Translog cost function estimation (sfpanel) and first derivative (margins)?

[Rudolf Lenger]

Hello everyone,

I am currently estimating a translog cost function from a unbalanced panel data set by using a stochastic frontier model (sfpanel command).

Code: sfpanel lntc lnglp llnglps lnpe_a lnfe_a lnae_a c.lnglp#c.lnpe_a c.lnglp#c.lnfe_a c.lnglp#c.lnae_a c.llnpe_as#c.llnfe_as c.llnpe_as#c.llnae_as c.llnae_as#c.llnfe_as i.year, model(bc95) dist(tn)

Where lntc is log of total cost, lnglp (log of gross loan portfolio) is my output variable and lnpe, lnfe and lnae are my input prices in log. The other variables are specified according to the formula (see attachment formula (3)) I am using e.g llnglps = 0.5*(lnglp)^2. I get (at least from my understanding good results when I run this. However in my next step I need to calculate the marginal costs for each firm, year observation in my dataset, I tried to use the margins command to calculate the first derivative of my translog cost function with respect to lnglp (my output variable) see also formula (4) in attachment.

Code: margins, dydx(lnglp) gen(MCF)

Now I am wondering if this is correct? Meaning if the command actually takes the derivative to each term where lnglp is involved or only to lnglp itself, as I pre specified the variables above to be called differently as e.g llnglps and I cant think of any other way to implement lnglp in general and then do the transformation in my translog cost function code directly. Would be very thankful for any help or any form of input / suggestions!

Best regards
Rudolf

[Andrew Musau]

sfpanel is from SSC (FAQ Advice #12).

Code: margins, dydx(lnglp) gen(MCF)

Now I am wondering if this is correct? Meaning if the command actually takes the derivative to each term where lnglp is involved or only to lnglp itself, as I pre specified the variables above to be called differently as e.g llnglps and I cant think of any other way to implement lnglp in general and then do the transformation in my translog cost function code directly.

No, margins has no way to know that "llnglps" is related to "lnglp", so your results would be incorrect. In any case, the marginal cost is the first derivative of the cost function with respect to quantity, as Eq. 4 in the attached picture shows. Work out the derivative and write it down, then use the -expression()- option of margins. You can then calculate the margins over observations (given that these define firm-year combinations in your panel).

[Rudolf Lenger]

Hi Andrew,

thank you very much for your answer and the hint with the -expression()- option of margins. I tried to implement it in my code and now it looks as follows.

Code: margins, expression((lntc/lnglp)*(_b[lnglp]+_b[llnglps]+_b[c.lnglp#c.lnpe_a]*lnpe_a+_b[c.lnglp#c.lnfe_a]*lnfe_a+_b[c.lnglp#c.lnae_a]*lnae_a)) dydx(lnglp) gen(MCF)

I am not 100% sure if this is correctly implemented now as I do not fully understand how the expression option of margins works. If you are familiar with it maybe you can let me know if this looks correct to you.
Thank you!

[Andrew Musau]

Here is an example:

1. Model

$$\text{price}= \beta_{0} + \beta_{1}\text{mpg} + \beta_{2}\text{mpg}^{2} +\beta_{3}\text{weight}+ \beta_{4}\text{mpg}\times\text{weight}$$

2. Taking the derivarive w.r.t. mpg:

$$\frac{\partial \text{price}}{\partial \text{mpg}} = \beta_1 + 2\beta_{2}\text{mpg}+ \beta_{4}\text{weight}$$

sysuse auto, clear
regress price c.mpg##c.mpg c.weight c.mpg#c.weight
gen obs=_n
*MARGINS CALCULATION
margins, dydx(mpg) over(obs)
*BY HAND USING EXPRESSION
margins, expression(_b[mpg]+ 2*_b[c.mpg#c.mpg]*mpg + _b[c.mpg#c.weight]*weight) over(obs)

Res.:

. margins, dydx(mpg) over(obs)

Average marginal effects                        Number of obs     =         74
Model VCE    : OLS

Expression   : Linear prediction, predict()
dy/dx w.r.t. : mpg
over         : obs

