In this article, the functionalityIn this article, the functionality of the commands xtsf3gpss1, xtsf3gpss2, xtsf3gpss3, and xtsf3gkss are showcased. The subset of banking is used:
Here are the specification and formula for first derivatives of the cost function with respect to input prices and outputs to check monotonicity assumptions and compute returns to scale.
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 year_c = 2001
global itermax = 1000xtsf3gpss1 fits the PSS Type I estimator of the stochastic frontier model for panel data, where some regressors are allowed to be correlated with effects.
It allows using factor variables (see fvvarlist). Unbalanced panels are supported.
The estimation goes in two stages. First, xtsf3gpss1 will go over the grid from gr0 to gr1 with a step/increment gri to find a bandwidth that results in the smallest MSE. In each of these grid points, xtsf3gpss1 will do reps bootstrap replications. This grid search is extremely time-consuming. If bandwidth has been found previously for this specification (and only this specification), there is an option to specify this bandwidth using the bandwidth. In the second step, the PSS Type I estimator is obtained using the optimal bandwidth. The second step is very fast.
timer clear 1
timer on 1
xtsf3gpss1 lnc $spetech if year > $year_c, cost gr0(0.1) gr1(0.9) gri(0.1) reps(9)
timer off 1
estadd scalar AIC = e(aic)
estadd scalar BIC = e(bic)
eststo M1
. timer clear 1
. timer on 1
. xtsf3gpss1 lnc $spetech if year > $year_c, cost gr0(0.1) gr1(0.9) gri(0.1) reps(9)
Description of the panel data:------------------------------------------------
id: 1155, 2040, ..., 3217331 n = 500
year: 2002, 2003, ..., 2007 T = 6
Delta(year) = 1 unit
Span(year) = 6 periods
(id*year uniquely identifies each observation)
Distribution of T_i: min 5% 25% 50% 75% 95% max
2 2 4 6 6 6 6
Freq. Percent Cum. | Pattern
---------------------------+---------
289 57.80 57.80 | 111111
34 6.80 64.60 | 11111.
33 6.60 71.20 | 1111..
27 5.40 76.60 | 111...
24 4.80 81.40 | 11....
15 3.00 84.40 | 11.111
14 2.80 87.20 | ..1111
11 2.20 89.40 | .11111
5 1.00 90.40 | .1111.
48 9.60 100.00 | (other patterns)
---------------------------+---------
500 100.00 | XXXXXX
Calculating optimal bandwidth for the PSS (some regressors are correlated with effects) estimator.
Please be patient!
It may take long time to compute estimates if the data size is large.
Choosing the bandwidth which has the smallest MSE for the PSS estimator.
Going over the grid, which contains 9 grid points
(in each grid point, 9 bootstrap replications are used)
----+--- 1 ---+--- 2 ---+--- 3 ---+--- 4 ---+--- 5
.........
Optimal bandwidth (for the within panel data estimator) is 0.1000
Sample:----------------------
Number of obs = 2546
Number of groups = 500
Diagnostics:-----------------
R-squared = 0.8214
Adj R-squared = 0.7763
AIC = -3.7229
BIC = -3.6885
Root MSE = 0.0937
-----------------------------
PSS Type I estimator: some regressors are correlated with effects
Park, Sickles, and Simar (1998), Journal of Econometrics, 84(2):273–301
Cost Stochastic Frontier
----------------------------------------------------------------------------------------
lnc | Coefficient Std. err. z P>|z| [95% conf. interval]
-----------------------+----------------------------------------------------------------
lny1 | 0.2596 0.0388 6.68 0.000 0.1834 0.3357
lny2 | -1.9301 0.0873 -22.10 0.000 -2.1013 -1.7590
lnw1 | -0.0077 0.0470 -0.16 0.870 -0.0998 0.0845
trend | -0.5000 0.0324 -15.45 0.000 -0.5634 -0.4366
|
c.half#c.lny1#c.lny1 | 0.0423 0.0014 30.82 0.000 0.0396 0.0449
|
c.half#c.lny2#c.lny2 | 0.2722 0.0068 39.93 0.000 0.2588 0.2855
|
c.half#c.lnw1#c.lnw1 | -0.0226 0.0042 -5.42 0.000 -0.0308 -0.0145
|
c.lny1#c.lny2 | -0.0487 0.0027 -17.76 0.000 -0.0540 -0.0433
|
c.lny1#c.lnw1 | -0.0010 0.0017 -0.57 0.572 -0.0044 0.0024
|
c.lny2#c.lnw1 | 0.0155 0.0035 4.40 0.000 0.0086 0.0225
|
c.trend#c.lny1 | 0.0023 0.0013 1.78 0.075 -0.0002 0.0049
|
c.trend#c.lny2 | 0.0103 0.0025 4.20 0.000 0.0055 0.0151
|
c.trend#c.lnw1 | -0.0105 0.0016 -6.50 0.000 -0.0137 -0.0073
|
c.trend#c.trend#c.half | 0.0768 0.0024 32.53 0.000 0.0722 0.0814
----------------------------------------------------------------------------------------
. timer off 1
. estadd scalar AIC = e(aic)
added scalar:
e(AIC) = -3.7229442
. estadd scalar BIC = e(bic)
added scalar:
e(BIC) = -3.6885239
. eststo M1xtsf3gpss2 PSS Type II estimator of the stochastic frontier model for panel data, where error follow AR(1).
