FMOLS

FMOLS, or Fully Modified Ordinary Least Squares, is an econometric estimator used to estimate the long-run relationship between integrated time series that are cointegrated. Introduced by Phillips and Ouliaris in 1990, FMOLS aims to provide unbiased and consistent estimates of the long-run slope in the presence of endogeneity and serial correlation that can bias ordinary least squares when the regressor is integrated.

Conceptually, FMOLS modifies the OLS estimator by applying nonparametric corrections for the contemporaneous correlation between the error term and the regressor, and for serial correlation in the error term. It uses an estimate of the long-run covariance matrix of the error term and the regressor—typically obtained via a kernel-based estimator—to adjust both the dependent variable and the regressor. The resulting estimator is asymptotically normal, enabling standard inference on the long-run coefficient.

FMOLS is commonly applied to bivariate or multivariate cointegration models with I(1) variables that share a common stochastic trend. It is designed to be robust to endogenous regressors and serially correlated disturbances, in contrast to standard OLS. The method can be extended to multivariate settings and is related to other long-run estimators such as Dynamic OLS (DOLS) and the Canonical Cointegrating Regression (CCR).

Compared with alternatives, FMOLS often offers improved small-sample properties relative to OLS in cointegrated systems, though it requires careful bandwidth selection for the long-run variance estimator and correct specification of the cointegration rank and the order of integration.