Estimate the Just-Pope variance (risk) function — estimate_risk

Step 3 of Koundouri & Nauges (2005). Given mean-function residuals, fits one of two functional forms for the variance function `h(x)`:

Usage

estimate_risk_function(
  residuals,
  input_data,
  input_vars,
  form = c("cobb_douglas", "exponential"),
  positive_only = TRUE,
  bootstrap_reps = 500,
  full_data = NULL,
  selection_args = NULL,
  mean_args = NULL,
  seed = NULL
)

Arguments

residuals: Numeric vector of mean-function residuals (Step 2).
input_data: Data frame aligned with `residuals`, holding input columns named in `input_vars`.
input_vars: Character vector. Inputs to enter the risk function.
form: Functional form: `"cobb_douglas"` (default) or `"exponential"`. See *Details*.
positive_only: Logical. Drop rows with zero residuals before estimation (default `TRUE`). Under `form = "cobb_douglas"` also drops rows with non-positive inputs (needed for the log transformation).
bootstrap_reps: Integer. Bootstrap replications (default 500; set 0 for OLS SEs only).
full_data, selection_args, mean_args: Optional. When all three are supplied, the full pipeline is rerun on each bootstrap draw. See [jp_fit()] which wires this up for you.
seed: Optional integer seed.

Value

List with the fitted `lm` object, a coefficient table, the bootstrap coefficient matrix, the post-filter sample size, and the functional form actually used.

Details

`"cobb_douglas"` (default): \(h(x) = \xi_0 \prod_j x_j^{\xi_j}\), estimated via \(\log|\hat w| = \xi_0 + \sum_j \xi_j \log x_j + \log\eta\). Coefficients are **variance elasticities** (a 1 changes output variance by \(\xi_j\) inputs.
`"exponential"`: \(h(x) = \exp(\xi_0 + \sum_j \xi_j x_j)\), estimated via \(\log|\hat w| = \xi_0 + \sum_j \xi_j x_j + \log\eta\). Coefficients are **variance semi-elasticities** (a 1-unit rise in input \(j\) changes log variance by \(\xi_j\)). Handles zero inputs because no log transformation is applied to \(x\). See Saha, Havenner & Talpaz (1997), Tveterås (1999).

Standard errors default to a 500-replication nonparametric bootstrap, matching the paper. If `full_data`, `selection_args`, and `mean_args` are supplied, the bootstrap resamples the entire pipeline (probit -> IMR -> mean function -> residuals -> risk function) on each replication so upstream parameter uncertainty propagates into the risk-function SEs.

Examples

if (FALSE) { # \dontrun{
fit <- jp_fit(data = simulate_kiti_data(),
              selection_var = "vegetables",
              selection_covariates = c("rainfall","irrigated","dist_town",
                                       "dist_coast","experience"),
              output_var   = "revenue",
              input_vars   = c("fertilizers","pesticides","labor","water"),
              shifter_vars = c("machinery","rainfall","irrigated",
                               "dist_town","dist_coast","experience"),
              bootstrap_reps = 100,
              risk_form    = "exponential")
fit$risk_with$coefficients
} # }