Multi step GM/IV estimation of a linear Cliff and Ord -type of model of the form:
$$y=\lambda W y + X \beta + u $$ $$u=\rho W u + e$$ with $$e ~ N(0,\sigma^2_i) $$
The model allows for spatial lag in the dependent variable and disturbances. The innovations in the disturbance process are assumed heteroskedastic of an unknown form.
gstslshet(formula, data = list(), listw, na.action = na.fail,
zero.policy = NULL, initial.value = 0.2, abs.tol = 1e-20,
rel.tol = 1e-10, eps = 1e-5, inverse = T, sarar = T)a description of the model to be fit
an object of class data.frame. An optional data frame containing the variables in the model.
an object of class listw created for example by nb2listw
a function which indicates what should happen when the data contains missing values. See lm for details.
See lagsarlm for details
The initial value for \(\rho\). It can be either numeric (default is 0.2) or
set to 'SAR', in which case the optimization will start from the estimated coefficient of a regression of the 2SLS
residuals over their spatial lag (i.e. a spatial AR model)
Absolute tolerance. See nlminb for details.
Relative tolerance. See nlminb for details.
Tolerance level for the approximation. See Details.
TRUE. If FALSE, an appoximated inverse is calculated. See Details.
TRUE. If FALSE, a spatial error model is estimated.
A list object of class sphet
Generalized Spatial two stage least squares coefficient estimates of \(\delta\) and GM estimator for \(\rho\).
variance-covariance matrix of the estimated coefficients
GS2SLS residuals variance
GS2SLS residuals
difference between GS2SLS residuals and response variable
the call used to create this object
the model matrix of data
'gs2slshac'
Wald test for both \(\rho\) and \(\lambda\) are zero
The procedure consists of two steps alternating GM and IV estimators. Each step consists of sub-steps. In step one \(\delta = [\beta',\lambda]'\) is estimated by 2SLS. The 2SLS residuals are first employed to obtain an initial (consistent but not efficient) GM estimator of \(\rho\) and then a consistent and efficient estimator (involving the variance-covariance matrix of the limiting distribution of the normalized sample moments). In step two, the spatial Cochrane-Orcutt transformed model is estimated by 2SLS. This corresponds to a GS2SLS procedure. The GS2SLS residuals are used to obtain a consistent and efficient GM estimator for \(\rho\).
The initial value for the optimization in step 1b is taken to be initial.value. The initial value in step 1c is the
optimal parameter of step 1b. Finally, the initial value for the optimization of step 2b is the optimal parameter of step 1c.
Internally, the object of class listw is transformed into a Matrix
using the function listw2dgCMatrix.
The expression of the estimated variance covariance matrix of the limiting
distribution of the normalized sample moments based on 2SLS residuals
involves the inversion of \(I-\rho W'\).
When inverse is FALSE, the inverse is calculated using the approximation
\(I +\rho W' + \rho^2 W'^2 + ...+ \rho^n W'^n\).
The powers considered depend on a condition.
The function will keep adding terms until the absolute value of the sum of all elements
of the matrix \(\rho^i W^i\) is greater than a fixed \(\epsilon\) (eps). By default eps
is set to 1e-5.
Arraiz, I. and Drukker, M.D. and Kelejian, H.H. and Prucha, I.R. (2007) A spatial Cliff-Ord-type Model with Heteroskedastic Innovations: Small and Large Sample Results, Department of Economics, University of Maryland'
Kelejian, H.H. and Prucha, I.R. (2007) Specification and Estimation of Spatial Autoregressive Models with Autoregressive and Heteroskedastic Disturbances, Journal of Econometrics, forthcoming.
Kelejian, H.H. and Prucha, I.R. (1999) A Generalized Moments Estimator for the Autoregressive Parameter in a Spatial Model, International Economic Review, 40, pages 509--533.
Kelejian, H.H. and Prucha, I.R. (1998) A Generalized Spatial Two Stage Least Square Procedure for Estimating a Spatial Autoregressive Model with Autoregressive Disturbances, Journal of Real Estate Finance and Economics, 17, pages 99--121.
# NOT RUN {
data(columbus, package = "spdep")
listw <- spdep::nb2listw(col.gal.nb)
res <- gstslshet(CRIME ~ HOVAL + INC, data = columbus, listw = listw)
summary(res)
# }
Run the code above in your browser using DataLab