It allows using factor variables (see fvvarlist). Unbalanced panels are supported.
The estimation is performed in two stages. First, xtsf3gpss2 will go over the grid from gr0 to gr1 with a step/increment gri to find a bandwidth that results in the smallest MSE. In each of these grid points, xtsf3gpss2 will do reps bootstrap replications. This grid search is extremely time-consuming. If bandwidth has been found previously for this specification (and only this specification), there is an option to specify this bandwidth using the bandwidth. In the second step, the PSS Type II estimator is obtained using the optimal bandwidth. The second step is very fast.
timer clear 2
timer on 2
xtsf3gpss2 lnc $spetech if year > $year_c, cost gr0(0.1) gr1(0.9) gri(0.1) reps(9)
timer off 2
estadd scalar AIC = e(aicW)
estadd scalar BIC = e(bicW)
eststo M2
. timer clear 2
. timer on 2
. xtsf3gpss2 lnc $spetech if year > $year_c, cost gr0(0.1) gr1(0.9) gri(0.1) reps(9)
Description of the panel data:------------------------------------------------
id: 1155, 2040, ..., 3217331 n = 500
year: 2002, 2003, ..., 2007 T = 6
Delta(year) = 1 unit
Span(year) = 6 periods
(id*year uniquely identifies each observation)
Distribution of T_i: min 5% 25% 50% 75% 95% max
2 2 4 6 6 6 6
Freq. Percent Cum. | Pattern
---------------------------+---------
289 57.80 57.80 | 111111
34 6.80 64.60 | 11111.
33 6.60 71.20 | 1111..
27 5.40 76.60 | 111...
24 4.80 81.40 | 11....
15 3.00 84.40 | 11.111
14 2.80 87.20 | ..1111
11 2.20 89.40 | .11111
5 1.00 90.40 | .1111.
48 9.60 100.00 | (other patterns)
---------------------------+---------
500 100.00 | XXXXXX
IDs with 2 or fewer observations have been excluded from estimation
id: 1155, 2040, ..., 3217331 n = 471
year: 2002, 2003, ..., 2007 T = 6
Delta(year) = 1 unit
Span(year) = 6 periods
(id*year uniquely identifies each observation)
Distribution of T_i: min 5% 25% 50% 75% 95% max
3 3 5 6 6 6 6
Freq. Percent Cum. | Pattern
---------------------------+---------
289 61.36 61.36 | 111111
34 7.22 68.58 | 11111.
33 7.01 75.58 | 1111..
27 5.73 81.32 | 111...
15 3.18 84.50 | 11.111
14 2.97 87.47 | ..1111
11 2.34 89.81 | .11111
5 1.06 90.87 | .1111.
5 1.06 91.93 | 1.1111
38 8.07 100.00 | (other patterns)
---------------------------+---------
471 100.00 | XXXXXX
Calculating optimal bandwidth for the PSS (AR(1) error) estimator
Please be patient!
Going over the grid, which contains 9 grid points
(in each grid point, 9 bootstrap replications are used)
----+--- 1 ---+--- 2 ---+--- 3 ---+--- 4 ---+--- 5
.........
