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sfa 1.2.0

A feature release. In brief:

• A Wald test for whether the endogeneity correction was needed, endogeneity_test(), from Hou, Ramalho and Roseta-Palma (2025). On an ivsfm() fit it tests H0: rho = 0 – that the noise is uncorrelated with the reduced-form errors, so a plain sfm() fit would have been consistent. The null is INTERIOR, since rho lives in the open unit ball, so this is an ordinary chi-square and not the chi-bar-square mixture inefficiency_test() needs. Size 0.046 at a nominal 5% over 1000 replications; power 1.000 at rho = 0.4 and 0.8.

• ivsfm() now reports standard errors for rho. They were NA, on the ground that rho = t/sqrt(1 + t't) has a non-diagonal Jacobian and a delta-method value computed as though it were diagonal would be wrong. The first half was right and the conclusion was not: the Jacobian is J = (I - rho rho')/s, so Var(rho) = J Var(t) J' exactly. Checked against a numerical Jacobian, and validated end-to-end by the size of the test above – a wrong Jacobian would show up there as a mis-sized test. The full covariance travels with the fit as $vcov_rho, which the joint test needs.

• ttsfm(model_name = "TTHN") no longer fabricates a log-density it cannot compute. The likelihood needs D = Phi2(x1, 0; rho1) - Phi2(x2, 0; rho2), a difference of two bivariate normal CDFs of the same order of magnitude, and in parts of the parameter space it cancels completely: at sigma_v = 0.3, sigma_u = 1, sigma_w = 0.2 on a 400-observation draw one observation’s D comes back as exactly 0 while both CDFs are O(0.1).

  pmax(D, .Machine$double.xmin) did not repair that – it invented log(D) = -708.4 for the observation, and that single observation moved the summed objective by 715 log-units in a surface whose real curvature is a few units per 0.1 step in log sigma. A cliff that size is what L-BFGS-B reports as ABNORMAL_TERMINATION_IN_LNSRCH. D is now rejected when it falls below the rounding error of its own subtraction, which leaves genuine small probabilities alone: a D of 6e-94 where both CDFs are also ~1e-94 has full significance and is kept.

  Three smaller repairs in the same branch: the exponentials for sigma_v/sigma_u/sigma_w are clipped at EXP_CLIP_UPPER as the TTNE branch already clipped its own; rho is held strictly inside (-1, 1), which sigma_v small enough does reach; and the substitution of -sqrt(.Machine$double.xmax / n) – about -9.5e152 at n = 200, 140 orders of magnitude past the finite penalty it sat beside – is gone, replaced by that penalty. It had also mapped a +Inf log-density to a large NEGATIVE one, so an unusable draw could be rewarded.

  Measured on eight matched fits: one that previously failed outright now converges, and one that previously returned sigma_w = 0.044 against a truth of 1 now returns 1.047 with a better likelihood. The other six are unchanged. sigma_v still collapses toward zero on some draws; that is a separate problem and is not fixed.

• ttsfm() no longer throws away a fit because of the optimizer’s exit code. It was the only entry point calling stop() on ANY non-zero convergence code, including L-BFGS-B’s code 52, ABNORMAL_TERMINATION_IN_LNSRCH, which it returns whenever its line search meets a discontinuity and usually having improved the objective. The guard was written when nothing checked opt$convergence at all; opt.optim() now rebuilds at the stage-2 point whenever the value or the Hessian is non-finite, so what was left of it only destroyed usable fits. Two of eight TTHN test fits returned code 52 with sound estimates where the stop() had aborted them.

• What skewness and kurtosis a pair of distributions can actually produce, moment_range(), from Papadopoulos and Parmeter (2021). spec_test() already shipped their formal statistic; this is the range check that comes before it. Because the variance share R lies in (0,1), the composed error can never be more skewed than u itself – Normal-Half-Normal is confined to gamma1 in (-0.995, 0), Normal-Exponential to (-2, 0) – so residual moments outside that set refute the pair with no test statistic involved. That is their Type II failure, alongside the Type I failure of wrong skew.

  Five noise and five inefficiency distributions, including the two whose shape is a free parameter: the truncated normal sweeps gamma1(u) in (0, 2) and gamma2(u) in (-0.243, 6) and is the only u here whose composed error can be platykurtic under normal noise, while the gamma is unbounded in both and so can never suffer a Type II failure. The kurtosis range is not the interval between the two components: with leptokurtic noise it dips below both, to 0.674 for Laplace noise with a half-normal u against the half-normal’s own 0.869.

  It is a diagnostic, not a test, and the print method says so. PP2021’s Tables 1-5 replicate – their population rows exactly, and Tables 2, 3 and 5 cell for cell – and they show the range check over-rejects: with correctly specified data at n = 200 and SNR 3, the sample excess kurtosis leaves its theoretical range 26% of the time. Use spec_test() for the p-value.

• A frontier that tolerates the wrong skewness, esfm(), from Hafner, Manner and Simar (2018). When the sample skewness of the residuals comes out positive – a small-sample accident, not evidence against the model – the classical MLE collapses: sigma_u is exactly zero, every firm is fully efficient and the fit says nothing. This model keeps the frontier and widens the distribution of u, with one parameter gamma whose SIZE is the scale of inefficiency and whose SIGN chooses the direction of skewness. The classical model is nested at gamma > 0.

  On a 50-observation wrongly skewed draw, sfm(model_name = "NHN") returns sigma_u = 0.0003 and mean efficiency 0.99976; esfm() returns gamma = -0.57 with the slopes still near truth.

  Because gamma = 0 is an interior point rather than a boundary, the LR test of “no inefficiency” is an ordinary chi2(1) – not the chi-bar-square mixture inefficiency_test() needs for the classical model. Size over 2000 replications on the paper’s design, at a nominal 5%: 0.178 / 0.082 / 0.054 / 0.061 / 0.058 for n = 50 / 100 / 200 / 500 / 1000, against the paper’s 0.152 / 0.076 / 0.061 / 0.054 / 0.053. Badly oversized below n = 100, usable from n = 200 – their own conclusion.

  It is not a free improvement everywhere. The paper reports smaller bias in mean efficiency than the classical model even when the population skewness has the correct sign, and that reproduces when inefficiency is not small: at gamma = 0.5 the bias is -0.0001 at n = 100 against the classical +0.021. But at gamma = 0.3, where 22% of samples are wrongly skewed, this model overshoots the other way (-0.040 against +0.036) and is the worse of the two at n = 50 (-0.080 against +0.044). A small true gamma lets it fit sizeable inefficiency of either sign to noise. Use it when the wrong skewness is the problem.

• Kim and Schmidt (2008) two-step test, uhet_test(). The procedure applied work actually runs – fit the frontier, regress the JLMS predictor on firm characteristics, test the slopes – is invalid as usually performed, because u-hat is a generated dependent variable and the first-step estimation error enters the second-step variance. The correction is not distribution-free, so "NHN" and "NE" are implemented, the two cases the paper works through.

  What it buys: with z independent of x the corrected and naive tests agree, as the theory requires. With z strongly correlated with x (rho = 0.9) the naive standard error is 52% too large at n = 800 and the naive test’s size collapses to 0.000 at a nominal 5%, while the corrected standard error matches the true sampling spread to within half a percent (ratio 0.995) and keeps size 0.032.

  Caveat, stated plainly: the corrected test is conservative in these simulations. On the paper’s own base case it rejects 0.024/0.036/0.028 at N = 200/500/1000 against a nominal 0.05, where the paper reports 0.0495 at N = 200. The naive branch does reproduce the paper’s figure for the uncorrected test (0.023-0.029 against their 0.0237). The implementation was checked against equations (5)-(13), the first-step scores agree with an analytic half-normal score to 1e-11, gamma-hat is unbiased and Gaussian, and an oracle test using the true sampling standard deviation has correct size (0.045) – so the gap is in the variance estimate, not in gamma-hat or the reference distribution. Two candidate explanations were tested and both refuted: the homoskedastic second-step variance (the paper’s “OLS” against its “OLS-H”) changes nothing, and neither does using the Hessian information rather than the outer product when forming r_i. The gap is unexplained rather than merely unverified. Read a rejection as meaningful and a non-rejection as weak evidence.

  Fits on the wrong-skew boundary are refused rather than answered: every E[u|eps] is then identical, so there is no estimated inefficiency for z to explain, and the information matrix is singular there (Waldman 1982).

• Chen and Wang (2012) centered-residuals moment test, cw_test(). Unlike gof_test(), it never evaluates the composed-error density or distribution function: it compares the empirical characteristic function of the centred residuals with its theoretical value, which follows from the characteristic functions of the two components separately – in closed form for many pairs whose convolution is not, the normal-gamma model being the standard example.

  Centring is what makes it work. A frontier’s intercept is not identified separately from E[u], so the least-squares intercept estimates alpha - E[u]; centring cancels it, which is why the test is valid off an ordinary regression rather than requiring maximum likelihood. The variance corrects for both nuisances – estimating the scales, and the centring itself. Omitting the second understates it and oversizes the test.

  Size at a nominal 5% over 500 replications, half-normal data:

  test	n	tau=1	tau=1.5	tau=2
  cosine	500	0.070	0.054	0.050
  cosine	2000	0.046	0.044	0.050
  sine	500	0.092	0.064	0.042
  sine	2000	0.068	0.062	0.064

  Power against exponential inefficiency at tau = 1 is 0.231 (n = 500) and 0.952 (n = 2000) for the cosine test against 0.108 and 0.706 for the sine test. Both of the authors’ conclusions reproduce: the cosine test is the less sensitive to tau and the more powerful, so it is the default. tau defaults to a single frequency: passing several at once gives rejection rates of 0.09-0.15 at a nominal 5%, because the cosine moments at nearby frequencies are near-collinear and the covariance matrix is then near-singular. The authors report the same for their combined tau.

  A wrongly signed third moment leaves the moment equations with no admissible solution, and that is an error rather than a number.

• The Gauss-Legendre rule is now cached. It depends on nothing but the node count and was rebuilt by an eigendecomposition on every call – hundreds of times per cw_test(), since numDeriv differences a quadrature-based moment function. Caching also keeps eigen() out of forked workers: macOS’s threaded BLAS is not fork-safe and mclapply() segfaulted inside it before this was added.

• Goodness-of-fit tests for the assumed inefficiency distribution, gof_test(), from Wang, Amsler and Schmidt (2011). The distribution of u is the assumption in this model least often defended and least often tested, and it is testable: hold the normality of v fixed and the assumed u implies a distribution for the composed error, so rejecting that is rejecting the assumed u. Kolmogorov-Smirnov and Pearson chi-square, both on the composed error rather than on u-hat – which is the same test, since u-hat = E[u|eps] is monotonic in eps, but far easier, and it avoids the trap the paper warns about: comparing the spread of u-hat with the assumed density of u is a mistake, not a diagnostic, because those are different distributions.

  Parameter estimation is not ignored. The default parametric bootstrap copies the estimation step exactly – each replication draws from the fitted model, rebuilds the response, refits, forms its own residuals and recomputes the statistic. The "asymptotic" option gives a chi-square p-value that the authors note is conservative at the MLE, and returns NA rather than a KS p-value, because with parameters estimated the Kolmogorov distribution does not apply and Bai’s (2003) transformation is not implemented.

  Size and power over 200 replications (half-normal data for size, exponential data fitted as half-normal for power), at nominal 0.10 / 0.05:

  	n	KS	chi2
  size	200	0.065 / 0.025	0.085 / 0.065
  size	800	0.070 / 0.040	0.065 / 0.010
  power	200	0.505 / 0.395	0.235 / 0.145
  power	800	0.985 / 0.965	0.810 / 0.655
  power	2000	1.000 / 1.000	0.990 / 0.985

  Both hold their size, mildly conservatively, and KS dominates chi-square at every sample size – the paper’s own conclusion, and the reason it is the better default.

• Coelli (1995)’s hypothesis tests, and his COLS standard errors. New inefficiency_test() tests H0: no technical inefficiency, reporting the one-sided LR test, the naive LR test, the Wald ratio and the third-moment test side by side. The point of the paper is that the two tests most often reported are the two with the wrong size: H0 puts gamma on the boundary, so the LR statistic follows the Gourieroux-Holly-Monfort mixture 0.5 chi2(0) + 0.5 chi2(1) (5% critical value 2.71, not 3.84) and the Wald ratio is not asymptotically normal at all. Measured here over 400 replications per cell at a nominal 5%, with sigma_u = 0:

  n	LR one-sided	LR naive	Wald	M3T
  100	0.060	0.030	0.230	0.055
  400	0.055	0.033	0.217	0.045
  800	0.060	0.035	0.182	0.062

  The one-sided LR and third-moment tests hold their size; the naive LR test rejects about half as often as it should; the Wald test rejects three to four times too often and does not improve with n. Under the alternative the one-sided LR test has the better power of the two valid tests at every sample size, which is Coelli’s recommendation. NTN fits get the two-restriction mixture 0.25/0.5/0.25 over chi2(0,1,2).

