Intr oduc tion to E c onome tr ic s Chapte r 4 E ze quie l Ur - PowerPoint PPT Presentation

Intr oduc tion to E c onome tr ic s Chapte r 4 E ze quie l Ur ie l Jimé ne z Unive r sity of Vale nc ia Vale nc ia, Se pte mbe r 2013

4 Hypothe sis te sting in the multiple r e gr e ssion mode l 4.1 Hypothe sis te sting: an ove r vie w e sting hypothe se s using the t te st 4.2 T ic tions using the F te st 4.3 T e sting multiple line ar r e str 4.4 T e sting without nor mality 4.5 Pr e dic tion E xe r c ise s

4 Hypothe sis te sting in the multiple r e gr e ssion mode l Motivation T e sting hypo the sis c an answe r the fo llo wing que stio ns: 1. I s the marg inal pro pe nsity to c o nsume smalle r than the ave rag e pro pe nsity to c o nsume ? 2. Has inc o me a ne g ative influe nc e o n infant mo rtality? 3. Do e s the rate o f c rime in an are a plays a ro le in the pric e s o f ho use s in that are a? 4. I s the e lastic ity e xpe nditure in fruit/ inc o me e qual to 1? I s fruit a luxury g o o d? 5. I s the Madrid sto c k e xc hang e marke t e ffic ie nt? 6. I s the rate o f re turn o f the Madrid Sto c k E xc hang e affe c te d by the rate o f re turn o f the T o kyo Sto c k E xc hang e ? 7. Are the re c o nstant re turns to sc ale in the c he mic al industry? 8. Adve rtising o rinc e ntive s? 9. I s the assumptio n o f ho mo g e ne ity admissible in the de mand fo r fish? 10. Have te nure and ag e jo intly a sig nific ant influe nc e o n wag e ? 11. I s the pe rfo rmanc e o f a c o mpany c ruc ial to se t the salarie s o f CE Os? All the se que stio ns are answe re d in this c hapte r [3]

4.1 Hypothe sis te sting: an ove r vie w 4 Hypothesis testing in the multiple regression T ABL E 4.1. Some distr ibutions use d in hypothe sis te sting. 1 or more 1 restriction restrictions model  Known N Chi-square 2  2 Unknown Student’s t Snedecor’s F [4]

4.1 Hypothe sis te sting: an ove r vie w 4 Hypothesis testing in the multiple regression Non Rejection Rejection Region RR Region NRR model  c W F IGURE 4.1. Hypothe sis te sting: c lassic al appr oac h. [5]

e sting hypothe se s using the t te st 4.2 T 4 Hypothesis testing in the multiple regression model mal and t for F IGURE 4. 2. De nsity func tions: nor diffe r e nt de gr e e s of fr e e dom. [6]

e sting hypothe se s using the t te st 4.2 T 4 Hypothesis testing in the multiple regression model IGURE 4.4. p-value using t : F IGURE 4.3. Re je c tion r e gion using F t : r ight- tail alte r native hypothe sis. r ight- tail alte r native hypothe sis. [7]

e sting hypothe se s using the t te st 4.2 T E XAMPL E 4.1 Is the mar ginal pr ope nsity to c onsume smalle r than the ave r age pr ope nsity to c onsume ? 4 Hypothesis testing in the multiple regression       cons inc u = + cons inc 0.41 0.843 1 2 i i (0.350) (0.062)   H      ˆ ˆ : 0 0 0 0.41 0 1     t 1 1 1 1.171   H   ˆ ˆ : 0 se se 0.35 ( ) ( ) 1 1 1 1 model xample 4.1: p-value F IGURE 4.5. E xample 4.1: Re je c tion F IGURE 4.6. E e gion using t with a r using t with r r ight- tail ight- tail alte r native alte r native hypothe sis. hypothe sis. [8]

e sting hypothe se s using the t te st 4.2 T 4 Hypothesis testing in the multiple regression Non rejected Rejected Rejection Non t  t  for for Region Rejection n k n k α >p-value ɑ <p-value RR Region NRR p-value  model t  t   0 0 ˆ j  n k IGURE 4.8. p-value using t : le ft- tail F IGURE 4.7. Re je c tion r e gion using F t: le ft- tail alte r native hypothe sis alte r native hypothe sis. [9]

e sting hypothe se s using the t te st 4.2 T 4.2 Has income a negative influence on infant mortality? E XAMPL E 4 Hypothesis testing in the multiple regression        deathun gnipc ilitrate u 5 1 2 3  = - + deathun gnipc ilitrate 5 27.91 0.000826 2.043 i i i (5.93) (0.00028) (0.183)    ˆ  H : 0 0.000826     t 0 2 2 2.966    ˆ H se 0.00028 : 0 ( ) 1 2 2 model xample 4.2: p-value F IGURE 4.9. E xample 4.2: Re je c tion F IGURE 4.10. E e gion using t with a le ft- tail using t with a le ft- tail alte r r native alte r native hypothe sis. hypothe sis. [10]

