Document Outline

maxiterlam

The algorithm to compute

stops if there is no convergence after "maxiterlam"

iterations (see

tol_obj

Tolerance for the gradiant of the objective function to compute

(see

optfct

Algorithm used for the parameter estimates

tol_mom

It is the tolerance for the moment condition

n
t

(

) = 0

, where

Dρ

(

< g

, 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

. The algorithm

Wu is only for

type="EL"

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

ATEgel

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.

verbose

If TRUE, a summary of the convergence is printed

tolConv

The tolerance for comparing moments between groups

...

More options to give to

Details

We want to estimate the model

treat

, where

is the treatment effect. GEL is

used to balance the sample based on the argument

above.

For example, if we want the sample mean of

and

to be balanced between the treated and

control, we set

~x1+x2

. If we want the sample mean of

x1*x2

x1^2

and

x2^2

, we set

~x1*x2 + I(x1^2) + I(x2^2)

Value

’gel’ returns an object of ’class’ ’"ategel"’

The functions ’summary’ is used to obtain and print a summary of the results.

The object of class "ategel" is a list containing the same elements contained in objects of class

http://www.jstatsoft.org/

References

Lee, Seojeong (2016), Asymptotic refinements of misspecified-robust bootstrap for GEL estimators,

Journal of Econometrics

192

, 86–104.

Schennach, Susanne, M. (2007), Point Estimation with Exponentially Tilted Empirical Likelihood.

Econometrica

, 634-672.

Wu, C. (2005), Algorithms and R codes for the pseudo empirical likelihood method in survey
sampling.

Survey Methodology

(2), page 239.

Chausse, P. (2010), Computing Generalized Method of Moments and Generalized Empirical Like-
lihood with R.

Journal of Statistical Software

(11), 1–35. URL

v34/i11/

Chausse, P and Luta, G. (2017) Estimating the Average Treatment Effect in Quasi-Experimental
Studies using the Generalized Empirical Likelihood, Work in progress.

Examples

data(nsw)

# Scale income

nsw$re78 <- nsw$re78/1000

nsw$re75 <- nsw$re75/1000

res <- ATEgel(re78~treat, ~age+ed+black+hisp+married+nodeg+re75,

data=nsw,type="ET")

summary(res)

chk <- checkConv(res)

res2 <- ATEgel(re78~treat, ~age+ed+black+hisp+married+nodeg+re75,

data=nsw,type="ET", momType="balSample")

bread

summary(res2)

chk2 <- checkConv(res2)

bread

Bread for sandwiches

Description

Computes the bread of the sandwich covariance matrix

Usage

## S3 method for class 'gmm'

bread(x, ...)

## S3 method for class 'gel'

bread(x, ...)

## S3 method for class 'tsls'

bread(x, ...)

Arguments

A fitted model of class

gmm

gel

...

Other arguments when

bread

is applied to another class object

Details

When the weighting matrix is not the optimal one, the covariance matrix of the estimated coeffi-
cients is:

(

W G

)

−

W V W G

(

W G

)

−

, where

g/dθ

is the matrix of weights, and

is the covariance matrix of the moment function. Therefore, the bread is

(

W G

)

−

, which is

the second derivative of the objective function.

The method if not yet available for

gel

objects.

Value

matrix (see details).

References

Zeileis A (2006), Object-oriented Computation of Sandwich Estimators.

Journal of Statistical Soft-

ware

(9), 1–16. URL

bwWilhelm

Examples

# See \code{\link{gmm}} for more details on this example.

# With the identity matrix

# bread is the inverse of (G'G)

n <- 1000

x <- rnorm(n, mean = 4, sd = 2)

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)

}

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)

}

res <- gmm(g, x, c(0, 0), grad = Dg,weightsMatrix=diag(3))

G <- Dg(res$coef, x)

bread(res)

solve(crossprod(G))

bwWilhelm

Wilhelm (2015) bandwidth selection

Description

It computes the optimal bandwidth for the HAC estimation of the covariance matrix of the moment
conditions. The bandwidth was shown by Wilhelm (2005) to be the one that minimizes the MSE of
the GMM estimator.

Usage

bwWilhelm(x, order.by = NULL, kernel = c("Quadratic Spectral",

"Bartlett", "Parzen", "Tukey-Hanning"), approx = c("AR(1)", "ARMA(1,1)"),

weights = NULL, prewhite = 1, ar.method = "ols", data = list())

Arguments

An object of class

gmm

order.by

Either a vector ’z’ or a formula with a single explanatory variable like ’~ z’. The
observations in the model are ordered by the size of ’z’. If set to ’NULL’ (the
default) the observations are assumed to be ordered (e.g., a time series).

bwWilhelm

kernel

type of kernel used to compute the covariance matrix of the vector of sample
moment conditions (see

for more details)

approx

A character specifying the approximation method if the bandwidth has to be
chosen by

bwAndrews

weights

numeric. A vector of weights used for weighting the estimated coefficients of
the approximation model (as specified by ’approx’). By default all weights are
1 except that for the intercept term (if there is more than one variable)

prewhite

logical or integer. Should the estimating functions be prewhitened? If

TRUE

greater than 0 a VAR model of order

as.integer(prewhite)

is fitted via

with method

"ols"

and

demean = FALSE

ar.method

character. The

method

argument passed to

for prewhitening.

data

an optional data frame containing the variables in the ’order.by’ model.

Value

The function ’bwWilhelm’ returns the optimal bandwidth.

Note

The function was written by Daniel Wilhelm and is based on

bwAndrews

References

Wilhelm, D. (2015), Optimal Bandwidth Selection for Robust Generalized Method of Moments
Estimation.

Econometric Theory

, 1054–1077

Zeileis A (2006), Object-oriented Computation of Sandwich Estimators.

Journal of Statistical Soft-

ware

(9), 1–16. URL

//academic2.american.edu/~jpnolan/stable/stable.html

Examples

data(Finance)

f1 <- Finance[1:300, "rm"]

f2 <- Finance[1:300, "hml"]

f3 <- Finance[1:300, "smb"]

y <- Finance[1:300,"WMK"]

## Silly example just to make it over-identified

###############################################

res <- gmm(y ~ f1, ~ f1 + f2 + f3)

summary(res)

## Set the bandwidth using the second step estimate

################################################

bw <- bwWilhelm(res)

res2 <- update(res, bw=bw)

summary(res2)

## Set the bandwidth using the first-step estimate as for bwAndrews

charStable

###################################################################

res3 <- gmm(y ~ f1, ~ f1 + f2 + f3, bw=bwWilhelm)

summary(res3)

charStable

The characteristic function of a stable distribution

Description

It computes the theoretical characteristic function of a stable distribution for two different parametriza-
tions. It is used in the vignette to illustrate the estimation of the parameters using GMM.

Usage

charStable(theta, tau, pm = 0)

Arguments

theta

Vector of parameters of the stable distribution. See details.

tau

A vector of numbers at which the function is evaluated.

The type of parametization. It takes the values 0 or 1.

Details

The function returns the vector

Ψ(

θ, τ, pm

)

defined as

(

ixτ

, where

is a vector of real numbers,

is the imaginary number,

is a stable random variable with parameters

(

α, β, γ, δ

)

and

is the

type of parametrization. The vector of parameters are the characteristic exponent, the skewness, the
scale and the location parameters, respectively. The restrictions on the parameters are:

∈

[

−

and

γ >

. For mode details see Nolan(2009).

Value

It returns a vector of complex numbers with the dimension equals to

length(tau)

References

Nolan J. P. (2009), Stable Disttributions.

Math/Stat Department, American University

. URL

http:

coef

Examples

# GMM is like GLS for linear models without endogeneity problems

pm <- 0

theta <- c(1.5,.5,1,0)

tau <- seq(-3, 3, length.out = 20)

char_fct <- charStable(theta, tau, pm)

coef

Coefficients of GEL or GMM

Description

It extracts the coefficients from

gel

gmm

objects.

Usage

## S3 method for class 'gmm'

coef(object, ...)

## S3 method for class 'gel'

coef(object, lambda = FALSE, ...)

Arguments

object

An object of class

gel

gmm

returned by the function

lambda

If set to TRUE, the lagrange multipliers are extracted instead of the vector of
coefficients

...

