gel {gmm}R Documentation

Generalized Empirical Likelihood estimation

Description

Function to estimate a vector of parameters based on moment conditions using the GEL method as presented by Newey-Smith(2004) and Anatolyev(2005).

Usage

gel(g, x, tet0, gradv = NULL, smooth = FALSE, type = c("EL","ET","CUE","ETEL"), 
    kernel = c("Truncated", "Bartlett"), bw = bwAndrews2, 
    approx = c("AR(1)", "ARMA(1,1)"), prewhite = 1, ar.method = "ols", tol_weights = 1e-7, 
    tol_lam = 1e-9, tol_obj = 1e-9, tol_mom = 1e-9, maxiterlam = 100, constraint = FALSE,
    optfct = c("optim", "optimize", "nlminb"), optlam = c("iter","numeric"), model = TRUE, 
    X = FALSE, Y = FALSE, TypeGel = "baseGel",...)

Arguments

g A function of the form g(theta,x) and which returns a n times q matrix with typical element g_i(theta,x_t) for i=1,...q and t=1,...,n. This matrix is then used to build the q sample moment conditions. It can also be a formula if the model is linear (see details below).
tet0 A k times 1 vector of starting values. If the dimension of theta is one, see the argument "optfct".
x The matrix or vector of data from which the function g(theta,x) is computed. If "g" is a formula, it is an n times Nh matrix of instruments (see details below).
gradv A function of the form G(theta,x) which returns a qtimes k matrix of derivatives of bar{g}(theta) with respect to theta. By default, the numerical algorithm numericDeriv is used. It is of course strongly suggested to provide this function when it is possible. This gradiant is used compute the asymptotic covariance matrix of hat{theta}. If "g" is a formula, the gradiant is not required (see the details below).
smooth If set to TRUE, the moment function is smoothed as proposed by Kitamura(1997)
type "EL" for empirical likelihood, "ET" for exponential tilting, "CUE" for continuous updated estimator and "ETEL" for exponentially tilted empirical likelihood of Schennach(2007).
kernel type of kernel used to compute the covariance matrix of the vector of sample moment conditions (see HAC for more details) and to smooth the moment conditions if "smooth" is set to TRUE. Only two types of kernel are available. The truncated implies a Bartlett kernel for the HAC matrix and the Bartlett implies a Parzen kernel (see Smith 2004).
bw The method to compute the bandwidth parameter. By default it is bwAndrews2 which is proposed by Andrews (1991). The alternative is bwNeweyWest2 of Newey-West(1994).
prewhite logical or integer. Should the estimating functions be prewhitened? If TRUE or greater than 0 a VAR model of order as.integer(prewhite) is fitted via ar with method "ols" and demean = FALSE.
ar.method character. The method argument passed to ar for prewhitening.
approx a character specifying the approximation method if the bandwidth has to be chosen by bwAndrews2.
tol_weights numeric. Weights that exceed tol are used for computing the covariance matrix, all other weights are treated as 0.
tol_lam Tolerance for λ between two iterations. The algorithm stops when |λ_i -λ_{i-1}| reaches tol_lamb (see getLamb)
maxiterlam The algorithm to compute λ stops if there is no convergence after "maxiterlam" iterations (see getLamb).
tol_obj Tolerance for the gradiant of the objective function to compute λ (see getLamb).
optfct Only when the dimension of theta is 1, you can choose between the algorithm optim or optimize. In that case, the former is unreliable. If optimize is chosen, "t0" must be 1times 2 which represents the interval in which the algorithm seeks the solution.It is also possible to choose the nlminb algorithm. In that case, borns for the coefficients can be set by the options upper= and lower=.
constraint If set to TRUE, the constraint optimization algorithm is used. See constrOptim to learn how it works. In particular, if you choose to use it, you need to provide "ui" and "ci" in order to impose the constraint ui theta - ci >=q 0.
tol_mom It is the tolerance for the moment condition sum_{t=1}^n p_t g(theta(x_t)=0, where p_t=frac{1}{n}Drho(<g_t,λ>) is the implied probability. It adds a penalty if the solution diverges from its goal.
optlam The default is "iter" which solves for λ using the Newton iterative method getLamb. If set to "numeric", the algorithm optim is used to compute λ instead.
model, X, Y logicals. If TRUE the corresponding components of the fit (the model frame, the model matrix, the response) are returned if g is a formula.
TypeGel The name of the class object created by the method getModel. It allows developers to extand the package and create other GEL methods.
... More options to give to optim, optimize or constrOptim.

Details

weightsAndrews2 and bwAndrews2 are simply modified version of weightsAndrews and bwAndrews from the package sandwich. The modifications have been made so that the argument x can be a matrix instead of an object of class lm or glm. The details on how is works can be found on the sandwich manual.

