Stochastic Frontier (SFA) model not converging

[Fanny Oegren]

Hi everyone,

I'm trying to estimate the eco-efficiency on industry level for 9 countries using panel data and Stochastic Frontier Analysis (SFA) by running the following model:

sfpanel logGDPGHGratio logLabor logCapital logRenew logNonrenew Year, model(tfe) dist(tn) ort(o)

logGDPGHGratio = The log of the ratio of value added and greenhouse gas emissions
logLabor = the log of the n.o. employed people
logCapital = The log of Fixed assets
logRenew = The log of Renewable energy consumption
logNonrenew = The log of Non-renewable energy consumption
Year = Year


However when I run the code it looks like this:

initial: Log likelihood = -<inf> (could not be evaluated)
feasible: Log likelihood = -817.8733
Iteration 0: Log likelihood = -817.8733 (not concave)
Iteration 1: Log likelihood = -386.18149 (not concave)
Iteration 2: Log likelihood = -154.2575 (not concave)
Iteration 3: Log likelihood = 20.408539
Iteration 4: Log likelihood = 34.537644 (not concave)
Iteration 5: Log likelihood = 153.05269
Iteration 6: Log likelihood = 179.65474
Iteration 7: Log likelihood = 197.76452
Iteration 8: Log likelihood = 209.66292 (not concave)
Iteration 9: Log likelihood = 210.85545
Iteration 10: Log likelihood = 211.04231
Iteration 11: Log likelihood = 211.07419
Iteration 12: Log likelihood = 211.15698 (not concave)
Iteration 13: Log likelihood = 211.15855
Iteration 14: Log likelihood = 211.18896
Iteration 15: Log likelihood = 211.2324
Iteration 16: Log likelihood = 211.30521
Iteration 17: Log likelihood = 211.3262
Iteration 18: Log likelihood = 211.34639
Iteration 19: Log likelihood = 211.36486
Iteration 20: Log likelihood = 211.38176 (not concave)
Iteration 21: Log likelihood = 211.38997 (not concave)
Iteration 22: Log likelihood = 211.39067
Iteration 23: Log likelihood = 211.39592
Iteration 24: Log likelihood = 211.40235
Iteration 25: Log likelihood = 211.41253
Iteration 26: Log likelihood = 211.41627
Iteration 27: Log likelihood = 211.4218
Iteration 28: Log likelihood = 211.42454
Iteration 29: Log likelihood = 211.43018
Iteration 30: Log likelihood = 211.43228
Iteration 31: Log likelihood = 211.43565
Iteration 32: Log likelihood = 211.44079
Iteration 33: Log likelihood = 211.44176
Iteration 34: Log likelihood = 211.4436
Iteration 35: Log likelihood = 211.44467
Iteration 36: Log likelihood = 211.44679
Iteration 37: Log likelihood = 211.44735
Iteration 38: Log likelihood = 211.44834
Iteration 39: Log likelihood = 211.44988
Iteration 40: Log likelihood = 211.45023
Iteration 41: Log likelihood = 211.45065
Iteration 42: Log likelihood = 211.45066 (backed up)
Iteration 43: Log likelihood = 211.45066 (backed up)
Iteration 44: Log likelihood = 211.45066 (backed up)
Iteration 45: Log likelihood = 211.45066 (backed up)
Iteration 46: Log likelihood = 211.45066 (backed up)
Iteration 47: Log likelihood = 211.45066 (backed up)
Iteration 48: Log likelihood = 211.45066 (backed up)
Iteration 49: Log likelihood = 211.45066 (backed up)
Iteration 50: Log likelihood = 211.45066 (backed up)
Iteration 51: Log likelihood = 211.45066 (backed up)
Iteration 52: Log likelihood = 211.45066 (backed up)
Iteration 53: Log likelihood = 211.45066 (backed up)
Iteration 54: Log likelihood = 211.45066 (backed up)
Iteration 55: Log likelihood = 211.45066 (backed up)
Iteration 56: Log likelihood = 211.45066 (backed up)
Iteration 57: Log likelihood = 211.45066 (backed up)
Iteration 58: Log likelihood = 211.45066 (backed up)
Iteration 59: Log likelihood = 211.45066 (backed up)
Iteration 60: Log likelihood = 211.45066 (backed up)
Iteration 61: Log likelihood = 211.45066 (backed up)
Iteration 62: Log likelihood = 211.45066 (backed up)
Iteration 63: Log likelihood = 211.45066 (backed up)
Iteration 64: Log likelihood = 211.45066 (backed up)
Iteration 65: Log likelihood = 211.45066 (backed up)
Iteration 66: Log likelihood = 211.45066 (backed up)
Iteration 67: Log likelihood = 211.45066 (backed up)
Iteration 68: Log likelihood = 211.45066 (backed up)
Iteration 69: Log likelihood = 211.45066 (backed up)
Iteration 70: Log likelihood = 211.45066 (backed up)
Iteration 71: Log likelihood = 211.45066 (backed up)
Iteration 72: Log likelihood = 211.45066 (backed up)
Iteration 73: Log likelihood = 211.45066 (backed up)
Iteration 74: Log likelihood = 211.45066 (backed up)
Iteration 75: Log likelihood = 211.45066 (backed up)
Iteration 76: Log likelihood = 211.45066 (backed up)
Iteration 77: Log likelihood = 211.45066 (backed up)
Iteration 78: Log likelihood = 211.45066 (backed up)
Iteration 79: Log likelihood = 211.45066 (backed up)
Iteration 80: Log likelihood = 211.45066 (backed up)
Iteration 81: Log likelihood = 211.45066 (backed up)
Iteration 82: Log likelihood = 211.45066 (backed up)
Iteration 83: Log likelihood = 211.45066 (backed up)
Iteration 84: Log likelihood = 211.45066 (backed up)
Iteration 85: Log likelihood = 211.45066 (backed up)
Iteration 86: Log likelihood = 211.45066 (backed up)
Iteration 87: Log likelihood = 211.45066 (backed up)
Iteration 88: Log likelihood = 211.45066 (backed up)
Iteration 89: Log likelihood = 211.45066 (backed up)
Iteration 90: Log likelihood = 211.45066 (backed up)
Iteration 91: Log likelihood = 211.45066 (backed up)
Iteration 92: Log likelihood = 211.45066 (backed up)
Iteration 93: Log likelihood = 211.45066 (backed up)
Iteration 94: Log likelihood = 211.45066 (backed up)
Iteration 95: Log likelihood = 211.45066 (backed up)
Iteration 96: Log likelihood = 211.45066 (backed up)
Iteration 97: Log likelihood = 211.45066 (backed up)
Iteration 98: Log likelihood = 211.45066 (backed up)
Iteration 99: Log likelihood = 211.45066 (backed up)
Iteration 100: Log likelihood = 211.45066 (backed up)

