## CAPM test with GMM
data(Finance)
r <- Finance[1:300, 1:10]
rm <- Finance[1:300, "rm"]
rf <- Finance[1:300, "rf"]
z <- as.matrix(r-rf)
t <- nrow(z)
zm <- rm-rf
h <- matrix(zm, t, 1)
res <- gmm(z ~ zm, x = h)
summary(res)
## linear tests can be performed using linearHypothesis from the car package
## The CAPM can be tested as follows:
library(car)
linearHypothesis(res,cbind(diag(10),matrix(0,10,10)),rep(0,10))
# The CAPM of Black
g <- function(theta, x) {
e <- x[,2:11] - theta[1] - (x[,1] - theta[1]) %*% matrix(theta[2:11], 1, 10)
gmat <- cbind(e, e*c(x[,1]))
return(gmat) }
x <- as.matrix(cbind(rm, r))
res_black <- gmm(g, x = x, t0 = rep(0, 11))
summary(res_black)$coefficients
## APT test with Fama-French factors and GMM
f1 <- zm
f2 <- Finance[1:300, "hml"] - rf
f3 <- Finance[1:300, "smb"] - rf
h <- cbind(f1, f2, f3)
res2 <- gmm(z ~ f1 + f2 + f3, x = h)
coef(res2)
summary(res2)$coefficients
## Same result with x defined as a formula:
res2 <- gmm(z ~ f1 + f2 + f3, ~ f1 + f2 + f3)
coef(res2)
## The following example has been provided by Dieter Rozenich (see details).
# It generates normal random numbers and uses the GMM to estimate
# mean and sd.
#-------------------------------------------------------------------------------
# Random numbers of a normal distribution
# First we generate normally distributed random numbers and compute the two parameters:
n <- 1000
x <- rnorm(n, mean = 4, sd = 2)
# Implementing the 3 moment conditions
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)
}
# Implementing the jacobian
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)
}
# Now we want to estimate the two parameters using the GMM.
gmm(g, x, c(0, 0), grad = Dg)
# Two-stage-least-squares (2SLS), or IV with iid errors.
# The model is:
# Y(t) = b[0] + b[1]C(t) + b[2]Y(t-1) + e(t)
# e(t) is an MA(1)
# The instruments are Z(t)={1 C(t) y(t-2) y(t-3) y(t-4)}
getdat <- function(n) {
e <- arima.sim(n,model=list(ma=.9))
C <- runif(n,0,5)
Y <- rep(0,n)
Y[1] = 1 + 2*C[1] + e[1]
for (i in 2:n){
Y[i] = 1 + 2*C[i] + 0.9*Y[i-1] + e[i]
}
Yt <- Y[5:n]
X <- cbind(1,C[5:n],Y[4:(n-1)])
Z <- cbind(1,C[5:n],Y[3:(n-2)],Y[2:(n-3)],Y[1:(n-4)])
return(list(Y=Yt,X=X,Z=Z))
}
d <- getdat(5000)
res4 <- gmm(d$Y~d$X-1,~d$Z-1,vcov="iid")
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