lagsarlm: Spatial simultaneous autoregressive lag model estimation

Description

The lagsarlm function provides Maximum likelihood estimation of spatial simultaneous autoregressive lag and spatial Durbin (mixed) models of the form:

$$y = \rho W y + X \beta + \varepsilon$$

where $rho$ is found by optimize() first, and $beta$ and other parameters by generalized least squares subsequently (one-dimensional search using optim performs badly on some platforms). In the spatial Durbin (mixed) model, the spatially lagged independent variables are added to X. Note that interpretation of the fitted coefficients should use impact measures, because of the feedback loops induced by the data generation process for this model. With one of the sparse matrix methods, larger numbers of observations can be handled, but the interval= argument may need be set when the weights are not row-standardised.

The spBreg_lag function is an early-release version of the Matlab Spatial Econometrics Toolbox function sar_g.m, using drawing by inversion, and not accommodating heteroskedastic disturbances.

Usage

lagsarlm(formula, data = list(), listw, 
	na.action, type="lag", method="eigen", quiet=NULL, 
	zero.policy=NULL, interval=NULL, tol.solve=1.0e-10, trs=NULL, 
	control=list())
spBreg_lag(formula, data = list(), listw, na.action, type="lag", zero.policy=NULL, control=list())

Value

A list object of class sarlmThe internal sar.lag.mixed.* functions return the value of the log likelihood function at $rho$.

Control arguments

Extra Bayesian control arguments

Details

The asymptotic standard error of $rho$ is only computed when method=eigen, because the full matrix operations involved would be costly for large n typically associated with the choice of method="spam" or "Matrix". The same applies to the coefficient covariance matrix. Taken as the asymptotic matrix from the literature, it is typically badly scaled, and with the elements involving $rho$ being very small, while other parts of the matrix can be very large (often many orders of magnitude in difference). It often happens that the tol.solve argument needs to be set to a smaller value than the default, or the RHS variables can be centred or reduced in range.

Versions of the package from 0.4-38 include numerical Hessian values where asymptotic standard errors are not available. This change has been introduced to permit the simulation of distributions for impact measures. The warnings made above with regard to variable scaling also apply in this case.

Note that the fitted() function for the output object assumes that the response variable may be reconstructed as the sum of the trend, the signal, and the noise (residuals). Since the values of the response variable are known, their spatial lags are used to calculate signal components (Cressie 1993, p. 564). This differs from other software, including GeoDa, which does not use knowledge of the response variable in making predictions for the fitting data.

References

Cliff, A. D., Ord, J. K. 1981 Spatial processes, Pion; Ord, J. K. 1975 Estimation methods for models of spatial interaction, Journal of the American Statistical Association, 70, 120-126; Anselin, L. 1988 Spatial econometrics: methods and models. (Dordrecht: Kluwer); Anselin, L. 1995 SpaceStat, a software program for the analysis of spatial data, version 1.80. Regional Research Institute, West Virginia University, Morgantown, WV; Anselin L, Bera AK (1998) Spatial dependence in linear regression models with an introduction to spatial econometrics. In: Ullah A, Giles DEA (eds) Handbook of applied economic statistics. Marcel Dekker, New York, pp. 237-289; Cressie, N. A. C. 1993 Statistics for spatial data, Wiley, New York; LeSage J and RK Pace (2009) Introduction to Spatial Econometrics. CRC Press, Boca Raton.

Roger Bivand, Gianfranco Piras (2015). Comparing Implementations of Estimation Methods for Spatial Econometrics. Journal of Statistical Software, 63(18), 1-36. http://www.jstatsoft.org/v63/i18/.

Bivand, R. S., Hauke, J., and Kossowski, T. (2013). Computing the Jacobian in Gaussian spatial autoregressive models: An illustrated comparison of available methods. Geographical Analysis, 45(2), 150-179.

