Nothing
Illustration
In snreg: Regression with Skew-Normally Distributed Error Term
Illustration
Data
is a data frame containing selected variables for 500 U.S. commercial
banks, randomly sampled from approximately 5000 banks, based on the
dataset of Koetter et al. (2012) for year 2007. The dataset is
provided solely for illustration and pedagogical purposes and is not
suitable for empirical research.
library(snreg)
data(banks07, package = "snreg")
head(banks07)
#R> id year TC Y1 Y2 W1 W2 W3 ER
#R> 1 152440 2007 4659.505 8661.84 58393.04 38.55422 37.75000 4.146092 0.1117789
#R> 2 544335 2007 10848.960 19492.91 146789.00 30.66785 45.13934 2.263886 0.1139121
#R> 3 548203 2007 14523.650 18630.49 219705.70 27.38797 56.02857 4.312672 0.1342984
#R> 4 568238 2007 1644.808 11902.50 18675.68 32.84768 50.47368 1.541316 0.1029891
#R> 5 651158 2007 3054.240 29322.21 38916.14 42.20033 52.23529 3.192056 0.1421004
#R> 6 705444 2007 6168.736 36568.03 67179.16 11.95771 41.46667 3.181241 0.1449880
#R> LA TA ZSCORE ZSCORE3 SDROA LLP lnzscore lnzscore3
#R> 1 0.7781611 75039.78 29.49198 11.93898 0.4532092 732.4904 3.384118 2.479809
#R> 2 0.7679300 191148.92 62.97659 20.66493 0.2298561 305.9889 4.142763 3.028438
#R> 3 0.8880010 247416.05 38.99969 30.88642 0.3874364 1534.6521 3.663554 3.430317
#R> 4 0.5050412 36978.53 30.68124 24.53655 0.4063778 0.0000 3.423651 3.200164
#R> 5 0.5264002 73928.81 30.02672 28.42257 0.5496323 0.0000 3.402088 3.347184
#R> 6 0.5222276 128639.62 110.67650 37.66002 0.1502981 0.0000 4.706612 3.628599
#R> lnsdroa scope ms_county
#R> 1 -0.7914014 0.4801792 0.3466652
#R> 2 -1.4703018 0.5254947 0.8830606
#R> 3 -0.9482036 0.8897177 1.1430008
#R> 4 -0.9004720 0.3917692 0.1708316
#R> 5 -0.5985058 0.5523984 0.3415328
#R> 6 -1.8951346 0.4073012 0.5942832
Specification
Define the specification (formula) that will be used:
# Translog cost function specification
spe.tl <- log(TC) ~ (log(Y1) + log(Y2) + log(W1) + log(W2))^2 +
I(0.5 * log(Y1)^2) + I(0.5 * log(Y2)^2) +
I(0.5 * log(W1)^2) + I(0.5 * log(W2)^2)
Linear Regression via MLE
To estimate simple OLS using MLE
# -------------------------------------------------------------
# Specification 1: homoskedastic noise (ln.var.v = NULL)
# -------------------------------------------------------------
formSV <- NULL
m1 <- lm.mle(
formula = spe.tl,
data = banks07,
ln.var.v = formSV
)
#R> theta0:
#R> (Intercept) log(Y1) log(Y2)
#R> -4.386991731 0.415874408 0.370050861
#R> log(W1) log(W2) I(0.5 * log(Y1)^2)
#R> 0.524152369 1.342097991 0.047454756
#R> I(0.5 * log(Y2)^2) I(0.5 * log(W1)^2) I(0.5 * log(W2)^2)
#R> 0.077484315 0.034996665 -0.226281101
#R> log(Y1):log(Y2) log(Y1):log(W1) log(Y1):log(W2)
#R> -0.057609631 -0.020462972 -0.005676474
#R> log(Y2):log(W1) log(Y2):log(W2) log(W1):log(W2)
#R> 0.013639077 0.018840745 -0.154925076
#R> lnVARv0i_(Intercept)
#R> -3.687587268
#R>
#R> -------------- Regression with normal errors: ---------------
#R>
#R> initial value -212.427550
#R> iter 1 value -212.427550
#R> final value -212.427550
#R> converged
#R> Estimate Std.Err Z value Pr(>z)
#R> (Intercept) -4.3869917 3.0631686 -1.4322 0.1520939
#R> log(Y1) 0.4158744 0.1494204 2.7832 0.0053817 **
#R> log(Y2) 0.3700509 0.2827874 1.3086 0.1906756
#R> log(W1) 0.5241524 0.2828127 1.8534 0.0638314 .
#R> log(W2) 1.3420980 1.2504546 1.0733 0.2831419
#R> I(0.5 * log(Y1)^2) 0.0474548 0.0060002 7.9088 2.665e-15 ***
#R> I(0.5 * log(Y2)^2) 0.0774843 0.0225416 3.4374 0.0005873 ***
#R> I(0.5 * log(W1)^2) 0.0349967 0.0273035 1.2818 0.1999249
#R> I(0.5 * log(W2)^2) -0.2262811 0.3426730 -0.6603 0.5090349
#R> log(Y1):log(Y2) -0.0576096 0.0113060 -5.0955 3.479e-07 ***
#R> log(Y1):log(W1) -0.0204630 0.0121495 -1.6843 0.0921316 .
