2  group and pivot data by name

... x2 , x1 x2 has zero mean and is uncorrelated with x1 and x2 ) d Why is equation (2.47) generally more useful than equation (2.48)? 2.4 For random scalars u and v and a random vector x, suppose ... that xj is continuous The right-hand side of equation (2.8) 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 ... cover the most general statement needed in this book Suppose that w is a random vector and y is a random variable Let x be a random vector that is some function of w, say x ¼ fðwÞ (The vector x could...

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... Examples of Unobserved E¤ects Panel Data Models Estimating Unobserved E¤ects Models by Pooled OLS Random E¤ects Methods 10.4.1 Estimation and Inference under the Basic Random E¤ects Assumptions 10.4.2 ... semiparametric and nonparametric methods General surveys of semiparametric and nonparametric methods are available in Volume of the Handbook of Econometrics—see Powell (1994) and Hardle and Linton ... 11.2.1 A Random Trend Model 11.2.2 General Models with Individual-Specific Slopes GMM Approaches to Linear Unobserved E¤ects Models 11.3.1 Equivalence between 3SLS and Standard Panel Data Estimators...

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... of full data matrices Because we will rely heavily on random sampling, it is important to know what it allows and what it rules out Random sampling is often reasonable for cross section data, where, ... 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 data methods, we must choose a stochastic ... treat explanatory variables as random for modern econometric analysis We should not confuse the random sampling assumption with so-called experimental data Experimental data fall under the fixed explanatory...

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... Op ð1Þ and op ð1Þ Á op ð1Þ ¼ op ð1Þ All of the previous definitions apply element by element to sequences of random vectors or matrices For example, if fxN g is a sequence of random K  random ... y and AvarðyÞ using the delta method  g  À:4 ^ ^ c If, for a sample of data, y ¼ ðÀ1:5; :5Þ and AvarðyÞ is estimated as , À:4 ^ find the asymptotic standard error of g pffiffiffiffiffi ^ ~ 3.9 Let y and ... 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 viewed as random...

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... (4.29) (except ability, of course) Find y1 and its ^ standard error; find y2 and its standard error when Deduc ¼ 4.2 a Show that, under random sampling and the zero conditional mean as^ sumption ... ð b ^ expectations exist and that b À b ¼ op ð1Þ Assume that all necessary expectations exist and are finite.] 4.5 Let y and z be random scalars, and let x be a  K random vector, where one element ... White standard errors or Huber standard errors, or some hyphenated combination of the names Eicker, Huber, and White It is probably best to just call them heteroskedasticity-robust standard errors,...

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... method proposed by Griliches and Mason (1972) and also used by Blackburn and Neumark (1992) Problems 5.1 In this problem you are to establish the algebraic equivalence between 2SLS and OLS estimation ... b¼ N i i¼1 i i i¼1 where Z and X are N Â K data matrices and Y is the N Â data vector on the yi The consistency of this estimator is immediate from equation (5.11) and the law of large numbers ... ðy1 À y0 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...

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... variables and interactions In the simplest case, we have two time periods, say year and year There are also two groups, which we will call a control group and an experimental group or treatment group ... the data in HPRICE.RAW, which is a subset of the data used by Kiel and McClain (1995) The file contains housing prices and characteristics for two years, 1978 and 1981, for homes sold in North Andover, ... that educ and u1 may be correlated The instruments for educ are parents’ education and husband’s education So, we first regress educ on 1, exper, exper , motheduc, fatheduc, and huseduc and obtain...

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... with G fixed and N tending to 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 ... SOLS.1 and SOLS.2, and therefore under Assumptions SGLS.1 and SOLS.2 (In what follows, we assume that PN Assumptions SOLS.2 and SGLS.2 both hold.) By the WLLN, plimðN À1 i¼1 ui ui0 Þ ¼ W, and so ... between males and females in both earnings and benefits But what about between whites and nonwhites, and married and unmarried people? The F-type statistic for joint significance of married and white...

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... system but where xi1 and xi2 can contain endogenous as well as exogenous variables Because xi1 and xi2 are generally correlated with ui1 and ui2 , estimation of these equations by OLS or FGLS, as ... more e‰cient estimators by estimating b and b jointly, 186 Chapter that is, by using a system procedure The e‰ciency gains are analogous to the gains that can be realized by using feasible GLS ... WÞZgÀ1 Z Y ð8:36Þ ^ By Theorems 8.1 and 8.2, b is consistent and asymptotically normal under Assumptions SIV.1–SIV.3 Assumption SIV.3 requires EðZi0 WZi Þ to be nonsingular, a standard assumption...

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... w is the dummy argument in the labor supply and labor demand functions We assume that observed hours, h, and observed wage, w, equate supply and demand: Simultaneous Equations Models 213 h ¼ h ... (9.38), and applying nonlinear GMM estimation See Lahiri and Schmidt (1978) and Hausman, Newey, and Taylor (1987) Simultaneous Equations Models 9.4.3 229 Subtleties Concerning Identification and ... was discovered independently by Hansen (1982) and White (1982b), and it follows from the discussion in Section 8.6 Expanding the IV list to arbitrary functions of z and applying full GMM is not...

