Instrumental-Variable Regression by 2SLS, 2SM, or 2SMM Estimation

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"),
  ...
)

Arguments

formula, instruments	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).
data	an optional data frame containing the variables in the model. By default the variables are taken from the environment of the formula.
subset	an optional vector specifying a subset of observations to be used in fitting the model.
na.action	a function that indicates what should happen when the data contain NAs. The default is set by the na.action option.
weights	an optional vector of weights to be used in the fitting process.
offset	an optional offset that can be used to specify an a priori known component to be included during fitting.
contrasts	an optional list. See the contrasts.arg of model.matrix.default.
model, x, y	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.
method	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.

Value

ivreg returns an object of class "ivreg" that inherits from class "lm", with the following components:

coefficients

parameter estimates, from the stage-2 regression.

residuals

vector of model residuals.

residuals1

matrix of residuals from the stage-1 regression.

residuals2

vector of residuals from the stage-2 regression.

fitted.values

vector of predicted means for the response.

weights

either the vector of weights used (if any) or NULL (if none).

offset

either the offset used (if any) or NULL (if none).

estfun

a matrix containing the empirical estimating functions.

n

number of observations.

nobs

number of observations with non-zero weights.

p

number of columns in the model matrix x of regressors.

q

number of columns in the instrumental variables model matrix z

rank

numeric rank of the model matrix for the stage-2 regression.

df.residual

residual degrees of freedom for fitted model.

cov.unscaled

unscaled covariance matrix for the coefficients.

sigma

residual standard deviation.

qr

QR decomposition for the stage-2 regression.

qr1

QR decomposition for the stage-1 regression.

rank1

numeric rank of the model matrix for the stage-1 regression.

coefficients1

matrix of coefficients from the stage-1 regression.

df.residual1

residual degrees of freedom for the stage-1 regression.

exogenous

columns of the "regressors" matrix that are exogenous.

endogenous

columns of the "regressors" matrix that are endogenous.

instruments

columns of the "instruments" matrix that are instruments for the endogenous variables.

method

the method used for the stage 1 and 2 regressions, one of "OLS", "M", or "MM".

rweights

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".

hatvalues

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.

df.residual

residual degrees of freedom for fitted model.

call

the original function call.

formula

the model formula.

na.action

function applied to missing values in the model fit.

terms

a list with elements "regressors" and "instruments" containing the terms objects for the respective components.

levels

levels of the categorical regressors.

contrasts

the contrasts used for categorical regressors.

model

the full model frame (if model = TRUE).

y

the response vector (if y = TRUE).

x

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.

Details

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 diagnotics, 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.

References

Greene, W.H. (1993) Econometric Analysis, 2nd ed., Macmillan.

See also

ivreg.fit, ivregDiagnostics, ivregMethods, lm, lm.fit

Examples

## 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    
#> 
#> Diagnostic tests:
#>                  df1 df2 statistic  p-value    
#> Weak instruments   1  45    45.158 2.65e-08 ***
#> Wu-Hausman         1  44     1.102      0.3    
#> Sargan             0  NA        NA       NA    
#> ---
#> 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    
#> 
#> Diagnostic tests:
#>                  df1 df2 statistic p-value    
#> Weak instruments   1  45    47.713 1.4e-08 ***
#> Wu-Hausman         1  44     1.287   0.263    
#> Sargan             0  NA        NA      NA    
#> ---
#> 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:     2 on NA DF,  p-value: NA 
#> 

## 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