#R> log(Y1):log(W2) -0.0056765 0.0386938 -0.1467 0.8833669
#R> log(Y2):log(W1) 0.0136391 0.0161416 0.8450 0.3981301
#R> log(Y2):log(W2) 0.0188407 0.0596386 0.3159 0.7520666
#R> log(W1):log(W2) -0.1549251 0.0695098 -2.2288 0.0258257 *
#R> lnVARv0i_(Intercept) -3.6875873 0.0632455 -58.3059 < 2.2e-16 ***
#R> ---
#R> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#R> _____________________________________________________________
coef(m1)
#R> (Intercept) log(Y1) log(Y2)
#R> -4.386991731 0.415874408 0.370050861
#R> log(W1) log(W2) I(0.5 * log(Y1)^2)
#R> 0.524152369 1.342097991 0.047454756
#R> I(0.5 * log(Y2)^2) I(0.5 * log(W1)^2) I(0.5 * log(W2)^2)
#R> 0.077484315 0.034996665 -0.226281101
#R> log(Y1):log(Y2) log(Y1):log(W1) log(Y1):log(W2)
#R> -0.057609631 -0.020462972 -0.005676474
#R> log(Y2):log(W1) log(Y2):log(W2) log(W1):log(W2)
#R> 0.013639077 0.018840745 -0.154925076
#R> lnVARv0i_(Intercept)
#R> -3.687587268
# -------------------------------------------------------------
# Specification 2: heteroskedastic noise (variance depends on TA)
# -------------------------------------------------------------
formSV <- ~ log(TA)
m2 <- lm.mle(
formula = spe.tl,
data = banks07,
ln.var.v = formSV
)
#R> theta0:
#R> (Intercept) log(Y1) log(Y2)
#R> -4.386991731 0.415874408 0.370050861
#R> log(W1) log(W2) I(0.5 * log(Y1)^2)
#R> 0.524152369 1.342097991 0.047454756
#R> I(0.5 * log(Y2)^2) I(0.5 * log(W1)^2) I(0.5 * log(W2)^2)
#R> 0.077484315 0.034996665 -0.226281101
#R> log(Y1):log(Y2) log(Y1):log(W1) log(Y1):log(W2)
#R> -0.057609631 -0.020462972 -0.005676474
#R> log(Y2):log(W1) log(Y2):log(W2) log(W1):log(W2)
#R> 0.013639077 0.018840745 -0.154925076
#R> lnVARv0i_(Intercept) lnVARv0i_log(TA)
#R> -3.687587268 0.000000000
#R>
#R> -------------- Regression with normal errors: ---------------
#R>
#R> initial value -212.427550
#R> iter 2 value -212.435637
#R> iter 2 value -212.435637
#R> iter 2 value -212.435638
#R> final value -212.435638
#R> converged
#R> Estimate Std.Err Z value Pr(>z)
#R> (Intercept) -4.38699173 3.07286157 -1.4277 0.1533907
#R> log(Y1) 0.41587442 0.14997873 2.7729 0.0055561 **
#R> log(Y2) 0.37005087 0.29216784 1.2666 0.2053093
#R> log(W1) 0.52415237 0.28241172 1.8560 0.0634555 .
#R> log(W2) 1.34209799 1.25258014 1.0715 0.2839596
#R> I(0.5 * log(Y1)^2) 0.04745481 0.00604526 7.8499 4.219e-15 ***
#R> I(0.5 * log(Y2)^2) 0.07748436 0.02254101 3.4375 0.0005871 ***
#R> I(0.5 * log(W1)^2) 0.03499667 0.02724489 1.2845 0.1989592
#R> I(0.5 * log(W2)^2) -0.22628109 0.34575683 -0.6545 0.5128209
#R> log(Y1):log(Y2) -0.05760953 0.01151446 -5.0032 5.638e-07 ***
#R> log(Y1):log(W1) -0.02046294 0.01213895 -1.6857 0.0918485 .
#R> log(Y1):log(W2) -0.00567644 0.03861266 -0.1470 0.8831243
#R> log(Y2):log(W1) 0.01363911 0.01617084 0.8434 0.3989832
#R> log(Y2):log(W2) 0.01884078 0.06006921 0.3137 0.7537860
#R> log(W1):log(W2) -0.15492507 0.06936169 -2.2336 0.0255105 *
#R> lnVARv0i_(Intercept) -3.68758724 1.00965557 -3.6523 0.0002599 ***
#R> lnVARv0i_log(TA) -0.00036448 0.08630781 -0.0042 0.9966305
#R> ---
#R> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#R> _____________________________________________________________
coef(m2)
#R> (Intercept) log(Y1) log(Y2)
#R> -4.3869917305 0.4158744170 0.3700508703
#R> log(W1) log(W2) I(0.5 * log(Y1)^2)
#R> 0.5241523719 1.3420979943 0.0474548059
#R> I(0.5 * log(Y2)^2) I(0.5 * log(W1)^2) I(0.5 * log(W2)^2)
#R> 0.0774843635 0.0349966695 -0.2262810941
#R> log(Y1):log(Y2) log(Y1):log(W1) log(Y1):log(W2)
#R> -0.0576095323 -0.0204629424 -0.0056764377
#R> log(Y2):log(W1) log(Y2):log(W2) log(W1):log(W2)
#R> 0.0136391061 0.0188407818 -0.1549250651
#R> lnVARv0i_(Intercept) lnVARv0i_log(TA)
#R> -3.6875872440 -0.0003644791
Linear Regression with Skew-Normal Errors
snreg fits a linear regression model where the disturbance term
follows a skew-normal distribution.
