Further Reading
All the topics dealt with in this chapter are treated at greater length and depth in Cameron and Trivedi (1998) which also provides a comprehensive bibliography. Winkelmann (1997) also provides …
Basic Model
Suppose we have a set of N cross-sections with T time series observations on each. The classic data introduced in Zellner's (1962) initial work comprised firm - level investment data …
General linear structural equation models
Up till now, we have discussed several models that specify linear relations among observed and/or latent variables. Such models are called (linear) structural equation models. A general formulation of structural …
Other diagnostics
The expression for the variance of the least squares estimator in equation (12.2), for the regression model with two explanatory variables, can be extended to the multiple regression context, for …
Estimation
2.2 Maximum likelihood estimation Maximum likelihood (ML) estimation of spatial lag and spatial error regression models was first outlined by Ord (1975).21 The point of departure is an assumption of …
Sample Selection Bias
1.1 Self-selection The problem of selection bias in economics arises when sampling observations are generated from the population by rules other than simple random sampling. Consequently, the sample representation of …
Sample selection models with a tobit selection rule
For the semiparametric estimation of a sample selection model with a discrete choice equation, the exclusion restriction of a relevant regressor in the choice equation from the outcome equation provides …
Large sample properties of limited information estimators
We now consider the large sample asymptotic behavior of the limited information estimators described in Section 2. Because of space limitation, theorems are stated without proof. We start with a …
Dependent processes
The following weak LLN follows immediately from Corollary 1. In contrast to the above LLNs this theorem does not require the variables to be independently distributed, but only requires uncorrelatedness. …
Closing Remarks
We conclude by pointing out the main lessons of this essay. First, we have tools, the Belsley, Kuh, and Welsch (1980) collinearity diagnostics, which allow us to determine the form …
Continuous mixture models
The negative binomial model can be obtained in many different ways. The following justification using a mixture distribution is one of the oldest and has wide appeal. Suppose the distribution …
Hypothesis Testing
Much has been written on the problem of testing for serial correlation, particularly in the disturbances of the linear regression model. Given the nonexperimental nature of almost all economic data …
Measurement Error. and Latent Variables
Tom Wansbeek and Erik Meijer* 1 Introduction Traditionally, an assumption underlying econometric models is that the regressors are observed without measurement error. In practice, however, economic observations, micro and macro, …
Optimal Moments and Nearly Uninformative Moments
Throughout the analysis of the GMM estimator, we have taken the population moment condition as given. However, in practice, a researcher is typically faced with a large set of alternatives …
Practical Problems
In Section 5 we noted that the motivation for the Cox test statistic was based upon the observation that unless two models, say f(-) and g( ) are nonnested then …
Panel Data Models
Cheng Hsiao* 1 Introduction A panel (or longitudinal or temporal cross-sectional) data set is one which follows a number of individuals over time. By providing sequential observations for a number …
Stochastic Specification
Many of the recent developments in estimation of the parameters of the SUR model have been motivated by the need to allow for more general stochastic specifications. These developments are …
Diagnostic Testing in Cross Section Contexts
To obtain a unified view of diagnostic testing, it is important to use a modern perspective. This requires facility with concepts from basic probability, such as conditional expectations and conditional …
Collinearity-influential observations
One or two observations can make a world of difference in a data set, substantially improving, or worsening, the collinearity in the data. Can we find these "collinearity-influential" observations? If …
Spatial two-stage least squares
The endogeneity of the spatially lagged dependent variable can also be addressed by means of an instrumental variables or two-stage least squares (2SLS) approach (Anselin, 1980, 1988a, 1990; Kelejian and …
Some conventional sample selection models
The tobit model assumes that the censoring threshold is deterministic and known. A generalization of the tobit model assumes that the censoring threshold is an unobservable stochastic variable. This generalization …
Identification and estimation of counterfactual outcomes
As counterfactual outcomes are important objects of inference, one may be interested in the identification and estimation of counterfactual outcomes. The possible identification of counterfactual outcomes follows from model structures …
Large sample properties of full information estimators
Theorem 8. For a linear simultaneous equations model satisfying the assumptions in Theorem 2 and with a nonsingular error covariance matrix, the following asymptotic properties of the 3SLS estimator hold:
Uniform laws of large numbers
In the previous sections we have been concerned with various notions of convergence for sequences of random variables and random vectors. Sometimes one is confronted with sequences of random functions, …
Nonnested Hypothesis. Testing: An Overview
M. Hashem Pesaran and Melvyn Weeks 1 Introduction This chapter focuses on the hypotheses testing problem when the hypotheses or models under consideration are "nonnested" or belong to "separate" families …
Finite mixture models
The mixture model in the previous subsection was a continuous mixture model, as the mixing random variable v was assumed to have continuous distribution. An alternative approach instead uses a …
Testing disturbances in the dynamic linear regression model
Finally we turn our attention to the problem of testing for autocorrelation in the disturbances of the dynamic regression model (3.19). Whether the DW test can be used in these …
The Linear Regression Model with Measurement Error
The standard linear multiple regression model can be written as у = SP + e, (8.1) where у is an observable N-vector, e an unobservable N-vector of random variables, the …
Nearly uninformative moment conditions
While it is desirable to base estimation on the optimal moment conditions, this is not necessary. Even if the population moment condition is sub-optimal, the GMM framework can be used …
Resampling the likelihood ratio statistic: bootstrap methods
The bootstrap is a data based simulation method for statistical inference. The bootstrap approach involves approximating the distribution of a function of the observed data by the bootstrap distribution of …
Linear Models
Suppose there are observations of 1 + k1 + k2 variables (yit, x't, §'t) of N crosssectional units over T time periods, where i = 1,..., N, and t = …
Testing linear restrictions
Under the standard assumptions of the basic SUR model, Atkinson and Wilson (1992) prove that: var[S(9)] > var[$(9)] > E[X'(9-1 ® fT)X]-1 (5.11) where 9 is any unbiased estimator of …
Diagnostic testing in nonlinear models of conditional means
Much of what we discussed in Section 2.1 carries over to nonlinear models. With a nonlinear conditional mean function, it becomes even more important to state hypotheses in terms of …
Detecting harmful collinearity
We can determine the number of collinear relations, their severity, and the variables involved using the diagnostics in Section 3. This does not end the diagnostic process, because we must …
Method of moments estimators
Recently, a number of approaches have been outlined to estimate the coefficients in a spatial error model as an application of general principles underlying the method of moments. Kelejian and …