------------------------------------------------------------------------------
             |            Delta-method
             |      dy/dx   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
mpg          |
         obs |
          1  |  -201.8247   107.4765    -1.88   0.065    -416.2345    12.58509
          2  |  -332.5317   167.1118    -1.99   0.051    -665.9106    .8471334
          3  |  -164.3203   116.7984    -1.41   0.164    -397.3266    68.68607
          4  |   -273.765   126.0757    -2.17   0.033    -525.2792   -22.25082
          5  |  -457.4955   182.5663    -2.51   0.015    -821.7053   -93.28575
          6  |  -358.6379   147.1334    -2.44   0.017     -652.161   -65.11474
          7  |  -50.18457    87.8117    -0.57   0.570    -225.3641     124.995
          8  |  -277.6448   125.7817    -2.21   0.031    -528.5723   -26.71722
          9  |  -416.3524   170.6884    -2.44   0.017    -756.8664   -75.83833
         10  |  -308.4419   136.6048    -2.26   0.027    -580.9609   -35.92287
         11  |   -505.105   195.0107    -2.59   0.012    -894.1407   -116.0693
         12  |   -449.495   199.1151    -2.26   0.027    -846.7187   -52.27124
         13  |  -392.9856    169.445    -2.32   0.023    -731.0191   -54.95202
         14  |   11.16866   88.59901     0.13   0.900    -165.5815    187.9189
         15  |  -391.7805   173.2067    -2.26   0.027    -737.3183   -46.24264
         16  |  -234.1561   107.3493    -2.18   0.033    -448.3122   -20.00003
         17  |  -239.3291   108.0372    -2.22   0.030    -454.8574   -23.80092
         18  |    -147.99   91.79161    -1.61   0.111    -331.1092    35.12926
         19  |  -312.3217   136.3338    -2.29   0.025    -584.3002   -40.34317
         20  |   25.15346   97.92599     0.26   0.798    -170.2035    220.5105
         21  |  -349.5851    147.281    -2.37   0.020    -643.4027   -55.76747
         22  |  -380.1412   175.3342    -2.17   0.034    -729.9234   -30.35894
         23  |  -382.9687   158.8217    -2.41   0.019    -699.8094   -66.12807
         24  |   35.98155   93.88042     0.38   0.703    -151.3048    223.2679
         25  |  -180.8916   132.1242    -1.37   0.175    -444.4721    82.68895
         26  |  -601.6172   222.2961    -2.71   0.009    -1045.086   -158.1486
         27  |  -586.0981   220.7133    -2.66   0.010    -1026.409    -145.787
         28  |  -440.4422   200.9237    -2.19   0.032     -841.274   -39.61038
         29  |  -156.5608   119.7936    -1.31   0.196    -395.5425    82.42099
         30  |  -470.1871   196.1553    -2.40   0.019    -861.5063   -78.86788
         31  |  -410.9383   187.8866    -2.19   0.032    -785.7618   -36.11484
         32  |  -319.8402   150.8035    -2.12   0.038    -620.6849   -18.99544
         33  |  -479.2399   195.3888    -2.45   0.017    -869.0298   -89.44996
         34  |  -219.4483   139.3446    -1.57   0.120     -497.433    58.53652
         35  |  -363.2407   151.2261    -2.40   0.019    -664.9285   -61.55288
         36  |  -296.8026   137.9301    -2.15   0.035    -571.9657   -21.63953
         37  |  -295.5093   138.1242    -2.14   0.036    -571.0596    -19.9591
         38  |  -361.2244   147.1716    -2.45   0.017    -654.8238   -67.62499
         39  |  -304.5621   136.9616    -2.22   0.029    -577.7931   -31.33122
         40  |  -145.4035   91.84726    -1.58   0.118    -328.6337    37.82678
         41  |  -435.7512   170.6498    -2.55   0.013    -776.1883   -95.31408
         42  |  -152.8338   151.7216    -1.01   0.317    -455.5101    149.8425
         43  |   127.6499   142.1627     0.90   0.372    -155.9569    411.2566
         44  |  -61.58286   98.68554    -0.62   0.535    -258.4551    135.2894
         45  |    -87.689   84.05339    -1.04   0.300    -255.3709    79.99292
         46  |  -314.6672   151.8759    -2.07   0.042    -617.6514   -11.68296
         47  |  -362.5176   147.2041    -2.46   0.016    -656.1819    -68.8534
         48  |  -332.7728   148.7086    -2.24   0.028    -629.4383   -36.10717
         49  |    -283.87   140.2812    -2.02   0.047    -563.7234   -4.016691
         50  |  -282.5768   140.5656    -2.01   0.048    -562.9975   -2.156075
         51  |  -311.0284   136.4145    -2.28   0.026     -583.168   -38.88884
         52  |  -140.2305   92.12985    -1.52   0.133    -324.0245    43.56357
         53  |  -265.2824   194.2952    -1.37   0.177    -652.8907    122.3258
         54  |  -75.32664   137.6672    -0.55   0.586    -349.9651    199.3118
         55  |  -119.7794   86.91863    -1.38   0.173    -293.1773    53.61853
         56  |  -114.1243   116.2523    -0.98   0.330    -346.0413    117.7927
         57  |   114.4763   173.5277     0.66   0.512    -231.7019    460.6545
         58  |  -87.20696   106.9888    -0.82   0.418    -300.6438    126.2299
         59  |  -193.8242   126.9578    -1.53   0.131    -447.0981    59.44982
         60  |  -113.6423   169.2806    -0.67   0.504    -451.3478    224.0633
         61  |  -66.75588   96.58715    -0.69   0.492     -259.442    125.9302
         62  |   41.15457   95.98544     0.43   0.669    -150.3311    232.6403
         63  |   43.25905   94.29183     0.46   0.648     -144.848    231.3661
         64  |  -387.4187   217.2454    -1.78   0.079    -820.8114    45.97399
         65  |   1.545669   110.5922     0.01   0.989    -219.0798    222.1712
         66  |   110.5965   175.6155     0.63   0.531    -239.7469    460.9399
         67  |  -195.6876   205.2911    -0.95   0.344    -605.2322     213.857
         68  |   30.08547    114.948     0.26   0.794    -199.2295    259.4005
         69  |  -229.3123   185.7326    -1.23   0.221    -599.8388    141.2142
         70  |  -86.96594   130.6616    -0.67   0.508    -347.6287    173.6968
         71  |   203.5581   297.0198     0.69   0.495    -388.9801    796.0963
         72  |  -26.66495   116.6412    -0.23   0.820    -259.3579     206.028
         73  |  -34.42448   112.1621    -0.31   0.760    -258.1817    189.3327
         74  |  -309.2531   174.7066    -1.77   0.081    -657.7833    39.27703
------------------------------------------------------------------------------

