Binary Dependent Variables—Logit or Probit? What's Special in Panel Data?

1. Opening: Two Commands, Nearly Identical Output—Why Bother Choosing?

You ran a binary choice model—studying "whether firms export." You first used Logit:

logit exporter productivity size age

Then you used Probit:

probit exporter productivity size age

You glanced at the AME:

* Logit's AME
margins, dydx(*) post
* → productivity's AME = 0.047
 
* Probit's AME
margins, dydx(*) post
* → productivity's AME = 0.046

A difference of 0.001—far smaller than the standard error (0.012). You realize: on the AME, the most economically meaningful quantity, Logit and Probit give nearly identical answers.

Then you asked yourself three questions:

1. If the AMEs are nearly identical—then where exactly do Logit and Probit differ? Why do two different models exist?
2. Under what circumstances would journal reviewers care which one you chose?
3. If your data are a panel—the same firm observed repeatedly across years—does the Logit vs. Probit choice still look the same?

The third question is especially important—because you are about to work with a panel binary choice model, and you have discovered that in panels, the divergence between Logit and Probit is larger than in cross-sections: fixed effects Logit exists and is consistent with small T, while fixed effects Probit is inconsistent with small T.

Core message: In cross-sectional binary choice, the only difference between Logit and Probit lies in the distributional assumption on the latent error—Logit assumes a Logistic distribution (variance $\pi^2/3$), Probit assumes a normal distribution (variance 1). This difference is almost always negligible for the AME (average marginal effect)—the predicted probabilities from the two models typically differ by less than 0.02 across the range of X, and the difference in AMEs is far smaller than sampling error. Therefore, the choice in cross-sections is primarily practical rather than statistical: Logit offers the intuitive tool of odds ratios, making it more mainstream in economics than Probit. The real divergence emerges in fixed effects models with panel data: Logit can eliminate the individual heterogeneity $u_i$ through conditional likelihood (Conditional Logit)—yielding a consistent $\hat{\beta}$, even with small T. Probit has no such conditional likelihood convenience—FE Probit suffers from the incidental parameters problem with small T, and $\hat{\beta}$ is inconsistent when T is fixed and N → ∞. Therefore, if you need panel fixed effects to control for time-invariant individual heterogeneity, Logit is the only binary choice model that is consistent with small T. But FE Logit carries its own costs: only individuals whose Y varies over time enter the estimation (the "switcher" sample); the coefficient interpretation shifts from "comparisons between individuals" to "changes within an individual"; and estimating the AME is more involved than in cross-sections—you cannot directly apply the cross-sectional margins logic, because the distribution of $u_i$ may be correlated with X and is typically assumed to be unknown.

2. The Core Difference Between Logit and Probit in Cross-Sections—Only One Place Differs

2.1 The Shared Latent Variable Framework—The Divergence Occurs Only at the Final Step

Logit and Probit share exactly the same model structure:

$Y_i^* = X_i \beta + \varepsilon_i$

$Y_i = \begin{cases} 1 & \text{if } Y_i^* > 0 \\ 0 & \text{if } Y_i^* \leq 0 \end{cases}$

The mean of $\varepsilon_i$ is set to 0—because any nonzero mean would be absorbed into the intercept. The variance of $\varepsilon_i$ is normalized to a fixed value—because $\beta$ and $\sigma_\varepsilon$ are only identified as the ratio $\beta/\sigma_\varepsilon$ in the latent variable framework, and they cannot be separately estimated.

The point of divergence: the distributional assumption on $\varepsilon_i$.

• Probit: $\varepsilon_i \sim N(0, 1)$ — standard normal distribution.
• Logit: $\varepsilon_i \sim \text{Logistic}(0, \pi^2/3)$ — Logistic distribution with variance $\pi^2/3 \approx 3.29$.

Both distributions are symmetric, bell-shaped, and mean zero—they are very similar in the center but differ in the tails: the Logistic distribution has heavier tails than the normal (decaying as $e^{-|x|}$ rather than $e^{-x^2}$).

