Robust instrumental variables estimator

Description

Finds robust instrumental variables estimator using high breakdown point multivariate location and scatter matrix S-estimators.

Usage

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riv(Y, Xend, Xex=NULL, Zinst, dummies=NULL,
    method = c('S-est', 'SD-est', 'MCD-est', 'classical'))

Arguments

Y

vector of responses.

Xend

matrix of the endogenous variables, i.e. covariates that are correlated with the regression's error term.

Xex

matrix of the exogenous variables, i.e. covariates that are uncorrelated with the regression's error term. Default = NULL, i.e. no exogenous variables are present in the model.

Zinst

matrix of instruments, variables correlated with the endogenous covariates, but uncorrelated with the error term. The number of instrumental variables needs to be larger than or equal to the number of endogenous covariates.

dummies

matrix of exogenous dummy covariates, i.e., where each D_i are 0–1 vectors.

method

the method to be used. The "S-est" method (default) is based on the S-estimator of multivariate location and covariance, and "classical" method is based on the sample mean and covariance and the resulting estimator is equivalent to the two-stage least squares estimator (2SLS). See Details section.

Details

For method "S-est", RIV is constructed using the robust multivariate location and scatter S-estimator based on the Tukey's biweight function (see CovSest).

If RIV is computed using the S-estimator, its variance-covariance matrix is estimated based on the empirical influence function. See references for more details.

For method "SD-est", RIV is constructed using the Stahel-Donoho's robust multivariate location and scatter estimator (see CovSde).

For method "MCD-est", RIV is constructed using the Minimum Covariance Determinant (MCD) robust multivariate location and scatter estimator (see CovMcd).

For method "classical", the estimator is the classical instrumental variables estimator based on the sample mean and sample variance-covariance matrix (also known as the two-stage least squares estimator, 2SLS).

If the model contains dummy variables (i.e., dummies != NULL), RIV is computed using an iterative algorithm called "L_1-RIV". Briefly, L_1-RIV estimates the coefficients of the dummies using an L_1-estimator and the coefficients of the continuous covariates using the original RIV. See Cohen Freue et al. for more details.

Value

A list with components:

Summary.Table

Matrix of information available about the estimator. It contains regression coefficients, and, for method = "S-est" and "classical" only, columns for the standard error, t-statistic, and p-value.

VC

estimated variance-covariance matrix, computed only if method = "S-est" or "classical"

MD

Squared Mahalanobis distances of each observation to the multivariate location S-estimator with respect to the scatter S-estimator (only computed if method = "S-est").

MSE

vector of three components, computed only if method = "S-est" or "classical":

  • sigma.hat1: the mean square error estimation;

  • sigma.hat2: the mean square error estimation taking into account the weights associated to each observation (only computed if method = "S-est" and coefficents of endogenous variables are exactly identified, i.e., the number of instruments is equal to the number of endogenous variables);

  • sigma.hat3: the squared median absolute deviation (only computed if method = "S-est").

weight

the weights assigned by RIV to each observation (only computed if method = "S-est" and the number of instruments is equal to the number of endogenous variables.

References

LOPUHAA H.P. (1989). On the Relation between S-estimators and M-estimators of Multivariate Location and Covariance. Ann. Statist. 17 1662-1683.

COHEN-FREUE, G.V., ORTIZ-MOLINA, H., and ZAMAR, R.H. (2012) A Natural Robustification of the Ordinary Instrumental Variables Estimator. Submitted to Biometrics.

See Also

CovSest, CovSde, CovMcd

Examples

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## load data earthquake: the first column contains the response (Y), the
## second the endogenous variable (X) and the third column is the
## instrument (W).
data(earthquake) 
riv.eq <- riv(earthquake$Y,earthquake$X,NULL,earthquake$W)

## plot of the RIV estimates and the outlying observations are
## identified by filled points
plot(earthquake$X,earthquake$Y,xlab="X",ylab="Y",cex=1.5)
abline(riv.eq$Summary.Table[,1])
outliers <- which(sqrt(riv.eq$MD)>sqrt(qchisq(0.99, 3)))
text(earthquake[outliers,2],
     earthquake[outliers,1],
     outliers,
     pos=c(4,4,4,2))
     
points(earthquake[outliers,2],
       earthquake[outliers,1],
       cex=1.5,pch=19)

## Weights given by RIV to each observation as a function of the square
## root of the Mahalanobis distances (d) of each observation to the
## multivariate location and covariance S-estimator (computed with
## CovSest in rrcov)
plot(sqrt(riv.eq$MD),riv.eq$weight,xlab="d",ylab="RIV's Weights",cex = 1.5)
abline(h=sqrt(qchisq(0.99, 3)))
text(sqrt(riv.eq$MD)[outliers],
     riv.eq$weight[outliers],
     outliers,
     pos=c(2, 1, 1, 4))
points(sqrt(riv.eq$MD)[outliers],
       riv.eq$weight[outliers],
       cex=1.5, pch=19)


## load data mortality
data(mortality) 
Y <- as.matrix(mortality[,1])            ## M070
Xex <- as.matrix(mortality[,c(2,3,5,6)]) ## MAGE,CI68,DENS,NONW 
Xend <- as.matrix(mortality[,4])         ## MDOC  
colnames(Xend) <- colnames(mortality)[4] 
Zinst <- as.matrix(mortality[,7:8])      ## EDUC,IN69

## Classical instrumental variables estimator
riv(Y, Xend, Xex, Zinst, method="classical")