We will work with the standard triangular representation of a regression specification and assume the existence of a single cointegrating vector as in Hansen (1992). Consider a panel structure for the dimensional time series vector process , with cointegrating equation

for cross-sections and periods , where are deterministic trend regressors and the stochastic regressors are governed by the system of equations:

The -vector of regressors enter into both the cointegrating equation and the regressors equations, while the -vector of are deterministic trend regressors which are included in the regressors equations but excluded from the cointegrating equation (see “Cointegrating Regression” for further discussion).

Notice that the cointegrating relationship between and is assumed to be homogeneous across cross-sections, and that the specification allows for cross-section specific deterministic effects.

Following Phillips and Moon (1999), we define the long run covariance matrices for the errors in cross-section are strictly stationary and ergodic with zero mean, contemporaneous covariance matrix , one-sided long-run covariance matrix , and long-run covariance matrix , each of which we partition conformably with

and define what Phillips and Moon term the long-run average covariance matrices , and .

Given this basic structure, we may define panel estimators of the cointegrating relationship coefficient using extensions of single-equation FMOLS and DOLS methods. There are different variants for each of the estimators depending on the assumptions that one wishes to make about the long-run covariances and how one wishes to use the panel structure of the data.