MISSPECIFICATION in terms of Regressors, Functional Forms, and Measurement

1. MISSPECIFICATIONin terms ofRegressors, Functional Forms, and Measurement Dr. C. Ertuna

2. DEFINITION Misspecification means that either functional form, or regressors, or measured data in an equation is incorrect. In other words: • Omitting relevant variable(s) • Including irrelevant variable(s) • Incorrect functional form • Measurement errors Dr. C. Ertuna

3. CONSEQUENCES of Omitting Relevant Variables • E(u) ≠ 0 • If omitted variable is correlated with the regressors in the equation, then coefficients are biased and inconsistent. • If a variable is omitted because it cannot be observed (measured) then we need to use a proxy variable to reduce effects of omission. Dr. C. Ertuna

4. CONSEQUENCES of Including Irrelevant Variables OLS estimators will be: • Unbiased, • Consistent, • However, not fully efficient. Dr. C. Ertuna

5. Dropping Irrelevant Variables If a variable is irrelevant, then after dropping it from the equation: • RSS will remain more or less unchanged, • will fall, • No sign change for coefficients, • No (appreciable) magnitude change for coefficients, • t-Statistics of remaining variables will not be seriously affected. Dr. C. Ertuna

6. CONSEQUENCES of Incorrect Functional Form • Coefficients will be incorrect estimators of true population parameters. Dr. C. Ertuna

7. DIAGNOSIS of Misspecification • Observe regression residuals (including checking for Normality) • RESET test (Regression Specification Error Test) Dr. C. Ertuna

8. Normality Test Analyze / Descriptive Statistics / Explore > Residuals (or variable under analysis) into “Dependent List” > Plots: check“Normality plots with Tests” Continue / Okay Dr. C. Ertuna

9. RESET Tests for MisspecificationF-Form Step-1: Run OLSregression (Afterdetermining which variables should be in the model) and save predicted values Step-2: Run an augmented regression where or andare added as regressors (ENTER). Step-3: Set Ho: Model correctly specified. Dr. C. Ertuna

10. RESET Tests for MisspecificationF-Form Step-3: Set Ho: Model correctly specified. Step-4: Consider the auxiliary regression as unrestricted model and the original model as restricted one. Step-5: Get RSS and k of Restricted Model (the model with fewer variables) and get RSS and k and N of Unrestricted Model (the model with more variables) Step-6: Apply the F-form of the Likelihood Ratio test (get organized and use Excel for computation). F(, N - ) = p-value = FDIST(F-value; ; N - ) Dr. C. Ertuna

11. Definition of Test Parameters = Residual Sum of Squares for Restricted Model (Model with fewer variables) = Residual Sum of Squares for Unrestricted Model (Model with fewer variables) = Number of parameters (included the intercept) in the Unrestricted model. = Number of parameters (included the intercept) in the Unrestricted model. N = Number of observations of the unrestricted model.

12. RESET Tests for MisspecificationF-Form Step-7: If p-value < alpha then conclude that there is significant evidence that the original model is misspecified. Note: Since RESET test is an omnibus test we don’t know the source or type of misspecification. Dr. C. Ertuna

13. RESET Tests for MisspecificationLM Form Step-1: Run OLS regression (Afterdetermining which variables should be in the model) and save residuals and predicted values Step-2: Run an auxiliary regression where the dependent variable is and independent variables are the original regressors plus or and(ENTER). Step-3: Set Ho: Model correctly specified. Dr. C. Ertuna

14. RESET Tests for MisspecificationLM Form Step-3: Set Ho: Model correctly specified. Step-4: Compute LM = n* , where n = number of observations in auxiliary regression = coefficient of determination of the auxiliary regression. Step-5: Compute p-value by p-value = CHIDIST(LM;df) where df = h, h= number of additional regressors in the auxiliary regression. Step-6: If p-value < alpha then conclude that there is significant evidence that the original model is misspecified. Note: Since RESET test is an omnibus test we don’t know the source or type of misspecification. Dr. C. Ertuna

15. Box-Cox Procedure for Functional Form Test • When comparing the linear with the log-linear (or log-log) forms, we cannot compare the R2because R2 is the ratio of explained variance to the total variance and the variances of y and log y are different. • One solution to this problem is to consider a more general model of which both the linear and log-linear forms are special cases. Dr. C. Ertuna

16. Box-Cox Procedure for Functional Form Test • Consider the regression model with Box-Cox transformation: • For this is a log-linear model and • For this is a linear model Dr. C. Ertuna

17. Box-Cox Procedure for Functional Form Test Consider two regression models: Ln Step-1: Compute Geometric Mean of → Step-2: Transform by Analyze / Reports / Case Summaries → Geometric Mean Dr. C. Ertuna

18. Box-Cox Procedure for Functional Form Test Step-3: Run regressions: Ln Step-4: Save and Step-5: Compute Box-Cox Test Statistics Dr. C. Ertuna

19. Box-Cox Procedure for Functional Form Test Step-6: Set Ho: One functional form is not superior over the other. Step-7: If p-value < alpha then conclude that there is significant evidence that one functional form is better than the other. Choose the functional form with lowest RSS as the better one. Dr. C. Ertuna

20. END Next, Chapter 10: Dummy Variables

21. RESET Tests for Misspecification Step-3: Set Ho: No significant difference between two functional form. Step-4: Compute LM = n* , where n = number of observations in auxiliary regression = coefficient of determination of the auxiliary regression. Step-5: Compute p-value by p-value = CHIDIST(LM;df) where df = k-1, k = number of parameters in the auxiliary regression. Step-6: If p-value < alpha than conclude that there is significant evidence that the functional forms differ. Choose the functional form with lowest RSS. Dr. C. Ertuna