• sfm(estimator = "cols") now reports Coelli’s analytic standard errors. It previously returned NA for sigv and sigu unless a bootstrap was requested, because the OLS errors are not appropriate for them: those parameters are non-linear functions of the residual moments m2 and m3, not regression coefficients. Coelli’s Appendix 1 (A15-A16) gives the correct delta-method variance and it is now filled in for NHN; the intercept additionally carries the error in its E[u] shift. NE and NG keep NA and the bootstrap, because the moment inversion – and therefore the variance – is distribution-specific.

  Checked against the nonparametric bootstrap that was already there, on the same data: the analytic/bootstrap ratio for sigu is 1.047 at n = 200, 1.011 at n = 800 and 0.984 at n = 3200, which is an asymptotic approximation behaving as one should.

  Coelli’s A13 inversion turned out to be the moment inversion the package already carried as .gtre_two_step() (equal to ten decimal places, not merely similar), and his A15 variance is the delta method whose GTRE_SEQ2 implementation is corrected elsewhere in this release – including the factor of 2 on the covariance term, which A15 independently confirms, and A16’s mu6 - mu3^2 - 6 mu2 mu4 + 9 mu2^3, which confirms the other two repairs. The central moments now live in one shared helper, .nhn_central_moments(), rather than being derived twice.

• A composed-error CDF for every cross-sectional model, pcomposed_model() and composed_cdf(). pcomposed() covers the half-normal only, which is enough for copulas but not for goodness-of-fit testing, where the CDF is needed for whichever inefficiency distribution the model assumes. All thirteen are covered: NHN, NTN, NE, NR, NU, NGE, NLN, NW, NG, NNAK, TSL, THT and tHN.

  Three parameterizations do not read the way their names suggest and are taken from each likelihood rather than from the label: THT lists sigu before sigv, the only model that does; NG’s sigu is the gamma scale and its mu the shape; NNAK’s sigu is the Nakagami spread, so Omega = sigu^2; and NLN’s mu is a meanlog.

  THT is not an independent convolution. Tancredi’s composed error is skew-t, a scale mixture in which v and u are divided by the same sqrt(V/a). Treating it as t noise plus an independent half-normal – which is exactly what tHN is – gets the log density wrong by up to 1.74. The first draft did precisely that, and the check below is what caught it.

  Verified three ways: against pcomposed() for the half-normal, agreeing to 3.5e-15 relative including a lower tail at log F = -209; against a four-million-draw simulation for all thirteen models, maximum absolute error 4e-4 against a Monte Carlo standard error of 7.5e-4; and by numerically differentiating the CDF and comparing with each model’s own stored log-density, which agrees to about 1e-8 for eleven of them. NLN and NW differ by ~3e-3 because their likelihoods are simulated, not because the models disagree.

• psfm() died with “system is exactly singular” when a factor was only partly collinear between individuals. Reported from use: factor(year) on an unbalanced panel, where the between-individual means of the year dummies are rank deficient. The guard added in 1.1.5 detected this correctly and printed an accurate warning, then did nothing, because it could only express the fix by deleting whole formula terms – and 11 of factor(year)’s 24 dummies being collinear names no term that can be deleted without discarding the 13 identified ones too. plm::ercomp() was then handed the same singular design and failed inside solve(). The starting-value design is now reduced at column granularity (.re_start_design()), so the identified dummies survive, the dropped ones take their starting values from a pooled OLS fit as documented, and the likelihood still uses the formula as written. This was a no-op for every partly-collinear factor, which is the common case – a fully collinear term was the only shape the old path could handle.

  Two consequences of that path are fixed with it. collinear_action = "warn_drop" silently dropped nothing in the same situation, and now says so instead. And "GTRE_SEQ1"/"GTRE_SEQ2" took the frontier standard errors from the starting-value fit by position, which errored on the length mismatch once the design was reduced; they are matched by name, with NA for columns that fit could not identify rather than a neighbour’s value.

• "GTRE_SEQ1" and "GTRE_SEQ2" are inconsistent for fixed T, and now say so. Both hand plm’s random-effects output to a second stage that treats it as draws from the latent composite errors. It is not: alpha_hat is a shrunken BLUP and eps_hat a quasi-demeaned residual, which attenuates the third central moment by about 0.81 at on the transient side and scales the persistent side by and . No factor tends to 1 as grows at fixed , so both converge to the wrong constants – settles near 0.68 against a true 0.80. Feeding the same two second stages the latent draws instead recovers the truth at the rate, which puts the fault in the shared first stage rather than in the likelihood or the moment inversion. Each call now warns and points at model_name = "GTRE". This is a documentation and diagnosis change: the estimators are unchanged, and are the large-T procedures they always were.

• "GTRE_SEQ2"’s standard errors were wrong in four ways. The variance of the third central moment used mu_3^3 for and 9 * mu_2 for (also dimensionally inconsistent); the delta-method cross-terms were missing their factor of 2; beta_0’s error added a standard error to a sum of variances instead of its square; and both derivatives were garbled, squaring neither denominator and carrying where belongs. Against the Monte Carlo sampling standard deviation the reported error for gamma_uv was 16 times too large and sigmaSq_uv 1.37 times; corrected, both land within 2%. The algebra now lives in .gtre_two_step_se() so it is unit-tested, and returns NA at the wrong-skew boundary where it is not differentiable rather than Inf. "GTRE_SEQ2" also reported a lambda computed from a different (likelihood) decomposition than the sigmas printed beside it; it now comes from its own.

• ttsfm() returned an infinite penalty on unusable parameter draws, in seven places. TTNE, TTHN and TTNLS each returned .Machine$double.xmax when a draw produced non-finite contributions. That is the construct sfm() was converted away from earlier in this release after "NGE" died on 3 of 45 fits: optim() differences the objective to get its gradient, and differencing 1.8e308 overflows to a non-finite value, which does not steer the search away – it aborts the fit. TTHN was failing at with ABNORMAL_TERMINATION_IN_LNSRCH, which is what L-BFGS-B does when handed gradient information it cannot use. All seven now return a finite penalty.

  The regression test that exists to catch exactly this construct only ever read sfm.R, which is how ttsfm.R kept all seven for as long as the test existed. It now reads every file carrying a likelihood closure.

• ttsfm() no longer calls the bivariate normal CDF once per observation. The TTHN density was evaluated with mapply() over observations, so one mnormt::pmnorm() call per observation, twice per likelihood evaluation – and with a numerically differenced gradient, calls per optimiser iteration. The correlation argument varies only through and , so under homoskedasticity it is one value repeated and the whole vector is a single call. Grouped accordingly, with a fallback to the per-observation form when the correlation genuinely varies per observation – there, grouping is slower, measured at 0.60x. Results are bit-identical to the previous form. The gain is about 1.4x, not more: pmnorm() still loops internally, so the call count falls but the work does not. TTNLS separately computed each of its two bivariate terms twice; those are now computed once.

• selsfm() gains a second model, model_name = "kts": Kumbhakar, Tsionas and Sipilainen (2009), joint estimation of technology choice and technical efficiency. Not a variant of Greene (2010), which this entry point already fitted, but a different model. Greene has one frontier observed only for the selected, with selection correlated with the noise. Here both technologies are observed, each with its own frontier and its own two scales, and the choice depends on inefficiency itself – u lowers output through the frontier and shifts the adoption decision through the choice equation. Neither two-step order works, which is the paper’s own argument for single-step maximum likelihood: the choice equation cannot be a probit because u is unobserved, and fitting the frontiers first ignores the endogeneity of the choice. Because u’s distribution depends on the regime while the regime’s probability depends on u, the marginal density of u is a mixture of the two half-normals and the observation’s density is the same mixture of two one-dimensional integrals. n_nodes controls the quadrature.

• copsfm() goes from two copula families to fifteen: Frank, Clayton, Gumbel and Joe join Gaussian and FGM, each of the last three with 90/180/270 rotations, since Clayton, Gumbel and Joe carry only positive dependence and nothing rules out a negative association between noise and inefficiency. Every density is verified against the second mixed partial of its own CDF, it integrates to 1 over the unit square, and it is exactly 1 at the independence parameter.

  copsfm() now warns when you choose a family that does not recover its own dependence parameter, and most of them do not. Fitting each family to 25 samples generated from itself at n = 400, the share of fits returning the independence boundary was 0% for Frank and Clayton but 36% for Gumbel, 40% for Joe, 56% for Clayton rotated 270 and 60% for Gumbel rotated 90. On data from a Gumbel copula with Spearman 0.685 at n = 2000, every family including the true one returns the boundary and their log-likelihoods differ by less than 0.04: the likelihood is close to flat in the dependence parameter. That is a property of the model, not of the implementation. The warning quotes the measured rate; the rotations that were not measured say so. Prefer "frank" or "clayton".

• zsfm(model_name = "ZISF") could be trapped at the boundary where the mass point vanishes, returning a much worse fit without saying so. Since P(fully efficient) = exp(-gamma), gamma -> Inf is the boundary at which the model collapses to an ordinary frontier, and it carries its own local optimum. On the package’s convergence design the default start fell into it on roughly 1 replication in 5000: one fit returned gamma = 9.93 (p = 5e-5) with a log-likelihood 188.7 points below what a start from anywhere in the interior reaches. Nothing in the returned object flagged it.

  zsfm() now tries several starting values for gamma and keeps the best attained objective, so it can only improve a fit. The guard applies to "ZISF" only: "ZISF_Z" has no scalar gamma – its layout is (sigv, sigu, beta, z-block) with the mixing parameterised through the z coefficients – so the same perturbation there would move the noise scale.

  Worth knowing if you have compared this model’s convergence behaviour before: that single fit inflated its sample size’s mean squared error 144-fold and flattened the log-MSE-on-log-n slope for gamma to zero, while four of the five sample sizes were textbook (MSE 0.0047, 0.0019, 0.00095, –, 0.00065).

• The sigma_u -> 0 repair listed further down opened a second catastrophic cancellation, at sigma_v -> 0, which is also fixed here. To be clear about the scope, since it affects whether anyone need re-check old results: this defect was introduced by that repair and existed only in this release’s own development. No published version of the package has it – 1.1.5 and earlier have the sigma_u defect instead, and .log_phi_tilt() does not exist in them. It is recorded because the mechanism is worth knowing and because the fix changes NE’s numerics.

  That repair works through .log_phi_tilt(), which returns log Phi(z) + z^2/2. For an observation with eps < 0 and sigma_v small, z goes large positive, log Phi(z) is ~0, and so the returned value is essentially z^2/2 – which the NE branch then subtracted eps^2/(2 sigma_v^2) from. At sigma_v = 1e-8 both are about 5e15, where consecutive doubles are ~1 apart, so the result was rounding noise: the summed negative log-likelihood read -5188 where the true value is +4398, an apparently excellent fit in a narrow band that a minimiser walks straight into. The domain guard did not catch it, firing only at sigma_v <= 0 while the band sits at 1e-9 to 1e-7.

  The consequence was not a visible failure. nlminb found the band, its “only accept an improvement” guard passed the point through because the objective had improved, and bobyqa and optim could not escape the corner; optim’s bounds, rebuilt around the corrupted point, then pinned the frontier coefficients. Such fits came back as ordinary sfareg objects with convergence = 0 – correctly, since a bound-constrained stationary point is a legitimate success – while carrying a gradient in the hundreds.

  The subtraction is analytic: z^2/2 = eps^2/(2 sv^2) + eps/su + sv^2/(2 su^2) identically, so above NE_TILT_SWITCH the expression collapses to the 1.1.5 tilt with no large intermediate formed. The two forms agree to 1e-13 across the switch, and below it the tilt form – and hence the sigma_u -> 0 fix – is untouched. Over 500 replications at lambda = 3.33, n = 1000, the rate of fits landing more than 0.3 from the truth falls from 3.8% to 1.0%, the log-likelihood improves on 18 and worsens on none.

  The root-n convergence sweep was then re-run in full, 1000 replications at each of n = 1000 to 5000, and "NE" now passes on the mean basis, which is what it used to fail: slopes -1.023 (sigma_v), -1.044 (sigma_u), -1.025, -1.033 and -1.009 (the frontier coefficients), every R-squared at least 0.989, and no failures in 5000 fits. Before the fix those slopes were -0.488, +0.262 – positive, i.e. error growing with the sample – -0.745, -0.443 and -0.165.

  This also resolves a dilemma the convergence notes had recorded and left open. They found that reverting the likelihood to its 1.1.5 form fixed the tail but would reintroduce the sigma_u -> 0 defect, and concluded “do not revert, fix the interaction instead”. Above the switch this fix is the 1.1.5 algebra, so the re-run reproduces that configuration’s slopes digit for digit while keeping the sigma_u protection. Both boundaries are now covered.

  "NGE" shares .log_phi_tilt() and was checked for the same defect; it does not have it.

