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:

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

2) higher moments estimation (Lewbel 1997)

3) heteroskedastic error approach (Lewbel 2012)

4) joint estimation using copula (Park and Gupta 2012)

5) 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
can be either
,
,
or
and should be specified in the **g** argument of the
third RHS of the formula, as **x2, x3, lnx** or **1/x**. In case of
internal instruments built only from the endogenous regressor,
e.g. **p2**, or from the response and the endogenous regressor, like
for example in **yp**, there is no need to specify **g** or the set of
exogenous regressors in the **IIV()** part of the formula. The function
returns a set of tests for checking the validity of the instruments and
the endogeneity assumption.

```
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")
```

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