Omitted-variable bias

In statistics, omitted-variable bias (OVB) occurs when a statistical model leaves out one or more relevant variables. The bias results in the model attributing the effect of the missing variables to those that were included. More specifically, OVB is the bias that appears in the estimates of parameters in a regression analysis, when the assumed specification is incorrect in that it omits an independent variable that is a determinant of the dependent variable and correlated with one or more of the included independent variables. Suppose the true cause-and-effect relationship is given by: with parameters a, b, c, dependent variable y, independent variables x and z, and error term u. We wish to know the effect of x itself upon y (that is, we wish to obtain an estimate of b). Two conditions must hold true for omitted-variable bias to exist in linear regression: the omitted variable must be a determinant of the dependent variable (i.e., its true regression coefficient must not be zero); and the omitted variable must be correlated with an independent variable specified in the regression (i.e., cov(z,x) must not equal zero). Suppose we omit z from the regression, and suppose the relation between x and z is given by with parameters d, f and error term e. Substituting the second equation into the first gives If a regression of y is conducted upon x only, this last equation is what is estimated, and the regression coefficient on x is actually an estimate of (b + cf ), giving not simply an estimate of the desired direct effect of x upon y (which is b), but rather of its sum with the indirect effect (the effect f of x on z times the effect c of z on y). Thus by omitting the variable z from the regression, we have estimated the total derivative of y with respect to x rather than its partial derivative with respect to x. These differ if both c and f are non-zero. The direction and extent of the bias are both contained in cf, since the effect sought is b but the regression estimates b+cf.