GM estimation of a Cliff-Ord type model with Heteroskedastic Innovations

Description

Multi step GM/IV estimation of a linear Cliff and Ord -type of model of the form:

y=λ W y + X β + u

u=ρ W u + e

with

e ~ N(0,σ^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.

Usage

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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)
## S3 method for class 'gstsls'
impacts(obj, ..., tr, R = NULL, listw = NULL,
tol = 1e-06, empirical = FALSE, Q=NULL)

Arguments

formula

a description of the model to be fit

data

an object of class data.frame. An optional data frame containing the variables in the model.

listw

an object of class listw created for example by nb2listw

na.action

a function which indicates what should happen when the data contains missing values. See lm for details.

zero.policy

See lagsarlm for details

initial.value

The initial value for ρ. 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)

abs.tol

Absolute tolerance. See nlminb for details.

rel.tol

Relative tolerance. See nlminb for details.

eps

Tolerance level for the approximation. See Details.

inverse

TRUE. If FALSE, an appoximated inverse is calculated. See Details.

sarar

TRUE. If FALSE, a spatial error model is estimated.

obj

A gstsls spatial regression object created by gstslshet

...

Arguments passed through to methods in the coda package

tr

A vector of traces of powers of the spatial weights matrix created using trW, for approximate impact measures; if not given, listw must be given for exact measures (for small to moderate spatial weights matrices); the traces must be for the same spatial weights as were used in fitting the spatial regression

R

If given, simulations are used to compute distributions for the impact measures, returned as mcmc objects

tol

Argument passed to mvrnorm: tolerance (relative to largest variance) for numerical lack of positive-definiteness in the coefficient covariance matrix

empirical

Argument passed to mvrnorm (default FALSE): if true, the coefficients and their covariance matrix specify the empirical not population mean and covariance matrix

Q

default NULL, else an integer number of cumulative power series impacts to calculate if tr is given

Details

The procedure consists of two steps alternating GM and IV estimators. Each step consists of sub-steps. In step one δ = [β',λ]' is estimated by 2SLS. The 2SLS residuals are first employed to obtain an initial (consistent but not efficient) GM estimator of ρ 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 ρ.

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-ρ W'. When inverse is FALSE, the inverse is calculated using the approximation I +ρ W' + ρ^2 W'^2 + ...+ ρ^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 ρ^i W^i is greater than a fixed ε (eps). By default eps is set to 1e-5.

Value

A list object of class sphet

coefficients

Generalized Spatial two stage least squares coefficient estimates of δ and GM estimator for ρ.

var

variance-covariance matrix of the estimated coefficients

s2

GS2SLS residuals variance

residuals

GS2SLS residuals

yhat

difference between GS2SLS residuals and response variable

call

the call used to create this object

model

the model matrix of data

method

'gs2slshac'

W

Wald test for both ρ and λ are zero

Author(s)

Gianfranco Piras gpiras@mac.com

References

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.

See Also

stslshac

Examples

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library(spdep)
data(columbus)
listw<-nb2listw(col.gal.nb)
res<-gstslshet(CRIME~HOVAL + INC, data=columbus, listw=listw)
summary(res)