Generalized Empirical Likelihood estimation

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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

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gel(g, x, tet0 = NULL, gradv = NULL, smooth = FALSE,
    type = c("EL","ET","CUE","ETEL","HD","ETHD"), 
    kernel = c("Truncated", "Bartlett"), bw = bwAndrews, 
    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("nlminb", "optim", "iter"), data,
    Lambdacontrol = list(), model = TRUE, X = FALSE, Y = FALSE,
    TypeGel = "baseGel", alpha = NULL, eqConst = NULL,
    eqConstFullVcov = FALSE, ...)
evalGel(g, x, tet0, gradv = NULL, smooth = FALSE,
        type = c("EL", "ET", "CUE", "ETEL", "HD", "ETHD"),
        kernel = c("Truncated", "Bartlett"), bw = bwAndrews,
        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, optlam = c("nlminb", "optim",
        "iter"), data, Lambdacontrol = list(), model = TRUE, X = FALSE,
        Y = FALSE, alpha = NULL, ...)

Arguments

g

A function of the form g(θ,x) and which returns a n \times q matrix with typical element g_i(θ,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 θ is one, see the argument "optfct". In the linear case, if tet0=NULL, the 2-step gmm estimator is used as starting value. However, it has to be provided when eqConst is not NULL

x

The matrix or vector of data from which the function g(θ,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(θ,x) which returns a q\times k matrix of derivatives of \bar{g}(θ) with respect to θ. 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{θ}. 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, "ETEL" for exponentially tilted empirical likelihood of Schennach(2007), "HD" for Hellinger Distance of Kitamura-Otsu-Evdokimov (2013), and "ETHD" for the exponentially tilted Hellinger distance of Antoine-Dovonon (2015).

kernel

type of kernel used to compute the covariance matrix of the vector of sample moment conditions (see kernHAC 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 bwAndrews which is proposed by Andrews (1991). The alternative is bwNeweyWest 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 bwAndrews.

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 θ 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 1\times 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 θ - ci ≥q 0.

tol_mom

It is the tolerance for the moment condition ∑_{t=1}^n p_t g(θ(x_t)=0, where p_t=\frac{1}{n}Dρ(<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.

data

A data.frame or a matrix with column names (Optional).

Lambdacontrol

Controls for the optimization of the vector of Lagrange multipliers used by either optim, nlminb or constrOptim

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.

alpha

Regularization coefficient for discrete CGEL estimation (experimental). By setting alpha to any value, the model is estimated by CGEL of type specified by the option type. See Chausse (2011)

eqConst

Either a named vector (if "g" is a function), a simple vector for the nonlinear case indicating which of the θ_0 is restricted, or a qx2 vector defining equality constraints of the form θ_i=c_i. See gmm for an example.

eqConstFullVcov

If FALSE, the constrained coefficients are assumed to be fixed and only the covariance of the unconstrained coefficients is computed. If TRUE, the covariance matrix of the full set of coefficients is computed.

...

More options to give to optim, optimize or constrOptim.

Details

If we want to estimate a model like Y_t = θ_1 + X_{2t}θ_2 + ... + X_{k}θ_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 ∑_{t=1}^n g(θ,x_t) is replaced by: ∑_{t=1}^n g^k(θ,x_t), where g^k(θ,x_t)=∑_{i=-r}^r k(i) g(θ,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{θ} = \arg\min ≤ft[\arg\max_λ \frac{1}{n}∑_{t=1}^n ρ(<g(θ,x_t),λ>) - ρ(0) \right]

evalGel generates the object of class "gel" for a fixed vector of parameters. There is no estimation for θ, but the optimal vector of Lagrange multipliers λ is computed. The objective function is then the profiled likelihood for a given θ. It can be used to construct a confidence interval by inverting the likelihood ratio test.

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

k\times 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.

Kitamura, Y. and Otsu, T. and Evdokimov, K. (2013), Robustness, Infinitesimal Neighborhoods and Moment Restrictions. Econometrica, 81, 1185-1201.

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/.

Chausse (2010), Computing Generalized Method of Moments and Generalized Empirical Likelihood with R. Journal of Statistical Software, 34(11), 1–35. URL http://www.jstatsoft.org/v34/i11/.

Chausse (2011), Generalized Empirical likelihood for a continumm of moment conditions. Working Paper, Department of Economics, University of Waterloo.

Examples

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# 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)

# Using evalGel to create the object without estimation

res <- evalGel(g, x, res$coefficients)