2.2 Why Are the AMEs Almost Always the Same?

Given $Y_i^* = X_i \beta + \varepsilon_i$, the probability of observing $Y_i = 1$ is:

$\Pr(Y_i = 1 \mid X_i) = \Pr(\varepsilon_i > -X_i \beta) = F(X_i \beta)$

where $F(\cdot)$ is the cumulative distribution function (CDF) of $\varepsilon_i$:

• Probit: $F(X\beta) = \Phi(X\beta)$ — standard normal CDF.
• Logit: $F(X\beta) = \frac{1}{1 + \exp(-X\beta)}$ — Logistic CDF.

Key fact: $\Phi(\cdot)$ and $\text{Logistic}(\cdot)$ are extremely close across the entire real line. Over the range Xβ ∈ [-2, +2] (which covers the predicted index for the vast majority of observations), the maximum difference between the two functions is less than 0.02. The difference only widens slightly in the tails as Xβ → ±∞—but most observations have Xβ within this core range.

Since the probabilities themselves are nearly identical—the marginal effects ($f(X\beta) \cdot \beta$) are naturally nearly identical as well. This is why the AME is almost insensitive to the choice between Logit and Probit.

2.3 What About β̂?—The "1.6 Rule"

Logit coefficients are approximately 1.6–1.8 times larger than Probit coefficients:

$\hat{\beta}_{\text{Logit}} \approx 1.6 \times \hat{\beta}_{\text{Probit}}$

Why? Because the error variance for Logit ($\pi^2/3 \approx 3.29$) is larger than that for Probit (1)—and $\beta = \partial Y^*/\partial X$ and $\sigma_\varepsilon$ are only identified as a ratio. A larger variance → the coefficients must be numerically larger to produce probability effects of the same magnitude.

But this does not mean you can use the "1.6 rule" to precisely convert coefficients between models—it is only an approximation, and it breaks down when Xβ is far from 0. Compare on the AME scale—not on the coefficient scale. This is also why you should never compare Logit and Probit without computing the AME.

2.4 Choosing Between Logit and Probit in Cross-Sections—A Practical Decision Framework

Rule of thumb in economics: with cross-sectional data—use Logit (then report the AME). With panel data, if you need fixed effects—use Logit (conditional Logit). If you need random effects or CRE (Correlated Random Effects)—Logit and Probit perform similarly in practice, but follow the convention in your field.

3. Binary Choice in Panel Data—Why Things Get Complicated?

3.1 Model Specification for Panel Binary Choice

In cross-sectional models, heterogeneity is subsumed into $\varepsilon_i$. In panels, you have two unobservable components:

$Y_{it}^* = X_{it} \beta + u_i + \varepsilon_{it}$

$Y_{it} = 1 \text{ if } Y_{it}^* > 0$

• $u_i$: time-invariant heterogeneity for individual i (ability, culture, risk preferences…).
• $\varepsilon_{it}$: time-varying error.

The core challenge of panel binary models: the relationship between $u_i$ and X. If $\text{Cov}(X_{it}, u_i) \neq 0$ (almost always true in observational data), ignoring $u_i$ or assuming it is uncorrelated with X (RE) leads to inconsistent estimates. But the way $u_i$ is handled—in binary panels—is fundamentally different from linear panel FE (where the within transformation directly eliminates $u_i$).

3.2 The Incidental Parameters Problem—FE Probit Is Inconsistent with Small T

Why does linear panel FE work? Because the within transformation $Y_{it} - \bar{Y}_i$ eliminates $u_i$—$u_i$ does not need to be directly estimated. The estimator for β is consistent as N → ∞ with T fixed.

In nonlinear models, the within transformation cannot eliminate $u_i$—you must directly estimate each individual's $u_i$. In FE Probit, you need to estimate a $u_i$ for each individual i—that is, in addition to the K parameters of β, you add N more parameters.

When N → ∞ with T fixed, the total number of parameters (N + K) grows at the same rate as the sample size (N × T). Each $u_i$ has only T observations for estimation—with T fixed, $\hat{u}_i$ is inconsistent. These N inconsistent $\hat{u}_i$'s contaminate the estimation of β—making $\hat{\beta}$ inconsistent even as N → ∞. This is the incidental parameters problem.

