# Identification using heteroscedasticity

### Description

This function estimates the model parameters and associated standard errors for a linear regression model with one or more endogenous regressors. Identification is achieved through heteroscedastic covariance restrictions within the triangular system.

### Usage

1 |

### Arguments

`formula` |
an object of class “formula” (or one that can be coerced to that class). |

`data` |
the data frame containing these data. This argument must be used. |

`clustervar` |
a character value naming the cluster on which to adjust the standard errors and test statistics. |

`robust` |
if |

### Details

The formula follows a four-part specification. Each part is separated by a vertical bar character “|”. The following formula is an example: `y2 ~ y1 | x1 + x2 + x3 | x1 + x2 | z1`

. Here, `y2`

is the dependent variable and `y1`

is the endogenous regressor. The code `x1 + x2 + x3`

represents the exogenous regressors whereas the third part `x1 + x2`

specifies the exogenous heteroscedastic variables from which the instruments are derived. The final part `z1`

is optional, allowing the user to include tradtional instrumental variables. If both `robust=TRUE`

and `!is.null(clustervar)`

the function overrides the robust command and computes clustered standard errors and test statistics adjusted to account for clustering. This function computes partial F-statistics that indicate potentially weak identification. In cases where there is more than one endogenous regressor the Angrist-Pischke (2009) method for multivariate first-stage F-statistics is employed.

### Value

`coef.est` |
a coefficient matrix with columns containing the estimates, associated standard errors, test statistics and p-values. |

`call` |
the matched call. |

`num.obs` |
the number of observations. |

`j.test` |
J-test for overidentifying restrictions. |

`f.test.stats` |
Partial F-test statistics for weak IV detection. |

### References

Angrist, J. and Pischke, J.S. (2009). Mostly Harmless Econometrics: An Empiricist's Companion, Princeton University Press.

Lewbel, A. (2012). Using heteroscedasticity to identify and estimate mismeasured and endogenous regressor models. Journal of Business & Economic Statistics, 30(1), 67-80.

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 | ```
set.seed(1234)
n = 1000
x1 = rnorm(n, 0, 1)
x2 = rnorm(n, 0, 1)
u = rnorm(n, 0, 1)
s1 = rnorm(n, 0, 1)
s2 = rnorm(n, 0, 1)
ov = rnorm(n, 0, 1)
z1 = rnorm(n, 0 ,1)
e1 = u + exp(x1)*s1 + exp(x2)*s1
e2 = u + exp(-x1)*s2 + exp(-x2)*s2
y1 = 1 + x1 + x2 + ov + e2 + 2*z1
y2 = 1 + x1 + x2 + y1 + 2*ov + e1
data = data.frame(y2, y1, x1, x2, z1)
lewbel(formula = y2 ~ y1 | x1 + x2 | x1 + x2, data = data)
lewbel(formula = y2 ~ y1 | x1 + x2 | x1 + x2 | z1, data = data)
``` |