---
title: "Standard SFA Metafrontier (groupType = \"sfacross\")"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Standard SFA Metafrontier (groupType = "sfacross")}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r setup, include=FALSE}
knitr::opts_chunk$set(
  collapse  = TRUE,
  comment   = "#>",
  eval      = TRUE
)
```

## Overview

When observed group labels are available (e.g., farm size, region, ownership type),
`groupType = "sfacross"` fits a separate cross-sectional stochastic frontier for each
group using `sfaR::sfacross()`. The group-specific results are then used to estimate the
common metafrontier using any of four methods.

## Data Preparation

We use the `ricephil` dataset from `sfaR`, which contains 344 Filipino rice farms.
We create three technology groups based on farm area terciles.

```{r data}
library(smfa)
data("ricephil", package = "sfaR")

ricephil$group <- cut(
  ricephil$AREA,
  breaks        = quantile(ricephil$AREA, probs = c(0, 1/3, 2/3, 1), na.rm = TRUE),
  labels        = c("small", "medium", "large"),
  include.lowest = TRUE
)

table(ricephil$group)
#>  small medium  large
#>    125    104    115
```

## Method 1: LP Metafrontier

The **linear programming (LP)** envelope minimises the sum of absolute deviations from
group frontier predictions while satisfying a convexity constraint. No stochastic
parameters are estimated for the metafrontier itself.

```{r lp}
meta_lp <- smfa(
  formula    = log(PROD) ~ log(AREA) + log(LABOR) + log(NPK),
  data       = ricephil,
  group      = "group",
  S          = 1,
  udist      = "hnormal",
  groupType  = "sfacross",
  metaMethod = "lp"
)
summary(meta_lp)
```

> **Note:** Since the LP metafrontier is estimated via linear programming, no estimated
> parameters are returned for the metafrontier level. The LP envelope is fully determined
> by the group frontier predictions.

## Method 2: QP Metafrontier

The **quadratic programming (QP)** envelope minimises the sum of *squared* deviations
from group frontier predictions. Unlike LP, QP produces a smooth envelope that is
differentiable everywhere, and it returns estimated coefficients with standard errors.

```{r qp}
meta_qp <- smfa(
  formula    = log(PROD) ~ log(AREA) + log(LABOR) + log(NPK),
  data       = ricephil,
  group      = "group",
  S          = 1,
  udist      = "hnormal",
  groupType  = "sfacross",
  metaMethod = "qp"
)
summary(meta_qp)
```

## Method 3: Stochastic Metafrontier — Huang et al. (2014)

The **two-stage stochastic metafrontier** of Huang, Huang & Liu (2014) uses the
group-specific *fitted frontier values* as the dependent variable in a second-stage
pooled SFA. The technology gap U and noise V are estimated stochastically, which
naturally bounds the metatechnology ratio MTR ∈ (0, 1].

```{r huang}
meta_huang <- smfa(
  formula     = log(PROD) ~ log(AREA) + log(LABOR) + log(NPK),
  data        = ricephil,
  group       = "group",
  S           = 1,
  udist       = "hnormal",
  groupType   = "sfacross",
  metaMethod  = "sfa",
  sfaApproach = "huang"
)
summary(meta_huang)
```

## Method 4: Stochastic Envelope — O'Donnell et al. (2008)

The **O'Donnell et al. (2008)** approach uses the LP deterministic envelope as the
dependent variable in the second-stage SFA, rather than the group-specific fitted values.
This mixed deterministic–stochastic approach embeds the envelope within an SFA framework.

> **Warning:** MTR values > 1 can arise with this method when the second-stage SFA
> estimates near-zero inefficiency. If this occurs, consider using `metaMethod = "lp"` or
> `sfaApproach = "huang"` instead.

```{r odonnell}
meta_ordonnell <- smfa(
  formula     = log(PROD) ~ log(AREA) + log(LABOR) + log(NPK),
  data        = ricephil,
  group       = "group",
  S           = 1,
  udist       = "hnormal",
  groupType   = "sfacross",
  metaMethod  = "sfa",
  sfaApproach = "ordonnell"
)
summary(meta_ordonnell)
```

## Comparing Methods

All four methods use identical group-level estimates. The differences arise only in how the
metafrontier is computed:

| Method | Metafrontier Coefficients Returned | MTR Bounded ≤ 1? |
|--------|------------------------------------|------------------|
| LP | No (envelope rule) | Yes |
| QP | Yes (with SE) | Yes |
| SFA (huang) | Yes (with SE) | Yes |
| SFA (ordonnell) | Yes (with SE) | Not guaranteed |

## Extracting Efficiencies

All models return firm-level efficiency estimates via `efficiencies()`:

```{r eff}
eff <- efficiencies(meta_lp)
head(eff)

# Subset for a specific group
eff_small <- eff[eff$group == "small", ]
summary(eff_small[, c("TE_group_BC", "TE_meta_BC", "MTR_BC")])
```

## Other Extractors

```{r other}
coef(meta_qp)          # metafrontier coefficients
vcov(meta_qp)          # variance-covariance matrix
logLik(meta_lp)        # log-likelihood
ic(meta_lp)            # AIC, BIC, HQIC
nobs(meta_lp)          # number of observations
fitted(meta_lp)        # fitted values
residuals(meta_lp)     # residuals
```