Likelihood Estimation Under Ignorable Panel Missingness

Why Incomplete Panels Can Still Inform Estimation

A panel may include units observed at different waves, or units with gaps in their outcome histories. Discarding every unit with at least one missing outcome can waste information. Likelihood estimation offers another route: specify a joint model for the outcomes, then calculate the probability density of what was actually observed by integrating over the unobserved outcomes. The resulting likelihood can use a unit’s observed measurements without filling in a fabricated value for a missing wave.

This approach is valid only under assumptions about both the outcome model and the missingness process. In particular, the missingness must be ignorable for inference about the outcome-model parameters. “Ignorable” is a technical condition, not a synonym for “unimportant” or “random-looking.”

Observed-Data Likelihood for a Panel

For unit $i$, let $Y_i=(Y_{i1},\ldots,Y_{iT})$ denote the full outcome vector, $X_i$ the observed covariates, and $R_{it}$ an indicator equal to 1 when $Y_{it}$ is observed and 0 otherwise. Write $Y_{i,\mathrm{obs}}$ and $Y_{i,\mathrm{mis}}$ for the observed and missing components. Suppose the full outcome model has density $f(y_i\mid x_i;\theta)$, where $\theta$ is the parameter of interest.

Definition: The observed-data likelihood contribution for unit $i$ is the density of that unit’s observed outcomes under the outcome model: $$ L_i(\theta)=f(y_{i,\mathrm{obs}}\mid x_i;\theta) =\int f(y_{i,\mathrm{obs}},y_{i,\mathrm{mis}}\mid x_i;\theta)\,dy_{i,\mathrm{mis}}. $$ The integral is over every missing outcome component. If outcomes are discrete, the integral is replaced by a sum over the possible missing values.

Thus, an incomplete unit contributes a marginal likelihood, not the full-data density evaluated at an invented completion. Under independent sampling of units conditional on covariates, the observed-outcome likelihood is $L(\theta)=\prod_i L_i(\theta)$. Dependence among a unit’s outcomes across waves remains part of its joint model and therefore affects the marginal density of its observed components.

To explain when the missingness mechanism can be left out, let $g(r_i\mid y_i,x_i;\psi)$ be a model for the observation indicators, with parameter $\psi$. A missingness mechanism is missing at random (MAR) if, for each realized pattern, its probability depends on the outcomes only through those observed under that pattern: $g(r_i\mid y_i,x_i;\psi)=g(r_i\mid y_{i,\mathrm{obs}},x_i;\psi)$. The parameters are distinct if the allowable values of $\theta$ and $\psi$ can vary independently; in particular, the missingness model imposes no restriction on $\theta$.

Theorem: If the outcome and missingness models have distinct parameters and the missingness mechanism is MAR, then the joint likelihood factors as $$ \begin{aligned} L_i(\theta,\psi) &=\int f(y_{i,\mathrm{obs}},y_{i,\mathrm{mis}}\mid x_i;\theta) g(r_i\mid y_{i,\mathrm{obs}},y_{i,\mathrm{mis}},x_i;\psi)\,dy_{i,\mathrm{mis}}\\ &=g(r_i\mid y_{i,\mathrm{obs}},x_i;\psi) f(y_{i,\mathrm{obs}}\mid x_i;\theta). \end{aligned} $$ The first factor does not involve $\theta$. Consequently, maximizing the likelihood for $\theta$ can use the observed-data likelihood alone.

The second equality follows because MAR makes $g$ constant with respect to the integration variable $y_{i,\mathrm{mis}}$, so it can be taken outside the integral. This is the key justification for ignoring the missingness model when estimating $\theta$. It does not mean that missingness is unrelated to observed outcomes: for example, a later wave’s response probability may depend on an earlier observed outcome. Nor does it mean that a misspecified outcome model becomes valid merely because MAR holds.

