# Two-Stage Least Squares Instrumental Variables Regression

Source:`R/estimatr_iv_robust.R`

`iv_robust.Rd`

This formula estimates an instrumental variables regression using two-stage least squares with a variety of options for robust standard errors

## Usage

```
iv_robust(
formula,
data,
weights,
subset,
clusters,
fixed_effects,
se_type = NULL,
ci = TRUE,
alpha = 0.05,
diagnostics = FALSE,
return_vcov = TRUE,
try_cholesky = FALSE
)
```

## Arguments

- formula
an object of class formula of the regression and the instruments. For example, the formula

`y ~ x1 + x2 | z1 + z2`

specifies`x1`

and`x2`

as endogenous regressors and`z1`

and`z2`

as their respective instruments.- data
A

`data.frame`

- weights
the bare (unquoted) names of the weights variable in the supplied data.

- subset
An optional bare (unquoted) expression specifying a subset of observations to be used.

- clusters
An optional bare (unquoted) name of the variable that corresponds to the clusters in the data.

- fixed_effects
An optional right-sided formula containing the fixed effects that will be projected out of the data, such as

`~ blockID`

. Do not pass multiple-fixed effects with intersecting groups. Speed gains are greatest for variables with large numbers of groups and when using "HC1" or "stata" standard errors. See 'Details'.- se_type
The sort of standard error sought. If

`clusters`

is not specified the options are "HC0", "HC1" (or "stata", the equivalent), "HC2" (default), "HC3", or "classical". If`clusters`

is specified the options are "CR0", "CR2" (default), or "stata". Can also specify "none", which may speed up estimation of the coefficients.- ci
logical. Whether to compute and return p-values and confidence intervals, TRUE by default.

- alpha
The significance level, 0.05 by default.

- diagnostics
logical. Whether to compute and return instrumental variable diagnostic statistics and tests.

- return_vcov
logical. Whether to return the variance-covariance matrix for later usage, TRUE by default.

- try_cholesky
logical. Whether to try using a Cholesky decomposition to solve least squares instead of a QR decomposition, FALSE by default. Using a Cholesky decomposition may result in speed gains, but should only be used if users are sure their model is full-rank (i.e., there is no perfect multi-collinearity)

## Value

An object of class `"iv_robust"`

.

The post-estimation commands functions `summary`

and `tidy`

return results in a `data.frame`

. To get useful data out of the return,
you can use these data frames, you can use the resulting list directly, or
you can use the generic accessor functions `coef`

, `vcov`

,
`confint`

, and `predict`

.

An object of class `"iv_robust"`

is a list containing at least the
following components:

- coefficients
the estimated coefficients

- std.error
the estimated standard errors

- statistic
the t-statistic

- df
the estimated degrees of freedom

- p.value
the p-values from a two-sided t-test using

`coefficients`

,`std.error`

, and`df`

- conf.low
the lower bound of the

`1 - alpha`

percent confidence interval- conf.high
the upper bound of the

`1 - alpha`

percent confidence interval- term
a character vector of coefficient names

- alpha
the significance level specified by the user

- se_type
the standard error type specified by the user

- res_var
the residual variance

- nobs
the number of observations used

- k
the number of columns in the design matrix (includes linearly dependent columns!)

- rank
the rank of the fitted model

- vcov
the fitted variance covariance matrix

- r.squared
the \(R^2\) of the second stage regression

- adj.r.squared
the \(R^2\) of the second stage regression, but penalized for having more parameters,

`rank`

- fstatistic
a vector with the value of the second stage F-statistic with the numerator and denominator degrees of freedom

- firststage_fstatistic
a vector with the value of the first stage F-statistic with the numerator and denominator degrees of freedom, useful for a test for weak instruments

- weighted
whether or not weights were applied

- call
the original function call

- fitted.values
the matrix of predicted means

We also return `terms`

with the second stage terms and `terms_regressors`

with the first stage terms, both of which used by `predict`

. If `fixed_effects`

are specified, then we return `proj_fstatistic`

, `proj_r.squared`

, and `proj_adj.r.squared`

, which are model fit statistics that are computed on the projected model (after demeaning the fixed effects).

We also return various diagnostics when ``diagnostics` == TRUE`

. These are stored in `diagnostic_first_stage_fstatistic`

, `diagnostic_endogeneity_test`

, and `diagnostic_overid_test`

. They have the test statistic, relevant degrees of freedom, and p.value in a named vector. See 'Details' for more. These are printed in a formatted table when the model object is passed to `summary()`

.

## Details

This function performs two-stage least squares estimation to fit
instrumental variables regression. The syntax is similar to that in
`ivreg`

from the `AER`

package. Regressors and instruments
should be specified in a two-part formula, such as
`y ~ x1 + x2 | z1 + z2 + z3`

, where `x1`

and `x2`

are
regressors and `z1`

, `z2`

, and `z3`

are instruments. Unlike
`ivreg`

, you must explicitly specify all exogenous regressors on
both sides of the bar.

The default variance estimators are the same as in `lm_robust`

.
Without clusters, we default to `HC2`

standard errors, and with clusters
we default to `CR2`

standard errors. 2SLS variance estimates are
computed using the same estimators as in `lm_robust`

, however the
design matrix used are the second-stage regressors, which includes the estimated
endogenous regressors, and the residuals used are the difference
between the outcome and a fit produced by the second-stage coefficients and the
first-stage (endogenous) regressors. More notes on this can be found at
the mathematical appendix.

