Applied Econometrics

1. Applied Econometrics William Greene Department of Economics Stern School of Business

2. Applied Econometrics 7. Estimating the Variance of the Least Squares Estimator

3. Context The true variance of b|X is ?2(X?X)-1 . We consider how to use the sample data to estimate this matrix. The ultimate objectives are to form interval estimates for regression slopes and to test hypotheses about them. Both require estimates of the variability of the distribution. We then examine a factor which affects how "large" this variance is, multicollinearity.

4. Estimating ?2 Using the residuals instead of the disturbances: The natural estimator: e?e/n as a sample surrogate for ???/n Imperfect observation of ?i = ei + (? - b)?xi Downward bias of e?e/n. We obtain the result E[e?e|X] = (n-K)?2

5. Expectation of e’e

6. Method 1:

7. Estimating s2 The unbiased estimator is s2 = e?e/(n-K). “Degrees of freedom correction” Therefore, the unbiased estimator is s2 = e?e/(n-K) = ??M?/(n-K).

8. Method 2: Some Matrix Algebra

9. Decomposing M

10. Example: Characteristic Roots of a Correlation Matrix

11. Gasoline Data

12. X’X and its Roots

13. Var[b|X] Estimating the Covariance Matrix for b|X The true covariance matrix is ?2 (X’X)-1 The natural estimator is s2(X’X)-1 “Standard errors” of the individual coefficients are the square roots of the diagonal elements.

14. Regression Results

16. Bootstrapping Some assumptions that underlie it - the sampling mechanism Method: 1. Estimate using full sample: --> b 2. Repeat R times: Draw n observations from the n, with replacement Estimate ? with b(r). 3. Estimate variance with V = (1/R)?r [b(r) - b][b(r) - b]’

17. Bootstrap Application --> matr;bboot=init(3,21,0.)$ Store results here --> name;x=one,y,pg$ Define X --> regr;lhs=g;rhs=x$ Compute b --> calc;i=0$ Counter --> Proc Define procedure --> regr;lhs=g;rhs=x$ … Regression --> matr;{i=i+1};bboot(*,i)=b$... Store b(r) --> Endproc Ends procedure --> exec;n=20;bootstrap=b$ 20 bootstrap reps --> matr;list;bboot' $ Display results

18. Results of Bootstrap Procedure

19. Bootstrap Replications

20. OLS vs. Least Absolute Deviations

21. Multicollinearity Not “short rank,” which is a deficiency in the model. A characteristic of the data set which affects the covariance matrix. Regardless, ? is unbiased. Consider one of the unbiased coefficient estimators of ?k. E[bk] = ?k Var[b] = ?2(X’X)-1 . The variance of bk is the kth diagonal element of ?2(X’X)-1 . We can isolate this with the result in your text. Let [ [X,z] be [Other xs, xk] = [X1,x2] (a convenient notation for the results in the text). We need the residual maker, M1. The general result is that the diagonal element we seek is [x2?M1x2]-1 , which we know is the reciprocal of the sum of squared residuals in the regression of x2 on X1. The remainder of the derivation of the result we seek is in your text. The estimated variance of bk is

22. Variance of a Least Squares Coefficient Estimator Estimated Var[bk] =

23. Gasoline Market

24. Gasoline Market

25. Multicollinearity Clearly, the greater the fit of the regression in the regression of x2 on X1, the greater is the variance. In the limit, a perfect fit produces an infinite variance. There is no “cure” for collinearity. Estimating something else is not helpful (principal components, for example). There are “measures” of multicollinearity, such as the condition number of X. Best approach: Be cognizant of it. Understand its implications for estimation. What is better: Include a variable that causes collinearity, or drop the variable and suffer from a biased estimator? Mean squared error would be the basis for comparison. Some generalities.

26. Specification and Functional Form:1Nonlinearity

27. Log Income Equation

28. Specification and Functional Form:2Interaction Effect

29. Interaction Effect