Fully Modified OLS (FMOLS) Estimator
Also known as: fully modified OLS, Phillips-Hansen FMOLS, Tam Düzeltilmiş OLS (FMOLS)
Fully Modified OLS, introduced by Phillips and Hansen (1990), estimates the long-run coefficients of a cointegrating relationship among I(1) variables. It applies a semi-parametric correction to ordinary least squares to remove the bias that endogeneity and serial correlation otherwise induce in cointegrated time series or panel data.
Key highlights
- Removes the endogeneity and serial-correlation bias that distorts plain OLS in cointegrated regressions, yielding asymptotically unbiased long-run coefficients.
- Requires no choice of lead and lag terms, unlike DOLS, because the correction is semi-parametric via a kernel estimate of the long-run covariance.
- Extends naturally to heterogeneous cointegrated panels through Pedroni's group-mean panel FMOLS estimator.
Intuition
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How it works
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When to use it
Use FMOLS when you need the long-run coefficients of a cointegrating relationship and your variables are integrated of order one, I(1), and confirmed to be cointegrated (for example after a Johansen or ARDL bounds test, or panel cointegration tests for panels). It needs a reasonable sample, at least about 50 observations, and is designed for single-equation long-run estimation in time series or, in the Pedroni group-mean form, heterogeneous cointegrated panels. It is not appropriate for stationary data, non-cointegrated I(1) variables, or systems with more than one cointegrating vector.
Strengths & limitations
- Removes the endogeneity and serial-correlation bias that distorts plain OLS in cointegrated regressions, yielding asymptotically unbiased long-run coefficients.
- Requires no choice of lead and lag terms, unlike DOLS, because the correction is semi-parametric via a kernel estimate of the long-run covariance.
- Extends naturally to heterogeneous cointegrated panels through Pedroni's group-mean panel FMOLS estimator.
- Only valid when the variables are I(1) and genuinely cointegrated; applying it otherwise gives meaningless long-run estimates.
- Performance depends on the kernel and bandwidth used to estimate the long-run covariance, and it can be less reliable than DOLS in small samples.
- Mature software support is limited; it is typically run in R, Stata, or EViews rather than core Python libraries.
Common pitfalls
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Applications
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Frequently asked
How is FMOLS different from DOLS?
Both estimate long-run cointegrating coefficients while correcting for endogeneity and serial correlation. FMOLS does so semi-parametrically, with a kernel estimate of the long-run covariance and no lead/lag terms. DOLS does it parametrically, by adding leads and lags of the differenced regressors, and often performs better in small samples.
What does it mean that variables must be cointegrated?
The variables must each be non-stationary and integrated of order one, I(1), yet share a stable long-run linear combination that is stationary. You confirm this with cointegration tests such as Johansen, the ARDL bounds test, or panel cointegration tests before applying FMOLS.
Can FMOLS be used with panel data?
Yes. Pedroni's (2001) group-mean panel FMOLS estimates a long-run coefficient for each cross-section and averages them, allowing heterogeneity across units. It does not, however, handle cross-sectional dependence; for that, estimators like CCEMG are more appropriate.
What is the role of the long-run covariance matrix?
The correction that makes OLS 'fully modified' is built from an estimate of the long-run covariance matrix of the errors, computed with a kernel such as the Bartlett kernel. It absorbs the endogeneity and serial-correlation effects so the long-run slope becomes asymptotically unbiased.
Sources
- 1.Phillips, P. C. B. & Hansen, B. E. (1990). Statistical Inference in Instrumental Variables Regression with I(1) Processes. Review of Economic Studies, 57(1), 99–125.
- 2.Pedroni, P. (2001). Fully Modified OLS for Heterogeneous Cointegrated Panels. Advances in Econometrics, 15, 93–130.
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Cite this page
ScholarGate. (2026, June 1). FMOLS Estimator. ScholarGate. https://scholargate.app/econometrics/fmols-estimator