.
. *BY HAND USING EXPRESSION

.
. margins, expression(_b[mpg]+ 2*_b[c.mpg#c.mpg]*mpg + _b[c.mpg#c.weight]*weight) over(obs)
Warning: expression() does not contain predict() or xb().

Predictive margins                              Number of obs     =         74
Model VCE    : OLS

Expression   : _b[mpg]+ 2*_b[c.mpg#c.mpg]*mpg + _b[c.mpg#c.weight]*weight
over         : obs

------------------------------------------------------------------------------
             |            Delta-method
             |     Margin   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         obs |
          1  |  -201.8247   107.4765    -1.88   0.060    -412.4748    8.825407
          2  |  -332.5317   167.1118    -1.99   0.047    -660.0648   -4.998676
          3  |  -164.3203   116.7984    -1.41   0.159    -393.2409    64.60029
          4  |   -273.765   126.0757    -2.17   0.030    -520.8689   -26.66113
          5  |  -457.4955   182.5663    -2.51   0.012    -815.3189   -99.67218
          6  |  -358.6379   147.1334    -2.44   0.015    -647.0141   -70.26167
          7  |  -50.18457    87.8117    -0.57   0.568    -222.2924    121.9232
          8  |  -277.6448   125.7817    -2.21   0.027    -524.1723   -31.11725
          9  |  -416.3524   170.6884    -2.44   0.015    -750.8955   -81.80925
         10  |  -308.4419   136.6048    -2.26   0.024    -576.1823    -40.7015
         11  |   -505.105   195.0107    -2.59   0.010     -887.319    -122.891
         12  |   -449.495   199.1151    -2.26   0.024    -839.7534   -59.23657
         13  |  -392.9856    169.445    -2.32   0.020    -725.0917   -60.87945
         14  |   11.16866     88.599     0.13   0.900    -162.4822    184.8195
         15  |  -391.7805   173.2067    -2.26   0.024    -731.2593   -52.30166
         16  |  -234.1561   107.3493    -2.18   0.029     -444.557   -23.75526
         17  |  -239.3291   108.0372    -2.22   0.027    -451.0781   -27.58021
         18  |    -147.99   91.79161    -1.61   0.107    -327.8982    31.91826
         19  |  -312.3217   136.3338    -2.29   0.022     -579.531   -45.11232
         20  |   25.15346   97.92599     0.26   0.797     -166.778    217.0849
         21  |  -349.5851    147.281    -2.37   0.018    -638.2506   -60.91958
         22  |  -380.1412   175.3342    -2.17   0.030      -723.79   -36.49238
         23  |  -382.9687   158.8217    -2.41   0.016    -694.2536   -71.68388
         24  |   35.98155   93.88042     0.38   0.702    -148.0207    219.9838
         25  |  -180.8916   132.1242    -1.37   0.171    -439.8503    78.06706
         26  |  -601.6172   222.2961    -2.71   0.007     -1037.31   -165.9248
         27  |  -586.0981   220.7133    -2.66   0.008    -1018.688   -153.5079
         28  |  -440.4422   200.9237    -2.19   0.028    -834.2454   -46.63898
         29  |  -156.5608   119.7936    -1.31   0.191    -391.3519    78.23043
         30  |  -470.1871   196.1553    -2.40   0.017    -854.6445   -85.72967
         31  |  -410.9383   187.8866    -2.19   0.029    -779.1893   -42.68738
         32  |  -319.8402   150.8035    -2.12   0.034    -615.4096   -24.27076
         33  |  -479.2399   195.3888    -2.45   0.014    -862.1948   -96.28494
         34  |  -219.4483   139.3446    -1.57   0.115    -492.5586    53.66205
