Intr oduc tion to E c onome tr ic s Chapte r 2 E ze quie l Ur - PowerPoint PPT Presentation

Intr oduc tion to E c onome tr ic s Chapte r 2 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

2 T he simple r e gr e ssion mode l: e stimation and pr ope r tie s 2.1 Some de finitions in the simple r e gr e ssion mode l 2.2 Obtaining the Or dinar y L e ast Squar e s E stimate s 2.3 Some c har ac te r istic s of O LS e stimator s 2.4 Units of me asur e me nt and func tional for m 2.5 Assumptions and statistic al pr ope r tie s of O LS E xe r c ise s Anne x 2.1 Case study: E nge l c ur ve for de mand of dair y pr oduc ts Appe ndixe s

2.1 Some de finitions in the simple r e gr e ssion mode l y y x    2  2 The simple regression model  1    i             x x am. . F E 2.1. T he population F E 2.2. T he sc atte r diagr IGUR IGUR r e gr e ssion func tion. (PR F ) [3]

2.1 Some de finitions in the simple r e gr e ssion mode l y y   μ y x 2 The simple regression model  y ˆ i  2  x  ˆ    i  1    2  ˆ   i 1  y ˆ i   y i    y  i u  ˆ i   u  i  y  ˆ i     μ yi       x i x x i x F E 2.3. T he population F E 2.4. T he sample r e gr e ssion IGUR IGUR r e gr e ssion func tion and the sc atte r func tion and the sc atte r diagr am. diagr am. [4]

2.2 Obtaining the Or dinar y L e ast Squar e s E stimate s y 2 The simple regression model x x x 1 x 2 x 3 x F E 2.5. T he pr oble ms of c r ite r ion 1. IGUR [5]

2.2 Obtaining the Or dinar y L e ast Squar e s E stimate s E XAMPL E 2.1 E stimation of the c onsumption func tion      cons inc u i 1 2 2 The simple regression model T E 2.1. Data and c alc ulations to e stimate the c onsumption func tion. ABL  cons cons ( ) i cons     cons inc cons cons inc inc 2 inc inc  2 ( ) inc inc Observ. i i i i i i i i  inc inc ( ) i 1 5 6 30 36 -4 -5 20 25 2 7 9 63 81 -2 -2 4 4 3 8 10 80 100 -1 -1 1 1 4 10 12 120 144 1 1 1 1 5 11 13 143 169 2 2 4 4 6 13 16 208 256 4 5 20 25 Sums 54 66 644 786 0 0 50 60    54 66 644 9 66        ˆ cons inc 9 11 (2-17) : 0.83   2 6 6 786 11 66    50          ˆ ˆ (2-18) : 0.83 9 0.83 11 0.16 2 1 [6] 60

2.3 Some c har ac te r istic s of O LS e stimator s aic implic ations and c alc ulating R 2 in the E XAMPL E 2.2 F ulfilling alge br c onsumption func tion 41.67             TSS ESS RSS R 2 0.992 42 o r, alternatively, 0.33 2 The simple regression model R   2 0.992 42 T E 2.2. Data and c alc ulations to e stimate the c onsumption func tion. ABL  2    2   cons cons 2  cons Observ. u u inc ( ) - ˆ i ´ cons 2 cons cons cons ˆ i cons u ( ) ˆ i i i i i i i i 1 4.83 0.17 1 0.81 25 16 23.36 17.36 2 7.33 -0.33 -3 -2.44 49 4 53.78 2.78 3 8.17 -0.17 -1.67 -1.36 64 1 66.69 0.69 4 9.83 0.17 2 1.64 100 1 96.69 0.69 5 10.67 0.33 4.33 3.56 121 4 113.78 2.78 6 13.17 -0.17 -2.67 -2.19 169 16 173.36 17.36 54 0 0 0 528 42 527.67 41.67 [7]

2.3 Some c har ac te r istic s of O LS e stimator s y     2 The simple regression model            x F E 2.6. A r e gr e ssion thr ough the or igin. IGUR [8]

2.4 Units of me asur e me nt and func tional for m E XAMPL E 2.3  = + ´ cons inc (2-39) : 0.2 0.85 i i 2 The simple regression model   ince inc 1000     cons ince 0.2 0.00085 i i E XAMPL E 2.4   conse cons 1000     conse inc 200 850 i i [9]

2.4 Units of me asur e me nt and func tional for m E XAMPL E 2.5    inc incd inc inc 20 i i 2 The simple regression model           cons inc incd (0.2 0.85 20) 0.85 ( 20) 17.2 0.85 i i i E XAMPL E 2.6    cons consd cons cons 15 i i       cons inc 15 0.2 15 0.85 i i      consd inc 14.8 0.85 i i [10]