If `fixed_effects`

are specified, both the outcome, regressors, and instruments
are centered using the method of alternating projections (Halperin 1962; Gaure 2013). Specifying
fixed effects in this way will result in large speed gains with standard error
estimators that do not need to invert the matrix of fixed effects. This means using
"classical", "HC0", "HC1", "CR0", or "stata" standard errors will be faster than other
standard error estimators. Be wary when specifying fixed effects that may result
in perfect fits for some observations or if there are intersecting groups across
multiple fixed effect variables (e.g. if you specify both "year" and "country" fixed effects
with an unbalanced panel where one year you only have data for one country).

If `diagnostics`

are requested, we compute and return three sets of diagnostics.
First, we return tests for weak instruments using first-stage F-statistics (`diagnostic_first_stage_fstatistic`

). Specifically,
the F-statistics reported compare the model regressing each endogeneous variable on both the
included exogenous variables and the instruments to a model where each endogenous variable is
regressed only on the included exogenous variables (without the instruments). A significant F-test
for weak instruments provides evidence against the null hypothesis that the instruments are weak.
Second, we return tests for the endogeneity of the endogenous variables, often called the Wu-Hausman
test (`diagnostic_endogeneity_test`

). We implement the regression test from Hausman (1978), which allows for robust variance estimation.
A significant endogeneity test provides evidence against the null that all the variables are exogenous.
Third, we return a test for the correlation between the instruments and the error term (`diagnostic_overid_test`

). We implement
the Wooldridge (1995) robust score test, which is identical to Sargan's (1958) test with classical
standard errors. This test is only reported if the model is overidentified (i.e. the number of
instruments is greater than the number of endogenous regressors), and if no weights are specified.

## References

Gaure, Simon. 2013. "OLS with multiple high dimensional category variables." Computational Statistics & Data Analysis 66: 8-1. doi:10.1016/j.csda.2013.03.024

Halperin, I. 1962. "The product of projection operators." Acta Scientiarum Mathematicarum (Szeged) 23(1-2): 96-99.

## Examples

```
library(fabricatr)
dat <- fabricate(
N = 40,
Y = rpois(N, lambda = 4),
Z = rbinom(N, 1, prob = 0.4),
D = Z * rbinom(N, 1, prob = 0.8),
X = rnorm(N),
G = sample(letters[1:4], N, replace = TRUE)
)
# Instrument for treatment `D` with encouragement `Z`
tidy(iv_robust(Y ~ D + X | Z + X, data = dat))
#> term estimate std.error statistic p.value conf.low conf.high
#> 1 (Intercept) 3.32899931 0.3532949 9.42272101 2.261759e-11 2.6131558 4.0448428
#> 2 D 0.31231384 0.6515096 0.47936952 6.344971e-01 -1.0077700 1.6323977
#> 3 X 0.03953936 0.4343723 0.09102642 9.279626e-01 -0.8405826 0.9196613
#> df outcome
#> 1 37 Y
#> 2 37 Y
#> 3 37 Y
# Instrument with Stata's `ivregress 2sls , small rob` HC1 variance
tidy(iv_robust(Y ~ D | Z, data = dat, se_type = "stata"))
#> term estimate std.error statistic p.value conf.low conf.high df
#> 1 (Intercept) 3.3157895 0.3419679 9.6962023 7.990568e-12 2.623512 4.008067 38
#> 2 D 0.3184211 0.6522180 0.4882126 6.282047e-01 -1.001925 1.638767 38
#> outcome
#> 1 Y
#> 2 Y
# With clusters, we use CR2 errors by default
dat$cl <- rep(letters[1:5], length.out = nrow(dat))
tidy(iv_robust(Y ~ D | Z, data = dat, clusters = cl))
#> term estimate std.error statistic p.value conf.low conf.high
#> 1 (Intercept) 3.3157895 0.4457303 7.4390043 0.002360625 2.0374558 4.594123
#> 2 D 0.3184211 0.4682124 0.6800782 0.533811304 -0.9820125 1.618855
#> df outcome
#> 1 3.698715 Y
#> 2 3.996357 Y
# Again, easy to replicate Stata (again with `small` correction in Stata)
tidy(iv_robust(Y ~ D | Z, data = dat, clusters = cl, se_type = "stata"))
#> term estimate std.error statistic p.value conf.low conf.high df
#> 1 (Intercept) 3.3157895 0.4414454 7.5112102 0.001681356 2.0901405 4.541438 4
#> 2 D 0.3184211 0.4634526 0.6870629 0.529805076 -0.9683296 1.605172 4
#> outcome
#> 1 Y
#> 2 Y
# We can also specify fixed effects, that will be taken as exogenous regressors
# Speed gains with fixed effects are greatests with "stata" or "HC1" std.errors
tidy(iv_robust(Y ~ D | Z, data = dat, fixed_effects = ~ G, se_type = "HC1"))
#> term estimate std.error statistic p.value conf.low conf.high df outcome
#> 1 D 0.2509696 0.6728668 0.3729855 0.7114087 -1.115023 1.616962 35 Y
```