Fit instrumental-variable regression by two-stage least squares (2SLS). This is equivalent to direct instrumental-variables estimation when the number of instruments is equal to the number of regressors. Alternative robust-regression estimators are also provided, based on M-estimation (2SM) and MM-estimation (2SMM).
ivreg(
formula,
instruments,
data,
subset,
na.action,
weights,
offset,
contrasts = NULL,
model = TRUE,
y = TRUE,
x = FALSE,
method = c("OLS", "M", "MM"),
...
)formula specification(s) of the regression
relationship and the instruments. Either instruments is missing and
formula has three parts as in y ~ x1 + x2 | z1 + z2 + z3
(recommended) or formula is y ~ x1 + x2 and instruments
is a one-sided formula ~ z1 + z2 + z3 (only for backward
compatibility).
an optional data frame containing the variables in the model.
By default the variables are taken from the environment of the
formula.
an optional vector specifying a subset of observations to be used in fitting the model.
a function that indicates what should happen when the data
contain NAs. The default is set by the na.action option.
an optional vector of weights to be used in the fitting process.
an optional offset that can be used to specify an a priori known component to be included during fitting.
an optional list. See the contrasts.arg of
model.matrix.default.
logicals. If TRUE the corresponding components of
the fit (the model frame, the model matrices, the response) are returned. These
components are necessary for computing regression diagnostics.
the method used to fit the stage 1 and 2 regression:
"OLS" for traditional 2SLS regression (the default),
"M" for M-estimation, or "MM" for MM-estimation, with the
latter two robust-regression methods implemented via the rlm
function in the MASS package.
further arguments passed to ivreg.fit.
ivreg returns an object of class "ivreg" that inherits from
class "lm", with the following components:
parameter estimates, from the stage-2 regression.
vector of model residuals.
matrix of residuals from the stage-1 regression.
vector of residuals from the stage-2 regression.
vector of predicted means for the response.
either the vector of weights used (if any) or NULL (if none).
either the offset used (if any) or NULL (if none).
a matrix containing the empirical estimating functions.
number of observations.
number of observations with non-zero weights.
number of columns in the model matrix x of regressors.
number of columns in the instrumental variables model matrix z
numeric rank of the model matrix for the stage-2 regression.
residual degrees of freedom for fitted model.
unscaled covariance matrix for the coefficients.
residual standard deviation.
QR decomposition for the stage-2 regression.
QR decomposition for the stage-1 regression.
numeric rank of the model matrix for the stage-1 regression.
matrix of coefficients from the stage-1 regression.
residual degrees of freedom for the stage-1 regression.
columns of the "regressors" matrix that are exogenous.
columns of the "regressors" matrix that are endogenous.
columns of the "instruments" matrix that are
instruments for the endogenous variables.
the method used for the stage 1 and 2 regressions, one of "OLS",
"M", or "MM".
a matrix of robustness weights with columns for each of the stage-1
regressions and for the stage-2 regression (in the last column) if the fitting method is
"M" or "MM", NULL if the fitting method is "OLS".
a matrix of hatvalues. For method = "OLS", the matrix consists of two
columns, for each of the stage-1 and stage-2 regression; for method = "M" or "MM",
there is one column for each stage=1 regression and for the stage-2 regression.
residual degrees of freedom for fitted model.
the original function call.
the model formula.
function applied to missing values in the model fit.
a list with elements "regressors" and "instruments"
containing the terms objects for the respective components.
levels of the categorical regressors.
the contrasts used for categorical regressors.
the full model frame (if model = TRUE).
the response vector (if y = TRUE).
a list with elements "regressors", "instruments", "projected",
containing the model matrices from the respective components (if x = TRUE).
"projected" is the matrix of regressors projected on the image of the instruments.
ivreg is the high-level interface to the work-horse function
ivreg.fit. A set of standard methods (including print,
summary, vcov, anova, predict, residuals,
terms, model.matrix, bread, estfun) is available
and described in ivregMethods. For methods related to regression
diagnostics, see ivregDiagnostics.
Regressors and instruments for ivreg are most easily specified in a
formula with two parts on the right-hand side, e.g., y ~ x1 + x2 | z1
+ z2 + z3, where x1 and x2 are the explanatory variables and z1,
z2, and z3 are the instrumental variables. Note that exogenous regressors
have to be included as instruments for themselves.
For example, if there is
one exogenous regressor ex and one endogenous regressor en
with instrument in, the appropriate formula would be y ~ en +
ex | in + ex. Alternatively, a formula with three parts on the right-hand
side can also be used: y ~ ex | en | in. The latter is typically more convenient, if
there is a large number of exogenous regressors.
Moreover, two further equivalent specification strategies are possible that are
typically less convenient compared to the strategies above. One option is to use
an update formula with a . in the second part of the formula is used:
y ~ en + ex | . - en + in. Another option is to use a separate formula
for the instruments (only for backward compatibility with earlier versions):
formula = y ~ en + ex, instruments = ~ in + ex.
Internally, all specifications are converted to the version with two parts on the right-hand side.
