... Expectations and Related Concepts in Econometrics 15 Eðy j x1 ; x2 Þ ¼ b0 þ b1 x1 þ b2 x2 þ b x2 2: 3Þ Eðy j x1 ; x2 Þ ¼ b0 þ b1 x1 þ b2 x2 þ b x1 x2 2: 4Þ Eðy j x1 ; x2 Þ ¼ exp½b þ b1 logðx1 Þ þ b2 x2 ... E½Eð y j xÞ j w 2: 20Þ Note how the positions of x and w have been switched on the right-hand side of equation (2. 20) compared with equation (2. 19) The result in equation (2. 20) follows easily ... b3 z 2: 23Þ but where z is unobserved By the LIE, and the linearity of the CE operator, Eðy j x1 ; x2 Þ ¼ Eðb0 þ b x1 þ b2 x2 þ b3 z j x1 ; x2 Þ ¼ b þ b1 x1 þ b x2 þ b3 Eðz j x1 ; x2 Þ 2: 24Þ Now,...
... deal with econometrically For example, retirement saving of employees within a firm may be correlated because of common (often unobserved) characteristics of workers within a firm or because of features ... detailed summary of properties of conditional expectations 1 .2 1 .2. 1 The Stochastic Setting and Asymptotic Analysis Data Structures In order to give proper treatment to modern cross section and panel ... Pesaran and Smith (1995), Kao (1999), and Phillips and Moon (1999) 1 .2. 2 Asymptotic Analysis Throughout this book we focus on asymptotic properties, as opposed to finite sample properties, of estimators...
... repeated application of Lemma 3 .2 (see Problem 3 .2) lemma 3.3: Let fZN : N ¼ 1; 2; g be a sequence of J Â K matrices such that ZN ¼ op ð1Þ, and let fxN g be a sequence of J Â random vectors such ... square root of the appropriate diagonal ele^N ment of V =N The asymptotic standard errors can be loosely thought of as estimating ^ the standard deviations of the elements of yN , and they are ... terms of g, and use g and seð^Þ to test g H0 What you conclude? pffiffiffiffiffi ^ ^ ^ 3.8 Let y ¼ ðy1 ; y2 Þ be a N -asymptotically normal estimator for y ¼ ðy1 ; y2 Þ , ^ ^ ^ with y2 0 Let g ¼ y1 =y2 be...
... variables) of x1 on Eð y j x1 ; x2 Þ, and let a2 be the same for x2 Find a1 and a2 in terms of the bj and mj b Rewrite the regression function so that a1 and a2 appear directly (Note that m1 and m2 ... x1 þ b2 x2 þ Á Á Á þ bK xK þ u ð4 :21 Þ 62 Chapter u gq þ v ð4 :22 Þ The error u in equation (4 .21 ) consists of two parts Under equation (4 .20 ), v has zero mean and is uncorrelated with x1 ; x2 ; ... x2 Þ ¼ b0 þ b1 x1 þ b2 x2 þ b x1 x2 þ b x2 Let m1 Eðx1 Þ and m2 Eðx2 Þ be the population means of the explanatory variables a Let a1 denote the average partial e¤ect (across the distribution of...
... would d1 and a1 be consistent? [Hint: Let y be the population linear projection of y2 on z2 , and let a2 be the projection error: y2 ¼ z2 l2 þ a2 , Eðz2 a2 Þ ¼ For simplicity, pretend that l2 is ... i¼1 and, using Assumptions 2SLS.1 and 2SLS .2, apply the law of large numbers to each term along with Slutsky’s theorem 5 .2. 2 Asymptotic Normality of 2SLS pffiffiffiffiffi ^ The asymptotic normality of N ... about 107 with a standard error of about 014 Thus, the 2SLS estimate is notably below the OLS estimate and has a larger standard error 5 .2. 3 Asymptotic E‰ciency of 2SLS The appeal of 2SLS comes...
