Misspecification

Types of Misspecification

Models may be incorrectly specified in one or more of three basic ways: (a) with respect to the set of regressors selected, (b) in the measurement of model variables, and (c) in the specification of the stochastic term or error, ui. First, estimation problems created by incorrect selection of regressors will vary depending on whether the researcher has (a) omitted an important variable from the model (see omitted variable), (b) included an extraneous variable, or (c) misspecified the functional form of the relationship between Xi and Y. Under most circumstances, the first condition—omission of a relevant variable from the model—results in the most serious consequences for model estimates. Specifically, if some variable Zi is responsible for a portion of the variation in Y, yet Zi is omitted from the model, then parameter estimates will be both biased and inconsistent, unless Z is perfectly independent of the included variables. The extent and direction of bias depend on the CORRELATION between the omitted Zi and the included variables Xk. Inclusion of an extraneous variable (i.e., “overfitting” the model) is generally less damaging because all model parameters remain unbiased and consistent. However, a degree of inefficiency in those estimates is introduced in that the calculated standard errors for each parameter take account of the correlation between each “true” variable and the extraneous Xk. Because that correlation is likely to be nonzero in the sample, estimated standard errors are inflated unnecessarily. Misspecification of the functional form linking Y to X can be seen as a mixed case of overfitting as well as underfitting. Imagine a “true” model, Y = β0 + β1X1 + β2X22 + ui. If we now estimate a model in which Y is a simple linear function of X1 and X2 (rather than the square of X2 found in the true model)—that is, Y = α0 + α1X1 + α2X2 + ei—then parameter estimates and standard errors for α0, α1, and α2 will be vitiated by the inclusion of the extraneous X2 as well as by the omission of the squared term X22.

Beyond the incorrect choice of regressors, measurement errors in the dependent and independent variables may be viewed as a form of model misspecification. This is true simply because such measurement error may result in the same variety of problems in parameter estimation as discussed above (i.e., estimated parameters that are biased, inconsistent, or inefficient). Finally, proper identification of the process creating the stochastic errors, ui, can be seen as a model specification issue. To see this point, begin with the common assumption that the errors are distributed normally with mean zero and variance σ2 (i.e., ui ∼ N(0,σ2)). Now imagine that the true errors are generated by any other process (e.g., perhaps the errors may exhibit HETEROSKEDASTICITY or they are distributed log normal). In this instance, we will again face issues of bias, inconsistency, or inefficiency depending on the particular form of misspecification of the error term.

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