Methodology: Just-Pope with Heckman correction • JPselection

The Just-Pope production function

Just and Pope (1978) decompose output into a deterministic mean plus a stochastic component whose magnitude itself depends on inputs:

$y_l = f(\mathbf{x}, \boldsymbol{\beta}_l) + h(\mathbf{x}, \boldsymbol{\xi}_l)\,\eta_l, \qquad \eta_l \sim N(0, 1).$

$f(\cdot)$ is the mean function, which maps inputs to expected output.
$h(\cdot)$ is the risk (variance) function, which maps inputs to output variability. Negative coefficients in $h$ mean an input is risk-decreasing; positive ones mean risk-increasing.

The risk function is the headline output: it tells whether labour, water, fertiliser, etc. reduce or amplify yield uncertainty.

Why crop choice creates a selectivity problem

Producers choose among $L$ candidate crops by comparing expected profits. Whether a farm specialises in vegetables vs. cereals depends partly on observables (soil, climate, water access) and partly on unobservables correlated with productivity. Estimating the production function $f$ on the chosen sub-sample alone therefore biases the estimates. This is a textbook Heckman problem.

The Heckman correction

For each candidate crop, a probit predicts the choice indicator $D_l$ :

$D_l = \mathbf{1}\big[g(\mathbf{z}, \boldsymbol{\lambda}_l) + v_l > 0\big],$

and the Inverse Mills Ratio is computed as

$M_l = \frac{\phi(g(\mathbf{z}, \hat{\boldsymbol{\lambda}}_l))}{\Phi(g(\mathbf{z}, \hat{\boldsymbol{\lambda}}_l))},$

where $\phi$ and $\Phi$ are the standard-normal pdf and cdf. Adding $M_l$ as a regressor in the production function absorbs the average unobserved “productivity” of the selected sample, making the remaining coefficients consistent.

The three estimation steps

Step	What is estimated	Function
1	Probit on $D_l$ , then compute $M_l$	`estimate_selection()`
2	Mean fn: $y_l = f(\mathbf{x},\boldsymbol{\beta}_l) + \sigma_l M_l + w_l$ with $f$ linear-quadratic	`estimate_mean_function()`
3	Risk fn via $\log\\|\hat w_l\\| = \xi_0 + \sum_j \xi_j \log x_j + \log\eta_l$ (Cobb-Douglas $h$ )	`estimate_risk_function()`

Step 2 fits the linear-quadratic mean function

$f(\mathbf{x}) = \beta_0 + \sum_j \beta_j x_j + \sum_j \beta_{2j} x_j^2 + \sum_{j < k} \beta_{jk} x_j x_k.$

Step 3 fits the Cobb-Douglas risk function $h(\mathbf{x}) = \xi_0 \prod_j x_j^{\xi_j}$ , so each coefficient $\xi_j$ is the variance elasticity of input $j$ : a 1% rise in the input changes output variance by $\xi_j$ %. Negative $\xi_j$ → risk-decreasing input; positive $\xi_j$ → risk-increasing.

Alternative mean-function forms

The default mean function is the linear-quadratic form used in Koundouri & Nauges (2005). Two alternatives are available through jp_fit(..., mean_form = ...):

"linear_quadratic" (default): $\beta_0 + \sum \beta_j x_j + \sum \beta_{jj} x_j^2 + \sum_{j<k} \beta_{jk} x_j x_k$ .
"quadratic": drops the pairwise interaction terms. Useful when the sample is small relative to the number of inputs.
"cobb_douglas": log-log specification $\log y = \beta_0 + \sum \beta_j \log x_j + \text{shifters} + \sigma M + w$ . Requires strictly positive output and inputs. Shankar & Nelson (1999) showed that this Cobb-Douglas mean + Cobb-Douglas variance pairing is robust to input endogeneity in the JP framework.

fit_q  <- jp_fit(..., mean_form = "quadratic")
fit_cd <- jp_fit(..., mean_form = "cobb_douglas")

Alternative risk-function form: exponential

Some applications need to handle zero-valued inputs (e.g. farms that report no pesticide use), which the Cobb-Douglas form cannot because it requires $\log(x_j)$ . jp_fit() therefore accepts risk_form = "exponential", fitting

$h(\mathbf{x}) = \exp(\xi_0 + \sum_j \xi_j x_j),$

estimated via $\log|\hat w| = \xi_0 + \sum_j \xi_j x_j + \log\eta$ . Coefficients are now variance semi-elasticities (a 1-unit rise in input $j$ changes log output variance by $\xi_j$ ). This form appears in Saha, Havenner & Talpaz (1997) and the Norwegian salmon applications of Tveterås (1999, 2000). The default stays risk_form = "cobb_douglas" to match Koundouri & Nauges (2005).

fit_exp <- jp_fit(..., risk_form = "exponential")

Translog and other forms with multiplicative interaction between the mean and variance functions are deliberately not supported, because they break the additive identification structure that Just-Pope requires (Koundouri & Nauges, 2005, footnote 5). Passing mean_form = "translog" or risk_form = "translog" raises an informative error.

Comparing specifications side by side

Functional form is a research-design choice, not a property of the data. To see how sensitive the conclusion is, use jp_compare():

cmp <- jp_compare(
  data = farms, selection_var = "vegetables", ...,
  mean_forms = c("linear_quadratic", "quadratic", "cobb_douglas"),
  risk_forms = c("cobb_douglas", "exponential")
)
cmp$summary        # adjusted R^2, IMR p-value, selection-bias flag per combo
cmp$coefficients   # long-format risk-function coefs with stars per combo

The summary table makes it easy to see whether selection bias is detected under each specification and whether the risk-function conclusions agree across forms.

Why the with-vs-without comparison matters

If selectivity bias is present ( $\sigma_l \neq 0$ in Step 2), the Step-3 risk-function coefficients estimated without the Mill’s ratio are biased. Koundouri & Nauges’ main finding is that ignoring selectivity can flip the sign or kill the significance of risk-function coefficients, exactly the gap that print(fit) and plot(fit) make visible side by side. The Mill’s ratio coefficient $\sigma_l$ in Step 2 is itself a direct test for selection bias: if it’s significant, the corrected column is the one you should report.

Standard errors

Step 3 SEs come from a 500-replication nonparametric bootstrap that resamples the full pipeline (probit, IMR, mean function, residuals, risk function) on every replication, so upstream parameter uncertainty propagates correctly. Set bootstrap_reps lower for quicker runs.

References

Heckman, J. (1979). Sample selection bias as a specification error. Econometrica, 47, 153-161.
Just, R. E. and Pope, R. D. (1978). Stochastic representation of production functions and econometric implications. Journal of Econometrics, 7, 67-86.
Koundouri, P. and Nauges, C. (2005). On Production Function Estimation with Selectivity and Risk Considerations. Journal of Agricultural and Resource Economics, 30(3), 597-608.
Saha, A., Havenner, A. and Talpaz, H. (1997). Stochastic production function estimation: small sample properties of ML versus FGLS. Applied Economics, 29(4), 459-469.
Shankar, B. and Nelson, C. H. (1999). Joint risk preference-technology estimation with a primal system: comment. American Journal of Agricultural Economics, 81(1), 241-244.
Tveterås, R. (1999). Production risk and productivity growth: some findings for Norwegian salmon aquaculture. Journal of Productivity Analysis, 12(2), 161-179.
Tveterås, R. (2000). Flexible panel data models for risky production technologies with an application to salmon aquaculture. Econometric Reviews, 19(3), 367-389.