Other arguments when

coef

is applied to an other class object

Value

Vector of coefficients

Examples

#################

n = 500

phi<-c(.2,.7)

thet <- 0

sd <- .2

x <- matrix(arima.sim(n=n,list(order=c(2,0,1),ar=phi,ma=thet,sd=sd)),ncol=1)

y <- x[7:n]

ym1 <- x[6:(n-1)]

ym2 <- x[5:(n-2)]

confint

H <- cbind(x[4:(n-3)], x[3:(n-4)], x[2:(n-5)], x[1:(n-6)])

g <- y ~ ym1 + ym2

x <- H

t0 <- c(0,.5,.5)

res <- gel(g, x, t0)

coef(res)

coef(res, lambda = TRUE)

###################

res <- gmm(g, x)

coef(res)

confint

Confidence intervals for GMM or GEL

Description

It produces confidence intervals for the coefficients from

gel

gmm

estimation.

Usage

## S3 method for class 'gel'

confint(object, parm, level = 0.95, lambda = FALSE,

type = c("Wald", "invLR", "invLM", "invJ"),

fact = 3, corr = NULL, ...)

## S3 method for class 'gmm'

confint(object, parm, level = 0.95, ...)

## S3 method for class 'ategel'

confint(object, parm, level = 0.95, lambda = FALSE,

type = c("Wald", "invLR", "invLM", "invJ"), fact = 3,

corr = NULL, robToMiss=TRUE, ...)

## S3 method for class 'confint'

print(x, digits = 5, ...)

Arguments

object

An object of class

gel

gmm

returned by the function

parm

A specification of which parameters are to be given confidence intervals, either
a vector of numbers or a vector of names. If missing, all parameters are consid-
ered.

level

The confidence level

lambda

If set to TRUE, the confidence intervals for the Lagrange multipliers are pro-
duced.

type

’Wald’ is the usual symetric confidence interval. The thee others are based on
the inversion of the LR, LM, and J tests.

confint

fact

This parameter control the span of search for the inversion of the test. By default
we search within plus or minus 3 times the standard error of the coefficient
estimate.

corr

This numeric scalar is meant to apply a correction to the critical value, such as
a Bartlett correction. This value depends on the model (See Owen; 2001)

An object of class

confint

produced by

confint.gel

and

confint.gmm

digits

The number of digits to be printed

robToMiss

TRUE

, the confidence interval is based on the standard errors that are robust to

misspecification

...

Other arguments when

confint

is applied to another classe object

Value

It returns a matrix with the first column being the lower bound and the second the upper bound.

References

Hansen, L.P. (1982), Large Sample Properties of Generalized Method of Moments Estimators.

Econometrica

, 1029-1054, Hansen, L.P. and Heaton, J. and Yaron, A.(1996), Finit-Sample

Properties of Some Alternative GMM Estimators.

Journal of Business and Economic Statistics

262-280. Owen, A.B. (2001), Empirical Likelihood.

Monographs on Statistics and Applied

Probability 92, Chapman and Hall/CRC

Examples

#################

n = 500

phi<-c(.2,.7)

thet <- 0

sd <- .2

x <- matrix(arima.sim(n = n, list(order = c(2,0,1), ar = phi, ma = thet, sd = sd)), ncol = 1)

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

t0 <- c(0,.5,.5)

resGel <- gel(g, x, t0)

confint(resGel)

confint(resGel, level = 0.90)

confint(resGel, lambda = TRUE)

########################

resGmm <- gmm(g, x)

estfun

confint(resGmm)

confint(resGmm, level = 0.90)

## Confidence interval with inversion of the LR, LM or J test.

##############################################################

set.seed(112233)

x <- rt(40, 3)

y <- x+rt(40,3)

# Simple interval on the mean

res <- gel(x~1, ~1, method="Brent", lower=-4, upper=4)

confint(res, type = "invLR")

confint(res)

# Using a Bartlett correction

k <- mean((x-mean(x))^4)/sd(x)^4

s <- mean((x-mean(x))^3)/sd(x)^3

a <- k/2-s^2/3

corr <- 1+a/40

confint(res, type = "invLR", corr=corr)

# Interval on the slope

res <- gel(y~x, ~x)

confint(res, "x", type="invLR")

confint(res, "x")

estfun

Extracts the empirical moment function

Description

It extracts the matrix of empirical moments so that it can be used by the

function.

Usage

## S3 method for class 'gmmFct'

estfun(x, y = NULL, theta = NULL, ...)

## S3 method for class 'gmm'

estfun(x, ...)

## S3 method for class 'gel'

estfun(x, ...)

## S3 method for class 'tsls'

estfun(x, ...)

## S3 method for class 'tsls'

model.matrix(object, ...)

Arguments

A function of the form

(

θ, y

)

or a

matrix with typical element

(

θ, y

)

for

= 1

, ...q

and

= 1

, ..., n

or an object of class

gmm

. See

for more details.

For

tsls

, it is an object of class

tsls

estfun

object

An object of class

tsls

The matrix or vector of data from which the function

(

θ, y

)

is computed if

a function.

theta

Vector of parameters if

is a function.

...

Other arguments when

estfun

is applied to another class object

Details

For

estfun.gmmFct

, it returns a

matrix with typical element

(

θ, y

)

for

= 1

, ...q

and

= 1

, ..., n

. It is only used by

gmm

to obtain the estimates.

For

estfun.gmm

, it returns the matrix of first order conditions of

min

, which is a

matrix with the

row being

(

θ, y

)

W G

, where

g/dθ

. It allows to compute the sandwich

covariance matrix using

vcovHAC

when

is not the optimal matrix.

The method if not yet available for

gel

objects.

For tsls, model.matrix and estfun are used by

vcov()

to compute different covariance matrices

using the

sandwich

package. See

vcov.tsls

model.matrix

returns the fitted values frin the first

stage regression and

esfun

the residuals.

Value

matrix (see details).

References

Zeileis A (2006), Object-oriented Computation of Sandwich Estimators.

Journal of Statistical Soft-

ware

(9), 1–16. URL

http://mba.tuck.dartmouth.edu/pages/faculty/ken.french

Examples

n = 500

phi<-c(.2,.7)

thet <- 0

sd <- .2

x <- matrix(arima.sim(n=n,list(order=c(2,0,1),ar=phi,ma=thet,sd=sd)),ncol=1)

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 <- gmm(g, x,weightsMatrix = diag(5))

gt <- res$gt

G <- res$G

foc <- gt

foc2 <- estfun(res)

Finance

foc[1:5,]

foc2[1:5,]

Finance

Returns on selected stocks

Description

Daily returns on selected stocks, the Market portfolio and factors of Fama and French from 1993-
01-05 to 2009-01-30 for CAPM and APT analysis

Usage

data(Finance)

Format

A data frame containing 24 time series. Dates are reported as rownames(). In the following descrip-
tion, company symboles are used.

WMK

Returns of WEIS MARKETS INC

UIS

Returns of UNISYS CP NEW

ORB

Returns of ORBITAL SCIENCES CP

MAT

Returns of Mattel, Inc.

ABAX

Returns of ABAXIS, Inc.

Returns of AT&T INC.

EMR

Returns of EMERSON ELEC CO

JCS

Returns of Communications Systems Inc.

VOXX

Returns of Audiovox Corp.

ZOOM

Returns of ZOOM Technologies Inc.

TDW

Returns of TIDEWATER INC

ROG

Returns of Rogers Corporation

GGG

Returns of Graco Inc.

Returns of Panasonic Corporation

GCO

Returns of Genesco Inc.

EBF

Returns of ENNIS, INC

Returns of FORD MOTOR CO

FNM

Returns of FANNIE MAE

NHP

Returns of NATIONWIDE HLTH PROP

Returns of ALCOA INC

Risk-free rate of Fama-French

Return of the market portfolio of Fama-French

hml

Factor High-Minus-Low of Fama-French

smb

Factor Small-Minus-Big of Fama-French

FinRes

Source

Yahoo Finance and

FinRes

Method to finalize the result of the momentEstim method

Description

It computes the final results that will be needed to create the object of class

gmm

.).

Usage

## S3 method for class 'baseGmm.res'

FinRes(z, object, ...)

Arguments

An object of class determined by the method

momentEstim

object

An object produced my

getModel

...

Other argument to be passed to other

FinRes

methods.

Value

It returns an object of class

gmm

. See

for more details.

References

Hansen, L.P. (1982), Large Sample Properties of Generalized Method of Moments Estimators.

Econometrica

, 1029-1054,

Hansen, L.P. and Heaton, J. and Yaron, A.(1996), Finit-Sample Properties of Some Alternative
GMM Estimators.

Journal of Business and Economic Statistics

262-280.

fitted

Fitted values of GEL and GMM

Description

Method to extract the fitted values of the model estimated by

Usage

## S3 method for class 'gel'

fitted(object, ...)

## S3 method for class 'gmm'

fitted(object, ...)

fitted

Arguments

object

An object of class

gel

returned by the function

...

