lagmess: Matrix exponential spatial lag model
In spdep: Spatial Dependence: Weighting Schemes, Statistics and Models

Description Usage Arguments Details Value Author(s) References See Also Examples

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

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 class 'lagmess'
summary(object, ...)
## S3 method for class 'lagmess'
print(x, ...)
## S3 method for class 'summary.lagmess'
print(x, digits = max(5, .Options$digits - 3),
    signif.stars = FALSE, ...)
## S3 method for class 'lagmess'
residuals(object, ...)
## S3 method for class 'lagmess'
deviance(object, ...)
## S3 method for class 'lagmess'
coef(object, ...)
## S3 method for class 'lagmess'
fitted(object, ...)
## S3 method for class 'lagmess'
logLik(object, ...)

`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

The underlying spatial lag model:

y = rho W y + X beta + e

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 + e

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.

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

Roger Bivand Roger.Bivand@nhh.no and Eric Blankmeyer

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.

lagsarlm, optim

data(baltimore)
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)
data(boston)
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)