xtsf1g2g Illustration and Replication
In this article, the functionality of the commands xtsf1g and xtsf2g is showcased. At the same time, the code provided below can be used to replicate the results in the chapter:
Stochastic frontier analysis in Stata: using existing and coding new commands in “Handbook of Research Methods and Applications in Empirical Microeconomics” (edited by Nigar Hashimzade and Michael A. Thornton), 2021, Chapter 17, Edward Elgar Publishing, DOI
Throughout the article, the subset of the banking data (which can be found here) will be used:
Define specifications
Here are the specification and formulas for the first derivatives of the cost function with respect to input prices and outputs to check monotonicity assumptions and compute returns to scale are defined:
global spetech "lny1 lny2 lnw1 trend c.half#(c.lny1#c.lny1 c.lny2#c.lny2 c.lnw1#c.lnw1) c.lny1#(c.lny2 c.lnw1) c.lny2#c.lnw1 c.trend#(c.lny1 c.lny2 c.lnw1 c.trend#c.half)"
global monotonW1 = "_b[lnw1] + _b[c.half#c.lnw1#c.lnw1] *lnw1 + _b[c.lny1#c.lnw1] *lny1 + _b[c.lny2#c.lnw1] *lny2 + _b[c.trend#c.lnw1] *trend"
global monotonY1 = "_b[lny1] + _b[c.half#c.lny1#c.lny1] *lny1 + _b[c.lny1#c.lny2] *lny2 + _b[c.lny1#c.lnw1] *lnw1 + _b[c.trend#c.lny1] *trend"
global monotonY2 = "_b[lny2] + _b[c.half#c.lny2#c.lny2] *lny2 + _b[c.lny1#c.lny2] *lny1 + _b[c.lny2#c.lnw1] *lnw1 + _b[c.trend#c.lny2] *trend"
global year_c = 2001
global itermax = 1000
global decimalopt "cformat(%9.3f) pformat(%5.2f) sformat(%8.2f)"No determinants of inefficiency
In the first 8 models the specification models don’t include determinants of inefficiency.
The inefficiency follow the uit = β(t)ui. In the 4 models below, β(t) is defined as follows:
- β(t) = 1 (time-invariant inefficiency)
- β(t) = (1+exp(γt+δt2))−1 (time-variant inefficiency)
- β(t) = 1 + γ(t−Ti) + δ(t−Ti)2 (time-variant inefficiency)
- β(t) = exp (−γ(t−Ti)) (time-variant inefficiency)
Models 1-4: Normal-half normal
Here are the models with normal-half normal assumptions.
Model 1 (1st generation, time-invariant inefficiency)
We start by the simplest one where uit = ui.
Code for Model 1 is detailed. For the rest of the models, fewer details are shown
timer clear 1
timer on 1
xtsf1g lnc $spetech if year > $year_c, cost iterate($itermax) $decimalopt
timer off 1
estadd scalar AIC = e(aic)
estadd scalar BIC = e(bic)
eststo M1
quietly predictnl M1_mW1 = $monotonW1 if e(sample)
quietly predictnl M1_mW2 = 1-M1_mW1 if e(sample)
quietly predictnl M1_eY1 = $monotonY1 if e(sample)
quietly predictnl M1_eY2 = $monotonY2 if e(sample)
quietly generate double M1_rts = 1/(M1_eY1 + M1_eY2)
tabstat M1_*, stat(min p25 mean p75 max sd) format(%9.4f)
. timer clear 1
. timer on 1
. xtsf1g lnc $spetech if year > $year_c, cost iterate($itermax) $decimalopt
Log-likelihood maximization using optimize()
iter 0: Log-likelihood = 685.89191
iter 1: Log-likelihood = 688.21588
iter 2: Log-likelihood = 1222.8627
iter 3: Log-likelihood = 1428.5402
iter 4: Log-likelihood = 1430.2777
iter 5: Log-likelihood = 1430.2788
iter 6: Log-likelihood = 1430.2788
Sample:----------------------
Number of obs = 2546
Number of groups = 500
Diagnostics:-----------------
R-squared = 0.9987
Adj R-squared = 0.9984
AIC = -2.3400
BIC = -2.3033
Root MSE = 0.1871
-----------------------------
The first-generation estimator of
the cost stochastic frontier model for panel data,
where effciency is time-invariant
--------------------------------------------------------------------------------------
lnc | Coefficient Std. err. z P>|z| [95% conf. interval]
---------------------+----------------------------------------------------------------
lny1 | 0.397 0.142 2.79 0.01 0.118 0.675
lny2 | -0.619 0.399 -1.55 0.12 -1.401 0.163
lnw1 | 0.385 0.166 2.32 0.02 0.059 0.711
trend | -0.561 0.040 -14.00 0.00 -0.640 -0.483
|
c.half#c.lny1#c.lny1 | 0.049 0.005 10.29 0.00 0.040 0.059
|
c.half#c.lny2#c.lny2 | 0.178 0.032 5.54 0.00 0.115 0.241
|
c.half#c.lnw1#c.lnw1 | -0.069 0.014 -4.96 0.00 -0.096 -0.042
|
c.lny1#c.lny2 | -0.067 0.011 -5.86 0.00 -0.089 -0.045
|
c.lny1#c.lnw1 | -0.003 0.006 -0.45 0.65 -0.013 0.008
|
c.lny2#c.lnw1 | -0.009 0.013 -0.68 0.50 -0.035 0.017
|
c.trend#c.lny1 | 0.005 0.002 3.11 0.00 0.002 0.009
|
c.trend#c.lny2 | 0.012 0.003 3.95 0.00 0.006 0.018
|
c.trend#c.lnw1 | -0.007 0.003 -2.60 0.01 -0.012 -0.002
|
c.trend#c.trend#|
c.half | 0.076 0.002 42.70 0.00 0.073 0.080
|
_cons | 5.383 2.642 2.04 0.04 0.205 10.561
---------------------+----------------------------------------------------------------
ln[var(vit)] |
_cons | -4.422 0.032 -138.37 0.00 -4.485 -4.359
---------------------+----------------------------------------------------------------
ln[var(ui)] |
_cons | -2.522 0.084 -30.11 0.00 -2.686 -2.358
--------------------------------------------------------------------------------------
. timer off 1
. estadd scalar AIC = e(aic)
added scalar:
e(AIC) = -2.3400425
. estadd scalar BIC = e(bic)
added scalar:
e(BIC) = -2.3033274
. eststo M1
. quietly predictnl M1_mW1 = $monotonW1 if e(sample)
. quietly predictnl M1_mW2 = 1-M1_mW1 if e(sample)
. quietly predictnl M1_eY1 = $monotonY1 if e(sample)
. quietly predictnl M1_eY2 = $monotonY2 if e(sample)
. quietly generate double M1_rts = 1/(M1_eY1 + M1_eY2)
. tabstat M1_*, stat(min p25 mean p75 max sd) format(%9.4f)
Stats | M1_mW1 M1_mW2 M1_eY1 M1_eY2 M1_rts
---------+--------------------------------------------------
Min | -0.1766 0.8766 -0.1627 0.3993 0.8636
p25 | -0.0165 0.9660 0.1332 0.6781 1.0478
Mean | 0.0067 0.9933 0.1615 0.7475 1.1062
p75 | 0.0340 1.0165 0.1942 0.8131 1.1583
Max | 0.1234 1.1766 0.3200 1.2801 1.4827
SD | 0.0408 0.0408 0.0537 0.1074 0.0824
------------------------------------------------------------Model 2 (2nd generation, time-varying inefficiency using the Kumbhakar (1990) specification)
This specification is by Kumbhakar (1990)
timer clear 2
timer on 2
xtsf2g lnc $spetech if year > $year_c, cost iterate($itermax) $decimalopt
timer off 2
estadd scalar AIC = e(aic)
estadd scalar BIC = e(bic)
eststo M2
quietly predictnl M2_mW1 = $monotonW1 if e(sample)
quietly predictnl M2_mW2 = 1-M2_mW1 if e(sample)
quietly predictnl M2_eY1 = $monotonY1 if e(sample)
quietly predictnl M2_eY2 = $monotonY2 if e(sample)
quietly generate double M2_rts = 1/(M2_eY1 + M2_eY2)
tabstat M2_*, stat(min p25 mean p75 max sd) format(%9.4f)
. timer clear 2
. timer on 2
. xtsf2g lnc $spetech if year > $year_c, cost iterate($itermax) $decimalopt
Log-likelihood maximization using optimize()
iter 0: Log-likelihood = 716.15322 (not concave)
iter 1: Log-likelihood = 1171.048 (not concave)
iter 2: Log-likelihood = 1300.934 (not concave)
iter 3: Log-likelihood = 1372.054
iter 4: Log-likelihood = 1444.2813
iter 5: Log-likelihood = 1454.265
iter 6: Log-likelihood = 1480.6447
iter 7: Log-likelihood = 1489.7689
iter 8: Log-likelihood = 1491.8923
iter 9: Log-likelihood = 1492.2628
iter 10: Log-likelihood = 1492.3095
iter 11: Log-likelihood = 1492.3115
iter 12: Log-likelihood = 1492.3115
Sample:----------------------
Number of obs = 2546
Number of groups = 500
Diagnostics:-----------------
R-squared = 0.9986
Adj R-squared = 0.9983
AIC = -2.3091
BIC = -2.2724
Root MSE = 0.1900
-----------------------------
The second-generation estimator of
the cost stochastic frontier model for panel data,
where effciency is time-varying
--------------------------------------------------------------------------------------
lnc | Coefficient Std. err. z P>|z| [95% conf. interval]
---------------------+----------------------------------------------------------------
lny1 | 0.429 0.140 3.06 0.00 0.154 0.703
lny2 | -0.483 0.393 -1.23 0.22 -1.254 0.287
lnw1 | 0.497 0.162 3.06 0.00 0.179 0.815
trend | -0.656 0.040 -16.25 0.00 -0.735 -0.577
|
c.half#c.lny1#c.lny1 | 0.047 0.005 10.17 0.00 0.038 0.056
|
c.half#c.lny2#c.lny2 | 0.171 0.031 5.45 0.00 0.110 0.233
|
c.half#c.lnw1#c.lnw1 | -0.062 0.013 -4.60 0.00 -0.088 -0.035
|
c.lny1#c.lny2 | -0.069 0.011 -6.14 0.00 -0.090 -0.047
|
c.lny1#c.lnw1 | 0.000 0.005 0.02 0.98 -0.010 0.011
|
c.lny2#c.lnw1 | -0.024 0.013 -1.80 0.07 -0.049 0.002
|
c.trend#c.lny1 | 0.005 0.002 3.23 0.00 0.002 0.009
|
c.trend#c.lny2 | 0.013 0.003 4.36 0.00 0.007 0.019
|
c.trend#c.lnw1 | -0.006 0.003 -2.29 0.02 -0.011 -0.001
|
c.trend#c.trend#|
c.half | 0.094 0.003 32.03 0.00 0.088 0.099
|
_cons | 4.497 2.617 1.72 0.09 -0.633 9.627
---------------------+----------------------------------------------------------------
ln[var(vit)] |
_cons | -4.498 0.032 -139.00 0.00 -4.561 -4.434
---------------------+----------------------------------------------------------------
ln[var(ui)] |
_cons | -2.315 0.089 -26.04 0.00 -2.489 -2.141
---------------------+----------------------------------------------------------------
beta[t] |
gamma | -5.000 1.898 -2.63 0.01 -8.721 -1.280
delta | 0.819 0.315 2.60 0.01 0.203 1.436
--------------------------------------------------------------------------------------
Note, the function of inefficiency change over time is
beta[t] = ( 1 + exp(gamma*t + delta*t^2) )^-1:
the larger is the beta[t] the larger is the inefficiency
. timer off 2
. estadd scalar AIC = e(aic)
added scalar:
e(AIC) = -2.3090757
. estadd scalar BIC = e(bic)
added scalar:
e(BIC) = -2.2723607
. eststo M2
. quietly predictnl M2_mW1 = $monotonW1 if e(sample)
