##
## Protocol for estimating "known variance" FGLS regression model for
## estimated dependent variables in R.
##
## As described in:
## Jeffrey B. Lewis and Drew A. Linzer. 2005. "Estimating Regression 
## 	Models in which the Dependent Variable Is Based on Estimates," 
##      Political Analysis. 13(4): 345-364.
##
## Jeffrey Lewis (jblewis@ucla.edu)
## Drew Linzer (dlinzer@ucla.edu)
## Department of Political Science
## University of California, Los Angeles
## 
## The "known-variance" FGLS estimator simply requires some matrix algebra to 
## calculate weights, and then a weighted least squares regression.
##
## The first thing to do is produce estimated values of sigma-hat squared, 
## following the second equation at the top of page 352 in our article.
##
## The command to do this in R is somewhat lengthy because of all the matrix 
## multiplication; the way it's written here, ystar is the dependent variable, 
## x is a matrix of the independent variables, and omegasq is a vector of the 
## "known" variances of the dependent variable.  Note that the x matrix should 
## not have a leading column of 1's; the command cbind(1,x) affixes the 1's 
## as needed for the matrix multiplication.  Also be sure that you have 
## installed library(MASS) and library(lasso2) prior to running these commands,
## or else R will give you an error message.
##

library(MASS)
library(lasso2)
sigmahatsq <- (sum((residuals(lm(ystar~x)))^2) - sum(omegasq) + 
		tr(ginv(t(cbind(1,x)) %*% cbind(1,x)) %*% t(cbind(1,x)) %*% 
		(diag(N) * omegasq) %*% cbind(1,x)))/(N-ncol(x)-1)

##
## We then suggest doing a quick check to set sigmahatsq to 0 if its 
## estimated value comes back as negative.
##

if (sigmahatsq < 0) sigmahatsq <- 0

##
## Finally, the FGLS model may be estimated as a weighted least squares 
## as follows.  (In our article, the equation for the weights is the 
## third equation down on page 352.)
##

model.FGLS <- lm(ystar~x,weights=(1/(omegasq+sigmahatsq)))