... that y2 is actually exogenous Hausman (1978) suggested comparing the OLS and 2SLS estimators of b 1 ðd1 ; a1 Þ as a formal test of endogeneity: if y2 is uncorrelated with u1 , the OLS and 2SLS ... element of y2 and regress it on z to obtain the RF residuals; then collect these in the row vector ^2 ) Now, estimate the model v v y1 ¼ z1 d1 þ y2 a1 þ ^2 r1 þ error ð6:18Þ and a standard F test of ... part a to argue that 2 Avarð^1 Þ ¼ Avarð~1 Þ þ b3 Avarðx2 Þ ¼ Avarð~1 Þ þ b ðs2 =NÞ a a a ~ where a1 ¼ b1 þ b3 m2 , a1 is the estimator of a1 if we knew m2 , and s2 ¼ Varðx2 Þ ^ c How would you...
... (SUR withCross Equation Restrictions): population model Consider the two-equation y1 ¼ g10 þ g11 x11 þ g 12 x 12 þ a1 x13 þ a2 x14 þ u1 ð7:54Þ y2 ¼ g20 þ g21 x21 þ a1 x 22 þ a2 x23 þ g24 x24 þ u2 ... strong 22 When G ¼ 2, W contains three distinct elements, s1 ¼ Eðui1 Þ, s2 ¼ Eðui2 Þ, and s 12 ¼ Eðui1 ui2 Þ These elements are not restricted by the assumptions we have made (The inequality js 12 ... theorem theorem 7 .2 (Asymptotic Normality of SOLS): SOLS .2, equation (7 .21 ) holds Under Assumptions SOLS.1 and 1 52 Chapter ^ The asymptotic variance of b is ^ Avarð b Þ ¼ AÀ1 BAÀ1 =N ð7 :22 Þ ^ so that...
... education, experience, and amount of job training) and unobserved attributes u2 , we write the wage o¤er function as w o ðhÞ ¼ g2 h þ z2 d2 þ u2 ð8 :2 Again, for given z2 and u2 , w o ðhÞ gives the wage ... matrix of u i ðui1 ; ui2 Þ , and write 11 s 12 s WÀ1 ¼ s 12 s 22 a Find EðZi0 WÀ1 u i Þ and show that it is not necessarily zero under the orthogonality 0 conditions Eðzi1 ui1 Þ ¼ and Eðzi2 ui2 ... functions of z1 , z2 , u1 , u2 , and the parameters; we consider this topic generally in Chapter Further, if z1 and z2 are exogenous in the sense that Eðu1 j z1 ; z2 Þ ¼ Eðu2 j z1 ; z2 Þ ¼ then,...
... Then y2 ¼ p21 z1 þ p 22 z2 þ v2 , where v2 is a linear combination of u1 and u2 Squaring this reduced form and using Eðv2 j zÞ ¼ gives 222 Eðy2 j zÞ ¼ p21 z1 þ p 22 z2 þ 2p21 p 22 z1 z2 þ Eðv2 j ... instruments, and let d 22 be the estimator of d 22 Then, estimate ^ y1 À d 22 z2 ¼ g 12 y2 þ d11 z1 þ d13 z3 þ error ^ p by 2SLS using ðz1 ; z2 ; z3 Þ as instruments Since d 22 ! d 12 when d 12 ¼ d 22 0, this ... g 12 y2 þ d11 z1 þ d 12 z2 þ d13 z3 þ u1 ð9:31Þ y2 ¼ g21 y1 þ d21 z1 þ d 22 z2 þ u2 ð9: 32 22 6 Chapter where each zj is uncorrelated with u1 and u2 (z1 can be unity to allow for an intercept) Without...
... variances and covariances of the elements of vi Under Assumption RE.1a, Eðci uit Þ ¼ 0, t ¼ 1; 2; ; T, and so 22 Eðvit Þ ¼ Eðci2 Þ þ 2Eðci uit Þ þ Eðuit Þ ¼ sc þ su where sc ¼ Eðci2 Þ Also, ... 1; 2; ; Tg be homoskedastic u € across t and serially uncorrelated The variance of uit can be computed as Eð€it Þ ¼ E½ðuit À ui Þ ¼ Eðuit Þ þ Eðui2 Þ À 2Eðuit ui Þ u2 22 ¼ su þ su =T À 2su ... dropped because it does not vary over time for any of the firms in the sample The estimated equation with standard errors is logð^crapÞ ¼ À:080 d88 À :24 7 d89 À :25 2 grant À : 422 grantÀ1 s ð :21 0Þ...