Other arguments when

fitted

is applied to an other class object

Value

It returns a matrix of the estimated mean

g=y~x

as it is done by

fitted.lm

Examples

# GEL can deal with endogeneity problems

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)

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))

plot(y, main = "Fitted ARMA with GEL")

lines(fitted(res), col = 2)

# GMM is like GLS for linear models without endogeneity problems

set.seed(345)

n = 200

phi<-c(.2,.7)

thet <- 0

sd <- .2

x <- matrix(arima.sim(n = n, list(order = c(2,0,1), ar = phi, ma = thet, sd = sd)), ncol = 1)

y <- 10 + 5*rnorm(n) + x

res <- gmm(y ~ x, x)

plot(x, y, main = "Fitted model with GMM")

lines(x, fitted(res), col = 2)

legend("topright", c("Y","Yhat"), col = 1:2, lty = c(1,1))

formula

Formula method for gel and gmm objects

Description

Method to extract the formula from

gel

gmm

objects.

Usage

## S3 method for class 'gel'

formula(x, ...)

## S3 method for class 'gmm'

formula(x, ...)

Arguments

An object of class

gel

gmm

returned by the function

...

Other arguments to pass to other methods

Examples

## GEL ##

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)

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))

formula(res)

# GMM is like GLS for linear models without endogeneity problems

set.seed(345)

n = 200

phi<-c(.2,.7)

thet <- 0

sd <- .2

x <- matrix(arima.sim(n = n, list(order = c(2,0,1), ar = phi, ma = thet, sd = sd)), ncol = 1)

y <- 10 + 5*rnorm(n) + x

gel

res <- gmm(y ~ x, x)

formula(res)

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"),

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"),

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

A function of the form

(

θ, x

)

and which returns a

matrix with typical

element

(

θ, x

)

for

= 1

, ...q

and

= 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

vector of starting values. If the dimension of

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

The matrix or vector of data from which the function

(

θ, x

)

is computed. If "g"

is a formula, it is an

N h

matrix of instruments (see details below).

gel

gradv

A function of the form

(

θ, x

)

which returns a

matrix of derivatives of

(

)

with respect to

. 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

. 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

kernel

type of kernel used to compute the covariance matrix of the vector of sample
moment conditions (see

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 im-
plies a Parzen kernel (see Smith 2004).

The method to compute the bandwidth parameter. By default it is

bwAndrews

which is proposed by Andrews (1991). The alternative is

bwNeweyWest

Newey-West(1994).

prewhite

logical or integer. Should the estimating functions be prewhitened? If

TRUE

greater than 0 a VAR model of order

as.integer(prewhite)

is fitted via

with method

"ols"

and

demean = FALSE

ar.method

character. The

method

argument passed to

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

between two iterations. The algorithm stops when

−

reaches

tol_lamb

(see

maxiterlam

The algorithm to compute

stops if there is no convergence after "maxiterlam"

iterations (see

tol_obj

Tolerance for the gradiant of the objective function to compute

(see

optfct

Only when the dimension of

is 1, you can choose between the algorithm

. In that case, the former is unreliable. If

is chosen, "t0"

must be

which represents the interval in which the algorithm seeks the

solution.It is also possible to choose the

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

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

uiθ

−

≥

tol_mom

It is the tolerance for the moment condition

n
t

(

) = 0

, where

Dρ

(

< g

, 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

. The algorithm

Wu is only for

type="EL"

gel

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

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 devel-

opers to extand the package and create other GEL methods.

alpha

Regularization coefficient for discrete CGEL estimation (experimental). By set-
ting 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

is restricted, or a qx2 vector defining equality

constraints of the form

. See

for an example.

eqConstFullVcov

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

onlyCoefficients

TRUE

, only the vector of coefficients and Lagrange multipliers are returned

...

More options to give to

Details

If we want to estimate a model like

...

using the moment conditions

Cov

(

) = 0

, where

is a vector of

N h

instruments, than we can define "g" like we do for

. We would have

g = y~x2+x3+...+xk

and the argument "x" above would become the matrix

H of instruments. As for

can be a

N y

vector which would imply that

N h

N y

The intercept is included by default so you do not have to add a column of ones to the matrix

You do not need to provide the gradiant in that case since in that case it is embedded in

. 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

n
t

(

θ, x

)

is replaced by:

n
t

(

θ, x

)

where

(

θ, x

) =

r
i

−

(

)

(

θ, x

)

, where

is a truncated parameter that depends on the

bandwidth and

(

)

are normalized weights so that they sum to 1.

The method solves

= arg min

arg max

n
t

(

< g

(

θ, x

)

, λ >

)

−

(0)

evalGel

generates the object of class "gel" for a fixed vector of parameters. There is no estimation

for

, but the optimal vector of Lagrange multipliers

is computed. The objective function is then

the profiled likelihood for a given

. 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:

gel

coefficients

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

vector of Lagrange multipliers.

vcov_par

the covariance matrix of "coefficients"

vcov_lambda

the covariance matrix of "lambda"

The implied probabilities

objective

the value of the objective function

conv_lambda

Convergence code for "lambda" (see

conv_mes

Convergence message for "lambda" (see

conv_par

Convergence code for "coefficients" (see

terms

the

object used when g is a formula.

call

the matched call.

if requested, the response used (if "g" is a formula).

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

983-1002.

Andrews DWK (1991), Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Esti-
mation.

Econometrica

, 817–858.

Kitamura, Yuichi (1997), Empirical Likelihood Methods With Weakly Dependent Processes.

The

Annals of Statistics

, 2084-2102.

Kitamura, Y. and Otsu, T. and Evdokimov, K. (2013), Robustness, Infinitesimal Neighborhoods and
Moment Restrictions.

Econometrica

, 1185-1201.

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

Econometrica

, 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 Auto-
correlation Consistent Covariance Matrix.

Econometrica

, 703–708.

Newey WK & West KD (1994), Automatic Lag Selection in Covariance Matrix Estimation.

Review

of Economic Studies

, 631-653.

Schennach, Susanne, M. (2007), Point Estimation with Exponentially Tilted Empirical Likelihood.

Econometrica

, 634-672.

Wu, C. (2005), Algorithms and R codes for the pseudo empirical likelihood method in survey
sampling.

Survey Methodology

(2), page 239.

Zeileis A (2006), Object-oriented Computation of Sandwich Estimators.

Journal of Statistical Soft-

ware

(9), 1–16. URL

http://www.jstatsoft.org/

gel

Chausse (2010), Computing Generalized Method of Moments and Generalized Empirical Likeli-
hood with R.

Journal of Statistical Software

(11), 1–35. URL

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)

getDat

Extracting data from a formula

Description

It extract the data from a formula y~z with instrument h and put everything in a matrix. It helps
redefine the function

(

θ, x

)

that is required by

and

Usage

getDat(formula, h, data, error=TRUE)

Arguments

formula

A formula that defines the linear model to be estimated (see details).

matrix of intruments(see details).

data

A data.frame or a matrix with colnames (Optionnal).

error

If FALSE, the data is generated without giving any error message

Details

The model to be estimated is based on the moment conditions

< h,

(

−

zθ

)

= 0

. It adds a column

of ones to z and h by default. They are removed if -1 is added to the formula. The error argument
has been added for

sysGmm

with common coefficients because the check is only valid for equation

by equation identification.

Value

x: A

matrix, where

ncol

(

) +

ncol

(

) +

ncol

(

) + 2

if "intercept" is TRUE and

ncol

(

) +

ncol

(

) +

xcol

(

)

if "intercept" is FALSE.

nh: dimension of h

k: dimension of z

ny: dimension of y

Examples

n = 500

phi<-c(.2, .7)

thet <- 0.2

sd <- .2

x <- matrix(arima.sim(n = n, list(order = c(2, 0, 1), ar = phi, ma = thet, sd = sd)), ncol = 1)

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)])

getLamb

x <- getDat(y ~ ym1 + ym2, H)

getLamb

Solving for the Lagrange multipliers of Generalized Empirical Likeli-
hood (GEL)

Description

It computes the vector of Lagrange multipliers, which maximizes the GEL objective function, using
an iterative Newton method.

Usage

getLamb(gt, l0, type = c("EL","ET","CUE", "ETEL", "HD","ETHD"),

tol_lam = 1e-7, maxiterlam = 100,

tol_obj = 1e-7, k = 1, method = c("nlminb", "optim", "iter", "Wu"),

control = list())

Arguments

matrix with typical element

(

θ, x

)

Vector of starting values for lambda

type

"EL" for empirical likelihood, "ET" for exponential tilting, "CUE" for continu-
ous updated estimator, and "HD" for Hellinger Distance. See details for "ETEL"
and "ETHD".

tol_lam

Tolerance for

between two iterations. The algorithm stops when

−

reaches

tol_lam

maxiterlam

The algorithm stops if there is no convergence after "maxiterlam" iterations.

tol_obj

Tolerance for the gradiant of the objective function. The algorithm returns a
non-convergence message if

max(

gradiant

)

does not reach

tol_obj

. It helps

the

gel

algorithm to select the right space to look for

It represents the ratio k1/k2, where

1 =

∞

−∞

(

)

and

2 =

∞

−∞

(

)

See Smith(2004).

method

The iterative procedure uses a Newton method for solving the FOC. It i however
recommended to use

optim

nlminb

. If type is set to "EL" and method to

"optim",

is called to prevent

log

−

)

from producing NA.