. quietly predictnl M2_mW2 = 1-M2_mW1 if e(sample)
. quietly predictnl M2_eY1 = $monotonY1 if e(sample)
. quietly predictnl M2_eY2 = $monotonY2 if e(sample)
. quietly generate double M2_rts = 1/(M2_eY1 + M2_eY2)
. tabstat M2_*, stat(min p25 mean p75 max sd) format(%9.4f)
Stats | M2_mW1 M2_mW2 M2_eY1 M2_eY2 M2_rts
---------+--------------------------------------------------
Min | -0.1651 0.8692 -0.1501 0.4225 0.8582
p25 | -0.0112 0.9664 0.1276 0.6782 1.0534
Mean | 0.0091 0.9909 0.1561 0.7474 1.1129
p75 | 0.0336 1.0112 0.1882 0.8137 1.1644
Max | 0.1308 1.1651 0.3162 1.2823 1.4384
SD | 0.0371 0.0371 0.0525 0.1077 0.0834
------------------------------------------------------------Model 3 (2nd generation, time-varying inefficiency using the modified Kumbhakar (1990) specification)
timer clear 3
timer on 3
xtsf2g lnc $spetech if year > $year_c, model(K1990modified) cost iterate($itermax) $decimalopt
timer off 3
predict M3_te, te_jlms_mean
estadd scalar AIC = e(aic)
estadd scalar BIC = e(bic)
eststo M3
quietly predictnl M3_mW1 = $monotonW1 if e(sample)
quietly predictnl M3_mW2 = 1-M3_mW1 if e(sample)
quietly predictnl M3_eY1 = $monotonY1 if e(sample)
quietly predictnl M3_eY2 = $monotonY2 if e(sample)
generate double M3_rts = 1/(M3_eY1 + M3_eY2)
tabstat M3_*, stat(min p25 mean p75 max sd) format(%9.4f)
. timer clear 3
. timer on 3
. xtsf2g lnc $spetech if year > $year_c, model(K1990modified) cost iterate($itermax) $decimalopt
Log-likelihood maximization using optimize()
iter 0: Log-likelihood = 685.89191
iter 1: Log-likelihood = 1134.4188 (not concave)
iter 2: Log-likelihood = 1295.1247 (not concave)
iter 3: Log-likelihood = 1327.2692 (not concave)
iter 4: Log-likelihood = 1337.2619 (not concave)
iter 5: Log-likelihood = 1346.7248 (not concave)
iter 6: Log-likelihood = 1353.5166 (not concave)
iter 7: Log-likelihood = 1360.746 (not concave)
iter 8: Log-likelihood = 1377.9014
iter 9: Log-likelihood = 1419.4682
iter 10: Log-likelihood = 1443.3139
iter 11: Log-likelihood = 1443.6859
iter 12: Log-likelihood = 1443.6866
iter 13: Log-likelihood = 1443.6866
Sample:----------------------
Number of obs = 2546
Number of groups = 500
Diagnostics:-----------------
R-squared = 0.9987
Adj R-squared = 0.9984
AIC = -2.3335
BIC = -2.2968
Root MSE = 0.1877
-----------------------------
The second-generation estimator of
the cost stochastic frontier model for panel data,
where effciency is time-varying
--------------------------------------------------------------------------------------
lnc | Coefficient Std. err. z P>|z| [95% conf. interval]
---------------------+----------------------------------------------------------------
lny1 | 0.475 0.142 3.35 0.00 0.197 0.753
lny2 | -0.563 0.394 -1.43 0.15 -1.336 0.210
lnw1 | 0.393 0.165 2.39 0.02 0.071 0.716
trend | -0.592 0.041 -14.62 0.00 -0.672 -0.513
|
c.half#c.lny1#c.lny1 | 0.047 0.005 10.12 0.00 0.038 0.057
|
c.half#c.lny2#c.lny2 | 0.180 0.032 5.68 0.00 0.118 0.242
|
c.half#c.lnw1#c.lnw1 | -0.066 0.014 -4.82 0.00 -0.093 -0.039
|
c.lny1#c.lny2 | -0.073 0.011 -6.42 0.00 -0.095 -0.050
|
c.lny1#c.lnw1 | 0.000 0.005 0.01 0.99 -0.011 0.011
|
c.lny2#c.lnw1 | -0.013 0.013 -0.98 0.33 -0.039 0.013
|
c.trend#c.lny1 | 0.005 0.002 2.75 0.01 0.001 0.008
|
c.trend#c.lny2 | 0.013 0.003 4.26 0.00 0.007 0.019
|
c.trend#c.lnw1 | -0.006 0.003 -2.35 0.02 -0.011 -0.001
|
c.trend#c.trend#|
c.half | 0.082 0.002 35.30 0.00 0.078 0.087
|
_cons | 4.705 2.631 1.79 0.07 -0.450 9.861
---------------------+----------------------------------------------------------------
ln[var(vit)] |
_cons | -4.435 0.032 -138.89 0.00 -4.498 -4.373
---------------------+----------------------------------------------------------------
ln[var(ui)] |
_cons | -2.757 0.100 -27.63 0.00 -2.953 -2.561
---------------------+----------------------------------------------------------------
beta[t] |
gamma | -0.125 0.028 -4.54 0.00 -0.179 -0.071
delta | -0.020 0.005 -3.77 0.00 -0.030 -0.009
--------------------------------------------------------------------------------------
Note, the function of inefficiency change over time is
beta[t] = 1 + gamma*(t-T_i) + delta*(t-T_i)^2:
the larger is the beta[t] the larger is the inefficiency
. timer off 3
. predict M3_te, te_jlms_mean
(872 missing values generated)
. estadd scalar AIC = e(aic)
added scalar:
e(AIC) = -2.3335014
. estadd scalar BIC = e(bic)
added scalar:
e(BIC) = -2.2967864
. eststo M3
. quietly predictnl M3_mW1 = $monotonW1 if e(sample)
. quietly predictnl M3_mW2 = 1-M3_mW1 if e(sample)
. quietly predictnl M3_eY1 = $monotonY1 if e(sample)
. quietly predictnl M3_eY2 = $monotonY2 if e(sample)
. generate double M3_rts = 1/(M3_eY1 + M3_eY2)
(872 missing values generated)
. tabstat M3_*, stat(min p25 mean p75 max sd) format(%9.4f)
Stats | M3_te M3_mW1 M3_mW2 M3_eY1 M3_eY2 M3_rts
---------+------------------------------------------------------------
Min | 1.0071 -0.1710 0.8828 -0.1547 0.4048 0.8383
p25 | 1.1348 -0.0143 0.9658 0.1307 0.6817 1.0432
Mean | 1.2764 0.0076 0.9924 0.1597 0.7534 1.1011
p75 | 1.3506 0.0342 1.0143 0.1928 0.8209 1.1513
Max | 2.8145 0.1172 1.1710 0.3257 1.3157 1.4546
SD | 0.2080 0.0391 0.0391 0.0540 0.1114 0.0818
----------------------------------------------------------------------Model 4 (2nd generation, time-varying inefficiency using the BC specification)
timer clear 4
timer on 4
xtsf2g lnc $spetech if year > $year_c, model(BC1992) cost iterate($itermax) $decimalopt
timer off 4
estadd scalar AIC = e(aic)
estadd scalar BIC = e(bic)
eststo M4
quietly predictnl M4_mW1 = $monotonW1 if e(sample)
quietly predictnl M4_mW2 = 1-M4_mW1 if e(sample)
quietly predictnl M4_eY1 = $monotonY1 if e(sample)
quietly predictnl M4_eY2 = $monotonY2 if e(sample)
generate double M4_rts = 1/(M4_eY1 + M4_eY2)
tabstat M4_*, stat(min p25 mean p75 max sd) format(%9.4f)
. timer clear 4
. timer on 4
. xtsf2g lnc $spetech if year > $year_c, model(BC1992) cost iterate($itermax) $decimalopt
Log-likelihood maximization using optimize()
iter 0: Log-likelihood = 685.89191
iter 1: Log-likelihood = 1198.9156
iter 2: Log-likelihood = 1400.7342
iter 3: Log-likelihood = 1434.2974
iter 4: Log-likelihood = 1435.2554
iter 5: Log-likelihood = 1435.2562
iter 6: Log-likelihood = 1435.2562
Sample:----------------------
Number of obs = 2546
Number of groups = 500
Diagnostics:-----------------
R-squared = 0.9987
Adj R-squared = 0.9984
AIC = -2.3349
BIC = -2.2982
Root MSE = 0.1876
-----------------------------
The second-generation estimator of
the cost stochastic frontier model for panel data,
where effciency is time-varying
--------------------------------------------------------------------------------------
lnc | Coefficient Std. err. z P>|z| [95% conf. interval]
---------------------+----------------------------------------------------------------
lny1 | 0.459 0.143 3.22 0.00 0.180 0.739
lny2 | -0.477 0.399 -1.20 0.23 -1.258 0.304
lnw1 | 0.417 0.165 2.52 0.01 0.092 0.741
trend | -0.556 0.040 -13.99 0.00 -0.634 -0.478
|
c.half#c.lny1#c.lny1 | 0.048 0.005 10.18 0.00 0.039 0.057
|
c.half#c.lny2#c.lny2 | 0.171 0.032 5.37 0.00 0.109 0.234
|
c.half#c.lnw1#c.lnw1 | -0.066 0.014 -4.79 0.00 -0.093 -0.039
|
c.lny1#c.lny2 | -0.072 0.011 -6.30 0.00 -0.094 -0.049
|
c.lny1#c.lnw1 | -0.001 0.005 -0.23 0.82 -0.012 0.009
|
c.lny2#c.lnw1 | -0.014 0.013 -1.04 0.30 -0.040 0.012
|
c.trend#c.lny1 | 0.005 0.002 3.04 0.00 0.002 0.009
|
c.trend#c.lny2 | 0.012 0.003 4.06 0.00 0.006 0.018
|
c.trend#c.lnw1 | -0.006 0.003 -2.45 0.01 -0.011 -0.001
|
c.trend#c.trend#|
c.half | 0.076 0.002 42.52 0.00 0.073 0.080
|
_cons | 4.190 2.657 1.58 0.11 -1.018 9.397
---------------------+----------------------------------------------------------------
ln[var(vit)] |
_cons | -4.429 0.032 -138.49 0.00 -4.492 -4.366
---------------------+----------------------------------------------------------------
ln[var(ui)] |
_cons | -2.638 0.092 -28.53 0.00 -2.819 -2.457
---------------------+----------------------------------------------------------------
beta[t] |
gamma | 0.029 0.009 3.16 0.00 0.011 0.048
--------------------------------------------------------------------------------------
Note, the function of inefficiency change over time is
beta[t] = exp( -gamma*(t-T_i) ):
the larger is the beta[t] the larger is the inefficiency
. timer off 4
. estadd scalar AIC = e(aic)
added scalar:
e(AIC) = -2.3348946
. estadd scalar BIC = e(bic)
added scalar:
e(BIC) = -2.2981796
. eststo M4
. quietly predictnl M4_mW1 = $monotonW1 if e(sample)
. quietly predictnl M4_mW2 = 1-M4_mW1 if e(sample)
. quietly predictnl M4_eY1 = $monotonY1 if e(sample)
. quietly predictnl M4_eY2 = $monotonY2 if e(sample)
. generate double M4_rts = 1/(M4_eY1 + M4_eY2)
(872 missing values generated)
. tabstat M4_*, stat(min p25 mean p75 max sd) format(%9.4f)
Stats | M4_mW1 M4_mW2 M4_eY1 M4_eY2 M4_rts
---------+--------------------------------------------------
Min | -0.1709 0.8793 -0.1587 0.4184 0.8521
p25 | -0.0141 0.9655 0.1305 0.6800 1.0495
Mean | 0.0081 0.9919 0.1601 0.7494 1.1050
p75 | 0.0345 1.0141 0.1932 0.8137 1.1541
Max | 0.1207 1.1709 0.3255 1.2968 1.4286
SD | 0.0391 0.0391 0.0542 0.1077 0.0784
------------------------------------------------------------Results of Models 1-4
Use the estout command for this. Note that parameters δ and γ have different interpretation in models 2-4.
