1. MOTIVATION AND RESULTS

Mundlak (1978) considered a panel data regression model with error component disturbances

where the individual effects are a linear function of the averages of all the explanatory variables across time

where ε_i ∼ IIN(0,σ_ε²), ν_it ∼ IIN(0,σ_ν²), and X_i.′ is 1 × K vector of observations on the explanatory variables averaged over time. Mundlak showed that generalized least squares (GLS) on the resulting model,

yields

and

with

where P is a matrix that averages the observation across time for each individual and Q = I_NT − P is a matrix that obtains the deviations from individual means. This note gives an alternative derivation of this result using system estimation. Arellano (1993) applied system estimation to obtain an alternative derivation of the Hausman (1978) test. In fact, Arellano (1993) used the forward orthogonal deviations operator. Here, we use the usual fixed effects transformation. In particular, we write the panel model in vector form as

where η = Z_με + ν, Z_μ = I_N [otimes ] ι_T with I_N denoting an identity matrix of dimension N and ι_T a vector of ones of dimension T. Here P is the projection matrix on Z_μ, i.e., P = Z_μ(Z_μ′Z_μ)⁻¹Z_μ′ = I_N [otimes ] J_T where J_T is a matrix of ones of dimension T and J_T = J_T /T. Premultiplying (7) by P one gets

because P² = P and PZ_μ = Z_μ. Note that ordinary least squares (OLS) or GLS on (8) yields

, which is the usual between estimator of y on X. Similarly, premultiplying (7) by Q one gets

because QP = 0. OLS or GLS on (9) yields

, which is the usual within or fixed effects estimator of y on X. Stacking the system of equations (8) and (9), we get

and the system error vector has mean 0 and variance-covariance matrix given by

where σ₁² = Tσ_ε² + σ_ν². This system estimation has been useful in deriving error components two-stage least squares (EC2SLS) and error components three-stage least squares (EC3SLS) (see Baltagi, 1981). It has also been used to derive GMM estimators for dynamic panel data models (see Arellano and Bover, 1995, and Blundell and Bond, 1998). For the Mundlak case, there is no need for partitioned inversion. In fact, the OLS normal equations on (10) yield

and

because P + Q = I_NT. Subtracting (13) from (12) one gets X′Qy = (X′QX)β, which yields

.

Solving (13) yields

. Similarly, the GLS normal equations on (10) yield

and

Equation (15) yields

. Subtracting (15) from (14) one gets X′Qy = (X′QX)β, which yields

. This proves that system OLS or GLS on (10) yields the same results that Mundlak found by applying GLS to (3).

In fact, one can prove that the Zyskind (1967) necessary and sufficient condition for OLS to be equivalent to GLS on the system of equations (10) is satisfied. This calls for P_ZΣ = ΣP_Z, where

is the matrix of regressors in (10) and Σ is the variance-covariance matrix of its disturbances. It is straightforward to show that

from which it follows that

Note that the Hausman (1978) specification test based on the between minus within estimators is basically a test for H₀,π = 0 in (3), and this is based upon

The

can be obtained from the GLS variance-covariance matrix of (10). This is given by the inverse of

which can be easily shown by partitioned inversion to be

Note that the second diagonal matrix is exactly the same as that given by (6), which completes the proof.