#### Functional form

A functional form refers to the algebraic
form of a relationship between a dependent variable and regressors
or explanatory 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:

- Semi-log. Either the dependent variable
or the independent variables are transformed using the natural
logarithm transformation.
- Double-log. Variables are transformed
using the natural logarithm transformation.
- 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.

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##### 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.