The gradient and hessian is provided to

nlminb

which speed up the convergence.

The latter is therefore the default value. "Wu" is for "EL" only. It uses the
algorithm of Wu (2005).

control

Controls to send to

getModel

Details

It solves the problem

max

n
t

(

)

. For the type "ETEL", it is only used by

. In

that case

is obtained by maximizing

n
t

(

)

, using

(

) =

−

exp

(so ET) and

minimizing the same equation but with

(

)

−

log (1

−

)

. To avoid NA’s,

is used

with the restriction

. The type "ETHD" is experimental and proposed by Antoine-Dovonon

(2015). The paper is not yet available.

Value

lambda: A

vector of Lagrange multipliers which solve the system of equations given above.

conv

: Details on the type of convergence.

References

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

Econometrica

, 219-255.

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

Working paper, CEMMAP

Wu, C. (2005), Algorithms and R codes for the pseudo empirical likelihood method in survey
sampling.

Survey Methodology

(2), page 239.

Examples

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)

}

n = 500

phi<-c(.2, .7)

thet <- 0.2

sd <- .2

x <- matrix(arima.sim(n = n, list(order = c(2, 0, 1), ar = phi, ma = thet, sd = sd)), ncol = 1)

gt <- g(c(0,phi),x)

getLamb(gt, type = "EL",method="optim")

getModel

Method for setting the properties of a model

Description

It collects what is needed by the method

momentEstim

(see details).

gmm

Usage

## S3 method for class 'baseGmm'

getModel(object, ...)

## S3 method for class 'sysGmm'

getModel(object, ...)

## S3 method for class 'baseGel'

getModel(object, ...)

## S3 method for class 'constGel'

getModel(object, ...)

## S3 method for class 'constGel'

getModel(object, ...)

## S3 method for class 'tsls'

getModel(object, ...)

## S3 method for class 'ateGel'

getModel(object, ...)

Arguments

object

An object of class

baseGmm

...

Other arguments when

getModel

is applied to another class object

Value

It returns an object of the right class which determines how the method

momentEstim

will treat it.

For example, if

is a formula and

type

is set to "cue", it creates an object of class

baseGmm.cue.formula

It this case,

momentEstim

, applied to this object, computes the continuously updated GMM of a lin-

ear model. It allows more flexibility this way. For example, it could be easy to add a GMM method
which is robust in presence of weak identification simply by creating a new class of model and the
associated

momentEstime

method.

gmm

Generalized method of moment estimation

Description

Function to estimate a vector of parameters based on moment conditions using the GMM method
of Hansen(82).

Usage

gmm(g,x,t0=NULL,gradv=NULL, type=c("twoStep","cue","iterative"),

wmatrix = c("optimal","ident"), vcov=c("HAC","MDS","iid","TrueFixed"),

kernel=c("Quadratic Spectral","Truncated", "Bartlett", "Parzen", "Tukey-Hanning"),

crit=10e-7,bw = bwAndrews, prewhite = 1, ar.method = "ols", approx="AR(1)",

tol = 1e-7, itermax=100,optfct=c("optim","optimize","nlminb", "constrOptim"),

model=TRUE, X=FALSE, Y=FALSE, TypeGmm = "baseGmm", centeredVcov = TRUE,

gmm

weightsMatrix = NULL, traceIter = FALSE, data, eqConst = NULL,

eqConstFullVcov = FALSE, mustar = NULL, onlyCoefficients=FALSE, ...)

evalGmm(g, x, t0, tetw=NULL, gradv=NULL, wmatrix = c("optimal","ident"),

vcov=c("HAC","iid","TrueFixed"), kernel=c("Quadratic Spectral","Truncated",

"Bartlett", "Parzen", "Tukey-Hanning"),crit=10e-7,bw = bwAndrews,

prewhite = FALSE, ar.method = "ols", approx="AR(1)",tol = 1e-7,

model=TRUE, X=FALSE, Y=FALSE, centeredVcov = TRUE, weightsMatrix = NULL,

data, mustar = NULL)

gmmWithConst(obj, which, value)

Arguments

A function of the form

(

θ, x

)

and which returns a

matrix with typical

element

(

θ, x

)

for

= 1

, ...q

and

= 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).

The matrix or vector of data from which the function

(

θ, x

)

is computed. If

"g" is a formula, it is an

N h

matrix of instruments or a formula (see details

below).

vector of starting values. It is required only when "g" is a function

because only then a numerical algorithm is used to minimize the objective func-
tion. If the dimension of

is one, see the argument "optfct".

tetw

vector to compute the weighting matrix.

gradv

A function of the form

(

θ, x

)

which returns a

matrix of derivatives of

(

)

with respect to

. By default, the numerical algorithm

numericDeriv

is used.

It is of course strongly suggested to provide this function when it is possible.
This gradient is used to compute the asymptotic covariance matrix of

and to

obtain the analytical gradient of the objective function if the method is set to
"CG" or "BFGS" in

and if "type" is not set to "cue". If "g" is a formula,

the gradiant is not required (see the details below).

type

The GMM method: "twostep" is the two step GMM proposed by Hansen(1982)
and the "cue" and "iterative" are respectively the continuous updated and the
iterative GMM proposed by Hansen, Eaton et Yaron (1996)

wmatrix

Which weighting matrix should be used in the objective function. By default, it
is the inverse of the covariance matrix of

(

θ, x

)

. The other choice is the identity

matrix which is usually used to obtain a first step estimate of

vcov

Assumption on the properties of the random vector x. By default, x is a weakly
dependant process. The "iid" option will avoid using the HAC matrix which will
accelerate the estimation if one is ready to make that assumption. The option
"TrueFixed" is used only when the matrix of weights is provided and it is the
optimal one.

kernel

type of kernel used to compute the covariance matrix of the vector of sample
moment conditions (see

for more details)

crit

The stopping rule for the iterative GMM. It can be reduce to increase the preci-
sion.

gmm

The method to compute the bandwidth parameter in the HAC weighting matrix.
The default is

link{bwAndrews}

(as proposed in Andrews (1991)), which mini-

mizes the MSE of the weighting matrix. Alternatives are

link{bwWilhelm}

(as

proposed in Wilhelm (2015)), which minimizes the mean-square error (MSE) of
the resulting GMM estimator, and

link{bwNeweyWest}

(as proposed in Newey-

West(1994)).

prewhite

logical or integer. Should the estimating functions be prewhitened? If

TRUE

greater than 0 a VAR model of order

as.integer(prewhite)

is fitted via

with method

"ols"

and

demean = FALSE

ar.method

character. The

method

argument passed to

for prewhitening.

approx

A character specifying the approximation method if the bandwidth has to be
chosen by

bwAndrews

tol

Weights that exceed

tol

are used for computing the covariance matrix, all other

weights are treated as 0.

itermax

The maximum number of iterations for the iterative GMM. It is unlikely that the
algorithm does not converge but we keep it as a safety.

optfct

Only when the dimension of

is 1, you can choose between the algorithm

. In that case, the former is unreliable. If

is chosen,

"t0" must be

which represents the interval in which the algorithm seeks

the solution. It is also possible to choose the

algorithm. In that case,

boundaries for the coefficients can be set by the options

upper=

and

lower=

The

is only available for nonlinear models for now. The standard

errors may have to be corrected if the estimtes reach the boundary set by ui and
ci.

model, X, Y

logical. If

TRUE

the corresponding components of the fit (the model frame, the

model matrix, the response) are returned if g is a formula.

