CHOICE OF FUNCTIONAL FORM
FUNCTIONAL FORM SPECIFICATION ERROR
To specify the classical linear regression model, we must choose a specific functional form. We
can choose any functional form that is linear in parameters. If we choose the incorrect functional
form, then the model is misspecified. It the model is misspecified then it may not be a reasonable
approximation of the true data generation process. We make a functional form specification error
when we choose the wrong functional form.
DIFFERENT TYPES OF LINEAR IN PARAMETERS FUNCTIONAL FORMS
There are a large number of linear in parameters functional forms from which we can choose. I
will briefly discuss some of those that are most often used. I will do so using one dependent
variable (Y) and one explanatory variable (X). This allows me to illustrate the function
graphically. However, these functional forms can be easily extended to more than one
explanatory variable.
Linear in Variables Functional Form
Functional form:
Graph:
Marginal effect:
Elasticity:
Y = 1 + 2X
m = 2
 = 2 (X/Y)
Note that to obtain an estimate of the elasticity of Y with respect to X, X and Y are evaluated at
their sample mean values. To calculate an estimate of the standard error for the elasticity
estimate, we can treat (X/Y) as a constant. Thus, s.e.( ) is given by the square root of Var( ) =
(X/Y)2Var(2).
Double Log Functional Form
Functional form:
Graph:
Marginal effect:
Elasticity:
lnY = 1 + 2lnX
m = 2(Y/X)
 = 2
For this functional form, the slope parameter is a direct measure of elasticity. To estimate this
functional form using OLS, we would first transform the data for Y and X into logarithmic form.
We would then run a regression of the log of Y on the log of X.
Linear-Log Functional Form
Functional form:
Graph:
Marginal effect:
Elasticity:
Y = 1 + 2lnX
m = 2/X
 = 2/Y
When calculating estimates of the marginal effect and elasticity, X and Y are evaluated at their
sample mean values. Treating X and Y as constants, estimates of the standard errors are given by
the square roots of the variances: Var(m) = (1/X)2Var(2); Var( ) = (1/Y)2Var(2). To
estimate this functional form using OLS, we would first transform the data for X into logarithmic
form. We would then run a regression of Y on the log of X.
Log-Linear Functional Form
Functional form:
Graph:
Marginal effect:
Elasticity:
lnY = 1 + 2X
m = 2Y
 = 2X
For this functional form, the slope parameter 2 has a useful interpretation. When X changes by
one unit, Y will change by approximately 2*100 percent. The smaller the absolute value of 2
the closer the approximation. When calculating estimates of the marginal effect and elasticity, Y
and X are evaluated at their sample mean values. Estimates of the standard errors of the estimates
of the marginal effect and elasticity are given by the square roots of the variances: Var(m) =
Y2Var(2); Var( ) = X2Var(2). To estimate this functional form using OLS, we would first
transform the data for Y into logarithmic form. We would then run a regression of the log of Y
on X.
Quadratic Functional Form
Functional form:
Graph:
Marginal effect:
Elasticity:
Y = 1 + 2X + 3X2
m = 2 + 23X
 = (2 + 23X)(X/Y)
If 2 < 0 and 3 > 0, then the curve has a U-shape. If  2 > 0 and 3 < 0, then the curve has a hill
shape. When calculating estimates of the marginal effect and elasticity, X and Y are evaluated at
their sample mean values. The estimate of the standard error of the estimate of the marginal effect
is given by the square root of the variance: Var(m) = Var(2) + (2X)2Var(3) + 2(2X)Cov(2,
3). To estimate this functional form using OLS, we would first create a new variable X2. We
would then run a regression of the Y on X and X2.
Varying Parameter Functional Form
There are many situations where one or more of the parameters of a statistical model may not be
constant. Rather, the value of the parameter depends on one or more other variables. For
example, suppose we have the model
Y =  + X
We might believe that the slope parameter is not constant, but rather depends upon some other
variable Z. Assuming that the slope parameter is a linear function of Z yields
 = 1 + 2Z
Substituting we get
or equivalently,
Y =  + (1 + 2Z)X
Y =  +  1X + 2ZX
The new variable Z*X is called an interaction term. This is because it captures the interaction
between the variables Z and X. To estimate this model, we create a new variable Z = A*X. We
then run an OLS regression of visits on hours worked and the new variable. This is called a
varying parameters model. However, it can be view as a specific type of functional form.
