Lecture 3 Specification

Transcription

1 Lecture 3 Specfcaton 1 OLS Estmaton - Assumptons CLM Assumptons (A1) DGP: y = X + s correctly specfed. (A) E[ X] = 0 (A3) Var[ X] = σ I T (A4) X has full column rank rank(x)=k-, where T k. Q: What happens when (A1) s not correctly specfed? In ths lecture, we look at (A1), always n the contet of lnearty. Are we omttng a relevant regressor? Are we ncludng an rrelevant varable? What happens when we mpose restrctons n the DGP? 1

2 Specfcaton Errors: Omtted Varables Omttng relevant varables: Suppose the correct model s y = X X + -.e., wth two sets of varables. But, we compute OLS omttng X. That s, y = X <= the short regresson. Some easly proved results: (1) E[b 1 X] = E [(X 1 X 1 ) -1 X 1 y]= 1 + (X 1 X 1 ) -1 X 1 X 1. So, unless X 1 X =0, b 1 s based. The bas can be huge. It can reverse the sgn of a prce coeffcent n a demand equaton. () Var[b 1 X] Var[b 1. X]. (The latter s the northwest submatr of the full covarance matr.) The proof uses M, the resdual maker. We get a smaller varance when we omt X. Specfcaton Errors: Omtted Varables We get a smaller varance when we omt X. Interpretaton: Omttng X amounts to usng etra nformaton -- = 0. Even f the nformaton s wrong, t reduces the varance. (3) MSE b 1 may be more precse. Precson = Mean squared error = varance + squared bas. Smaller varance but postve bas. If bas s small, may stll favor the short regresson. Note: Suppose X 1 X = 0. Then the bas goes away. Interpretaton, the nformaton s not rght, t s rrelevant. b 1 s the same as b 1..

3 Omtted Varables Eample: Gasolne Demand We have a lnear model for the demand for gasolne: G = PG 1 + Y +, Q: What happens when you wrongly eclude Income (Y)? E[b 1 X] = 1 + Cov[ Prce, Income] Var[ Prce] In tme seres data, 1 < 0, > 0 (usually) Cov[Prce,Income] > 0 n tme seres data. => The short regresson wll overestmate the prce coeffcent. In a smple regresson of G (demand) on a constant and PG, the Prce Coeffcent ( 1 ) should be negatve. Estmaton of a Demand Equaton (Greene): Shouldn t the Prce Coeffcent be Negatve? 3

4 Estmaton of a Demand Equaton (Greene): Multple Regresson - Theory Works. Ordnary least squares regresson... LHS=G Mean = Standard devaton = Number of observs. = 36 Model sze Parameters = 3 Degrees of freedom = 33 Resduals Sum of squares = Standard error of e = Ft R-squared = Adjusted R-squared =.9856 Model test F[, 33] (prob) = 987.1(.0000) Varable Coeffcent Standard Error t-rato P[ T >t] Constant *** Y.0369*** PG *** Note: Income s helpng us to dentfy a demand equaton.e., wth a negatve slope for the prce varable. Specfcaton Errors: Irrelevant Varables Irrelevant varables. Suppose the correct model s y = X e., wth one set of varables. But, we estmate y = X X + <= the long regresson. Some easly proved results: Includng rrelevant varables just reverse the results: It ncreases varance -the cost of not usng nformaton-; but does not create bases. => Snce the varables n X are truly rrelevant, then = 0, so E[b 1. X] = 1. 4

5 Specfcaton Errors: Irrelevant Varables A smple eample Suppose the correct model s: y = 1 + X + But, we estmate: y = 1 + X + 3 X 3 + Unbased: gven that 3 =0 =>E[b X]= Effcency: b 1 X 1 r X X, X X X 3 Note: These are the results n general. Note that f X and X 3 are uncorrelated, there wll be no loss of effcency after all. 9 Other Models Lookng ahead to nonlnear models: nether of the precedng results etend beyond the lnear regresson model. Omttng relevant varables from a model s always costly. (No eceptons.) The bengn result above almost never carres over to more nvolved nonlnear models. (Greene) 5

6 Specfcaton and Functonal Form: Non-lnearty In the contet of OLS estmaton, we can ntroduce some nonlneartes: quadratc, cubc and nteracton effects can be easly estmated by OLS. For eample: y = 1 + X + 3 X + 4 X X 3 + Partal effects, y/x, (and standard errors) can be dfferent. In the above model y/x = + 3 X + 4 X 3 Note: Recall that n a smple lnear model: y = 1 + X + 3 X 3 + the partal effect s equal to the coeffcent: y/x =. Specfcaton and Functonal Form: Non-lnearty The estmator of partal effects and ther varances are dfferent from b and Var[b X] n the presence of non-lneartes Eample: Quadratc Effect Populaton Estmators y 1 3 4z yˆ b1 b b3 b4z E[ y, z] ˆ 3 b b3 Estmator of the varance of ˆ ˆ r[ b3] 4 Cov[ b, b3] EstVar. [ ] Var[ b ] 4 Va Note: Now, the partal effect and the varance are a functon of the data! Usually, an average s used n the estmaton. 6

7 Applcaton (Greene): Log Income Equaton Ordnary least squares regresson... LHS=LOGY Mean = Estmated Cov[b1,b] Standard devaton = Number of observs. = 73 Model sze Parameters = 7 Degrees of freedom = 7315 Resduals Sum of squares = Standard error of e = Ft R-squared = Varable Coeffcent Standard Error b/st.er. P[ Z >z] Mean of X AGE.065*** AGESQ ***.448D Constant *** MARRIED.3153*** HHKIDS *** FEMALE EDUC.0554*** At Average Age = = Estmated Partal effect =.0665 (.00074) = Estmated Varance e-6 + 4(43.57) *( e-10) + 4(43.57)* (-5.185e-8) = e-08. Estmated standard error = Specfcaton and Functonal Form: Non-lnearty Eample: Interactve Effect Populaton Estmators y z z yˆ b b b z b z E[ y, z] ˆ 4z b b4z Estm ator of the varance of ˆ ˆ r[ b4 ] zcov[ b, b4 ] EstVar. [ ] Var[ b ] z Va 7

