# Metafrontier Methods: Theory and Computation

    library(metafrontier)

This vignette provides a detailed exposition of the metafrontier methods implemented in the package, linking the econometric theory to the computational approach used at each step.

## 1. The metafrontier framework

### 1.1 Group-specific stochastic frontiers

Consider \(J\) groups of firms, where group \(j\) contains \(n_j\) firms. For group \(j\), the stochastic frontier model is:

\[\ln y_{ij} = x_{ij}'\beta_j + v_{ij} - u_{ij}, \quad i = 1, \ldots, n_j\]

where \(y_{ij}\) is the output of firm \(i\) in group \(j\), \(x_{ij}\) is a vector of (logged) inputs including a constant, \(\beta_j\) is the group-specific parameter vector, \(v_{ij} \sim N(0, \sigma_{v,j}^2)\) is symmetric noise, and \(u_{ij} \ge 0\) is one-sided inefficiency.

The group-specific technical efficiency is:

\[TE_{ij} = \exp(-u_{ij}) \in (0, 1]\]

estimated by default via the Battese and Coelli (1988) conditional expectation estimator \(E[\exp(-u_{ij}) \mid \varepsilon_{ij}]\) (`estimator = "bc88"`). The Jondrow et al. (1982) estimator \(\exp(-E[u_{ij} \mid \varepsilon_{ij}])\) is also computed and stored, and `efficiencies(fit, estimator = "jlms")` switches between the two without refitting.

### 1.2 The metafrontier

The metafrontier is defined as a function \(f^*(x) = \exp(x'\beta^*)\) such that:

\[x'\beta^* \ge x'\beta_j \quad \text{for all } x \text{ and all } j\]

That is, the metafrontier weakly dominates all group frontiers. It is the boundary of the industry metatechnology \(T^*\): the set of all input-output combinations that are technically feasible in the industry (Battese, Rao and O’Donnell, 2004; O’Donnell, Rao and Battese, 2008). Each group frontier bounds a restricted subset \(T_j \subseteq T^*\) of this metatechnology, where the restrictions reflect regulation, the physical environment, resource endowments, or the cost of switching production systems.

### 1.3 The efficiency decomposition

For each firm, efficiency relative to the metafrontier decomposes as:

\[TE^*_{ij} = TE_{ij} \times TGR_{ij}\]

where the technology gap ratio is:

\[TGR_{ij} = \frac{\exp(x_{ij}'\beta_j)}{\exp(x_{ij}'\beta^*)} = \exp\left(x_{ij}'(\beta_j - \beta^*)\right) \in (0, 1]\]

A \(TGR\) of 1 means the group frontier coincides with the metafrontier at that input mix; values below 1 indicate a technology gap.

## 2. Deterministic metafrontier (Battese, Rao, and O’Donnell, 2004)

### 2.1 Estimation

After obtaining group estimates \(\hat\beta_j\) in Stage 1, the metafrontier parameters \(\hat\beta^*\) are chosen so that the metafrontier lies as close as possible to the group frontiers while enveloping all of them. Battese, Rao and O’Donnell (2004) propose two optimisation criteria, both implemented in the package via the `objective` argument.

Minimum sum of absolute deviations (`objective = "lp"`, the default):

\[\min_{\beta^*} \sum_{j=1}^{J} \sum_{i=1}^{n_j} \left|x_{ij}'\beta^* - x_{ij}'\hat\beta_j\right|\] \[\text{subject to: } x_{ij}'\beta^* \ge x_{ij}'\hat\beta_j \quad \forall\, i, j\]

Because the envelopment constraints force every deviation to be non-negative, the absolute values can be dropped and the problem reduces to a linear programme (O’Donnell, Rao and Battese, 2008, Eqs. 23-25). This LP is the primary presentation in O’Donnell, Rao and Battese (2008) and is the package default; it is solved with `lpSolveAPI`.

