Regression Analysis > Model Misspecification

## What is Model Misspecification?

Model Misspecification is where the model you made with regression analysis is in error. In other words, it doesn’t account for everything it should. Models that are misspecified can have biased coefficients and error terms, and tend to have biased parameter estimations.

## Left Out Variables

Important variables can be missed for a variety of reasons, including by mistake or on purpose. The following models show (1) the correct model and (2) a model without x_{k+1t}:

(1)y_{1} = Β_{1}x_{1t} + Β_{2}x_{2t} +…Β_{k}x_{kt} + Β_{k+1}x_{k+1} + η

(2) y_{1} = Β_{1}x_{1t} + Β_{2}x_{2t} +…Β_{k}x_{kt} + η

The opposite of a left-out variable is an *irrelevant variable.*

## Irrelevant Variables

Irrelevant variables shouldn’t have been put in the model in the first place. The following models show (1) the correct model and (2) a model with an irrelevant variable x_{k+1t}:

(1) y_{1} = Β_{1}x_{1t} + Β_{2}x_{2t} +…Β_{k}x_{kt} + η

(2)y_{1} = Β_{1}x_{1t} + Β_{2}x_{2t} +…Β_{k}x_{kt} + Β_{k+1}x_{k+1} + η

## Functional Form Misspecification

When a model has the appropriate explanatory variables, but still fails to account for the relationship between the explanatory and response variables, then the model has *functional form misspecification*. Examples of function form misspecification include leaving out a squared variable or constraining dy/dx to be constant (Wooldridge, 1994).

## Tests for Model Misspecification

These tests are usually performed using software:

**The Ramsey Regression Specification Error Test (RESET)**: a general misspecification test for linear regression models.**Davidson and MacKinnon J Test**: a test for non-nested model specification.

**References:**

Davidson, R. and J.G. MacKinnon (1981). “Several tests for model specification in the presence of alternative hypotheses,” Econometrica 49, 781–793.

Ramsey, J.B. (1969). “Tests for specification errors in classical linear least-squares analysis,” Journal of the Royal Statistical Society, Series B, 71, 350–371.

Rao, Potluri. “Some Notes on Misspecification in Multiple Regressions.” The American Statistician 25, no. 5 (1971): 37-39. doi:10.2307/2686082.

Wooldridge, J.M. (1994). “A simple specification test for the predictive ability of transformation models,” Review of Economics and Statistics 76, 59–65.