Applied Econometrics

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Applied Econometrics. 3. Linear Least Squares. Vocabulary. Some terms to be used in the discussion.Population characteristics and entities vs. sample quantities and analogsResiduals and disturbancesPopulation regression line and sample regressionObjective: Learn about the conditional mean func

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Applied Econometrics

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

2. Applied Econometrics 3. Linear Least Squares

3. Vocabulary Some terms to be used in the discussion. Population characteristics and entities vs. sample quantities and analogs Residuals and disturbances Population regression line and sample regression Objective: Learn about the conditional mean function. Estimate ? and ?2 First step: Mechanics of fitting a line (hyperplane) to a set of data

4. Fitting Criteria The set of points in the sample Fitting criteria - what are they: LAD Least squares and so on Why least squares? (We do not call it ‘ordinary’ at this point.) A fundamental result: Sample moments are “good” estimators of their population counterparts We will spend the next few weeks using this principle and applying it to least squares computation.

5. An Analogy Principle In the population E[y | X ] = X? so E[y - X? |X] = 0 Continuing E[xi ?i] = 0 Summing, Si E[xi ?i] = Si 0 = 0 Exchange Si and E E[Si xi ?i] = E[ X?? ] = 0 E[ X? (y - X?) ] = 0 Choose b, the estimator of ? to mimic this population result: i.e., mimic the population mean with the sample mean Find b such that As we will see, the solution is the least squares coefficient vector.

6. Population and Sample Moments We showed that E[?i|xi] = 0 and Cov[xi,?i] = 0. If it is, and if E[y|X] = X?, then ? = (Var[xi])-1 Cov[xi,yi]. This will provide a population analog to the statistics we compute with the data.

8. Least Squares Example will be, yi = Gi on xi = [a constant, PGi and Yi] = [1,Pgi,Yi] Fitting criterion: Fitted equation will be yi = b1xi1 + b2xi2 + ... + bKxiK. Criterion is based on residuals: ei = yi - b1xi1 + b2xi2 + ... + bKxiK Make ei as small as possible. Form a criterion and minimize it.

9. Fitting Criteria Sum of residuals: Sum of squares: Sum of absolute values of residuals: Absolute value of sum of residuals We focus on now and later

10. Least Squares Algebra

11. Least Squares Normal Equations

12. Least Squares Solution

13. Second Order Conditions

14. Does b Minimize e’e?

15. Sample Moments - Algebra

16. Positive Definite Matrix

17. Algebraic Results - 1

18. Residuals vs. Disturbances

19. Algebraic Results - 2 A “residual maker” M = (I - X(X’X)-1X’) e = y - Xb= y - X(X’X)-1X’y = My MX = 0 (This result is fundamental!) How do we interpret this result in terms of residuals? (Therefore) My = MXb + Me = Me = e (You should be able to prove this. y = Py + My, P = X(X’X)-1X’ = (I - M). PM = MP = 0. Py is the projection of y into the column space of X.

20. The M Matrix M = I- X(X’X)-1X’ is an nxn matrix M is symmetric – M = M’ M is idempotent – M*M = M (just multiply it out) M is singular – M-1 does not exist. (We will prove this later as a side result in another derivation.)

21. Results when X Contains a Constant Term X = [1,x2,…,xK] The first column of X is a column of ones Since X’e = 0, x1’e = 0 – the residuals sum to zero.

22. Least Squares Algebra

23. Least Squares

24. Residuals

25. Least Squares Residuals

26. Least Squares Algebra-3

27. Least Squares Algebra-4

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