estout M1 M2 M3 M4, ///
cells(b(star fmt(%9.4f)) se(par)) ///
stats(AIC BIC shat RSS ll N sumTi, ///
labels("AIC" "BIC" "\$\hat\sigma\$" "RSS" "log-likelihood" "\$N\$" "\$\sum T_{i}\$") ///
fmt(%9.4f %9.4f %9.4f %9.2f %9.2f %9.0f %9.0f)) ///
starlevels(* 0.10 ** 0.05 *** 0.01) ///
varlabels(_cons Constant ) ///
substitute("_ " "Frontier") ///
legend label collabels(none) mlabels(none) replace
------------------------------------------------------------------------------------
Frontier
lny1 0.3966*** 0.4286*** 0.4747*** 0.4592***
(0.1422) (0.1403) (0.1418) (0.1427)
lny2 -0.6190 -0.4833 -0.5632 -0.4770
(0.3989) (0.3933) (0.3945) (0.3985)
lnw1 0.3852** 0.4970*** 0.3933** 0.4166**
(0.1663) (0.1625) (0.1646) (0.1654)
trend -0.5613*** -0.6561*** -0.5925*** -0.5562***
(0.0401) (0.0404) (0.0405) (0.0398)
half # lny1 # lny1 0.0495*** 0.0468*** 0.0475*** 0.0482***
(0.0048) (0.0046) (0.0047) (0.0047)
half # lny2 # lny2 0.1783*** 0.1713*** 0.1796*** 0.1713***
(0.0322) (0.0314) (0.0316) (0.0319)
half # lnw1 # lnw1 -0.0687*** -0.0616*** -0.0664*** -0.0659***
(0.0139) (0.0134) (0.0138) (0.0138)
lny1 # lny2 -0.0669*** -0.0686*** -0.0727*** -0.0718***
(0.0114) (0.0112) (0.0113) (0.0114)
lny1 # lnw1 -0.0025 0.0001 0.0000 -0.0013
(0.0055) (0.0053) (0.0055) (0.0055)
lny2 # lnw1 -0.0091 -0.0237* -0.0130 -0.0139
(0.0134) (0.0131) (0.0133) (0.0133)
trend # lny1 0.0054*** 0.0054*** 0.0048*** 0.0053***
(0.0017) (0.0017) (0.0017) (0.0017)
trend # lny2 0.0122*** 0.0131*** 0.0130*** 0.0125***
(0.0031) (0.0030) (0.0031) (0.0031)
trend # lnw1 -0.0068*** -0.0058** -0.0061** -0.0064**
(0.0026) (0.0025) (0.0026) (0.0026)
trend # trend # half 0.0765*** 0.0937*** 0.0823*** 0.0762***
(0.0018) (0.0029) (0.0023) (0.0018)
Constant 5.3832** 4.4970* 4.7054* 4.1896
(2.6418) (2.6172) (2.6305) (2.6571)
------------------------------------------------------------------------------------
ln[var(vit)]
Constant -4.4220*** -4.4975*** -4.4352*** -4.4290***
(0.0320) (0.0324) (0.0319) (0.0320)
------------------------------------------------------------------------------------
ln[var(ui)]
Constant -2.5216*** -2.3149*** -2.7570*** -2.6381***
(0.0837) (0.0889) (0.0998) (0.0925)
------------------------------------------------------------------------------------
beta[t]
gamma -5.0001*** -0.1252*** 0.0295***
(1.8983) (0.0276) (0.0093)
delta 0.8193*** -0.0197***
(0.3145) (0.0052)
------------------------------------------------------------------------------------
AIC -2.3400 -2.3091 -2.3335 -2.3349
BIC -2.3033 -2.2724 -2.2968 -2.2982
$\hat\sigma$ 0.1871 0.1900 0.1877 0.1876
RSS 228.96 250.63 227.45 229.70
log-likelihood 1430.28 1492.31 1443.69 1435.26
$N$ 500 500 500 500
$\sum T_{i}$ 2546 2546 2546 2546
------------------------------------------------------------------------------------
* p<0.10, ** p<0.05, *** p<0.01Models 5-8: Normal-truncated normal
Here are the models with normal-truncated normal assumptions.
Model 5 (1g)
timer clear 5
timer on 5
xtsf1g lnc $spetech if year > $year_c, cost iterate($itermax) distribution(t) $decimalopt
timer off 5
estadd scalar AIC = e(aic)
estadd scalar BIC = e(bic)
eststo M5
quietly predictnl M5_mW5 = $monotonW1 if e(sample)
quietly predictnl M5_mW2 = 1-M5_mW5 if e(sample)
quietly predictnl M5_eY5 = $monotonY1 if e(sample)
quietly predictnl M5_eY2 = $monotonY2 if e(sample)
quietly generate double M5_rts = 1/(M5_eY5 + M5_eY2)
tabstat M5_*, stat(min p25 mean p75 max sd) format(%9.4f)
. timer clear 5
. timer on 5
. xtsf1g lnc $spetech if year > $year_c, cost iterate($itermax) distribution(t) $decimalopt
Log-likelihood maximization using optimize()
iter 0: Log-likelihood = 499.06419 (not concave)
iter 1: Log-likelihood = 1248.0912
iter 2: Log-likelihood = 1377.9019
iter 3: Log-likelihood = 1425.5041
iter 4: Log-likelihood = 1473.1026
iter 5: Log-likelihood = 1489.2684
iter 6: Log-likelihood = 1496.1943
iter 7: Log-likelihood = 1496.4395
iter 8: Log-likelihood = 1496.4627
iter 9: Log-likelihood = 1496.4649
iter 10: Log-likelihood = 1496.4652
iter 11: Log-likelihood = 1496.4652
Sample:----------------------
Number of obs = 2546
Number of groups = 500
Diagnostics:-----------------
R-squared = 0.9905
Adj R-squared = 0.9881
AIC = -2.3421
BIC = -2.3054
Root MSE = 0.1869
-----------------------------
The first-generation estimator of
the cost stochastic frontier model for panel data,
where effciency is time-invariant
--------------------------------------------------------------------------------------
lnc | Coefficient Std. err. z P>|z| [95% conf. interval]
---------------------+----------------------------------------------------------------
lny1 | 0.208 0.141 1.47 0.14 -0.070 0.485
lny2 | -1.735 0.398 -4.36 0.00 -2.516 -0.955
lnw1 | 0.009 0.178 0.05 0.96 -0.341 0.358
trend | -0.515 0.039 -13.32 0.00 -0.591 -0.439
|
c.half#c.lny1#c.lny1 | 0.048 0.005 10.06 0.00 0.038 0.057
|
c.half#c.lny2#c.lny2 | 0.257 0.032 7.95 0.00 0.194 0.321
|
c.half#c.lnw1#c.lnw1 | -0.030 0.015 -2.02 0.04 -0.059 -0.001
|
c.lny1#c.lny2 | -0.050 0.011 -4.46 0.00 -0.072 -0.028
|
c.lny1#c.lnw1 | 0.000 0.005 0.04 0.97 -0.010 0.011
|
c.lny2#c.lnw1 | 0.014 0.014 0.97 0.33 -0.014 0.041
|
c.trend#c.lny1 | 0.004 0.002 2.64 0.01 0.001 0.008
|
c.trend#c.lny2 | 0.009 0.003 3.15 0.00 0.004 0.015
|
c.trend#c.lnw1 | -0.009 0.003 -3.37 0.00 -0.014 -0.004
|
c.trend#c.trend#|
c.half | 0.077 0.002 44.66 0.00 0.073 0.080
|
_cons | 12.540 3.188 3.93 0.00 6.292 18.787
---------------------+----------------------------------------------------------------
ln[var(vit)] |
_cons | -4.512 0.031 -143.66 0.00 -4.574 -4.451
---------------------+----------------------------------------------------------------
ln[var(ui)] |
_cons | -3.646 0.070 -51.84 0.00 -3.784 -3.508
---------------------+----------------------------------------------------------------
E[ui|z] |
_cons | 0.797 1.831 0.44 0.66 -2.792 4.386
--------------------------------------------------------------------------------------
. timer off 5
. estadd scalar AIC = e(aic)
added scalar:
e(AIC) = -2.3420887
. estadd scalar BIC = e(bic)
added scalar:
e(BIC) = -2.3053737
. eststo M5
. quietly predictnl M5_mW5 = $monotonW1 if e(sample)
. quietly predictnl M5_mW2 = 1-M5_mW5 if e(sample)
. quietly predictnl M5_eY5 = $monotonY1 if e(sample)
. quietly predictnl M5_eY2 = $monotonY2 if e(sample)
. quietly generate double M5_rts = 1/(M5_eY5 + M5_eY2)
. tabstat M5_*, stat(min p25 mean p75 max sd) format(%9.4f)
Stats | M5_mW5 M5_mW2 M5_eY5 M5_eY2 M5_rts
---------+--------------------------------------------------
Min | -0.0684 0.9204 -0.1494 0.2310 0.8575
p25 | 0.0091 0.9590 0.1253 0.6525 1.0330
Mean | 0.0239 0.9761 0.1482 0.7411 1.1409
p75 | 0.0410 0.9909 0.1778 0.8319 1.2252
Max | 0.0796 1.0684 0.2884 1.2766 2.1699
