Regression using mixed annual and quarterly data

By consideration of either Generalised Least-Squares of Theil-Goldberger estimates of equations for which data on the dependent variable is available for a subset of the sample period on only an annual basis, it is shown that best estimates of the equation coefficients may be obtained by Ordinary Least-Squares estimation over the complete set of quarterly observations with quarterly averages substituted for the missing quarterly data, and symmetrically for the corresponding observations on the regressor variables. The approach is extended to the Two-Stage Least-Squares case, and to the (structurally or stochastically) autoregressive case.