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.