How large is the bias in FE Probit? Monte Carlo simulations show that with T = 5, the bias in β for FE Probit can reach 40%–60%. Even with T = 20, the bias remains around 10%–20%. FE Probit is unusable in short-T panels.

3.3 Conditional Logit (FE Logit)—Logit's Rescue

The Logit model has a special property that Probit lacks: in Logit, $\sum_t Y_{it}$ (the number of times Y = 1 for individual i across all periods) is a sufficient statistic for $u_i$ (Chamberlain, 1980).

This means—by conditioning on the total $\sum_t Y_{it}$—you can eliminate $u_i$ without directly estimating it. The conditional likelihood function depends only on β, not on $u_i$—conceptually parallel to the within transformation in linear panel FE (both eliminate $u_i$ through a mathematical operation rather than estimating it).

$\text{Conditional likelihood: } \Pr(Y_{i1}, \dots, Y_{iT} \mid \sum_t Y_{it}, X_i, u_i) = \frac{\exp(\sum_t Y_{it} X_{it}\beta)}{\sum_{d \in S_i} \exp(\sum_t d_t X_{it}\beta)}$

where $S_i$ is the set of all 0/1 sequences d satisfying $\sum_t d_t = \sum_t Y_{it}$—that is, all possible ways of arranging the time periods with Y = 1 and Y = 0, holding the total fixed.

Intuition: Conditional Logit does not compare "whether Y is more likely to be 1 for individual A versus individual B"—but rather "for the same individual i, in which periods is Y more likely to be 1 (relative to other periods)." Identification comes from within-individual time variation in Y—exactly corresponding to the "within variation identification" of linear FE.

* Conditional Logit (FE Logit)—uses only individuals with time variation in Y
xtlogit y x1 x2, fe
* Equivalent to
clogit y x1 x2, group(id)

Three key costs of Conditional Logit:

Cost one: only "switchers" are included. If an individual's Y is 0 in all periods or 1 in all periods ($\sum_t Y_{it} = 0$ or T), that individual contributes zero to the conditional likelihood—these individuals are automatically dropped. If your dependent variable has little variation (e.g., "whether listed" or "whether bankrupt"—most firms do not change status during the sample period), your estimation sample may shrink dramatically. In your paper, explicitly reporting "the sample size entering the FE Logit estimation fell from N to N_switcher" is essential.

Cost two: the interpretation of coefficients changes. Cross-sectional Logit coefficients reflect odds differences corresponding to X differences between individuals. Conditional Logit coefficients reflect odds changes corresponding to X changes within the same individual across periods—this is a "within effect," not a "between effect." This is consistent with the interpretation of linear panel FE coefficients—but in binary models, this interpretation is more easily mistaken for the cross-sectional one.

Cost three: the AME can no longer be directly computed in FE Logit. In cross-sectional Logit, you can average marginal effects over all individuals in the sample—because you do not need the specific values of $u_i$, only an approximation using the average predicted probability. But in FE Logit, although $u_i$ is eliminated by conditioning, if you want the AME, you need the values of $u_i$—and FE Logit does not estimate $u_i$. Using margins directly after FE Logit requires assuming $u_i = 0$—which is not a good assumption. Correctly computing the AME in FE Logit requires additional steps (see Section 4.3 below).

4. Empirical Strategies for Panel Binary Choice—FE Logit Is Not the Only Path

4.1 Strategy Map—Four Approaches and Their Assumptions

4.2 Strategy A: Pooled Logit—Ignoring the Panel Structure

When might this be considered?

• Your panel is a "pseudo-panel"—N is large but T is extremely small (T = 2 or 3), and within-individual variation in Y is minimal.
• Your key explanatory variable is itself nearly time-invariant (e.g., gender, race, years of education)—you are not really doing panel identification anyway.
• But even in these cases, you should at least cluster standard errors at the individual level—acknowledging that observations from the same firm in different years are not independent.

4.3 Strategy B: FE Logit (Conditional Logit)—The Strongest Way to Control for $u_i$

When to use it?

