gmm.tmvnorm {tmvtnorm}

GMM Estimation for the Truncated Multivariate Normal Distribution

Description

Generalized Method of Moments (GMM) Estimation for the Truncated Multivariate Normal Distribution

Usage

gmm.tmvnorm(X, 
  lower = rep(-Inf, length = ncol(X)), 
  upper = rep(+Inf, length = ncol(X)), 
  start = list(mu = rep(0, ncol(X)), sigma = diag(ncol(X))), 
  fixed = list(),
  method=c("ManjunathWilhelm","Lee"),
  cholesky = FALSE,
  ...)

Arguments

X

Matrix of quantiles, each row is taken to be a quantile.

lower

Vector of lower truncation points, default is rep(-Inf, length = ncol(X)).

upper

Vector of upper truncation points, default is rep( Inf, length = ncol(X)).

start

Named list with elements mu (mean vector) and sigma (covariance matrix). Initial values for optimizer.

fixed

Named list. Parameter values to keep fixed during optimization.

method

Which set of moment conditions used, possible methods are "ManjunathWilhelm" (default) and "Lee".

cholesky

if TRUE, we use the Cholesky decomposition of sigma as parametrization

...

Further arguments to pass to gmm

Details

This method performs an estimation of the parameters mean and sigma of a truncated multinormal distribution using the Generalized Method of Moments (GMM), when the truncation points lower and upper are known. gmm.tmvnorm() is a wrapper for the general GMM method gmm, so one does not have to specify the moment conditions.

Manjunath/Wilhelm moment conditions
Because the first and second moments can be computed thanks to the mtmvnorm function, we can set up a method-of-moments estimator by equating the sample moments to their population counterparts. This way we have an exactly identified case.

Lee (1979,1983) moment conditions
The recursive moment conditions presented by Lee (1979,1983) are defined for l=0,1,2,\ldots as

\sigma^{iT} E(x_i^l \textbf{x}) = \sigma^{iT} \mu E(x_i^l) + l E(x_i^{l-1}) + \frac{a_i^l F_i(a_i)}{F} - \frac{b_i^l F_i(b_i)}{F}

where E(x_i^l) and E(x_i^l \textbf{x}) are the moments of x_i^l and x_i^l \textbf{x} respectively and F_i(c)/F is the one-dimensional marginal density in variable i as calculated by dtmvnorm.marginal. \sigma^{iT} is the i-th column of the inverse covariance matrix \Sigma^{-1}.

This method returns an object of class gmm, for which various diagnostic methods are available, like profile(), confint() etc. See examples.

Value

An object of class gmm

Author(s)

Stefan Wilhelm [e-mail]

References

Tallis, G. M. (1961). The moment generating function of the truncated multinormal distribution. Journal of the Royal Statistical Society, Series B, 23, 223–229

Lee, L.-F. (1979). On the first and second moments of the truncated multi-normal distribution and a simple estimator. Economics Letters, 3, 165–169

Lee, L.-F. (1983). The determination of moments of the doubly truncated multivariate normal Tobit model. Economics Letters, 11, 245–250

Manjunath B G and Wilhelm, S. (2009). Moments Calculation For the Double Truncated Multivariate Normal Density. Working Paper. Available at SSRN: https://www.ssrn.com/abstract=1472153

See Also

gmm

Examples

## Not run: 
set.seed(1.234)

# the actual parameters
lower <- c(-1, -2)
upper <- c(3, Inf)
mu    <- c(0, 0)
sigma <- matrix(c(1, 0.8,
                0.8, 2), 2, 2)
               
# generate random samples               
X <- rtmvnorm(n=500, mu, sigma, lower, upper)

# estimate mean vector and covariance matrix sigma from random samples X
# with default start values
gmm.fit1 <- gmm.tmvnorm(X, lower=lower, upper=upper)

# diagnostic output of the estimated parameters
summary(gmm.fit1)
vcov(gmm.fit1)

# confidence intervals
confint(gmm.fit1)

# choosing a different start value
gmm.fit2 <- gmm.tmvnorm(X, lower=lower, upper=upper, 
  start=list(mu=c(0.1, 0.1), 
  sigma=matrix(c(1, 0.4, 0.4, 1.8),2,2)))
summary(gmm.fit2)

# GMM estimation with Lee (1983) moment conditions
gmm.fit3 <- gmm.tmvnorm(X, lower=lower, upper=upper, method="Lee")
summary(gmm.fit3)
confint(gmm.fit3)

# MLE estimation for comparison
mle.fit1 <- mle.tmvnorm(X, lower=lower, upper=upper)
confint(mle.fit1)

## End(Not run)