Bayesian methods
While increased computational power has had important implications for many areas of econometrics, the impact has probably been most dramatic in the area of Bayesian econometrics. It has long been …
Modes of Convergence for Sequences of Random Vectors
In this section we define and discuss various modes of convergence for sequences of random vectors taking their values in Rk. 1.1 Convergence in probability, almost surely, and in rth …
Estimation methods designed specifically for collinear data
A number of estimation methods have been developed to improve upon the least squares estimator when collinearity is present. We will briefly discuss two, ridge regression and principal components regression, …
Poisson Regression
The Poisson is the starting point for count data analysis, though it is often inadequate. In Sections 2.1-2.3 we present the Poisson regression model and estimation by maximum likelihood, interpretation …
Misspecification and tests
In the sample selection model (18.1), the data on observed outcomes y are censored and least squares estimators of в obtained using y suffer from selectivity bias when the disturbances …
Serial Correlation
Maxwell L. King 1 Introduction In its most general form, serial correlation involves the correlation of successive time series observations. It has been the subject of much research in econometrics …
Basic Concepts
Consider a structure s that describes the probability distribution function Ps(y) of a random vector Y. The set of all a priori possible structures is called a model. We assume …
The Estimator and a Fundamental Decomposition
The introduction provides the essence of GMM. In this section, we discuss the form of the estimator in more detail, and also describe how estimation effects a fundamental decomposition of …
The Cox procedure
This procedure focuses on the loglikelihood ratio statistic, and in the case of the above regression models is given by (using the notations of Section 2): T ln 2 zT …
Semiparametric models
By semiparametric models we mean partially parametric models that have an infinite-dimensional component. One example is optimal estimation of the regression parameters p, when p, = exp(x'P) is assumed but …
An Empirical Example
In this section, we use annual UK data to estimate the demand for money (i. e. equation (19.12) extended to include one additional explanatory variable), with and without concomitants, over …
Testing for heteroskedasticity
Assuming that hi(a) is such that h;(0) = 1, tests for heteroskedasticity can be formulated in terms of the hypotheses H0 : a = 0 H1 : a Ф 0. …
Panel data
Panel data are repeated measurements over time for a set of cross-sectional units (e. g. households, firms, regions). The additional time dimension allows for consistent estimation when there is measurement …
Diagnosing collinearity using the singular value decomposition
The singular-value decomposition is a factorization of X. The matrix X may be decomposed as X = UA1/2C', where U'U = C' C = CC = IK and A1/2 is …
Aymptotics in spatial stochastic processes
As in time series analysis, the properties of estimators and tests for spatial series are derived from the asymptotics for stochastic processes. However, these properties are not simply extensions to …
Panel Data with Qualitative Variables
Early reviews of models of panel data with qualitative variables are in Heckman (1981) and Chamberlain (1980, 1984). This work in the early 1980s is reviewed in Maddala (1987) which …
Improved estimation
In a series of papers Hill, Cartwright, and Arbaugh (1990, 1991, 1992) consider the performance of conventional FGLS compared to various improved estimators when applied to a basic SUR model. …
Convergence in distribution
Let 0n be an estimator for a real-valued parameter 0 and assume 0n -— 0. If Gn denotes the cumulative distribution function (CDF) of 0n, i. e., Gn(z) = P(0n …
Artificial orthogonalization
Consider a regression model containing two explanatory variables, y = в1 + в,*2 + вз*э + e, (12.17) where the regressors x2 and x3 are highly correlated. To "purge" the …
Interpretation of regression coefficients
For linear models, with E[y |x] = x'P, the coefficients в are readily interpreted as the effect of a one-unit change in regressors on the conditional mean. For nonlinear models …
Semiparametric and Nonparametric Approaches
1.3 Semiparametric two-stage estimation Manski (1975) showed that a parametric distribution is not necessary for consistent estimation of discrete choice models, and thus originated the semiparametric estimation literature in microeconometrics. …
The Box-Jenkins class of models
The simplest time series model of serial correlation is the first-order autoregressive (AR(1)) process, which for a time series, yt, t = 1,..., n, with mean zero can be written …
The Jacobian Matrix Criterion
The advantage of Theorem 2 is that we do not have to operate on the joint probability distribution of the underlying random variables directly when analyzing the identification of a …
Asymptotic Properties
At the beginning, we presented an intuitive justification for the GMM estimation framework. In this section, we provide a more rigorous argument by establishing the consistency and asymptotic normality of …
The comprehensive approach
Another approach closely related to the Cox's procedure is the comprehensive approach advocated by Atkinson (1970) whereby tests of nonnested models are based upon a third comprehensive model, artificially constructed …
Time Series, Multivariate and Panel Data
In this section we very briefly present extension from cross section to other types of count data (see Cameron and Trivedi, 1998, for further detail). For time series and multivariate …
Other extensions
Rilestone (1991) has compared the relative efficiency of semiparametric and parametric estimators of в under different types of heteroskedasticity, whereas Surekha and Griffiths (1984) compare the relative efficiency of some …
Latent Variable Models
Consider again the bivariate measurement error model (8.7), where the unobservable random variables En, є n, and vn are assumed to be mutually independent with expectation zero. The variables yn …
Collinearity and the least squares predictor
Another bit of conventional wisdom is that while collinearity may affect the precision of the least squares estimator, it need not affect the reliability of predictions based on it, if …
Spatial Regression Models
2.1 Spatial lag and spatial error models In the standard linear regression model, spatial dependence can be incorporated in two distinct ways: as an additional regressor in the form of …
Semiparametric Estimation
We have seen that in the estimation of QRM the assumption is made about the error term being distributed according to some known distribution (i. e. logistic or normal). The …
Simultaneous. Equation Model. Estimators: Statistical. Properties and. Practical Implications
Roberto S. Mariano 1 The Linear Simultaneous Equations Model This chapter deals with the statistical properties of estimators in simultaneous equation models. The discussion covers material that extends from the …
Convergence properties and transformations
We are often interested in the convergence properties of transformed random vectors or variables. In particular, suppose Zn converges to Z in a certain mode, then given a function g …
Nonlinear Models
Assessing the severity and consequences of collinearity in nonlinear models is more complicated than in linear models. To illustrate, we first discuss its detection in nonlinear regression, and then in …
Truncation and censoring
In some studies, inclusion in the sample requires that sampled individuals have been engaged in the activity of interest. Then the count data are truncated, as the data are observed …