         35  |  -363.2407   151.2261    -2.40   0.016    -659.6384   -66.84299
         36  |  -296.8026   137.9301    -2.15   0.031    -567.1407   -26.46452
         37  |  -295.5093   138.1242    -2.14   0.032    -566.2278   -24.79088
         38  |  -361.2244   147.1716    -2.45   0.014    -649.6755   -72.77327
         39  |  -304.5621   136.9616    -2.22   0.026    -573.0019   -36.12233
         40  |  -145.4035   91.84726    -1.58   0.113    -325.4208    34.61383
         41  |  -435.7512   170.6498    -2.55   0.011    -770.2187   -101.2837
         42  |  -152.8338   151.7216    -1.01   0.314    -450.2027     144.535
         43  |   127.6499   142.1627     0.90   0.369    -150.9838    406.2836
         44  |  -61.58286   98.68554    -0.62   0.533     -255.003    131.8372
         45  |    -87.689   84.05339    -1.04   0.297    -252.4306    77.05261
         46  |  -314.6672   151.8759    -2.07   0.038    -612.3385   -16.99579
         47  |  -362.5176   147.2041    -2.46   0.014    -651.0325   -74.00281
         48  |  -332.7728   148.7086    -2.24   0.025    -624.2363   -41.30921
         49  |    -283.87   140.2812    -2.02   0.043    -558.8162   -8.923928
         50  |  -282.5768   140.5656    -2.01   0.044    -558.0803   -7.073261
         51  |  -311.0284   136.4145    -2.28   0.023     -578.396   -43.66082
         52  |  -140.2305   92.12985    -1.52   0.128    -320.8016    40.34073
         53  |  -265.2824   194.2952    -1.37   0.172    -646.0939    115.5291
         54  |  -75.32664   137.6672    -0.55   0.584    -345.1493     194.496
         55  |  -119.7794   86.91863    -1.38   0.168    -290.1368      50.578
         56  |  -114.1243   116.2523    -0.98   0.326    -341.9746     113.726
         57  |   114.4763   173.5277     0.66   0.509    -225.6317    454.5842
         58  |  -87.20696   106.9888    -0.82   0.415    -296.9012    122.4873
         59  |  -193.8242   126.9578    -1.53   0.127     -442.657    55.00865
         60  |  -113.6423   169.2806    -0.67   0.502    -445.4261    218.1416
         61  |  -66.75588   96.58716    -0.69   0.489    -256.0632    122.5515
         62  |   41.15457   95.98544     0.43   0.668    -146.9734    229.2826
         63  |   43.25905   94.29183     0.46   0.646    -141.5495    228.0676
         64  |  -387.4187   217.2454    -1.78   0.075    -813.2118    38.37444
         65  |   1.545669   110.5921     0.01   0.989    -215.2109    218.3022
         66  |   110.5965   175.6155     0.63   0.529    -233.6036    454.7967
         67  |  -195.6876   205.2911    -0.95   0.340    -598.0508    206.6756
         68  |   30.08547    114.948     0.26   0.794    -195.2085    255.3794
         69  |  -229.3123   185.7326    -1.23   0.217    -593.3416     134.717
         70  |  -86.96594   130.6616    -0.67   0.506     -343.058    169.1261
         71  |   203.5581   297.0197     0.69   0.493    -378.5899    785.7061
         72  |  -26.66495   116.6412    -0.23   0.819    -255.2776    201.9477
         73  |  -34.42448   112.1621    -0.31   0.759    -254.2581    185.4091
         74  |  -309.2531   174.7066    -1.77   0.077    -651.6718    33.16554
------------------------------------------------------------------------------

