Two-Stage Least Squares Instrumental Variables RegressionSource:
This formula estimates an instrumental variables regression using two-stage least squares with a variety of options for robust standard errors
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 )
an object of class formula of the regression and the instruments. For example, the formula
y ~ x1 + x2 | z1 + z2specifies
x2as endogenous regressors and
z2as their respective instruments.
the bare (unquoted) names of the weights variable in the supplied data.
An optional bare (unquoted) expression specifying a subset of observations to be used.
An optional bare (unquoted) name of the variable that corresponds to the clusters in the data.
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'.
The sort of standard error sought. If
clustersis not specified the options are "HC0", "HC1" (or "stata", the equivalent), "HC2" (default), "HC3", or "classical". If
clustersis specified the options are "CR0", "CR2" (default), or "stata". Can also specify "none", which may speed up estimation of the coefficients.
logical. Whether to compute and return p-values and confidence intervals, TRUE by default.
The significance level, 0.05 by default.
logical. Whether to compute and return instrumental variable diagnostic statistics and tests.
logical. Whether to return the variance-covariance matrix for later usage, TRUE by default.
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)
An object of class
The post-estimation commands functions
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
An object of class
"iv_robust" is a list containing at least the
the estimated coefficients
the estimated standard errors
the estimated degrees of freedom
the p-values from a two-sided t-test using
the lower bound of the
1 - alphapercent confidence interval
the upper bound of the
1 - alphapercent confidence interval
a character vector of coefficient names
the significance level specified by the user
the standard error type specified by the user
the residual variance
the number of observations used
the number of columns in the design matrix (includes linearly dependent columns!)
the rank of the fitted model
the fitted variance covariance matrix
the \(R^2\) of the second stage regression
the \(R^2\) of the second stage regression, but penalized for having more parameters,
a vector with the value of the second stage F-statistic with the numerator and denominator degrees of freedom
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
whether or not weights were applied
the original function call
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
fixed_effects are specified, then we return
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_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
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
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
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.
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).
diagnostics are requested, we compute and return three sets of diagnostics.
First, we return tests for weak instruments using first-stage F-statistics (
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
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.
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.
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