SD | 0.0234 0.0234 0.0481 0.1322 0.1430
------------------------------------------------------------Model 6 (2g)
timer clear 6
timer on 6
xtsf2g lnc $spetech if year > $year_c, cost iterate($itermax) distribution(t) $decimalopt
timer off 6
estadd scalar AIC = e(aic)
estadd scalar BIC = e(bic)
eststo M6
quietly predictnl M6_mW1 = $monotonW1 if e(sample)
quietly predictnl M6_mW2 = 1-M6_mW1 if e(sample)
quietly predictnl M6_eY1 = $monotonY1 if e(sample)
quietly predictnl M6_eY2 = $monotonY2 if e(sample)
quietly generate double M6_rts = 1/(M6_eY1 + M6_eY2)
tabstat M6_*, stat(min p25 mean p75 max sd) format(%9.4f)
. timer clear 6
. timer on 6
. xtsf2g lnc $spetech if year > $year_c, cost iterate($itermax) distribution(t) $decimalopt
Log-likelihood maximization using optimize()
iter 0: Log-likelihood = 649.6318 (not concave)
iter 1: Log-likelihood = 1241.5285 (not concave)
iter 2: Log-likelihood = 1322.5145 (not concave)
iter 3: Log-likelihood = 1355.508 (not concave)
iter 4: Log-likelihood = 1392.307 (not concave)
iter 5: Log-likelihood = 1421.9537 (not concave)
iter 6: Log-likelihood = 1433.7357
iter 7: Log-likelihood = 1450.4961 (not concave)
iter 8: Log-likelihood = 1483.9436 (not concave)
iter 9: Log-likelihood = 1488.1369
iter 10: Log-likelihood = 1521.1974
iter 11: Log-likelihood = 1554.7545
...
iter 77: Log-likelihood = 1623.0228
iter 78: Log-likelihood = 1623.0229
iter 79: Log-likelihood = 1623.023
iter 80: Log-likelihood = 1623.023
Sample:----------------------
Number of obs = 2546
Number of groups = 500
Diagnostics:-----------------
R-squared = -0.7572
Adj R-squared = -1.2019
AIC = 1.1384
BIC = 1.1752
Root MSE = 1.0652
-----------------------------
The second-generation estimator of
the cost stochastic frontier model for panel data,
where effciency is time-varying
--------------------------------------------------------------------------------------
lnc | Coefficient Std. err. z P>|z| [95% conf. interval]
---------------------+----------------------------------------------------------------
lny1 | 0.178 0.136 1.31 0.19 -0.089 0.444
lny2 | -1.712 0.382 -4.48 0.00 -2.461 -0.963
lnw1 | 0.073 0.169 0.43 0.66 -0.258 0.405
trend | -4.351 1.593 -2.73 0.01 -7.475 -1.228
|
c.half#c.lny1#c.lny1 | 0.048 0.004 10.82 0.00 0.039 0.057
|
c.half#c.lny2#c.lny2 | 0.255 0.031 8.28 0.00 0.195 0.316
|
c.half#c.lnw1#c.lnw1 | -0.023 0.014 -1.68 0.09 -0.051 0.004
|
c.lny1#c.lny2 | -0.048 0.011 -4.46 0.00 -0.069 -0.027
|
c.lny1#c.lnw1 | 0.001 0.005 0.27 0.79 -0.008 0.011
|
c.lny2#c.lnw1 | 0.005 0.013 0.39 0.70 -0.021 0.032
|
c.trend#c.lny1 | 0.005 0.002 2.94 0.00 0.002 0.008
|
c.trend#c.lny2 | 0.010 0.003 3.55 0.00 0.004 0.015
|
c.trend#c.lnw1 | -0.009 0.002 -3.69 0.00 -0.013 -0.004
|
c.trend#c.trend#|
c.half | 0.831 0.315 2.64 0.01 0.214 1.448
|
_cons | 10.760 3.428 3.14 0.00 4.042 17.478
---------------------+----------------------------------------------------------------
ln[var(vit)] |
_cons | -4.638 0.031 -147.71 0.00 -4.699 -4.576
---------------------+----------------------------------------------------------------
ln[var(ui)] |
_cons | -2.781 0.122 -22.89 0.00 -3.019 -2.543
---------------------+----------------------------------------------------------------
E[ui|z] |
_cons | 17.054 10.102 1.69 0.09 -2.745 36.852
---------------------+----------------------------------------------------------------
beta[t] |
gamma | -0.594 0.129 -4.61 0.00 -0.847 -0.341
delta | 0.096 0.021 4.69 0.00 0.056 0.136
--------------------------------------------------------------------------------------
Note, the function of inefficiency change over time is
beta[t] = ( 1 + exp(gamma*t + delta*t^2) )^-1:
the larger is the beta[t] the larger is the inefficiency
. timer off 6
. estadd scalar AIC = e(aic)
added scalar:
e(AIC) = 1.1384354
. estadd scalar BIC = e(bic)
added scalar:
e(BIC) = 1.1751504
. eststo M6
. quietly predictnl M6_mW1 = $monotonW1 if e(sample)
. quietly predictnl M6_mW2 = 1-M6_mW1 if e(sample)
. quietly predictnl M6_eY1 = $monotonY1 if e(sample)
. quietly predictnl M6_eY2 = $monotonY2 if e(sample)
. quietly generate double M6_rts = 1/(M6_eY1 + M6_eY2)
. tabstat M6_*, stat(min p25 mean p75 max sd) format(%9.4f)
Stats | M6_mW1 M6_mW2 M6_eY1 M6_eY2 M6_rts
---------+--------------------------------------------------
Min | -0.0483 0.9271 -0.1466 0.2418 0.8629
p25 | 0.0119 0.9603 0.1268 0.6539 1.0308
Mean | 0.0251 0.9749 0.1495 0.7411 1.1395
p75 | 0.0397 0.9881 0.1792 0.8326 1.2245
Max | 0.0729 1.0483 0.2913 1.2670 2.1383
SD | 0.0199 0.0199 0.0480 0.1315 0.1436
------------------------------------------------------------Model 7 (2g)
timer clear 7
timer on 7
xtsf2g lnc $spetech if year > $year_c, model(K1990modified) cost iterate($itermax) distribution(t) $decimalopt
timer off 7
predict M7_te, te_jlms_mean
estadd scalar AIC = e(aic)
estadd scalar BIC = e(bic)
eststo M7
quietly predictnl M7_mW1 = $monotonW1 if e(sample)
quietly predictnl M7_mW2 = 1-M7_mW1 if e(sample)
quietly predictnl M7_eY1 = $monotonY1 if e(sample)
quietly predictnl M7_eY2 = $monotonY2 if e(sample)
generate double M7_rts = 1/(M7_eY1 + M7_eY2)
tabstat M7_*, stat(min p25 mean p75 max sd) format(%9.4f)
. timer on 7
. xtsf2g lnc $spetech if year > $year_c, model(K1990modified) cost iterate($itermax) distribution(t) $decimalopt
Log-likelihood maximization using optimize()
iter 0: Log-likelihood = 499.06419 (not concave)
iter 1: Log-likelihood = 1239.3495
iter 2: Log-likelihood = 1314.4895 (not concave)
iter 3: Log-likelihood = 1436.7572 (not concave)
iter 4: Log-likelihood = 1447.0897
iter 5: Log-likelihood = 1471.5866
iter 6: Log-likelihood = 1495.7501
iter 7: Log-likelihood = 1500.0676
iter 8: Log-likelihood = 1501.2273
iter 9: Log-likelihood = 1501.3099
iter 10: Log-likelihood = 1501.3116
iter 11: Log-likelihood = 1501.3116
Sample:----------------------
Number of obs = 2546
Number of groups = 500
Diagnostics:-----------------
R-squared = 0.9936
Adj R-squared = 0.9920
AIC = -2.3338
BIC = -2.2971
Root MSE = 0.1877
-----------------------------
The second-generation estimator of
the cost stochastic frontier model for panel data,
where effciency is time-varying
--------------------------------------------------------------------------------------
lnc | Coefficient Std. err. z P>|z| [95% conf. interval]
---------------------+----------------------------------------------------------------
lny1 | 0.231 0.141 1.63 0.10 -0.046 0.508
lny2 | -1.787 0.398 -4.48 0.00 -2.568 -1.006
lnw1 | 0.042 0.179 0.23 0.82 -0.309 0.392
trend | -0.535 0.041 -13.16 0.00 -0.614 -0.455
|
c.half#c.lny1#c.lny1 | 0.047 0.005 9.99 0.00 0.038 0.056
|
c.half#c.lny2#c.lny2 | 0.265 0.032 8.19 0.00 0.201 0.328
|
c.half#c.lnw1#c.lnw1 | -0.030 0.015 -2.03 0.04 -0.059 -0.001