Optimal bandwidth (for the within estimator) is .6
Optimal bandwidth (for the GLS estimator) is .4
Sample:----------------------
Number of obs = 2488
Number of groups = 471
Diagnostics:-----------------
Within:----------------------
R-squared = 0.8108
Adj R-squared = 0.7651
AIC = -2.2908
BIC = -2.2557
Root MSE = 0.1918
GLS:-------------------------
R-squared = 0.8163
Adj R-squared = 0.7719
AIC = -2.3204
BIC = -2.2853
Root MSE = 0.1890
-----------------------------
PSS Type 2 estimator: AR(1) error
Park, Sickles, and Simar (2003), Journal of Econometrics, 117(2):279–309
Cost Stochastic Frontier
----------------------------------------------------------------------------------------
lnc | Coefficient Std. err. z P>|z| [95% conf. interval]
-----------------------+----------------------------------------------------------------
Within |
lny1 | -0.1337 0.1166 -1.15 0.251 -0.3622 0.0947
lny2 | -2.9294 0.3208 -9.13 0.000 -3.5582 -2.3005
lnw1 | -0.0834 0.1486 -0.56 0.575 -0.3747 0.2079
trend | -0.5785 0.0263 -21.96 0.000 -0.6302 -0.5269
|
c.half#c.lny1#c.lny1 | 0.0514 0.0039 13.32 0.000 0.0438 0.0589
|
c.half#c.lny2#c.lny2 | 0.3360 0.0260 12.94 0.000 0.2851 0.3868
|
c.half#c.lnw1#c.lnw1 | -0.0270 0.0122 -2.22 0.027 -0.0508 -0.0031
|
c.lny1#c.lny2 | -0.0230 0.0092 -2.50 0.012 -0.0410 -0.0050
|
c.lny1#c.lnw1 | 0.0061 0.0045 1.36 0.175 -0.0027 0.0150
|
c.lny2#c.lnw1 | 0.0170 0.0113 1.51 0.132 -0.0051 0.0391
|
c.trend#c.lny1 | 0.0030 0.0011 2.63 0.009 0.0008 0.0053
|
c.trend#c.lny2 | 0.0107 0.0020 5.34 0.000 0.0068 0.0146
|
c.trend#c.lnw1 | -0.0099 0.0018 -5.57 0.000 -0.0134 -0.0064
|
c.trend#c.trend#c.half | 0.0913 0.0013 68.17 0.000 0.0886 0.0939
-----------------------+----------------------------------------------------------------
GLS |
lny1 | -0.0566 0.0523 -1.08 0.279 -0.1592 0.0460
lny2 | -2.8499 0.1577 -18.08 0.000 -3.1589 -2.5409
lnw1 | -0.0521 0.0715 -0.73 0.466 -0.1921 0.0880
trend | -0.5407 0.0121 -44.66 0.000 -0.5644 -0.5169
|
c.half#c.lny1#c.lny1 | 0.0538 0.0017 31.19 0.000 0.0505 0.0572
|
c.half#c.lny2#c.lny2 | 0.3374 0.0129 26.15 0.000 0.3121 0.3627
|
c.half#c.lnw1#c.lnw1 | -0.0139 0.0060 -2.33 0.020 -0.0257 -0.0022
|
c.lny1#c.lny2 | -0.0299 0.0041 -7.21 0.000 -0.0380 -0.0217
|
c.lny1#c.lnw1 | -0.0016 0.0020 -0.79 0.427 -0.0056 0.0024
|
c.lny2#c.lnw1 | 0.0169 0.0055 3.10 0.002 0.0062 0.0276
|
c.trend#c.lny1 | 0.0041 0.0005 7.79 0.000 0.0031 0.0051
|
c.trend#c.lny2 | 0.0090 0.0009 9.63 0.000 0.0072 0.0108
|
c.trend#c.lnw1 | -0.0096 0.0008 -11.97 0.000 -0.0112 -0.0080
|
c.trend#c.trend#c.half | 0.0843 0.0006 143.53 0.000 0.0831 0.0854
----------------------------------------------------------------------------------------
. timer off 2
. estadd scalar AIC = e(aicW)
added scalar:
e(AIC) = -2.2908232
. estadd scalar BIC = e(bicW)
added scalar:
e(BIC) = -2.2557394
. estadd scalar shat = e(shatW)
added scalar:
e(shat) = .191812
. estadd scalar RSS = e(RSSW)
added scalar:
e(RSS) = 24.312897
. eststo M2xtsf3gpss3 PSS Type II estimator of the stochastic frontier model for panel data, where error follow AR(1).
It allows using factor variables (see fvvarlist). Unbalanced panels are supported.
