Functional form
A functional form refers to the algebraic form of a relationship between a
dependent variable and regressors oexplanatory variables.
The simplest functional form is the linear functional form, where the
relationship between the dependent variable and an independent variable is
graphically represented by a straight line.
Other useful functional forms in regression analysis include:
(1) Semi-log. Either the dependent variable or the independent variables are
transformed using the natural logarithm transformation.
(2) Double-log. Variables are transformed using the natural logarithm
transformation.
(3) Reciprocal. Independent variables (one or more) are represented as the
reciprocal (that is, for variable x, the transformation is 1/x).
These functional forms allow the analyst to represent a wide range of shapes.
Interpretation
The interpretation of coefficients is different in alternative functional
forms. In the following formulations Y represents the dependent variable, x
the independent variable, a is the y-intercept, b is the slope coefficient,
ln(y) and ln(x) represent the natural logarithm of y and x, respectively.
and e is an error term.
(1) Linear: y = a + b x + e
In this functional form b represents the change in y (in units of y) that
will occurs as x changes one unit.
(2) Semi-log: ln(y) = a + b x + e
In this functional form b is interpreted as follows.
A one unit change in x will cause a b(100)% change in y, e.g., if the
estimated coefficient is 0.05 that means that a one unit increase in x
will generate a 5% increase in y.
(3) Double-log: ln(y) = a + b ln(x) + e
In this functional form b is the elasticity coefficient.
A one one percent change in x will cause a b% change in y, e.g., if the
estimated coefficient is -2 that means that a 1% increase in x will
generate a -2% decrease in y.