TypeGmm

The name of the class object created by the method

getModel

. It allows devel-

opers to extend the package and create other GMM methods.

centeredVcov

Should the moment function be centered when computing its covariance matrix.
Doing so may improve inference.

weightsMatrix

It allows users to provide

gmm

with a fixed weighting matrix. This matrix must

, symmetric and strictly positive definite. When provided, the

type

option becomes irrelevant.

traceIter

Tracing information for GMM of type "iter"

data

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

eqConst

Either a named vector (if "g" is a function), a simple vector for the nonlinear
case indicating which of the

is restricted, or a qx2 vector defining equality

constraints of the form

. See below for an example.

which, value

The equality constraint is of the form which=value. "which" can be a vector of
type characters with the names of the coefficients being constrained, or a vector
of type numeric with the position of the coefficient in the whole vector.

obj

Object of class "gmm"

gmm

eqConstFullVcov

mustar

If not null, it must be a vector with the number of elements being equal to the
number of moment conditions. In that case, the vector is subtracted from the
sample moment vector before minimizing the objective function. It is useful to
do a bootstrap procedure.

onlyCoefficients

If set to

TRUE

, the function only returns the coefficient estimates. It may be of

interest when the standard errors are not needed

...

More options to give to

Details

If we want to estimate a model like

· · ·

using the moment conditions

Cov

(

) = 0

, where

is a vector of

N h

instruments, than we can define "g" like we do for

We would have

2 +

3 +

· · ·

and the argument "x" above would become the matrix H

of instruments. As for

can be a

N y

vector which would imply that

N h

N y

. The

intercept is included by default so you do not have to add a column of ones to the matrix

. You do

not need to provide the gradiant in that case since in that case it is embedded in

. 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. It is also possible to express "x" as a formula. For example, if the instruments are

{

, z

}

, we can set "x" to

1 +

2 +

. By default, a column of ones is added. To remove

it, set "x" to

1 +

2 +

−

The following explains the last example bellow. Thanks to Dieter Rozenich, a student from the
Vienna University of Economics and Business Administration. He suggested that it would help to
understand the implementation of the Jacobian.

For the two parameters of a normal distribution

(

µ, σ

)

we have the following three moment condi-

tions:

−

(

−

)

−

(

+ 3

)

, m

can be directly obtained by the definition of

(

µ, σ

)

. The third moment condition comes

from the third derivative of the moment generating function (MGF)

(

) =

exp

µt

evaluated at

(

= 0)

Note that we have more equations (3) than unknown parameters (2).

The Jacobian of these two conditions is (it should be an array but I can’t make it work):

−

+ 2

gmm

−

µσ

gmmWithConst()

re-estimates an unrestricted model by adding an equality constraint.

evalGmm()

creates an object of class ’"gmm"’ for a given parameter vector. If no vector "tetw" is provided and
the weighting matrix needs to be computed, "t0" is used.,

Value

’gmm’ returns an object of ’class’ ’"gmm"’

The functions ’summary’ is used to obtain and print a summary of the results. It also compute the
J-test of overidentying restriction

The object of class "gmm" is a list containing at least:

coefficients

vector of coefficients

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.

vcov

the covariance matrix of the coefficients

objective

the value of the objective function

var

(¯

)

−

terms

the

object used when g is a formula.

call

the matched call.

if requested, the response used (if "g" is a formula).

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.

algoInfo

Information produced by either

related to the convergence if

"g" is a function. It is printed by the

summary.gmm

method.

References

Zeileis A (2006), Object-oriented Computation of Sandwich Estimators.

Journal of Statistical Soft-

ware

(9), 1–16. URL

http://fhayashi.fc2web.com/datasets.htm

Pierre Chausse (2010), Computing Generalized Method of Moments and Generalized Empirical
Likelihood with R.

Journal of Statistical Software

(11), 1–35. URL

http://www.jstatsoft.

org/v34/i11/

Andrews DWK (1991), Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Esti-
mation.

Econometrica

, 817–858.

Newey WK & West KD (1987), A Simple, Positive Semi-Definite, Heteroskedasticity and Auto-
correlation Consistent Covariance Matrix.

Econometrica

, 703–708.

Newey WK & West KD (1994), Automatic Lag Selection in Covariance Matrix Estimation.

Review

of Economic Studies

, 631-653.

Hansen, L.P. (1982), Large Sample Properties of Generalized Method of Moments Estimators.

Econometrica

, 1029-1054,

Hansen, L.P. and Heaton, J. and Yaron, A.(1996), Finite-Sample Properties of Some Alternative
GMM Estimators.

Journal of Business and Economic Statistics

262-280.

gmm

Examples

## 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"]

f3 <- Finance[1:300, "smb"]

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.

#-------------------------------------------------------------------------------

gmm

# 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")

res4

### Examples with equality constraint

######################################

# Random numbers of a normal distribution

## Not run:

gmm

# The following works but produces warning message because the dimension of coef is 1

# Brent should be used

# without named vector

# Method Brent is used because the problem is now one-dimensional

gmm(g, x, c(4, 0), grad = Dg, eqConst=1, method="Brent", lower=-10,upper=10)

# with named vector

gmm(g, x, c(mu=4, sig=2), grad = Dg, eqConst="sig", method="Brent", lower=-10,upper=10)

## End(Not run)

gmm(g, x, c(4, 0), grad = Dg, eqConst=1,method="Brent",lower=0,upper=6)

gmm(g, x, c(mu=4, sig=2), grad = Dg, eqConst="sig",method="Brent",lower=0,upper=6)

# Example with formula

# first coef = 0 and second coef = 1

# Only available for one dimensional yt

z <- z[,1]

res2 <- gmm(z ~ f1 + f2 + f3, ~ f1 + f2 + f3, eqConst = matrix(c(1,2,0,1),2,2))

res2

# CUE with starting t0 requires eqConst to be a vector

res3 <- gmm(z ~ f1 + f2 + f3, ~ f1 + f2 + f3, t0=c(0,1,.5,.5), type="cue", eqConst = c(1,2))

res3

### Examples with equality constraints, where the constrained coefficients is used to compute

### the covariance matrix.

### Useful when some coefficients have been estimated before, they are just identified in GMM

### and don't need to be re-estimated.

### To use with caution because the covariance won't be valid if the coefficients do not solve

### the GMM FOC.

######################################

res4 <- gmm(z ~ f1 + f2 + f3, ~ f1 + f2 + f3, t0=c(0,1,.5,.5), eqConst = c(1,2),

eqConstFullVcov=TRUE)

summary(res4)

### Examples with equality constraint using gmmWithConst

###########################################################

res2 <- gmm(z ~ f1 + f2 + f3, ~ f1 + f2 + f3)

gmmWithConst(res2,c("f2","f3"),c(.5,.5))

gmmWithConst(res2,c(2,3),c(.5,.5))

## Creating an object without estimation for a fixed parameter vector

###################################################################

res2_2 <- evalGmm(z ~ f1 + f2 + f3, ~ f1 + f2 + f3,

t0=res2$coefficients, tetw=res2$coefficients)

summary(res2_2)

Growth

Growth Data

Description

Panel of Macroeconomic data for 125 countries from 1960 to 1985 constructed by Summers and
Heston (1991))

Usage

data(Growth)

Format

A data frame containing 9 vectors.

Country_ID

Country identification number

COM

1 if the country is in a communist regime, 0 otherwise

OPEC

1 if the country is part of the OPEC, 0 otherwise

Year

GDP

Per capita GDP (in thousands) in 1985 U.S. dollars.

LagGDP

GDP of the previous period

SavRate

Saving rate measured as the ratio of real investment to real GDP

LagSavRate

SavRate of the previous period

Country

Country names

Pop

Population in thousands

LagPop

Population of the previous period

Source

KTest

Compute the K statistics of Kleibergen

Description

The test is proposed by Kleibergen (2005). It is robust to weak identification.

Usage

KTest(obj, theta0 = NULL, alphaK = 0.04, alphaJ = 0.01)

## S3 method for class 'gmmTests'

print(x, digits = 5, ...)

Arguments

obj

Object of class "gmm" returned by

theta0

The null hypothesis being tested. See details.

alphaK, alphaJ

The size of the J and K tests when combining the two. The overall size is
alphaK+alphaJ.

An object of class

gmmTests

returned by

KTest

digits

The number of digits to be printed

...

Other arguments when

is applied to another class object

Details

The function produces the J-test and K-statistics which are robust to weak identification. The test is
either

0 :

theta

, in which case theta0 must be provided, or

, where

= (

, β

)

, and

is assumed to be identified. In the latter case, theta0 is NULL and obj is a restricted estimation in

which

is fixed to

. See

and the option "eqConst" for more details.

Value

Tests and p-values

References

Keibergen, F. (2005), Testing Parameters in GMM without assuming that they are identified.

Econo-

metrica

, 1103-1123,

Examples

library(mvtnorm)

sig <- matrix(c(1,.5,.5,1),2,2)

n <- 400

e <- rmvnorm(n,sigma=sig)

x4 <- rnorm(n)

marginal

w <- exp(-x4^2) + e[,1]

y <- 0.1*w + e[,2]

h <- cbind(x4, x4^2, x4^3, x4^6)

g3 <- y~w

res <- gmm(g3,h)

# Testing the whole vector:

KTest(res,theta0=c(0,.1))

# Testing a subset of the vector (See \code{\link{gmm}})

res2 <- gmm(g3, h, eqConst=matrix(c(2,.1),1,2))

res2

KTest(res2)

marginal

Marginal effects Summary

Description

It produces the summary table of marginal effects for GLM estimation with GEL. Only imple-
mented for ATEgel.