• Robust estimation gains its missing half. sfm(robust = ) already fitted the Normal–Half-Normal frontier by maximum -likelihood, the power-Psi criterion and minimum density-power divergence, but the tuning parameter had to be supplied by hand with no guidance and no diagnostic. Four new functions close that, from Bernstein, Parmeter and Wright (2026):

  • hscore_select() chooses the tuning parameter by minimising the Hyvarinen score (Sugasawa and Yonekura, 2021) over a coarse-to-fine grid that includes the maximum-likelihood endpoint, so the criterion is free to decline robustification. It guards two silent failure modes: candidates lost to numerical overflow, and warm-start capture in the degenerate sigma_u -> 0 basin.
  • hscore() evaluates the criterion, in logarithms. The natural-scale expression raises the fitted density to the power c - 2, so one observation whose density underflows makes the score non-finite; because the surviving candidates are those closest to maximum likelihood, the failure biases selection toward too little robustness without any warning. stable = FALSE reproduces the naive evaluation for comparison.
  • calibrate_c() gives the fixed alternative: the tuning value at which a residual three scale units out retains a stated share of its clean-data influence. The influence ratio is not monotone for the Fisher-consistency-corrected criteria – Psi and MDPD cross the target twice – so every root is found and reported in a "roots" attribute rather than resolved silently.
  • density_weights() returns the per-observation weight the estimator applies, documented with what it can and cannot detect: it responds to response contamination and not to a mis-recorded regressor, which can be highly influential and still receive a weight near one.

  The power-Psi and MDPD objectives satisfy obj_MDPD = (1 + c) * obj_Psi identically, so they share a maximiser and are the same estimator at a common c. Both remain available as peer methods and calibrate_c() returns the same value for either.

• influence_sfa(), an influence-function diagnostic for any "sfareg" fit made with keep_objective = TRUE, after Stead, Wheat and Greene (2023). The standard outlier rules do not transfer to this model – they ignore the asymmetry of the composed error, so a genuinely inefficient firm reads as an outlier and is discarded. It reports the empirical influence function, a Cook’s-distance-shaped case influence, and two sensitivities. Compare models on sensitivity_std, the self-standardised one: the raw sup-norm depends on the parameterisation, and on a clean sample reads 72.6 for "NHN" against 1322.5 for "tHN", which says nothing about robustness. On the standardised scale, contaminating one response takes "NHN" from 13.5 to 32.3 and "tHN" from 13.1 to 17.2.

• A fifth model-fitting entry point, lcsfm() — the latent class stochastic frontier, with n_class coexisting technologies.

• New accessors. efficiency() returns technical efficiency with a choice of predictor ("bc", "jlms", "mode") and of scale for the dependent variable (logDepVar); the mode is the only one that reaches exactly 1, which it does wherever the posterior mean of inefficiency is negative. meanefficiency() returns the model-implied E[exp(-U)] for ten distributions – seven in closed form – together with mean efficiency among the most efficient p, which is a property of the fitted model rather than of the sample. simulation_se() reports how much of a simulated-ML standard error is simulation noise rather than sampling noise.

• sfm() gains four arguments. weights/wscale for observation weights (a weight of 2 gives the same estimates as including the row twice); start_from to seed a hard model from a simpler fitted one, matched by parameter name; scaling for the Wang-Schmidt scaling-property model on "NTN"; and shapehet, an experimental covariate-dependent shape for "NG"/"NNAK" whose limitations are set out in ?sfm.

• vcov(fit, type = "bhhh") returns the outer-product-of-gradients covariance, which is defined even where the Hessian is not.

• Regression tables. texreg::screenreg(), texreg() and htmlreg() now render "sfareg" fits directly.

• Two fixes to things that were shipped but did not work. The sandwich methods were defined without being registered, so sandwich::estfun(), vcovCL() and lmtest::coeftest() all failed on an installed package – every test passed because devtools::load_all() exports everything. And sim_seed silently did nothing under the default sim_type = "halton", leaving the cross-sectional simulated-ML models with unrandomized draws; they are now shifted like the panel models. Fixing the second is what makes simulation_se() meaningful.

• "NNAK" no longer fails outright at larger shape values. gsl::hyperg_U throws above order ~16 and the vectorized call meant one bad element nulled the whole vector. There is now a fallback that evaluates the parabolic cylinder function by its integral representation, agreeing with the route it replaces to 1e-12 where both work.

• A seventh entry point, ivsfm() — stochastic frontier estimation when a regressor is endogenous, in the sense of being correlated with the noise. Three estimators from Amsler, Prokhorov and Schmidt (2016): "IVLIML" (full maximum likelihood over the frontier and the reduced form jointly), "IVCF" (the two-step control function of Kutlu 2010) and "C2SLS" (corrected 2SLS, which is moment-based and so reports no log-likelihood). Supplying uhet gives the model of Amsler, Prokhorov and Schmidt (2017), in which environmental variables entering the inefficiency scale may themselves be endogenous. Ignoring endogeneity is not a small matter: on simulated data with a correlation of 0.6 between the noise and a regressor’s reduced-form error, an ordinary sfm() fit carries a bias of 0.133 in that regressor’s coefficient where ivsfm() carries 0.002. Note that "IVCF"’s conventional standard errors are understated unless the correlation is zero, and that the case where the regressor is correlated with the inefficiency rather than the noise is not covered.

• A sixth entry point, selsfm() — Greene’s (2010) stochastic frontier with a correction for sample selection, for the case where the units in the sample are there because of something correlated with the frontier’s own noise. The selection and frontier equations are supplied as separate selection and frontier arguments. Estimation is Greene’s two-step scheme: probit on the full sample, then maximum simulated likelihood on the selected subsample, with efficiency scores from the simulated Bayes rule over the same draws. Note that the standard errors are conditional on the first-stage estimate and do not carry the Murphy–Topel correction; bootstrap over both stages when inference on rho matters.

• Six new panel estimators, all classical rather than maximum likelihood and none assuming a distribution for inefficiency: "CSS" (Cornwell-Schmidt-Sickles 1990), "LS" (Lee-Schmidt 1993), "KSS" (Kneip-Sickles-Song 2012), and "SSRE"/"SSCRE", which complete the Schmidt-Sickles family that "SSFE" already began.

• Heteroskedasticity in more than one error component, as named formula arguments to sfm(): vhet, uhet and muhet. The last, with model_name = "NTN", is Battese-Coelli (1995).

• Three corrections that change numeric output: "NE"/"NGE" could return a positive log-likelihood with sigma_u at zero; nobs() counted rows supplied rather than rows used, so BIC() used the wrong n; and "NGE" aborted outright on about 7% of small samples.

• Plus z_link for sfm() and ttsfm(), marginal_effects() for panel fits, efficiency_ci(), and sfm(model_name = "TSL").

Nothing in this release changes the meaning of an existing argument or model_name. Every addition is a new argument defaulting to the previous behaviour, or a new model_name value.

• Heteroskedasticity in more than one component: new vhet, uhet and muhet arguments to sfm(). sigma_v, sigma_u and (for "NTN") the pre-truncation mean of u may now each be driven by their own covariates, in any combination:

  sfm(y ~ x1 + x2 | z_u, vhet = ~ z_v, model_name = "NHN_Z")
  sfm(y ~ x1 + x2, muhet = ~ z_mu, model_name = "NTN")   # Battese-Coelli 1995

  Named formulas rather than further pipe segments, because pipe POSITION already means different things in different families – the second segment is sigma_w in ttsfm() but sigma_h in psfm()’s GTRE_Z – and a fourth position would not be readable. The existing | z keeps its meaning, so nothing that worked before changes. uhet is the same specification as | z and refuses if both are given; it exists so that "NTN", which takes no pipe segment, can still have a heteroskedastic sigma_u.

  Available for NHN, NHN_Z, NE, NE_Z and NTN. muhet requires NTN, the only family in which a pre-truncation mean exists. Recovery on simulated data with all three blocks non-constant, n = 1500: every one of the six parameters within one standard error of truth, and the two z-links agree to 1e-6 in log-likelihood with deltas differing by exactly the factor of two the reparameterization implies.

  Two reasons this matters beyond completeness. Ignored heteroskedasticity in v biases the estimated FRONTIER, not just its precision (Caudill, Ford and Gropper 1995), because the composed error’s mean depends on the scales. And vhet is one of the few specifications that can absorb apparent wrong skewness without forcing sigma_u onto the zero boundary.

  These fits carry every positive quantity on the log scale, so the parameter vector is unconstrained and no variance can be driven onto a boundary by the optimizer. marginal_effects() reads them with no extra arguments. estimator = "cols" and the robust divergences are moment-based and homoskedastic respectively; both refuse rather than silently ignoring the new arguments.

• print() and summary() now say when a variance scale is on the zero boundary, for sfm()’s sigma_u under wrong skewness, for "tHN"’s sigma_u absorbed by heavy-tailed noise, and for psfm()’s two persistent scales. Each says which of those it is, since they mean different things.

  The condition was already warned about at fit time, but a warning is a one-off: it does not survive saveRDS(), a fresh session, or a loop that suppressed warnings to keep the console readable. What people look at afterwards is the printed table, and there a collapsed scale appears as an ordinary coefficient of 0.0009 with nothing to say that it is a boundary solution, that the split it belongs to is unidentified in this sample, or that the intercept moved to absorb the difference.

• psfm_bootstrap() warns when asked to bootstrap a boundary fit. A parametric bootstrap resamples from the fitted model, so a fit whose persistent scale has collapsed resamples from a data-generating process in which that component is absent – and the bootstrap is inconsistent on the boundary of the parameter space in any case. The intervals would understate the uncertainty in the persistent split rather than represent it, while looking perfectly ordinary.

  Since one of GTRE’s two persistent scales collapses in roughly a third of samples, this is expected to fire in ordinary use. That is the point.

• psfm(model_name = "GTRE") now reports a persistent scale that has collapsed to zero, in $sigh_at_bound and $sigr_at_bound and in a warning naming the surviving scale. Under both estimators: "fiml", which is the default and packs its results in a separate branch, and "sml".

  GTRE has two persistent components and the likelihood cannot always separate them in a given sample. When it cannot it merges them: one goes to zero, the other absorbs its variation and comes back inflated. On simulated data with both genuinely present, one of the two collapsed in 37% of 87 replications, and sigr -> 0 was three to five times commoner than sigh -> 0.

  Whether firms or periods help depends on the estimator, and the two answers are opposite. Crossed (N, T) designs on the same DGP, regressing the collapse rate on log N and log T:

  estimator	fits	log N	log T
  "fiml" (the default)	600	-1.66 (0.18)	-0.93 (0.20)
  "sml"	251	-0.68 (0.26)	-1.07 (0.27)

  Under the default, firms buy about 1.8 times what periods do; under simulated ML it is the other way round. Restricting the "fiml" design to N <= 200 – cell for cell the grid the "sml" one used – leaves it at -1.64 (0.22), so this is the estimator and not the design; the difference in the N coefficient is significant (z = 2.8) and the difference in the T coefficient is not.

  So the practical advice under the default is add firms: at T = 4 the collapse rate falls 68%, 36%, 18%, 6% across N = 50, 100, 200, 400 and reaches 0% at N = 400, T = 16. Under "sml" that same row is flat (60%, 39%, 41%), which is why a short panel is a dead end there however many firms it has. ?psfm now gives both.

  A user-facing consequence: you can tell in advance. The quantity that governs whether a panel supports the split is not sigma_h/sigma_r – that only decides which scale dies – but the persistent variance in the firm means against the transient variance still in them, [sigr^2 + sigh^2 (1-2/pi)] / ([sigu^2 (1-2/pi) + sigv^2] / T). Across six designs varying all four scales it orders the collapse rate perfectly (Spearman -1 under both estimators; logistic coefficient on log(S/N) of -1.15, SE 0.20): 54% of fits merge at S/N = 0.67, 31% at 1.30, 6% at 11.7. Below about 1, expect half your fits to merge; it takes roughly S/N > 5 to get under 10%. ?psfm and the vignette give the formula and a helper.

  One limit, stated because it is easy to over-read: those elasticities were measured at sigma_r = 0.20 against sigma_h = 0.40. On a design where the random effect dominates instead (0.50 against 0.40, N = 100, T = 6) the level ordering reverses – "fiml" collapses on 33% of replications and "sml" on 20%. Which estimator handles the split better depends on how much persistent inefficiency there is to find, not on the estimator alone.

  This is usually the CORRECT maximum likelihood estimate, not a failure: on one such replication the boundary solution beat the true parameter vector by 3.66 log units. It is the panel counterpart of the cross-sectional wrong-skewness result (Waldman 1982). So it is reported rather than prevented – bounding either scale away from zero would corrupt exactly the samples where the boundary is the answer.

  Your intercept moved with it, and the warning now says so. One rule covers what a collapse does: it conserves the persistent variance and relocates the persistent mean into the intercept. Those are two different destinations – h is half-normal, carrying both a mean E[h] = sigh*sqrt(2/pi) and a variance Var(h) = sigh^2 (1 - 2/pi), and the model has a separate place for each: the intercept takes the mean, since E[y] = beta0 - E[h] - E[u], and the surviving persistent scale takes the variance. So the survivor does not come back at sqrt(sigh^2 + sigr^2); that shortcut predicts 0.447 where the answer is 0.313.

  	surviving scale	intercept
  sigh -> 0	sqrt(sigr^2 + sigh^2(1-2/pi)) = 0.3133	-E[h] = -0.3192
  sigr -> 0	sqrt(sigh^2 + sigr^2/(1-2/pi)) = 0.5197	+0.0955

  Measured over 1000 replications at each of five sample sizes: 0.2973, -0.3035, 0.5058, +0.0931 – within 2.5% to 5.1% with nothing fitted, and all four slightly below prediction, which is what selection produces rather than a wrong constant. The practical consequence is that a collapsed fit does not only mislabel the persistent split; it displaces any level read off coef() by a known and substantial amount.

