Appendix 2A. Simple regression and multiple regression By Cheng Few Lee Joseph Finnerty John Lee Alice C Lee Donald Wort

Appendix 2A. Simple regression and multiple regression 2. A.1 INTRODUCTION 2. A.2 SIMPLE REGRESSION Variance of Multiple Regression 2

Appendix 2A. Simple regression and multiple regression (2.A.1a) (2.A.1b) (2.A.2a) (2.A.2b) 3

Appendix 2A. Simple regression and multiple regression (2.A.3) (2.A.4) (2.A.5a) (2.A.5b) 4

Appendix 2A. Simple regression and multiple regression (2.A.6a) (2.A.6b) 5

Appendix 2A. Simple regression and multiple regression (2.A.7) (2.A.7a) 6

Appendix 2A. Simple regression and multiple regression (2.A.8) (2.A.8a) 7

Variance of Equation (2.A.7a) implies that: (2.A.7b) Where 8

Variance of (2.A.7c) (2.A.9) 9

Variance of 10

Variance of (2.A.10) (2.A.11) (2.A.12) 11

Multiple Regression (2.A.13a) The error sum of squares can be defined as: Where 12

Multiple Regression (2.A.14a) (2.A.14b) (2.A.14c) 13

Multiple Regression 0 = na + b(0) + c(0), (2.A.15a) (2.A.15b) (2.A.15c) 14

Multiple Regression (2.A.16a) (2.A.16b) (2.A.17) 15

Multiple Regression (2.A.13b) (2.A.18) (2.A.19) 16

Multiple Regression (2.A.20) where TSS = Total sum of squares; ESS = Residual sum of squares; and RSS = Regression sum of squares. 17

Multiple Regression (2.A.21) (2.A.22) Where and k = the number of independent variables. 18

Multiple Regression (2.A.23) where F(k-1, n-k) represents F-statistic with k － 1 and n － k degrees of freedom. 19

Appendix 2B. Instrumental variables and two- stage least squares By Cheng Few Lee Joseph Finnerty John Lee Alice C Lee Donald Wort

Appendix 2B. Instrumental variables and two-stage least squares 2. B.1 ERRORS-IN-VARIABLE PROBLEM 2. B.2 INSTRUMENTAL VARIABLES 2. B.3 TWO-STAGE, LEAST-SQUARE 21

2. B.1 ERRORS-IN-VARIABLE PROBLEM (2.B.1) (2.B.2) (2.B.3) 22

2. B.1 ERRORS-IN-VARIABLE PROBLEM (2.B.4) (2.B.5) 23

2. B.2 INSTRUMENTAL VARIABLES (2.B.6) (2.B.7) (2.B.8a) (2.B.8b) 24

2. B.2 INSTRUMENTAL VARIABLES (2.B.9a) (2.B.9b) (2.B.10a) (2.B.10b) 25

2.B.3 TWO-STAGE, LEAST-SQUARE (2.B.11a) (2.B.11b) (2.B.10′a) (2.B.10′b) 26

2.B.3 TWO-STAGE, LEAST-SQUARE (2.B.12a) (2.B.12b) 27