Frisch–Waugh–Lovell theorem

Frisch–Waugh–Lovell theorem (FWL theorem) is a fundamental result in the field of econometrics, which is the application of statistical methods to economic data. The theorem, named after Ragnar Frisch, Frederick V. Waugh, and Michael C. Lovell, who were instrumental in its development, provides a simplified method for estimating the coefficients of one or more variables of interest in a linear regression model, while controlling for the effects of other variables.

Overview

The FWL theorem states that in a linear regression model, the estimated coefficients obtained from regressing the dependent variable on a subset of independent variables, after both the dependent and the selected independent variables have been orthogonalized with respect to the other independent variables, are identical to the coefficients obtained from the full regression model. This result holds true regardless of whether the other independent variables in the model are orthogonal to each other.

Mathematical Formulation

Consider a linear regression model:

\[Y = X\beta + Z\gamma + \epsilon\]

where \(Y\) is the dependent variable, \(X\) and \(Z\) are matrices of independent variables, \(\beta\) and \(\gamma\) are vectors of coefficients, and \(\epsilon\) is the error term. The FWL theorem shows that the estimate of \(\beta\), denoted as \(\hat{\beta}\), from the full model can be obtained by first regressing \(Y\) on \(Z\) to get the residuals \(M_Z Y\), and then regressing \(X\) on \(Z\) to get the residuals \(M_Z X\), where \(M_Z\) is the residual maker matrix for \(Z\). Finally, \(\hat{\beta}\) is obtained by regressing \(M_Z Y\) on \(M_Z X\).

Applications

The FWL theorem is widely used in econometrics for various purposes, including:

• Simplifying the estimation of regression coefficients when the model includes a large number of control variables.
• Facilitating the interpretation of coefficients by isolating the effect of variables of interest from other confounding variables.
• Enabling the use of partitioned regression techniques to test hypotheses about the significance of subsets of coefficients.

Limitations

While the FWL theorem provides a powerful tool for econometric analysis, it has limitations. It assumes that the model is correctly specified and that the error terms are homoscedastic and uncorrelated with the independent variables. Violations of these assumptions can lead to biased and inconsistent estimates.

See Also

• Linear regression
• Econometrics
• Ragnar Frisch
• Frederick V. Waugh
• Michael C. Lovell