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... r1 ¼ À:5, which is implied by the standard random and fixed e¤ects assumption that the uit are serially uncorrelated An alternative to computing robust standard errors and test statistics is to ... nutritional programs and prenatal care.) b Estimate the preceding equation by pooled OLS, and discuss the results You should report the usual standard errors and serial correlation–robust standard errors ... 10.4 10.4.1 Random E¤ects Methods Estimation and Inference under the Basic Random E¤ects Assumptions As with pooled OLS, a random e¤ects analysis puts ci into the error term In fact, random e¤ects...

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... measurement error, can also be solved by eliminating ci and finding appropriate IVs Measurement error in panel data was studied by Solon (1985) and Griliches and Hausman (1986) It is widely believed ... estimating a and b; see also Lemieux (1998) 11.3 11.3.1 GMM Approaches to Linear Unobserved E¤ects Models Equivalence between 3SLS and Standard Panel Data Estimators Random e¤ects, fixed e¤ects, and first ... K2 and assume that Eðzi1 ci Þ ¼ and Eðxit1 ci Þ ¼ 0; all t ð11:68Þ We still maintain assumption (11.66), so that zi and xis are uncorrelated with uit for all t and s Assumptions (11.66) and...

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... , 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 ... for example, Davidson and MacKinnon (1993), Bera and McKenzie (1986), Orme (1990), and Chesher and Spady (1991) PN ~ ~ The Hessian form of the LM statistic uses M ¼ i¼1 Hi , and it has a few drawbacks: ... 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: the parameter...

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... implicitly by taking z t to be nonrandom In Chapter we saw that strict exogeneity played no role in getting consistent, asymptotically normal estimators in linear panel data models by pooled OLS, and ... y is estimated, as usual, by AÀ1 BAÀ1 =N for the chosen estimators A and B The asymptotic standard errors come directly from this matrix, and Wald tests for linear and nonlinear hypotheses can ... mean and homoskedasticity (see Section 7.8) If the panel data probit model is dynamically complete, any software package that does standard probit can be used to obtain valid standard errors and...

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... 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 ... problem, pointed out by Bartik (1987), Epple (1987), and Kahn and Lang (1988): because of matching that happens between individual buyers and sellers, x i2 is correlated with ui1 , and x i1 is correlated ... struc- Generalized Method of Moments and Minimum Distance Estimation 433 tural demand and supply functions, we can use these as instruments in both the demand and supply equations Specifically, we...

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... yi0 ; z i Þ and with a and sa replacing c and sc , respectively Conveniently, this result means that we can use standard random e¤ects probit software to estimate c, d, r, x0 , x, and sa2 : we ... typically called the random e¤ects probit estimator, and the theory in Section 13.9 can be applied directly to obtain asymptotic standard errors and test statistics Since b and sc can be estimated, ... coe‰cient by The biggest di¤erence between the LPM model on one hand, and the logit and probit models on the other, is that the LPM assumes constant marginal e¤ects for educ, kidslt6, and so on,...

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... 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 to compute elasticities, and ... =2Þ ^ ^ g and these are easily estimated given b , s , and ^ We cannot obtain the Tobit model as a special case of the model (16.37) and (16.38) by imposing parameter restrictions, and this inability ... Regression to Panel Data and Cluster Samples We now cover Tobit methods for panel data and cluster samples The treatment is very similar to that for probit models in Section 15.8, and so we make it...

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... could be estimated by OLS If, in addition, u1 and u2 were uncorrelated, equation (17.44) could be estimated by censored Tobit Correlation between u1 and u2 could be handled by the methods of Section ... Š ð17:47Þ Under joint normality of u1 and u2 , v is normally distributed Therefore, if b were known, b and a2 could be estimated by standard Tobit using z and z1 b as regressors Operationalizing ... j, denote this random sample by fwij : i ¼ 1; 2; ; Nj g The strata sample sizes Nj are nonrandom Therefore, the total sample size, N ¼ N1 þ Á Á Á þ NJ , is also nonrandom A randomly drawn observation...

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... typically accounts for the overdispersion and serial dependence implied by assumptions (19.57) and (19.58) For count data, the Poisson random e¤ects model is given by yit j x i ; ci @ Poisson½ci mðx ... unobserved e¤ects count data models was done by Hausman, Hall, and Griliches (1984) (HHG), who were interested in explaining patent applications by firms in terms of spending on research and development ... interpret the standard errors with caution when they are obtained under assumption (19.2) The GLM standard errors are easily obtained by multiplying the pffiffiffiffiffi ^ ^ Poisson standard errors by s s The...

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... ‘ Data and Econometric Tools for Development Analysis,’’ in Handbook of Development Economics, Volume 3A, ed J Berhman and T N Srinivasan Amsterdam: North Holland, 1785– 1882 Dehejia, R H., and ... (20.16) However, Cox’s method is intended to be applied to flow data as opposed to grouped data If we apply Cox’s methods to grouped data, we must confront the practically important issue of individuals ... for Within-Groups Estimators,’’ Oxford Bulletin of Economics and Statistics 49, 431–434 Arellano, M., and S R Bond (1991), ‘‘Some Specification Tests for Panel Data: Monte Carlo Evidence and an Application...

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