Usage

## S3 method for class 'ategel'

marginal(object, ...)

Arguments

object

An object of class

ategel

returned by the function

ATEgel

...

Other arguments for other methods

Value

It returns a matrix with the marginal effects, the standard errors based on the Delta method when
the link is nonlinear, the t-ratios, and the pvalues.

References

Owen, A.B. (2001), Empirical Likelihood.

Monographs on Statistics and Applied Probability 92,

Chapman and Hall/CRC

momentEstim

Examples

## We create some artificial data with unbalanced groups and binary outcome

genDat <- function(n)

{

eta=c(-1, .5, -.25, -.1)

Z <- matrix(rnorm(n*4),ncol=4)

b <- c(27.4, 13.7, 13.7, 13.7)

bZ <- c(Z%*%b)

Y1 <- as.numeric(rnorm(n, mean=210+bZ)>220)

Y0 <- as.numeric(rnorm(n, mean=200-.5*bZ)>220)

etaZ <- c(Z%*%eta)

pZ <- exp(etaZ)/(1+exp(etaZ))

T <- rbinom(n, 1, pZ)

Y <- T*Y1+(1-T)*Y0

X1 <- exp(Z[,1]/2)

X2 <- Z[,2]/(1+exp(Z[,1]))

X3 <- (Z[,1]*Z[,3]/25+0.6)^3

X4 <- (Z[,2]+Z[,4]+20)^2

data.frame(Y=Y, cbind(X1,X2,X3,X4), T=T)

}

dat <- genDat(200)

res <- ATEgel(Y~T, ~X1+X2+X3+X4, data=dat, type="ET", family="logit")

summary(res)

marginal(res)

momentEstim

Method for estimating models based on moment conditions

Description

It estimates a model which is caracterized by the method

getModel

(see details).

Usage

## S3 method for class 'baseGmm.twoStep'

momentEstim(object, ...)

## S3 method for class 'baseGmm.twoStep.formula'

momentEstim(object, ...)

## S3 method for class 'sysGmm.twoStep.formula'

momentEstim(object, ...)

## S3 method for class 'tsls.twoStep.formula'

momentEstim(object, ...)

## S3 method for class 'baseGmm.iterative.formula'

momentEstim(object, ...)

## S3 method for class 'baseGmm.iterative'

momentEstim(object, ...)

nsw

## S3 method for class 'baseGmm.cue.formula'

momentEstim(object, ...)

## S3 method for class 'baseGmm.cue'

momentEstim(object, ...)

## S3 method for class 'baseGmm.eval'

momentEstim(object, ...)

## S3 method for class 'baseGel.mod'

momentEstim(object, ...)

## S3 method for class 'baseGel.modFormula'

momentEstim(object, ...)

## S3 method for class 'baseGel.eval'

momentEstim(object, ...)

Arguments

object

An object created by the method

getModel

...

Other arguments when

momentEstim

is applied to an other class object

Value

It returns an object of class determined by the argument "TypeGMM" of

http://users.nber.org/~rdehejia/

. By default, it is of

class

baseGmm.res

. It estimates the model and organize the results that will be finalized by the

method

FinRes

. More methods can be created in order to use other GMM methods not yet included

in the package.

References

Hansen, L.P. (1982), Large Sample Properties of Generalized Method of Moments Estimators.

Econometrica

, 1029-1054,

Hansen, L.P. and Heaton, J. and Yaron, A.(1996), Finit-Sample Properties of Some Alternative
GMM Estimators.

Journal of Business and Economic Statistics

262-280.

nsw

Lalonde subsample of the National Supported Work Demonstration
Data (NSW)

Description

This data was collected to evaluate the National Supported Work (NSW) Demonstration project in
Lalonde (1986).

Usage

data(nsw)

plot

Format

A data frame containing 9 variables.

treat

Treatment assignment

age

Age

Years of Education

black

1 if Black, 0 otherwise

hisp

1 if Hispanic 0 otherwise

married

1 if married 0 otherwise

nodeg

1 if no college degree 0 otherwise

re75

1975 earnings

re78

1978 earnings

Details

The dataset was obtained from the ATE package (see reference).

Source

"NSW Data Files" from Rajeev Dehejia’s website. URL:

data/nswdata2.html

"National Supported Work Evaluation Study, 1975-1979: Public Use Files." from the Interuniversity
Consortium for Political and Social Research. URL:

http://www.icpsr.umich.edu/icpsrweb/

ICPSR/studies/7865

References

Lalonde, R. (1986). "Evaluating the Econometric Evaluations of Training Programs," American
Economic Review, 76(4), 604-620.

Dehejia R. and Wahba S. (1999). "Causal Effects in Non-Experimental Studies: Re-Evaluating the
Evaluation of Training Programs," JASA 94 (448), 1053-1062.

Asad Haris and Gary Chan (2015). ATE: Inference for Average Treatment Effects using Covariate
Balancing. R package version 0.2.0.

https://CRAN.R-project.org/package=ATE

plot

Plot Diagnostics for gel and gmm objects

Description

It is a plot method for

gel

gmm

objects.

plot

Usage

## S3 method for class 'gel'

plot(x, which = c(1L:4),

main = list("Residuals vs Fitted values", "Normal Q-Q",

"Response variable and fitted values","Implied probabilities"),

panel = if(add.smooth) panel.smooth else points,

ask = prod(par("mfcol")) < length(which) && dev.interactive(), ...,

add.smooth = getOption("add.smooth"))

## S3 method for class 'gmm'

plot(x, which = c(1L:3),

main = list("Residuals vs Fitted values", "Normal Q-Q",

"Response variable and fitted values"),

panel = if(add.smooth) panel.smooth else points,

ask = prod(par("mfcol")) < length(which) && dev.interactive(), ...,

add.smooth = getOption("add.smooth"))

Arguments

gel

gmm

object, typically result of

which

if a subset of the plots is required, specify a subset of the numbers

1:4

for

gel

1:3

for

gmm

main

Vector of titles for each plot.

panel

panel function. The useful alternative to

points

panel.smooth

can be chosen

add.smooth = TRUE

ask

logical; if

TRUE

, the user is

ask

ed before each plot, see

par

(ask=.)

...

other parameters to be passed through to plotting functions.

add.smooth

logical indicating if a smoother should be added to most plots; see also

panel

above.

Details

It is a beta version of a plot method for

gel

objects. It is a modified version of

plot.lm

. For now, it

is available only for linear models expressed as a formula. Any suggestions are welcome regarding
plots or options to include. The first two plots are the same as the ones provided by

plot.lm

, the

third is the dependant variable

with its mean

(the fitted values) and the last plots the implied

probabilities with the empirical density

Examples

# GEL #

n = 500

phi<-c(.2,.7)

thet <- 0

sd <- .2

x <- matrix(arima.sim(n = n,list(order = c(2,0,1), ar = phi, ma = thet, sd = sd)), ncol = 1)

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

t0 <- c(0,.5,.5)

res <- gel(g, x, t0)

plot(res, which = 3)

plot(res, which = 4)

# GMM #

res <- gmm(g, x)

plot(res, which = 3)

Printing a gmm or gel object

Description

It is a printing method for

gmm

gel

objects.

Usage

## S3 method for class 'gmm'

print(x, digits = 5, ...)

## S3 method for class 'gel'

print(x, digits = 5, ...)

## S3 method for class 'sysGmm'

print(x, digits = 5, ...)

Arguments

An object of class

gmm

gel

returned by the function

digits

The number of digits to be printed

...

Other arguments when print is applied to an other class object

Value

It prints some results from the estimation like the coefficients and the value of the objective function.

residuals

Examples

# GMM #

n = 500

phi<-c(.2,.7)

thet <- 0

sd <- .2

x <- matrix(arima.sim(n = n, list(order = c(2,0,1), ar = phi, ma = thet, sd = sd)), ncol = 1)

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 <- gmm(g, x)

print(res)

# GEL #

t0 <- c(0,.5,.5)

res <- gel(g,x,t0)

print(res)

residuals

Residuals of GEL or GMM

Description

Method to extract the residuals of the model estimated by

gmm

gel

Usage

## S3 method for class 'gel'

residuals(object, ...)

## S3 method for class 'gmm'

residuals(object, ...)