Minimum sum of squared deviations (`objective = "qp"`):

\[\min_{\beta^*} \sum_{j=1}^{J} \sum_{i=1}^{n_j} \left(x_{ij}'\beta^* - x_{ij}'\hat\beta_j\right)^2\]

subject to the same envelopment constraints. This is a convex quadratic programme, solved exactly with `quadprog` when that package is available, with an adaptive-barrier fallback via `constrOptim()` from base R otherwise.

The two criteria weight deviations differently (the squared criterion penalises large gaps between the metafrontier and a group frontier more heavily), but in practice they typically produce very similar coefficients.

### 2.2 Properties

- The deterministic metafrontier is a point estimate with no associated standard errors (since Stage 2 is a deterministic optimisation, not a statistical model).
- The enveloping constraints guarantee \(TGR_{ij} \le 1\) for all observations in the estimation sample.
- The metafrontier coefficients depend on the observed input range; they are global only within the sample support.

### 2.3 Example

    sim <- simulate_metafrontier(
      n_groups = 2, n_per_group = 300,
      tech_gap = c(0, 0.4),
      sigma_u = c(0.2, 0.35),
      seed = 123
    )

    fit_det <- metafrontier(
      log_y ~ log_x1 + log_x2,
      data = sim$data,
      group = "group",
      meta_type = "deterministic"
    )

    # Metafrontier coefficients (no standard errors)
    coef(fit_det, which = "meta")
    #> (Intercept)      log_x1      log_x2
    #>   1.0416500   0.4950084   0.1926936

    # Group coefficients for comparison
    coef(fit_det, which = "group")
    #> $G1
    #> (Intercept)      log_x1      log_x2
    #>   1.0416500   0.4950084   0.1926936
    #>
    #> $G2
    #> (Intercept)      log_x1      log_x2
    #>   0.5814439   0.5287600   0.2015517

The metafrontier intercept should be at least as large as all group intercepts:

    meta_b0 <- coef(fit_det, which = "meta")[1]
    group_b0 <- sapply(coef(fit_det, which = "group"), `[`, 1)
    meta_b0 >= group_b0
    #> G1.(Intercept) G2.(Intercept)
    #>           TRUE           TRUE

To use the minimum sum of squared deviations criterion instead, set `objective = "qp"`:

    fit_qp <- metafrontier(
      log_y ~ log_x1 + log_x2,
      data = sim$data,
      group = "group",
      meta_type = "deterministic",
      objective = "qp"
    )

    # LP and QP criteria typically give very similar coefficients
    cbind(
      LP = coef(fit_det, which = "meta"),
      QP = coef(fit_qp, which = "meta")
    )
    #>                    LP        QP
    #> (Intercept) 1.0416500 1.0416500
    #> log_x1      0.4950084 0.4950084
    #> log_x2      0.1926936 0.1926936

## 3. Stochastic metafrontier (Huang, Huang, and Liu, 2014)

### 3.1 Estimation

Huang, Huang, and Liu (2014) propose treating the technology gap as a stochastic variable. In Stage 2, the fitted group frontier values become the dependent variable in a second SFA:

\[\ln \hat{f}(x_{ij}; \hat\beta_j) = x_{ij}'\beta^* + v^*_{ij} - u^*_{ij}\]

where \(u^*_{ij} \ge 0\) captures the technology gap and \(v^*_{ij}\) is a noise term. This is estimated via MLE, yielding:

- Point estimates \(\hat\beta^*\) with standard errors
- A variance-covariance matrix for inference
- A distributional \(\widehat{TGR}\) with associated uncertainty

### 3.2 Advantages over the deterministic approach

1. Inference: Standard errors, confidence intervals, and hypothesis tests on metafrontier parameters are available.
2. Robustness: The noise term \(v^*_{ij}\) absorbs sampling variation from Stage 1, preventing overfitting.
3. Consistency: The metafrontier need not strictly envelop all group frontiers in finite samples, which can be more realistic.

### 3.3 Caveat: the generated-regressor problem

The stochastic metafrontier is a two-stage estimator. In Stage 2, the dependent variable \(\ln \hat{f}(x_{ij}; \hat\beta_j)\) is itself an estimate from Stage 1 – it is a generated regressor (Murphy and Topel, 1985). The standard errors reported by the package are derived from the Stage 2 Hessian alone and do not account for the sampling uncertainty in the Stage 1 group frontier estimates.