# -------------------------------------------------------------
# Specification 1: homoskedastic & symmetric noise
# -------------------------------------------------------------
formSV <- NULL # variance equation
formSK <- NULL # skewness equation
m1 <- snreg(
formula = spe.tl,
data = banks07,
ln.var.v = formSV,
skew.v = formSK
)
#R> theta0:
#R> (Intercept) log(Y1) log(Y2)
#R> -4.386991731 0.415874408 0.370050861
#R> log(W1) log(W2) I(0.5 * log(Y1)^2)
#R> 0.524152369 1.342097991 0.047454756
#R> I(0.5 * log(Y2)^2) I(0.5 * log(W1)^2) I(0.5 * log(W2)^2)
#R> 0.077484315 0.034996665 -0.226281101
#R> log(Y1):log(Y2) log(Y1):log(W1) log(Y1):log(W2)
#R> -0.057609631 -0.020462972 -0.005676474
#R> log(Y2):log(W1) log(Y2):log(W2) log(W1):log(W2)
#R> 0.013639077 0.018840745 -0.154925076
#R> lnVARv0i_(Intercept) Skew_v0i_(Intercept)
#R> -3.687587268 3.000000000
#R>
#R> -------------- Regression with skewed errors: ---------------
#R>
#R> Iteration 0 (at starting values): log likelihood = 5.146960520337
#R>
#R> Iteration 1 (hessian is provided, 1 in total): log likelihood = 201.5619797261
#R> Iteration 2 (hessian is provided, 2 in total): log likelihood = 218.5157385369
#R> Iteration 3 (hessian is provided, 3 in total): log likelihood = 219.8304053615
#R> Iteration 4 (hessian is provided, 4 in total): log likelihood = 221.1570395072
#R> Iteration 5 (hessian is provided, 5 in total): log likelihood = 221.4173977326
#R> Iteration 6 (hessian is provided, 6 in total): log likelihood = 221.45910305
#R> Iteration 7 (hessian is provided, 7 in total): log likelihood = 221.4742381631
#R> Iteration 8 (hessian is provided, 8 in total): log likelihood = 221.4742892007
#R> Iteration 9 (hessian is provided, 9 in total): log likelihood = 221.4772909678
#R>
#R> Convergence given g inv(H) g' = 1.147956e-05 < lmtol
#R>
#R> Final log likelihood = 221.4772909678
#R>
#R> Estimate Std.Err Z value Pr(>z)
#R> (Intercept) -5.88439482 3.00884171 -1.9557 0.050500 .
#R> log(Y1) 0.43986913 0.15724371 2.7974 0.005152 **
#R> log(Y2) 0.50497847 0.27574767 1.8313 0.067055 .
#R> log(W1) 0.54680112 0.27757332 1.9699 0.048846 *
#R> log(W2) 1.62221319 1.19336826 1.3594 0.174034
#R> I(0.5 * log(Y1)^2) 0.04152960 0.00579831 7.1624 7.929e-13 ***
#R> I(0.5 * log(Y2)^2) 0.06775246 0.02251267 3.0095 0.002617 **
#R> I(0.5 * log(W1)^2) 0.03985513 0.02624261 1.5187 0.128833
#R> I(0.5 * log(W2)^2) -0.28421570 0.32037935 -0.8871 0.375013
#R> log(Y1):log(Y2) -0.05595449 0.01162258 -4.8143 1.477e-06 ***
#R> log(Y1):log(W1) -0.02195339 0.01251279 -1.7545 0.079349 .
#R> log(Y1):log(W2) 0.00032291 0.04106930 0.0079 0.993727
#R> log(Y2):log(W1) 0.01294569 0.01577160 0.8208 0.411747
#R> log(Y2):log(W2) 0.00975499 0.05798118 0.1682 0.866391
#R> log(W1):log(W2) -0.15803381 0.06620501 -2.3870 0.016985 *
#R> lnVARv0i_(Intercept) -3.01076847 0.11366937 -26.4871 < 2.2e-16 ***
#R> Skew_v0i_(Intercept) 1.86603587 0.32264117 5.7836 7.311e-09 ***
#R> ---
#R> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#R> _____________________________________________________________
coef(m1)
#R> (Intercept) log(Y1) log(Y2)
#R> -5.884394817 0.439869125 0.504978469
#R> log(W1) log(W2) I(0.5 * log(Y1)^2)
#R> 0.546801118 1.622213193 0.041529597
#R> I(0.5 * log(Y2)^2) I(0.5 * log(W1)^2) I(0.5 * log(W2)^2)
#R> 0.067752458 0.039855132 -0.284215702
#R> log(Y1):log(Y2) log(Y1):log(W1) log(Y1):log(W2)
#R> -0.055954492 -0.021953387 0.000322907
#R> log(Y2):log(W1) log(Y2):log(W2) log(W1):log(W2)
#R> 0.012945689 0.009754988 -0.158033814
#R> lnVARv0i_(Intercept) Skew_v0i_(Intercept)
#R> -3.010768472 1.866035868
# -------------------------------------------------------------
# Specification 2: heteroskedastic + skewed noise
# -------------------------------------------------------------
formSV <- ~ log(TA) # heteroskedasticity in v
formSK <- ~ ER # skewness driven by equity ratio
m2 <- snreg(
formula = spe.tl,
data = banks07,
ln.var.v = formSV,