• You have key time-invariant confounders ($u_i$), and $\text{Cov}(X, u_i) \neq 0$ is theoretically unavoidable.
• Your Y varies over time for a sufficient number of individuals (enough "switchers" to maintain an effective sample size).
• The coefficient you care about is the within effect of X—the effect of within-individual changes in X over time on the odds of Y.

How to compute the AME after FE Logit?

This is a widely discussed but often overlooked practical issue. The conditional likelihood of FE Logit does not estimate $u_i$—so you cannot directly apply the cross-sectional Logit AME computation (which requires predicted probabilities, which in turn require $u_i$). Two viable approaches exist:

Approach one (imperfect but common): assume $u_i = 0$ to compute predicted probabilities—this gives an approximation of the marginal effect for a "typical individual." This approach is usually reasonable in magnitude (because the distribution of $u_i$ is centered in the population), but it is inaccurate at the individual level. You can note in the paper that "the AME is based on the assumption $u_i = 0$—it should be interpreted as the marginal effect for an individual randomly drawn with $u_i = 0$."

Approach two (more rigorous): use the CRE (Correlated Random Effects) approach—add $\bar{X}_i$ in the RE framework, obtain an estimate of $\sigma_u^2$, and then compute the AME using this $\hat{\sigma}_u^2$ and the contribution of $\bar{X}_i$ to construct individual-level predictions. This approach is more complex but more honest.

In practice, many empirical papers choose to report odds ratios (OR) rather than the AME after FE Logit—because the OR does not require estimating $u_i$ (the OR depends only on β̂, which is consistent in conditional Logit). This is a pragmatic trade-off: using the OR to provide economic intuition without making unreliable approximations for the AME.

4.4 Strategy C: CRE (Correlated Random Effects)—The Third Path

The CRE approach of Mundlak (1978) and Chamberlain (1984) is particularly valuable in binary choice panels—because it allows $u_i$ to be correlated with X (through individual means $\bar{X}_i$), while avoiding the incidental parameters problem.

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

where $v_i \sim N(0, \sigma^2_v)$, and $\varepsilon_{it}$ is Logistic or normal. The part of $u_i$ that is linearly related to X is explicitly absorbed by $\bar{X}_i \gamma$—the remaining part $v_i$ is uncorrelated with X (under Mundlak's assumption).

Advantages of CRE over FE Logit:

• Does not drop "non-switchers"—all individuals (including those whose Y is always 0 or 1) enter the estimation. The sample size is larger.
• Coefficients on time-invariant variables can be estimated—$\bar{X}_i$ enters the model, so between effects of time-invariant X can be incorporated and estimated.
• AME computation is more direct—you have an estimate of $\sigma_v^2$ that can be used to construct predicted probabilities and marginal effects, which is more honest than the $u_i = 0$ approximation in FE Logit.

* CRE Probit (Mundlak approach)
bysort id: egen x1_mean = mean(x1)
bysort id: egen x2_mean = mean(x2)
xtprobit y x1 x2 x1_mean x2_mean, re
margins, dydx(*) predict(pu0)

Limitations of CRE:

• It only allows linear correlation between $u_i$ and X—if the true correlation is nonlinear, CRE cannot fully address it.
• $\bar{X}_i$ is a noisy proxy for small-T panels—which may weaken the precision of the control for $u_i$.

4.5 Strategy D: FE Probit—When Might It Be Considered?

Answer: when T is sufficiently large (T ≥ 15–20, and your number of individuals is not too large). With larger T, each individual's $u_i$ has enough information to be consistently estimated—the severity of the incidental parameters problem diminishes as T grows. Monte Carlo studies show that with T ≈ 20, the bias in FE Probit is typically acceptable (< 10%). But in micro-panels with T < 10—which is the T range for the vast majority of panels—FE Probit should not be used as the primary identification strategy.

If a reviewer asks for FE Probit with T < 10—you should explain the incidental parameters problem and present conditional Logit (FE Logit) as a robustness check.

5. Choosing Between Logit and Probit in Panels—A Recalibrated Decision Framework

5.1 From Cross-Sections to Panels—A Fundamental Shift in the Choice Logic

In cross-sections, the choice between Logit and Probit is primarily about "convenience"—the AMEs are the same, Logit's odds ratios are more intuitive, so Logit wins.

