Conditional Fixed-Effects Logit for Panels

Why Condition on Each Unit’s Outcome Total?

Panel data follow the same units over multiple periods. A unit may have a persistent, unobserved propensity for a binary outcome: some people may be more likely to participate in every period, or some firms may be more likely to adopt a practice throughout the observation window. A fixed-effects logit model represents that persistent difference with a separate intercept for each unit. Directly estimating one intercept per unit can be problematic when the number of units is large. Conditional fixed-effects logit removes these intercepts from the slope likelihood by conditioning on each unit’s total number of positive outcomes.

This conditioning also makes the source of slope information explicit. A unit whose outcome is always zero or always one has only one possible outcome pattern once its total is fixed. Its conditional likelihood is therefore one for every slope value. Such units can inform neither the estimated slopes nor their conditional likelihood precision.

Model and assumptions: For unit $i=1,\ldots,N$ observed in periods $t=1,\ldots,T_i$, let $Y_{it}$ be binary and $x_{it}$ a vector of observed covariates. The fixed-effects logit model is $$ P(Y_{it}=1\mid x_{i1},\ldots,x_{iT_i},\alpha_i)= \frac{\exp(\alpha_i+x_{it}'\beta)}{1+\exp(\alpha_i+x_{it}'\beta)}. $$ Here $\alpha_i$ is an unrestricted unit-specific intercept and $\beta$ is the common slope vector. The conditional likelihood construction assumes that, given the covariates and $\alpha_i$, outcomes are independent Bernoulli variables with these probabilities. For the usual within-unit interpretation, the covariate and outcome specification must also be appropriate: in particular, relevant time-varying omitted factors should not be changing with the covariates in a way that confounds their association with the outcome. Units are treated as independent for the likelihood across $i$.

The model allows $\alpha_i$ to be associated with the covariate histories; it does not require the unit effect to be independent of the regressors. The fixed intercept captures stable unit differences, but it does not automatically control for time-varying confounding, misspecification, or dependence beyond that represented in the model.

Deriving the Conditional Likelihood

Write $m_i=\sum_{t=1}^{T_i}Y_{it}$ for unit $i$’s total number of positive outcomes. Under the model’s conditional independence assumption, the probability of a particular outcome sequence $y_i=(y_{i1},\ldots,y_{iT_i})$, conditional on the covariates and intercept, is

P(Y_i=y_i\mid X_i,\alpha_i) = \frac{\exp\left(\alpha_i\sum_t y_{it}+\beta'\sum_t x_{it}y_{it}\right)} {\prod_{t=1}^{T_i}\left[1+\exp(\alpha_i+x_{it}'\beta)\right]}.

For a fixed total $m_i$, the denominator is the same for every outcome sequence with that total. The term involving $\alpha_i$ in the numerator is also the same, because $\sum_t y_{it}=m_i$. Thus, when we condition on $m_i$, the factors involving $\alpha_i$ cancel. The possible sequences are all binary vectors with exactly $m_i$ ones.

Definition: The conditional probability of the observed outcome sequence, given the unit’s outcome total and covariate history, is $$ P(Y_i=y_i\mid m_i,X_i) = \frac{\exp\left(\beta'\sum_t x_{it}y_{it}\right)} {\displaystyle\sum_{a_i:\,\sum_t a_{it}=m_i} \exp\left(\beta'\sum_t x_{it}a_{it}\right)}. $$ The conditional likelihood for $\beta$ is the product of these probabilities over units. Its denominator sums over every possible allocation of the $m_i$ positive outcomes across that unit’s periods.

The individual intercept has disappeared, while the covariate information remains through comparisons among outcome allocations with the same total. For example, if a unit has exactly one positive outcome, the conditional model compares the periods in which that outcome could have occurred. If it has two positive outcomes, it compares all pairs of periods that could have received them.

Worked Example: A Unit With One Positive Outcome

Suppose one hypothetical unit is observed for three periods, with covariate values $x_{i1}=0$, $x_{i2}=1$, and $x_{i3}=2$. The observed outcomes are $(0,1,0)$, so $m_i=1$. For simplicity, there is one slope $\beta$.

Form the conditional likelihood. Since exactly one outcome is positive, the possible outcome sequences place the one in period 1, 2, or 3. Their covariate-weighted sums are respectively 0, 1, and 2. The probability of the observed sequence is therefore $$ P((0,1,0)\mid m_i=1,X_i) = \frac{\exp(\beta)} {1+\exp(\beta)+\exp(2\beta)}. $$

Evaluate at a stated slope. At $\beta=\log(2)$, the three weights are $1$, $2$, and $4$. Thus the observed sequence has conditional probability $$ \frac{2}{1+2+4}=\frac{2}{7}\approx 0.2857. $$ This is the probability of this particular outcome allocation conditional on the unit having one positive outcome and on its covariates, at the specified slope. It is not an unconditional probability of a positive outcome in period 2.

Which Units Identify the Slopes?

When $m_i=0$, the only possible sequence is all zeros. When $m_i=T_i$, the only possible sequence is all ones. In either case, the conditional probability of the observed sequence is $1$, regardless of $\beta$. Its log-likelihood contribution is $\log(1)=0$, its score is zero, and its information about the slopes is zero. The same conclusion holds when $T_i=1$: fixing the outcome total leaves no alternative sequence to compare.

Worked Example: No Outcome Change, No Conditional Slope Information

Consider two hypothetical units, each observed for three periods. Unit A has outcomes $(0,0,0)$ and covariates $(0,1,3)$; unit B has outcomes $(1,1,1)$ and covariates $(2,1,0)$. Each has a different covariate history, but neither outcome changes within the unit.