|
c.lny1#c.lny2 | -0.052 0.011 -4.61 0.00 -0.073 -0.030
|
c.lny1#c.lnw1 | 0.001 0.005 0.23 0.82 -0.009 0.012
|
c.lny2#c.lnw1 | 0.010 0.014 0.69 0.49 -0.018 0.037
|
c.trend#c.lny1 | 0.004 0.002 2.53 0.01 0.001 0.008
|
c.trend#c.lny2 | 0.010 0.003 3.31 0.00 0.004 0.016
|
c.trend#c.lnw1 | -0.008 0.003 -3.33 0.00 -0.013 -0.003
|
c.trend#c.trend#|
c.half | 0.081 0.003 31.08 0.00 0.076 0.086
|
_cons | 12.816 2.624 4.88 0.00 7.673 17.958
---------------------+----------------------------------------------------------------
ln[var(vit)] |
_cons | -4.519 0.031 -143.72 0.00 -4.580 -4.457
---------------------+----------------------------------------------------------------
ln[var(ui)] |
_cons | -3.728 0.077 -48.57 0.00 -3.878 -3.577
---------------------+----------------------------------------------------------------
E[ui|z] |
_cons | 0.619 0.084 7.38 0.00 0.455 0.784
---------------------+----------------------------------------------------------------
beta[t] |
gamma | -0.038 0.014 -2.80 0.01 -0.065 -0.011
delta | -0.005 0.002 -2.10 0.04 -0.009 -0.000
--------------------------------------------------------------------------------------
Note, the function of inefficiency change over time is
beta[t] = 1 + gamma*(t-T_i) + delta*(t-T_i)^2:
the larger is the beta[t] the larger is the inefficiency
. timer off 7
. predict M7_te, te_jlms_mean
(872 missing values generated)
. estadd scalar AIC = e(aic)
added scalar:
e(AIC) = -2.3337688
. estadd scalar BIC = e(bic)
added scalar:
e(BIC) = -2.2970538
. eststo M7
. quietly predictnl M7_mW1 = $monotonW1 if e(sample)
. quietly predictnl M7_mW2 = 1-M7_mW1 if e(sample)
. quietly predictnl M7_eY1 = $monotonY1 if e(sample)
. quietly predictnl M7_eY2 = $monotonY2 if e(sample)
. generate double M7_rts = 1/(M7_eY1 + M7_eY2)
(872 missing values generated)
. tabstat M7_*, stat(min p25 mean p75 max sd) format(%9.4f)
Stats | M7_te M7_mW1 M7_mW2 M7_eY1 M7_eY2 M7_rts
---------+------------------------------------------------------------
Min | 1.0519 -0.0628 0.9206 -0.1460 0.2248 0.8420
p25 | 1.7413 0.0097 0.9597 0.1235 0.6545 1.0278
Mean | 1.9336 0.0241 0.9759 0.1469 0.7455 1.1378
p75 | 2.0620 0.0403 0.9903 0.1761 0.8388 1.2234
Max | 3.8968 0.0794 1.0628 0.2888 1.2980 2.1940
SD | 0.3052 0.0228 0.0228 0.0478 0.1363 0.1466
----------------------------------------------------------------------Model 8 (2g)
timer clear 8
timer on 8
xtsf2g lnc $spetech if year > $year_c, model(BC1992) cost iterate($itermax) distribution(t) $decimalopt
timer off 8
estadd scalar AIC = e(aic)
estadd scalar BIC = e(bic)
eststo M8
quietly predictnl M8_mW1 = $monotonW1 if e(sample)
quietly predictnl M8_mW2 = 1-M8_mW1 if e(sample)
quietly predictnl M8_eY1 = $monotonY1 if e(sample)
quietly predictnl M8_eY2 = $monotonY2 if e(sample)
generate double M8_rts = 1/(M8_eY1 + M8_eY2)
tabstat M8_*, stat(min p25 mean p75 max sd) format(%9.4f)
. timer clear 8
. timer on 8
. xtsf2g lnc $spetech if year > $year_c, model(BC1992) cost iterate($itermax) distribution(t) $decimalopt
Log-likelihood maximization using optimize()
iter 0: Log-likelihood = 499.06419 (not concave)
iter 1: Log-likelihood = 1245.8926
iter 2: Log-likelihood = 1374.1779
iter 3: Log-likelihood = 1433.7164
iter 4: Log-likelihood = 1479.5883
iter 5: Log-likelihood = 1493.3381
iter 6: Log-likelihood = 1497.8862
iter 7: Log-likelihood = 1498.517
iter 8: Log-likelihood = 1498.5502
iter 9: Log-likelihood = 1498.5507
iter 10: Log-likelihood = 1498.5507
Sample:----------------------
Number of obs = 2546
Number of groups = 500
Diagnostics:-----------------
R-squared = 0.9927
Adj R-squared = 0.9909
AIC = -2.3343
BIC = -2.2976
Root MSE = 0.1876
-----------------------------
The second-generation estimator of
the cost stochastic frontier model for panel data,
where effciency is time-varying
--------------------------------------------------------------------------------------
lnc | Coefficient Std. err. z P>|z| [95% conf. interval]
---------------------+----------------------------------------------------------------
lny1 | 0.231 0.142 1.63 0.10 -0.046 0.509
lny2 | -1.714 0.397 -4.31 0.00 -2.493 -0.936
lnw1 | 0.025 0.178 0.14 0.89 -0.324 0.375
trend | -0.505 0.039 -13.02 0.00 -0.581 -0.429
|
c.half#c.lny1#c.lny1 | 0.047 0.005 9.99 0.00 0.038 0.056
|
c.half#c.lny2#c.lny2 | 0.258 0.032 8.01 0.00 0.195 0.321
|
c.half#c.lnw1#c.lnw1 | -0.028 0.015 -1.91 0.06 -0.057 0.001
|
c.lny1#c.lny2 | -0.052 0.011 -4.62 0.00 -0.074 -0.030
|
c.lny1#c.lnw1 | 0.001 0.005 0.15 0.88 -0.010 0.011
|
c.lny2#c.lnw1 | 0.011 0.014 0.80 0.42 -0.016 0.039
|
c.trend#c.lny1 | 0.005 0.002 2.73 0.01 0.001 0.008
|
c.trend#c.lny2 | 0.009 0.003 3.15 0.00 0.004 0.015
|
c.trend#c.lnw1 | -0.009 0.003 -3.44 0.00 -0.014 -0.004
|
c.trend#c.trend#|
c.half | 0.076 0.002 44.49 0.00 0.073 0.080
|
_cons | 12.319 2.616 4.71 0.00 7.192 17.445
---------------------+----------------------------------------------------------------
ln[var(vit)] |
_cons | -4.516 0.031 -143.61 0.00 -4.578 -4.455
---------------------+----------------------------------------------------------------
ln[var(ui)] |
_cons | -3.697 0.075 -49.51 0.00 -3.843 -3.550
---------------------+----------------------------------------------------------------
E[ui|z] |
_cons | 0.674 0.111 6.09 0.00 0.457 0.891
---------------------+----------------------------------------------------------------
beta[t] |
gamma | 0.014 0.007 2.02 0.04 0.000 0.027
--------------------------------------------------------------------------------------
Note, the function of inefficiency change over time is
beta[t] = exp( -gamma*(t-T_i) ):
the larger is the beta[t] the larger is the inefficiency
. timer off 8
. estadd scalar AIC = e(aic)
added scalar:
e(AIC) = -2.3343331
. estadd scalar BIC = e(bic)
added scalar:
e(BIC) = -2.297618
. eststo M8
. quietly predictnl M8_mW1 = $monotonW1 if e(sample)
. quietly predictnl M8_mW2 = 1-M8_mW1 if e(sample)
. quietly predictnl M8_eY1 = $monotonY1 if e(sample)
. quietly predictnl M8_eY2 = $monotonY2 if e(sample)
. generate double M8_rts = 1/(M8_eY1 + M8_eY2)
(872 missing values generated)
. tabstat M8_*, stat(min p25 mean p75 max sd) format(%9.4f)
Stats | M8_mW1 M8_mW2 M8_eY1 M8_eY2 M8_rts
---------+--------------------------------------------------
Min | -0.0618 0.9213 -0.1467 0.2353 0.8485
p25 | 0.0105 0.9591 0.1239 0.6552 1.0313
Mean | 0.0247 0.9753 0.1472 0.7442 1.1380
p75 | 0.0409 0.9895 0.1768 0.8348 1.2215
Max | 0.0787 1.0618 0.2894 1.2883 2.1433
SD | 0.0225 0.0225 0.0480 0.1331 0.1417
------------------------------------------------------------Results of Models 5-8
Use the estout command for this. Note that parameters δ and γ have different interpretation in models 6-8.