  And it is wrong skewness, measured between firms. Average the OLS residuals within each firm – that average is the persistent part of the error – and look at their skew. Over 1000 replications the samples whose between-firm skew came out the wrong way for a production frontier lost sigma_h to the boundary 69% of the time against 3.6% for the rest, with an ROC area of 0.955 for that one statistic, and not one of them collapsed sigma_r instead. So a collapsed sigma_h is not an accident to retry with different starting values: the sample carries no between-firm evidence of persistent inefficiency and the estimate says so.

  Read the sign, not a p-value. The firm mean of residuals contains -ubar_i as well as -h_i, and the mean of T half-normals is skewed the same way, so the between-firm skew leans negative even at sigma_h = 0 – 82% of the time at T = 4, 63% at T = 10, 53% at T = 30. skewness_test() is therefore not applicable to between-firm residuals as it stands, and ?psfm says so.

  psfm() also gains keep_objective, which retains the likelihood for the simulated-ML panel models so it can be evaluated away from the optimum. That is what distinguishes a weakly identified parameter from a badly estimated one, and it is how the above was established. ?psfm has a worked example.

• Robust and clustered standard errors, via sandwich. vcov() returned the inverse Hessian and nothing else, which is valid only if the likelihood is correctly specified and the observations are independent. Neither is safe to assume in applied frontier work, and clustered errors are routine on firm-level panels.

  bread() and estfun() methods are now registered for "sfareg" when sandwich is installed, which is all that package needs:

  fit <- sfm(y ~ x1 + x2, model_name = "NHN", data = d, keep_objective = TRUE)
  lmtest::coeftest(fit, vcov. = sandwich::vcovCL(fit, cluster = d$firm))

  Two requirements, both of which error clearly rather than returning something wrong. The fit needs keep_objective = TRUE, since the scores are differenced from the retained likelihood; and the robust divergence estimators are excluded, because they do not maximise a log-likelihood and so have no score. sandwich::vcovHC() does not work and is not expected to – its corrections are built from hat values, which are undefined for a nonlinear likelihood whose parameters include variance components.

  The scores are central differences with a per-parameter step. Validated two ways at the optimum: against NHN’s hand-derived analytic gradient (agreeing to 3e-07) and against numDeriv (2.9e-07), with the column sums vanishing relative to their own scale, which is the identity that makes them scores.

  sandwich and lmtest are in Suggests; neither is required to use the package.

• New function skewness_test(): a p-value for wrong skewness. sfm() already detected the condition – $wrong_skew, $sigma_u_at_bound, and a warning – but could not say whether the skew was wrong by more than sampling noise. Wrong skew is the single most common reason an applied SFA fit is meaningless, so that gap mattered.

  Two tests: D’Agostino’s (1970), which is the one Schmidt and Lin (1984) appeal to, and Coelli’s (1995) M3T. Returns an ordinary "htest", so it prints like t.test() and carries m3, nobs and wrong_skew alongside the usual fields. Takes a fitted sfm() object – using the OLS residuals the fit recorded, which is what the tests are defined on, rather than the composed residuals – or a bare numeric vector.

  "agostino" is the default because it holds its size where the asymptotic form does not. One-sided at a nominal 5%, 4,000 replications of symmetric residuals:

  n	25	50	100	400
  coelli	0.031	0.037	0.043	0.050
  agostino	0.045	0.046	0.046	0.051

  Coelli’s is conservative below about n = 100 and the two agree by n = 400. The D’Agostino implementation agrees with moments::agostino.test to within 1e-10 at n = 30/60/200/1000.

  sfm() fits now also carry $ols_residuals, which is what makes the test exact rather than re-derived from the recorded call.

• lcsfm() – the latent class stochastic frontier (Greene 2005; Orea and Kumbhakar 2004; Caudill 2003), a new exported entry point alongside sfm(), psfm(), zsfm() and ttsfm(). J distinct technologies coexist in one sample, each with its own frontier beta_j and its own sigma_v,j and sigma_u,j, and which one a firm operates is unobserved. Class membership follows a multinomial logit with class J as the reference, constant under "LCM" and parameterized by the second formula segment under "LCM_Z". The number of classes is n_class, default 2.

  Every firm contributes to every class weighted by its class probability, so this is not the same as fitting separate frontiers to known subgroups – the groups are not known, and the uncertainty about them is carried into the efficiency predictions. post.prob is the n x J matrix of posterior class probabilities, jlms_class the class-conditional JLMS scores, and jlms their posterior-weighted average.

  It is its own function rather than a model_name inside zsfm(). The two are related – zsfm()’s "ZISF" is the restricted two-class case in which one class has no inefficiency at all – but a user looking for a latent class model would not think to look inside a function named for zero inefficiency, and n_class means nothing to ZISF, whose two components are not free.

  Recovery on simulated two-class data, n = 800, true shares 0.60/0.40: beta_1 = (1.00, 0.48, 0.51) against a true (1.0, 0.5, 0.5), beta_2 = (2.97, 0.98, 0.21) against (3.0, 1.0, 0.2), scales 0.51/0.98 against 0.5/1.0, fitted shares 0.598/0.402, and 92% of firms assigned to their true class. On three classes at n = 1200 the figure is 95.5% after matching labels.

  Class labels are identified only up to permutation. Starting values come from splitting the OLS residuals at their J-quantiles and refitting within each group, which makes a run reproducible on the same data and avoids the saddle point that identical starting components would sit at, but it does not pin labels to any external ordering. Match classes before comparing two fits. The likelihood-ratio test of J against J+1 is not valid here – the null puts a class probability on the boundary – so use AIC()/BIC() and interpretability instead.

• psfm() gave every firm the SAME simulation draws. A single R x 2 Halton block was built once and recycled across all N firms, in three separate places. sfm() already does the opposite and says why: Train (2002, p. 228) attributes Halton’s advantage to its coverage and to the negative correlation it induces across observations, and the second half only exists when observations get different blocks. Sharing one block also makes every firm’s simulation error the same realization, so it cannot average out as N grows. Firm i now gets its own contiguous block.

  Measured, GTRE by simulated ML, 30 replications at each of N = 50/100/200, old and new on identical seeds so the comparison is paired: sigh improves on 62 of 87 pairs (Wilcoxon p = 0.015) and sigr on 48 of 87 (p = 0.048), while beta1 – the control, since the frontier slope is not what the draws integrate over – is unaffected at 46 of 87 (p = 0.41). Collapses of sigh to exactly zero halved, 11/87 to 6/87.

  The improvement grows with N, which is the point: 16/28 at N = 50 (p = 0.58), 21/29 at N = 100 (p = 0.25), 25/30 at N = 200 (p = 0.012). That is the shape the mechanism predicts.

  It does not fix the GTRE convergence failure. RMSE(N=50)/RMSE(N=200) for sigh moves from 0.94 to 1.08 where root-n requires 2.00. Shared draws were a contributor, not the cause.

• rand.gtre randomized the draws in a way that destroyed what they were for. The old code drew 9999 random permutations of Halton dimension 1 and kept whichever correlated least with dimension 2. A two-dimensional Halton sequence is worth having because the pairs cover the unit square evenly; permuting one column preserves each margin, randomizes the pairing, and leaves joint coverage no better than random. Measured at R = 150 with primes 2 and 3 and 1000 discarded: joint discrepancy 0.0217 before the shuffle and 0.0550 after, against 0.1400 for purely random pairing – to remove a correlation of 0.008 that was never a problem.

  rand.gtre now applies a uniform shift modulo 1 (Tuffin 1996; Train 9.3.4), which moves the lattice without disturbing its structure: same measurement, 0.0250. Results change for anyone who passed rand.gtre.

• psfm(model_name = "KSS") – Kneip, Sickles and Song (2012). The firm effect is a smooth function of time lying in an L-dimensional space whose basis is estimated from the data: alpha_it = sum_r theta_ir * g_r(t). Nothing on CRAN implements this, so it is a differentiator rather than catch-up.

  It makes explicit what the other time-varying estimators are special cases of: SSFE is L = 1 with a constant basis, LS is L = 1 with the basis free, CSS is L = 3 with the basis fixed to {1, t, t^2}, and KSS estimates both. Follows the original’s three steps – cross-sectional centering by period, then smoothing of each firm’s residual trajectory followed by an eigendecomposition of their empirical covariance, then loadings by least squares – iterated with beta. Balanced panels only, which is what the estimator is defined on; it refuses rather than quietly fitting something else.

  Recovers the true rank exactly at L = 1, 2, 3 on simulated data (n = 110, T = 10), with the estimated basis spanning the true factor space and sigma_v recovered to within 3%.

  Both tuning choices are exposed. kss_smooth defaults to GCV; kss_L defaults to the Bai and Ng (2002) IC_p2 criterion, which is not the original paper’s threshold rule – said plainly in ?psfm because a different rule can select a different L. $kss$eigenvalues is returned so the choice can be inspected.

  The automatic search is capped at floor(T/2), and warns if it selects that cap. IC_p2 works because the residual variance flattens out once the real factors are in; on a short panel, letting L reach T - 1 lets the factors span nearly the whole time dimension, so the variance collapses and no penalty competes. Measured at n = 100 over five seeds and two true ranks, an uncapped criterion returned its maximum on all ten designs at T = 6 and again at T = 8. With the cap it recovers the true rank exactly from T = 8 upward. An explicit kss_L is the user’s own call and is not subject to the cap.

  One trap worth knowing: KSS’s period centering means its alpha_hat carries no period mean while CSS’s and LS’s do, so the two differ by a per-period constant by construction. Correlating them directly understates the agreement badly (0.93 against 0.99 on a design where they agree). Compare u_hat, which is a within-period contrast and invariant to the shift.

• psfm(model_name = "SSRE") and "SSCRE" complete the Schmidt-Sickles

  (1984) family that SSFE starts: the GLS estimator, and its correlated-random-effects correction. SSRE is more efficient than SSFE and identifies time-invariant regressors, but assumes the effects are uncorrelated with the regressors – in a frontier model, that a firm’s inefficiency is unrelated to its input choices.

  SSCRE adds the within-firm means of the time-varying regressors (Mundlak 1978), which models that correlation rather than assuming it away. Its slopes then equal SSFE’s within slopes exactly, which the tests pin at 1e-6; the coefficients on the added means (reported with a .mean_ prefix) are the Mundlak form of the Hausman test.

• Two classical time-varying panel estimators: psfm(model_name = "CSS") and psfm(model_name = "LS"). Cornwell, Schmidt and Sickles (1990) and Lee and Schmidt (1993). Both sit between "SSFE", which holds inefficiency fixed over time, and the ML panel models, which let it move only by assuming a distribution for u and a decay path for it: these let the firm effect itself vary with time and read inefficiency off it, assuming no distribution at all.

  They differ in what they spend on flexibility. CSS gives every firm its own quadratic in time (3N parameters; firms may overtake one another; T_i >= 4 required before a firm contributes to beta). LS imposes one common temporal pattern scaled per firm, alpha_it = delta_t * alpha_i (N + T - 1 parameters; the ranking of firms cannot change, only the spread). Cross-over versus no cross-over is the substantive choice between them.

  CSS residualizes on each firm’s own (1, t, t^2) rather than building 3N dummy columns, and refuses a regressor spanned by those quadratics (a pure time trend, say) instead of reporting it as a small coefficient. LS is a rank-one factor model and is fitted by alternating least squares, so an unbalanced panel needs no special-casing. On simulated data at N = 90, T = 8, LS recovers the temporal pattern with correlation > 0.99 to truth and the firm scales at 0.995; both recover the frontier slopes and the noise standard deviation.

  Neither is maximum likelihood, so neither carries an $opt component and logLik()/AIC()/BIC() return NA with a warning, as for SSFE.

• The wrong-skew boundary report fired on a knife-edge. sfm() flags sigma_u_at_bound when the one-sided scale collapses relative to the residual SD, and the threshold was 1e-3. On the design used to test it the collapsed sample sits at 8.6e-4 and the interior ones at 0.47-0.71, so the threshold cleared the case that matters by a factor of 1.16 and the others by 470. An optimizer’s stopping point moves by more than 16% between BLAS implementations: the same fit reported TRUE locally and FALSE on another platform. Now 1e-2, which sits in the middle of a gap spanning nearly three orders of magnitude and leaves every sample at least 11x clear.

  This only ever changes the SECOND of two conditions – the warning also requires wrong skew (m3 >= 0), which is a deterministic function of the data – so the practical effect is that the report is stable across platforms rather than that it fires more often.

• nobs() counted rows SUPPLIED rather than rows USED, so BIC() was computed against the wrong n whenever any row was dropped for missingness. nobs.sfareg() re-evaluated the data argument of the recorded call, which of course returns the full frame; data_proc2() has always dropped incomplete cases before fitting. On a 300-row frame with 7 missing values nobs() returned 300 against 293 actually used. The fitted object now records the effective count directly, and nobs() falls back to the length of a stored per-observation vector before it resorts to re-evaluating the call.

