Editing Orthogonality (section)

==Statistics, econometrics, and economics==
When performing statistical analysis, [[Dependent and independent variables|independent variables]] that affect a particular [[Dependent and independent variables|dependent variable]] are said to be orthogonal if they are uncorrelated,<ref>{{cite book |title=Probability, Random Variables and Stochastic Processes |author1=Athanasios Papoulis |author2=S. Unnikrishna Pillai |year=2002 |pages=211 |publisher=McGraw-Hill |isbn= 0-07-366011-6}}</ref> 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 linear regression|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 [[econometrics|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 the explanatory variables and model residuals.