skew.v = formSK
)
#R> theta0:
#R> (Intercept) log(Y1) log(Y2)
#R> -4.386991731 0.415874408 0.370050861
#R> log(W1) log(W2) I(0.5 * log(Y1)^2)
#R> 0.524152369 1.342097991 0.047454756
#R> I(0.5 * log(Y2)^2) I(0.5 * log(W1)^2) I(0.5 * log(W2)^2)
#R> 0.077484315 0.034996665 -0.226281101
#R> log(Y1):log(Y2) log(Y1):log(W1) log(Y1):log(W2)
#R> -0.057609631 -0.020462972 -0.005676474
#R> log(Y2):log(W1) log(Y2):log(W2) log(W1):log(W2)
#R> 0.013639077 0.018840745 -0.154925076
#R> lnVARv0i_(Intercept) lnVARv0i_lnVARv0i_log(TA) Skew_v0i_(Intercept)
#R> -3.687587268 0.000000000 3.000000000
#R> Skew_v0i_Skew_v0i_ER
#R> 0.000000000
#R>
#R> ----------------- Regression with skewed errors: -----------------
#R>
#R> Iteration 0 (at starting values): log likelihood = 5.146960520337
#R>
#R> Iteration 1 (hessian is provided, 1 in total): log likelihood = 204.6988401214
#R> Iteration 2 (hessian is provided, 2 in total): log likelihood = 221.6390613234
#R> Iteration 3 (hessian is provided, 3 in total): log likelihood = 224.8771023643
#R> Iteration 4 (hessian is provided, 4 in total): log likelihood = 225.6867346145
#R> Iteration 5 (hessian is provided, 5 in total): log likelihood = 225.9380532273
#R> Iteration 6 (hessian is provided, 6 in total): log likelihood = 226.0484166119
#R> Iteration 7 (hessian is provided, 7 in total): log likelihood = 226.0721225638
#R> Iteration 8 (hessian is provided, 8 in total): log likelihood = 226.0820635056
#R> Iteration 9 (hessian is provided, 9 in total): log likelihood = 226.0835072457
#R> Iteration 10 (hessian is provided, 10 in total): log likelihood = 226.0909506421
#R>
#R> Convergence given g inv(H) g' = 9.357213e-05 < lmtol
#R>
#R> Final log likelihood = 226.0909506421
#R>
#R> Estimate Std.Err Z value Pr(>z)
#R> (Intercept) -5.71501334 2.98991742 -1.9114 0.055950 .
#R> log(Y1) 0.45216186 0.15708195 2.8785 0.003996 **
#R> log(Y2) 0.64671047 0.28153250 2.2971 0.021613 *
#R> log(W1) 0.48917765 0.27664356 1.7683 0.077018 .
#R> log(W2) 1.15530138 1.17400483 0.9841 0.325082
#R> I(0.5 * log(Y1)^2) 0.04070906 0.00590546 6.8935 5.445e-12 ***
#R> I(0.5 * log(Y2)^2) 0.06170967 0.02265802 2.7235 0.006459 **
#R> I(0.5 * log(W1)^2) 0.04271441 0.02667730 1.6012 0.109343
#R> I(0.5 * log(W2)^2) -0.15999322 0.31994376 -0.5001 0.617028
#R> log(Y1):log(Y2) -0.05757860 0.01197968 -4.8064 1.537e-06 ***
#R> log(Y1):log(W1) -0.01887894 0.01276125 -1.4794 0.139035
#R> log(Y1):log(W2) 0.00165155 0.04117676 0.0401 0.968006
#R> log(Y2):log(W1) 0.00516054 0.01583258 0.3259 0.744467
#R> log(Y2):log(W2) -0.00022261 0.05922730 -0.0038 0.997001
#R> log(W1):log(W2) -0.13107647 0.06626509 -1.9781 0.047922 *
#R> lnVARv0i_(Intercept) -0.89452888 1.03829604 -0.8615 0.388943
#R> lnVARv0i_lnVARv0i_log(TA) -0.17730037 0.08736480 -2.0294 0.042415 *
#R> Skew_v0i_(Intercept) 3.10567750 0.76059557 4.0832 4.442e-05 ***
#R> Skew_v0i_Skew_v0i_ER -9.55208958 4.87607389 -1.9590 0.050116 .
#R> ---
#R> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#R> __________________________________________________________________
coef(m2)
#R> (Intercept) log(Y1) log(Y2)
#R> -5.7150133374 0.4521618623 0.6467104718
#R> log(W1) log(W2) I(0.5 * log(Y1)^2)
#R> 0.4891776489 1.1553013814 0.0407090640
#R> I(0.5 * log(Y2)^2) I(0.5 * log(W1)^2) I(0.5 * log(W2)^2)
#R> 0.0617096653 0.0427144069 -0.1599932243
#R> log(Y1):log(Y2) log(Y1):log(W1) log(Y1):log(W2)
#R> -0.0575785975 -0.0188789363 0.0016515541
#R> log(Y2):log(W1) log(Y2):log(W2) log(W1):log(W2)
#R> 0.0051605363 -0.0002226092 -0.1310764688
#R> lnVARv0i_(Intercept) lnVARv0i_lnVARv0i_log(TA) Skew_v0i_(Intercept)
#R> -0.8945288842 -0.1773003673 3.1056774984
#R> Skew_v0i_Skew_v0i_ER
#R> -9.5520895773
Stochastic Frontier Model with a Skew-Normally Distributed Error Term
snsf performs maximum likelihood estimation of the parameters and
technical or cost efficiencies in a Stochastic Frontier Model with a
skew-normally distributed error term.