• sfm(model_name = "NE") could return a positive log-likelihood with sigma_u = 0 and a divergent sigma_v. The log-density was built as -log(sigma_u) + log Phi(z) + eps/sigma_u + sigma_v^2/(2 sigma_u^2). Using pnorm(log.p = TRUE) for the middle term is correct as far as it goes, but that term and the tilt both diverge like z^2/2 with opposite signs as sigma_u -> 0, and their sum is a catastrophic cancellation. At sigma_v = 587, sigma_u = 1e-7 the two are -/+1.7252e19, where consecutive doubles are 2048 apart, so the sum came back as rounding noise – positive noise, which the optimizer then maximized by running sigma_u to its lower bound.

  This was not rare and it was silent: scanning 150 samples at lambda = 0.75, N = 100, one returned sigma_v = 587.4, sigma_u = 0, logLik = +468992, and another sigma_v = 3.4e15, logLik = +7.9e30, both as ordinary sfareg objects with no error and no warning. Across a 12-cell design at 1,500 replications the rate was 0.74%, reaching 4.4% at lambda = 0.5, N = 100.

  New .log_phi_tilt() (matrix_utils.R) does the cancellation analytically instead. Because z^2/2 = eps^2/(2 sigma_v^2) + eps/sigma_u + sigma_v^2/(2 sigma_u^2), the tail expansion of log Phi(z) cancels the tilt exactly and leaves only -eps^2/(2 sigma_v^2) - log(2 pi)/2 - log(-z) + log1p(-1/z^2 + 3/z^4 - ...), in which no large intermediate is ever formed. It agrees with the previous expression to 8.5e-13 over 75,000 evaluations wherever the previous one was trustworthy, integrates to 1, and takes this failure rate to 0 in 2,200 fits.

  To be precise about what that last number does and does not cover: it is the rate of the catastrophic-cancellation failure – a positive log-likelihood with sigma_u at its bound – and nothing else. It is not a claim that NE fits are uniformly well behaved. convergence/CONVERGENCE_STATUS.md records a separate and still-open defect at the other end of the parameter space: at lambda = 3.33 the root-n sweep shows a tail of 2-4% of replications landing far from the truth on some parameter, and unlike the rest it does not shrink with n. There the scales are near the truth and the frontier SLOPES break, with beta1 and beta2 trading against each other, so it is a different failure from this one and is not addressed by this fix. The two live in different regions – cancellation bites at LOW lambda, the tail at high – and the median fit is unaffected in both.

• "NGE" carried the identical defect and is fixed the same way. Its likelihood is a difference of two exponentially tilted Gaussians, both with the same structure as "NE"; both now go through .log_phi_tilt(), in the likelihood and in the post-estimation efficiency block. Agreement with the previous expression is 7.1e-10. "TSL", "TTNE" and "NE_Z" share the structure and have not been checked.

• "NE" starting values now come from a bias-corrected moment estimator (new R/ne_start.R) rather than the flat sigma_u = sigma_v = 0.1 that start_cs() hands every cross-sectional model. Minus the mean negative OLS residual has an exact asymptotic bias factor h(lambda); .ne_start() divides it out, shrunk by the share of MSE the bias accounts for. Over an 88-cell design at 2,000 replications, total MSE of the start against the truth: 5.49 for the uncorrected version, 2.86 for the third-moment (COLS) inversion, 1.88 for their minimum, 1.82 for this one. Effect on the fitted MLE is smaller and confined to lambda >= 1.5, where it lowers MSE of sigma_u-hat by 5-51%; below lambda = 1 it changes nothing. Derived in Bernstein, Parmeter and Wright, “Starting Values for the Normal-Exponential Stochastic Frontier Model”.

  Note this leaves the flat 0.1 start in place for every other cross-sectional model, which PROJECT_STATUS.md still lists as open.

• sfm() now reports when sigma_u sits on the zero boundary under wrong skewness, instead of saying nothing. A one-sided scale at zero when the OLS residuals are skewed the wrong way is the correct maximum likelihood estimate – the Type I failure of Olson, Schmidt and Waldman (1980) – not a numerical problem. The estimator = "cols" path has warned about this for some time; the likelihood path did not, so a user got a boundary fit with no explanation.

  New $wrong_skew, $sigma_u_at_bound and $residual_m3 components, plus a warning that says what the boundary means and that the efficiency scores are uninformative there. Measured at lambda = 0.75, N = 100 over 600 fits: 13.2% of samples are wrongly skewed, 17.7% of those put sigma_u on the boundary, and not one correctly skewed sample does. That is why the fix is to report the boundary rather than bound sigma_u away from zero – a bound would corrupt precisely the samples where the boundary is the answer.

• "NE" no longer emits a stream of NaNs produced warnings. Its likelihood had no guard against a non-positive scale, so the optimizer probing sigma_u <= 0 evaluated log(sigma_u) and warned – 17 times in a single fit on a wrongly skewed sample, burying the warning that mattered. It now returns a large finite penalty for out-of-domain parameters, as "NLN" and "NW" already do. No estimate changes: this bounds the objective’s domain, not the estimate.

• "NGE" aborted outright on about 7% of small samples, and "NU" could have. Both guarded their scale parameters with return(.Machine$double.xmax). optim() differences the objective to form a gradient, and differencing 1.8e308 overflows to a non-finite value, so the fit died with non-finite finite-difference value instead of the optimizer being steered away. Measured before the change: 3 of 45 "NGE" fits at N = 150 failed this way, including at sigma_u = 1, sigma_v = 0.3. Both now use the same finite penalty the other branches use – 0 of 45 failures after, and "NU" estimates are bit-identical across 45 fits, since the penalty only ever applies where the parameters are already inadmissible.

• ttsfm() gains z_link too, completing the cross-sectional half of the variance-determinant inconsistency. Two-tier models placed z'delta on the standard deviation, like sfm() and unlike psfm() and the competing packages; z_link = "var" aligns them. The default is "sd", and fits under it are byte-identical to before – checked against the previous build with the particle swarm seeded, for both "TTNE" and "TTHN".

  Both tiers move together: delta on zu and on zw each halve exactly between the links, while the maximised log-likelihood and the frontier coefficients are unchanged.

• marginal_effects() now works on panel fits, psfm(model_name = "TRE_Z") and psfm(model_name = "GTRE_Z"). These place z'delta on the variance, and the link is read from the fit rather than assumed, so their effects are directly comparable with an sfm() fit – including one fitted with the new z_link = "var".

  "GTRE_Z" separates persistent inefficiency from transient, and both are reported: component = "u" for the transient block and component = "h" for the persistent one. Each block is located by name rather than by position, because the sigma_h coefficients are the trailing ones – a positional rule would have silently reported sigma_h effects under a sigma_u label. Output columns carry the component (dE_u.dz against dE_h.dzp), so a table cannot be misread once separated from the call that produced it. The default is "u", so nothing existing changes.

• sfm() gains z_link, which fixes a real trap in comparing _Z fits across the package. sfm()’s "NHN_Z"/"NE_Z" put the variance-determinant linear predictor on the standard deviation, sigma_u = exp(z'delta), while psfm()’s "TRE_Z"/"GTRE_Z" put it on the variance, sigma_u = sqrt(exp(z'delta)) – as do the competing packages. Since exp(eta) = exp(eta/2)^2 the two fit the same model and return the same sigma_u, the same log-likelihood and the same marginal effects, but delta under the SD link is exactly half delta under the variance link. Reading a delta from one family as if it came from the other therefore doubles or halves every reported effect.

  z_link = "var" puts an sfm() fit on the same footing as psfm() and sfaR. The default is "sd", so no existing result changes. The efficiency predictor and the z_spec that marginal_effects() reads both follow whichever link was used.

• sfm()’s simulated-ML models "NLN" and "NW" now estimate the composed density from two proposals at once, which fixes an accuracy defect that no single-proposal scheme can. The integrand is a product of a normal kernel of width sigma_v centred at u = -eps and the inefficiency density. Drawing from the inefficiency quantile – what both models did through 1.1.5 – misses the kernel whenever it is the narrow factor: at eps = -4.7 with sigma_v = 0.3 the spike needs a draw beyond the 0.9999 quantile, while 400 draws reach only about 0.9975, and five observations in 800 carried 72% of "NW"’s total error. Drawing from the noise instead cures that case and breaks the mirror one, wherever the inefficiency density is the narrow factor.

  Both draws are now taken – half the count each – and combined by the balance heuristic of Veach and Guibas (1995), each draw weighted by which proposal was likelier to have produced it. There is no selection rule to get wrong. Total error against adaptive quadrature, over an 80-cell parameter grid on a 300-observation sample, restricted to the cells an optimizer can reach:

  cells within … of the optimum	1.1.5 scheme	now
  "NW", 200	220, worst cell 172	16, worst cell 5.7
  "NW", 500	3633, worst cell 1910	40, worst cell 11
  "NW", 1000	17900, worst cell 8553	128, worst cell 64
  "NLN", 200	233, worst cell 120	11, worst cell 1.0
  "NLN", 500	3564, worst cell 1553	49, worst cell 8.8
  "NLN", 1000	15962, worst cell 6639	205, worst cell 63

  (Each scheme is measured at its own draw rule, so "NW"’s 1.1.5 column uses the 400 draws that version would have taken and the new column 200.)

  "NW"’s draw rule drops from max(400, 8*sqrt(n)) to the max(200, 3*sqrt(n)) "NLN" already used, so at n = 1500 an "NW" fit costs about what it did in 1.1.5 (roughly 50 seconds) while being far more accurate; "NLN" costs about 1.5x more, since its draw count did not fall. Nsim still means the total number of draws.

  The two models now also converge at the same rate, so the old advice that "NLN" needs far more draws than "NW" no longer applies – that was a property of the integration scheme, not the model. At n = 3000, measured against quadrature at the true parameters, total simulation error is 3.4 log-likelihood units for "NLN" and 3.0 for "NW" at the default, falling to 0.07 and 0.06 at Nsim = 6400. The comparable figure for the old scheme was 226.8.

  The accuracy gain shows up in the estimates. Over 12 replications at n = 800, bias and RMSE against the true parameters:

  	sigma_v	sigma_u	shape	x1
  "NW" 1.1.5	+0.036 / 0.066	-0.149 / 0.237	-0.160 / 0.293	-0.021 / 0.043
  "NW" now	-0.006 / 0.044	-0.024 / 0.112	+0.001 / 0.111	+0.001 / 0.045
  "NLN" 1.1.5	-0.012 / 0.022	+0.005 / 0.101	+0.012 / 0.166	+0.003 / 0.037
  "NLN" now	-0.004 / 0.035	+0.003 / 0.080	+0.005 / 0.131	+0.002 / 0.038

  "NW" is the clear case: its sigma_u and shape bias essentially vanish and their RMSE roughly halves. "NLN" improves more modestly, and its sigma_v RMSE is slightly worse – at 12 replications that difference is within noise and should not be read either way.

  The efficiency predictor reuses the weights the likelihood computed, so u_hat cannot drift away from the density that was maximised.

• sfm() gains user control over its simulated-ML draws, following the practice in Train (2002, ch. 9) and matching the options sfaR exposes: sim_type ("halton", "sobol", "torus", "uniform"), antithetics, sim_burn, sim_scrambling, sim_prime and sim_seed. These affect the models fitted by simulation, "NLN" and "NW".

  "sobol" with sim_scrambling 1-3 is the scrambled sequence of Bhat (2003), which removes cross-dimension correlation while keeping the coverage. "uniform" exists as a baseline to measure against, not as a recommendation – Bhat (2001), quoted by Train, found 100 Halton draws more precise than 1000 pseudorandom ones. antithetics = TRUE takes half the draws from the sequence and creates the rest as mirror images (Hammersley and Morton 1956), which costs nothing.

  Defaults reproduce the previous behaviour exactly – Halton, no antithetics, 1000 leading elements discarded – and this is asserted by a test that rebuilds the old construction and compares byte for byte, so no existing result moves.

• The draw construction is now one shared internal function, .sml_draws(), rather than being written inline at each site. That matters beyond tidiness: the sites had drifted apart, with sfm() giving each observation its own contiguous block of the sequence while psfm() hands every firm the same one. Train (ch. 9) attributes the value of a low-discrepancy sequence to its coverage and to the negative correlation it induces across observations, and the second only exists when units get different blocks. The shared constructor always blocks by unit and takes a dim argument for the multi-dimensional panel case, so the other entry points can adopt it directly.

• New model sfm(model_name = "TSL"): the normal / truncated skew-Laplace frontier (Wang 2012). Inefficiency has the signed-mixture density f(u) = ((1+lambda)/(sigma_u(2*lambda+1))) [2 exp(-u/sigma_u) - exp(-(1+lambda) u/sigma_u)] on u >= 0 – the second exponential enters negatively – so it nests the exponential model as lambda -> 0 while allowing a non-monotonic inefficiency density. Reports (sigv, sigu, lambda) plus the frontier, with both u_hat and exp_u_hat.

  The composed density is evaluated as a difference in log space rather than by subtracting the raw terms. Both carry sigma_v^2/(2 sigma_u^2), which overflows once sigma_u is small relative to sigma_v – on a routine grid spanning sigma_u in {0.05, 0.1, 0.3} and sigma_v in {0.3, 1} the direct form returns NaN or Inf at 9 of 30 points, including sigma_u = 0.05, sigma_v = 1, eps = 0, which is not an extreme point. The log form is finite throughout.

