Editing Omitted-variable bias (section)

=== Intuition ===
Suppose the true cause-and-effect relationship is given by:

:<math>y=a+bx+cz+u</math>

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

:<math>z=d+fx+e</math>

with parameters ''d'', ''f'' and error term ''e''. Substituting the second equation into the first gives

:<math>y=(a+cd)+(b+cf)x+(u+ce).</math>

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''&nbsp;+&nbsp;''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&nbsp;''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''. The extent of the bias is the absolute value of ''cf'', and the direction of bias is upward (toward a more positive or less negative value) if ''cf'' > 0 (if the direction of correlation between ''y'' and ''z'' is the same as that between ''x'' and ''z''), and it is downward otherwise.