myprod <- FALSE
# -------------------------------------------------------------
# Specification 1: homoskedastic & symmetric
# -------------------------------------------------------------
formSV <- NULL # variance equation
formSK <- NULL # skewness equation
formSU <- NULL # inefficiency equation (unused here)
m1 <- snsf(
formula = spe.tl,
data = banks07,
prod = myprod,
ln.var.v = formSV,
skew.v = formSK
)
#R>
#R> -------------- Regression with skewed errors: ---------------
#R>
#R> initial value -105.392823
#R> iter 2 value -105.561746
#R> iter 3 value -105.568000
#R> iter 4 value -105.836407
#R> iter 5 value -106.255928
#R> iter 6 value -107.989401
#R> iter 7 value -129.390401
#R> iter 8 value -148.614164
#R> iter 8 value -148.614164
#R> iter 9 value -148.627674
#R> iter 10 value -148.628304
#R> iter 10 value -148.628304
#R> iter 10 value -148.628304
#R> final value -148.628304
#R> converged
#R> Estimate Std.Err Z value Pr(>z)
#R> X(Intercept) -4.40500089 3.80254827 -1.1584 0.246687
#R> Xlog(Y1) 0.45706967 0.18307828 2.4966 0.012540 *
#R> Xlog(Y2) 0.31560584 0.33532057 0.9412 0.346599
#R> Xlog(W1) 0.48827077 0.38637994 1.2637 0.206335
#R> Xlog(W2) 1.29215252 1.70829901 0.7564 0.449411
#R> XI(0.5 * log(Y1)^2) 0.06252731 0.00718640 8.7008 < 2.2e-16 ***
#R> XI(0.5 * log(Y2)^2) 0.19680294 0.03130033 6.2876 3.225e-10 ***
#R> XI(0.5 * log(W1)^2) 0.05041646 0.03723206 1.3541 0.175700
#R> XI(0.5 * log(W2)^2) -0.28522159 0.46697561 -0.6108 0.541342
#R> Xlog(Y1):log(Y2) -0.14258698 0.00610987 -23.3372 < 2.2e-16 ***
#R> Xlog(Y1):log(W1) 0.00029592 0.01705705 0.0173 0.986158
#R> Xlog(Y1):log(W2) 0.17610934 0.05558196 3.1685 0.001532 **
#R> Xlog(Y2):log(W1) 0.01454480 0.02162340 0.6726 0.501175
#R> Xlog(Y2):log(W2) -0.09354141 0.07710129 -1.2132 0.225043
#R> Xlog(W1):log(W2) -0.21197450 0.09475124 -2.2372 0.025275 *
#R> lnVARv0i_(Intercept) -2.66540968 0.11013951 -24.2003 < 2.2e-16 ***
#R> Skew_v0i_(Intercept) 0.98854923 0.03434255 28.7850 < 2.2e-16 ***
#R> sv 0.26376286 0.00766250 34.4226 < 2.2e-16 ***
#R> ---
#R> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#R> _____________________________________________________________
#R> theta0:
#R> (Intercept) log(Y1) log(Y2)
#R> -4.4050008900 0.4570696682 0.3156058426
#R> log(W1) log(W2) I(0.5 * log(Y1)^2)
#R> 0.4882707672 1.2921525200 0.0625273146
#R> I(0.5 * log(Y2)^2) I(0.5 * log(W1)^2) I(0.5 * log(W2)^2)
#R> 0.1968029394 0.0504164557 -0.2852215911
#R> log(Y1):log(Y2) log(Y1):log(W1) log(Y1):log(W2)
#R> -0.1425869800 0.0002959179 0.1761093362
#R> log(Y2):log(W1) log(Y2):log(W2) log(W1):log(W2)
#R> 0.0145448021 -0.0935414140 -0.2119744969
#R> lnVARv0i_(Intercept) Skew_v0i_(Intercept) lnVARu0i_(Intercept)
#R> -4.2020745955 0.3982617494 -4.5984104618
#R>
#R> ---------------------- The main model: ----------------------
#R>
#R> Iteration 0 (at starting values): log likelihood = 21.64346032248
#R>
#R> Iteration 1 (hessian is provided, 1 in total): log likelihood = 186.4880063145
#R> Iteration 2 (hessian is provided, 2 in total): log likelihood = 214.3430643968
#R> Iteration 3 (hessian is provided, 3 in total): log likelihood = 220.957921695
#R> Iteration 4 (hessian is provided, 4 in total): log likelihood = 223.1040419712
#R> Iteration 5 (hessian is provided, 5 in total): log likelihood = 224.3725358921
#R> Iteration 6 (hessian is provided, 6 in total): log likelihood = 224.6133195507
#R> Iteration 7 (hessian is provided, 7 in total): log likelihood = 224.7096271178
#R> Iteration 8 (hessian is provided, 8 in total): log likelihood = 224.7298349674
#R> Iteration 9 (hessian is provided, 9 in total): log likelihood = 224.8005650898
#R> Iteration 10 (hessian is provided, 10 in total): log likelihood = 224.8011109934
#R> Iteration 11 (hessian is provided, 11 in total): log likelihood = 224.8019248405
#R> Iteration 12 (hessian is provided, 12 in total): log likelihood = 224.8021079368
#R> Iteration 13 (hessian is provided, 13 in total): log likelihood = 224.8054321542
#R> Iteration 14 (hessian is provided, 14 in total): log likelihood = 224.8420044777
#R> Iteration 15 (hessian is provided, 15 in total): log likelihood = 225.4135588864
#R> Iteration 16 (hessian is provided, 16 in total): log likelihood = 225.5282232035
#R> Iteration 17 (hessian is provided, 17 in total): log likelihood = 225.533351752
#R> Iteration 18 (hessian is provided, 18 in total): log likelihood = 225.5368333384
#R> Iteration 19 (hessian is provided, 19 in total): log likelihood = 225.5371105355
#R> Iteration 20 (hessian is provided, 20 in total): log likelihood = 225.539300346
#R> Iteration 21 (hessian is provided, 21 in total): log likelihood = 225.539317247
#R> Iteration 22 (hessian is provided, 22 in total): log likelihood = 225.539324279
#R>
#R> Convergence given g inv(H) g' = 1.027857e-06 < lmtol
#R>
#R> Final log likelihood = 225.539324279
#R>
#R>
#R> Log likelihood maximization completed in
#R> 0.18 seconds
#R> _____________________________________________________________
#R>
#R> Cross-sectional stochastic (cost) frontier model
#R>
#R> Distributional assumptions
#R>
#R> Component Distribution Assumption
#R> 1 Random noise: skew normal homoskedastic
#R> 2 Inefficiency: exponential homoskedastic
#R>
#R> Number of observations = 500
#R>
#R> -------------------- Estimation results: --------------------
#R>
#R> Coef. SE z P>|z|
#R> _____________________________________________________________
#R> Frontier
#R>
#R> (Intercept) -6.3606 3.0602 -2.08 0.0377 *
#R> log(Y1) 0.4425 0.1767 2.50 0.0123 *
#R> log(Y2) 0.5660 0.2753 2.06 0.0398 *
#R> log(W1) 0.4983 0.2835 1.76 0.0788 .
