VGAM (version 1.1-14)

SURff: Seemingly Unrelated Regressions Family Function

Description

Fits a system of seemingly unrelated regressions.

Usage

SURff(mle.normal = FALSE,
      divisor = c("n", "n-max(pj,pk)", "sqrt((n-pj)*(n-pk))"),
      parallel = FALSE, Varcov = NULL, matrix.arg = FALSE)

Value

An object of class "vglmff" (see vglmff-class). The object is used by modelling functions such as

vglm and vgam.

Details

Proposed by Zellner (1962), the basic seemingly unrelated regressions (SUR) model is a set of LMs (\(M > 1\) of them) tied together at the error term level. Each LM's model matrix may potentially have its own set of predictor variables.

Zellner's efficient (ZEF) estimator (also known as Zellner's two-stage Aitken estimator) can be obtained by setting maxit = 1 (and possibly divisor = "sqrt" or divisor = "n-max").

The default value of maxit (in vglm.control) probably means iterative GLS (IGLS) estimator is computed because IRLS will probably iterate to convergence. IGLS means, at each iteration, the residuals are used to estimate the error variance-covariance matrix, and then the matrix is used in the GLS. The IGLS estimator is also known as Zellner's iterative Aitken estimator, or IZEF.

References

Zellner, A. (1962). An Efficient Method of Estimating Seemingly Unrelated Regressions and Tests for Aggregation Bias. J. Amer. Statist. Assoc., 57(298), 348--368.

Kmenta, J. and Gilbert, R. F. (1968). Small Sample Properties of Alternative Estimators of Seemingly Unrelated Regressions. J. Amer. Statist. Assoc., 63(324), 1180--1200.

See Also

uninormal, gew.

Examples

# Obtain some of the results of p.1199 of Kmenta and Gilbert (1968)
clist <- list("(Intercept)" = diag(2),
              "capital.g"   = rbind(1, 0),
              "value.g"     = rbind(1, 0),
              "capital.w"   = rbind(0, 1),
              "value.w"     = rbind(0, 1))
zef1 <- vglm(cbind(invest.g, invest.w) ~
             capital.g + value.g + capital.w + value.w,
             SURff(divisor = "sqrt"), maxit = 1,
             data = gew, trace = TRUE, constraints = clist)

round(coef(zef1, matrix = TRUE), digits = 4)  # ZEF
zef1@extra$ncols.X.lm
zef1@misc$divisor
zef1@misc$values.divisor
round(sqrt(diag(vcov(zef1))),    digits = 4)  # SEs
nobs(zef1, type = "lm")
df.residual(zef1, type = "lm")


mle1 <- vglm(cbind(invest.g, invest.w) ~
             capital.g + value.g + capital.w + value.w,
             SURff(mle.normal = TRUE),
             epsilon = 1e-11,
             data = gew, trace = TRUE, constraints = clist)
round(coef(mle1, matrix = TRUE), digits = 4)  # MLE
round(sqrt(diag(vcov(mle1))),    digits = 4)  # SEs