Econometrics file 2.doc - Specification error One of the...

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Specification error One of the assumptions of the classical linear regression model, is that the regression model used in the analysis is “correctly” specified: If the model is not “correctly” specified, we encounter the problem of model specification error or model specification bias. MODEL SELECTION CRITERIA According to Hendry and Richard, a model chosen for empirical analysis should satisfy the following criteria: 1. Be data admissible ; that is, predictions made from the model must be logically possible. 2. Be consistent with theory ; that is, it must make good economic sense. For example, if Milton Friedman’s permanent income hypothesis holds, the intercept value in the regression of permanent consumption on permanent income is expected to be zero. 3. Have weakly exogenous regressors ; that is, the explanatory variables, or regressors, must be uncorrelated with the error term. 4. Exhibit parameter constancy ; that is, the values of the parameters should be stable. Otherwise, forecasting will be difficulty.. 5. Exhibit data coherency ; that is, the residuals estimated from the model must be purely random (technically, white noise). In other words, if the regression model is adequate, the residuals from this model must be white noise. If that is not the case, there is some specification error in the model. 6. Be encompassing ; that is, the model should encompass or include all the rival models in the sense that it is capable of explaining their results. In short, other models cannot be an improvement over the chosen model. TYPES OF SPECIFICATION ERRORS In developing an empirical model, one is likely to commit one or more of the following specification errors: 1. Omission of a relevant variable(s) 2. Inclusion of an unnecessary variable(s) 3. Adopting the wrong functional form 4. Errors of measurement 5. Incorrect specification of the stochastic error term CONSEQUENCES OF MODEL SPECIFICATION ERRORS Inclusion of an Irrelevant Variable (Overfitting a Model) Now let us assume that Yi = β 1 + β 2 X 2 i + ui is the truth, but we fit the following model: Yi = α 1 + α 2 X 2 i + α 3 X 3 i + vi and thus commit the specification error of including an unnecessary variable in the model. The consequences of this specification error are as follows: 1. The OLS estimators of the parameters of the “incorrect” model are all unbiased and consistent, that is, E ( α 1) = β 1, E α 2) = β 2, and E α 3) = β 3 = 0. 2. The error variance σ 2 is correctly estimated. 3. The usual confidence interval and hypothesis-testing procedures remain valid. 4. However, the estimated α ’s will be generally inefficient, that is, their variances will be generally larger than those of the ˆ β ’s of the true model. From the usual OLS formula we know that and that is, the variance of ˆ α 2 is generally greater than the variance of ˆ β 2 The implication of this finding is that the inclusion of the unnecessary variable X 3 makes the variance of ˆ α 2 larger than necessary, thereby making ˆ α 2 less precise. This is also true of ˆ α 1.
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