From Structural Equations to an Observable-Data Likelihood
A structural model specifies how outcomes are generated, often through equations involving unobserved shocks or latent variables. Estimation, however, uses the data that are recorded. The likelihood must therefore describe the probability density or probability mass of those observations—not merely the distribution assigned to the shocks. The distinction matters whenever the equations transform shocks into outcomes, latent variables are unobserved, or an observation rule such as censoring changes what is recorded.
The basic task is to start with a structural model and its maintained assumptions, then derive the distribution of the observables. For a continuous outcome, a likelihood contribution is a density evaluated at the observed value; for a discrete event or censored value, it is generally a probability. A single data set can require both kinds of contributions.
Throughout, likelihoods are conditional on observed exogenous covariates unless stated otherwise. When observations are independent conditional on those covariates, the sample likelihood is the product of the individual contributions. Independence is an assumption about the sampling model, not a consequence of writing structural equations.
Invertible Structural Systems and the Jacobian
Suppose a \(k\)-dimensional observed outcome \(Y\) is generated by \(Y=T_\theta(\varepsilon)\), where the \(k\)-dimensional structural shock \(\varepsilon\) has density \(f_\varepsilon(\cdot;\theta)\). Assume, for the outcome and parameter values under consideration, that the mapping is one-to-one, differentiable, and has a differentiable inverse \(\varepsilon=h_\theta(y)\), with nonzero Jacobian determinant. The change-of-variables formula gives the observable density.
The determinant adjusts for how the transformation stretches or compresses volume. Omitting it generally gives the shock density evaluated at the implied shocks, not the density of the observed outcomes. In a structural system written implicitly as \(F(y,\varepsilon;\theta)=0\), the same formula applies after solving for \(\varepsilon=h_\theta(y)\), provided the required inverse exists. A sample likelihood is the product of these observable densities under the stated conditional independence assumption.
Worked Example: A Simultaneous Linear Equilibrium
Consider the hypothetical two-equation system \(Y_1=0.4Y_2+U_1\) and \(Y_2=0.5Y_1+U_2\). Assume that, conditional on the model parameters, \(U_1\) and \(U_2\) are independent standard normal shocks. For this example, restrict the coefficients so that \(1-(0.4)(0.5)=0.8\ne0\); the system then has a unique solution for each shock pair. Find the observable density at the hypothetical observation \((y_1,y_2)=(1,2)\).
Step 1: Recover the shocks from the observed outcomes. The equations imply \(u_1=y_1-0.4y_2\) and \(u_2=y_2-0.5y_1\). Thus at \((1,2)\), \(u_1=1-0.4(2)=0.2\) and \(u_2=2-0.5(1)=1.5\).
Step 2: Calculate the Jacobian. The inverse mapping from outcomes to shocks has derivative matrix \(\begin{pmatrix}1&-0.4\\-0.5&1\end{pmatrix}\), whose determinant is \(1-(0.4)(0.5)=0.8\). Its absolute value is the density adjustment.
Step 3: Evaluate the density. Since each shock has standard normal density, the joint observable density is \(0.8\,\phi(0.2)\phi(1.5)\), where \(\phi(u)=(2\pi)^{-1/2}\exp(-u^2/2)\). Substituting gives \(0.8(2\pi)^{-1}\exp[-(0.2^2+1.5^2)/2]=0.0405\), to three significant figures.
The value \(0.0405\) is a joint density at the observed outcome pair, not the probability of observing that exact pair. The equations determine which shocks are implied by the pair; the Jacobian converts the shock density into the density in outcome units.
Latent Variables: Integrate, Do Not Condition on the Unobserved Value
If a latent variable \(Z\) affects the outcome but is not observed, the observable-data density averages over its possible values according to the model. In the continuous case, if the model supplies a conditional density \(f(y\mid z;\theta)\) and latent-variable density \(f_Z(z;\theta)\), then \(f_Y(y;\theta)=\int f(y\mid z;\theta)f_Z(z;\theta)\,dz\). For a discrete latent variable, the integral is replaced by a sum over its support. This is marginalization: the likelihood is for the observed \(y\), not for \(y\) together with an unobserved realization of \(z\).
The conditional density and latent distribution must be compatible with the structural model. For instance, if a structural equation implies \(Y=m(Z;\theta)+\varepsilon\), the conditional density \(f(y\mid z;\theta)\) is obtained from the assumed distribution of \(\varepsilon\) and that equation. If a change of variables is needed to obtain that conditional density, its Jacobian belongs inside the integral.
Worked Example: Integrating Out a Latent Normal Effect
Suppose a hypothetical observation follows \(Y=\theta+Z+\varepsilon\), where the unobserved effect \(Z\sim N(0,\tau^2)\), the structural shock \(\varepsilon\sim N(0,\sigma^2)\), and \(Z\) and \(\varepsilon\) are independent. Assume \(\theta=0.5\), \(\tau=0.6\), \(\sigma=0.8\), and that the observed value is \(y=1.2\). Derive and evaluate its likelihood contribution.
Conditional on \(Z=z\), the outcome density is \(\frac{1}{\sigma}\phi((y-\theta-z)/\sigma)\), while the latent density is \(\frac{1}{\tau}\phi(z/\tau)\). Therefore the observable density is the integral \(\int \frac{1}{\sigma}\phi((y-\theta-z)/\sigma)\frac{1}{\tau}\phi(z/\tau)\,dz\).