True fixed-effects model (truncated-normal) Number of obs =
> 327
Group variable: ID Number of groups =
> 25
Time variable: Year Obs per group: min =
> 5
avg = 1
> 3.1
max =
> 14

Prob > chi2 = 0.0
> 000
Log likelihood = 211.4507 Wald chi2(5) = 102
> .07

---------------------------------------------------------------------------
> ---
logGDPGHGr~o | Coefficient Std. err. z P>|z| [95% conf. interv
> al]
-------------+-------------------------------------------------------------
> ---
Frontier |
logLabor | .3261576 .1169416 2.79 0.005 .0969564 .5553
> 589
logCapital | .0388976 .0918791 0.42 0.672 -.1411821 .2189
> 774
logRenew | .0043788 .014619 0.30 0.765 -.024274 .0330
> 316
logNonrenew | -.0466693 .0126807 -3.68 0.000 -.0715231 -.0218
> 155
Year | .01462 .002246 6.51 0.000 .0102179 .0190
> 221
-------------+-------------------------------------------------------------
> ---
Mu |
_cons | -98.94816 230.2629 -0.43 0.667 -550.2552 352.3
> 589
-------------+-------------------------------------------------------------
> ---
Usigma |
_cons | 1.945617 2.329127 0.84 0.404 -2.619388 6.510
> 622
-------------+-------------------------------------------------------------
> ---
Vsigma |
_cons | -4.48299 .1443512 -31.06 0.000 -4.765913 -4.200
> 067
-------------+-------------------------------------------------------------
> ---
sigma_u | 2.645363 3.080694 0.86 0.391 .2699026 25.92
> 768
sigma_v | .1062995 .0076722 13.86 0.000 .0922773 .1224
> 523
lambda | 24.88595 3.081093 8.08 0.000 18.84712 30.92
> 478
---------------------------------------------------------------------------


