In all regressions, the residuals consistently estimate corresponding regression errors. Therefore, to find a probability limit of the Durbin–Watson statistic, it suffices to compute the variance and first-order autocovariance of the errors across the stacked equations

where

and u_it's denote regression errors. Note that the errors are uncorrelated where the index i switches between individuals, hence summation from t = 2 in

.

Consider the original regression (1) where u_it = μ_i + v_it. Then

and

Thus

Consider the Within regression (2) where u_it = v_it − v_i·. Then

Thus

Consider the Between regression (3) where u_it = μ_i + v_i·. Then

Thus

The generalized least squares transformation orthogonalizes the errors; therefore,

Because all computed probability limits except that for DW_OLS do not depend on the variance components, the only way to construct an asymptotic test of H₀ : σ_μ² = 0 vs. H_A : σ_μ² > 0 is by using DW_OLS. Under H₀,

(estimation of β does not affect the limiting distribution). Under H_A, p lim_n→∞ DW_OLS < 2. Hence a one-sided asymptotic test for σ_μ² = 0 for a given level α is

where z_α is the α-quantile of the standard normal distribution.