8 Applcaton (Greene): Interacton Effect Ordnary least squares regresson... LHS=LOGY Mean = Standard devaton = Number of observs. = 73 Model sze Parameters = 4 Degrees of freedom = 7318 Resduals Sum of squares = Standard error of e = Ft R-squared = Adjusted R-squared = Model test F[ 3, 7318] (prob) = 8.4(.0000) Varable Coeffcent Standard Error b/st.er. P[ Z >z] Mean of X Constant -1.59*** AGE.007*** FEMALE.139*** AGE_FEM *** Do women earn more than men (n ths sample?) The coeffcent on FEMALE would suggest so. But, the female dfference.e., partal effect- s: *Age. At average Age, the effect s: (43.57) = OLS Subject to Restrctons Restrctons: Theory mposes certan restrctons on parameters. Eamples: (1) Droppng varables from the equaton. That s, certan coeffcents n b forced to equal 0. (Is varable 3 =sze sgnfcant? ) () Addng up condtons: Sums of certan coeffcents must equal fed values. Addng up condtons n demand systems. Constant returns to scale n producton functons (α+β=1 n a Cobb-Douglas producton functon). (3) Equalty restrctons: Certan coeffcents must equal other coeffcents. Usng real vs. nomnal varables n equatons. Common formulaton: Mn b {S(, θ) =Σ e = e e = (y- Xb) (y- Xb)} s.t. Rb = q 8

9 Restrcted Least Squares In practce, restrctons can usually be mposed by solvng them out. (1) Droppng varables.e., force a coeffcent to equal zero. n Problem: mn y b b b s. t. b 0 mn b b 1 n y b 1 1 b. () Addng up. Do least squares subject to b 1 +b +b 3 =1. Then, b 3 = 1- b 1 -b. Make the substtuton so (y- 3 ) = b 1 ( 1-3 ) + b ( - 3 ) + e. n Problem: Mn b ( y ) b ( ) b ( 1 n ) (3) Equalty. If b 3 = b, then y= b 1 1 b +b 3 +e = b 1 1 +b ( + 3 )+e Problem: Mn b y b1 1 b ( 3) 3 Restrcted Least Squares Theoretcal results provde nsghts and the foundaton of several tests. Programmng problem: Mnmze wrt b L * = (y - Xb)(y - Xb) subject to Rb = q Quadratc programmng problem => Mnmze a quadratc crteron s.t. a set of lnear restrctons. - Concave programmng problem, all bndng constrants. No need for Kuhn-Tucker - Solve usng a Lagrangean formulaton The Lagrangean approach Mn b, L* = (y - Xb)(y - Xb) + (Rb q) --the s for convenence. 9

10 Restrcted Least Squares The Lagrangean approach Mn b, L* = (y - Xb)(y - Xb) + (Rb q) f.o.c: L*/b = -X(y-Xb*) + R = 0 => -X(y-Xb*) + R = 0 L*/ = (Rb* - q) = 0. => (Rb * - q) = 0 Then, from the 1 st equaton (and assumng full rank for X): -Xy + XXb* + R = 0 => b * = (XX) -1 Xy - (XX) -1 R = b - (XX) -1 R Premultply both sdes by R and then subtract q Rb* - q = Rb - R(XX) -1 R - q 0 = - R(XX) -1 R + (Rb q) Solvng for => = [R(XX) -1 R] -1 (Rb - q) Substtutng n b* => b*= b - (XX) -1 R[R(XX) -1 R] -1 (R b q) Lnear Restrctons Q: How do lnear restrctons affect the propertes of the least squares estmator? Model ( DGP): y = X + Theory (nformaton): R - q = 0 Restrcted LS estmator: b* = b -(XX) -1 R[R(XX) -1 R] -1 (Rb - q) 1. Unbased? E[b* X] = -(XX) -1 R[R(XX) -1 R] -1 E[(Rb - q) X] =. Effcency? Var[b* X] = (XX) -1 - (XX) -1 R[R(XX) -1 R] -1 R(XX) -1 Var[b* X] =Var[b X] a nonnegatve defnte matr<var[b X] 3. b* may be more precse. Precson = Mean squared error = varance + squared bas. 10

11 Lnear Restrctons 1. b* = b - Cm, m = the dscrepancy vector Rb - q. Note: If m = 0 => b* = b. (Q: What does m = 0 mean?). =[R(XX) -1 R] -1 (Rb - q) = [R(XX) -1 R] -1 m When does = 0? What does ths mean? 3. Combnng results: b* = b -(XX) -1 R 4. Recall: ee = (y -Xb)(y-Xb) e*e* = (y Xb*)(y-Xb*) => Restrctons cannot ncrease R => R R * Lnear Restrctons - Interpretaton Two cases - Case 1: Theory s correct: R - q = 0 (restrctons hold). b* s unbased & Var[b* X] Var[b X] - Case : Theory s ncorrect: R - q 0 (restrctons do not hold). b* s based & Var[b* X] Var[b X]. Interpretaton - The theory gves us nformaton. Bad nformaton produces bas (away from the truth. ) Any nformaton, good or bad, makes us more certan of our answer. In ths contet, any nformaton reduces varance. 11

12 Lnear Restrctons - Interpretaton - What about gnorng nformaton (theory)? Not usng the correct nformaton does not produce bas. Not usng nformaton foregoes the varance reducton. 1