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 spdep::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 logarithm 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
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

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