Should I be worried that the model doesn't seem to converge and that the estimated coefficients may be uncertain (because of the "(not concave)" and "(backed up)" messages on the iterations)? What can be done to try and remedy this issue?

Worth mentioning is that I get reasonable values on the estimated eco-efficiency scores. To obtain these scores I use the code snippet predict te, jlms.

Many thanks in advance!

[Andrew Musau]

sfpanel is from SSC (FAQ Advice #12). Most commands would exit with the error "convergence not achieved" if the ML algorithm failed to converge. I'm unsure if sfpanel is programmed to stop iterations after reaching 100 by default and output the estimates. However, if this is the case, you'll need to discard those estimates as they do not represent the maximized likelihood. There is a simple way to check. After you get the results, run

mat b= e(b)
sfpanel logGDPGHGratio logLabor logCapital logRenew logNonrenew Year, model(tfe) dist(tn) ort(o) from(b)

If the estimation does not stop immediately and keeps on iterating, then convergence was not achieved the first time around. If this is the case, you may want to email the authors of the command and ask them to change this default.

[Fanny Oegren]

Hi Andrew, thanks for the reply.

When running the suggested code I get an error message saying:
Option from() not allowed. Use sveqnname() options.
invalid syntax

When trying to use sveqnname(b)instead of from(b), Stata iterates a hundred times as before, giving me the same estimates.

Is there any other way of determining if the model fails to converge? Thank you!

[Andrew Musau]

When running the suggested code I get an error message saying:
Option from() not allowed. Use sveqnname() options.
invalid syntax

You're correct, in sfpanel you specify starting values using the following:

svfrontier() specifies a 1 X k vector of starting values for the frontier. The vector must have the same length as there are parameters to estimate in the frontier
equation.

svusigma() specifies a starting value for the parameter of the variance of the inefficiency term.

svvsigma() specifies a starting value for the parameter of the variance of the idiosyncratic term.

Is there any other way of determining if the model fails to converge?

It is clear that your model did not converge, so you cannot use those estimates obtained after 100 iterations. A suggestion is to start with Greene's True Random Effects (TRE) model. If it converges and you get estimates, use these for the TFE model. Here is how to do it showing indeed that the starting values are picked up by the TFE model.

webuse xtfrontier1, clear
*GREENE'S TRE MODEL
sfpanel lnwidgets lnmachines lnworkers, model(tre) dist(hnormal) usigma(lnworkers) difficult rescale nsim(50) simtype(genhalton)

*COEFFICIENTS MATRIX
mat l e(b)

*EXTRACT THE 3 MATRICES
mat b_f= e(b)[1, 1..`e(df_m)']
mat b_u= e(b)[1, "Usigma:"]
mat b_v= e(b)[1, "Vsigma:"]

mat l b_f
mat l b_u
mat l b_v

*RUN TFE MODEL WITH NO ITERATIONS TO SHOW STARTING VALUES PICKED UP
sfpanel lnwidgets lnmachines lnworkers, model(tfe) dist(exp) usigma(lnworkers)  svfrontier(b_f) svusigma(b_u) svvsigma(b_v) iter(0)

*NOW RUN THE MODEL WITH THE SPECIFIED STARTING VALUES
sfpanel lnwidgets lnmachines lnworkers, model(tfe) dist(exp) usigma(lnworkers)  svfrontier(b_f) svusigma(b_u) svvsigma(b_v)

Res.:

