Parametric Estimation
1.2 Two-stage estimation Consider the estimation of the model (18.1). Assume that u and e are jointly normally distributed with zero-means and the variance of e being normalized to be …
Structure of Limited Information Estimators as Regression Functions
The structure of the SEM estimators is discussed at length in Hendry (1976) and Hausman (1983), and Phillips (1983). In this section, we develop a perspective on the structure of …
Central Limit Theorems
Let Zt, t Є N, be a sequence of iid random variables with EZ t = p and var(Z t) = a2, 0 < a2 < ro. Let Zn = …
Examples of Nonnested Models
We start with examples of unconditional nonnested models. One such example, originally discussed by Cox (1961) is that of testing a lognormal versus an exponential distribution. Hf : f(y I …
Modified count models
The leading motivation for modified count models is to solve the so-called problem of excess zeros, the presence of more zeros in the data than predicted by count models such …
Random Coefficient. Models
P. A.V. B. Swamy and George S. Tavlas* 1 Introduction Random coefficient models (RCMs) grew out of Zellner's path-breaking article (1969) on aggregation and have undergone considerable modification over time.1 …
Model Selection
A difficult question when modeling economic behavior is to decide on what lags should be in the ARIMA model, the ARMA disturbance model, or the dynamic regression model. It is …
Inconsistency and bias of the OLS estimators
Given the setup, the probability limits of b and s2, for both the structural and the functional model, are к = plim b = p - XX‘Qp = X^p. (8.4) …
Finite Sample Behavior
The foregoing discussion has rested upon asymptotic theory. In finite samples, such theory can only provide an approximation. It is therefore important to assess the quality of this approximation in …
Spatial Econometrics
Luc Anselin* 1 Introduction Spatial econometrics is a subfield of econometrics that deals with spatial interaction (spatial autocorrelation) and spatial structure (spatial heterogeneity) in regression models for cross-sectional and panel …
Dynamic Models
When xit and/or zit contain lagged dependent variables, because a typical panel contains a large number of cross-sectional units followed over a short period of time, it turns out that …
Diagnostic testing
A cornerstone of the SUR model is the presence of contemporaneous correlation. When X is diagonal, joint estimation is not required, which simplifies computations. Shiba and Tsurumi (1988) provide a …
Diagnostic Testing in Time Series Contexts
All of the tests we covered in Section 2 can be applied in time series contexts. However, because we can no longer assume that the observations are independent of one …
What to Do?
In this section we address the question of what to do when harmful collinearity is found with respect to the important parameters in a regression model. This section is like …
Specification Tests
2.4 Moran's I The most commonly used specification test for spatial autocorrelation is derived from a statistic developed by Moran (1948) as the two-dimensional analog of a test for univariate …
Polychotomous choice sample selection models
A polychotomous choice model with m-alternatives specifies the latent utility values Uj = гу + v, j = 1,..., m, and the alternative j is chosen if and only if …
Finite Sample Properties of Estimators
2.1 Arbitrary number of included endogenous variables Let us first consider the available results dealing with the general case where there are no further restrictions on the linear system nor …
Further Readings
There is a large number of books available that provide further in-depth discussions of the material (or parts of the material) presented in this article. The list of such books …
Model Selection Versus Hypothesis Testing
Hypothesis testing and model selection are different strands in the model evaluation literature. However, these strands differ in a number of important respects which are worth emphasizing here.10 Model selection …
Discrete choice models
Count data can be modeled by discrete choice model methods, possibly after some grouping of counts to limit the number of categories. For example, the categories may be 0, 1, …
Some First-Generation RCMs
When considering situations in which the parameters of a regression model are thought to change - perhaps as frequently as every observation - the parameter variation must be given structure …
Heteroskedasticity
William E. Griffiths* 1 Introduction A random variable y is said to be heteroskedastic if its variance can be different for different observations. Conversely, it is said to be homoskedastic …
Bounds on the parameters
Let us return to the bivariate regression model (no intercept, all variables having mean zero), written in scalar notation: Vn = №n + є n (8.7a) Xn = £ n …
Collinearity
R. Carter Hill and Lee C. Adkins* Multicollinearity is God's will, not a problem with OLS or statistical techniques in general. Blanchard (1987, p. 49) 1 Introduction Collinearity, a devilish …
Spatial autocorrelation
In a regression context, spatial effects pertain to two categories of specifications. One deals with spatial dependence, or its weaker expression, spatial autocorrelation, and the other with spatial heterogeneity.3 The …
Sample Attrition and Sample Selection
Missing observations occur frequently in panel data. If individuals are missing randomly, most estimation methods for the balanced panel can be extended in a straightforward manner to the unbalanced panel …
Other Developments
5.1 Unequal observations and missing data Extending the standard SUR model to allow for an unequal number of observations in different equations causes some problems for estimation of the disturbance …
Omnibus tests on the errors in time series regression models
Omnibus tests on the errors in time series regression models have recently become popular. A good example is the so-called BDS test (Brock, Dechert, LeBaron, and Scheinkman, 1996), which has …
Methods for introducing exact nonsample information
The most familiar method for introducing nonsample information into a regression model is to use restricted least squares (RLS). The restricted least squares estimator, which we denote as b*, is …
Implementation Issues
To date, spatial econometric methods are not found in the main commercial econometric and statistical software packages, although macro and scripting facilities may be used to implement some estimators (Anselin …
Simulation estimation
The multinomial probit model has long been recognized as a useful discrete choice model. But, because its choice probability does not have a closed-form expression and its computation involves multiple …
Practical Implications
Instead of going into more complicated formulas for pdfs or cdfs or moments of estimators, we devote this section to a discussion of the practical implications of the finite sample …
Generalized Method. of Moments
Alastair R. Hall* 1 Introduction Generalized method of moments (GMM) was first introduced into the econometrics literature by Lars Hansen in 1982. Since then, GMM has had considerable impact on …
Alternative Approaches to Testing Nonnested Hypotheses
with Application to Linear Regression Models To provide an intuitive introduction to concepts which are integral to an understanding of nonnested hypothesis tests we consider testing of linear regression models …
Partially Parametric Models
By partially parametric models we mean that we focus on modeling the data via the conditional mean and variance, and even these may not be fully specified. In Section 4.1 …
