Matrix exponential spatial lag model

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)

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

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.

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

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

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.

Author

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

Examples