cross section and mri through the middle of the right leg

... analysis Volume of the Handbook of Econometrics and Volume 11 of the Handbook of Statistics contain nice surveys of these topics (Hajivassilou and Ruud, 1994; Hall, 1994; Hajivassilou, 1993; and Keane, ... available in Volume of the Handbook of Econometrics—see Powell (1994) and Hardle and Linton (1994)—as well as in ¨ Volume 11 of the Handbook of Statistics—see Horowitz (1993) and Ullah and Vinod (1993) ... refer to sections throughout the book first by chapter number followed by section number and, sometimes, subsection number Therefore, Section 6.3 refers to Section in Chapter 6, and Section 13.8.3...

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... presence of within-cluster correlation Another important issue is that cross section samples often are, either intentionally or unintentionally, chosen so that they are not random samples from the ... cover—although the estimation methods we cover can be usually used—is seen when the cross section dimension and time series dimensions are roughly of the same magnitude, such as when the sample consists of ... the post–World War II period In this case it makes little sense to fix the time series dimension and let the cross section dimension grow The research on asymptotic analysis with these kinds of...

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... expectation operator, and several additional properties that are consequences of the randomness of mðxÞ Some of the statements we make are proven in the appendix, but general proofs of other assertions ... shows that the elasticity is a function of x When y and x are random, it makes sense to use the right- hand side of equation (2.8), but where f ðxÞ is the conditional mean, mðxÞ Therefore, the (partial) ... know the outcome of w, then we know the outcome of x The most general statement of the LIE that we will need is Eðy j xÞ ¼ E½Eð y j wÞ j xŠ ð2:19Þ In other words, if we write m1 ðwÞ Eð y j wÞ and...

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... is the 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 ... ð3:2Þ ^ for any value of y, then we say yN is a consistent estimator of y Because there are other notions of convergence, in the theoretical literature condition (3.2) is often referred to as ... p ^ ^N jth diagonal vjj , and V ! V, then the asymptotic standard error of yNj , denoted ^Nj Þ, is ð^Njj =NÞ 1=2 seðy v In other words, the asymptotic standard error of an estimator, which is...

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... a1 denote the average partial e¤ect (across the distribution of the explanatory 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 ... (Consistency of OLS): Under Assumptions OLS.1 and OLS.2, the ^ OLS estimator b obtained from a random sample following the population model (4.5) is consistent for b The simplicity of the proof of Theorem ... simple way ^ to determine the sign, and perhaps the magnitude, of the inconsistency in bK If g > and xK and q are positively correlated, the asymptotic bias is positive The other combinations are...

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... Þ=ðx1 À x0 Þ where y0 and x0 are the sample averages of yi and xi over the part of the sample with zi ¼ 0, and y1 and x1 are the sample averages of yi and xi over the part of the sample with zi ... define the linear combination of interest, say y a1 b1 þ a2 b2 þ Á Á Á þ aK bK , and then to write one of the bj in terms of y and the other elements of b Then, substitute into the equation of interest ... 500,000 observations! The problem is that the instruments— representing quarters of birth and various interactions of these with year of birth and state of birth—are very weak, and they are too numerous...

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... elements of x i and xi0 x i : the levels, squares, and cross products of the regressors in the conditional mean The Breusch-Pagan and White tests have degrees of freedom that depend on the number of ... Rather than comparing the OLS and 2SLS estimates of a particular linear combination of the parameters—as the original Hausman test does—it often makes sense to compare just the estimates of the ... instruments The tools needed to make the proof rigorous are introduced in Chapter 12, but the key components of the proof can be given here in the context of the linear model Write the model as...

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... infinity In the household demand example, we are interested in a set of three demand functions, and the unit of obser- Estimating Systems of Equations by OLS and GLS 145 vation is the family Therefore, ... will often need to indicate the equation for a generic cross section unit i When we study the asymptotic properties of various estimators of the bg , the asymptotics is done with G fixed and N ... parameter vector of interest and ui is a G Â vector of unobservables Equation (7.9) explains the G variables yi1 ; ; yiG in terms of Xi and the unobservables ui Because of the random sampling...

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... or unconditional) The reason we can allow this generality is that we fix the row dimension of Zi and u i and let N ! y Therefore, we are assuming that N, the size of the cross section, is large ... periods for the same crosssectional unit (so G ¼ T, the total number of time periods) Therefore, the following analysis applies to panel data models where T is small relative to the cross section ... no di¤erence whether the first-stage u ^ residuals ^i are used in place of ^ i The square roots of diagonal elements of this u u matrix are the asymptotic standard errors of the optimal GMM estimator...

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... restrictions, the matrix R1 consists only of zeros and ones, and the number of rows in R1 equals the number of excluded right- hand-side endogenous variables, G À G1 À 1, plus the number of excluded ... that the linear projection of the square is the square of the linear projection What the 2SLS estimator does in the first stage is project each of y2 and y2 onto the original exogenous variables and ... visitation rights and the other exogenous variables finc (father’s income), fremarr (binary indicator if father remarried), and dist (miles currently between the mother and father) Similarly, the second...

