed, LPM's t-Tests Are Invalid

In large samples, the OLS estimator is asymptotically normal (by the central limit theorem)—even if the errors are not normal. This asymptotic property does not depend on the normality assumption for the errors. Combined with robust standard errors, LPM inference is valid in large samples.

In other words: the violation of the normality assumption is the least of your worries in LPM.

6.3 Misconception Three: Logit/Probit Is Always "More Correct" Than LPM

Logit/Probit assume an S-shaped functional form—this is an assumption, just as LPM's straight-line assumption is also an assumption. If the true probability function happens not to be the S-shape of logit or probit (for example, it is approximately linear over some range), then Logit/Probit is also wrong—and wrong in a way that is less transparent than LPM.

No model is the "true model." Both LPM and Logit/Probit are different approximations to the true conditional expectation function. Which one you choose depends on your trade-off over approximation quality—whether you want a purely linear approximation that is easy to interpret but not guaranteed to be reasonable at extreme values, or a nonlinear approximation that stays within [0, 1] everywhere but is more complex to interpret and implement.

7. Summary

Four core takeaways about LPM:

1. LPM is simply running OLS with a 0/1 dependent variable treated as continuous. The coefficients directly give "for each one-unit change in X, the probability that Y = 1 changes by X percentage points"—this is LPM's core advantage over Logit/Probit.

2. The three sins: predicted probabilities can fall outside [0, 1] (a problem when X is extreme, not when X is concentrated), structural heteroskedasticity (solved with robust standard errors), and the linear form assuming constant marginal effects (a problem in scenarios where an S-shape is more appropriate).

3. Scenarios where LPM is irreplaceable: when using instrumental variables (IV), panel fixed effects (FE), and when you need simple interpretation of interaction effects—LPM is simpler, more transparent, and more reliable than Logit/Probit.

4. Best practice: LPM + robust standard errors is the baseline. If you are concerned about predictions outside [0, 1] or non-constant marginal effects, use Logit/Probit in the main text and LPM in the appendix as a robustness check—let the evidence from both point in the same direction.

One-sentence conclusion:

"LPM is not a 'wrong method'—it is a 'crude but honest' method. It honestly tells you: if I use a straight line to fit 0s and 1s, here are the coefficients, the standard errors have been corrected, the predicted values fall within [0, 1] in most places, and please treat the extreme values with caution. Logit/Probit are more refined—but they add a layer of shell that you need margins to peel away and translate. Which one you choose depends on whether you fear 'negative predicted probabilities' more or 'coefficients that cannot be directly explained to others' more."

8. Presentation Suggestions for Bilibili / WeChat Official Account

• Bilibili video: Use "forcibly stuffing 0/1 into a straight line" as the visual opening. Visual: the vertical axis has only two tick marks—0 and 1, with all scatter points falling on these two horizontal lines. A straight line (the OLS regression line) enters from below 0 and exits above 1. Narration: "When your dependent variable is only 0 and 1—you force a straight line into it. It fits reasonably well in the middle, but at the two ends—the line's tips exit the legal territory of probability—running to −0.3 and 1.4. This is the linear probability model." Then unfold in three acts. Act One, "The Three Sins": animations reveal each in turn—predicted values out of bounds (the line exits the [0, 1] region, alarms flashing), heteroskedasticity (the dispersion of residuals is largest at 0.5 and shrinks toward 0/1—a trumpet shape collapsing into two cones), constant marginal effects (an S-shaped curve overlaid on the straight line, showing the S flattening at the ends while the line keeps rising). Act Two, "Why Still Used": three scenarios—IV (an instrumental variable arrow inserted into the model, labeled "2SLS + LPM = simple and direct"), panel FE (multiple individual dummy variables entering the model, labeled "Logit's incidental parameters problem—LPM does not have it"), interaction effects (margins produces results in one line, labeled "interactions in nonlinear models = hell mode"). Act Three, "Selection Guide": a decision-tree animation—question branches ("Have IV?" → LPM; "Have panel FE?" → LPM; "Need interactions?" → LPM; "Need precise probability predictions?" → Logit). Closing visual: two paths—the left signpost says "LPM: crude but straightforward," the right signpost says "Logit: refined but needs translation," with narration: "It is not about who is right or wrong—it is about what problem you are trying to solve."
• WeChat Official Account: For LPM's three sins, make each sin into an "evidence card" (symptom diagram + mathematical cause + solution). Turn the LPM vs. Logit/Probit selection decision tree into a vertical flowchart. Present the five-scenario recommendation table as a matrix infographic. Put Angrist & Pischke's classic advice ("LPM + robust standard errors is good enough") in a pull quote box. For the histogram of predicted probability distribution (annotating the proportion with min < 0 or max > 1), include a real Stata output example.
• Recommended titles:
  • Main title: 《Can You Run OLS Directly with a Dummy Variable as the Dependent Variable? — The Three Sins of the Linear Probability Model and Its Defense》
  • Alternative title: 《Y Is Only 0 and 1—Use reg or logit? — LPM's Fatal Flaws and Its Irreplaceable Advantages》
  • New media title: 《Your Dependent Variable Is 0/1—Why Do Economists Still Use OLS Instead of Logit?》
• Key quotes:

  "LPM forces your 0s and 1s into a straight line. It fits well in the middle, and at the ends it can slip outside the legal territory of probability without warning. But its coefficients need no translation—'one more unit of X, and the probability of Y=1 rises by β percentage points'—Logit cannot say this without first translating through margins."

  "LPM's heteroskedasticity is not because your data are dirty—it is built into the mathematical structure of 0/1. No need to rerun, no need to switch models—just add robust and you are done."

  "If you need instrumental variables, unit fixed effects, or simply interpretable interaction effects—Logit either cannot do these things or makes you suffer through them. LPM says: I can."

  "LPM and Logit are not about right versus wrong—they are different approximations to the same unknown function. LPM's approximation is a straight line—crude but transparent. Logit's approximation is an S-curve—refined but requiring translation. Which one you choose depends on your data, your methods, and the audience you need to communicate with."