. *GREENE'S TRE MODEL

.
. sfpanel lnwidgets lnmachines lnworkers, model(tre) dist(hnormal) usigma(lnworkers) difficult rescale nsim(50) simtype
> (genhalton)


Initial:      Log simulated-likelihood = -1755.9634
Rescale:      Log simulated-likelihood = -1755.9634
Rescale eq:   Log simulated-likelihood = -1503.2923
Iteration 0:  Log simulated-likelihood = -1503.2923  
Iteration 1:  Log simulated-likelihood = -1485.8234  (not concave)
Iteration 2:  Log simulated-likelihood = -1479.4207  
Iteration 3:  Log simulated-likelihood = -1478.0449  
Iteration 4:  Log simulated-likelihood = -1477.9694  
Iteration 5:  Log simulated-likelihood = -1477.9678  
Iteration 6:  Log simulated-likelihood = -1477.9678  

True random-effects model (half-normal)              Number of obs =       948
Group variable: id                                Number of groups =        91
Time variable: t                                Obs per group: min =         6
                                                               avg =      10.4
                                                               max =        14

                                                     Prob > chi2   =    0.0000
Log simulated-likelihood = -1477.9678                Wald chi2(2)  =    398.66

Number of Randomized Halton Sequences = 50
Base for Randomized Halton Sequences  = 7
------------------------------------------------------------------------------
   lnwidgets | Coefficient  Std. err.      z    P>|z|     [95% conf. interval]
-------------+----------------------------------------------------------------
Frontier     |
  lnmachines |   .2906754   .0165098    17.61   0.000     .2583168     .323034
   lnworkers |   .2639837   .0290145     9.10   0.000     .2071163    .3208511
       _cons |   1.722812   .2276866     7.57   0.000     1.276555     2.16907
-------------+----------------------------------------------------------------
Usigma       |
   lnworkers |  -.0802731   .0709493    -1.13   0.258    -.2193313     .058785
       _cons |  -.4036188   .5552325    -0.73   0.467    -1.491854    .6846168
-------------+----------------------------------------------------------------
Vsigma       |
       _cons |  -.3090683   .1838835    -1.68   0.093    -.6694732    .0513367
-------------+----------------------------------------------------------------
Theta        |
       _cons |    1.33327    .094778    14.07   0.000     1.147508    1.519031
-------------+----------------------------------------------------------------
  E(sigma_u) |   .8670305                                 .8616404    .8724205
     sigma_v |   .8568142    .078777    10.88   0.000     .7155265    1.026001
------------------------------------------------------------------------------

.
.
.
. *COEFFICIENTS MATRIX

.
. mat l e(b)

e(b)[1,7]
      Frontier:   Frontier:   Frontier:     Usigma:     Usigma:     Vsigma:      Theta:
    lnmachines   lnworkers       _cons   lnworkers       _cons       _cons       _cons
y1    .2906754   .26398369   1.7228125  -.08027314  -.40361883  -.30906827   1.3332696

.
.
.
. *EXTRACT THE 3 MATRICES

.
. mat b_f= e(b)[1, 1..`e(df_m)']

.
. mat b_u= e(b)[1, "Usigma:"]

.
. mat b_v= e(b)[1, "Vsigma:"]

.
.
.
. mat l b_f

b_f[1,2]
      Frontier:   Frontier:
    lnmachines   lnworkers
y1    .2906754   .26398369

.
. mat l b_u

b_u[1,2]
        Usigma:     Usigma:
     lnworkers       _cons
y1  -.08027314  -.40361883

.
. mat l b_v

symmetric b_v[1,1]
        Vsigma:
         _cons
y1  -.30906827

.
.
.
. *RUN TFE MODEL WITH NO ITERATIONS TO SHOW STARTING VALUES PICKED UP

.
. sfpanel lnwidgets lnmachines lnworkers, model(tfe) dist(exp) usigma(lnworkers)  svfrontier(b_f) svusigma(b_u) svvsigm
> a(b_v) iter(0)


Iteration 0:  Log likelihood = -3399.9534  (not concave)

