Endogeneity arises when the independence assumption between an explanatory variable and the error in a statistical model is violated. Among its most common causes are omitted variable bias (e.g. like ability in the returns to education estimation), measurement error (e.g. survey response bias), or simultaneity (e.g. advertising and sales).

Instrumental variable estimation is a common treatment when
endogeneity is of concern. However valid, strong external instruments
are difficult to find. Consequently, statistical methods to correct for
endogeneity without external instruments have been advanced. They are
called **internal instrumental variable models (IIV)**.

REndo implements the following instrument-free methods:

latent instrumental variables approach (Ebbes, Wedel, Boeckenholt, and Steerneman 2005)

higher moments estimation (Lewbel 1997)

heteroskedastic error approach (Lewbel 2012)

joint estimation using copula (Park and Gupta 2012)

multilevel GMM (Kim and Frees 2007)

The new version of **REndo** comes with a lot of
improvements in terms of code optimization as well as different syntax
for all functions.

Below, we present the syntax for each of the 5 implemented instrument-free methods:

`latentIV(y ~ P, data, start.params=c()) `

The first argument is the formula of the model to be estimated,
**y ~ P**, where **y** is the response and
**P** is the endogenous regressor. The second argument is
the name of the dataset used and the last one,
**start.params=c()**, which is optional, is a vector with
the initial parameter values. When not indicated, the initial parameter
values are taken to be the coefficients returned by the OLS estimator of
**y** on **P**.

`copulaCorrection( y ~ X1 + X2 + P1 + P2 | continuous(P1) + discrete(P2), data, start.params=c(), num.boots)`

The first argument is a two-part formula of the model to be
estimated, with the second part of the RHS defining the endogenous
regressor, here **continuous(P1) + discrete(P2)**. The
second argument is the name of the data, the third argument of the
function, **start.params**, is optional and represents the
initial parameter values supplied by the user (when missing, the OLS
estimates are considered); while the fourth argument,
**num.boots**, also optional, is the number of bootstraps
to be performed (the default is 1000). Of course, defining the
endogenous regressors depends on the number of endogenous regressors and
their assumed distribution. Transformations of the explanatory
variables, such as I(X), ln(X) are supported.

`higherMomentsIV(y ~ X1 + X2 + P | P | IIV(iiv = gp, g= x2, X1, X2) + IIV(iiv = yp) | Z1, data)`

Here, **y** is the response; the first RHS of the
formula, **X1 + X2 + P**, is the model to be estimated; the
second part, **P**, specifies the endogenous regressors;
the third part, **IIV()**, specifies the format of the
internal instruments; the fourth part, **Z1**, is optional,
allowing the user to add any external instruments available.

Regarding the third part of the formula, **IIV()**, it
has a set of three arguments:

**iiv**- specifies the form of the instrument,**g**- specifies the transformation to be done on the exogenous regressors,- the set of exogenous variables from which the internal instruments should be built (it can be one or all of the exogenous variables).

A set of six instruments can be constructed, which should be
specified in the **iiv** argument of
**IIV()**:

**g**- for ,**gp**- for ,**gy**- for ,**yp**- for ,**p2**- for ,**y2**- for .

where

` hetErrorsIV(y ~ X1 + X2 + X3 + P | P | IIV(X1,X2) | Z1, data)`

Here, **y** is the response variable, **X1 + X2 +
X3 + P** represents the model to be estimated; the second part,
**P**, specifies the endogenous regressors, the third part,
**IIV(X1, X2)**, specifies the exogenous heteroskedastic
variables from which the instruments are derived, while the final part
**Z1** is optional, allowing the user to include additional
external instrumental variables. Like in the higher moments approach,
allowing the inclusion of additional external variables is a convenient
feature of the function, since it increases the efficiency of the
estimates. Transformation of the explanatory variables, such as I(X),
ln(X) are possible both in the model specification as well as in the
IIV() specification.

`multilevelIV(y ~ X11 + X12 + X21 + X22 + X23 + X31 + X33 + X34 + (1|CID) + (1|SID) | endo(X12), data) `

The call of the function has a two-part formula and an argument for
data specification. In the formula, the first part is the model
specification, with fixed and random parameter specification, and the
second part which specifies the regressors assumed endogenous, here
**endo(X12)**. The function returns the parameter estimates
obtained with fixed effects, random effects and the GMM estimator
proposed by Kim and Frees (2007), such that a comparison across models
can be done.

Install the stable version from CRAN:

`install.packages("REndo")`

Install the development version from GitHub:

`devtools::install_github("mmeierer/REndo", ref = "development")`