#R> log(W2) 1.6557 1.1889 1.39 0.1637
#R> I(0.5 * log(Y1)^2) 0.0449 0.0066 6.76 0.0000 ***
#R> I(0.5 * log(Y2)^2) 0.0650 0.0222 2.93 0.0034 **
#R> I(0.5 * log(W1)^2) 0.0363 0.0258 1.41 0.1597
#R> I(0.5 * log(W2)^2) -0.3099 0.3170 -0.98 0.3283
#R> log(Y1):log(Y2) -0.0586 0.0117 -4.99 0.0000 ***
#R> log(Y1):log(W1) -0.0227 0.0127 -1.78 0.0750 .
#R> log(Y1):log(W2) -0.0006 0.0420 -0.01 0.9895
#R> log(Y2):log(W1) 0.0117 0.0152 0.77 0.4393
#R> log(Y2):log(W2) 0.0106 0.0563 0.19 0.8505
#R> log(W1):log(W2) -0.1375 0.0682 -2.02 0.0438 *
#R> -------------------------------------------------------------
#R> Variance of the random noise component: log(sigma_v^2)
#R>
#R> lnVARv0i_(Intercept) -3.8207 0.2255 -16.94 0.0000 ***
#R> -------------------------------------------------------------
#R> Skewness of the random noise component: `alpha`
#R>
#R> Skew_v0i_(Intercept) -1.5007 0.8755 -1.71 0.0865 .
#R> -------------------------------------------------------------
#R> Inefficiency component: log(sigma_u^2)
#R>
#R> lnVARu0i_(Intercept) -4.3422 0.2398 -18.11 0.0000 ***
#R> _____________________________________________________________
#R> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#R>
#R> --------------- Summary of cost efficiencies: ---------------
#R>
#R> JLMS:= exp(-E[ui|ei])
#R>
#R> TE_JLMS
#R> Obs 500
#R> NAs 0
#R> Mean 0.8955
#R> StDev 0.0715
#R> IQR 0.0645
#R> Min 0.4559
#R> 5% 0.7420
#R> 10% 0.7959
#R> 25% 0.8780
#R> 50% 0.9208
#R> 75% 0.9425
#R> 90% 0.9520
#R> Max 0.9691
coef(m1)
#R> (Intercept) log(Y1) log(Y2)
#R> -6.3606134654 0.4425312174 0.5660453675
#R> log(W1) log(W2) I(0.5 * log(Y1)^2)
#R> 0.4983319531 1.6557355502 0.0448519992
#R> I(0.5 * log(Y2)^2) I(0.5 * log(W1)^2) I(0.5 * log(W2)^2)
#R> 0.0649995029 0.0362747554 -0.3099280819
#R> log(Y1):log(Y2) log(Y1):log(W1) log(Y1):log(W2)
#R> -0.0586056701 -0.0226917826 -0.0005509121
#R> log(Y2):log(W1) log(Y2):log(W2) log(W1):log(W2)
#R> 0.0117430476 0.0106095399 -0.1375404433
#R> lnVARv0i_(Intercept) Skew_v0i_(Intercept) lnVARu0i_(Intercept)
#R> -3.8206601915 -1.5006628040 -4.3422317628
# -------------------------------------------------------------
# Specification 2: heteroskedastic + skewed noise
# -------------------------------------------------------------
formSV <- ~ log(TA) # heteroskedastic variance
formSK <- ~ ER # skewness driver
formSU <- ~ LA + ER # inefficiency
m2 <- snsf(
formula = spe.tl,
data = banks07,
prod = myprod,
ln.var.v = formSV,
skew.v = formSK
)
#R>
#R> -------------- Regression with skewed errors: ---------------
#R>
#R> initial value -105.392823
#R> iter 2 value -105.561746
#R> iter 3 value -105.568000
#R> iter 4 value -105.836407
#R> iter 5 value -106.255928
#R> iter 6 value -107.989401
#R> iter 7 value -129.390401
#R> iter 8 value -148.614164
#R> iter 8 value -148.614164
#R> iter 9 value -148.627674
#R> iter 10 value -148.628304
#R> iter 10 value -148.628304
#R> iter 10 value -148.628304
#R> final value -148.628304
#R> converged
#R> Estimate Std.Err Z value Pr(>z)
#R> X(Intercept) -4.40500089 3.80254827 -1.1584 0.246687
#R> Xlog(Y1) 0.45706967 0.18307828 2.4966 0.012540 *
#R> Xlog(Y2) 0.31560584 0.33532057 0.9412 0.346599
#R> Xlog(W1) 0.48827077 0.38637994 1.2637 0.206335
#R> Xlog(W2) 1.29215252 1.70829901 0.7564 0.449411
#R> XI(0.5 * log(Y1)^2) 0.06252731 0.00718640 8.7008 < 2.2e-16 ***