In panels, the choice logic shifts to "feasibility"—because with short T you have only one consistent panel FE binary model: conditional Logit.

Panel data, need to control for u_i, $\text{Cov}(X,u_i) \neq 0$:
│
├── T < 10 (typical micro panel)
│   ├── Y varies over time for enough individuals → Conditional Logit (FE Logit)
│   │   ├── Focus on within effects → report OR (avoid the u_i problem for AME)
│   │   └── Need AME → consider CRE as a robustness check (provides AME)
│   │
│   └── Y varies over time for very few individuals → CRE (RE + Mundlak)
│       └── Uses all individuals, AME can be estimated
│
└── T ≥ 15–20 (macro panel or long panel)
    ├── Conditional Logit remains available and consistent
    └── FE Probit bias is acceptable → can be considered, but Logit is still recommended as the primary approach

5.2 Interpretation and Explanation in Panels—Three Key Differences from Cross-Sections

Difference one: your coefficients are "within effects," not "between effects."

In cross-sectional Logit, β reflects "the odds difference in Y between individual A (high X) vs. individual B (low X)." In FE Logit (conditional Logit), β reflects "the odds difference in Y for the same individual in high-X years vs. low-X years." These two quantities are usually economically different—cross-sectional variation in X may come from long-run structural differences, while time variation in X comes from short-run shocks or adjustments. In your paper, be explicit that your coefficients are "within effects"—they answer "how do the odds of Y change when X changes within an individual"—not "individuals with high X are more likely to have Y=1 than those with low X."

Difference two: your sample has changed—from "all individuals" to "switchers."

In FE Logit, individuals whose Y is always 0 or always 1 are dropped. You need to report the sample size entering the estimation and discuss whether this sample selection may compromise external validity—if "switchers" are systematically different from "non-switchers" in observable characteristics, external validity may be compromised. Showing descriptive statistics comparing the two groups in your paper is standard practice.

Difference three: the AME cannot directly follow the cross-sectional logic.

In cross-sections, you can reasonably assume that $u_i$ is absorbed into $\varepsilon_i$—the AME comes out directly after margins. In panel FE Logit, you cannot—because although $u_i$ is eliminated, it is not estimated and cannot be assumed to be zero. If your paper needs the AME (rather than just the OR)—consider CRE as the main model or at least as a supplementary analysis.

6. Common Misconceptions

6.1 Misconception One: "Logit and Probit give different conclusions—I should run a test to choose one"

In cross-sections, the AMEs are nearly identical—no test is needed to choose. If you run Logit and Probit on the same data and get substantively different AMEs, the problem is not model choice—it is that something may be wrong with your data or model specification (such as perfect separation, extremely rare Y, or model misspecification). Inconsistent AMEs between Logit and Probit are themselves a diagnostic signal—not a question of "which to choose," but "why is my model unstable under the Logit and Probit assumptions?"

6.2 Misconception Two: "FE Logit only uses switchers → there is sample selection bias → it cannot be used"

This sample reduction is a mathematical consequence of the conditional likelihood—it does not affect internal validity (your β remains consistent for the subpopulation of individuals who experienced at least one transition of Y from 0 to 1 or from 1 to 0). But it does affect external validity—can your conclusions be generalized to all individuals? Discuss this candidly in your paper rather than avoiding it.

6.3 Misconception Three: "FE Probit runs in Stata → it must be usable"

Stata's xtprobit, fe command will run—but it does not solve the incidental parameters problem. With small T (T < 10), the output appears, but $\hat{\beta}$ has systematic bias. Stata can run it ≠ the model is correct. In short panels, conditional Logit is the only consistent binary FE model.

6.4 Misconception Four: "My panel has T = 5—I use RE Logit because the Hausman test is not significant"

In cross-sectional linear panels, if the Hausman test is not significant → RE and FE are indistinguishable → RE is more efficient, so RE can be used. But in binary panels: the comparison between conditional Logit (FE) and RE Logit is not so simple—because the model structures differ (FE's conditional likelihood ≠ RE Logit with dummies added). More importantly, the assumption of RE Logit that $\text{Cov}(X, u_i) = 0$ rests on fragile ground in observational data—even if some tests are not significant, theoretical arguments usually do not support this assumption. In binary panels, using CRE (Mundlak-Chamberlain) to relax the RE assumption is more standard than relying on tests to switch between RE and FE.

