This book aims at meeting the growing demand in the field by introducing the basic spatial econometrics methodologies to a wide variety of researchers. It provides a practical guide that illustrates the potential of spatial econometric modelling, discusses problems and solutions and interprets empirical results.
non-stochastic exogenous regressors including a constant term,
a vector of k unknown parameters to be estimated and
a
vector of stochastic disturbances. We will assume throughout the book that the n observations refer to territorial units such as regions or countries.
The classical linear regression model assumes normality, identicity and independence of the stochastic disturbances conditional upon the k regressors. In short
Δ | X â i.i.d.N (0, Ï2Δn In) (1.2)
n In being an n-by-n identity matrix. Equation (1.2) can also be written as:
E(Δ | X) = 0(1.3)
E(ΔΔT | X) = Ï2Δn In (1.4)
Equation (1.3) corresponds to the assumption of exogeneity, Equation (1.4) to the assumption of spherical disturbances (Greene, 2011).
Furthermore it is assumed that the k regressors are not perfectly dependent on one another (full rank of matrix X). Under this set of hypotheses the Ordinary Least Squares fitting criterion (OLS) leads to the best linear unbiased estimators (BLUE) of the vector of parameters ÎČ, say
OLS =
. In fact the OLS criterion requires:
S(ÎČ) = eT e = min (1.5)
where e = y â X
are the observed errors and eT indicates the transpose of e.
From Equation (1.5) we have:
whence:
OLS = (XT X)-1XT y (1.6)
As said the OLS estimator is unbiased
E(
OLS | X) = ÎČ (1.7)
with a variance
Var(
OLS | X) = (XT X)â1Ï2Δ (1.8)
which achieves the minimum among all possible linear estimators (full efficiency) and tends to zero when n tends to infinity (weak consistency).
From the assumption of normality of the stochastic disturbances, normality of the estimators also follows:
OLS | X â N[ÎČ; (XT X)â1Ï2Δ] (1.9)
Furthermore, from the assumption of normality of the stochastic disturbances, it also follows that the alternative estimators, based on the Maximum Likelihood criterion (ML), coincide with the OLS solution.
In fact, the single stochastic disturbance is distributed as:
f being a density function, and consequently the likelihood of the observed sample is:
(1.10)
from the assumption of independence of the disturbances. From (1.1) we have that
Δ = y â XÎČ (1.11)
hence (1.10) can be written as:
(1.12)
and the log-likelihood as:
(1.13)
The scores functions are defined as:
(1.14)
and solving the system of k + 1 equations, we have:
(1.15)
Thus, under the hypothesis of normality of residuals, the ML estimator of ÎČ coincides with the OLS estimator. The ML estimator of
on the contrary differs from the unbiased estimator
and it is biased, but asymptotically unbiased.
To ensure that the solution obtained is a maximum we consider the second derivatives:
(1.16)
which can be arranged in the Fisherâs Information Matrix:
(1.17)
which is positive definite.
The equivalence between the ML and the OLS estimators ensures that the solution found enjoys all the large sample properties of the ML estimators, that is to say: asymptotic normality, consistency, asymptotic unbiasedness, full efficiency with respect to a larger class of estimators other than the linear...
Table des matiĂšres
Cover
Title
Copyright
Contents
List of Figures
List of Examples
Foreword by William Greene
Preface and Acknowledgements
1 The Classical Linear Regression Model
2 Some Important Spatial Definitions
3 Spatial Linear Regression Models
4 Further Topics in Spatial Econometrics
5 Alternative Model Specifications for Big Datasets
6 Conclusions: Whatâs Next?
Solutions to the Exercises
Index
Normes de citation pour A Primer for Spatial Econometrics
APA 6 Citation
Arbia, G. (2014). A Primer for Spatial Econometrics ([edition unavailable]). Palgrave Macmillan UK. Retrieved from https://www.perlego.com/book/3486551/a-primer-for-spatial-econometrics-with-applications-in-r-pdf (Original work published 2014)
Chicago Citation
Arbia, G. (2014) 2014. A Primer for Spatial Econometrics. [Edition unavailable]. Palgrave Macmillan UK. https://www.perlego.com/book/3486551/a-primer-for-spatial-econometrics-with-applications-in-r-pdf.
Harvard Citation
Arbia, G. (2014) A Primer for Spatial Econometrics. [edition unavailable]. Palgrave Macmillan UK. Available at: https://www.perlego.com/book/3486551/a-primer-for-spatial-econometrics-with-applications-in-r-pdf (Accessed: 15 October 2022).
MLA 7 Citation
Arbia, G. A Primer for Spatial Econometrics. [edition unavailable]. Palgrave Macmillan UK, 2014. Web. 15 Oct. 2022.