1 / 33

Econometrics I

Econometrics I. Professor William Greene Stern School of Business Department of Economics. Gauss-Markov Theorem. A theorem of Gauss and Markov: Least Squares is the minimum variance linear unbiased estimator (MVLUE) 1. Linear estimator 2. Unbiased: E[ b | X ] = β

moanna
Download Presentation

Econometrics I

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Econometrics I Professor William Greene Stern School of Business Department of Economics

  2. Gauss-Markov Theorem A theorem of Gauss and Markov: Least Squares is the minimum variance linear unbiased estimator (MVLUE) 1. Linear estimator 2. Unbiased: E[b|X] = β Theorem: Var[b*|X] – Var[b|X] is nonnegative definite for any other linear and unbiased estimator b* that is not equal to b. Definition: b is efficientin this class of estimators.

  3. Implications of Gauss-Markov • Theorem: Var[b*|X] – Var[b|X] is nonnegative definite for any other linear and unbiased estimator b* that is not equal to b. Implies: • bk = the kth particular element of b.Var[bk|X] = the kth diagonal element of Var[b|X]Var[bk|X] < Var[bk*|X] for each coefficient. • cb = any linear combination of the elements of b.Var[cb|X] <Var[cb*|X] for any nonzero c and b* that is not equal to b.

  4. Summary: Finite Sample Properties of b • Unbiased: E[b]= • Variance: Var[b|X] = 2(XX)-1 • Efficiency: Gauss-Markov Theorem with all implications • Distribution: Under normality, b|X ~ N[, 2(XX)-1 (Without normality, the distribution is generally unknown.)

  5. Comparação de modelos • Podemos comparar modelos sem usar testes estatsticos, mas sim medidas conhecidas como criterios de informação que traduzem a qualidade de ajustamento de um modelo. • Para estes indicadores a variável principal acaba por ser uma medida do valor absoluto dos erros.

  6. Medida de Ajuste R2 = bXM0Xb/yM0y (Very Important Result.) R2 is bounded by zero and one only if: (a) There is a constant term in X and (b) The line is computed by linear least squares.

  7. Comparing fits of regressions Make sure the denominator in R2 is the same - i.e., same left hand side variable. Example, linear vs. loglinear. Loglinear will almost always appear to fit better because taking logs reduces variation.

  8. Adjusted R Squared • Adjusted R2 (for degrees of freedom) = 1 - [(n-1)/(n-K)](1 - R2) • includes a penalty for variables that don’t add much fit. Can fall when a variable is added to the equation.

  9. Adjusted R2 What is being adjusted? The penalty for using up degrees of freedom. = 1 - [ee/(n – K)]/[yM0y/(n-1)] uses the ratio of two ‘unbiased’ estimators. Is the ratio unbiased? = 1 – [(n-1)/(n-K)(1 – R2)] Will rise when a variable is added to the regression? is higher with z than without z if and only if the t ratio on z is in the regression when it is added is larger than one in absolute value.

  10. Full Regression (Without PD) ---------------------------------------------------------------------- Ordinary least squares regression ............ LHS=G Mean = 226.09444 Standard deviation = 50.59182 Number of observs. = 36 Model size Parameters = 9 Degrees of freedom = 27 Residuals Sum of squares = 596.68995 Standard error of e = 4.70102 Fit R-squared = .99334 <********** Adjusted R-squared = .99137 <********** Info criter. LogAmemiya Prd. Crt. = 3.31870 <********** Akaike Info. Criter. = 3.30788 <********** Model test F[ 8, 27] (prob) = 503.3(.0000) --------+------------------------------------------------------------- Variable| Coefficient Standard Error t-ratio P[|T|>t] Mean of X --------+------------------------------------------------------------- Constant| -8220.38** 3629.309 -2.265 .0317 PG| -26.8313*** 5.76403 -4.655 .0001 2.31661 Y| .02214*** .00711 3.116 .0043 9232.86 PNC| 36.2027 21.54563 1.680 .1044 1.67078 PUC| -6.23235 5.01098 -1.244 .2243 2.34364 PPT| 9.35681 8.94549 1.046 .3048 2.74486 PN| 53.5879* 30.61384 1.750 .0914 2.08511 PS| -65.4897*** 23.58819 -2.776 .0099 2.36898 YEAR| 4.18510** 1.87283 2.235 .0339 1977.50 --------+-------------------------------------------------------------

