Spatial simultaneous autoregressive lag model estimation
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.
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.
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())
- a symbolic description of the model to be fit. The details
of model specification are given for
- an optional data frame containing the variables in the model. By default the variables are taken from the environment which the function is called.
listwobject created for example by
- a function (default
options("na.action")), can also be
na.excludewith 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
nb2listwmay be subsetted.
- default "lag", may be set to "mixed"; when "mixed", the lagged intercept is dropped for spatial weights style "W", that is row-standardised weights, but otherwise included; Durbin may be used instead of mixed
- "eigen" (default) - the Jacobian is computed as the product
of (1 - rho*eigenvalue) using
eigenw, and "spam" or "Matrix_J" for strictly symmetric weights lists of styles "B" and "C", or made symmetric by similarity (Ord, 1975, Appendix C) if possible for styles "W" and "S", using code from the spam or Matrix packages to calculate the determinant; Matrix and spam_update provide updating Cholesky decomposition methods; "LU" provides an alternative sparse matrix decomposition approach. In addition, there are "Chebyshev" and Monte Carlo "MC" approximate log-determinant methods; the Smirnov/Anselin (2009) trace approximation is available as "moments". Three methods: "SE_classic", "SE_whichMin", and "SE_interp" are provided experimentally, the first to attempt to emulate the behaviour of Spatial Econometrics toolbox ML fitting functions. All use grids of log determinant values, and the latter two attempt to ameliorate some features of "SE_classic".
- default NULL, use !verbose global option value; if FALSE, reports function values during optimization.
- default NULL, use global option value; if TRUE assign zero to the lagged value of zones without
neighbours, if FALSE (default) assign NA - causing
lagsarlm()to terminate with an error
- default is NULL, search interval for autoregressive parameter
- the tolerance for detecting linear dependencies in the columns of matrices to be inverted - passed to
solve()(default=1.0e-10). This may be used if necessary to extract coefficient standard errors (for instance lowering to 1e-12), but errors in
solve()may constitute indications of poorly scaled variables: if the variables have scales differing much from the autoregressive coefficient, the values in this matrix may be very different in scale, and inverting such a matrix is analytically possible by definition, but numerically unstable; rescaling the RHS variables alleviates this better than setting tol.solve to a very small value
- default NULL, if given, a vector of powered spatial weights matrix traces output by
trW; when given, insert the asymptotic analytical values into the numerical Hessian instead of the approximated values; may be used to get around some problems raised when the numerical Hessian is poorly conditioned, generating NaNs in subsequent operations; the use of trs is recommended
- list of extra control arguments - see section below
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
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.
A list object of class
sarlmThe internal sar.lag.mixed.* functions return the value of the log likelihood function at $rho$.
Extra Bayesian control arguments
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.
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)