A COMPANION TO Theoretical Econometrics

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 a spatially lagged dependent variable (Wy), or in the error structure (Е[є;є;] Ф 0). The former is referred to as a spatial lag model and is appropriate when the focus of interest is the assessment of the existence and strength of spatial interaction. This is inter­preted as substantive spatial dependence in the sense of being directly related to a spatial model (e. g. a model that incorporates spatial interaction, yardstick com­petition, etc.). Spatial dependence in the regression disturbance term, or a spatial error model is referred to as nuisance dependence. This is appropriate when the concern is with correcting for the potentially biasing influence of the spatial autocorrelation, due to the use of spatial data (irrespective of whether the model of interest is spatial or not).

Formally, a spatial lag model, or a mixed regressive, spatial autoregressive model is expressed as

y = pWy + Xp + e, (14.9)

where p is a spatial autoregressive coefficient, e is a vector of error terms, and the other notation is as before.16 Unlike what holds for the time series counterpart of this model, the spatial lag term Wy is correlated with the disturbances, even when the latter are iid. This can be seen from the reduced form of (14.9),

y = (I - pW)-1Xp + (I - pW)-1E, (14.10)

in which each inverse can be expanded into an infinite series, including both the explanatory variables and the error terms at all locations (the spatial multiplier). Consequently, the spatial lag term must be treated as an endogenous variable and proper estimation methods must account for this endogeneity (OLS will be biased and inconsistent due to the simultaneity bias).

A spatial error model is a special case of a regression with a non-spherical error term, in which the off-diagonal elements of the covariance matrix express the structure of spatial dependence. Consequently, OLS remains unbiased, but it is no longer efficient and the classical estimators for standard errors will be biased. The spatial structure can be specified in a number of different ways, and (except for the non-parametric approaches) results in a error variance-covariance matrix of the form

Подпись: (14.11)E[ee'] = Q(0),

where 0 is a vector of parameters, such as the coefficients in an SAR error

process.

Добавить комментарий

A COMPANION TO Theoretical Econometrics

Normality tests

Let us now consider the fundamental problem of testing disturbance normality in the context of the linear regression model: Y = Xp + u, (23.12) where Y = (y1, ..., …

Univariate Forecasts

Univariate forecasts are made solely using past observations on the series being forecast. Even if economic theory suggests additional variables that should be useful in forecasting a particular variable, univariate …

Further Research on Cointegration

Although the discussion in the previous sections has been confined to the pos­sibility of cointegration arising from linear combinations of I(1) variables, the literature is currently proceeding in several interesting …

Как с нами связаться:

Украина:
г.Александрия
тел./факс +38 05235  77193 Бухгалтерия
+38 050 512 11 94 — гл. инженер-менеджер (продажи всего оборудования)

+38 050 457 13 30 — Рашид - продажи новинок
e-mail: msd@msd.com.ua
Схема проезда к производственному офису:
Схема проезда к МСД

Партнеры МСД

Контакты для заказов шлакоблочного оборудования:

+38 096 992 9559 Инна (вайбер, вацап, телеграм)
Эл. почта: inna@msd.com.ua

За услуги или товары возможен прием платежей Онпай: Платежи ОнПай