... discussion and engaging men as codrivers of social change Through documenting and expanding our understanding ofand approaches to the intersection of sexual and reproductive health and rights and HIV, ... knowledge and understanding ofthe needs of women and girls33 to underpin the complex and nuanced intersections between HIV prevention, treatment and care, and sexual and reproductive health and rights ... (UNAIDS) andThe ATHENA Network All rights reserved The designations employed andthe presentation ofthe material in this publication not imply the expression of any opinion whatsoever on the part of...
... analysis Volume ofthe Handbook of Econometrics and Volume 11 ofthe Handbook of Statistics contain nice surveys of these topics (Hajivassilou and Ruud, 1994; Hall, 1994; Hajivassilou, 1993; and Keane, ... available in Volume ofthe Handbook of Econometrics—see Powell (1994) and Hardle and Linton (1994)—as well as in ¨ Volume 11 ofthe 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, andSection 13.8.3...
... context, the ui are always independent and identically distributed, regardless of how they are related to the xi Assuming that the population mean ofthe error is zero is without loss of generality ... But the random sampling assumption, properly stated, does allow for temporal correlation What we will is assume random sampling in thecross Introduction section dimension The dependence in the ... cover—although the estimation methods we cover can be usually used—is seen when thecrosssection dimension and time series dimensions are roughly ofthe same magnitude, such as when the sample consists of...
... expectation operator, and several additional properties that are consequences ofthe randomness of mðxÞ Some ofthe 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) ... how the notion of statistical independence relates to conditional expectations If u is a random variable independent ofthe random vector x, then Eðu j xÞ ¼ EðuÞ, so that if EðuÞ ¼ and u and...
... is the square root ofthe appropriate diagonal ele^N ment of V =N The asymptotic standard errors can be loosely thought of as estimating ^ the standard deviations ofthe elements of yN , and they ... Limiting Behavior of Estimators and Test Statistics In this section, we apply the previous concepts to sequences of estimators Because estimators depend on the random outcomes of data, they are properly ... terms ofthe asymptotic variance of g ^ ^ ^ c Suppose that, for a sample of data, y ¼ and seðyÞ ¼ What is g and its (asymptotic) standard error? d Consider the null hypothesis H0 : y ¼ What is the...
... a1 denote the average partial e¤ect (across the distribution ofthe 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 ofthe 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 ofthe proof of Theorem ... simple way ^ to determine the sign, and perhaps the magnitude, ofthe inconsistency in bK If g > and xK and q are positively correlated, the asymptotic bias is positive The other combinations are...
... Þ=ðx1 À x0 Þ where y0 and x0 are the sample averages of yi and xi over the part ofthe sample with zi ¼ 0, and y1 and x1 are the sample averages of yi and xi over the part ofthe 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 ofthe bj in terms of y andthe other elements of b Then, substitute into the equation of interest ... larger than the OLS standard errors Not suprisingly, the magnitudes ofthe 2SLS standard errors depend, among other things, on the quality ofthe instrument(s) used in estimation For the following...
... elements of x i and xi0 x i : the levels, squares, andcross products ofthe 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 ofthe parameters—as the original Hausman test does—it often makes sense to compare just the estimates ofthe ... instruments The tools needed to make the proof rigorous are introduced in Chapter 12, but the key components ofthe proof can be given here in the context ofthe linear model Write the model as...
... will often need to indicate the equation for a generic crosssection unit i When we study the asymptotic properties of various estimators ofthe bg , the asymptotics is done with G fixed and N tending ... infinity In the household demand example, we are interested in a set of three demand functions, andthe unit of obser- Estimating Systems of Equations by OLS and GLS 145 vation is the family Therefore, ... 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 andthe unobservables ui Because ofthe random sampling...
... dependence (conditional 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 ofthecross ... 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 thecrosssection ... 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 ofthe optimal GMM estimator...
... restrictions, the matrix R1 consists only of zeros and ones, andthe 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 ... solve for either crime or spending (or any other transformation of them) in terms of z, u1 , u2 , andthe parameters And there is no way to define y1 and y2 to yield a linear SEM in two endogenous ... that the linear projection ofthe square is the square ofthe linear projection What the 2SLS estimator does in the first stage is project each of y2 and y2 onto the original exogenous variables and...
... normality ofthe uit ), we can test the equality ofthe 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 ... between the composite errors vit and vis 2 does not 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 ofthe variance of ci to the ... variance ofthe composite error, and it is useful as a measure ofthe relative importance ofthe unobserved e¤ect ci Assumptions (10.28) and (10.29) are special to random e¤ects For e‰ciency of feasible...
... that the variance matrix W ofthe 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 ... in a single crosssection on the seconddi¤erenced data, so that if the second-di¤erence error is homoskedastic conditional on xi , the standard OLS analysis on thecrosssectionof second di¤erences ... 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 crosssection observation...
... 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 ofthe appropriate diagonal element of matrix ... is simply on ui~i1 , ui~i2 The number of regressors in the final regression ofthe robust test is always the same as the degrees of freedom ofthe test ~ Finally, thesepffiffiffiffi procedures are easily ... depends on Ao , 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 ofthe Wald and LM statistics and...
... ðy Þ; and i¼1 t¼1 ^ ^ s it ðy Þs it ðy Þ ð13:50Þ i¼1 t¼1 The validity ofthe second of these follows from a standard iterated expectations argument, andthe last of these follows from the conditional ... lit ðy Þ There are several important features of these formulas First, the matrix A o is just the sum across t of minus the expected Hessian Second, the matrix Bo generally depends on the correlation ... correlation in the score The first term on the right- hand side of equation (13.51) can be replaced by one ofthe other two estimators in equation (13.50) The asymptotic variance of ^ ^ ^^ ^ ^ y...
... with L À P degrees of freedom under the conditions of Theorem 14.2 Therefore, the value ofthe 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 ... a linear demand and supply system for G 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...
... analysis The bj , their standard errors, andthe value ofthe likelihood func^ tion are reported by all software packages that binary response analysis The bj give the signs ofthe partial e¤ects of ... from probit of y on x and z (the unrestricted model), and let Lr denote the value ofthe likelihood function from probit of y on x (the restricted model) Then the likelihood ratio test of H0 : g ... Models 459 the sign ofthe e¤ect is given by the sign of bj Also, the relative e¤ects not depend on x: for continuous variables xj and xh , the ratio ofthe partial e¤ects is constant and qpðxÞ=qxj...
... between zero and one Therefore, the sign of bj is the same as the sign ofthe 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 andthe mean values ofthe xj (but where we square exper rather than use the average ofthe experi2 ... about whether the median or mean is more relevant for determining the e¤ects ofthe xj on y It depends on the context and is somewhat a matter of taste In some cases a quantile other than the median...
... exper, and exper The results of OLS on the selected sample andthe Heckit method are given in Table 17.1 The di¤erences between the OLS and Heckit estimates are practically small, andthe inverse ... rather than a probit, equation The analysis ofthe models in this section comes from Wooldridge (1998) The model in Section 17.5.1 is a special case ofthe model studied by Vella (1992) in the ... it These first two assumptions ensure consistency of FE on the unbalanced panel In the case of a randomly rotating panel, and in other cases where selection is entirely random, s i is independent...
... depends on the distribution of x i That is, in addition to assumption (19.7), identification of bo requires some restrictions on the distribution of explanatory variables, and these depend on the ... 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 ... and Trivedi (1994) andthe references therein discuss the problems of data censoring, truncation, and two-tier or hurdle models 19.5.1 Endogeneity We approach the problem of endogenous explanatory...