  Verified two ways: against numerical convolution of the implied u density with the normal noise (agreeing to 8 decimals across lambda in {0.5, 1.5, 4}), and against an independent implementation, which it matches to an identical log-likelihood and identical coefficients on 8 of 8 samples. lambda is a shape parameter and is the least sharply identified of the three – read its t-value before interpreting it.

• data_gen_cs() gains lam_tsl and the matching u_tsl / y_pcs_tsl columns. The signed mixture cannot be drawn by picking a component, so it is drawn by rejection off its first exponential; the acceptance probability never falls below 1/2.

• New function efficiency_ci(): Horrace and Schmidt (1996) confidence intervals for individual inefficiency. sfa has always reported point predictions of u_i – u_hat (Jondrow et al. 1982) and exp_u_hat (Battese and Coelli 1988) – and nothing about how sharply either is pinned down. Both are posterior means of u given the composed residual, and the posterior is a truncated normal in closed form, so the interval costs no estimation beyond the fit itself. Returns a data frame of u_lower/u_hat/u_upper and the corresponding technical-efficiency bounds, at any level.

  Available for sfm()’s "NHN", "NHN_Z", "NE" and "NTN", whose posterior really is a truncated normal. Other models have a posterior of a different shape, and the function says so by name rather than returning a number the formulas do not support.

  These intervals condition on the fitted parameters: they do not narrow as n grows, because they measure the irreducible difficulty of splitting one residual into noise and inefficiency, not estimation uncertainty. That is worth knowing before reading an efficiency ranking closely – on a routine 200-observation half-normal fit the median 90% interval for a single unit spans about 0.49 in efficiency, against a total spread of 0.85 across all the point predictions.

• npsfm(method = "SZ") now solves its DEA step in the package. The output-oriented envelopment program is computed directly, as one linear program per unit, by the new internal .dea_out(), covering all four returns-to-scale settings. Suggests gains lpSolve, a general-purpose linear-programming solver (pure C, no system libraries).

  Results are unchanged, and were verified against an independent reference implementation over all four returns-to-scale settings at n = 30, 80 and 150, with one to three inputs and one to two outputs: agreement to 3e-12 or better throughout.

• New function marginal_effects(): what the _Z models’ delta coefficients actually mean for inefficiency. For a fit whose inefficiency scale depends on covariates, it returns the per-observation d E[u]/d z_k and d Var[u]/d z_k, with the average marginal effects attached. These are what applied papers report; a delta on its own is a coefficient in a log link for a scale parameter and is not interpretable in the units of either u or z.

  This also defuses a long-standing trap. sfm()’s "NHN_Z"/"NE_Z" put the linear predictor on the standard deviation and psfm()’s "TRE_Z"/ "GTRE_Z" put it on the variance, so the same delta means different things in the two families – the marginal effects differ by a factor of two. The effect is on the scale of u either way, so reporting it rather than the coefficient removes the ambiguity. The returned object records which convention was used.

  Currently covers sfm()’s "NHN_Z" and "NE_Z"; psfm()’s panel _Z models are not wired up yet. Standard errors are deliberately not reported – see ?marginal_effects for why.

• "NHN_Z" and "NE_Z" fits now carry a $z_spec component (the variance-determinant design, its coefficients, and the link convention), which is what marginal_effects() reads.

• sfm(model_name = "NNAK") now starts from the method of moments. The normal-Nakagami likelihood has sigma_u -> 0 as a genuine attractor, and the previous hard-coded start of sigma_u = sigma_v = 0.1 sat next to it. Over twelve samples at n = 3000 with a true sigma_u of 1, the old start drove sigma_u to 0.0013 and to 0.0000 on two of them – inefficiency vanishing altogether, with the frontier intercept pushed negative – for a log-likelihood 44 and 45 points worse than the new start reaches. The new start was never worse on any of the twelve and was strictly better on seven, for a mean gain of 7.9 log-likelihood points.

  The construction follows FronPy (Stead 2024, Journal of Productivity Analysis, the paper these closed forms come from): invert the half-normal moment equations for the two scales and shift the frontier intercept up by the implied E[u]. The half-normal is the right auxiliary because it is the m = 1/2 member of the Nakagami family. The shape still starts at 0.5, as before. When the residuals are skewed the wrong way and the moment equations have no admissible solution, the old constants are used as before.

• sfm(estimator = "mols") is accepted as a synonym for estimator = "cols". What the package computes under that name is the modified OLS moment estimator of Olson, Schmidt and Waldman (1980) – it inverts the second and third central moments of the OLS residuals – and not Winsten’s corrected OLS, which shifts the intercept by the largest residual and estimates no variance parameters. The documentation now names it MOLS and draws the distinction. "cols" is unchanged and not deprecated.

• Fitted "sfareg" objects from the four models above carry a new $u_posterior component (mu_star, sigma_star), which is what efficiency_ci() reads.

sfa 1.1.5

• The intro_to_psfm vignette now builds in a fraction of the time. At CRAN’s request, the models it fits are sized as toy illustrations rather than as estimation exercises: the simulated panel is 70 firms over 6 periods instead of 100 over 10, the simulated-ML fits draw 50 Halton points via halton_num rather than the default ceiling(sqrt(nrow(data))) + 100, and the psfm_bootstrap() example uses 5 replications on a 30-firm panel instead of 10 on 60. Vignette rebuild time falls by roughly a factor of five. The reported estimates therefore differ from previous versions, and the vignette now says plainly that they are not to be read as a serious fit.

• Every model fit in that vignette now sets rand.gtre and rand.psoptim. With PSopt = TRUE the particle-swarm stage draws from the session’s RNG, so the vignette’s results previously changed from one build to the next; they are now reproducible.

• No change to any R code, to NAMESPACE, or to the documented interface.

sfa 1.1.4

Breaking change

• psfm(model_name = "GTRE") now defaults to full information maximum likelihood. The four ways of fitting the four-component GTRE model were four separate model_name values, which made them look like four different models rather than four routes to the same one. They are now selected with an estimator argument, in the same spirit as sfm()’s estimator = c("mle", "cols"):

  • "fiml" (the default) – full information ML through the closed-skew-normal representation. Deterministic; requires a balanced panel.
  • "sml" – simulated ML over Halton draws. Handles unbalanced panels. This is what "GTRE" meant through 1.1.3.
  • "seq1", "seq2" – the two-step moment estimators.

  Scripts that pass model_name = "GTRE" therefore get a different estimator than they did, and are warned once per call. Pass estimator explicitly to silence it. The names "GTRE_FML", "GTRE_SEQ1" and "GTRE_SEQ2" are unchanged and still select the same three routes directly.

  On an unbalanced panel "fiml" cannot be fitted. Taking the default warns and falls back to "sml", because erroring would make "GTRE" unusable by default on a whole class of data; asking for "fiml" explicitly errors instead of silently fitting something else.

New models

• psfm(model_name = "PL80_MVTN") – Pitt and Lee’s (1981) Model III, the multivariate truncated normal panel likelihood from their Appendix 2. Inefficiency varies over time and is correlated within a firm: u_i = (u_i1,...,u_iT)' is drawn from a T-variate normal truncated to the negative orthant, where the existing "PL80" holds inefficiency fixed over time.

  Pitt and Lee derived this likelihood and then set it aside, writing that it “is difficult to evaluate since the quantities P0 and P(y_i - x_i beta) involve T-dimensional numerical integrals”, and estimating Model III by Zellner SUR instead. Those quantities are multivariate normal orthant probabilities; mnormt::sadmvn() evaluates one in about 3 ms at T = 6, so a likelihood evaluation costs roughly N+1 of them – about 0.3 s at N = 100, and about 12 s for a whole fit at N = 80, T = 4. What was intractable in 1981 is merely slow now.

  Sigma is parameterized as equicorrelated, sigma_u^2[(1-rho)I + rho 11'], costing two parameters. Unrestricted Sigma costs T(T+1)/2 – 21 at T = 6, 55 at T = 10 – every one identified only through orthant probabilities. The equicorrelated form captures what the general Sigma was for: dependence of a firm’s inefficiency across periods, with rho = 0 giving independence over time and rho -> 1 approaching the time-invariant "PL80". Because the form is equicorrelated, every matrix quantity is closed form (Sherman-Morrison, verified against brute force to 1e-17), so only the orthant probabilities are numerical.

  Requires a balanced panel with T >= 2; an unbalanced one errors and points at "PL80". data_gen_p() gains y_pl_mvtn and u_mvtn columns and a rho_mvtn argument to test it, generated last so the RNG stream feeding every existing column is untouched. Registered in the convergence framework as PL80_MVTN. See ?PL80_MVTN.

  Validated before wiring: profiling the likelihood one parameter at a time puts the minimum of the negative log-likelihood at the truth for all five parameters, and maximizing it recovers (sigma_v, sigma_u, rho) = (0.32, 0.78, 0.45) against a truth of (0.30, 0.80, 0.50) at N = 300.

• npsfm(), nonparametric stochastic frontier models. A fifth entry point, alongside sfm(), psfm(), zsfm() and ttsfm(), for frontiers whose shape is estimated by kernel regression rather than assumed linear. Two estimators:

  • method = "FLW" – Fan, Li and Weersink (1996). Fits E[y|x] nonparametrically, then recovers the scale parameters from the residuals, by maximizing their concentrated likelihood in lambda for dist = "hn" and by inverting central moments for dist = "exp", "gamma" and "unif".
  • method = "SVKZ" – Simar, Van Keilegom and Zelenyuk (2017). Three local-linear regressions give sigma_u(x) and sigma_v(x) pointwise, so both variance components vary with the covariates. No optimizer runs. Normal-half normal only.
  • method = "PSZ" (alias "KPST") – Park, Simar and Zelenyuk. Local maximum likelihood: the frontier and both log variance components get local-linear expansions and the kernel-weighted normal-half normal likelihood is maximized in those 3(k+1) parameters, once per observation.
  • method = "MY" – Martins-Filho and Yao. Iterative local likelihood, alternating local frontier fits with a global update of (lambda, sigma).
  • method = "SZ" – Simar and Zelenyuk (2011). Passes an existing smooth frontier through an output-oriented DEA to impose monotonicity and convexity.

  "PSZ" and "MY" run one numerical optimization per observation (for "MY", per observation per iteration), so they are one to two orders of magnitude slower than "FLW". Both are seeded from an "FLW" fit.

  Ported from Christopher Parmeter’s research scripts. Results return as class "npsfareg" rather than "sfareg": there is no parameter vector with standard errors, so coef(), vcov() and logLik() would have nothing to return. fitted(), residuals(), nobs(), print() and summary() are provided.

  Kernel regression comes from the np package, added to Suggests rather than Imports – nothing else in sfa needs it, and npsfm() checks for it and stops with an install instruction if it is absent.

  A correction worth recording, because it is easy to repeat: the two local-likelihood estimators maximize the composed-error likelihood, in which the local intercept is the frontier m(x) itself. They therefore take no half-normal mean shift, unlike the least-squares methods, whose first stage estimates E[y|x] = m(x) - E[u] and does need one. Applying the shift to "PSZ"/"MY" biases the whole frontier up by about E[u]; mean absolute frontier error at n = 300 fell from 0.372 to 0.116 ("PSZ") and 0.410 to 0.070 ("MY") once it was removed.

  Against a simulated nonlinear frontier with sigma_u = 0.6, sigma_v = 0.25, both least-squares estimators converge as n grows (6 replications at each size): FLW recovers sigma_u = 0.560, 0.539, 0.595 at n = 150, 300, 600, and SVKZ 0.477, 0.506, 0.559, with mean absolute frontier error falling from 0.102 to 0.043 and 0.206 to 0.079 respectively. SVKZ’s downward bias at small n is the wrong-skew floor: the share of observations whose local third moment has the wrong sign, and whose sigma_u(x) is therefore set to zero, falls from 21.6% to 0.3% over that range.

New methods and arguments

• sfa_diagnostics(), plot() for "sfareg", and convergence reporting. The numerical hardening was already in place – staged minimizer, clipping constants, analytic gradients where they exist – but nothing was reported back. A fit carried optim()’s convergence code, message, evaluation counts and Hessian, and print()/summary() showed none of it, so a fit that stopped on the iteration cap printed exactly like a converged one.

  sfa_diagnostics() returns the convergence code and what it means, the eigenvalue spectrum and condition number of the Hessian, whether it is positive definite, which parameters load on its flattest direction, the implied parameter correlations, and – with keep_objective = TRUE – the gradient at the reported optimum. plot() draws the Hessian spectrum, the correlation matrix, a likelihood slice per parameter, and the gradient. print() and summary() now report the convergence code.

  The code by itself is not diagnostic and is not treated as though it were. Across NHN, NE and NTN at n = 150, 500 and 1500, code 52 (“ABNORMAL_TERMINATION_IN_LNSRCH”) appears routinely alongside a maximum relative gradient of ~1e-6 and a positive definite Hessian: the staged minimizer had already converged and L-BFGS-B could not step away from the optimum. The same code on NTN at n = 150 came with a relative gradient of 5e+07 and an indefinite Hessian, a real failure. The verdict therefore combines the code with the gradient and the Hessian, and distinguishes benign from unverified (a line-search code with no objective retained, so no gradient to settle it) from failure. Code 1, the iteration limit, is never treated as benign.