2.4 Units of me asur e me nt and func tional for m T E 2.3. E xample s of pr opor tional c hange and c hange in logar ithms. ABL x 1 202 210 220 240 300 2 The simple regression model x 0 200 200 200 200 200 Proporti onal c hange i n % 1% 5,0% 10,0% 20,0% 50,0% Change i n l ogari thms i n % 1% 4,9% 9,5% 18,2% 40,5% [11]

2.4 Units of me asur e me nt and func tional for m E XAMPL E 2.7 Quantity sold of c offe e as a func tion of its pr ic e . L ine ar mode l (file c offe e 1)      coffqty coffpric u 1 2      - coffqty coffpric R n 2 693.33 0.95 2 The simple regression model T E 2.4. Data on quantitie s and pr ic e s of c offe e . ABL week coffpric coffqty 1 1.00 89 2 1.00 86 3 1.00 74 4 1.00 79 5 1.00 68 6 1.00 84 7 0.95 139 8 0.95 122 9 0.95 102 10 0.85 186 11 0.85 179 [12] 12 0.85 187

2.4 Units of me asur e me nt and func tional for m E XAMPL E 2.8 E xplaining mar ke t c apitalization of Spanish banks. L ine ar mode l (file bolmad95)   + marktval bookval 29.42 1.219 2 The simple regression model = = R n 2 0.836 20 E XAMPL E 2.9 Quantity sold of c offe e as a func tion of its pr ic e . L og- log mode l (Continuation e xample 2.7) (file c offe e 1)    - coffqty coffpric ln( ) 5.132ln( )    R 2 n 0.90 [13]

2.4 Units of me asur e me nt and func tional for m E XAMPL E 2.10 E xplaining mar ke t c apitalization of Spanish banks. L og-log mode l (Continuation e xample 2.8) (file bolmad95)   + marktval bookval ln( ) 0.6756 0.938ln( ) = = R n 2 0.928 20 2 The simple regression model T E 2.5. Inte r pr e tation of in diffe r e nt mode ls.. ABL If x increases by then y will increase by Model  ˆ linear 1 unit units 2  ˆ ( /100) linear-log 1% units 2  ˆ log-linear 1 unit (100 )% 2  ˆ % log-log 1% 2 [14]

2.5 Assumptions and statistic al pr ope r tie s of O LS 2 The simple regression model F( u ) F( u ) y y µ y µ y x 1 x 1       y  y  i  i  1 x 1 x 2 2 i i x 2 x 2 x i x i x x a) b) E 2. 7. Random disturbances: F IGUR a) homoscedastic; b) heteroskedastic. [15]

2.5 Assumptions and statistic al pr ope r tie s of O LS ( ) 2 The simple regression model f b 2 ˆ ( )  f b 2 ( ) b 2 ( )     b 2 ˆ b = b ˆ b 2 1 E b 2 b 2 2 b 2 b 2 1 ˆ E b 2 2 ˆ ( ) ( ) ( ) ( ) 2 2 F E 2.8. Unbiase d e stimator . F E 2.9. Biase d e stimator . IGUR IGUR [16]

2.5 Assumptions and statistic al pr ope r tie s of O LS 2 The simple regression model ( ) ( )  f b 2 ˆ f b 2 b 2 b 2    b 2 3 ˆ b 2 4 ˆ b 2 ˆ b 2 4 b 2 3 b 2 ( ) ( ) ( ) ( ) F E 2.10. E stimator with small F E 2.11. E stimator with big IGUR IGUR var ianc e . var ianc e . [17]

2.5 Assumptions and statistic al pr ope r tie s of O LS 2 The simple regression model E stimator E stimator E stimator E stimator E stimator E stimator L inear L inear L inear L inear L inear L inear L inear U nbiased U nbiased U nbiased U nbiased U nbiased U nbiased U nbiased U nbiased U nbiased U nbiased U nbiased U nbiased U nbiased U nbiased U nbiased U nbiased U nbiased U nbiased U nbiased U nbiased the B est                             ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ   ˆ ˆ , BLUE 1 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 F E 2.12. T he O LS e stimator is the BL UE . IGUR [18]

2.5 Assumptions and statistic al pr ope r tie s of O LS 2 The simple regression model E stimator U nbiased M inimum V ariance   ˆ ˆ , 1 2 MVUE F E 2.13. T he O LS e stimator is the MVUE . IGUR [19]