The estimation is performed in two stages. First, xtsf3gpss3 will go over the grid from gr0 to gr1 with a step/increment gri to find a bandwidth that results in the smallest MSE. In each of these grid points reps reps bootstrap replications will be performed. This grid search is extremely time-consuming. If bandwidth has been found previously for this specification (and only this specification), there is an option to specify this bandwidth using the bw. In the second step, the PSS Type III estimator is obtained using the optimal bandwidth. The second step is very fast.
timer clear 3
timer on 3
xtsf3gpss3 lnc $spetech if year > $year_c, cost gr0(0.1) gr1(0.9) gri(0.1) reps(9)
timer off 3
estadd scalar AIC = e(aic)
estadd scalar BIC = e(bic)
eststo M3
. timer clear 3
. timer on 3
. xtsf3gpss3 lnc $spetech if year > $year_c, cost gr0(0.1) gr1(0.9) gri(0.1) reps(9)
Description of the panel data:------------------------------------------------
id: 1155, 2040, ..., 3217331 n = 500
year: 2002, 2003, ..., 2007 T = 6
Delta(year) = 1 unit
Span(year) = 6 periods
(id*year uniquely identifies each observation)
Distribution of T_i: min 5% 25% 50% 75% 95% max
2 2 4 6 6 6 6
Freq. Percent Cum. | Pattern
---------------------------+---------
289 57.80 57.80 | 111111
34 6.80 64.60 | 11111.
33 6.60 71.20 | 1111..
27 5.40 76.60 | 111...
24 4.80 81.40 | 11....
15 3.00 84.40 | 11.111
14 2.80 87.20 | ..1111
11 2.20 89.40 | .11111
5 1.00 90.40 | .1111.
48 9.60 100.00 | (other patterns)
---------------------------+---------
500 100.00 | XXXXXX
IDs with 4 or fewer observations have been excluded from estimation
id: 1155, 2040, ..., 3153297 n = 359
year: 2002, 2003, ..., 2007 T = 6
Delta(year) = 1 unit
Span(year) = 6 periods
(id*year uniquely identifies each observation)
Distribution of T_i: min 5% 25% 50% 75% 95% max
5 5 6 6 6 6 6
Freq. Percent Cum. | Pattern
---------------------------+---------
289 80.50 80.50 | 111111
34 9.47 89.97 | 11111.
15 4.18 94.15 | 11.111
11 3.06 97.21 | .11111
5 1.39 98.61 | 1.1111
4 1.11 99.72 | 1111.1
1 0.28 100.00 | 111.11
---------------------------+---------
359 100.00 | XXXXXX
Calculating optimal bandwidth for the PSS (dynamic panel data model) estimator.
Please be patient!
It may take long time to compute estimates if the data size is large.
Choosing the bandwidth which has the smallest MSE for the PSS estimator.
Going over the grid, which contains 9 grid points
(in each grid point, 9 bootstrap replications are used)
----+--- 1 ---+--- 2 ---+--- 3 ---+--- 4 ---+--- 5
.........
Optimal bandwidth (for the dynamic panel data estimator) is 0.5000
Sample:----------------------
Number of obs = 2084
Number of groups = 359
Diagnostics:-----------------
R-squared = 0.9066
Adj R-squared = 0.8864
AIC = -3.1583
BIC = -3.1177
Root MSE = 0.1242
-----------------------------
PSS Type 3 estimator: dynamic panel data model
Park, Sickles, and Simar (2007), Journal of Econometrics, 136(1):281–301
Cost Stochastic Frontier
----------------------------------------------------------------------------------------
lnc | Coefficient Std. err. z P>|z| [95% conf. interval]
-----------------------+----------------------------------------------------------------
lny1 | 0.4531 0.1484 3.05 0.002 0.1622 0.7440
lny2 | -2.8233 0.5082 -5.56 0.000 -3.8193 -1.8272
lnw1 | 0.0246 0.2276 0.11 0.914 -0.4214 0.4706
trend | -0.5087 0.0384 -13.23 0.000 -0.5840 -0.4334
|
c.half#c.lny1#c.lny1 | 0.0088 0.0047 1.87 0.062 -0.0004 0.0179
|
c.half#c.lny2#c.lny2 | 0.3408 0.0416 8.18 0.000 0.2592 0.4224
|
c.half#c.lnw1#c.lnw1 | -0.0605 0.0177 -3.42 0.001 -0.0951 -0.0259
|
c.lny1#c.lny2 | -0.0500 0.0119 -4.20 0.000 -0.0734 -0.0267
|
c.lny1#c.lnw1 | 0.0134 0.0061 2.21 0.027 0.0015 0.0252
|
c.lny2#c.lnw1 | 0.0005 0.0173 0.03 0.976 -0.0334 0.0345
|
c.trend#c.lny1 | 0.0028 0.0016 1.70 0.088 -0.0004 0.0060
|
c.trend#c.lny2 | -0.0010 0.0030 -0.34 0.734 -0.0068 0.0048
|
c.trend#c.lnw1 | 0.0042 0.0026 1.65 0.098 -0.0008 0.0093
|
c.trend#c.trend#c.half | 0.0866 0.0017 49.98 0.000 0.0832 0.0900
|
lnc |
L1. | 0.5126 . . . . .