True fixed-effects model (exponential)               Number of obs =       948
Group variable: id                                Number of groups =        91
Time variable: t                                Obs per group: min =         6
                                                               avg =      10.4
                                                               max =        14

                                                     Prob > chi2   =    0.0000
Log likelihood = -3399.9534                          Wald chi2(2)  =     60.13

------------------------------------------------------------------------------
   lnwidgets | Coefficient  Std. err.      z    P>|z|     [95% conf. interval]
-------------+----------------------------------------------------------------
Frontier     |
  lnmachines |   .2906754   .0406166     7.16   0.000     .2110683    .3702825
   lnworkers |   .2639837   .0532677     4.96   0.000     .1595809    .3683865
-------------+----------------------------------------------------------------
Usigma       |
   lnworkers |  -.0802731   .0848607    -0.95   0.344    -.2465971    .0860509
       _cons |  -.4036188   .1709012    -2.36   0.018    -.7385791   -.0686586
-------------+----------------------------------------------------------------
Vsigma       |
       _cons |  -.3090683   .2617274    -1.18   0.238    -.8220446    .2039081
-------------+----------------------------------------------------------------
  E(sigma_u) |   .8670305                                 .8616404    .8724205
     sigma_v |   .8568142   .1121259     7.64   0.000     .6629721    1.107333
------------------------------------------------------------------------------

.
.
.
. *NOW RUN THE MODEL WITH THE SPECIFIED STARTING VALUES

.
. sfpanel lnwidgets lnmachines lnworkers, model(tfe) dist(exp) usigma(lnworkers)  svfrontier(b_f) svusigma(b_u) svvsigm
> a(b_v)


Initial:      Log likelihood = -3399.9534
Iteration 0:  Log likelihood = -3399.9534  (not concave)
Iteration 1:  Log likelihood = -1571.2884  (not concave)
Iteration 2:  Log likelihood = -1483.9504  (not concave)
Iteration 3:  Log likelihood = -1445.5861  
Iteration 4:  Log likelihood = -1331.6921  
Iteration 5:  Log likelihood = -1309.6402  
Iteration 6:  Log likelihood =  -1301.774  
Iteration 7:  Log likelihood =  -1301.092  
Iteration 8:  Log likelihood = -1300.9274  
Iteration 9:  Log likelihood = -1300.9124  
Iteration 10: Log likelihood = -1300.9119  
Iteration 11: Log likelihood = -1300.9119  

True fixed-effects model (exponential)               Number of obs =       948
Group variable: id                                Number of groups =        91
Time variable: t                                Obs per group: min =         6
                                                               avg =      10.4
                                                               max =        14

                                                     Prob > chi2   =    0.0000
Log likelihood = -1300.9119                          Wald chi2(2)  =    454.34

------------------------------------------------------------------------------
   lnwidgets | Coefficient  Std. err.      z    P>|z|     [95% conf. interval]
-------------+----------------------------------------------------------------
Frontier     |
  lnmachines |   .2923237   .0157401    18.57   0.000     .2614737    .3231736
   lnworkers |   .2720863    .027266     9.98   0.000      .218646    .3255266
-------------+----------------------------------------------------------------
Usigma       |
   lnworkers |  -.1365877   .1366951    -1.00   0.318    -.4045052    .1313297
       _cons |  -2.435708   1.051787    -2.32   0.021    -4.497173    -.374243
-------------+----------------------------------------------------------------
Vsigma       |
       _cons |  -.2196069    .122049    -1.80   0.072    -.4588185    .0196048
-------------+----------------------------------------------------------------
  E(sigma_u) |   .3290061                                 .3252884    .3327237
     sigma_v |   .8960102   .0546786    16.39   0.000     .7950031    1.009851
------------------------------------------------------------------------------

If you do not get improvements with this, go back with your supervisor to the drawing board and look at your model specification and data. There are some diagnostics to check whether your data meets the assumptions of the to be estimated model. You are using a truncated normal distribution for the inefficiency. Here, there is a chance to model the inefficiency as a linear function of a set of covariates using the -emean()- option as well.

