ols.est {SteinIV}

The Ordinary Least Squares (OLS) Estimator.

Description

Compute the OLS estimator of a multiple regression, as well as the set of standard errors for the individual vector entries, and the estimate of the asymptotic variance/covariance matrix.

Usage

  ols.est(y,X,SE=FALSE)

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.

SE

Logical: If TRUE, then the function also returns the standard errors of the individual TSLS estimator, and a sample estimate of its asymptotic variance/covariance matrix.

Details

The OLS estimator is computed for a standard one-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 formula for the OLS estimator is then obtained in the standard fashion by the following equation,

\hat\beta_{OLS} := (X^TX)^{-1}(X^{T}y),

with variance/covariance matrix given by

\hat\Sigma_{OLS} := \hat\sigma^{2}(X^{T}X)^{-1},

in which the sample residual sum of squares is \hat\sigma^{2}:=(y - X\hat\beta_{OLS})^{T}(y - X\hat\beta_{OLS})/(n-k).

Value

list

A list with one or three arguments, depending on whether the user has activated the SE flag. The first element (est) in the list is the TSLS estimate of the model in vector format. The second element (se) is the vector of standard errors; and the third element (var) is the sample estimate of the asymptotic variance/covariance matrix.

Author(s)

Cedric E. Ginestet <[e-mail]>

References

Cameron, A. and Trivedi, P. (2005). Microeconometrics: Methods and Applications. Cam- bridge University press, Cambridge.

Davidson, R. and MacKinnon, J.G. (1993). Estimation and inference in econometrics. OUP Catalogue.

Wooldridge, J. (2002). Econometric analysis of cross-section and panel data. MIT press, London.

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 OLS estimator with SEs and variance/covariance matrix.
print(ols.est(y,X))
print(ols.est(y,X,SE=TRUE))