sps.est {SteinIV}

Semi-parametric Stein-like (SPS) estimator.

Description

Computes the SPS estimator for a two-stage structural model, as well as the set of standard errors for each individual estimator, and the sample estimate of the asymptotic variance/covariance matrix.

Usage

  sps.est(y,X,Z,SE=FALSE,ALPHA=TRUE,REF="TSLS",n.bt=100,n.btj=10)

Arguments

y

Numeric: A vector of observations, representing the outcome variable.

X

Numeric: A matrix of observations, whose number of columns corresponds to the number of predictors in the model, and the number of rows should be conformal with the number of entries in y. This matrix may contain both endogenous and exogenous variables.

Z

Numeric: A matrix of observations representing the intrumental variables (IVs) in the first-stage structural equation. The number of IVs should be at least as large as the number of endogenous variables in X.

SE

Logical: If TRUE, then the function also returns the standard errors of the individual SPS estimators, and a sample (or bootstrap, if JIVE is selected as a reference estimator) estimate of its asymptotic variance/covariance matrix.

ALPHA

Logical: If TRUE, the function returns the value of the sample estimate of the parameter controlling the respective contribution of the reference estimator (by default, this is the TSLS estimator), and the one of the alternative estimator (by default, this is the OLS estimator).

REF

Character: Controls the choice of the reference estimator in the SPS framework. This can accept two values: "TSLS" or "JIVE", with the former being the default option. The alternative estimator is always the OLS estimator.

n.bt

Numeric: The number of bootstrap samples performed, when the sample variance/covariance matrix is estimated using the bootstrap. This automatically occurs, whenever the user selects the JIVE as the reference estimator.

n.btj

Numeric: The number of boostrap iterations performed, when computing the SPS estimator, when using the JIVE as reference estimator. This option is only relevant, when JIVE has been selected as the reference estimator. These iterations are used to compute the various components entering in the calculation of the SPS estimator.

Details

The SPS estimator is applied to a two-stage structural model. We here adopt the terminology commonly used in econometrics. See, for example, the references below for Cameron and Trivedi (2005), Davidson and MacKinnon (1993), as well as Wooldridge (2002). The second-stage equation is thus modelled as follows,

y = X\beta + \epsilon,

in which y is a vector of n observations representing the outcome variable, X is a matrix of order n\times k denoting the predictors of the model, and comprised of both exogenous and endogenous variables, \beta is the k-dimensional vector of parameters of interest; whereas \epsilon is an unknown vector of error terms. The first-stage level of the model is given by a multivariate multiple regression. That is, this is a linear modle with a multivariate outcome variable, as well as multiple predictors. This first-stage model is represented in this manner,

X = Z\Gamma + \Delta,

where X is the matrix of predictors from the second-stage equation, Z is a matrix of instrumental variables (IVs) of order n \times l, \Gamma is a matrix of unknown parameters of order l\times k; whereas \Delta denotes an unknown matrix of order n\times k of error terms.

As for the TSLS estimator, whenever certain variables in X are assumed to be exogenous, these variables should be incorporated into Z. That is, all the exogneous variables are their own instruments. Moreover, it is also assumed that the model contains at least as many instruments as predictors, in the sense that l\geq k, as commonly donein practice (Wooldridge, 2002). Also, the matrices, X^TX, Z^TX, and Z^TZ are all assumed to be full rank. Finally, both X and Z should comprise a column of one's, representing the intercept in each structural equation.

The formula for the SPS estimator is then obtained as a weigthed combination of the OLS and TSLS estimators (using the default options), such that

\hat\beta_{SPS}(\alpha) := \alpha\hat\beta_{OLS} + (1-\alpha)\hat\beta_{TSLS},

for every \alpha. The proportion parameter, \alpha, controls the respective contributions of the OLS and TSLS estimators. (Despite our choice of name, however, note that \alpha needs not be bounded between 0 and 1.) This parameter is selected in order to minimize the trace of the theoretical MSE of the corresponding SPS estimator,

MSE(\hat\beta_{SPS}(\alpha)) = E[(\bar\beta(\alpha)-\beta)(\hat\beta(\alpha)-\beta)^{T}] = Var(\hat\beta(\alpha)) + Bias^{2}(\hat\beta(\alpha)),

where \beta\in R^{k} is the true parameter of interest and the MSE is a k\times k matrix. It is particularly appealing to combine these two estimators, because the asymptotic unbiasedness of the TSLS estimator guarantees that the resulting SPS is asymptotically unbiased. Thus, the MSE automatically strikes a trade-off between the unbiasedness of the TSLS estimator and the efficiency of the OLS estimator.

Value

list

A list with one or four arguments, depending on whether the user has activated the SE flag, and the ALPHA flag. The first element (est) in the list is the SPS estimate of the model in vector format. The second element (se) is the vector of standard errors; the third element (var) is the sample estimate of the asymptotic variance/covariance matrix; the fourth element (alpha) is a real number representing the estimate of the contribution of the OLS to the combined SPS estimator.

Author(s)

Cedric E. Ginestet <[e-mail]>

References

Judge, G.G. and Mittelhammer, R.C. (2004). A semiparametric basis for combining esti- mation problems under quadratic loss. Journal of the American Statistical Association, 99(466), 479–487.

Judge, G.G. and Mittelhammer, R.C. (2012a). An information theoretic approach to econo- metrics. Cambridge University Press.

Judge, G. and Mittelhammer, R. (2012b). A risk superior semiparametric estimator for over-identified linear models. Advances in Econometrics, 237–255.

Judge, G. and Mittelhammer, R. (2013). A minimum mean squared error semiparametric combining estimator. Advances in Econometrics, 55–85.

Mittelhammer, R.C. and Judge, G.G. (2005). Combining estimators to improve structural model estimation and inference under quadratic loss. Journal of econometrics, 128(1), 1–29.

Examples

### Generate a simple example with synthetic data, and no intercept. 
n <- 100; k <- 3; l <- 3;
Ga<- diag(rep(1,l)); be <- rep(1,k);
Z <- matrix(0,n,l); for(j in 1:l) Z[,j] <- rnorm(n); 
X <- matrix(0,n,k); for(j in 1:k) X[,j] <- Z[,j]*Ga[j,j] + rnorm(n); 
y <- X%*%be + rnorm(n);

### Compute SPS estimator with SEs and variance/covariance matrix.
print(sps.est(y,X,Z))
print(sps.est(y,X,Z,SE=TRUE));