As a result:

- Standard errors, confidence intervals (`confint()`), and hypothesis tests may be understated (confidence intervals narrower than their nominal coverage warrants).
- This issue does not affect point estimates of \(\hat\beta^*\) or efficiency scores, only inference.
- The bias is negligible when group sample sizes are large relative to the number of frontier parameters.
- The Murphy–Topel (1985) correction is available via `vcov(fit, correction = "murphy-topel")` and `confint(fit, correction = "murphy-topel")`. This adjusts the Stage 2 variance-covariance matrix to account for Stage 1 estimation uncertainty.

### 3.4 Example

    fit_sto <- metafrontier(
      log_y ~ log_x1 + log_x2,
      data = sim$data,
      group = "group",
      meta_type = "stochastic"
    )

    summary(fit_sto)
    #>
    #> Metafrontier Model Summary
    #> ==========================
    #>
    #> Call:
    #> metafrontier(formula = log_y ~ log_x1 + log_x2, data = sim$data,
    #>     group = "group", meta_type = "stochastic")
    #>
    #> Method:        sfa
    #> Metafrontier:  stochastic
    #>
    #> --- Group: G1 (n = 300) ---
    #>              Estimate Std. Error z value Pr(>|z|)
    #> (Intercept)  1.041650   0.054009  19.287  < 2e-16 ***
    #> log_x1       0.495008   0.009173  53.966  < 2e-16 ***
    #> log_x2       0.192694   0.008907  21.635  < 2e-16 ***
    #> log_sigma_v -1.698990   0.114126 -14.887  < 2e-16 ***
    #> log_sigma_u -1.524665   0.228427  -6.675 2.48e-11 ***
    #> ---
    #> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
    #> Log-likelihood: 22.685
    #>
    #> --- Group: G2 (n = 300) ---
    #>             Estimate Std. Error z value Pr(>|z|)
    #> (Intercept)  0.58144    0.05155   11.28   <2e-16 ***
    #> log_x1       0.52876    0.01019   51.88   <2e-16 ***
    #> log_x2       0.20155    0.01037   19.44   <2e-16 ***
    #> log_sigma_v -2.00652    0.16307  -12.30   <2e-16 ***
    #> log_sigma_u -0.87260    0.08019  -10.88   <2e-16 ***
    #> ---
    #> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
    #> Log-likelihood: -36.696
    #>
    #> --- Metafrontier ---
    #>             Estimate Std. Error z value Pr(>|z|)
    #> (Intercept) 0.813734   0.208469   3.903 9.49e-05 ***
    #> log_x1      0.513997   0.005155  99.716  < 2e-16 ***
    #> log_x2      0.197514   0.004989  39.588  < 2e-16 ***
    #> ---
    #> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
    #> Log-likelihood: 181.35
    #>
    #> --- Efficiency Decomposition ---
    #>  Group Mean_TE Mean_TGR Mean_TE_star
    #>     G1  0.8477   1.1839       1.0036
    #>     G2  0.7372   0.8306       0.6124
    #>
    #> --- Technology Gap Ratio Summary ---
    #>  Group   N   Mean     SD    Min     Q1 Median     Q3    Max
    #>     G1 300 1.1839 0.0320 1.1222 1.1576 1.1861 1.2087 1.2464
    #>     G2 300 0.8306 0.0181 0.7949 0.8148 0.8300 0.8445 0.8693
    #>
    #> --- Convergence ---
    #> All estimation stages converged.