Arguments

object

An object of class

gmm

gel

returned by the function

...

Other arguments when

residuals

is applied to an other classe object

Value

It returns the matrix of residuals

(

−

)

g=y~x

as it is done by

residuals.lm

smoothG

Examples

# GEL can deal with endogeneity problems

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)

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))

e <- residuals(res)

plot(e, type = 'l', main = "Residuals from an ARMA fit using GEL")

# GMM is like GLS for linear models without endogeneity problems

set.seed(345)

n = 200

phi<-c(.2,.7)

thet <- 0

sd <- .2

x <- matrix(arima.sim(n = n, list(order = c(2,0,1), ar = phi, ma = thet, sd = sd)), ncol = 1)

y <- 10 + 5*rnorm(n) + x

res <- gmm(y ~ x, x)

plot(x, residuals(res), main = "Residuals of an estimated model with GMM")

smoothG

Kernel smoothing of a matrix of time series

Description

It applies the required kernel smoothing to the moment function in order for the GEL estimator to
be valid. It is used by the

gel

function.

Usage

smoothG(x, bw = bwAndrews, prewhite = 1, ar.method = "ols", weights = weightsAndrews,

kernel = c("Bartlett", "Parzen", "Truncated", "Tukey-Hanning"),

approx = c("AR(1)", "ARMA(1,1)"), tol = 1e-7)

smoothG

Arguments

matrix of time series, where n is the sample size.

The method to compute the bandwidth parameter. By default, it uses the band-
width proposed by Andrews(1991). As an alternative, we can choose bw=bwNeweyWest
(without "") which is proposed by Newey-West(1996).

prewhite

logical or integer. Should the estimating functions be prewhitened? If

TRUE

greater than 0 a VAR model of order

as.integer(prewhite)

is fitted via

with method

"ols"

and

demean = FALSE

ar.method

character. The

method

argument passed to

for prewhitening.

weights

The smoothing weights can be computed by

weightsAndrews

of it can be pro-

vided manually. If provided, it has to be a

vector (see details).

approx

a character specifying the approximation method if the bandwidth has to be
chosen by

bwAndrews

tol

numeric. Weights that exceed

tol

are used for computing the covariance matrix,

all other weights are treated as 0.

kernel

The choice of kernel

Details

The sample moment conditions

n
t

(

θ, x

)

is replaced by:

n
t

(

θ, x

)

, where

(

θ, x

) =

r
i

−

(

)

(

θ, x

)

, where

is a truncated parameter that depends on the bandwidth and

(

)

are normalized weights so that they sum to 1.

If the vector of weights is provided, it gives only one side weights. For exemple, if you provide the
vector (1,.5,.25),

(

)

will become

(

, .

25)

(

25 +

5 + 1 +

5 +

25) = (

, .

Value

smoothx: A

matrix containing an estimator of the asymptotic variance of

√

, where

vector with typical element

n
j

. This function is called by

but can also be

used by itself.

kern_weights

: Vector of weights used for the smoothing.

References

Anatolyev, S. (2005), GMM, GEL, Serial Correlation, and Asymptotic Bias.

Econometrica

983-1002.

Andrews DWK (1991), Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Esti-
mation.

Econometrica

, 817–858.

Kitamura, Yuichi (1997), Empirical Likelihood Methods With Weakly Dependent Processes.

The

Annals of Statistics

, 2084-2102.

Zeileis A (2006), Object-oriented Computation of Sandwich Estimators.

Journal of Statistical Soft-

ware

(9), 1–16. URL

specTest

Examples

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)

}

n = 500

phi<-c(.2, .7)

thet <- 0.2

sd <- .2

x <- matrix(arima.sim(n = n, list(order = c(2, 0, 1), ar = phi, ma = thet, sd = sd)), ncol = 1)

gt <- g(c(0, phi), x)

sgt <- smoothG(gt)$smoothx

plot(gt[,1])

lines(sgt[,1])

specTest

Compute tests of specification

Description

Generic function for testing the specification of estimated models. It computes the J-test from

gmm

objects and J-test, LR-test and LM-test from

gel

objects.

Usage

## S3 method for class 'gmm'

specTest(x, ...)

## S3 method for class 'gel'

specTest(x, ...)

## S3 method for class 'specTest'

print(x, digits = 5, ...)

specTest(x, ...)

Arguments

A fitted model object.

digits

The number of digits to be printed.

...

Arguments passed to methods.

Value

Tests and p-values

summary

References

Hansen, L.P. (1982), Large Sample Properties of Generalized Method of Moments Estimators.

Econometrica

, 1029-1054,

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

CeMMAP working papers, Insti-

tute for Fiscal Studies

Examples

#################

n = 500

phi<-c(.2,.7)

thet <- 0

sd <- .2

x <- matrix(arima.sim(n=n,list(order=c(2,0,1),ar=phi,ma=thet,sd=sd)),ncol=1)

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

t0 <- c(0,.5,.5)

res <- gel(g, x, t0)

specTest(res)

###################

res <- gmm(g, x)

specTest(res)

summary

Method for object of class gmm or gel

Description

It presents the results from the

gmm

gel

estimation in the same fashion as

summary

does for the

class objects for example. It also compute the tests for overidentifying restrictions.

Usage

## S3 method for class 'gmm'

summary(object, ...)

## S3 method for class 'sysGmm'

summary(object, ...)

## S3 method for class 'gel'

summary(object, ...)

summary

## S3 method for class 'ategel'

summary(object, robToMiss = TRUE, ...)

## S3 method for class 'tsls'

summary(object, vcov = NULL, ...)

## S3 method for class 'summary.gmm'

print(x, digits = 5, ...)

## S3 method for class 'summary.sysGmm'

print(x, digits = 5, ...)

## S3 method for class 'summary.gel'

print(x, digits = 5, ...)

## S3 method for class 'summary.tsls'

print(x, digits = 5, ...)

Arguments

object

An object of class

gmm

gel

returned by the function

An object of class

summary.gmm

summary.gel

returned by the function

summary.gmm

summary.gel

digits

The number of digits to be printed

vcov

An alternative covariance matrix computed with

vcov.tsls

robToMiss

TRUE

, it computes the robust to misspecification covariance matrix

...

Other arguments when summary is applied to another class object

Value

It returns a list with the parameter estimates and their standard deviations, t-stat and p-values. It
also returns the J-test and p-value for the null hypothesis that

(

θ, X

) = 0

References

Hansen, L.P. (1982), Large Sample Properties of Generalized Method of Moments Estimators.

Econometrica

, 1029-1054,

Hansen, L.P. and Heaton, J. and Yaron, A.(1996), Finit-Sample Properties of Some Alternative
GMM Estimators.

Journal of Business and Economic Statistics

262-280.

Anatolyev, S. (2005), GMM, GEL, Serial Correlation, and Asymptotic Bias.

Econometrica

983-1002.

Kitamura, Yuichi (1997), Empirical Likelihood Methods With Weakly Dependent Processes.

The

Annals of Statistics

, 2084-2102.

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

Econometrica

, 219-255.

Examples

# GMM #

set.seed(444)

n = 500

sysGmm

phi<-c(.2,.7)

thet <- 0

sd <- .2

x <- matrix(arima.sim(n = n, list(order = c(2,0,1), ar = phi, ma = thet, sd = sd)), ncol = 1)

y <- x[7:n]

ym1 <- x[6:(n-1)]

ym2 <- x[5:(n-2)]

ym3 <- x[4:(n-3)]

ym4 <- x[3:(n-4)]

ym5 <- x[2:(n-5)]

ym6 <- x[1:(n-6)]

g <- y ~ ym1 + ym2

x <- ~ym3+ym4+ym5+ym6

res <- gmm(g, x)

summary(res)

# GEL #

t0 <- res$coef

res <- gel(g, x, t0)

summary(res)

# tsls #

res <- tsls(y ~ ym1 + ym2,~ym3+ym4+ym5+ym6)

summary(res)

sysGmm

Generalized method of moment estimation for system of equations

Description

Functions to estimate a system of equations based on GMM.

