4-11 Hedonic house price results Dependent variable: Canadian Dollars per month Variable Coefficient t-ratio a priori sign expected Intercept 282.21 56.09 + Lnage -53.10 59.71 NBROOMS 48.47 104.81 AREABYRM 3.97 2999 ELEVATOR 4504 BASEMENT 15.90 -11.32 OUTPARK 7.17 707 +++-++ INDPARK 73.76 31.25 NOLEASE 16.99 -762 LndIStCBD 584 SINGLPAR -4.27 38.88 DSHOPCNTR 10.04 -5.97 VACDIFFI 0.29 598 Notes: Adjusted R=0.651; regression F-statistic =2082.27. Source: Des Rosiers and Therialt (1996). Reprinted with permission of the American Real Estate Society
4-11 Hedonic House Price Results Dependent Variable: Canadian Dollars per Month Variable Coefficient t-ratio A priori sign expected Intercept 282.21 56.09 + LnAGE -53.10 -59.71 - NBROOMS 48.47 104.81 + AREABYRM 3.97 29.99 + ELEVATOR 88.51 45.04 + BASEMENT -15.90 -11.32 - OUTPARK 7.17 7.07 + INDPARK 73.76 31.25 + NOLEASE -16.99 -7.62 - LnDISTCBD 5.84 4.60 - SINGLPAR -4.27 -38.88 - DSHOPCNTR -10.04 -5.97 - VACDIFF1 0.29 5.98 - Notes: Adjusted R 2 = 0.65l; regression F-statistic = 2082.27. Source: Des Rosiers and Thérialt (1996). Reprinted with permission of the American Real Estate Society
4-12 3 Tests of Non-nested Hypotheses All of the hypothesis tests concluded thus far have been in the context of“ nested”, models But what if we wanted to compare between the following models? Model 1: y X +u 1+221 Model 2: y,=B,+B, xX2+1 3t We could use R2 or adjusted R, but what if the number of explanatory variables were different across the 2 models? ·还有一些其他的“信息准则” ·但很多情况下,不同的准则可能导致选取不同的模型。 An alternative approach is an encompassing test, based on examination of the hybrid model: Model 3:y,=n,+r,x,,+r3xa,+
4-12 3 Tests of Non-nested Hypotheses • All of the hypothesis tests concluded thus far have been in the context of “nested” models. • But what if we wanted to compare between the following models? • We could use R2 or adjusted R2 , but what if the number of explanatory variables were different across the 2 models? • 还有一些其他的“信息准则”。 • 但很多情况下，不同的准则可能导致选取不同的模型。 • An alternative approach is an encompassing test, based on examination of the hybrid model: t t t t t t y x v y x u = + + = + + 1 2 3 1 2 2 Model 2 : Model1: t t t wt Model 3: y = 1 + 2 x2 + 3 x3 +
4-13 Tests of Non-nested Hypotheses There are 4 possible outcomes when model 3 is estimated: n is significant but y is not, model 1 y3 is significant but n is not, model 2 n and y3 are both statistically significant, Model 3 Neither n nor ya are significant, none; other method Problems with encompassing approach Hybrid model may be meaningless Possible high correlation between x, and x3
4-13 Tests of Non-nested Hypotheses • There are 4 possible outcomes when Model 3 is estimated: – 2 is significant but 3 is not，model 1 – 3 is significant but 2 is not, model 2 – 2 and 3 are both statistically significant, Model 3 – Neither 2 nor 3 are significant, none; other method • Problems with encompassing approach – Hybrid model may be meaningless – Possible high correlation between x2 and x3
4 Violation of the assumptions of the *14 CLRM Recall that we assumed of the clrm disturbance terms: l.E(u)=0 2.Ⅴar(u)=a2<0 3. Cov(u, M, )=0 4. The X matrix is non-stochastic or fixed in repeated samples 5.l2~N(0,a3)
4-14 4 Violation of the Assumptions of the CLRM • Recall that we assumed of the CLRM disturbance terms: 1. E(ut ) = 0 2. Var(ut ) = 2 < 3. Cov (ui ,uj ) = 0 4. The X matrix is non-stochastic or fixed in repeated samples 5. ut N(0, 2 )
4-15 Investigating Violations of the Assumptions of the clrm We will now study these assumptions further How we test for violations Causes Consequences in general we could encounter any combination of 3 problems: the coefficient estimates are wrong the associated standard errors are wrong the distribution that we assumed for the test statistics will be inappropriate Solutions the assumptions are no longer violated we work around the problem so that we use alternative techniques which are still valid
4-15 Investigating Violations of the Assumptions of the CLRM • We will now study these assumptions further : - How we test for violations - Causes - Consequences in general we could encounter any combination of 3 problems: - the coefficient estimates are wrong - the associated standard errors are wrong - the distribution that we assumed for the test statistics will be inappropriate - Solutions - the assumptions are no longer violated - we work around the problem so that we use alternative techniques which are still valid