e sting hypothe se s using the t te st 4.2 T 4 Hypothesis testing in the multiple regression model IGURE 4.12. p-value using t : two- F IGURE 4.11. Re je c tion r e gion using F t: two- tail alte r native hypothe sis. tail alte r native hypothe sis. [11]

e sting hypothe se s using the t te st 4.2 T 4.3 Has the rate of crime play a role in the price of houses in an area? E XAMPL E          price rooms lowstat crime u 1 2 3 4  = - + - - price rooms lowstat crime 4 Hypothesis testing in the multiple 15694 6788 268.2 3854 i i i i (8022) (1210) (80.7) (960) T E 4.2. Standar d output in the r e gr e ssion e xplaining house pr ic e . n=55. ABL regression model Variable Coefficient Std. Error t-Statistic Prob. C -15693.61 8021.989 -1.956324 0.0559 rooms 6788.401 1210.72 5.60691 0.0000 lowstat -268.1636 80.70678 -3.32269 0.0017 crime -3853.564 959.5618 -4.015962 0.0002  ˆ    H 3854 : 0     t 4 0 4 4.016  ˆ   se H 960 ( ) : 0 4 1 4 [12]

e sting hypothe se s using the t te st 4.2 T 4.3 Has the rate of crime play a role in the price of houses in an area? E XAMPL E (Continuation) 4 Hypothesis testing in the multiple regression model xample 4.3: p-value using t with a two- tail alte r F IGURE 4.13. E native hypothe sis. [13]

e sting hypothe se s using the t te st 4.2 T E XAMPL E 4.4 Is the e lastic ity e xpe nditur e in fr uit/ inc ome e qual to 1? Is fr uit a luxur y good? 4 Hypothesis testing in the multiple regression          fruit inc househsize punders u ln( ) ln( ) 1 2 3 4  = - + - - fruit inc househsize punder ln( ) 9.768 2.005ln( ) 1.205 0.018 5 i i i i (3.701) (0.512) (0.179) (0.013) uit. n =40. T E 4.3. Standar d output in a r e gr e ssion e xplaining e xpe nditur e in fr ABL model Variable Coefficient Std. Error t-Statistic Prob. C -9.767654 3.701469 -2.638859 0.0122 ln(inc) 2.004539 0.51237 3.912286 0.0004 househsize -1.205348 0.178646 -6.747147 0.0000 punder5 -0.017946 0.013022 -1.378128 0.1767   H : 1 0 2       ˆ ˆ 0 1   2.005 1 H : 1     t 2 2 2 1.961 1 2   ˆ ˆ se se 0.512 ( ) ( ) 2 2   H : 1 [14] 1 2

e sting hypothe se s using the t te st 4.2 T E XAMPL E 4.5 Is the Madr id stoc k e xc hange mar ke t e ffic ie nt? D + + P D A  RA t t t Rate o f to tal re turn: t P 4 Hypothesis testing in the multiple regression t - 1 Rate o f re turn due to inc re ase in quo tatio n D P  D RA P 2 ln  Pro po rtio nal c hang e : t Chang e in lo g arithms: RA 1 t t t P - t 1    + rmad rmad 92 0.0004 0.1267 92      rmad rmad u 92 92 - t t 1  t t t (0.0007) (0.0629) 1 2 1 = = R n 2 0.0163 247   H : 1 model 0 2  ˆ 0.1267      t H 2 : 1 2.02 1 2  ˆ se 0.0629 ( ) 2 E XAMPL E 4.6 Is the r ate of r e tur n of the Madr id Stoc k E xc hange affe c te d by the r ate of r e tur n of the T okyo Stoc k E xc hange ?    + rmad rtok 92 0.0005 0.1244 92 t t      rmad rtok u (0.0007) (0.0375) 92 92 t t t 1 2 = = R n 2 0.0452 235   H : 1 0 2  ˆ 0.1244      H t 2 : 1 3.32 1 2  ˆ [15] se 0.0375 ( ) 2

ope nsity to c onsume in e sting hypothe se s using the t te st 1.011 0.968 0.947 ginal pr e xample 4.1. mar 0,99 0,95 0,90 0.843 vals for IGURE 4.14. Confide nc e inte r 0.739 0.718 4.2 T 0.675 F model [16] 4 Hypothesis testing in the multiple regression

e sting hypothe se s using the t te st 4.2 T E XAMPL E 4.7 Ar e the r e c onstant r e tur ns to sc ale in the c he mic al industr y?        output labor capital u ln( ) ln( ) ln( ) 1 2 3 4 Hypothesis testing in the multiple  = + + output labor capital ln( ) 1.170 0.603ln( ) 0.376ln( ) i i i (0.327) (0.126) (0.085) regression model T E 4.4. Standar d output of the e stimation of the pr oduc tion func tion: ABL mode l (4- 20). Variable Coefficient Std. Error t-Statistic Prob. constant 1.170644 0.326782 3.582339 0.0015 ln( labor ) 0.602999 0.125954 4.787457 0.0001 ln( capital ) 0.37571 0.085346 4.402204 0.0002     H : 1 0 2 3     H : 1 1 2 3 [17]