... (1981) (HT) partition zi and xit as zi ¼ ðzi1 ; zi2 Þ, xit ¼ ðxit1 ; xit2 Þ—where zi1 is  J1 , zi2 is  J2 , xit1 is  K1 , xit2 is  K2 and assume that Eðzi1 ci Þ ¼ and Eðxit1 ci Þ ¼ 0; all ... uncorrelated with yi; tÀh , h b Anderson and Hsiao (19 82) suggested pooled IV with instruments yi; t 2 or Dyi; t 2 , whereas Arellano and Bond (1991) proposed using the entire set of instruments in a GMM ... equation (11 .21 ) Papke estimates equation (11 .21 ) by 2SLS, using Dyi; t 2 as an instrument for Dyi; tÀ1 ; because of the lags used, equation (11 .21 ) can be estimated for six years of data The...
... these values exist, and we would like to estimate them Generic candidates for yo1 and yo2 are labeled y1 and y2 , and, without further information, y1 is any positive number and y2 is any real number: ... yÞ ¼ ½ y À mðx; yÞ =2 ð 12: 24Þ The score of equation ( 12. 24) can be written as sðw; yÞ ¼ À‘y mðx; yÞ ½y À mðx; yÞ ð 12: 25Þ where ‘y mðx; yÞ is the  P gradient of mðx; yÞ, and therefore ‘y mðx; ... theorem 12. 2 (Consistency of M-Estimators): Under the assumptions of Theorem 12. 1, assume that the identification assumption ( 12. 11) holds Then a random vector, ^ ^ p y, solves problem ( 12. 8), and...
... Y ð13 :22 Þ Ð since Y f ðy j x i ; y ÞnðdyÞ is unity for all y, and therefore the partial derivatives with respect to y must be identically zero But the right-hand side of equation (13 .22 ) can ... interchanges of derivative and integral in equations (13 .21 ) and (13 .25 ) hold for all y A intðYÞ; (d) the elements of ‘y2 lðy; x; yÞ are bounded in absolute value by a function bðy; xÞ with finite ... summed across i, to estimate r and se2 along with the ‘‘fixed e¤ects’’ ci ? b If ci j yi0 @ Normalða þ a1 yi0 ; sa2 Þ, where sa2 Varðai Þ and ci À a À a1 yi0 , write down the density of ðyi1 ;...
... (14.56), A2 ¼ ð1=rÞEðs 02 s1 Þ, and so V2 ¼ AÀ1 Eðs s 02 ÞðA 02 ÞÀ1 ¼ r ½Eðs s1 ÞÀ1 Eðs s 02 Þ½Eðs1 s 02 ÞÀ1 Generalized Method of Moments and Minimum Distance Estimation 449 Now we use the standard ... attributes of a good or service (see Epple, 1987; Kahn and Lang, 1988; and Wooldridge, 1996) The demand and supply system is written as demandg ¼ h1g þ wa1g þ x1 b 1g þ u1g ; supplyg ¼ h2g þ wa2g þ x2 ... h2g þ ðq i P þ x i3 GÞa2g þ x i2 b 2g þ ui2g ; g ¼ 1; ; G ð14:46Þ These two equations are linear in q i ; x i1 ; x i2 , and x i3 but nonlinear in the parameters Let u i1 be the G Â vector of...
... ðz1 ; y2 Þ, a consistent estimator of expression (15.45) is ^ ^ Fðz1 dy1 þ ay1 y2 Þ ð15:46Þ ^2 ^2 ^2 ^2 ^ ^ ^ ^ ^2 where dy1 dr1 =ðyr1 t2 þ 1Þ 1 =2 and ay1 ar1 =ðyr1 t2 þ 1Þ 1 =2 Note that t2 is ... Þ, t2 ¼ Varðv Þ, and e1 is independent of z and v (and therefore of y2 ) Because of joint normality of ðu1 ; v Þ, e1 is also normally 474 Chapter 15 2 distributed with Eðe1 Þ ¼ and Varðe1 Þ ¼ Varðu1 ... kind of expectation in Section 15.7.1 The same reasoning gives Ev ½Fðz1 dr1 þ ar1 y2 þ yr1 v Þ ¼ Fðz1 dy1 þ ay1 y2 Þ 22 where dy1 dr1 =ðyr1 t2 þ 1Þ 1 =2 and ay1 ar1 =ðyr1 t2 þ 1Þ 1 =2 , where t2...
