Specification (regression)

In regression analysis 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 $y = f(s,x)$ between personal income $y$ and human capital in terms of schooling $s$ and on-the-job experience $x$ as:^[1]

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

where $\varepsilon$ 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 $\rho$ and $\beta$ 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 an independent variable is correlated with the error term. 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 (omitted-variable bias);^[3]
• an irrelevant variable may be included in the model;
• the dependent variable may be part of a system of simultaneous equations (simultaneity bias);
• measurement errors may affect the independent variables.

Detection

The Ramsey RESET test can help test for specification error.

See also

• Model risk
• Regression model validation

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

• MacKinnon, James G. (1992). "Model Specification Tests and Artificial Regressions". Journal of Economic Literature 30 (1): 102–146. JSTOR 2727880.
• Thursby, Jerry G.; Schmidt, Peter (September 1977). "Some Properties of Tests for Specification Error in a Linear Regression Model". Journal of the American Statistical Association 72 (359): 635–641. doi:10.1080/01621459.1977.10480627. JSTOR 2286231.
• Sapra, Sunil (2005). "A regression error specification test (RESET) for generalized linear models" (PDF). Economics Bulletin 3 (1): 1–6.

Further reading

• Asteriou, Dimitrios; Hall, Stephen G. (2011). "Misspecification: Wrong Regressors, Measurement Errors and Wrong Functional Forms". Applied Econometrics (Second ed.). London: Palgrave MacMillan. pp. 172–197.
• Gujarati, Damodar N.; Porter, Dawn C. (2009). "Econometric Modeling: Model Specification and Diagnostic Testing". Basic Econometrics (Fifth ed.). New York: McGraw-Hill Irwin. pp. 467–522. ISBN 978-0-07-337577-9.
• Kmenta, Jan (1986). Elements of Econometrics (Second ed.). New York: Macmillan. pp. 442–455. ISBN 0-02-365070-2.
• Leamer, Edward E. (1978). Specification Searches: Ad hoc Inference with Nonexperimental Data. New York: Wiley. ISBN 0-471-01520-2.
• Maddala, G. S.; Lahiri, Kajal (2009). "Diagnostic Checking, Model Selection, and Specification Testing". Introduction to Econometrics (Fourth ed.). Chichester: Wiley. pp. 401–449. ISBN 978-0-470-01512-4.