#R> XI(0.5 * log(Y2)^2) 0.19680294 0.03130033 6.2876 3.225e-10 ***
#R> XI(0.5 * log(W1)^2) 0.05041646 0.03723206 1.3541 0.175700
#R> XI(0.5 * log(W2)^2) -0.28522159 0.46697561 -0.6108 0.541342
#R> Xlog(Y1):log(Y2) -0.14258698 0.00610987 -23.3372 < 2.2e-16 ***
#R> Xlog(Y1):log(W1) 0.00029592 0.01705705 0.0173 0.986158
#R> Xlog(Y1):log(W2) 0.17610934 0.05558196 3.1685 0.001532 **
#R> Xlog(Y2):log(W1) 0.01454480 0.02162340 0.6726 0.501175
#R> Xlog(Y2):log(W2) -0.09354141 0.07710129 -1.2132 0.225043
#R> Xlog(W1):log(W2) -0.21197450 0.09475124 -2.2372 0.025275 *
#R> lnVARv0i_(Intercept) -2.66540968 0.11013951 -24.2003 < 2.2e-16 ***
#R> Skew_v0i_(Intercept) 0.98854923 0.03434255 28.7850 < 2.2e-16 ***
#R> sv 0.26376286 0.00766250 34.4226 < 2.2e-16 ***
#R> ---
#R> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#R> _____________________________________________________________
#R> theta0:
#R> (Intercept) log(Y1) log(Y2)
#R> -4.405001e+00 4.570697e-01 3.156058e-01
#R> log(W1) log(W2) I(0.5 * log(Y1)^2)
#R> 4.882708e-01 1.292153e+00 6.252731e-02
#R> I(0.5 * log(Y2)^2) I(0.5 * log(W1)^2) I(0.5 * log(W2)^2)
#R> 1.968029e-01 5.041646e-02 -2.852216e-01
#R> log(Y1):log(Y2) log(Y1):log(W1) log(Y1):log(W2)
#R> -1.425870e-01 2.959179e-04 1.761093e-01
#R> log(Y2):log(W1) log(Y2):log(W2) log(W1):log(W2)
#R> 1.454480e-02 -9.354141e-02 -2.119745e-01
#R> lnVARv0i_(Intercept) lnVARv0i_log(TA) Skew_v0i_(Intercept)
#R> -4.202075e+00 2.212626e-16 3.982617e-01
#R> Skew_v0i_ER lnVARu0i_(Intercept)
#R> 0.000000e+00 -4.598410e+00
#R>
#R> ---------------------- The main model: ----------------------
#R>
#R> Iteration 0 (at starting values): log likelihood = 21.64346032248
#R>
#R> Iteration 1 (hessian is provided, 1 in total): log likelihood = 133.2018982456
#R> Iteration 2 (hessian is provided, 2 in total): log likelihood = 183.2824024606
#R> Iteration 3 (hessian is provided, 3 in total): log likelihood = 214.7412845581
#R> Iteration 4 (hessian is provided, 4 in total): log likelihood = 223.1763227676
#R> Iteration 5 (hessian is provided, 5 in total): log likelihood = 226.655477596
#R> Iteration 6 (hessian is provided, 6 in total): log likelihood = 226.7105002202
#R> Iteration 7 (hessian is provided, 7 in total): log likelihood = 226.8074129256
#R> Iteration 8 (hessian is provided, 8 in total): log likelihood = 226.8082101988
#R> Iteration 9 (hessian is provided, 9 in total): log likelihood = 226.8082151929
#R> Iteration 10 (hessian is provided, 10 in total): log likelihood = 226.808224564
#R>
#R> Convergence given g inv(H) g' = 2.529443e-06 < lmtol
#R>
#R> Final log likelihood = 226.808224564
#R>
#R>
#R> Log likelihood maximization completed in
#R> 0.103 seconds
#R> _____________________________________________________________
#R>
#R> Cross-sectional stochastic (cost) frontier model
#R>
#R> Distributional assumptions
#R>
#R> Component Distribution Assumption
#R> 1 Random noise: skew normal heteroskedastic
#R> 2 Inefficiency: exponential homoskedastic
#R>
#R> Number of observations = 500
#R>
#R> -------------------- Estimation results: --------------------
#R>
#R> Coef. SE z P>|z|
#R> _____________________________________________________________
#R> Frontier
#R>
#R> (Intercept) -5.9441 3.0205 -1.97 0.0491 *
#R> log(Y1) 0.4589 0.1647 2.79 0.0053 **
#R> log(Y2) 0.6135 0.2820 2.18 0.0296 *
#R> log(W1) 0.5168 0.2747 1.88 0.0599 .