Marginalizing a Joint Outcome Model

A useful special case is a multivariate normal outcome model. Partition $Y_i$ into observed and missing components, and partition its mean and covariance matrix in the same way. The marginal distribution of the observed components is $Y_{i,\mathrm{obs}}\mid X_i=x_i\sim N(\mu_{\mathrm{obs}},\Sigma_{\mathrm{obs},\mathrm{obs}})$. This follows by integrating the joint normal density over the missing coordinates; the observed block of the mean and covariance determines the marginal density. Cross-wave covariances still matter when they affect that observed block or when other outcomes are observed, but a missing coordinate itself is not assigned a value.

Worked Example: One Observed Wave from a Bivariate Normal Model

Suppose, hypothetically, a unit’s two outcomes have joint model $Y_i\sim N\left(\begin{pmatrix}1\\2\end{pmatrix}, \begin{pmatrix}4&1\\1&9\end{pmatrix}\right)$. The first outcome is observed as $y_{i1}=3$, while the second is missing. Find the unit’s observed-data likelihood contribution.

Marginalize. The first component has marginal distribution $Y_{i1}\sim N(1,4)$. The second outcome’s mean, variance, and unobserved value do not enter this one-dimensional marginal density. Therefore, $$ L_i=\frac{1}{\sqrt{2\pi(4)}}\exp\left\{-\frac{(3-1)^2}{2(4)}\right\}. $$

Substitute and calculate. The exponent is $-4/8=-0.5$, and $\sqrt{8\pi}\approx5.013$. Hence $L_i\approx (1/5.013)e^{-0.5}\approx0.1210$ per outcome unit. This is the density contribution at the observed first-wave value, not the probability that a continuous outcome equals exactly 3.

The joint normal model may describe dependence between waves, but no value for the missing second outcome is needed to evaluate this marginal contribution. If both outcomes were observed, their joint density would use the covariance between them.

Available Outcomes and Dependence Across Waves

The phrase “use available observations” needs care. Under a model with independent waves, each observed wave may contribute a univariate density. Under a model with within-unit dependence, the observed outcomes for a unit generally contribute their joint marginal density. Treating them as unrelated measurements would discard the dependence structure and could produce a different, misspecified likelihood.

Worked Example: A Random-Intercept Panel with One Missing Wave

Consider the hypothetical two-wave model $Y_{it}=\mu+b_i+e_{it}$, where $b_i\sim N(0,1)$ is a unit-specific random intercept, $e_{it}\sim N(0,1)$, and the random intercept and errors are mutually independent. For one unit, $Y_{i1}=2$ is observed and $Y_{i2}$ is missing. Evaluate the observed-data density when $\mu=1$.

Integrate out the missing outcome. Conditional on $b_i$, $Y_{i1}$ has distribution $N(\mu+b_i,1)$. Integrating over the random intercept gives $Y_{i1}\sim N(\mu,1+1)=N(\mu,2)$. Equivalently, integrating the full two-wave density over $Y_{i2}$ leaves this first-wave marginal. Thus, $$ L_i(\mu=1)=\frac{1}{\sqrt{2\pi(2)}}\exp\left\{-\frac{(2-1)^2}{2(2)}\right\} \approx\frac{1}{3.5449}e^{-0.25} \approx0.2197. $$

The observed value informs $\mu$ through the model’s marginal distribution, which includes variation from both the random intercept and the wave-specific error. The missing second wave is not replaced by its conditional mean or by a draw from a predictive distribution. If both waves were observed, their shared random intercept would induce covariance 1, and their joint likelihood would retain that dependence.

For a panel with several missingness patterns, each unit can have a different marginal density in the product likelihood. For example, units observed at both waves contribute a two-dimensional density, while units observed only at wave 1 contribute its wave-1 marginal. Estimation then maximizes the product of these pattern-specific contributions. It is not generally equivalent to fitting a model to a completed rectangular data set.

Worked Example: Maximum Likelihood from Unequal Observation Patterns

Suppose, hypothetically, that each wave’s outcome is independent with distribution $N(\mu,4)$, and the missingness is ignorable with distinct parameters. Three units have observed values: unit A has 4 at wave 1 and wave 2 missing; unit B has 5 at wave 1 and wave 2 missing; unit C has 7 at wave 2 and wave 1 missing. Find the maximum-likelihood estimate of $\mu$.