  On a single NNAK fit the report reproduces what the convergence sweeps found only across replications: mu and sigu correlated at 0.998 – the documented ridge – with the flattest Hessian direction loading on exactly that pair.

  When the Hessian is singular enough that some parameter has no usable variance, the correlation report drops those parameters and names them, rather than vanishing as a whole – a diagnostic for ill-conditioning should be most informative exactly when conditioning is worst, not least.

• sfm(keep_objective = TRUE) stores the likelihood on the fitted object so the gradient and likelihood slices can be computed after the fact. Off by default: a closure carries its enclosing environment, so a fit saved with one serializes the estimation data too (about 38 KB to 1.7 MB on a 200-observation example). Everything else sfa_diagnostics() reports works without it.

New features

• Model names are matched without regard to case. match.arg() is case sensitive, so psfm(model_name = "gtre") used to fail with a list of valid names that visibly contained what the user had just typed. All five entry points now fold case, for model_name and for npsfm()’s method. No entry point has a case collision among its choices – sfm()’s "THT" and "tHN" differ in more than case – so the canonical spelling is always recoverable.

  Exact matches beat partial ones, which matters because "GTRE" is a prefix of four other names and must resolve to itself. Genuinely ambiguous partials ("GTRE_S", between GTRE_SEQ1 and GTRE_SEQ2) are still rejected rather than guessed at, and an unrecognized name now suggests the two closest valid choices instead of listing everything.

sfa 1.1.3

Breaking change

• psfm()’s optimizer iteration defaults have been raised, from maxit.bobyqa = 100, maxit.psoptim = 10, maxit.optim = 10 to 5000, 100 and 1000. maxit.nlminb is now an argument (default 500); it was previously hard-coded at 200 in the "GTRE_FML" branch and 500 elsewhere, and could not be set from the call.

  The old caps were binding rather than merely economical. The "K1990"/"K1990modified" code already carried a note that 100 bobyqa evaluations left its seven-parameter fits several log-likelihood units short of the optimum purely on the iteration cap, and "GTRE_FML" at N = 500, T = 10 roughly halves its root-mean-square error against known true values once the caps are lifted (0.0080 to 0.0037 and 0.0179 to 0.0073 on two draws), for about 1.5 times the run time.

  Existing scripts will get more accurate estimates and slower fits. Pass the old values explicitly to restore the previous behaviour.

• psfm(model_name = "TFE") now fits a different estimator. Through version 1.1.2 the name "TFE" selected Chen, Schmidt and Wang’s (2014) within maximum-likelihood estimator. It now selects Greene’s (2005) true fixed effects estimator, which is what the name means in the literature. The Chen-Schmidt-Wang estimator is unchanged and is now model_name = "TFE_WMLE".

  Scripts written against 1.1.2 or earlier that pass "TFE" will silently get a different estimator, so psfm() issues a warning whenever "TFE" is used. To reproduce earlier results, change the name to "TFE_WMLE".

New features

• Corrected ordinary least squares, sfm(estimator = "cols"). The moment estimator of Olson, Schmidt and Waldman (1980). OLS is consistent for the slopes of a composed-error frontier whatever the one-sided distribution – only the intercept is biased, by E[u] – so COLS keeps the OLS slopes, inverts the central moments of the OLS residuals for the scale parameters, and shifts the intercept up by the implied E[u].

  Implemented for "NHN", "NE" and "NG"; other models error, because the moment inversion is distribution-specific. No optimizer runs and the result is deterministic, which makes it a natural robustness check against a maximum-likelihood fit that may have settled at a local optimum.

  Wrong-skew samples are reported rather than absorbed: a production frontier implies a negative third central moment, and when a sample comes out the other way the moment equations have no admissible solution. sfm() warns, reports sigu = 0 with the whole residual variance assigned to sigv, and returns no efficiency predictions – to be read as no evidence of inefficiency in these data, not as an estimate of zero.

  Standard errors: the OLS slope standard errors are reported and are valid as such. The scale parameters and the corrected intercept carry NA by default, since neither has a closed-form standard error here and the OLS intercept standard error would be wrong (it knows nothing about the sampling error of a third-moment estimate). Set cols_boot for a nonparametric bootstrap covering every parameter, with rand.cols to make it reproducible.

New models

• sfm(model_name = "tHN") – Student’s t–half-normal. Heavy-tailed noise (v ~ sigma_v * t_nu) with a conventional half-normal inefficiency term (u ~ |N(0, sigma_u^2)|), drawn independently.

  This is not THT. In THT (Tancredi 2002) both components come from one shared scale mixture, so both are t with the same degrees of freedom and the composed error is a closed-form skew-t. In tHN the tails differ, there is no closed form, and the density is the convolution f(e) = integral_0^Inf f_v(e+u) f_u(u) du, evaluated by Gauss-Legendre quadrature. tHN is therefore the natural parametric comparison for the density-power robust estimators (robust = "mlqe"/"psi"/"mdpd"), which THT cannot be, because its inefficiency term is heavy-tailed too. Parameters are reported as (sigv, sigu, nu) – the conventional order, not THT’s inverted one. Returns exp_u_hat and u_hat by a Bayes rule over the same quadrature nodes.

  Two documented properties, both surfaced rather than hidden. The degrees of freedom are weakly identified: on data simulated from the model at n = 1000 with a true nu = 5, the profile log-likelihood moves only about 0.24 across nu from 10 to 100, and peaks near 20. Profile over a grid of fixed nu rather than reporting one selected value. Because of that flat ridge every tHN fit runs from several widely separated starts, keeps the best, and reports the outcome in thn_starts, warning when the starts reach different optima. sigma_u can collapse onto zero on real data, the heavy noise tail absorbing the whole one-sided component and leaving mean predicted efficiency near one; sfm() warns and sets thn_sigma_u_at_bound rather than bounding sigma_u away from zero, because the collapse is a property of the model and is the thing a user needs to see.

  The quadrature node count scales with sigma_u/sigma_v and is not fixed. A fixed 96-node rule is accurate near lambda = 3 but carries 4% relative error at lambda = 20 and 60% at lambda = 62 – and the model does reach that region. Note that integrating the density to 1 does not detect this: the error redistributes across the support and integrates away, so total mass still reads 1.000 while the density is 40% wrong pointwise.

• data_gen_cs() gained y_pcs_thn (with v_thn and u_thn), the matching generator for tHN.

• psfm(model_name = "TFE") – Greene (2005) true fixed effects. The composed-error likelihood with one intercept per individual, estimated as a profile likelihood in (lambda, sigma, beta) with the firm effects concentrated out. Reports the same parameter layout as "TFE_WMLE", plus r_hat_m (the maximum-likelihood firm effects), exp_u_hat and u_hat.

  Note that this likelihood always has a supremum on the sigma_v = 0 boundary, because the individual effects are unrestricted. The new argument tfe_lambda_max (default 100) bounds the search accordingly, and a fit that pins at the bound warns. See ?psfm for the details; this is a property of the estimator and is one of the motivations for "TFE_WMLE".

• psfm(model_name = "K1990") and "K1990modified" – Kumbhakar (1990) time-varying inefficiency, B_it = (1 + exp(bt + ct^2))^-1 and B_it = 1 + d(t - T_i) + e(t - T_i)^2. These share one likelihood with "PL80" and "BC92", differing only in B_it. K1990’s b and c are weakly identified, so the fitted B_it path is more interpretable than either coefficient on its own.

• Four new inefficiency distributions in sfm(): "NU" (normal-uniform), "NGE" (normal-generalized exponential), "NLN" (normal-lognormal) and "NW" (normal-Weibull). The last two are estimated by simulated maximum likelihood over Halton draws.

New methods and arguments

• predict(), fitted() and residuals() methods for class "sfareg", alongside the existing coef(), vcov(), logLik() and nobs(). predict() accepts newdata.

• psfm() now accepts an ordinary data.frame (or tibble/data.table) as well as a plm::pdata.frame; the panel index is constructed internally from individual and the new time argument. Previously a plain data frame failed with an uninformative "empty model" error.

• psfm(collinear_action =) controls what happens when the between-individual design used to build starting values is rank deficient – the situation created by, for example, time dummies, which are estimable in a pooled specification but collapse onto the intercept once averaged within each unit. "start_only" (default) keeps the requested model and drops the offending columns from the starting-value regression only; "error" stops and names them; "warn_drop" removes them from the estimated model. Previously this surfaced as an opaque solve(crossprod(ZBeta)) LAPACK error inside plm.

• sfm(robust =) selects a divergence-based robust estimator – "mlqe", "psi" or "mdpd" – with sandwich standard errors. Currently implemented for model_name = "NHN"; other models error rather than silently ignoring the argument.

• sfm() gained use.nlminb and use.bobyqa (both "auto" by default) and maxit.nlminb, for control over the optimizer stack described below.

Estimation and performance

• nlminb now leads the optimizer stack for the models where it helps, ahead of the derivative-free stages, with an analytic gradient supplied for NHN. Model-by-model defaults are chosen automatically: NHN, NE, NTN and NU use it, while NR and NGE (where it degraded the fit) do not. Typical cross-sectional fits are several times faster with unchanged estimates.

• An nlminb stage was added to the PL80/BC92/K1990/K1990modified branch, worth up to +25 log-likelihood on the seven-parameter models, where psfm()’s low default maxit.bobyqa was stopping short of convergence. BC92 now matches or beats frontier::sfa() on parameter, variance and efficiency accuracy.

• PL80 and BC92 are now estimated by a native maximum-likelihood implementation instead of wrapping frontier::sfa(). Verified against frontier (coefficients, log-likelihood and predicted efficiencies) on balanced and unbalanced panels and on both production and cost specifications before the dependency was removed.

• Efficiency prediction (exp_u_hat) is now available for NE and NTN, which previously returned none.

Bug fixes

• The vignette still described "PL80" and "BC92" as wrappers around frontier::sfa(). They have been native closed-form maximum-likelihood implementations since 1.1.3, the release that dropped frontier as a dependency – so for four releases vignette("intro_to_psfm") advertised a dependency on a competing package that the package had deliberately removed, and that appears in neither Imports nor Suggests. ?psfm had it right throughout (“natively estimated … verified against frontier::sfa() before that dependency was removed”); only the vignette was stale.

  Two related corrections in the same file. The psfm_bootstrap() section gave the wrong reason for excluding PL80/BC92 – it is that they do not expose the $U/$H efficiency structure the function reads, not that they wrap anything – and its list of supported models omitted TFE_WMLE and GTRE_FML, the latter being what psfm(model_name = "GTRE") returns under its default estimator. And the model table, which says “eleven” choices, now says twenty-one and points at ?psfm rather than implying the table is exhaustive.

• psfm_bootstrap() refused the estimator psfm() reaches by default. Since 1.1.4, psfm(model_name = "GTRE") defaults to estimator = "fiml" and returns an object whose $model_name is "GTRE_FML". psfm_bootstrap() checked that name against a list that did not contain it, so the natural two-line sequence – fit a GTRE model, then bootstrap it – stopped with "psfm_bootstrap() does not support model_name = 'GTRE_FML'", and the error message’s own explanation named only the moment-based and wrapped models, none of which applied. Reaching the bootstrap at all required knowing to pass estimator = "sml" to the fit.

  "GTRE_FML" is now supported. The data-generating process is the one the "GTRE" arm already simulated – same four-component model, different estimator – so only the parameter layout differs: FIML reports the four raw scales (sigr, sigv, sigh, sigu) after the frontier block rather than "GTRE"’s lambda/sigma reparameterization. $H and the boundary warning come with it.

• psfm(OPG_calc = TRUE) returned NA for every OPG and sandwich standard error, and wrote a variable into the global environment. The OPG “meat” matrix was stored with a superassignment, OPG_meat <<- crossprod(score_mat), on the assumption that the surrounding tryCatch({...}) introduced a scope to escape from. It does not – tryCatch() evaluates its expression in the calling frame – so <<- began its search one frame further out, skipped the local OPG_meat <- NULL binding entirely, and assigned into globalenv(). The immediately following solve(OPG_meat) therefore still saw NULL and failed, as did the MASS::ginv() fallback, so the OPG standard errors were always NA; the sandwich errors, which reuse the same matrix, were NA with them. Both failures were reported through the existing handlers as "OPG matrix singular, using pseudoinverse" and "OPG matrix is singular", which pointed at a rank problem in the data rather than at the scoping bug. Changed to a plain <-. On a fixed-seed GTRE_Z fit the parameter estimates and Hessian standard errors are bit-identical before and after, the OPG and sandwich errors change from NA to finite values, the two warnings stop firing, and OPG_meat no longer appears in the user’s workspace.

• psfm_bootstrap() failed on Windows whenever sfa was not installed in a default library. The function distributes work over PSOCK cluster workers, which start as fresh R sessions holding the default library path rather than the parent session’s. Where the parent had found sfa somewhere else – a project library, renv/packrat, a user-set R_LIBS_USER, or the temporary sfa.Rcheck tree that R CMD check installs into – the workers’ library() call could not see it, and the bootstrap died in parallel:::checkForRemoteErrors() with there is no package called 'sfa'. On Unix the workers generally inherit R_LIBS from the parent’s environment, which hid the bug; on Windows they do not. Where the workers instead found a different, older copy of sfa in a default library, they silently ran the bootstrap against that version rather than the one the user had loaded. The parent’s .libPaths() is now pushed to the workers before any package is loaded, so both sessions resolve every package identically.