estout M5 M6 M7 M8, ///
cells(b(star fmt(%9.4f)) se(par)) ///
stats(AIC BIC shat RSS ll N sumTi, ///
labels("AIC" "BIC" "\$\hat\sigma\$" "RSS" "log-likelihood" "\$N\$" "\$\sum T_{i}\$") ///
fmt(%9.4f %9.4f %9.4f %9.2f %9.2f %9.0f %9.0f)) ///
starlevels(* 0.10 ** 0.05 *** 0.01) ///
varlabels(_cons Constant ) ///
substitute("_ " "Frontier") ///
legend label collabels(none) mlabels(none) replace
------------------------------------------------------------------------------------
Frontier
lny1 0.2077 0.1778 0.2311 0.2311
(0.1414) (0.1360) (0.1415) (0.1416)
lny2 -1.7352*** -1.7119*** -1.7872*** -1.7144***
(0.3981) (0.3822) (0.3985) (0.3973)
lnw1 0.0086 0.0735 0.0417 0.0254
(0.1781) (0.1693) (0.1789) (0.1782)
trend -0.5151*** -4.3514*** -0.5345*** -0.5053***
(0.0387) (1.5935) (0.0406) (0.0388)
half # lny1 # lny1 0.0476*** 0.0480*** 0.0468*** 0.0469***
(0.0047) (0.0044) (0.0047) (0.0047)
half # lny2 # lny2 0.2571*** 0.2553*** 0.2645*** 0.2578***
(0.0324) (0.0308) (0.0323) (0.0322)
half # lnw1 # lnw1 -0.0298** -0.0234* -0.0300** -0.0282*
(0.0148) (0.0139) (0.0148) (0.0147)
lny1 # lny2 -0.0499*** -0.0478*** -0.0515*** -0.0516***
(0.0112) (0.0107) (0.0112) (0.0112)
lny1 # lnw1 0.0002 0.0014 0.0012 0.0008
(0.0053) (0.0050) (0.0053) (0.0053)
lny2 # lnw1 0.0136 0.0052 0.0098 0.0113
(0.0140) (0.0135) (0.0141) (0.0141)
trend # lny1 0.0045*** 0.0046*** 0.0043** 0.0046***
(0.0017) (0.0016) (0.0017) (0.0017)
trend # lny2 0.0094*** 0.0099*** 0.0098*** 0.0093***
(0.0030) (0.0028) (0.0030) (0.0030)
trend # lnw1 -0.0086*** -0.0087*** -0.0084*** -0.0087***
(0.0025) (0.0024) (0.0025) (0.0025)
trend # trend # half 0.0766*** 0.8308*** 0.0810*** 0.0764***
(0.0017) (0.3148) (0.0026) (0.0017)
Constant 12.5398*** 10.7597*** 12.8159*** 12.3189***
(3.1876) (3.4275) (2.6238) (2.6156)
------------------------------------------------------------------------------------
ln[var(vit)]
Constant -4.5121*** -4.6378*** -4.5188*** -4.5163***
(0.0314) (0.0314) (0.0314) (0.0314)
------------------------------------------------------------------------------------
ln[var(ui)]
Constant -3.6457*** -2.7813*** -3.7279*** -3.6967***
(0.0703) (0.1215) (0.0768) (0.0747)
------------------------------------------------------------------------------------
E[ui|z]
Constant 0.7969 17.0536* 0.6192*** 0.6738***
(1.8311) (10.1015) (0.0839) (0.1107)
------------------------------------------------------------------------------------
beta[t]
gamma -0.5944*** -0.0380*** 0.0138**
(0.1290) (0.0136) (0.0069)
delta 0.0962*** -0.0048**
(0.0205) (0.0023)
------------------------------------------------------------------------------------
AIC -2.3421 1.1384 -2.3338 -2.3343
BIC -2.3054 1.1752 -2.2971 -2.2976
$\hat\sigma$ 0.1869 1.0652 0.1877 0.1876
RSS 1710.04 317650.01 1158.39 1317.51
log-likelihood 1496.47 1623.02 1501.31 1498.55
$N$ 500 500 500 500
$\sum T_{i}$ 2546 2546 2546 2546
------------------------------------------------------------------------------------
* p<0.10, ** p<0.05, *** p<0.01Determinants of inefficiency and heterskedasticity function
In this section time-invariant determinants of inefficiency are used.
Models 9-12: with determinants
The following specifications are used for the next models. The conditional mean of the truncated distribution can be made observation specific by imposing linear structure on it as in Kumbhakar et al (1991),
μi = E[ui|zu2i] = zu2iδu,
so that ui ∼ N+(zu2iδu,σu2) and only δu parameters are estimated instead of μi, i = 1, …, N.
With half-normal or truncated normal distribution, the inefficiency term is still i.i.d., which can introduce bias. If the heteroskedasticity function is known, bias can be eliminated by making σu2 observation specific (see Caudill, Ford and Gropper, 1995)
ln σui2 = zu1iγu
Besides, since E(ui) = (2/π)1/2σui = (2/π)1/2exp 0.5zu1iγu, the zu1i variables can be viewed as determinants of persistent inefficiency.
Similarly, the heteroskedasticity function of noise can be specified ln σvit2 = zvitγv.
In the following, μi is defined by uimean, ln σui2 is defined by uilnvariance, and ln σvit2 is defined by vitlnvariance.
Model 9
timer clear 9
timer on 9
xtsf1g lnc $spetech if year > $year_c, cost uimean(llp_ave la_ave, noconstant) uilnvariance(er_ave) vitlnvariance(lnta) $decimalopt
timer off 9
predict M9_te, te_jlms_mean
estadd scalar AIC = e(aic)
estadd scalar BIC = e(bic)
eststo M9
quietly predictnl M9_mW1 = $monotonW1 if e(sample)
quietly predictnl M9_mW2 = 1-M9_mW1 if e(sample)
quietly predictnl M9_eY1 = $monotonY1 if e(sample)
quietly predictnl M9_eY2 = $monotonY2 if e(sample)
generate double M9_rts = 1/(M9_eY1 + M9_eY2)
tabstat M9_*, stat(min p25 mean p75 max sd) format(%9.4f)
. timer clear 9
. timer on 9
. xtsf1g lnc $spetech if year > $year_c, cost uimean(llp_ave la_ave, noconstant) uilnvariance(er_ave) vitlnvariance(lnta) $decimalopt
Log-likelihood maximization using optimize()
iter 0: Log-likelihood = 530.64988 (not concave)
iter 1: Log-likelihood = 1274.0126 (not concave)
iter 2: Log-likelihood = 1423.8577
iter 3: Log-likelihood = 1502.7847
iter 4: Log-likelihood = 1521.1266
iter 5: Log-likelihood = 1523.9925
iter 6: Log-likelihood = 1524.0591
iter 7: Log-likelihood = 1524.0591
Sample:----------------------
Number of obs = 2546
Number of groups = 500
Diagnostics:-----------------
R-squared = 0.9976
Adj R-squared = 0.9970
AIC = -2.2954
BIC = -2.2587
Root MSE = 0.1913
-----------------------------
The first-generation estimator of
the cost stochastic frontier model for panel data,
where effciency is time-invariant
--------------------------------------------------------------------------------------
lnc | Coefficient Std. err. z P>|z| [95% conf. interval]
---------------------+----------------------------------------------------------------
lny1 | 0.302 0.141 2.15 0.03 0.026 0.578
lny2 | -1.249 0.392 -3.18 0.00 -2.017 -0.480
lnw1 | 0.167 0.161 1.04 0.30 -0.148 0.482
trend | -0.543 0.039 -14.11 0.00 -0.619 -0.468
|
c.half#c.lny1#c.lny1 | 0.053 0.005 11.09 0.00 0.044 0.062
|
c.half#c.lny2#c.lny2 | 0.219 0.032 6.91 0.00 0.157 0.281
|
c.half#c.lnw1#c.lnw1 | -0.044 0.014 -3.15 0.00 -0.071 -0.017
|
c.lny1#c.lny2 | -0.060 0.011 -5.42 0.00 -0.082 -0.038
|
c.lny1#c.lnw1 | -0.000 0.005 -0.02 0.98 -0.010 0.010
|
c.lny2#c.lnw1 | 0.002 0.013 0.15 0.88 -0.023 0.027
|
c.trend#c.lny1 | 0.004 0.002 2.21 0.03 0.000 0.007
|
c.trend#c.lny2 | 0.013 0.003 4.36 0.00 0.007 0.019
|
c.trend#c.lnw1 | -0.008 0.003 -3.12 0.00 -0.013 -0.003
|
c.trend#c.trend#|
c.half | 0.076 0.002 44.14 0.00 0.073 0.079
|
_cons | 9.823 2.582 3.80 0.00 4.763 14.883
---------------------+----------------------------------------------------------------
ln[var(vit)] |
lnta | 0.430 0.085 5.04 0.00 0.263 0.597
_cons | -9.502 0.996 -9.54 0.00 -11.453 -7.550
---------------------+----------------------------------------------------------------
ln[var(ui)] |
er_ave | -3.430 2.589 -1.32 0.19 -8.504 1.644
_cons | -3.399 0.282 -12.07 0.00 -3.951 -2.847
---------------------+----------------------------------------------------------------
E[ui|z] |
llp_ave | 0.000 0.000 11.81 0.00 0.000 0.000
la_ave | 0.400 0.048 8.33 0.00 0.306 0.495
--------------------------------------------------------------------------------------
. timer off 9
. predict M9_te, te_jlms_mean
(872 missing values generated)
. estadd scalar AIC = e(aic)
added scalar:
e(AIC) = -2.2954302
. estadd scalar BIC = e(bic)
added scalar:
e(BIC) = -2.2587152
. eststo M9
. quietly predictnl M9_mW1 = $monotonW1 if e(sample)
. quietly predictnl M9_mW2 = 1-M9_mW1 if e(sample)
. quietly predictnl M9_eY1 = $monotonY1 if e(sample)
. quietly predictnl M9_eY2 = $monotonY2 if e(sample)
. generate double M9_rts = 1/(M9_eY1 + M9_eY2)
(872 missing values generated)
. tabstat M9_*, stat(min p25 mean p75 max sd) format(%9.4f)
Stats | M9_te M9_mW1 M9_mW2 M9_eY1 M9_eY2 M9_rts
---------+------------------------------------------------------------
Min | 1.0096 -0.1121 0.9162 -0.1612 0.2442 0.9010
p25 | 1.2933 -0.0075 0.9717 0.1498 0.5997 1.0902
Mean | 1.4539 0.0086 0.9914 0.1768 0.6805 1.1781
p75 | 1.5453 0.0283 1.0075 0.2101 0.7585 1.2498
Max | 3.0491 0.0838 1.1121 0.3337 1.2298 1.9093
SD | 0.2417 0.0283 0.0283 0.0545 0.1206 0.1206
----------------------------------------------------------------------Model 10
timer clear 10
timer on 10
xtsf2g lnc $spetech if year > $year_c, cost uimean(llp_ave la_ave, noconstant) uilnvariance(er_ave) vitlnvariance(lnta) model("K1990") $decimalopt
timer off 10
predict M10_te, te_jlms_mean
estadd scalar AIC = e(aic)
estadd scalar BIC = e(bic)
eststo M10
quietly predictnl M10_mW1 = $monotonW1 if e(sample)
quietly predictnl M10_mW2 = 1-M10_mW1 if e(sample)
quietly predictnl M10_eY1 = $monotonY1 if e(sample)
quietly predictnl M10_eY2 = $monotonY2 if e(sample)
generate double M10_rts = 1/(M10_eY1 + M10_eY2)
tabstat M10_*, stat(min p25 mean p75 max sd) format(%9.4f)
. timer clear 10
. timer on 10
. xtsf2g lnc $spetech if year > $year_c, cost uimean(llp_ave la_ave, noconstant) uilnva
> riance(er_ave) vitlnvariance(lnta) model("K1990") $decimalopt
Log-likelihood maximization using optimize()
iter 0: Log-likelihood = 685.59629 (not concave)
iter 1: Log-likelihood = 1271.6597 (not concave)
iter 2: Log-likelihood = 1317.3539 (not concave)
iter 3: Log-likelihood = 1407.2609
iter 4: Log-likelihood = 1517.782 (not concave)
...