  11. PD added to the model. R2 rises, Adj. R2 falls ---------------------------------------------------------------------- Ordinary least squares regression ............ LHS=G Mean = 226.09444 Standard deviation = 50.59182 Number of observs. = 36 Model size Parameters = 10 Degrees of freedom = 26 Residuals Sum of squares = 594.54206 Standard error of e = 4.78195 Fit R-squared = .99336 Was 0.99334 Adjusted R-squared = .99107 Was 0.99137 --------+------------------------------------------------------------- Variable| Coefficient Standard Error t-ratio P[|T|>t] Mean of X --------+------------------------------------------------------------- Constant| -7916.51** 3822.602 -2.071 .0484 PG| -26.8077*** 5.86376 -4.572 .0001 2.31661 Y| .02231*** .00725 3.077 .0049 9232.86 PNC| 30.0618 29.69543 1.012 .3207 1.67078 PUC| -7.44699 6.45668 -1.153 .2592 2.34364 PPT| 9.05542 9.15246 .989 .3316 2.74486 PD| 11.8023 38.50913 .306 .7617 1.65056 (NOTE LOW t ratio) PN| 47.3306 37.23680 1.271 .2150 2.08511 PS| -60.6202** 28.77798 -2.106 .0450 2.36898 YEAR| 4.02861* 1.97231 2.043 .0514 1977.50 --------+-------------------------------------------------------------

  12. Outrasmedidas de Ajuste Para alternativasnãoaninhadas Incluipenalidade de grau de liberdade. Critério de Informação • Schwarz (BIC): n log(ee) + k(log(n)) • Akaike (AIC): n log(ee) + 2k Quando se quer decidir entre dois modelos não aninhados, o melhor é o que produz o menor valor do critério. A penalidade no BIC de incluir algo não relevante é maior que no AIC.

  13. Outrasmedidas de Ajuste • Tanto o AIC quanto o BIC aumentam conforme SQR aumenta. • Além disso, ambos critérios penalizam modelos com muitas variáveis sendo que valores menores de AIC e BIC são preferíveis. • Como modelos com mais variáveis tendem a produzir menor SQR mas usam mais parâmetros, a melhor escolha é balancear o ajuste com a quantidade de variáveis. • A penalidade no BIC de incluir algo não relevante é maior que no AIC.

  14. Multicolinearidade

  15. Formas funcionais

  16. Specification and Functional Form:Nonlinearity

  17. Log Income Equation ---------------------------------------------------------------------- Ordinary least squares regression ............ LHS=LOGY Mean = -1.15746 Estimated Cov[b1,b2] Standard deviation = .49149 Number of observs. = 27322 Model size Parameters = 7 Degrees of freedom = 27315 Residuals Sum of squares = 5462.03686 Standard error of e = .44717 Fit R-squared = .17237 --------+------------------------------------------------------------- Variable| Coefficient Standard Error b/St.Er. P[|Z|>z] Mean of X --------+------------------------------------------------------------- AGE| .06225*** .00213 29.189 .0000 43.5272 AGESQ| -.00074*** .242482D-04 -30.576 .0000 2022.99 Constant| -3.19130*** .04567 -69.884 .0000 MARRIED| .32153*** .00703 45.767 .0000 .75869 HHKIDS| -.11134*** .00655 -17.002 .0000 .40272 FEMALE| -.00491 .00552 -.889 .3739 .47881 EDUC| .05542*** .00120 46.050 .0000 11.3202 --------+------------------------------------------------------------- Average Age = 43.5272. Estimated Partial effect = .066225 – 2(.00074)43.5272 = .00018. Estimated Variance 4.54799e-6 + 4(43.5272)2(5.87973e-10) + 4(43.5272)(-5.1285e-8) = 7.4755086e-08. Estimated standard error = .00027341.

