Statistical 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 $y=f(s,x)$ between personal income $y$ and human capital, with the latter proxied by schooling $s$ and on-the-job experience $x,$ as

$\ln y=\ln y_{0}+\rho s+\beta _{1}x+\beta _{2}x^{2}+\varepsilon$

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.

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 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);
• 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.

See also

• Model risk
• Regression model validation

References

• 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.