Write the observed likelihood. Each observed value has variance 4, and independence makes the observed-data likelihood $$ L(\mu)=\prod_{y\in\{4,5,7\}}\frac{1}{\sqrt{2\pi(4)}}\exp\left\{-\frac{(y-\mu)^2}{8}\right\}. $$ The absent wave for each unit contributes no additional outcome-density factor.

Maximize. The log likelihood, apart from a constant not involving $\mu$, is $\ell(\mu)=-\frac{1}{8}\{(4-\mu)^2+(5-\mu)^2+(7-\mu)^2\}$. Differentiating gives $\ell'(\mu)=\frac{1}{4}\{(4-\mu)+(5-\mu)+(7-\mu)\}$. Setting this equal to zero yields $\hat{\mu}=(4+5+7)/3=16/3\approx5.333$ outcome units. Since $\ell''(\mu)=-3/4<0$, this stationary point is the maximum.

Here the observed-data likelihood uses all three recorded measurements and no imputed values. The calculation depends on the stated independent, common-mean normal model; with within-unit dependence, the appropriate observed joint marginals would replace the product of univariate densities.

What Ignorability Does—and Does Not—Guarantee

Under MAR and parameter distinctness, likelihood-based estimation can use the observed outcomes without explicitly modeling the response process for inference about $\theta$. This is an advantage when the observation process is complicated but the outcome model is credible. It also avoids treating imputed outcomes as if they were directly observed in the likelihood. However, the result does not guarantee that every available-case calculation is valid, nor that the model’s estimates target every possible population quantity. Those conclusions depend on the specified outcome model, the covariates conditioned on, and the estimand.

For instance, MAR may hold conditional on recorded history and covariates but fail when the response probability depends on an unobserved outcome after conditioning on that information. The response mechanism is then not ignorable under this likelihood argument. The observed-data likelihood for the outcome parameters cannot in general be obtained by simply dropping the missingness factor. Assessment of nonignorable attrition requires modeling or sensitivity analysis of the missingness process, rather than assuming the factorization above applies.

Specify the outcome model.
Define the joint distribution of the full repeated-outcome vector and its parameters, including any within-unit dependence.

Record each observed pattern.
For every unit, identify which outcome coordinates are observed and which are missing.

Form the pattern-specific marginal.
Integrate or sum the full-data density over that unit’s missing coordinates.

Check ignorability conditions.
Use the outcome likelihood alone for inference on its parameters only when MAR and parameter distinctness hold.

Maximize and interpret.
Maximize the product of observed-data contributions, while interpreting estimates under the chosen outcome model and target population.

Common Mistake: Treating Missing Outcomes as Completed Data

A tempting argument is: “The model predicts a missing value of 6, so I will insert 6 and fit the full-data likelihood.” This is not the observed-data likelihood. A predicted value is not an observation, and inserting a single prediction generally fails to account for uncertainty about the missing outcome. The correct likelihood contribution averages the full-data density over all possible missing values, which is the integral defining the observed-data density. In special cases that integral simplifies to a familiar marginal distribution, as in the normal examples, but it is still marginalization—not deterministic imputation.

Key takeaway: Under a correctly specified outcome model, MAR, and distinct outcome and missingness parameters, an incomplete panel unit contributes the marginal density of its observed outcomes. Likelihood estimation can use those outcomes without imputing missing values, while preserving dependence among measurements that are observed.

Check Your Understanding

Answer each question using the definitions and likelihood logic developed above.

Write the observed-data likelihood contribution for a unit with $Y_{i1}$ observed and $Y_{i2}$ missing in a two-outcome model.
State the two main conditions that allow the missingness mechanism to be ignored when estimating the outcome parameter.
In a multivariate normal panel model, which parts of the mean vector and covariance matrix determine the marginal likelihood when only a subset of outcomes is observed?
Why is inserting a predicted missing outcome generally not equivalent to integrating it out?
If missingness depends on an unobserved outcome even after conditioning on observed outcomes and covariates, does the MAR factorization justify using the outcome likelihood alone? Explain.