# Model specification

In regression analysis, model specification is the process of developing a regression model. This process consists of selecting an appropriate functional form for the model and choosing which variables to include. For instance, one may specify the functional relationship $\displaystyle{ y = f(s,x) }$ between personal income $\displaystyle{ y }$ and human capital, with the latter proxied by schooling $\displaystyle{ s }$ and on-the-job experience $\displaystyle{ x, }$ as[1]

$\displaystyle{ \ln y = \ln y_0 + \rho s + \beta_1 x + \beta_2 x^2 + \varepsilon }$

where $\displaystyle{ \varepsilon }$ is the unexplained error term that is supposed to be independent and identically distributed. If assumptions of the regression model are correct, the least squares estimates of the parameters $\displaystyle{ \rho }$ and $\displaystyle{ \beta }$ will be efficient and unbiased. Hence specification diagnostics usually involve testing the first to fourth moment of the residuals.[2]

## Specification error and bias

Specification error occurs when the functional form or the choice of independent variables does not coincide with that of the true underlying process. In particular, bias (the expected value of the difference of an estimated parameter and the true underlying value) occurs if an independent variable is correlated with the errors inherent in the underlying process. There are several different causes of specification error:

• An incorrect functional form could be employed;
• a variable omitted from the model may have a relationship with both the dependent variable and one or more of the independent variables (causing omitted-variable bias);[3]
• an irrelevant variable may be included in the model (although this does not create bias, it involves overfitting and so can lead to poor out-of-sample predictive performance);
• the dependent variable may be part of a system of simultaneous equations (giving simultaneity bias);
• measurement errors may affect the independent variables (while this is not a specification error, it creates statistical bias).

### Detection of misspecification

The Ramsey RESET test can help test for specification error.

## Model building

Building a model involves finding a set of relationships that coincide with the underlying process that is generating the data. This requires avoiding all the sources of misspecification mentioned above. It is best to start with a model in general form that relies on a theoretical understanding of the underlying process; then the model can be fit to the data and checked for the various sources of misspecification, in the process called regression validation. Theoretical understanding can then guide the modification of the model in such a way as to retain theoretical validity while removing the sources of misspecification. But if it proves impossible to find a theoretically acceptable specification that fits the data, the theoretical model may have to be rejected and replaced with another one.

## References

1. This particular example is known as Mincer earnings function.
2. Long, J. Scott; Trivedi, Pravin K. (1993). "Some Specification Tests for the Linear Regression Model". in Bollen, Kenneth A.; Long, J. Scott. Testing Structural Equation Models. London: Sage. pp. 66–110. ISBN 0-8039-4506-X.
3. Untitled