# A guide to estimating systems of equations This document is meant to describe how to create system of equations objects, estimating them and peforming hypothesis tests. ## Details Instread of repeating the same example for each method, we are going through all methods and classes for systems of equations. ## Examples ```r data(simData) ## first, we create an sysGmm object g1 <- y1~x1+x4; h1 <- ~x4+z1+z2+z3+z4 g2 <- y2~x1+x2+x3; h2 <- ~x3+z1+z2+z3+z4 g3 <- y3~x2+x3+x4; h3 <- ~x3+x4+z1+z2+z3+z4 g <- list(g1,g2,g3) h <- list(h1,h2,h3) smodel <- sysMomentModel(g, h, data=simData, vcov="MDS") ## The show or print method smodel ## The ']' method smodel[1:2] smodel[1] ## becomes a one equation model ## equation by equation 2SLS tsls(smodel) ## or manually lapply(1:3, function(i) coef(tsls(smodel[i]))) ## Fitting the model by two-step GMM res <- gmmFit(smodel) ## testing Overidentifying restrictions specTest(res) ## All info using the summary method ## which includes equation by equation measures of ## the instrument stengths summary(res) ### When the error id iid (homoscedastic), we have a ### FIVE estimator with 2SLS as the first step smodel <- sysMomentModel(g, h, data=simData, vcov="iid") gmmFit(smodel) ### When the error is iid (homoscedastic), ### all instruments are the same, and the first step is 2SLS, ### we have 3SLS smodel <- sysMomentModel(g, ~x4+z1+z2+z3+z4, data=simData, vcov="iid") gmmFit(smodel, initW='tsls') ### When the error is iid (homoscedastic), ### the instruments are the same and are the union of all regressors, ### we have SUR smodel <- sysMomentModel(g, NULL, data=simData, vcov="iid") gmmFit(smodel, initW='tsls') ############ Restricted models ################## ## unrestricted smodel <- sysMomentModel(g, h, data=simData, vcov="MDS") res <- gmmFit(smodel) ## no cross-equation restrictions R1 <- list(c("x1=-12*x4"), character(), c("x2=0.8", "x4=0.3")) rm1 <- restModel(smodel, R1) (res1 <- gmmFit(rm1)) ## Cross equation restrictions R2<- c("Eqn1.x1=1", "Eqn2.x1=Eqn3.x2") rm2 <- restModel(smodel, R2) (es2 <- gmmFit(rm2))## no longer expressed as a system ## testing the restriction hypothesisTest(res, res1, type="LR") hypothesisTest(res, res1, type="LM") hypothesisTest(res, res1, type="Wald") ```

systemGmm function