Lesson 11 Regressions Part II Ka-fu Wong C2007 ECON1003: Analysis of Economic Data Lesson 11-1
Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson11-1 Lesson 11: Regressions Part II
Does watching television rot your mind? Zavodny, Madeline(2006): Does watching television rot your mind? Estimates of the effect on test scores economias of education Review25(5):565-573 a Television is one of the most omnipresent features of Americans lives. The average american adult watches about 15 h of television per week, accounting for almost one -half of free time a The substantial amount of time that most individuals spend watching television makes it important to examine its effects on society including human capital accumulation and academic achievement Ka-fu Wong C2007 ECON1003: Analysis of Economic Data Lesson 11-2
Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson11-2 Does watching television rot your mind? ◼ Zavodny, Madeline (2006): “Does watching television rot your mind? Estimates of the effect on test scores,” Economics of Education Review, 25 (5): 565–573 ◼ Television is one of the most omnipresent features of Americans’ lives. The average American adult watches about 15 h of television per week, accounting for almost one-half of free time. ◼ The substantial amount of time that most individuals spend watching television makes it important to examine its effects on society, including human capital accumulation and academic achievement
Data regression model a This analysis uses three data sets to examine the relationship between television viewing and test scores: the national Longitudinal Survey of Youth 1979(NLSY the hsb survey and the NELS. Each survey includes test scores and a question about the number of hours of television watched by young adults Test Scorer a +hOurs of Tv +INdividual characteristics + Family Background in t dOther Uses of Timeit+t it Gil Test score of individual i at time t Ka-fu Wong C2007 ECON1003: Analysis of Economic Data Lesson 11-3
Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson11-3 Data & Regression model ◼ This analysis uses three data sets to examine the relationship between television viewing and test scores: the National Longitudinal Survey of Youth 1979 (NLSY), the HSB survey and the NELS. Each survey includes test scores and a question about the number of hours of television watched by young adults. Test score of individual i at time t
Summary of samples from data sets HSB NElS Number of individuals 2477 14.988 6255 Type of data Cross-section Panel Panel Time period 1980,1982 1988,1990,1992 Age of respondents 16-19 13-22 12-20 Sibling structure 239 sibling pairs 432 twin pairs None Test score AFQT Vocabulary, reading, math Reading, math TV variable Hours per week Hours per weekday Hours per weekday, weekend y Categorical or linear L Categorical (7) Categorical (6 or 7) Ka-fu Wong C2007 ECON1003: Analysis of Economic Data Lesson 11-4
Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson11-4 Summary of samples from data sets
Regression results <0<0.05<01 Test Scorei a+hOurs of TV +iNdividual Characteristics +2, Family Backgroundit dOther Uses of Timeit +tIt+ cir All Male F emale NLSY: AFOTscore (A)TⅤonl .007* 012 (.001) (.002) (.002) (B)A+individual 007 007 008* characteristics (.001) (.002 (.002 (C)B+family background . 003*3. .002 .005* (.001) (.002) (.001) (D)C+other uses of time -.003 -.002 (.001) (.002) 002 Number of observations 2477 1309 1168 Ka-fu Wong C2007 ECON1003: Analysis of Economic Data Lesson 11-5
Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson11-5 Regression results **p<0.01; *p<0.05; †p<0.1
Multiple Linear Regression Model a Relationship Between Variables Is a Linear Function Random yintercept Slope Error Y=β0+β11+阝2×2+β3X3+…+阝Ⅹ+E Dependent Independent (Response) Explanatory Variable Variable Ka-fu Wong C2007 ECON1003: Analysis of Economic Data Lesson 11-6
Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson11-6 Multiple Linear Regression Model ◼ Relationship Between Variables Is a Linear Function Y intercept Slope Random Error Dependent (Response) Variable Independent (Explanatory) Variable Y = b0 + b1X1 + b2X2 + b3X3 + … + bkXk + e
Finance Application: multifactor pricing model a It is assumed that rate of return on a stock(r) is linearly related to the rate of return on some factor and the rate of return on the overall market(rm) t=βo+βoiRt+阝Rmt+8 Rate of return on some major stock index Rate of return on a particular oil company stock i at The rate of return on time t crude oil price on date t Ka-fu Wong C2007 ECON1003: Analysis of Economic Data Lesson 11-7
Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson11-7 Finance Application: multifactor pricing model ◼ It is assumed that rate of return on a stock (R) is linearly related to the rate of return on some factor and the rate of return on the overall market (Rm). Rate of return on a particular oil company stock i at time t Rate of return on some major stock index The rate of return on crude oil price on date t Rit = b0 + boi Rot+ b1Rmt +e
Estimation by Method of moments Number of moment condition needed Y=βo+βX1+β22+β33+…+阝×k+E k+1 parameters to estimate. Need k+1 moment conditions ■ Assumption#1 ■E(6)=0 implies e(y)-βo-β1E(X)-β2E(X2)-….kE(xX)=0 ■ Assumption#2 (eX)=0 implies E[(y-βo-β11-…-β×1]=0 Since Cova, X, =E(Ex1)-EsE(X,= E(ex,), the assumption really imply s and x are uncorrelated a Assumption #3: E(EX2)=0 Assumption #4: E(Ex3)=0 Assumption #k+1: E(ex)=0 Ka-fu Wong C2007 ECON1003: Analysis of Economic Data Lesson 11-8
Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson11-8 Estimation by Method of moments Number of moment condition needed Y = b0 + b1X1 + b2X2 + b3X3 + … + bkXk + e k+1 parameters to estimate. Need k+1 moment conditions. ◼ Assumption #1 ◼ E(e) = 0 implies E(y) – b0 – b1 E(x1 ) – b2 E(x2 ) - … bk E(xk )= 0 ◼ Assumption #2 ◼ E(ex1 ) =0 implies E[(y – b0 – b1x1 - … - bkxk )x1 ]=0 ◼ Since Cov(e, x1 ) = E(ex1 ) – E(e)E(x1 ) = E(ex1 ), the assumption really imply e and x are uncorrelated. ◼ Assumption #3: E(ex2 ) =0 ◼ Assumption #4: E(ex3 ) =0 ◼ … ◼ Assumption #k+1: E(exk ) =0
Estimation ofβo阝1rβ2r…Pk Method of moments Two approaches 1. Solve theβo阝1,β2x…阝 k from the k+1 moment conditions,in terms of covariances, variances and means. Plug in to sample analog of these covariances, variances and means ro produce the sample estimate bo, bl, bye. b k 2. Assume bor b1, b2 k, solve them from the sample analog of the k+ 1 moment conditions Ka-fu Wong C2007 ECON1003: Analysis of Economic Data Lesson11-9
Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson11-9 Estimation of b0 , b1 , b2 ,…, bk Method of moments ◼ Two approaches: 1. Solve the b0 , b1 , b2 ,…, bk from the k+1 moment conditions, in terms of covariances, variances and means. Plug in to sample analog of these covariances, variances and means ro produce the sample estimate b0 , b1 , b2 ,…, bk 2. Assume b0 , b1 , b2 ,…, bk , solve them from the sample analog of the k+1 moment conditions
Estimation ofβoβ1β2r…βk Maximum likelihood a Assume G to be independent identically distributed with normal distribution of zero mean and variance o2, denote the normal density for e be f)=f(yβo-βX1阝2x2…x) normal density a Choose bor b1, b2,., bk to maximize the joint likelihood L(bo,b1,b2…,b)=f(e1)*fe2)*,*f(e f(e)=f(y-bo-b1X1-b2X2-.-bk Xx) Ka-fu Wong C2007 ECON1003: Analysis of Economic Data Lesson11-10
Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson11-10 Estimation of b0 , b1 , b2 ,…, bk Maximum Likelihood ◼ Assume ei to be independent identically distributed with normal distribution of zero mean and variance s2 . Denote the normal density for e be ◼ f(e)=f(y-b0 -b1x1 -b2x2 -…-bkxk ) f(e)= f(y-b0 -b1x1 -b2x2 -…-bkxk ) normal density ◼ Choose b0 , b1 , b2 , …, bk to maximize the joint likelihood: ◼ L(b0 , b1 , b2 , …, bk ) = f(e1 )*f(e2 )*…*f(en )
