Statistical model specification

In statistical inference, model specification is part of the process of building a statistical model: specification consists of selecting an appropriate functional form for the model and choosing which variables to include. For example, 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 follows:

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

where $\varepsilon$ is the unexplained error term that is supposed to comprise independent and identically distributed Gaussian variables.

Specification error and bias

Specification error occurs when the functional form or the choice of independent variables poorly represent relevant aspects of the true data-generating 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);
• 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.

In the example given above relating personal income and human capital, if the assumptions of the model are correct, then 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.

Model building

Building a model involves finding a set of relationships that represent 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 a process called statistical model 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.

See also

• All models are wrong
• Conceptual model
• Econometric model
• Information matrix test
• Model identification
• Model risk
• Model selection
• Regression validation
• Spurious relationship
• Statistical conclusion validity

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.
• MacKinnon, James G. (1992). "Model Specification Tests and Artificial Regressions". Journal of Economic Literature. 30 (1): 102–146. JSTOR 2727880.
• 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.
• 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.