Greene, W.H. (2003) Econometric Analysis, 5th ed., Upper Saddle River: Prentice Hall.
ivreg.fit, ivregDiagnostics, ivregMethods,
lm, lm.fit
## data
data("CigaretteDemand", package = "ivreg")
## model
m <- ivreg(log(packs) ~ log(rprice) + log(rincome) | salestax + log(rincome),
data = CigaretteDemand)
summary(m)
#>
#> Call:
#> ivreg(formula = log(packs) ~ log(rprice) + log(rincome) | salestax +
#> log(rincome), data = CigaretteDemand)
#>
#> Residuals:
#> Min 1Q Median 3Q Max
#> -0.611000 -0.086072 0.009423 0.106912 0.393159
#>
#> Coefficients:
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept) 9.4307 1.3584 6.943 1.24e-08 ***
#> log(rprice) -1.1434 0.3595 -3.181 0.00266 **
#> log(rincome) 0.2145 0.2686 0.799 0.42867
#> ---
#> Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#>
#> Residual standard error: 0.1896 on 45 degrees of freedom
#> Multiple R-Squared: 0.4189, Adjusted R-squared: 0.3931
#> Wald test: 6.534 on 2 and 45 DF, p-value: 0.003227
#>
summary(m, vcov = sandwich::sandwich, df = Inf)
#>
#> Call:
#> ivreg(formula = log(packs) ~ log(rprice) + log(rincome) | salestax +
#> log(rincome), data = CigaretteDemand)
#>
#> Residuals:
#> Min 1Q Median 3Q Max
#> -0.611000 -0.086072 0.009423 0.106912 0.393159
#>
#> Coefficients:
#> Estimate Std. Error z value Pr(>|z|)
#> (Intercept) 9.4307 1.2194 7.734 1.04e-14 ***
#> log(rprice) -1.1434 0.3605 -3.172 0.00151 **
#> log(rincome) 0.2145 0.3018 0.711 0.47729
#> ---
#> Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#>
#> Residual standard error: 0.1896 on Inf degrees of freedom
#> Multiple R-Squared: 0.4189, Adjusted R-squared: 0.3931
#> Wald test: 17.47 on 2 DF, p-value: 0.0001605
#>
## ANOVA
m2 <- update(m, . ~ . - log(rincome) | . - log(rincome))
anova(m, m2)
#> Analysis of Variance Table
#>
#> Model 1: log(packs) ~ log(rprice) + log(rincome) | salestax + log(rincome)
#> Model 2: log(packs) ~ log(rprice) | salestax
#> Res.Df RSS Df Sum of Sq F Pr(>F)
#> 1 45 1.6172
#> 2 46 1.6668 -1 -0.049558 0.6379 0.4287
car::Anova(m)
#> Analysis of Deviance Table (Type II tests)
#>
#> Response: log(packs)
#> Df F Pr(>F)
#> log(rprice) 1 10.1161 0.002662 **
#> log(rincome) 1 0.6379 0.428667
#> Residuals 45
#> ---
#> Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
## same model specified by formula with three-part right-hand side
ivreg(log(packs) ~ log(rincome) | log(rprice) | salestax, data = CigaretteDemand)
#>
#> Call:
#> ivreg(formula = log(packs) ~ log(rincome) | log(rprice) | salestax, data = CigaretteDemand)
#>
#> Coefficients:
#> (Intercept) log(rprice) log(rincome)
#> 9.4307 -1.1434 0.2145
#>
# Robust 2SLS regression
data("Kmenta", package = "ivreg")
Kmenta1 <- Kmenta
Kmenta1[20, "Q"] <- 95 # corrupted data
deq <- ivreg(Q ~ P + D | D + F + A, data=Kmenta) # demand equation, uncorrupted data
deq1 <- ivreg(Q ~ P + D | D + F + A, data=Kmenta1) # standard 2SLS, corrupted data
deq2 <- ivreg(Q ~ P + D | D + F + A, data=Kmenta1, subset=-20) # standard 2SLS, removing bad case
deq3 <- ivreg(Q ~ P + D | D + F + A, data=Kmenta1, method="MM") # 2SLS MM estimation
car::compareCoefs(deq, deq1, deq2, deq3)
#> Calls:
#> 1: ivreg(formula = Q ~ P + D | D + F + A, data = Kmenta)
#> 2: ivreg(formula = Q ~ P + D | D + F + A, data = Kmenta1)
#> 3: ivreg(formula = Q ~ P + D | D + F + A, data = Kmenta1, subset = -20)
#> 4: ivreg(formula = Q ~ P + D | D + F + A, data = Kmenta1, method = "MM")
#>
#> Model 1 Model 2 Model 3 Model 4
#> (Intercept) 94.63 117.96 92.42 91.09
#> SE 7.92 11.64 9.67 10.62
#>
#> P -0.2436 -0.4054 -0.2300 -0.2374
#> SE 0.0965 0.1417 0.1047 0.1135
#>
#> D 0.3140 0.2351 0.3233 0.3468
#> SE 0.0469 0.0690 0.0527 0.0569
#>
round(deq3$rweights, 2) # robustness weights
#> P stage_2
#> 1922 0.97 0.98
#> 1923 0.97 0.98
#> 1924 1.00 0.87
#> 1925 1.00 0.96
#> 1926 0.98 0.90
#> 1927 1.00 0.98
#> 1928 0.97 0.95
#> 1929 0.64 0.53
#> 1930 0.80 0.91
#> 1931 0.89 0.77
#> 1932 0.98 1.00
#> 1933 1.00 0.91
#> 1934 0.97 0.92
#> 1935 0.89 1.00
#> 1936 0.72 0.88
#> 1937 0.84 0.53
#> 1938 0.94 1.00
#> 1939 0.53 0.69
#> 1940 1.00 0.98
#> 1941 0.98 0.00