[Rudolf Lenger]

Hi Andrew,

thank you very much that was very helpful!

[Marghe Neri]

Hi Andrew, do you have a suggestion for exporting the results of your margins command with the expression() option (e.g. using esttab)?

[Andrew Musau]

estout (esttab) is from SSC.

sysuse auto, clear
regress price c.mpg##c.mpg c.weight c.mpg#c.weight
gen obs_no=_n
margins, expression(_b[mpg]+ 2*_b[c.mpg#c.mpg]*mpg + _b[c.mpg#c.weight]*weight) over(obs_no) post
mat toexport= r(table)[1..6, 1...]'
esttab mat(toexport), nomtitle

Res.:

. esttab mat(toexport), nomtitle

------------------------------------------------------------------------------------------
                        b           se            z       pvalue           ll           ul
------------------------------------------------------------------------------------------
1.obs_no        -201.8247     107.4765    -1.877849     .0604018    -412.4748     8.825407
2.obs_no        -332.5317     167.1118    -1.989876     .0466046    -660.0648    -4.998676
3.obs_no        -164.3203     116.7984    -1.406872     .1594655    -393.2409     64.60029
4.obs_no         -273.765     126.0757    -2.171433     .0298984    -520.8689    -26.66113
5.obs_no        -457.4955     182.5663    -2.505915     .0122135    -815.3189    -99.67218
6.obs_no        -358.6379     147.1334    -2.437501     .0147892    -647.0141    -70.26167
7.obs_no        -50.18457      87.8117    -.5715021     .5676594    -222.2924     121.9232
8.obs_no        -277.6448     125.7817    -2.207355     .0272893    -524.1723    -31.11725
9.obs_no        -416.3524     170.6884    -2.439254     .0147176    -750.8955    -81.80925
10.obs_no       -308.4419     136.6048    -2.257915      .023951    -576.1823     -40.7015
11.obs_no        -505.105     195.0107     -2.59014     .0095937     -887.319     -122.891
12.obs_no        -449.495     199.1151    -2.257463     .0239791    -839.7534    -59.23657
13.obs_no       -392.9856      169.445    -2.319251     .0203814    -725.0917    -60.87945
14.obs_no        11.16866       88.599     .1260586     .8996856    -162.4822     184.8195
15.obs_no       -391.7805     173.2067    -2.261925      .023702    -731.2593    -52.30166
16.obs_no       -234.1561     107.3493    -2.181253     .0291647     -444.557    -23.75526
17.obs_no       -239.3291     108.0372    -2.215249      .026743    -451.0781    -27.58021
18.obs_no         -147.99     91.79161    -1.612239       .10691    -327.8982     31.91826
19.obs_no       -312.3217     136.3338     -2.29086     .0219715     -579.531    -45.11232
20.obs_no        25.15346     97.92599      .256862     .7972853     -166.778     217.0849
21.obs_no       -349.5851      147.281    -2.373592      .017616    -638.2506    -60.91958
22.obs_no       -380.1412     175.3342    -2.168094     .0301515      -723.79    -36.49238
23.obs_no       -382.9687     158.8217    -2.411312     .0158952    -694.2536    -71.68388
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------------------------------------------------------------------------------------------

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