The stochastic metafrontier provides standard errors:

    # Variance-covariance matrix
    vcov(fit_sto)
    #>               (Intercept)        log_x1        log_x2
    #> (Intercept)  4.345943e-02 -8.269764e-05 -6.986322e-05
    #> log_x1      -8.269764e-05  2.657002e-05  1.867196e-06
    #> log_x2      -6.986322e-05  1.867196e-06  2.489219e-05

    # Log-likelihood of the metafrontier model
    logLik(fit_sto)
    #> 'log Lik.' 181.3504 (df=3)

### 3.5 A note on TGR values

Under the stochastic metafrontier, TGR values are not constrained to be \(\le 1\) in finite samples, since the metafrontier need not strictly envelop all group frontiers. Values slightly above 1 can occur and are consistent with the stochastic framework.

    tgr_vals <- efficiencies(fit_sto, type = "tgr")
    summary(tgr_vals)
    #>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
    #>  0.7949  0.8300  0.9958  1.0072  1.1861  1.2464

## 4. DEA-based metafrontier

### 4.1 Approach

For a nonparametric metafrontier:

1. Compute group-specific DEA efficiencies \(\hat\theta_{ij}^{group}\) using only observations from group \(j\).
2. Compute pooled DEA efficiencies \(\hat\theta_{ij}^{pool}\) using all observations.
3. The TGR is: \(TGR_{ij} = \hat\theta_{ij}^{pool} / \hat\theta_{ij}^{group}\).

The package solves the DEA linear programs using `lpSolveAPI`.

### 4.2 Returns to scale

The `rts` argument controls the technology assumption:

- `"crs"` (constant returns to scale): the standard CCR model
- `"vrs"` (variable returns to scale): the BCC model
- `"drs"` / `"irs"` (decreasing / increasing returns)

    # CRS metafrontier
    fit_crs <- metafrontier(
      log_y ~ log_x1 + log_x2,
      data = sim$data,
      group = "group",
      method = "dea",
      rts = "crs"
    )

    # VRS metafrontier
    fit_vrs <- metafrontier(
      log_y ~ log_x1 + log_x2,
      data = sim$data,
      group = "group",
      method = "dea",
      rts = "vrs"
    )

    # Compare mean TGR
    cbind(
      CRS = tapply(fit_crs$tgr, fit_crs$group_vec, mean),
      VRS = tapply(fit_vrs$tgr, fit_vrs$group_vec, mean)
    )
    #>          CRS       VRS
    #> G1 1.0000000 1.0000000
    #> G2 0.6014273 0.8180669

## 5. Comparing methods

The choice between deterministic, stochastic, and DEA metafrontiers involves trade-offs:

Feature | Deterministic SFA | Stochastic SFA | DEA
--- | --- | --- | ---
Functional form | Parametric | Parametric | Nonparametric
Noise handling | Stage 1 only | Both stages | None
Inference on TGR | No | Yes | No
TGR \(\le\) 1 guaranteed | Yes | No | Yes
Small sample performance | Moderate | Moderate | Poor
References | BRO (2004) | HHL (2014) | ORB (2008)

    # Compare TGR estimates across methods
    tgr_det <- tapply(fit_det$tgr, fit_det$group_vec, mean)
    tgr_sto <- tapply(fit_sto$tgr, fit_sto$group_vec, mean)
    tgr_dea <- tapply(fit_crs$tgr, fit_crs$group_vec, mean)
    true_tgr <- tapply(sim$data$true_tgr, sim$data$group, mean)

    comparison <- data.frame(
      True = true_tgr,
      Deterministic = tgr_det,
      Stochastic = tgr_sto,
      DEA_CRS = tgr_dea
    )
    round(comparison, 4)
    #>      True Deterministic Stochastic DEA_CRS
    #> G1 1.0000         1.000     1.1839  1.0000
    #> G2 0.6703         0.702     0.8306  0.6014

## 6. Choosing a method: practical guidance

Selecting between deterministic SFA, stochastic SFA, and DEA metafrontiers depends on the research question, data characteristics, and inferential requirements.

Use the deterministic SFA metafrontier (BRO 2004) when:

- You need guaranteed \(TGR \le 1\) (the metafrontier strictly envelops all group frontiers).
- Inference on metafrontier parameters is not required.
- The goal is descriptive decomposition of efficiency into within-group and between-group components.

