spdep (version 0.8-1)

lagmess: Matrix exponential spatial lag model

Description

The function fits a matrix exponential spatial lag model, using optim to find the value of alpha, the spatial coefficient.

Usage

lagmess(formula, data = list(), listw, zero.policy = NULL, na.action = na.fail,
 q = 10, start = -2.5, control=list(), method="BFGS", verbose=NULL,
 use_expm=FALSE)
# S3 method for lagmess
summary(object, ...)
# S3 method for lagmess
print(x, ...)
# S3 method for summary.lagmess
print(x, digits = max(5, .Options$digits - 3),
    signif.stars = FALSE, ...)
# S3 method for lagmess
residuals(object, ...)
# S3 method for lagmess
deviance(object, ...)
# S3 method for lagmess
coef(object, ...)
# S3 method for lagmess
fitted(object, ...)
# S3 method for lagmess
logLik(object, ...)

Arguments

formula

a symbolic description of the model to be fit. The details of model specification are given for lm()

data

an optional data frame containing the variables in the model. By default the variables are taken from the environment which the function is called.

listw

a listw object created for example by nb2listw

zero.policy

default NULL, use global option value; if TRUE assign zero to the lagged value of zones without neighbours, if FALSE assign NA - causing lagmess() to terminate with an error

na.action

a function (default options("na.action")), can also be na.omit or na.exclude with consequences for residuals and fitted values - in these cases the weights list will be subsetted to remove NAs in the data. It may be necessary to set zero.policy to TRUE because this subsetting may create no-neighbour observations. Note that only weights lists created without using the glist argument to nb2listw may be subsetted.

q

default 10; number of powers of the spatial weights to use

start

starting value for numerical optimization, should be a small negative number

control

control parameters passed to optim

method

default BFGS, method passed to optim

verbose

default NULL, use global option value; if TRUE report function values during optimization

use_expm

default FALSE; if TRUE use expm::expAtv instead of a truncated power series of W

x,object

Objects of classes lagmess or summary.lagmess to be passed to methods

digits

the number of significant digits to use when printing

signif.stars

logical. If TRUE, "significance stars" are printed for each coefficient.

…

further arguments passed to or from other methods

Value

The function returns an object of class lagmess with components:

lmobj

the lm object returned after fitting alpha

alpha

the spatial coefficient

alphase

the standard error of the spatial coefficient using the numerical Hessian

rho

the value of rho implied by alpha

bestmess

the object returned by optim

q

the number of powers of the spatial weights used

start

the starting value for numerical optimization used

na.action

(possibly) named vector of excluded or omitted observations if non-default na.action argument used

nullLL

the log likelihood of the aspatial model for the same data

Details

The underlying spatial lag model:

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

where \(\rho\) is the spatial parameter may be fitted by maximum likelihood. In that case, the log likelihood function includes the logartithm of cumbersome Jacobian term \(|I - \rho W|\). If we rewrite the model as:

$$S y = X \beta + \varepsilon$$

we see that in the ML case \(S y = (I - \rho W) y\). If W is row-stochastic, S may be expressed as a linear combination of row-stochastic matrices. By pre-computing the matrix \([y Wy, W^2y, ..., W^{q-1}y]\), the term \(S y (\alpha)\) can readily be found by numerical optimization using the matrix exponential approach. \(\alpha\) and \(\rho\) are related as \(\rho = 1 - \exp{\alpha}\), conditional on the number of matrix power terms taken q.

References

J. P. LeSage and R. K. Pace (2007) A matrix exponential specification. Journal of Econometrics, 140, 190-214; J. P. LeSage and R. K. Pace (2009) Introduction to Spatial Econometrics. CRC Press, Chapter 9.

See Also

lagsarlm, optim

Examples

# NOT RUN {
data(baltimore, package="spData")
baltimore$AGE <- ifelse(baltimore$AGE < 1, 1, baltimore$AGE)
lw <- nb2listw(knn2nb(knearneigh(cbind(baltimore$X, baltimore$Y), k=7)))
obj1 <- lm(log(PRICE) ~ PATIO + log(AGE) + log(SQFT),
 data=baltimore)
lm.morantest(obj1, lw)
lm.LMtests(obj1, lw, test="all")
system.time(obj2 <- lagmess(log(PRICE) ~ PATIO + log(AGE) + log(SQFT), data=baltimore, listw=lw))
summary(obj2)
system.time(obj2a <- lagmess(log(PRICE) ~ PATIO + log(AGE) + log(SQFT), data=baltimore, listw=lw,
 use_expm=TRUE))
summary(obj2a)
obj3 <- lagsarlm(log(PRICE) ~ PATIO + log(AGE) + log(SQFT), data=baltimore, listw=lw)
summary(obj3)
# }
# NOT RUN {
data(boston, package="spData")
lw <- nb2listw(boston.soi)
gp2 <- 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, lw, method="Matrix")
summary(gp2)
gp2a <- lagmess(CMEDV ~ CRIM + ZN + INDUS + CHAS + I(NOX^2) + I(RM^2)
 +  AGE + log(DIS) + log(RAD) + TAX + PTRATIO + B + log(LSTAT),
 data=boston.c, lw)
summary(gp2a)
# }