bread function

Bread for sandwiches

Computes the bread of the sandwich covariance matrix

## S3 method for class 'gmm'
bread(x, ...)
## S3 method for class 'gel'
bread(x, ...)
## S3 method for class 'tsls'
bread(x, ...)

Arguments

• x: A fitted model of class gmm or gel.
• ...: Other arguments when bread is applied to another class object

Details

When the weighting matrix is not the optimal one, the covariance matrix of the estimated coefficients is: $(G'WG)^{-1} G'W V W G(G'WG)^{-1}$, where $G=d\bar{g}/d\theta$, $W$ is the matrix of weights, and $V$ is the covariance matrix of the moment function. Therefore, the bread is $(G'WG)^{-1}$, which is the second derivative of the objective function.

The method if not yet available for gel objects.

Returns

A $k \times k$ matrix (see details).

References

Zeileis A (2006), Object-oriented Computation of Sandwich Estimators. Journal of Statistical Software, 16 (9), 1--16. URL tools:::Rd_expr_doi("10.18637/jss.v016.i09") .

Examples

# See \code{\link{gmm}} for more details on this example.
# With the identity matrix 
# bread is the inverse of (G'G)

n <- 1000
x <- rnorm(n, mean = 4, sd = 2)
g <- function(tet, x)
        {
        m1 <- (tet[1] - x)
        m2 <- (tet[2]^2 - (x - tet[1])^2)
        m3 <- x^3 - tet[1]*(tet[1]^2 + 3*tet[2]^2)
        f <- cbind(m1, m2, m3)
        return(f)
        }
Dg <- function(tet, x)
        {
        jacobian <- matrix(c( 1, 2*(-tet[1]+mean(x)), -3*tet[1]^2-3*tet[2]^2,0, 2*tet[2],
                                -6*tet[1]*tet[2]), nrow=3,ncol=2)
        return(jacobian)
        }

res <- gmm(g, x, c(0, 0), grad = Dg,weightsMatrix=diag(3))
G <- Dg(res$coef, x)
bread(res)
solve(crossprod(G))