Use the stochastic SFA metafrontier (HHL 2014) when:

- You need standard errors, confidence intervals, or hypothesis tests on the metafrontier parameters.
- You want a distributional framework for the technology gap ratio.
- Sample sizes per group are moderate to large (at least 50–100 observations per group is recommended).
- You are comfortable with the generated-regressor caveat (Section 3.3).

Use the DEA metafrontier when:

- You prefer a nonparametric approach with no functional form assumptions.
- Multiple inputs and/or multiple outputs are involved.
- Sample sizes are large enough to support DEA (a rough guideline: \(n \ge 3 \times (m + s)\) per group, where \(m\) is the number of inputs and \(s\) the number of outputs).
- The returns-to-scale assumption (`rts`) is well-justified by the application context.

In many applied studies, it is informative to estimate multiple methods and compare TGR estimates for robustness (as shown in Section 5).

## 7. Testing for technology heterogeneity

Before estimating a metafrontier, it is useful to test whether separate group frontiers are actually needed. The poolability test uses a likelihood ratio statistic:

\[LR = -2\left[LL_{pooled} - \sum_{j=1}^{J} LL_j\right] \sim \chi^2_{df}\]

where \(LL_{pooled}\) is the log-likelihood of a single frontier estimated on the pooled sample and \(LL_j\) are the group-specific log-likelihoods.

    poolability_test(fit_det)
    #>
    #>  Likelihood Ratio Test for Poolability of Group Frontiers
    #>
    #> data:  fit_det
    #> LR = 442.35, df = 5, p-value < 2.2e-16

A significant test (p < 0.05) confirms that the group frontiers differ, that is, the groups face different restrictions of the common metatechnology, and the metafrontier decomposition is warranted.

## 8. Simulation for Monte Carlo studies

The `simulate_metafrontier()` function generates data from a known DGP, enabling parameter recovery studies:

    # Monte Carlo: check parameter recovery over 100 replications
    set.seed(1)
    n_rep <- 100
    beta_hat <- matrix(NA, n_rep, 3)

    for (r in seq_len(n_rep)) {
      sim_r <- simulate_metafrontier(
        n_groups = 2, n_per_group = 200,
        tech_gap = c(0, 0.3),
        sigma_u = c(0.2, 0.3),
        sigma_v = 0.15
      )
      fit_r <- metafrontier(
        log_y ~ log_x1 + log_x2,
        data = sim_r$data,
        group = "group",
        meta_type = "deterministic"
      )
      beta_hat[r, ] <- coef(fit_r, which = "meta")
    }

    # Bias
    true_beta <- c(1.0, 0.5, 0.3)
    colMeans(beta_hat) - true_beta

The `simulate_metafrontier()` function supports:

- Arbitrary number of groups (`n_groups`)
- Unequal group sizes (`n_per_group` as a vector)
- Custom metafrontier coefficients (`beta_meta`)
- Group-specific technology gaps (`tech_gap`)
- Group-specific inefficiency dispersion (`sigma_u`)
- Reproducible results via `seed`

## References

- Battese, G.E. and Coelli, T.J. (1988). Prediction of firm-level technical efficiencies with a generalized frontier production function and panel data. Journal of Econometrics, 38(3), 387–399.

- Battese, G.E., Rao, D.S.P. and O’Donnell, C.J. (2004). A metafrontier production function for estimation of technical efficiencies and technology gaps for firms operating under different technologies. Journal of Productivity Analysis, 21(1), 91–103.

- Huang, C.J., Huang, T.-H. and Liu, N.-H. (2014). A new approach to estimating the metafrontier production function based on a stochastic frontier framework. Journal of Productivity Analysis, 42(3), 241–254.

- Jondrow, J., Lovell, C.A.K., Materov, I.S. and Schmidt, P. (1982). On the estimation of technical inefficiency in the stochastic frontier production function model. Journal of Econometrics, 19(2–3), 233–238.

- O’Donnell, C.J., Rao, D.S.P. and Battese, G.E. (2008). Metafrontier frameworks for the study of firm-level efficiencies and technology ratios. Empirical Economics, 34(2), 231–255.