6.5 Misconception Five: "For binary panels I just use the linear probability model (LPM) + FE—it's convenient anyway"

LPM + FE is a widely accepted pragmatic choice in binary panels—especially when N is large and T is small. It has no incidental parameters problem, the AME is just the OLS coefficient, and standard errors are easy to cluster. But LPM has its own problems (predicted probabilities outside [0,1], constant marginal effects assumption). If the incidence rate of Y is between 20%–80%—LPM typically performs quite well. If Y is extremely rare (< 5%) or extremely common (> 95%)—nonlinear models (Logit/Probit) may give more reasonable predictions and more reliable marginal effects. In your paper, if you use LPM + FE as the baseline, it is advisable to also present conditional Logit as a robustness check—giving reviewers evidence that your conclusions hold under a nonlinear specification.

7. Summary

Logit vs. Probit + Panel Binary Choice—Seven Core Takeaways:

1. In cross-sections, the AMEs of Logit and Probit are nearly identical. The only difference is the distributional assumption on the latent error—Logistic vs. normal. The difference in AMEs is far smaller than sampling error in the vast majority of empirical data. The choice is primarily practical—Logit's OR has an economic intuition advantage.

2. The "1.6 rule" operates at the coefficient level—do not compare at the coefficient level. Logit coefficients ≈ 1.6 × Probit coefficients—because of different variance normalizations. Compare at the AME level—you will see almost no difference.

3. The divergence point in panels—the incidental parameters problem. In nonlinear FE models, you must estimate N values of $u_i$—these estimates are inconsistent when T is fixed, which in turn contaminates the estimation of β. Logit is the only binary model that can eliminate $u_i$ through conditional likelihood without estimating it—this is why FE Logit is the only consistent option with small T.

4. Conditional Logit = only "switchers" are used + coefficients are within effects + the AME cannot be directly produced. These are the three costs of FE Logit. Make them explicit in your paper—do not hide the sample reduction, do not conflate within and between effects, and do not pretend you can give a cross-sectional-style AME without additional assumptions.

5. CRE (Mundlak-Chamberlain) is the "third path" for panel binary choice. It allows $u_i$ to be correlated with X (through $\bar{X}_i$), retains all individuals, and the AME computation is more direct. The cost is the assumption that the correlation between $u_i$ and X is linear. In many empirical settings, CRE as the main model is more flexible than FE Logit.

6. LPM + FE is an accepted pragmatic baseline. When the incidence rate of Y is not extreme (20%–80%) and N is large with T small—LPM + FE is simple, transparent, and generally accepted by reviewers. Use conditional Logit as a robustness check to strengthen the argument.

7. In panels, the choice between Logit and Probit shifts from "convenience" to "feasibility." In cross-sections, you can choose either. In panels, if you need fixed effects—Logit is the only consistent option (with short T). This fact should dominate your model choice—rather than the cross-sectional discussion of odds ratios vs. coefficient interpretation.

One-sentence conclusion:

"In cross-sections, Logit and Probit are twins—you cannot go wrong with either, but Logit's odds ratios make it more articulate. In panels, the twins go their separate ways—Logit can escape the vortex of the incidental parameters problem through conditional likelihood, while Probit is left behind—with small T, its β is doomed to be dragged down by N inconsistent $u_i$'s. So you choose Logit—not because its odds ratios tell a better story, but because in the uncertain waters of short panels, it is the only ship that will not sink. But remember—this ship carries only 'switchers,' it tells the story of 'within effects,' and you cannot easily compute the average marginal effect for everyone on board. Binary choice in panels is like this—you trade one advantage for new constraints. Good empirical work does not pretend these constraints do not exist—rather, it clearly tells the reader, within those constraints: what is the effect of X on Y, for whom, and under what assumptions."

