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gel

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 = NULL, gradv = NULL, smooth = FALSE,
    type = c("EL","ET","CUE","ETEL","HD","ETHD","RCUE"), 
    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", "Wu"), data,
    Lambdacontrol = list(), model = TRUE, X = FALSE, Y = FALSE,
    TypeGel = "baseGel", alpha = NULL, eqConst = NULL,
    eqConstFullVcov = FALSE, onlyCoefficients=FALSE, ...)
evalGel(g, x, tet0, gradv = NULL, smooth = FALSE,
        type = c("EL", "ET", "CUE", "ETEL", "HD", "ETHD","RCUE"),
        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", "Wu"), 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). "RCUE" is a restricted version of "CUE" in which the probabilities are bounded below by zero. In that case, an analytical Kuhn-Tucker method is used to find the solution.
`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`	Algorithm used to solve for the lagrange multiplier in `getLamb`. The algorithm Wu is only for `type="EL"`. The value of `optlam` is ignored for "CUE" because in that case, the analytical solution exists.
`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.
`onlyCoefficients`	If `TRUE`, only the vector of coefficients and Lagrange multipliers are returned
`...`	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.

Wu, C. (2005), Algorithms and R codes for the pseudo empirical likelihood method in survey sampling. Survey Methodology, 31(2), page 239.

Zeileis A (2006), Object-oriented Computation of Sandwich Estimators. Journal of Statistical Software, 16(9), 1–16. URL https://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 https://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

# 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)

gmm

Generalized Method of Moments and Generalized Empirical Likelihood

v1.6-6

GPL (>= 2)

Authors

Pierre Chausse <pchausse@uwaterloo.ca>

Initial release

2021-02-07

gel