Consider the following first structural equation of a simultaneous equation panel data model:

where Z₁ = [Y₁, X₁] and δ₁′ = (γ₁′, β₁′). As in the standard simultaneous equation literature, Y₁ is the set of g₁ right-hand-side endogenous variables, and X₁ is the set of k₁ included exogenous variables. Let X = [X₁, X₂] be the set of all exogenous variables in the system. Let k₂, the number of excluded exogenous variables from the first equation (X₂), be larger than or equal to g₁. The disturbance u₁ follows the one-way error component model

where Z_μ = (I_N [otimes ] ι_T) with I_N being an identity matrix of dimension N and ι_T a vector of ones of dimension T. In this case, μ₁′ = (μ₁₁,…, μ_N1) is a vector of random individual effects of dimension N and ν₁′ = (ν₁₁₁,…, ν_NT1) is a vector of remainder disturbances of dimension NT. These vectors have zero means and covariance matrix

Baltagi (1981) derives the error component two-stage least squares (EC2SLS) estimator of this model, which is the random effects (RE) counterpart of a classical error components panel data regression. Similarly, the fixed effects two-stage least squares (FE2SLS) estimator of this model is the fixed effects (FE) counterpart of a classical error components panel data regression. Both EC2SLS and FE2SLS allow for the endogeneity of Y₁.

Hausman (1978) suggests comparing the FE and RE estimators in the classical panel data regression. With endogenous right-hand-side regressors such as Y₁ this test can be generalized to test H₀ : E(u₁|Z₁) = 0 based on

.

(a) Show that under H₀ : E(u₁|Z₁) = 0,

and the asymptotic

.

Conclude that

, where var denotes the asymptotic variance. This is used in computing the Hausman test statistic given by

Under H₀, m₁ is asymptotically distributed as χ_r², where r denotes the dimension of the slope vector of the time varying variables in Z₁. This can be easily implemented using standard software packages, like STATA. The usual Hausman test can yield misleading inference in the presence of Y₁.