----------------------------------------------------------------------------------------
. timer off 3
. estadd scalar AIC = e(aic)
added scalar:
e(AIC) = -3.1583365
. estadd scalar BIC = e(bic)
added scalar:
e(BIC) = -3.1177268
. eststo M3xtsf3gkss fits the KSS estimator of the stochastic frontier model for panel data, where arbitrary temporal heterogeneity is allowed.
timer clear 4
timer on 4
xtsf3gkss lnc $spetech if year > $year_c, cost gr0(0.1) gr1(1.0) gri(0.1) li(3) ls(7) imean tmean level(99) cformat(%9.4f)
timer off 4
estadd scalar AIC = e(aic)
estadd scalar BIC = e(bic)
eststo M4
. timer clear 4
. timer on 4
. xtsf3gkss lnc $spetech if year > $year_c, cost gr0(0.1) gr1(1.0) gri(0.1) li(3) ls(7) imean tmean level(99) cformat(%9.4f)
Description of the panel data:------------------------------------------------
id: 1155, 2040, ..., 3217331 n = 500
year: 2002, 2003, ..., 2007 T = 6
Delta(year) = 1 unit
Span(year) = 6 periods
(id*year uniquely identifies each observation)
Distribution of T_i: min 5% 25% 50% 75% 95% max
2 2 4 6 6 6 6
Freq. Percent Cum. | Pattern
---------------------------+---------
289 57.80 57.80 | 111111
34 6.80 64.60 | 11111.
33 6.60 71.20 | 1111..
27 5.40 76.60 | 111...
24 4.80 81.40 | 11....
15 3.00 84.40 | 11.111
14 2.80 87.20 | ..1111
11 2.20 89.40 | .11111
5 1.00 90.40 | .1111.
48 9.60 100.00 | (other patterns)
---------------------------+---------
500 100.00 | XXXXXX
Select L such that Cl is less than 2.33
Please be patient!
It may take long time to compute estimates if the data size is large.
Going over the grid, which contains 5 grid points
----+--- 1 ---+--- 2 ---+--- 3 ---+--- 4 ---+--- 5
.
Sample:----------------------
Number of obs = 2546
Number of groups = 500
Diagnostics:-----------------
R-squared = 0.5480
Adj R-squared = 0.4338
AIC = -1.8907
BIC = -1.8563
Root MSE = 0.2343
-----------------------------
Kneip-Sickles-Song Estimator
Kneip, Sickles, and Song (2012), Econometric Theory, 28(3):590–628
Cost Stochastic Frontier
----------------------------------------------------------------------------------------
lnc | Coefficient Std. err. z P>|z| [95% conf. interval]
-----------------------+----------------------------------------------------------------
lny1 | -0.0813 0.1752 -0.46 0.643 -0.4247 0.2621
lny2 | -2.7225 0.6251 -4.36 0.000 -3.9476 -1.4974
lnw1 | -0.2761 0.2035 -1.36 0.175 -0.6749 0.1227
trend | -0.4539 0.0857 -5.30 0.000 -0.6218 -0.2859
|
c.half#c.lny1#c.lny1 | 0.0121 0.0054 2.23 0.026 0.0014 0.0228
|
c.half#c.lny2#c.lny2 | 0.2557 0.0519 4.92 0.000 0.1539 0.3575
|
c.half#c.lnw1#c.lnw1 | -0.0396 0.0145 -2.72 0.007 -0.0681 -0.0111
|
c.lny1#c.lny2 | 0.0052 0.0150 0.34 0.731 -0.0243 0.0346
|
c.lny1#c.lnw1 | 0.0203 0.0049 4.13 0.000 0.0107 0.0299
|
c.lny2#c.lnw1 | 0.0258 0.0175 1.47 0.141 -0.0085 0.0600
|
c.trend#c.lny1 | -0.0115 0.0033 -3.49 0.000 -0.0180 -0.0051
|
c.trend#c.lny2 | 0.0358 0.0064 5.60 0.000 0.0233 0.0483
|