Functional Form with Lags
There are many situations where the value of dependent variable this period may depend on the
value of an explanatory both this period and in one or more previous periods. For example,
suppose that Y is consumption and X is income. It may be the case that consumption in 1995
depends on 1995 income and 1994 income. Consumption in 1996 depends upon 1995 income and
1994 income. Etc. This model is written as
Yt = 1 + 2Xt + 3Xt-1
The variable Xt-1 is called a lagged variable. In this example, it is one period. If we desire, we
could lag it two periods, e.g., consumption in 1995 depends upon income in 1993, three periods,
etc. Note that if we lag a variable one period, we lose one observation and hence one degree of
freedom. For example, if we have annual data on consumption and income for the period 1980 to
1996 we have 17 observations. However, we lose the observation for 1980, so we are left with 16
observations. We can also use the lagged value of the dependent variable as an explanatory
variable. For example,
Yt = 1 + 2Xt + 3Xt-1 + 4Yt-1
This can be view as a specific functional form.
Mixed Functional Forms
It is possible to mix functional forms. For example, suppose we have the model
Y = 1 + 2lnH + 3lnL + 4A + 5A2 + 6G
Where Y is patient visits, H is physician hours worked, L is aides employed, A is physician age,
and G is gender. In this case, visits are a semi-log function of hours and aides, a quadratic
function of age, and a linear function of gender. The marginal effect and elasticity for each of
these variables is given by the formulas above.
Consequences of Choosing the Wrong Functional Form
The OLS estimator will be biased.
Detection and Correction of Functional Form Specification Errors
Two alternative methodologies used to choose a specific functional form for a model are the
following. 1) Maintained hypothesis methodology. 2) Theory/testing methodology.
Maintained Hypothesis Methodology
This methodology uses theory and/or tractability to choose a specific functional form. Once a
specific functional form is chosen, it is treated as a maintained hypothesis and not tested using the
sample data. Choosing functional form based on tractability is not good practice. Relying on
theory alone may not be good practice. This is because there are many situations when theory has
nothing to say about the appropriate functional form. If the wrong functional form is chosen, then
the parameter estimates will be biased and all tests of hypothesis, strictly speaking, will be
incorrect.
Theory/Testing Methodology
This methodology involves the following steps.
1. Identify the set of specific functional forms that are consistent with theory. If a specific
functional form is inconsistent with the theory being used to guide the specification of the
statistical model, then it should not be considered.
2. Conduct statistical tests to determine which specific functional form should be chosen.
3. To carry-out step #3, use one of two approaches. 1) Testing-down approach. 2) Testing-up
approach.
Testing-Down Approach
When using the testing-down approach, we begin with a general model and test-down to a more
specific model. To test for nonlinear terms and interaction terms, we begin with a general model
that includes one or more of these terms. For example, the general model might include nonlinear
terms such as X2 and/or lnX, or interaction terms such as X*A. We then use a t-Test and/or an Ftest to test whether these terms belong in the model.
General Functional Forms
A more systematic approach is to begin with a general functional form. This general functional
form has one or more specific functional forms as a special case. We use an F-test or a t-test to
test whether a specific functional form is the appropriate functional form.
Testing-up Approach
When using the testing-up approach, we begin with a specific model and test-up to a more
general model. To test for nonlinear terms and interaction terms, we begin with a specific model
that does not include one or more of these terms. For example, the specific model might not
include nonlinear terms such as X2 and/or lnX, or interaction terms such as X*A. We then use a
Lagrange multiplier test to test whether these terms should be added to the model.
Other Tests for Functional Form
A number of other criteria and statistical tests are also used to test for functional form. Some of
these are the following. 1) Adjusted R2. 2) Ramsey’s Reset Test. 3) Recursive Residual Test.