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 new syntax for each function call:

`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")`