Usage

sysGmm(g, h, wmatrix = c("optimal","ident"),

vcov=c("MDS", "HAC", "CondHom", "TrueFixed"),

kernel=c("Quadratic Spectral","Truncated", "Bartlett", "Parzen", "Tukey-Hanning"),

crit=10e-7,bw = bwAndrews, prewhite = FALSE, ar.method = "ols", approx="AR(1)",

tol = 1e-7, model=TRUE, X=FALSE, Y=FALSE, centeredVcov = TRUE,

weightsMatrix = NULL, data, crossEquConst = NULL, commonCoef = FALSE)

five(g, h, commonCoef = FALSE, data = NULL)

threeSLS(g, h, commonCoef = FALSE, data = NULL)

sur(g, commonCoef = FALSE, data = NULL)

randEffect(g, data = NULL)

sysGmm

Arguments

A possibly named list of formulas

A formula if the same instruments are used in each equation or a list of formulas.

wmatrix

Which weighting matrix should be used in the objective function. By default, it
is the inverse of the covariance matrix of

(

θ, x

)

. The other choice is the identity

matrix.

vcov

Assumption on the properties of the moment vector. By default, it is a martin-
gale difference sequence. "HAC" is for weakly dependent processes and "Cond-
Hom" implies conditional homoscedasticity. The option "TrueFixed" is used
only when the matrix of weights is provided and it is the optimal one.

kernel

type of kernel used to compute the covariance matrix of the vector of sample
moment conditions (see

for more details)

crit

The stopping rule for the iterative GMM. It can be reduce to increase the preci-
sion.

The method to compute the bandwidth parameter. By default it is

bwAndrews

which is proposed by Andrews (1991). The alternative is

bwNeweyWest

Newey-West(1994).

prewhite

logical or integer. Should the estimating functions be prewhitened? If

TRUE

greater than 0 a VAR model of order

as.integer(prewhite)

is fitted via

with method

"ols"

and

demean = FALSE

ar.method

character. The

method

argument passed to

for prewhitening.

approx

A character specifying the approximation method if the bandwidth has to be
chosen by

bwAndrews

tol

Weights that exceed

tol

are used for computing the covariance matrix, all other

weights are treated as 0.

model, X, Y

logical. If

TRUE

the corresponding components of the fit (the model frame, the

model matrix, the response) are returned if g is a formula.

centeredVcov

Should the moment function be centered when computing its covariance matrix.
Doing so may improve inference.

weightsMatrix

It allows users to provide

gmm

with a fixed weighting matrix. This matrix must

, symmetric and strictly positive definite. When provided, the

type

option becomes irrelevant.

data

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

commonCoef

If true, coefficients accross equations are the same

crossEquConst

Only used if the number of regressors are the same in each equation. It is a vector
which indicates which coefficient are constant across equations. The order is 1
for Intercept and 2 to k as it is formulated in the formulas

. Setting it to

1:k

equivalent to setting commonCoef to TRUE.

Details

This set of functions implement the estimation of system of equations as presented in Hayashi
(2000)

sysGmm

Value

’sysGmm’ returns an object of ’class’ ’"sysGmm"’

The functions ’summary’ is used to obtain and print a summary of the results. It also compute the
J-test of overidentying restriction

The object of class "sysGmm" is a list containing at least:

coefficients

list of vectors of coefficients for each equation

residuals

list of the residuals for each equation.

fitted.values

list of the fitted values for each equation.

vcov

the covariance matrix of the stacked coefficients

objective

the value of the objective function

var

(¯

)

−

terms

The list of

objects for each equation

call

the matched call.

If requested, a list of response variables.

if requested, a list of the model matrices.

model

if requested (the default), a list of the model frames.

References

Zeileis A (2006), Object-oriented Computation of Sandwich Estimators.

Journal of Statistical Soft-

ware

(9), 1–16. URL

Andrews DWK (1991), Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Esti-
mation.

Econometrica

, 817–858.

Newey WK & West KD (1987), A Simple, Positive Semi-Definite, Heteroskedasticity and Auto-
correlation Consistent Covariance Matrix.

Econometrica

, 703–708.

Newey WK & West KD (1994), Automatic Lag Selection in Covariance Matrix Estimation.

Review

of Economic Studies

, 631-653.

Hayashi, F. (2000), Econometrics.

Princeton University Press

Examples

data(wage)

eq1 <- LW~S+IQ+EXPR

eq2 <- LW80~S80+IQ+EXPR80

g2 <- list(Wage69=eq1, WAGE80=eq2)

h2 <- list(~S+EXPR+MED+KWW, ~S80+EXPR80+MED+KWW)

res <- sysGmm(g2, h2, data=wage, commonCoef=TRUE)

summary(res)

res2 <- sysGmm(g2, h2, data=wage)

summary(res2)

tsls

five(g2, h2, data=wage)

threeSLS(g2, h2[[1]], data=wage)

sur(g2, data=wage)

randEffect(g2, data=wage)

## Cross-Equation restrictions

## All but the intercept are assumed to be the same

res <- sysGmm(g2, h2, data=wage, crossEquConst = 2:4)

summary(res)

tsls

Two stage least squares estimation

Description

Function to estimate a linear model by the two stage least squares method.

Usage

tsls(g,x,data)

Arguments

A formula describing the linear regression model (see details below).

The matrix of instruments (see details below).

data

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

Details

The function just calls

with the option vcov="iid". It just simplifies the the implementation of

2SLS. The users don’t have to worry about all the options offered in

. The model is

In the first step,

is used to regress

on the set of instruments

. The second step also uses

to regress

on the fitted values of the first step.

Value

’tsls’ returns an object of ’class’ ’"tsls"’ which inherits from class ’"gmm"’.

The functions ’summary’ is used to obtain and print a summary of the results. It also compute the
J-test of overidentying restriction

The object of class "gmm" is a list containing at least:

vcov

coefficients

vector of coefficients

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.

vcov

the covariance matrix of the coefficients

objective

the value of the objective function

var

(¯

)

−

terms

the

object used when g is a formula.

call

the matched call.

if requested, the response used (if "g" is a formula).

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.

algoInfo

Information produced by either

related to the convergence if

"g" is a function. It is printed by the

summary.gmm

method.

References

Hansen, L.P. (1982), Large Sample Properties of Generalized Method of Moments Estimators.

Econometrica

, 1029-1054,

Examples

n <- 1000

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(C[5:n],Y[4:(n-1)])

Z <- cbind(C[5:n],Y[3:(n-2)],Y[2:(n-3)],Y[1:(n-4)])

res <- tsls(Yt~X,~Z)

res

vcov

Variance-covariance matrix of GMM or GEL

Description

It extracts the matrix of variances and covariances from

gmm

gel

objects.

vcov

Usage

## S3 method for class 'gmm'

vcov(object, ...)

## S3 method for class 'gel'

vcov(object, lambda = FALSE, ...)

## S3 method for class 'tsls'

vcov(object, type=c("Classical","HC0","HC1","HAC"),

hacProp = list(), ...)

## S3 method for class 'ategel'

vcov(object, lambda = FALSE, robToMiss = TRUE, ...)

Arguments

object

An object of class

gmm

returned by the function

lambda

If set to TRUE, the covariance matrix of the Lagrange multipliers is produced.

type

Type of covariance matrix for the meat

hacProp

A list of arguments to pass to

http://fhayashi.fc2web.com/datasets.htm

robToMiss

TRUE

, it computes the robust to misspecification covariance matrix

...

Other arguments when

vcov

is applied to another class object

Details

For tsls(), if vcov is set to a different value thand "Classical", a sandwich covariance matrix is
computed.

Value

A matrix of variances and covariances

Examples

# GMM #

n = 500

phi<-c(.2,.7)

thet <- 0

sd <- .2

x <- matrix(arima.sim(n = n,list(order = c(2,0,1), ar = phi, ma = thet, sd = sd)), ncol = 1)

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 <- gmm(g, x)

vcov(res)

wage

## GEL ##

t0 <- c(0,.5,.5)

res <- gel(g, x, t0)

vcov(res)

vcov(res, lambda = TRUE)

wage

Labor Data

Description

Data used to measure return to education by Griliches (1976)

Usage

data(wage)

Format

A data frame containing 20 cross-sectional vectors.

AGE, AGE80

Age in 1969 and 1980 respetively

EXPR, EXPR80

Working experience in 1969 and 1980 respetively

IQ measure of the individual

KWW

A test score

LW, LW80

Log wage in 1969 and 1980 respectively

MED

Mother education

MRT, MRT80

RNS, RNS80

S, S80

Schooling in 1969 and 1980 respetively

SMSA, SMSA80

TENURE, TENURE80

Tenure in 1969 and 1980 respetively

YEAR

Source

Index

∗

Topic

datasets

Finance

Growth

nsw

wage

ATEgel

bread

bwAndrews

bwNeweyWest

bwWilhelm

charStable

checkConv

(

ATEgel

)

coef

confint

constrOptim

estfun

evalGel

(

gel

)

evalGmm

(

gmm

)

Finance

FinRes

fitted

five

(

sysGmm

)

formula

gel

getDat

getLamb

getModel

gmm

gmmWithConst

(

gmm

)

Growth

kernHAC

KTest

marginal

model.matrix.tsls

(

estfun

)

momentEstim

nlminb

nsw

optim

optimize