Orthogonality - Statistics, Econometrics, and Economics

Statistics, Econometrics, and Economics

When performing statistical analysis, independent variables that affect a particular dependent variable are said to be orthogonal if they are uncorrelated, since the covariance forms an inner product. In this case the same results are obtained for the effect of any of the independent variables upon the dependent variable, regardless of whether one models the effects of the variables individually with simple regression or simultaneously with multiple regression. If correlation is present, the factors are not orthogonal and different results are obtained by the two methods. This usage arises from the fact that if centered (by subtracting the expected value (the mean)), uncorrelated variables are orthogonal in the geometric sense discussed above, both as observed data (i.e. vectors) and as random variables (i.e. density functions). One econometric formalism that is alternative to the maximum likelihood framework, the Generalized Method of Moments, relies on orthogonality conditions. In particular, the Ordinary Least Squares estimator may be easily derived from an orthogonality condition between predicted dependent variables and model residuals.