If we want to estimate a model like Y_t = theta_1 + X_{2t}theta_2 + ... + X_{k}theta_k + ε_t using the moment conditions Cov(ε_tH_t)=0, where H_t is a vector of Nh instruments, than we can define "g" like we do for lm. We would have g = y~x2+x3+...+xk and the argument "x" above would become the matrix H of instruments. As for lm, Y_t can be a Ny times 1 vector which would imply that k=Nh times Ny. The intercept is included by default so you do not have to add a column of ones to the matrix H. You do not need to provide the gradiant in that case since in that case it is embedded in gel. The intercept can be removed by adding -1 to the formula. In that case, the column of ones need to be added manually to H.

If "smooth" is set to TRUE, the sample moment conditions sum_{t=1}^n g(theta,x_t) is replaced by: sum_{t=1}^n g^k(theta,x_t), where g^k(theta,x_t)=sum_{i=-r}^r k(i) g(theta,x_{t+i}), where r is a truncated parameter that depends on the bandwidth and k(i) are normalized weights so that they sum to 1.

The method solves hat{theta} = argmin <=ft[argmax_λ frac{1}{n}sum_{t=1}^n rho(<g(theta,x_t),λ>) - rho(0) right]

Value

'gel' returns an object of 'class' '"gel"'
The functions 'summary' is used to obtain and print a summary of the results.
The object of class "gel" is a list containing at least the following:

coefficients ktimes 1 vector of parameters
residuals the residuals, that is response minus fitted values if "g" is a formula.
fitted.values the fitted mean values if "g" is a formula.
lambda q times 1 vector of Lagrange multipliers.
vcov_par the covariance matrix of "coefficients"
vcov_lambda the covariance matrix of "lambda"
pt The implied probabilities
objective the value of the objective function
conv_lambda Convergence code for "lambda" (see getLamb)
conv_mes Convergence message for "lambda" (see getLamb)
conv_par Convergence code for "coefficients" (see optim, optimize or constrOptim)
terms the terms object used when g is a formula.
call the matched call.
y if requested, the response used (if "g" is a formula).
x if requested, the model matrix used if "g" is a formula or the data if "g" is a function.
model if requested (the default), the model frame used if "g" is a formula.

References

Anatolyev, S. (2005), GMM, GEL, Serial Correlation, and Asymptotic Bias. Econometrica, 73, 983-1002.

Andrews DWK (1991), Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimation. Econometrica, 59, 817–858.

Kitamura, Yuichi (1997), Empirical Likelihood Methods With Weakly Dependent Processes. The Annals of Statistics, 25, 2084-2102.

Newey, W.K. and Smith, R.J. (2004), Higher Order Properties of GMM and Generalized Empirical Likelihood Estimators. Econometrica, 72, 219-255.

Smith, R.J. (2004), GEL Criteria for Moment Condition Models. Working paper, CEMMAP.

Newey WK & West KD (1987), A Simple, Positive Semi-Definite, Heteroskedasticity and Autocorrelation Consistent Covariance Matrix. Econometrica, 55, 703–708.

Newey WK & West KD (1994), Automatic Lag Selection in Covariance Matrix Estimation. Review of Economic Studies, 61, 631-653.

Schennach, Susanne, M. (2007), Point Estimation with Exponentially Tilted Empirical Likelihood. Econometrica, 35, 634-672.

Zeileis A (2006), Object-oriented Computation of Sandwich Estimators. Journal of Statistical Software, 16(9), 1–16. URL http://www.jstatsoft.org/v16/i09/.

Examples

# First, an exemple with the fonction g()

g <- function(tet, x)
        {
        n <- nrow(x)
        u <- (x[7:n] - tet[1] - tet[2]*x[6:(n-1)] - tet[3]*x[5:(n-2)])
        f <- cbind(u, u*x[4:(n-3)], u*x[3:(n-4)], u*x[2:(n-5)], u*x[1:(n-6)])
        return(f)
        }

Dg <- function(tet,x)
        {
        n <- nrow(x)
        xx <- cbind(rep(1, (n-6)), x[6:(n-1)], x[5:(n-2)])
        H  <- cbind(rep(1, (n-6)), x[4:(n-3)], x[3:(n-4)], x[2:(n-5)], x[1:(n-6)])
        f <- -crossprod(H, xx)/(n-6)
        return(f)
        }
n = 200
phi<-c(.2, .7)
thet <- 0.2
sd <- .2
set.seed(123)
x <- matrix(arima.sim(n = n, list(order = c(2, 0, 1), ar = phi, ma = thet, sd = sd)), ncol = 1)

res <- gel(g, x, c(0, .3, .6), grad = Dg)
summary(res)

# The same model but with g as a formula....  much simpler in that case

y <- x[7:n]
ym1 <- x[6:(n-1)]
ym2 <- x[5:(n-2)]

H <- cbind(x[4:(n-3)], x[3:(n-4)], x[2:(n-5)], x[1:(n-6)])
g <- y ~ ym1 + ym2
x <- H

res <- gel(g, x, c(0, .3, .6))
summary(res)


[Package gmm version 1.3-0 Index]