Here \(Y-\theta=Z+\varepsilon\). Independence and normality imply that \(Z+\varepsilon\) is normal with mean \(0\) and variance \(\tau^2+\sigma^2\). Equivalently, evaluating the convolution integral gives \(Y\sim N(\theta,\tau^2+\sigma^2)\). Thus the likelihood contribution at \(y=1.2\) is \(\frac{1}{\sqrt{\tau^2+\sigma^2}}\phi((y-\theta)/\sqrt{\tau^2+\sigma^2})\).
Substitute the values: \(\tau^2+\sigma^2=0.6^2+0.8^2=1\), so the contribution is \(\phi((1.2-0.5)/1)=\phi(0.7)=(2\pi)^{-1/2}\exp(-0.7^2/2)=0.312\). This is a density per unit of \(Y\). Treating \(Z\) as if its realized value were observed would produce a different, conditional likelihood and would not describe the observed-data density.
Equilibrium and Observation Rules Are Part of the Likelihood
A structural system may not determine a unique outcome for every shock. In that case, the equations alone do not define an observable likelihood. The model must additionally specify how an equilibrium is selected, or otherwise assign probabilities to possible equilibria. If a shock value \(\varepsilon\) can generate several solutions \(y_1,\ldots,y_J\), and the model specifies selection probabilities \(s_j(\varepsilon;\theta)\), the probability of observing a region \(A\) is obtained by integrating the shock density times the probability that the selected equilibrium lies in \(A\). In particular, the selection probabilities must be nonnegative and sum to one over the available equilibria. A deterministic selection rule is the special case in which one equilibrium is selected with probability one. If the outcome-to-shock transformation is many-to-one, density calculations must also account for all relevant inverse branches; using one branch alone is not generally valid.
Observation rules can similarly change the likelihood. If an outcome is censored, the recorded value may indicate only that the latent outcome lies in a region. The contribution for such a record is the probability of that region, whereas an exactly observed continuous outcome contributes a density. A threshold value recorded because of censoring should therefore not automatically be treated as an exact realization of the latent outcome.
Worked Example: A Left-Censored Outcome
Suppose, hypothetically, that the latent outcome for unit \(i\) is \(Y_i^*=x_i'\beta+\varepsilon_i\), with independent \(\varepsilon_i\sim N(0,2^2)\) conditional on the covariates. The recording rule is \(Y_i=\max(8,Y_i^*)\): outcomes at or below 8 are recorded as 8, while larger outcomes are recorded exactly. Assume the conditional mean \(x_i'\beta\) is 10 for each of two units. The first is recorded as 8 and the second as 11. Find their joint likelihood contribution.
For the first unit, the record \(Y_1=8\) means \(Y_1^*\le 8\), not that \(Y_1^*=8\). Its contribution is therefore \(P(Y_1^*\le8\mid x_1)=\Phi((8-10)/2)=\Phi(-1)=0.1587\).
The second unit is observed exactly above the threshold. Its contribution is the normal density at 11: \(\frac{1}{2}\phi((11-10)/2)=\frac{1}{2}\phi(0.5)=0.1760\). The factor \(1/2\) converts the standard normal density to a density in the units of \(Y^*\).
Under the assumed conditional independence, the joint contribution is \(0.1587(0.1760)=0.0279\). One factor is a probability and the other is a density; together they are the likelihood contribution for the mixed observed data. The censoring threshold and the rule defining what is recorded are essential parts of this calculation.
A Practical Construction Sequence
A reliable derivation keeps the structural model separate from the observation mechanism, then combines them only after both are explicit.
Specify the equations, parameter values being varied, shock distributions, and any dependence or independence assumptions.
Check whether the system has a unique solution, whether an inverse from outcomes to shocks exists, and whether any equilibrium-selection rule is needed.
Apply change of variables when appropriate, integrate out unobserved variables, and represent censoring or other recording rules using their implied probabilities or densities.
Combine individual contributions using the sampling dependence assumptions, conditional on observed covariates where relevant.
Why the Details Matter
A likelihood is only as appropriate as the mapping from the structural model to the observed data. A correct shock distribution does not by itself settle the observable density. A missing Jacobian changes that density; conditioning on an unobserved latent realization changes the target; and treating a censored threshold as an exact outcome changes a probability contribution into an inappropriate density contribution. Likewise, equilibrium equations with multiple solutions require an explicit selection mechanism before they define a complete data distribution.
These distinctions also clarify what likelihood-based estimation can and cannot establish. The likelihood records the implications of the specified structural restrictions and distributional assumptions for the data. It does not make those assumptions true, nor does a well-defined likelihood alone guarantee that distinct parameter values imply distinct observable distributions.
Check Your Understanding
For each question, identify which part of the structural-to-observable derivation is at issue.
- In an invertible system, what role does the absolute Jacobian determinant play in converting a shock density to an outcome density?
- A model has an unobserved binary state. Should the likelihood for an observed outcome condition on one state or sum over the possible states? What weights enter that sum?
- Why do structural equations with several equilibria require an additional specification before they determine an observable likelihood?
- Under the rule \(Y=\max(c,Y^*)\), what likelihood contribution corresponds to an observation recorded at \(c\)?
- When may individual likelihood contributions be multiplied to form a sample likelihood?