For unit A, $m_A=0$, so the only allocation in the conditional denominator is $(0,0,0)$. Its numerator and denominator are both $1$, giving a conditional likelihood contribution $1/1=1$.

For unit B, $m_B=3$, so the only allocation is $(1,1,1)$. Its numerator is $\exp\{\beta(2+1+0)\}$, and the denominator contains that same single term. The contribution is again $1$, for every $\beta$.

Consequently, neither unit changes the conditional likelihood as $\beta$ changes. Their covariate variation cannot compensate for the absence of outcome variation. By contrast, an outcome-changing unit can contribute information when its possible outcome allocations have different covariate-weighted sums. Outcome changes are necessary for contribution, though not always sufficient: if all relevant covariate contrasts are identical, that unit may still provide no information about a particular slope.

Time-invariant covariates also disappear from this conditional comparison. If a covariate $z_i$ has the same value in every period for unit $i$, its contribution to $\beta'\sum_t x_{it}y_{it}$ is proportional to $z_i\sum_t y_{it}=z_i m_i$. That quantity is constant across all allocations in the conditional denominator, so it cancels from the conditional probability. The model therefore cannot estimate a separate slope for a covariate that never changes within units.

A Score Justification for the Information Claim

The cancellation can also be seen from the conditional log-likelihood for one unit:

\ell_i(\beta) = \beta'\sum_t x_{it}y_{it} - \log\left[ \sum_{a_i:\,\sum_t a_{it}=m_i} \exp\left(\beta'\sum_t x_{it}a_{it}\right) \right].

Differentiating the log of the sum in the second term gives a weighted average over the possible allocations. The weights are their conditional probabilities under $\beta$. Therefore the score is

\frac{\partial \ell_i(\beta)}{\partial\beta} = \sum_t x_{it}y_{it} - E_\beta\left[\sum_t x_{it}Y_{it}\mid m_i,X_i\right].

The score compares the covariate-weighted sum for the observed allocation with its conditional expectation across all allocations having the same total. If there is only one possible allocation, that allocation equals its own expectation, so the score is zero. The conditional information is the conditional covariance of the allocation’s covariate-weighted sum; with only one allocation this covariance is zero as well. This gives a direct likelihood justification for why all-zero and all-one units contribute no slope information.

Worked Example: Interpreting a Within-Unit Slope

Suppose a hypothetical conditional fixed-effects logit fit for a time-varying exposure $x_{it}$, measured in one-unit increments, gives $\hat{\beta}=\log(1.5)\approx0.4055$. Consider two periods for the same unit that differ by one unit in exposure, with other modeled covariates held fixed.

In the logit model, the difference in the linear predictor is $\hat{\beta}\times 1=0.4055$. The corresponding conditional odds ratio is $$ \exp(\hat{\beta})=\exp\{\log(1.5)\}=1.5. $$ Thus, for a one-unit within-unit increase in exposure, the modeled odds of a positive outcome are multiplied by $1.5$, comparing periods in the conditional allocation framework and holding the other included covariates fixed.

To make the conditioning concrete, suppose the unit has exactly one positive outcome across two periods and the exposure values are 0 and 1. The conditional probability that the positive outcome is in the exposed period is $$ \frac{\exp(\hat{\beta})}{1+\exp(\hat{\beta})} = \frac{1.5}{2.5}=0.6000. $$ The odds of the positive outcome being in the exposed rather than unexposed period are $0.6000/0.4000=1.5$, as implied by the coefficient. This conditional probability is not a population-average probability change, and the odds ratio alone does not specify an absolute change in outcome probability. The estimate describes a within-unit association; a causal interpretation would require additional design and identification assumptions.

Estimation, Scope, and Common Pitfalls

In practice, the conditional likelihood is maximized over $\beta$, using only units with outcome totals that leave more than one possible allocation. The unit-specific intercepts are not estimated as part of this likelihood. As a result, this method is useful when the main target is the slope on time-varying covariates and unrestricted stable unit heterogeneity is a concern. Its limitations follow directly from the conditioning: it does not estimate time-invariant covariate effects or unit-specific intercepts, and the slope estimates are identified by within-unit outcome changes rather than by all observed units equally.

A common mistake is to assume that every panel unit contributes to the slope estimates just because it has observed covariates. That is not true here. An always-zero or always-one unit has no alternative outcome allocation once its total is fixed, so its conditional contribution is constant in $\beta$. Another mistake is to describe $\exp(\beta)$ as a risk ratio or as a fixed increase in probability. It is a conditional odds ratio for a specified within-unit covariate contrast. Probability changes depend on the covariate values and the relevant conditioning, and the coefficient does not by itself establish causation.

Key takeaway: Conditional fixed-effects logit eliminates an unrestricted unit intercept by conditioning on the unit’s total number of positive outcomes. The slope is learned from comparisons among within-unit outcome allocations, so outcome-changing units and time-varying covariate contrasts are central to identification.

Check Your Understanding

Try these questions before moving on.

For one unit observed in four periods with exactly two positive outcomes, how many possible outcome allocations appear in the conditional denominator?
Why does an all-zero unit have a conditional likelihood contribution of one for every value of $\beta$?
Explain why a covariate that is constant over time within each unit cannot have its own slope estimated by this conditional likelihood.
If a fitted slope is $0.20$, what is the conditional odds ratio for a one-unit increase in its covariate? State what the coefficient does not establish by itself.
Can a unit with outcome variation still provide no information about a particular slope? Give a reason based on its within-unit covariate contrasts.