8. Presentation Suggestions for Bilibili/WeChat

• Bilibili video: Use "the two fates of twins" as the core visual metaphor. Opening: in the cross-sectional world—two people who look almost identical (Logit and Probit) stand next to an S-shaped curve. One holds a sign reading "OR (odds ratio)" (Logit), the other holds nothing (Probit). Narration: "In cross-sections, Logit and Probit give nearly identical AMEs—they are twins standing behind the same S-shaped curve. Logit's only advantage is—it has the odds ratio tool in hand, able to tell the story of 'how many times more likely.'" Scene transition—enter the panel world, where a massive vortex appears (the "incidental parameters problem"). Probit is pulled into the vortex—labeled "FE Probit is inconsistent with small T." Logit grabs the rope of conditional likelihood and swings out of the vortex. Narration: "In panels, the fates of these twins diverge. Fixed effects Probit needs to estimate N values of $u_i$—with small T, these inconsistent $u_i$'s drag β down with them. Logit has the rope of conditional likelihood—it can eliminate $u_i$ without directly estimating them, thus surviving with short T." First act "The three costs of FE Logit": three scenes presented in sequence—(1) half the sample is discarded (labeled "only switchers remain"), (2) an arrow on the coefficient is engraved with "within effect," (3) the AME button cannot be pressed (red warning—"u_i not estimated, AME requires additional steps"). Narration: "FE Logit's escape is not free—your sample shrinks, your coefficient interpretation shifts from between to within, and your AME can no longer be computed as directly as in cross-sections." Second act "CRE—the third path": the figure of Mundlak appears—adding each person's individual mean of X (X̄_i) to the RE model. The endogenous part of $u_i$ is absorbed by X̄_i. All individuals (including non-switchers) remain in the model. The AME button can now be pressed (green). Narration: "The Mundlak-Chamberlain CRE approach—you do not have to choose between FE and RE. You put individual means into RE to absorb endogeneity—retaining all individuals and restoring the AME computation. The cost is that you assume the correlation between $u_i$ and X is linear. There is no perfect solution—but there is one whose limitations you can candidly discuss."

• WeChat article: The cross-sectional Logit vs. Probit comparison table (distributional assumptions, variance, coefficient relationship, AME differences, OR availability) as the core infographic. The comparison framework table of four panel strategies (Pooled, RE, FE Logit, CRE) as the core reference card. The three costs of conditional Logit (switcher sample, within effects, AME difficulty) as three warning cards. The Mundlak approach for CRE with code (xtprobit, re + X̄_i) as a code template card. The graphical explanation of the incidental parameters problem (N values of $u_i$ need to be estimated; with T fixed, each $u_i$ has only T observations → inconsistency) as a visual explanation card. The five common misconceptions as correction cards.

• Recommended titles:

  • Main title: 《Logit or Probit?—Nearly Irrelevant in Cross-Sections, Worlds Apart in Panels》
  • Alternative title: 《Conditional Logit, RE Logit, CRE—Three Strategies for Panel Binary Choice》
  • New media title: 《FE Probit Is Inconsistent in Short Panels—Why Your Panel Binary Choice Model Should Use Logit?》

• Key quotes:

  "In cross-sections, Logit and Probit are twins—the AMEs are nearly identical, and either choice works. Logit's odds ratio is the extra Swiss Army knife in its pocket—not essential, but handy. In panels, the twins are separated by the vortex of the incidental parameters problem—Logit has the rope of conditional likelihood, Probit is left in the vortex."

  "Conditional Logit is not a free lunch—you gain consistency for $u_i$, but pay three costs: your sample shrinks to only 'switchers,' your coefficients become within effects, and you can no longer compute the AME as easily as in cross-sections. Good empirical work does not pretend these costs do not exist—rather, it clearly tells the reader, while the costs remain, the scope within which the conclusions hold."

  "CRE is the third path for binary panels—you place individual means in RE, letting them absorb the endogenous part of $u_i$. You are not assuming $u_i$ and X are uncorrelated—you are assuming the relationship between $u_i$ and X is linear and can be captured by the individual means of X. This assumption is weaker than RE and more flexible than FE's mathematics—