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... normality of the uit ), we can test the equality of the ci using a standard F test for T of any size [The degrees of freedom are N À and NðT À 1Þ À K.] Unfortunately, the properties of this test ... depend on the di¤erence between t and s: Corrðvis ; vit Þ ¼ sc =ðsc þ su Þ b 0; s t This correlation is also the ratio of the variance of ci to the variance of the composite error, and it is ... intercept is often reported The overall intercept is ^ either for an arbitrary cross section unit or, more commonly, for the average of the ci across i ^ Sometimes it is useful to obtain the ci even...

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... that the variance matrix W of the composite error vi ¼ ci jT þ ui has the random e¤ects structure and Assumption SIV.5 from Section 8.3.4 holds Neither of these is necessary, but together they ... ÞÀ1 Zi0 ð11:45Þ the projection matrix onto the null space of Zi [the matrix Zi ðZi0 Zi ÞÀ1 Zi0 is the projection matrix onto the column space of Zi ] In other words, for each cross section observation ... in ^ has the random e¤ects structure, and IASW (1999, Theorem the special case that W 3.1) obtained the general case In expression (11.63) there are þ TK þ K parameters, and the matrix of instruments...

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... degrees -of- freedom adjustment, where P is the dimension of y.] As always, the asymptotic standard error of ^ each element of y is the square root of the appropriate diagonal element of matrix ... is simply on ui~i1 , ui~i2 The number of regressors in the final regression of the robust test is always the same as the degrees of freedom of the test ~ Finally, thesepffiffiffiffi procedures are easily ... Bo , Co , and , and can be estimated by using consistent estimators of Ao , Bo , and Co When we add assumption (12.53), then the special versions of the Wald and LM statistics and the QLR statistics...

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... ðy Þ; and i¼1 t¼1 ^ ^ s it ðy Þs it ðy Þ ð13:50Þ i¼1 t¼1 The validity of the second of these follows from a standard iterated expectations argument, and the last of these follows from the conditional ... correlation in the score The first term on the right- hand side of equation (13.51) can be replaced by one of the other two estimators in equation (13.50) The asymptotic variance of ^ ^ ^^ ^ ^ y ... derivatives of li ðy Þ with respect to each of the P parameters, but then we 398 Chapter 13 evaluate this vector of partials at the restricted estimates Then, from Section 12.6.2 and the information...

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... with L À P degrees of freedom under the conditions of Theorem 14.2 Therefore, the value of the objective function (properly standardized by the sample size) can be used as a test of any overidentifying ... Montgomery, Shaw, and Benedict, 1992; Hagy, 1998), where the assumption is that the demand and supply functions not change across region or industry but the type of matching does, and therefore pi ... 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 attribute demand disturbances and u i2 the G Â vector of attribute...

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... analysis The bj , their standard errors, and the value of the likelihood func^ tion are reported by all software packages that binary response analysis The bj give the signs of the partial e¤ects of ... from probit of y on x and z (the unrestricted model), and let Lr denote the value of the likelihood function from probit of y on x (the restricted model) Then the likelihood ratio test of H0 : g ... functions of some explanatory variables, such as natural logs or quadratics, there is the issue of using the log of the average versus the average of the log (and similarly with quadratics) To get the...

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... between zero and one Therefore, the sign of bj is the same as the sign of the partial e¤ect of xj Other functional forms are easily handled Suppose that x1 ¼ logðz1 Þ (and that this is the only ... estimates by the adjustment factors in equations (16.11) and (16.16), evaluated at the estimates and the mean values of the xj (but where we square exper rather than use the average of the experi2 ... because the unrestricted model is just standard Tobit 16.5 Reporting the Results ^ For data censoring applications, the quantities of interest are the bj and their standard errors (We might use these...

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... exper, and exper The results of OLS on the selected sample and the Heckit method are given in Table 17.1 The di¤erences between the OLS and Heckit estimates are practically small, and the inverse ... rather than a probit, equation The analysis of the models in this section comes from Wooldridge (1998) The model in Section 17.5.1 is a special case of the model studied by Vella (1992) in the ... where the final equality follows because the sj sum to unity Therefore, the expected value of the weighted objective function [over the distribution of ðw; hÞ] equals the expected value of qðw;...

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... place of b, and A i is the rights ^ hand side of equation (19.13) with b in place of bo This is the fully robust variance matrix estimator in the sense that it requires only assumption (19.7) and ... by putting other functions of v2 on the right- hand side, such as squares and cross products, but we not show these explicitly Note that y2 is exogenous if and only if r1 ¼ Under the maintained ... significance of logðcigpricÞ and logðincomeÞ with the QLR statistic in equation (19.17) Count Data and Related Models 679 e Compute the fully robust standard errors, and compare these with the GLM standard...

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... estimation of the l m leads to a wellknown estimator of the survivor function Rather than derive the MLE of the survivor function, it is easier to motivate the estimator from the representation of the ... ^ The estimate of a is 806, and the standard error of a leads to a strong rejection of H : a ¼ against H : a < Therefore, there is evidence of negative duration dependence, conditional on the ... time—usually, at the beginning of the spell and we not re-collect data on the covariates during the course of the spell Time-varying covariates are more naturally handled in the context of grouped...

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