[Fanny Oegren]

Hi again Andrew,

When trying Greene's True Random Effects (TRE) model, the model doesn't converge either and after a few iterations I get the following error message: "could not calculate numerical derivatives -- discontinuous region with
missing values encountered".

However, when I try to include the emean() option in the true fixed effects model and put the variable "Year" in it, it looks like this:

. sfpanel logGDPGHGratio logLabor logCapital logRenew logNonrenew Year, model(tfe) dist(tn) emean(Year) ort(o)


initial: Log likelihood = -<inf> (could not be evaluated)
feasible: Log likelihood = -817.8733
Iteration 0: Log likelihood = -817.8733 (not concave)
Iteration 1: Log likelihood = -382.52866 (not concave)
Iteration 2: Log likelihood = -216.0596 (not concave)
Iteration 3: Log likelihood = 22.360711 (not concave)
Iteration 4: Log likelihood = 119.88889
Iteration 5: Log likelihood = 161.12759
Iteration 6: Log likelihood = 202.39408 (not concave)
Iteration 7: Log likelihood = 218.01912
Iteration 8: Log likelihood = 221.4917
Iteration 9: Log likelihood = 221.90934
Iteration 10: Log likelihood = 221.9181
Iteration 11: Log likelihood = 221.92266 (not concave)
Iteration 12: Log likelihood = 221.92319
Iteration 13: Log likelihood = 221.92393
Iteration 14: Log likelihood = 221.92426
Iteration 15: Log likelihood = 221.92427

True fixed-effects model (truncated-normal) Number of obs =
> 327
Group variable: ID Number of groups =
> 25
Time variable: Year Obs per group: min =
> 5
avg = 1
> 3.1
max =
> 14

Prob > chi2 = 0.0
> 000
Log likelihood = 221.9243 Wald chi2(5) = 129
> .91

---------------------------------------------------------------------------
> ---
logGDPGHGr~o | Coefficient Std. err. z P>|z| [95% conf. interv
> al]
-------------+-------------------------------------------------------------
> ---
Frontier |
logLabor | .4054443 .1125838 3.60 0.000 .1847841 .6261
> 045
logCapital | .0603758 .0903203 0.67 0.504 -.1166487 .2374
> 003
logRenew | -.0031402 .0143083 -0.22 0.826 -.031184 .0249
> 036
logNonrenew | -.0407008 .0117756 -3.46 0.001 -.0637806 -.0176
> 211
Year | .0239321 .0031359 7.63 0.000 .0177859 .0300
> 784
-------------+-------------------------------------------------------------
> ---
Mu |
Year | .7161188 1.539471 0.47 0.642 -2.301189 3.733
> 426
_cons | -1447.7 3112.998 -0.47 0.642 -7549.064 4653.
> 664
-------------+-------------------------------------------------------------
> ---
Usigma |
_cons | -1.857554 2.264724 -0.82 0.412 -6.296332 2.581
> 224
-------------+-------------------------------------------------------------
> ---
Vsigma |
_cons | -4.391942 .0987163 -44.49 0.000 -4.585423 -4.198
> 462
-------------+-------------------------------------------------------------
> ---
sigma_u | .3950365 .4473244 0.88 0.377 .0429308 3.63
> 501
sigma_v | .1112505 .0054911 20.26 0.000 .1009923 .1225
> 506
lambda | 3.550875 .4471721 7.94 0.000 2.674434 4.427
> 317
---------------------------------------------------------------------------


But I'm not sure if I should use some other variable inside the emean() specification? What exactly is the interpretation of using "Year" here (a variable that only indicates which year the other variables' data points are from, ranging from 2008-2021)? I have seen someone in an example video using "imports shares" inside this specification, could that make sense for me to use as well?

Thank you in advance!

[Andrew Musau]

But I'm not sure if I should use some other variable inside the emean() specification?

This should come from theory. In other words, you have to answer the question of what factors determine inefficiency in your setting and come up with a theoretical model. There is some discussion in the Stata manual entry of frontier, where the corresponding option is -cm()-. See Example 3: The truncated-normal model. Also, look at what variables past researchers have used, but ultimately make sure that you can make a theoretical argument for your choice.