  18. Specification and Functional Form:Interaction Effect

  19. Interaction Effect ---------------------------------------------------------------------- Ordinary least squares regression ............ LHS=LOGY Mean = -1.15746 Standard deviation = .49149 Number of observs. = 27322 Model size Parameters = 4 Degrees of freedom = 27318 Residuals Sum of squares = 6540.45988 Standard error of e = .48931 Fit R-squared = .00896 Adjusted R-squared = .00885 Model test F[ 3, 27318] (prob) = 82.4(.0000) --------+------------------------------------------------------------- Variable| Coefficient Standard Error b/St.Er. P[|Z|>z] Mean of X --------+------------------------------------------------------------- Constant| -1.22592*** .01605 -76.376 .0000 AGE| .00227*** .00036 6.240 .0000 43.5272 FEMALE| .21239*** .02363 8.987 .0000 .47881 AGE_FEM| -.00620*** .00052 -11.819 .0000 21.2960 --------+------------------------------------------------------------- Do women earn more than men (in this sample?) The +.21239 coefficient on FEMALE would suggest so. But, the female “difference” is +.21239 - .00620*Age. At average Age, the effect is .21239 - .00620(43.5272) = -.05748.

  20. Quebra estrutural

  21. Linear Restrictions Context: How do linear restrictions affect the properties of the least squares estimator? Model: y = X +  Theory (information) R - q = 0 Restricted least squares estimator: b* = b - (XX)-1R[R(XX)-1R]-1(Rb - q) Expected value: E[b*] =  - (XX)-1R[R(XX)-1R]-1(Rb - q) Variance:2(XX)-1 - 2 (XX)-1R[R(XX)-1R]-1R(XX)-1 = Var[b] – a nonnegative definite matrix < Var[b] Implication: (As before) nonsample information reduces the variance of the estimator.

  22. Interpretation Case 1: Theory is correct: R - q = 0 (the restrictions do hold). b* is unbiased Var[b*] is smaller than Var[b] How do we know this? Case 2: Theory is incorrect: R - q  0 (the restrictions do not hold). b* is biased – what does this mean? Var[b*] is still smaller than Var[b]

  23. Linear Least Squares Subject to Restrictions Restrictions: Theory imposes certain restrictions on parameters. Some common applications  Dropping variables from the equation = certain coefficients in b forced to equal 0. (Probably the most common testing situation. “Is a certain variable significant?”)  Adding up conditions: Sums of certain coefficients must equal fixed values. Adding up conditions in demand systems. Constant returns to scale in production functions.  Equality restrictions: Certain coefficients must equal other coefficients. Using real vs. nominal variables in equations. General formulation for linear restrictions: Minimize the sum of squares, ee, subject to the linear constraintRb = q.

  24. Restricted Least Squares

  25. Restricted Least Squares Solution • General Approach: Programming ProblemMinimize for L = (y - X)(y - X) subject to R = qEach row of R is the K coefficients in a restriction.There are J restrictions: J rows • 3 = 0: R = [0,0,1,0,…] q = (0). • 2 =3: R = [0,1,-1,0,…] q = (0) • 2= 0,3 = 0: R = 0,1,0,0,… q = 0 0,0,1,0,… 0

  26. Solution Strategy • Quadratic program: Minimize quadratic criterion subject to linear restrictions • All restrictions are binding • Solve using Lagrangean formulation • Minimize over (,) L* = (y - X)(y - X) + 2(R-q)(The 2 is for convenience – see below.)

  27. Restricted LS Solution

  28. Restricted Least Squares

  29. Aspects of Restricted LS 1. b* = b - Cm where m = the “discrepancy vector” Rb - q. Note what happens if m = 0. What does m = 0 mean? 2. =[R(XX)-1R]-1(Rb - q) = [R(XX)-1R]-1m. When does  = 0. What does this mean? 3. Combining results: b* = b - (XX)-1R. How could b* = b?

More Related