Examples

Run this code

data(oldcol)
COL.lag.eig <- lagsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 nb2listw(COL.nb, style="W"), method="eigen", quiet=FALSE)
summary(COL.lag.eig, correlation=TRUE)
COL.lag.eig$fdHess
COL.lag.eig$resvar
W <- as(nb2listw(COL.nb), "CsparseMatrix")
trMatc <- trW(W, type="mult")
COL.lag.eig1 <- lagsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 nb2listw(COL.nb, style="W"), control=list(fdHess=TRUE), trs=trMatc)
COL.lag.eig1$fdHess
system.time(COL.lag.M <- lagsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 nb2listw(COL.nb), method="Matrix", quiet=FALSE))
summary(COL.lag.M)
impacts(COL.lag.M, listw=nb2listw(COL.nb))
## Not run: 
# system.time(COL.lag.sp <- lagsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
#  nb2listw(COL.nb), method="spam", quiet=FALSE))
# summary(COL.lag.sp)
# ## End(Not run)
COL.lag.B <- lagsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 nb2listw(COL.nb, style="B"))
summary(COL.lag.B, correlation=TRUE)
COL.mixed.B <- lagsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 nb2listw(COL.nb, style="B"), type="mixed", tol.solve=1e-9)
summary(COL.mixed.B, correlation=TRUE)
COL.mixed.W <- lagsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 nb2listw(COL.nb, style="W"), type="mixed")
summary(COL.mixed.W, correlation=TRUE)
NA.COL.OLD <- COL.OLD
NA.COL.OLD$CRIME[20:25] <- NA
COL.lag.NA <- lagsarlm(CRIME ~ INC + HOVAL, data=NA.COL.OLD,
 nb2listw(COL.nb), na.action=na.exclude, 
 control=list(tol.opt=.Machine$double.eps^0.4))
COL.lag.NA$na.action
COL.lag.NA
resid(COL.lag.NA)
## Not run: 
# data(boston)
# gp2mM <- lagsarlm(log(CMEDV) ~ CRIM + ZN + INDUS + CHAS + I(NOX^2) + 
# I(RM^2) +  AGE + log(DIS) + log(RAD) + TAX + PTRATIO + B + log(LSTAT), 
# data=boston.c, nb2listw(boston.soi), type="mixed", method="Matrix")
# summary(gp2mM)
# W <- as(nb2listw(boston.soi), "CsparseMatrix")
# trMatb <- trW(W, type="mult")
# gp2mMi <- lagsarlm(log(CMEDV) ~ CRIM + ZN + INDUS + CHAS + I(NOX^2) + 
# I(RM^2) +  AGE + log(DIS) + log(RAD) + TAX + PTRATIO + B + log(LSTAT), 
# data=boston.c, nb2listw(boston.soi), type="mixed", method="Matrix", 
# trs=trMatb)
# summary(gp2mMi)
# ## End(Not run)
summary(COL.lag.eig)
COL.lag.Bayes <- spBreg_lag(CRIME ~ INC + HOVAL, data=COL.OLD,
 nb2listw(COL.nb, style="W"))
summary(COL.lag.Bayes)
set.seed(1)
summary(impacts(COL.lag.Bayes, tr=trMatc), short=TRUE, zstats=TRUE)
## Not run: 
# data(elect80)
# lw <- nb2listw(e80_queen, zero.policy=TRUE)
# el_ml <- lagsarlm(log(pc_turnout) ~ log(pc_college) + log(pc_homeownership)
#  + log(pc_income), data=elect80, listw=lw, zero.policy=TRUE, method="LU")
# summary(el_ml)
# set.seed(1)
# el_B <- spBreg_lag(log(pc_turnout) ~ log(pc_college) + log(pc_homeownership)
#  + log(pc_income), data=elect80, listw=lw, zero.policy=TRUE)
# summary(el_B)
# el_ml$timings
# attr(el_B, "timings")
# ## End(Not run)

Run the code above in your browser using DataCamp Workspace