c.trend#c.lnw1 | 0.0008 0.0046 0.16 0.869 -0.0082 0.0097
|
c.trend#c.trend#c.half | 0.0400 0.0042 9.44 0.000 0.0317 0.0482
----------------------------------------------------------------------------------------
. timer off 4
. estadd scalar AIC = e(aic)
added scalar:
e(AIC) = -1.8907164
. estadd scalar BIC = e(bic)
added scalar:
e(BIC) = -1.8562961
. eststo M4Use estout command for this:
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
. estout M1 M2 M3 M4, ///
> cells(b(star fmt(%9.4f)) se(par)) ///
> stats(AIC BIC shat RSS N sumTi, ///
> labels("AIC" "BIC" "\$\hat\sigma\$" "RSS" "\$N\$" "\$\sum T_{i}\$") ///
> fmt(%9.4f %9.4f %9.4f %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
------------------------------------------------------------------------------------
main
lny1 0.2596*** -0.1337 0.4531*** -0.0813
(0.0388) (0.1166) (0.1484) (0.1752)
lny2 -1.9301*** -2.9294*** -2.8233*** -2.7225***
(0.0873) (0.3208) (0.5082) (0.6251)
lnw1 -0.0077 -0.0834 0.0246 -0.2761
(0.0470) (0.1486) (0.2276) (0.2035)
trend -0.5000*** -0.5785*** -0.5087*** -0.4539***
(0.0324) (0.0263) (0.0384) (0.0857)
half # lny1 # lny1 0.0423*** 0.0514*** 0.0088* 0.0121**
(0.0014) (0.0039) (0.0047) (0.0054)
half # lny2 # lny2 0.2722*** 0.3360*** 0.3408*** 0.2557***
(0.0068) (0.0260) (0.0416) (0.0519)
half # lnw1 # lnw1 -0.0226*** -0.0270** -0.0605*** -0.0396***
(0.0042) (0.0122) (0.0177) (0.0145)
lny1 # lny2 -0.0487*** -0.0230** -0.0500*** 0.0052
(0.0027) (0.0092) (0.0119) (0.0150)
lny1 # lnw1 -0.0010 0.0061 0.0134** 0.0203***
(0.0017) (0.0045) (0.0061) (0.0049)
lny2 # lnw1 0.0155*** 0.0170 0.0005 0.0258
(0.0035) (0.0113) (0.0173) (0.0175)
trend # lny1 0.0023* 0.0030*** 0.0028* -0.0115***
(0.0013) (0.0011) (0.0016) (0.0033)
trend # lny2 0.0103*** 0.0107*** -0.0010 0.0358***
(0.0025) (0.0020) (0.0030) (0.0064)
trend # lnw1 -0.0105*** -0.0099*** 0.0042* 0.0008
(0.0016) (0.0018) (0.0026) (0.0046)
trend # trend # half 0.0768*** 0.0913*** 0.0866*** 0.0400***
(0.0024) (0.0013) (0.0017) (0.0042)
L.lnc 0.5126
(.)
------------------------------------------------------------------------------------
GLS
lny1 -0.0566
(0.0523)
lny2 -2.8499***
(0.1577)
lnw1 -0.0521
(0.0715)
trend -0.5407***
(0.0121)
half # lny1 # lny1 0.0538***
(0.0017)
half # lny2 # lny2 0.3374***
(0.0129)
half # lnw1 # lnw1 -0.0139**
(0.0060)
lny1 # lny2 -0.0299***
(0.0041)
lny1 # lnw1 -0.0016
(0.0020)
lny2 # lnw1 0.0169***
(0.0055)
trend # lny1 0.0041***
(0.0005)
trend # lny2 0.0090***
(0.0009)
trend # lnw1 -0.0096***
(0.0008)
trend # trend # half 0.0843***
(0.0006)
------------------------------------------------------------------------------------
AIC -3.7229 -2.2908 -3.1583 -1.8907
BIC -3.6885 -2.2557 -3.1177 -1.8563
$\hat\sigma$ 0.0937 0.1918 0.1242 0.2343
RSS 512068.91 24.31 456358.91 1.26e+06
$N$ 500 471 359 500
$\sum T_{i}$ 2546 2488 2084 2546
------------------------------------------------------------------------------------
* p<0.10, ** p<0.05, *** p<0.01