iter 998: Log-likelihood = 1546.8553 (not concave)
iter 999: Log-likelihood = 1546.8778 (not concave)
iter 1000: Log-likelihood = 1546.9004 (not concave)
convergence not achieved
Sample:----------------------
Number of obs = 2546
Number of groups = 500
Diagnostics:-----------------
R-squared = 0.9979
Adj R-squared = 0.9974
AIC = -2.2840
BIC = -2.2473
Root MSE = 0.1924
-----------------------------
The second-generation estimator of
the cost stochastic frontier model for panel data,
where effciency is time-varying
--------------------------------------------------------------------------------------
lnc | Coefficient Std. err. z P>|z| [95% conf. interval]
---------------------+----------------------------------------------------------------
lny1 | 0.659 0.142 4.65 0.00 0.381 0.937
lny2 | -0.527 0.391 -1.35 0.18 -1.293 0.239
lnw1 | 0.427 0.159 2.68 0.01 0.115 0.739
trend | -0.673 0.040 -16.86 0.00 -0.751 -0.595
|
c.half#c.lny1#c.lny1 | 0.048 0.005 10.15 0.00 0.038 0.057
|
c.half#c.lny2#c.lny2 | 0.186 0.031 5.98 0.00 0.125 0.247
|
c.half#c.lnw1#c.lnw1 | -0.042 0.014 -3.06 0.00 -0.069 -0.015
|
c.lny1#c.lny2 | -0.087 0.011 -7.81 0.00 -0.109 -0.065
|
c.lny1#c.lnw1 | -0.002 0.005 -0.35 0.72 -0.012 0.008
|
c.lny2#c.lnw1 | -0.020 0.013 -1.55 0.12 -0.045 0.005
|
c.trend#c.lny1 | 0.004 0.002 2.19 0.03 0.000 0.007
|
c.trend#c.lny2 | 0.012 0.003 4.18 0.00 0.007 0.018
|
c.trend#c.lnw1 | -0.008 0.002 -3.35 0.00 -0.013 -0.003
|
c.trend#c.trend#|
c.half | 0.103 0.003 33.44 0.00 0.097 0.109
|
_cons | 3.826 2.605 1.47 0.14 -1.280 8.932
---------------------+----------------------------------------------------------------
ln[var(vit)] |
lnta | 0.443 0.086 5.16 0.00 0.275 0.612
_cons | -9.689 1.005 -9.64 0.00 -11.658 -7.720
---------------------+----------------------------------------------------------------
ln[var(ui)] |
er_ave | -4.248 2.663 -1.60 0.11 -9.468 0.971
_cons | -2.550 0.295 -8.63 0.00 -3.129 -1.970
---------------------+----------------------------------------------------------------
E[ui|z] |
llp_ave | 0.000 0.000 11.13 0.00 0.000 0.001
la_ave | 0.550 0.073 7.50 0.00 0.406 0.694
---------------------+----------------------------------------------------------------
beta[t] |
gamma | -0.818 . . . . .
delta | 0.133 0.003 44.49 0.00 0.127 0.139
--------------------------------------------------------------------------------------
Note, the function of inefficiency change over time is
beta[t] = ( 1 + exp(gamma*t + delta*t^2) )^-1:
the larger is the beta[t] the larger is the inefficiency
. timer off 10
. predict M10_te, te_jlms_mean
(872 missing values generated)
. estadd scalar AIC = e(aic)
added scalar:
e(AIC) = -2.2839873
. estadd scalar BIC = e(bic)
added scalar:
e(BIC) = -2.2472723
. eststo M10
. quietly predictnl M10_mW1 = $monotonW1 if e(sample)
. quietly predictnl M10_mW2 = 1-M10_mW1 if e(sample)
. quietly predictnl M10_eY1 = $monotonY1 if e(sample)
. quietly predictnl M10_eY2 = $monotonY2 if e(sample)
. generate double M10_rts = 1/(M10_eY1 + M10_eY2)
(872 missing values generated)
. tabstat M10_*, stat(min p25 mean p75 max sd) format(%9.4f)
Stats | M10_te M10_mW1 M10_mW2 M10_eY1 M10_eY2 M10_rts
---------+------------------------------------------------------------
Min | 1.0068 -0.1345 0.8841 -0.1484 0.3370 0.8216
p25 | 1.2592 -0.0084 0.9714 0.1368 0.6276 1.0857
Mean | 1.4119 0.0088 0.9912 0.1696 0.7054 1.1504
p75 | 1.5019 0.0286 1.0084 0.2058 0.7779 1.2075
Max | 3.0785 0.1159 1.1345 0.3579 1.3408 1.4940
SD | 0.2410 0.0304 0.0304 0.0579 0.1228 0.0938
----------------------------------------------------------------------Model 11
timer clear 11
timer on 11
xtsf2g lnc $spetech if year > $year_c, cost uimean(llp_ave la_ave, noconstant) uilnvariance(er_ave) vitlnvariance(lnta) model("K1990modified") $decimalopt
timer off 11
predict M11_te, te_jlms_mean
estadd scalar AIC = e(aic)
estadd scalar BIC = e(bic)
eststo M11
quietly predictnl M11_mW1 = $monotonW1 if e(sample)
quietly predictnl M11_mW2 = 1-M11_mW1 if e(sample)
quietly predictnl M11_eY1 = $monotonY1 if e(sample)
quietly predictnl M11_eY2 = $monotonY2 if e(sample)
generate double M11_rts = 1/(M11_eY1 + M11_eY2)
tabstat M11_*, stat(min p25 mean p75 max sd) format(%9.4f)
. timer clear 11
. timer on 11
. xtsf2g lnc $spetech if year > $year_c, cost uimean(llp_ave la_ave, noconstant) uilnva
> riance(er_ave) vitlnvariance(lnta) model("K1990modified") $decimalopt
Log-likelihood maximization using optimize()
iter 0: Log-likelihood = 530.64988 (not concave)
iter 1: Log-likelihood = 1275.4967 (not concave)
iter 2: Log-likelihood = 1418.682
iter 3: Log-likelihood = 1495.3091
iter 4: Log-likelihood = 1508.3066
iter 5: Log-likelihood = 1527.6979
iter 6: Log-likelihood = 1530.5061
iter 7: Log-likelihood = 1535.1122
iter 8: Log-likelihood = 1535.3355
iter 9: Log-likelihood = 1535.3362
iter 10: Log-likelihood = 1535.3362
Sample:----------------------
Number of obs = 2546
Number of groups = 500
Diagnostics:-----------------
R-squared = 0.9977
Adj R-squared = 0.9971
AIC = -2.2876
BIC = -2.2509
Root MSE = 0.1921
-----------------------------
The second-generation estimator of
the cost stochastic frontier model for panel data,
where effciency is time-varying
--------------------------------------------------------------------------------------
lnc | Coefficient Std. err. z P>|z| [95% conf. interval]
---------------------+----------------------------------------------------------------
lny1 | 0.361 0.141 2.57 0.01 0.086 0.637
lny2 | -1.244 0.393 -3.17 0.00 -2.013 -0.474
lnw1 | 0.191 0.159 1.21 0.23 -0.120 0.502
trend | -0.577 0.039 -14.64 0.00 -0.655 -0.500
|
c.half#c.lny1#c.lny1 | 0.051 0.005 10.94 0.00 0.042 0.060
|
c.half#c.lny2#c.lny2 | 0.223 0.031 7.09 0.00 0.161 0.284
|
c.half#c.lnw1#c.lnw1 | -0.042 0.014 -3.05 0.00 -0.070 -0.015
|
c.lny1#c.lny2 | -0.064 0.011 -5.81 0.00 -0.086 -0.043
|
c.lny1#c.lnw1 | 0.002 0.005 0.35 0.73 -0.008 0.012
|
c.lny2#c.lnw1 | -0.002 0.013 -0.19 0.85 -0.028 0.023
|
c.trend#c.lny1 | 0.003 0.002 1.71 0.09 -0.000 0.006
|
c.trend#c.lny2 | 0.016 0.003 5.18 0.00 0.010 0.021
|
c.trend#c.lnw1 | -0.007 0.003 -2.95 0.00 -0.012 -0.003
|
c.trend#c.trend#|
c.half | 0.080 0.002 33.13 0.00 0.075 0.085
|
_cons | 9.501 2.598 3.66 0.00 4.408 14.594
---------------------+----------------------------------------------------------------
ln[var(vit)] |
lnta | 0.412 0.085 4.85 0.00 0.246 0.579
_cons | -9.308 0.995 -9.36 0.00 -11.258 -7.359
---------------------+----------------------------------------------------------------
ln[var(ui)] |
er_ave | -3.854 2.655 -1.45 0.15 -9.057 1.349
_cons | -3.540 0.290 -12.20 0.00 -4.109 -2.971
---------------------+----------------------------------------------------------------
E[ui|z] |
llp_ave | 0.000 0.000 12.12 0.00 0.000 0.000
la_ave | 0.347 0.048 7.22 0.00 0.253 0.442
---------------------+----------------------------------------------------------------
beta[t] |
gamma | -0.078 0.021 -3.81 0.00 -0.119 -0.038
delta | -0.009 0.004 -2.40 0.02 -0.017 -0.002
--------------------------------------------------------------------------------------
Note, the function of inefficiency change over time is
beta[t] = 1 + gamma*(t-T_i) + delta*(t-T_i)^2:
the larger is the beta[t] the larger is the inefficiency
. timer off 11
. predict M11_te, te_jlms_mean
(872 missing values generated)
. estadd scalar AIC = e(aic)
added scalar:
e(AIC) = -2.2875846
. estadd scalar BIC = e(bic)
added scalar:
e(BIC) = -2.2508695
. eststo M11
. quietly predictnl M11_mW1 = $monotonW1 if e(sample)
. quietly predictnl M11_mW2 = 1-M11_mW1 if e(sample)
. quietly predictnl M11_eY1 = $monotonY1 if e(sample)
. quietly predictnl M11_eY2 = $monotonY2 if e(sample)
. generate double M11_rts = 1/(M11_eY1 + M11_eY2)
(872 missing values generated)
.