• stats::dlnorm was used without being imported. The rewritten "NLN" likelihood and its efficiency block call dlnorm(), but NAMESPACE did not import it, so the call resolved only via the search path rather than the package namespace. R CMD check --as-cran reported it as an undefined global. Now imported explicitly.

• zsfm()’s efficiency predictor used a different mixing probability from the likelihood it maximised. "ZISF"’s likelihood sets prob = exp(-abs(gamma)), which makes it exactly symmetric in gamma: +g and -g fit identically and the optimizer may return either. The JLMS block then used exp(-gamma), so a negative estimate produced a mixing “probability” above 1 and silently invalid post.prob and jlms. This was reachable from an ordinary starting value, not a pathology: seeding at the negated estimate returns gamma = -0.3015 with an identical log-likelihood, where the old code computed prob = 1.3519. Both places now use exp(-abs(gamma)).

• zsfm()’s two-component mixture is now formed on the log scale. It built prob*exp(f1) + (1-prob)*exp(f2) and took log(f + 1e-10). Both terms underflow to zero when an observation is unlikely under either regime, and the 1e-10 guard then floors that observation’s contribution at -23.03 – which does not merely protect the logarithm, it makes the objective flat across the whole region beyond the floor, exactly where the optimizer needs a gradient. A new internal helper, .log_add2(), computes log(exp(a) + exp(b)) without leaving the log scale, and post.prob is now a ratio taken in logs. Fitted coefficients are unchanged to within optimizer path noise (worst discrepancy 1.9e-3 over eight seeds, log-likelihoods agreeing to 1e-6).

• zsfm(logit = TRUE) now uses plogis() instead of exp(eta)/(1 + exp(eta)), which overflows to Inf/Inf = NaN once the linear predictor passes about 710 – a value the optimizer can reach while searching. Identical wherever the old form was finite.

• zsfm()’s efficiency block now branches on model_name rather than on is.na(n_z_vars). The likelihood already branched on the model, and the parameter layout is a property of the model; keying the predictor off a different condition meant the two could disagree, reading "ZISF"’s parameters under "ZISF_Z"’s layout if n_z_vars ever arrived as 0 rather than NA.

  Note left in place, not changed: the logit = FALSE branch computes pnorm(eta)/(1 + pnorm(eta)), which is bounded above by 0.5 and is not the probit link. Correcting it would change results for that option, which is a modelling decision rather than a cleanup.

• sfm(model_name = "NLN") was integrating a spike. Its simulated likelihood averaged the normal kernel over lognormal draws, but the kernel is only sigma_v wide in u while the lognormal spreads over decades, so nearly every draw landed where the kernel is numerically zero and a handful carried the whole integral. Measured against a reference verified two ways (adaptive quadrature and a 200,000-point Simpson rule, agreeing to 5e-9), the simulated log-likelihood at the true parameters was 226.8 units low at n = 3000 under the default draw count – which is precisely the unexplained 228-unit gap recorded against this model in the convergence notes. It was simulation error, not a defect in the likelihood.

  sfm() now substitutes u = sigma_v*t - e, turning the integral into a standard-normal expectation of the smooth lognormal density truncated to u > 0. The error at the same draw count is then 0.12, and 0.86 at Nsim = 50. Raising Nsim was not an alternative: the old error per observation was about -0.076 at both n = 1000 and n = 3000 under the 8*sqrt(n) rule, so the total bias grew linearly in n at the same rate as the log-likelihood itself, and closing it needed Nsim proportional to n – quadratic work per evaluation. "NW", which uses the same machinery but a far lighter-tailed inefficiency term, is unchanged.

• sfm(model_name = "NR") was started from a flat guess it could not recover from, and was documented as the wrong model. Two separate problems, one in the estimator and one in everything written about it.

  The estimator: start_cs() hard-codes sigma_u = sigma_v = 0.1 for the cross-sectional models, and from there "NR" converged to a point with a worse log-likelihood than the true parameter vector in 9 of 14 replications at n = 4000. It failed in two modes – a hard collapse to sigma_u = 1e-7 with sigma_v inflated to absorb the spread (a ~50 log-likelihood deficit), and a partial stall at sigma_u around 0.85 with the intercept ~0.4 too low. "NR" is now started by inverting the Rayleigh moment equations instead, which reaches the same optimum a truth-seeded run finds in all 14. Because the Rayleigh skewness is a constant, the third central moment of the residuals identifies Var(u) outright. Wrongly skewed residuals leave the moment equations with no admissible solution, and the old flat start is then used unchanged.

  The documentation: "NR" had been described in this package as “an alternative closed-form derivation of the same normal/half-normal composed error” as "NHN". That is wrong. "NR" is normal-Rayleigh – its coded density reproduces a numerical normal-Rayleigh convolution to 1e-8 and misses the half-normal one by 86%. The two are different families, not reparameterizations: the standardized skewness the inefficiency contributes is a different constant in each (-0.631 against -0.995), and no transformation of a two-scale family moves a standardized moment. The likelihood itself was correct throughout and is unchanged.

  data_gen_cs() gains u_r and y_pcs_r to test "NR" against its own data-generating process; it had previously been tested against y_pcs, which it cannot fit. The new columns are appended at the end of the function, so every existing column is bit-for-bit unchanged. sigma_u is on the convention E[u^2] = sigma_u^2, matching "NHN", so the Rayleigh scale is sigma_u/sqrt(2).

• sfm(model_name = "NG") and "NNAK" read their frontier coefficients from the wrong position in the parameter vector. The likelihood closure slices the coefficients out by a fixed offset – x[3:(n_x+2)] for models with two leading scale parameters, x[4:(n_x+3)] for three. NG and NNAK carry three (sigv, sigu, mu) but sat in the two-parameter group, so the slice took the right number of coefficients starting one slot too early.

  The consequences were severe and entirely silent. mu was used simultaneously as the gamma (or Nakagami) shape and as the intercept coefficient, so it could not move freely – which is why the shape appeared never to leave its starting value. Every remaining slope was shifted one place, and the last coefficient never entered the likelihood at all, so it simply kept whatever starting value it was given. The efficiency block used the correct offset throughout, so the two halves of the model disagreed about which number meant what.

  On the package’s own test DGP at n = 4000, NG returned sigma_u = 0 and stopped 710 log-likelihood units below the true parameter vector. Across 15 fits spanning n = 1000 to 5000, none reached the truth’s likelihood. After the fix, all 15 do, no fit collapses, and mean RMSE against the truth falls from 0.60 to 0.27 and declines with n. NNAK improves on the same fix – 6 of 7 fits now beat the truth and standard errors are finite, where previously the Hessian was singular in every replication – but it still produces occasional failures and is not yet considered repaired.

  The NG density itself was never wrong: it agrees with numerical convolution of the normal and gamma densities to 7e-14 across the whole residual range, and that check is now a test.

• sfm(model_name = "NG") also started in the wrong place. start_cs() hard-codes sigma_u = sigma_v = 0.1 and the NG start pinned the shape at 1, so the search began from E[u] = 0.1 against a true 1, with the intercept at the raw OLS value – itself E[u] below the frontier. NG now builds candidate starts from the residual moments and sweeps the shape along the E[u] = mu*sigma_u ridge, which is the direction the data leave weakly determined (Ritter and Simar, 1997), polishing the most promising few before choosing. The search is reported in $ng_starts.

• psfm(model_name = "GTRE_FML") started its search from the wrong intercept. It seeded beta_0 at the raw panel-regression intercept, while "GTRE" and "TRE" seed theirs at that intercept plus E[u] + E[h]. In a composed-error model the regression intercept sits below the frontier by (sigma_u + sigma_h) * sqrt(2/pi) – 1.12 at the package’s own test DGP – so the FIML search began a full unit low, in exactly the direction of the sigma_h = 0 boundary optimum where the model collapses to "TRE" and the intercept absorbs the missing E[h].

  Fits that fell in were not merely imprecise: on one of six test draws the reported solution had a lower log-likelihood than the true parameter vector (-5267.87 against -5267.80), with sigma_h = 2e-16, beta_0 = 0.155 against a true 0.5, and sigma_r inflated to 0.307 against a true 0.2 – a local optimum, not an estimate.

  "GTRE_FML" now also builds a second candidate start from the two-step moment estimator (the one "GTRE_SEQ2" reports), evaluates the likelihood at both and begins from the better, as Colombi (2010) and Colombi, Martini and Vittadini (2011) recommend for this likelihood. Across the six test draws, boundary collapses go from one to none, mean RMSE against the truth falls from 0.049 to 0.023, and all six fits now reach a higher likelihood than the truth. The chosen start is reported in start_search.

• sfm(model_name = "THT") used the wrong likelihood. The skew-t density of Tancredi (2002, eq. 4) has scale omega = sqrt(sigma_v^2 + sigma_u^2), but the implementation evaluated the Student-t factor at the raw residual and omitted the 1/omega Jacobian, which pins the scale at 1. Because 2*f(e)*G(w(e)) is a valid density for any symmetric f and odd w (Azzalini’s lemma), the wrong version still integrated to 1 and still produced plausible fits rather than an obvious failure – but sigma_u and sigma_v were then identified only through the skewing term, and the degrees of freedom a had to absorb the scale mismatch. Fitted a and sigma_v were consequently inconsistent. Fixed; THT now reproduces sn::dst() to machine precision.

• sfm(model_name = "THT") now reports efficiency. It previously returned no efficiency prediction at all. It now returns exp_u_hat = E[exp(-u)|e], u_hat = E[u|e], and sd_exp_u_hat, following Tancredi (2002, section 2.2). The conditional density in that paper’s eq. (7) is a Student-t truncated to the non-negative half-line with df = a + 1, location -e*sigma_u^2/omega^2 and scale sqrt((a + e^2/omega^2)*sigma_v^2*sigma_u^2/(omega^2*(a+1))), which reduces to the Jondrow et al. (1982) normal predictor as a grows. Note the behaviour this buys: for a large positive residual the half-normal model drives predicted efficiency to 1 with near-zero uncertainty, whereas the skew-t model reads the point as an outlier, so efficiency turns back down and sd_exp_u_hat widens.

• sfm(model_name = "THT") degrees of freedom are better started and bounded. The starting value was 1 – the Cauchy case, which has neither a mean nor a variance – and the lower bound was 1e-7. The start is now a moment estimate from the excess kurtosis of the OLS residuals (a ~ 4 + 6/kurtosis, clipped to [3, 30]), and the lower bound is 2.05, so the search stays where the composed error has both moments.

• data_gen_cs() gained y_pcs_st, which is the column THT should be tested against. The existing y_pcs_t draws its two error components as two independent rt() variates; that shares the degrees of freedom but not the mixing variable, so the composed error is not skew-t and THT cannot recover its own parameters from it. y_pcs_st uses the single common Gamma(a/2, a/2) mixing of Tancredi eq. (5), with lam_st, u_st and v_st also returned. y_pcs_t is retained unchanged, because renumbering the random draws would alter every column generated after it.

• nobs() on an sfm(), zsfm() or ttsfm() fit returned NA when called from inside a function, because the recorded data argument was re-evaluated in the caller’s frame rather than the one the model was fitted in. This propagated silently to BIC(), which needs the observation count, while AIC() kept working.

• logLik() on the estimators that are not maximum-likelihood (GTRE_SEQ1, GTRE_SEQ2, SSFE) returned an unclassed NA, which made AIC() and BIC() return numeric(0) – a missing value that disappeared instead of propagating. They now return NA as documented.

• psfm()’s TFE and SSFE models silently depended on the first two columns of data being the panel index, because plm() was called without an explicit index. Any data whose first two columns were something else produced "empty model".

• Fixed a checkSymmetricPositiveDefinite() failure in the GTRE models caused by asymmetric dimnames on an otherwise symmetric covariance matrix.

• The Halton draws used by the simulated-likelihood models were reshaped in column-major order, so each observation drew from a narrow, non-equidistributed slice of the sequence.

• A singular Hessian now yields NA standard errors rather than aborting the whole fit.

Internal changes

• All 76 stop() and warning() calls in R/ now pass call. = FALSE; 24 of them did not, which made the error output inconsistent between older and newer code paths.

• The seven sapply() calls in psfm.R – six extracting ridge/method diagnostics from the GTRE posterior solver, one building the transient efficiency vector – are now vapply() with explicit numeric(1) and character(1) templates, so a change in what the solver returns fails loudly instead of silently producing a list column.

Dependencies

• Removed the dependency on frontier, along with eight other packages. Imports went from 25 packages to 15.

• Lowered the R requirement from R (>= 4.4.0) to R (>= 4.0.0).

• Added a testthat suite covering the model branches, the numerical helpers, the S3 methods and the data generators.

sfa 1.0.4

• Version released on CRAN, 2026-01-21.

These binaries (installable software) and packages are in development.
They may not be fully stable and should be used with caution. We make no claims about them.