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In the next section we give an overview of the important features of the. Because Eđy j xỡ is an expectation, it can be obtained from the conditional density of y given x by integration, summation, or a combination of the two (depending on the nature of y). 0 đ2:5ỡ The model in equation (2.2) is linear in the explanatory...
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definition 3.1: (1) A sequence of nonrandom numbers fa N : N Ử 1. We write a N ! a as N ! y. y such that ja N j a b for all N Ử 1. Example 3.1: (1) If a N Ử 2 ợ 1=N, then a N ! 2. definition 3.2: (1) A sequence fa N g...
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4.1 Overview of the Single-Equation Linear Model. The error form of the model in equation (4.1) is useful for presenting a unified treatment of the statistical properties of various econometric procedures. Neverthe- less, the steps one uses for getting to equation (4.1) are just as important. From Chapter 2 we know that equation (4.1) and assumption (4.2) are equivalent to...
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So, we may be interested in a conditional expectation as in equation (4.18), but we do not observe q, and q is correlated with x K. As we saw in Chapter 4, OLS estimation of equation (5.1) generally results in in- consistent estimators of all the b j if Covðx K . endogenous, we need an observable variable, z 1...
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From this relation it is easily shown that the usual OLS assumption in the population—. Since ^ d d ^ is also obtained using data—usually the same sample of data—uncertainty in the estimate should be accounted for in the second step.. -limiting distribution of the OLS estimators from regression (6.2) is the same as the OLS estimators when q replaces...
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Then, we assume we have a random sample of firms that have data in each of the T years.) The panel data model covered here, while having many useful applications, does not fully exploit the replicability over time. Example 7.1 (Seemingly Unrelated Regressions): The population model is a set of G linear equations,. Even if equation (7.7) contains enough lags...
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The labor supply function can be derived from individual utility-maximizing behavior, and the nota- tion in equation (8.1) is intended to emphasize that, for given z 1 and u 1 , a labor supply function gives the desired hours worked at any possible wage đwỡ facing the worker. But the counterfactual reasoning underlying equation (8.1) is the proper way to...
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In fact, in the. the endogenous variables in the system are all choice variables of the same economic unit.. Usually there is some overlap in the exogenous variables. Assumption (9.2) implies that the exogenous variables appearing anywhere in the system are orthogonal to all the structural errors. Generally, in the system (9.1), the error u g in equation g will...
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10.1 Motivation: The Omitted Variables Problem. 2 ð10:5Þ One implication of condition (10.5) is. Then, di¤erencing equation (10.4) gives. Given a panel data set with two time periods, equation (10.7) is just a standard cross section equation. Consider condition (10.8) first. 10.2 Assumptions about the Unobserved E¤ects and Explanatory Variables. 10.2.1 Random or Fixed E¤ects?. 10.2.2 Strict Exogeneity Assumptions on...
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Given model (11.1), assumption (11.2) is equivalent to. Example 11.1 (Dynamic Unobserved E¤ects Model): An AR(1) model with addi- tional explanatory variables is. y i0 Þ, and the sequential exogeneity assumption (11.3) requires. In this example, assumption (11.5) is an example of dynamic completeness condi- tional on c i . Assumption (11.3) does not require that z i;tþ1. 0 under...
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12.1 Introduction. Given equation (12.2), equation (12.1) clearly holds. Conversely, given equation (12.1), we obtain equation (12.2) by defining the error to be u 1 y mðx. Thus, taking the expected value of equation (12.4) gives. yÞ 2 g ð12:5Þ Since the last term in equation (12.5) is nonnegative, it follows that. The NLS objective function in expression (12.7) is...
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13.1 Introduction. 13.2 Preliminaries and Examples. Example 13.1 (Probit): Suppose that the latent variable y i follows. (13.3) To be succinct, it is useful to write equations (13.2) and (13.3) in terms of the indi- cator function, denoted 1ơ . Thus, equations (13.2) and (13.3) are equivalently written as y i Ử 1ơy i >. Because e i is normally...
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14.1 Asymptotic Properties of GMM. theorem 14.1 (Consistency of GMM): Assume that (a) Y is compact. Then a random vector y y ^ exists that solves problem (14.2), and y y. This assumption essentially means that the moment conditions (14.1) are nonredundant. theorem 14.2 (Asymptotic Normality of GMM): In addition to the assumptions in Theorem 14.1, assume that (a) y...
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15.1 Introduction. 15.2 The Linear Probability Model for Binary Response. Equation (15.5) implies that, given a random sample, the OLS regression of y on 1. Equation (15.6) means that heteroskedasticity is present unless all of the slope co- e‰cients b 1. 15.3 Index Models for Binary Response: Probit and Logit We now study binary response models of the form. The...
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16.1 Introduction and Motivation. Example 16.1 (Top Coding of Wealth): In the population of all families in the United States, let wealth denote actual family wealth, measured in thousands of dollars. Example 16.2 (Charitable Contributions): Problem 15.1 shows how to derive a probit model from a utility maximization problem for charitable giving, using utility function util i ðc. Comparing equations...
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17.1 Introduction. Example 17.2 (Truncation Based on Wealth): We are interested in estimating the eấect of worker eligibility in a particular pension plan [for example, a 401(k) plan] on family wealth. Example 17.3 (Wage Oấer Function): Consider estimating a wage oấer equation for people of working age. 17.2 When Can Sample Selection Be Ignored?. 17.2.1 Linear Models: OLS and 2SLS....
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18.1 Introduction. 18.2 A Counterfactual Setting and the Self-Selection Problem. similarly, equation (18.2) becomes Eðy 1 y 0 j x. Using equation (18.3), we have Eð y j w ¼ 1Þ ¼ Eðy 1 j w ¼ 1Þ ¼ Eðy 1 Þ. Unfortunately, condition (18.6) is a strong assumption. 18.3 Methods Assuming Ignorability of Treatment. 18.3.1 Regression Methods. We can...
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19.1 Why Count Data Models?. In Section 19.2 we discuss the most popular model for count data, the Poisson re- gression model. 19.2 Poisson Regression Models with Cross Section Data. 19.2.1 Assumptions Used for Poisson Regression. In what follows we refer to assumption (19.2) as the Poisson variance assumption.. This assumption is used in the generalized linear models (GLM) literature,...
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20.1 Introduction. (We will discuss the exact nature of censoring in Sections 20.3 and 20.4.) The calendar dates on which units enter the initial state do not have to be the same. In Section 20.4 we study methods that allow flexible estimation of a hazard func- tion, both with time-constant and time-varying covariates. 20.2 Hazard Functions. 20.2.1 Hazard Functions without...
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Good theory helps you grasp what is going on in the world around you. Here are some of the major changes:. same time we have rewritten Chapter 14 as a free- standing introduction to the nature of the corpora- tion, to the major sources of corporate financing, and to financial markets and institutions. In fact we have revised and updated...