. tabstat M11_*, stat(min p25 mean p75 max sd) format(%9.4f)
Stats | M11_te M11_mW1 M11_mW2 M11_eY1 M11_eY2 M11_rts
---------+------------------------------------------------------------
Min | 1.0083 -0.1097 0.9189 -0.1551 0.2430 0.8731
p25 | 1.2865 -0.0059 0.9721 0.1460 0.6038 1.0845
Mean | 1.4430 0.0096 0.9904 0.1739 0.6879 1.1723
p75 | 1.5357 0.0279 1.0059 0.2073 0.7688 1.2440
Max | 3.1478 0.0811 1.1097 0.3327 1.2664 1.8977
SD | 0.2435 0.0271 0.0271 0.0542 0.1252 0.1219
----------------------------------------------------------------------Model 12
timer clear 12
timer on 12
xtsf2g lnc $spetech if year > $year_c, cost uimean(llp_ave la_ave, noconstant) uilnvariance(er_ave) vitlnvariance(lnta) model("BC1992") $decimalopt
timer off 12
predict M12_te, te_jlms_mean
estadd scalar AIC = e(aic)
estadd scalar BIC = e(bic)
eststo M12
quietly predictnl M12_mW1 = $monotonW1 if e(sample)
quietly predictnl M12_mW2 = 1-M12_mW1 if e(sample)
quietly predictnl M12_eY1 = $monotonY1 if e(sample)
quietly predictnl M12_eY2 = $monotonY2 if e(sample)
generate double M12_rts = 1/(M12_eY1 + M12_eY2)
tabstat M12_*, stat(min p25 mean p75 max sd) format(%9.4f)
. timer clear 12
. timer on 12
. xtsf2g lnc $spetech if year > $year_c, cost uimean(llp_ave la_ave, noconstant) uilnva
> riance(er_ave) vitlnvariance(lnta) model("BC1992") $decimalopt
Log-likelihood maximization using optimize()
iter 0: Log-likelihood = 530.64988 (not concave)
iter 1: Log-likelihood = 1274.1847 (not concave)
iter 2: Log-likelihood = 1420.8998
iter 3: Log-likelihood = 1505.2472
iter 4: Log-likelihood = 1526.9576
iter 5: Log-likelihood = 1531.4269
iter 6: Log-likelihood = 1531.6695
iter 7: Log-likelihood = 1531.6716
iter 8: Log-likelihood = 1531.6716
Sample:----------------------
Number of obs = 2546
Number of groups = 500
Diagnostics:-----------------
R-squared = 0.9976
Adj R-squared = 0.9970
AIC = -2.2825
BIC = -2.2458
Root MSE = 0.1926
-----------------------------
The second-generation estimator of
the cost stochastic frontier model for panel data,
where effciency is time-varying
--------------------------------------------------------------------------------------
lnc | Coefficient Std. err. z P>|z| [95% conf. interval]
---------------------+----------------------------------------------------------------
lny1 | 0.352 0.140 2.51 0.01 0.077 0.627
lny2 | -1.219 0.391 -3.12 0.00 -1.985 -0.453
lnw1 | 0.196 0.159 1.23 0.22 -0.116 0.507
trend | -0.549 0.038 -14.40 0.00 -0.624 -0.474
|
c.half#c.lny1#c.lny1 | 0.051 0.005 10.98 0.00 0.042 0.061
|
c.half#c.lny2#c.lny2 | 0.219 0.031 7.00 0.00 0.158 0.281
|
c.half#c.lnw1#c.lnw1 | -0.041 0.014 -2.96 0.00 -0.068 -0.014
|
c.lny1#c.lny2 | -0.063 0.011 -5.74 0.00 -0.085 -0.042
|
c.lny1#c.lnw1 | 0.001 0.005 0.17 0.87 -0.009 0.011
|
c.lny2#c.lnw1 | -0.002 0.013 -0.17 0.87 -0.027 0.023
|
c.trend#c.lny1 | 0.003 0.002 1.82 0.07 -0.000 0.006
|
c.trend#c.lny2 | 0.015 0.003 5.07 0.00 0.009 0.021
|
c.trend#c.lnw1 | -0.008 0.003 -3.12 0.00 -0.013 -0.003
|
c.trend#c.trend#|
c.half | 0.075 0.002 43.53 0.00 0.072 0.079
|
_cons | 9.342 2.585 3.61 0.00 4.276 14.408
---------------------+----------------------------------------------------------------
ln[var(vit)] |
lnta | 0.423 0.085 4.97 0.00 0.256 0.589
_cons | -9.429 0.993 -9.49 0.00 -11.376 -7.482
---------------------+----------------------------------------------------------------
ln[var(ui)] |
er_ave | -3.388 2.571 -1.32 0.19 -8.427 1.651
_cons | -3.542 0.280 -12.63 0.00 -4.092 -2.993
---------------------+----------------------------------------------------------------
E[ui|z] |
llp_ave | 0.000 0.000 12.32 0.00 0.000 0.000
la_ave | 0.374 0.045 8.23 0.00 0.285 0.463
---------------------+----------------------------------------------------------------
beta[t] |
gamma | 0.032 0.008 3.93 0.00 0.016 0.048
--------------------------------------------------------------------------------------
Note, the function of inefficiency change over time is
beta[t] = exp( -gamma*(t-T_i) ):
the larger is the beta[t] the larger is the inefficiency
. timer off 12
. predict M12_te, te_jlms_mean
(872 missing values generated)
. estadd scalar AIC = e(aic)
added scalar:
e(AIC) = -2.2824787
. estadd scalar BIC = e(bic)
added scalar:
e(BIC) = -2.2457637
. eststo M12
. quietly predictnl M12_mW1 = $monotonW1 if e(sample)
. quietly predictnl M12_mW2 = 1-M12_mW1 if e(sample)
. quietly predictnl M12_eY1 = $monotonY1 if e(sample)
. quietly predictnl M12_eY2 = $monotonY2 if e(sample)
. generate double M12_rts = 1/(M12_eY1 + M12_eY2)
(872 missing values generated)
. tabstat M12_*, stat(min p25 mean p75 max sd) format(%9.4f)
Stats | M12_te M12_mW1 M12_mW2 M12_eY1 M12_eY2 M12_rts
---------+------------------------------------------------------------
Min | 1.0090 -0.1067 0.9180 -0.1562 0.2445 0.8826
p25 | 1.3004 -0.0055 0.9720 0.1471 0.5997 1.0904
Mean | 1.4585 0.0100 0.9900 0.1749 0.6828 1.1778
p75 | 1.5551 0.0280 1.0055 0.2080 0.7624 1.2494
Max | 3.1618 0.0820 1.1067 0.3320 1.2529 1.8928
SD | 0.2464 0.0266 0.0266 0.0541 0.1233 0.1211
----------------------------------------------------------------------Results of Models 9-12
Use the estout command for this. Note that parameters δ and γ have different interpretation in models 2-4.
estout M9 M10 M11 M12, ///
cells(b(star fmt(%9.4f)) se(par)) ///
stats(AIC BIC shat RSS ll N sumTi, ///
labels("AIC" "BIC" "\$\hat\sigma\$" "RSS" "log-likelihood" "\$N\$" "\$\sum T_{i}\$") ///
fmt(%9.4f %9.4f %9.4f %9.2f %9.2f %9.0f %9.0f)) ///
starlevels(* 0.10 ** 0.05 *** 0.01) ///
varlabels(_cons Constant ) ///
substitute("_ " "Frontier") ///
legend label collabels(none) mlabels(none) replace
------------------------------------------------------------------------------------
Frontier
lny1 0.3022** 0.6570*** 0.3611** 0.3518**
(0.1407) (0.1432) (0.1406) (0.1404)
lny2 -1.2486*** -0.5262 -1.2435*** -1.2189***
(0.3923) (0.3977) (0.3927) (0.3909)
lnw1 0.1669 0.4350*** 0.1913 0.1955
(0.1606) (0.1602) (0.1587) (0.1589)
trend -0.5434*** -0.6384*** -0.5774*** -0.5488***
(0.0385) (0.0389) (0.0394) (0.0381)
half # lny1 # lny1 0.0531*** 0.0463*** 0.0512*** 0.0515***
(0.0048) (0.0047) (0.0047) (0.0047)
half # lny2 # lny2 0.2191*** 0.1878*** 0.2228*** 0.2195***
(0.0317) (0.0315) (0.0314) (0.0313)
half # lnw1 # lnw1 -0.0438*** -0.0470*** -0.0424*** -0.0408***
(0.0139) (0.0139) (0.0139) (0.0138)
lny1 # lny2 -0.0602*** -0.0872*** -0.0642*** -0.0634***
(0.0111) (0.0112) (0.0111) (0.0110)
lny1 # lnw1 -0.0001 0.0007 0.0018 0.0009
(0.0053) (0.0052) (0.0052) (0.0052)
lny2 # lnw1 0.0020 -0.0220* -0.0024 -0.0021
(0.0130) (0.0130) (0.0128) (0.0129)
trend # lny1 0.0038** 0.0048*** 0.0029* 0.0031*
(0.0017) (0.0017) (0.0017) (0.0017)
trend # lny2 0.0128*** 0.0124*** 0.0155*** 0.0152***
(0.0029) (0.0030) (0.0030) (0.0030)
trend # lnw1 -0.0079*** -0.0073*** -0.0075*** -0.0079***
(0.0025) (0.0025) (0.0025) (0.0025)
trend # trend # half 0.0761*** 0.0940*** 0.0800*** 0.0754***
(0.0017) (0.0025) (0.0024) (0.0017)
Constant 9.8233*** 3.7296 9.5008*** 9.3421***
(2.5817) (2.6552) (2.5984) (2.5848)
------------------------------------------------------------------------------------
ln[var(vit)]
lnta 0.4298*** 0.4423*** 0.4123*** 0.4228***
(0.0852) (0.0861) (0.0851) (0.0850)
Constant -9.5016*** -9.6649*** -9.3084*** -9.4288***
(0.9958) (1.0073) (0.9945) (0.9934)
------------------------------------------------------------------------------------
ln[var(ui)]
er_ave -3.4300 -5.3783* -3.8540 -3.3880
(2.5888) (2.8642) (2.6546) (2.5708)
Constant -3.3989*** -2.0839*** -3.5403*** -3.5422***
(0.2815) (0.3280) (0.2903) (0.2805)
------------------------------------------------------------------------------------
E[ui|z]
llp_ave 0.0003*** 0.0005*** 0.0003*** 0.0003***
(0.0000) (0.0000) (0.0000) (0.0000)
la_ave 0.4004*** 0.5800*** 0.3475*** 0.3743***
(0.0481) (0.0947) (0.0481) (0.0455)
------------------------------------------------------------------------------------
beta[t]
gamma -0.4181 -0.0784*** 0.0319***
(.) (0.0206) (0.0081)
delta 0.0722*** -0.0092**
(0.0027) (0.0038)
------------------------------------------------------------------------------------
AIC -2.2954 -2.2840 -2.2876 -2.2825
BIC -2.2587 -2.2473 -2.2509 -2.2458
$\hat\sigma$ 0.1913 0.1924 0.1921 0.1926
RSS 428.34 376.56 414.74 434.83
log-likelihood 1524.06 1539.31 1535.34 1531.67
$N$ 500 500 500 500
$\sum T_{i}$ 2546 2546 2546 2546
------------------------------------------------------------------------------------
* p<0.10, ** p<0.05, *** p<0.01