Estimating equations for Two-Stage Least Squares (2SLS)

Estimates the causal effect using two-stage least squares for instrumental variable analysis. The first stage regresses the treatment on the instruments (and exogenous variables), and the second stage regresses the outcome on predicted treatment (and exogenous variables).

Usage

ee_2sls(theta, y, A, Z, W = NULL, weights = NULL)

Arguments

theta

Numeric vector of length 1 + b + 2c, where b is the number of instruments in Z and c is the number of exogenous variables in W. The first 1 + c parameters are for the second-stage model; the remainder are for the first-stage model.

y

Numeric vector of n observed outcomes.

A

Numeric vector of n observed treatment values.

Z

Numeric n-by-b matrix of instrumental variable(s).

W

Optional n-by-c matrix of exogenous variables included in both stages. Default NULL.

weights

Optional numeric vector of n weights. Default NULL.

Value

A (1+b+2c)-by-n matrix of estimating equation contributions. The second-stage rows are named stage2_A for the fitted treatment and stage2_W_1 through stage2_W_c; the first-stage rows are named stage1_Z_1 through stage1_Z_b and stage1_W_1 through stage1_W_c. The columns of W appear in both stages, so each name carries the stage it belongs to.

Examples

# A continuous treatment confounded by an unmeasured U, a strong instrument
# Z, and one measured exogenous covariate. The true causal effect is 2.
set.seed(42)
n <- 500
W1 <- rnorm(n)
Z <- cbind(rbinom(n, 1, 0.5))
U <- rnorm(n)
A <- 1.5 * Z[, 1] + 0.3 * W1 + U + rnorm(n, sd = 0.5)
Y <- 2 * A - U + 0.5 * W1 + rnorm(n)
W <- cbind(1, W1)

# At a first stage of all zeros the fitted treatment is identically zero, so
# the second stage degenerates: its leading design column vanishes and the
# coefficient on it, the causal effect, has nothing left to be estimated
# from. Seed the starting values with the ordinary least squares fit of each
# stage instead: the first stage regresses A on the instrument and the
# exogenous covariates, the second regresses Y on the fitted A and the same
# covariates.
alpha_init <- as.numeric(coef(lm(A ~ cbind(Z, W) - 1)))
a_hat <- as.numeric(cbind(Z, W) %*% alpha_init)
beta_init <- as.numeric(coef(lm(Y ~ cbind(a_hat, W) - 1)))

psi <- function(theta) ee_2sls(theta, y = Y, A = A, Z = Z, W = W)

# theta holds the three second-stage coefficients followed by the three
# first-stage coefficients.
m <- m_estimate(
  stacked_equations = psi,
  init = c(beta_init, alpha_init)
)

# theta_1 is the coefficient on the fitted treatment, the causal effect.
coef(m)
#>    stage2_A  stage2_W_1  stage2_W_2  stage1_Z_1  stage1_W_1  stage1_W_2 
#>  1.91139044  0.03769661  0.54088789  1.63627118 -0.06199778  0.30488707