Direct estimation with discrete covariates

\(E(Y|X = x, Z = z) = G(x'\beta + z'\alpha)\) (2.45) with X continuous and Z discrete.

Usage

morie_direct_discrete(
  x,
  y,
  z = NULL,
  beta = NULL,
  c0 = NULL,
  c1 = NULL,
  bandwidth = NULL,
  n_grid = 60L
)

Arguments

x

continuous covariates.

y

numeric response.

z

discrete covariates; omit for the plain index case.

beta

index direction (the R mirror does not re-estimate it).

c0, c1

truncation levels of Assumption G, c0 < c1.

bandwidth

bandwidth for the within-stratum regressions.

n_grid

points for the integral defining J.

Value

list: beta, alpha, support_z, J, c0, c1, identified, average_derivative_can_estimate_alpha, n, d, dz, method.

Details

Average-derivative methods CANNOT estimate \(\alpha\): derivatives with respect to the discrete components do not exist, so no amount of averaging produces them. That absence, not a technical inconvenience, is why the section exists.

The construction goes around it. \(\beta\) is estimated stratum by stratum and combined by (2.46), a weighted average renormalised by its first component. \(\alpha\) then comes from a LINEAR system: under the weak monotonicity of Assumption G the truncated integral \(J(z)\) satisfies \(J[z^{(i)}] - J[z^{(1)}] = (c_1 - c_0)(z^{(i)} - z^{(1)})'\alpha\) (2.47), solved by \(\alpha = (c_1 - c_0)(W'W)^{-1}W'\Delta J\) (2.48). A shift in the discrete covariate shows up as a horizontal SHIFT of G, and integrating the truncated G recovers its size. Mirrors morie.fn.hrzdiscd.

References

Horowitz, Sec. 2.6.3, eqs. (2.45)-(2.51), Assumption G, Theorem 2.5; Horowitz and Hardle (1996).

Examples

set.seed(1)
n <- 300
x <- cbind(rnorm(n), rnorm(n))
z <- as.numeric(sample(0:2, n, TRUE))
y <- tanh(x %*% c(1, -0.6) + z * 0.8) + rnorm(n) * 0.2
morie_direct_discrete(x, y, z, beta = c(1, -0.6))$identified
#> [1] TRUE