... now allow one of the variables in the Tobit model to be endogenous The model is y1 ¼ maxð0; z1 d1 þ a1 y2 þ u1 Þ ð16 :26 Þ y2 ¼ zd2 þ v2 ¼ z1 d21 þ z2 d 22 þ v2 ð16 :27 Þ where ðu1 ; v2 Þ are zero-mean ... 15.7 .2 The density f ð y2 j zÞ is Normalðzd2 ; t2 Þ Further, from equation (16 .29 ), y1 given ðy2 ; zÞ follows a Tobit with latent mean z1 d1 þ a1 y2 þ y1 v2 ¼ z1 d1 þ a1 y2 þ ðh1 =t2 Þð y2 À zd2 ... di¤erences of ^2 ^2 ^2 ^ ^ mðz1 d1 þ a1 y2 ; y1 t2 þ t1 Þ ð16:31Þ ^2 ^2 where all estimates except t2 come from step b of the Smith-Blundell procedure; t2 is simply the usual estimate of the error...
... y2 ¼ j y1 ; xÞ ¼ Ff½xd2 þ s 12 s 2 ð y1 À x1 b Þð1 À s 12 s 2 ÞÀ1 =2 g 1 Combining all of these pieces [and noting the cancellation of Pðy2 ¼ j xÞ we get li ðy Þ ¼ ð1 À yi2 Þ log½1 À Fðx i d2 ... full density of y2 given x, which is f ðy2 j xÞ ¼ ½Fðxd2 Þ y2 ½1 À Fðxd2 Þ 1Ày2 , y2 ¼ 0, 1, we can only use the density f ðy1 j y2 ; xÞ when y2 ¼ To find f ð y1 j y2 ; xÞ at y2 ¼ 1, we can ... approach is fairly straightforward because, when y2 > 0, y1 j x; y2 @ Normal½x1 b þ 222 g1 ðy2 À xd2 Þ; h1 , where h1 ¼ s1 À s 12 =t2 , s1 ¼ Varðu1 Þ, and s 12 ¼ Covðu1 ; v Þ The log likelihood...
... À.0644 (.0063) À. 021 7 (.0 025 ) age 27 2 (.017) 337 (.009) age À.0019 (.0003) À.0041 (.0001) 6 82 (.0 52) 315 (. 021 ) urban À .22 8 (.046) À.086 (.019) electric À .26 2 (.076) À. 121 (.034) tv À .25 0 (.090) À.145 ... i ; bo Þ and note that, under assumptions (19.3) and (19.7), ui2 À so mðx i ; bo Þ is uncorrelated with any function of x i Let hðx i ; b Þ be a  Q vector of functions of x i and b, and consider ... assume that y2 has a linear reduced form satisfying certain assumptions Write y2 ¼ zP2 þ v2 ð19:38Þ where P2 is an L  G matrix of reduced form parameters and v2 is a  G vector of reduced form...
... constant across i Assumption (20 .22 ) holds for certain nonstandard censoring schemes, too For example, if an element of x i is education, assumption (20 .22 ) holds if, say, individuals with more ... Honour of John Vanderkamp, ed L N Christophides, E K Grant, and R Swidinsky Toronto: University of Toronto Press, 20 1 22 2 Case, A C., and L F Katz (1991), ‘‘The Company You Keep: The E¤ects of Family ... E¤ect of Community Colleges on Educational Attainment,’’ Journal of Business and Economic Statistics 13, 21 7 22 4 Rubin, D B (1974), ‘‘Estimating Causal E¤ects of Treatments in Randomized and Nonrandomized...