#R> log(W2) 1.2707 1.1934 1.06 0.2870
#R> I(0.5 * log(Y1)^2) 0.0414 0.0061 6.81 0.0000 ***
#R> I(0.5 * log(Y2)^2) 0.0614 0.0226 2.72 0.0065 **
#R> I(0.5 * log(W1)^2) 0.0439 0.0262 1.68 0.0938 .
#R> I(0.5 * log(W2)^2) -0.2078 0.3247 -0.64 0.5223
#R> log(Y1):log(Y2) -0.0575 0.0120 -4.79 0.0000 ***
#R> log(Y1):log(W1) -0.0215 0.0128 -1.68 0.0927 .
#R> log(Y1):log(W2) 0.0000 0.0417 0.00 0.9997
#R> log(Y2):log(W1) 0.0068 0.0158 0.43 0.6682
#R> log(Y2):log(W2) 0.0086 0.0589 0.15 0.8836
#R> log(W1):log(W2) -0.1368 0.0661 -2.07 0.0384 *
#R> -------------------------------------------------------------
#R> Variance of the random noise component: log(sigma_v^2)
#R>
#R> lnVARv0i_(Intercept) -1.8839 1.6595 -1.14 0.2563
#R> lnVARv0i_log(TA) -0.1558 0.1328 -1.17 0.2407
#R> -------------------------------------------------------------
#R> Skewness of the random noise component: `alpha`
#R>
#R> Skew_v0i_(Intercept) 2.0043 1.0468 1.91 0.0555 .
#R> Skew_v0i_ER -7.2632 5.0581 -1.44 0.1510
#R> -------------------------------------------------------------
#R> Inefficiency component: log(sigma_u^2)
#R>
#R> lnVARu0i_(Intercept) -4.6407 0.3416 -13.58 0.0000 ***
#R> _____________________________________________________________
#R> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#R>
#R> --------------- Summary of cost efficiencies: ---------------
#R>
#R> JLMS:= exp(-E[ui|ei])
#R>
#R> TE_JLMS
#R> Obs 500
#R> NAs 0
#R> Mean 0.9084
#R> StDev 0.0564
#R> IQR 0.0547
#R> Min 0.5206
#R> 5% 0.8096
#R> 10% 0.8379
#R> 25% 0.8903
#R> 50% 0.9241
#R> 75% 0.9450
#R> 90% 0.9568
#R> Max 0.9758
coef(m2)
#R> (Intercept) log(Y1) log(Y2)
#R> -5.944090e+00 4.589482e-01 6.134700e-01
#R> log(W1) log(W2) I(0.5 * log(Y1)^2)
#R> 5.167646e-01 1.270694e+00 4.138294e-02
#R> I(0.5 * log(Y2)^2) I(0.5 * log(W1)^2) I(0.5 * log(W2)^2)
#R> 6.141001e-02 4.385504e-02 -2.077538e-01
#R> log(Y1):log(Y2) log(Y1):log(W1) log(Y1):log(W2)
#R> -5.745504e-02 -2.151307e-02 1.351571e-05
#R> log(Y2):log(W1) log(Y2):log(W2) log(W1):log(W2)
#R> 6.752251e-03 8.627646e-03 -1.368348e-01
#R> lnVARv0i_(Intercept) lnVARv0i_log(TA) Skew_v0i_(Intercept)
#R> -1.883867e+00 -1.558322e-01 2.004266e+00
#R> Skew_v0i_ER lnVARu0i_(Intercept)
#R> -7.263169e+00 -4.640750e+00
Additional Resources
See
Oleg Badunenko and Daniel J. Henderson (2023). "Production analysis with asymmetric noise". Journal of Productivity Analysis, 61(1), 1–18. DOI 
for more details
Try the snreg package in your browser
Any scripts or data that you put into this service are public.
snreg documentation built on Feb. 6, 2026, 5:08 p.m.
R Package Documentation
Browse R Packages
We want your feedback!
Note that we can't provide technical support on individual packages. You should contact the package authors for that.
Illustration
Data
is a data frame containing selected variables for 500 U.S. commercial banks, randomly sampled from approximately 5000 banks, based on the dataset of Koetter et al. (2012) for year 2007. The dataset is provided solely for illustration and pedagogical purposes and is not suitable for empirical research.
library(snreg) data(banks07, package = "snreg") head(banks07) #R> id year TC Y1 Y2 W1 W2 W3 ER #R> 1 152440 2007 4659.505 8661.84 58393.04 38.55422 37.75000 4.146092 0.1117789 #R> 2 544335 2007 10848.960 19492.91 146789.00 30.66785 45.13934 2.263886 0.1139121 #R> 3 548203 2007 14523.650 18630.49 219705.70 27.38797 56.02857 4.312672 0.1342984 #R> 4 568238 2007 1644.808 11902.50 18675.68 32.84768 50.47368 1.541316 0.1029891 #R> 5 651158 2007 3054.240 29322.21 38916.14 42.20033 52.23529 3.192056 0.1421004 #R> 6 705444 2007 6168.736 36568.03 67179.16 11.95771 41.46667 3.181241 0.1449880 #R> LA TA ZSCORE ZSCORE3 SDROA LLP lnzscore lnzscore3 #R> 1 0.7781611 75039.78 29.49198 11.93898 0.4532092 732.4904 3.384118 2.479809 #R> 2 0.7679300 191148.92 62.97659 20.66493 0.2298561 305.9889 4.142763 3.028438 #R> 3 0.8880010 247416.05 38.99969 30.88642 0.3874364 1534.6521 3.663554 3.430317 #R> 4 0.5050412 36978.53 30.68124 24.53655 0.4063778 0.0000 3.423651 3.200164 #R> 5 0.5264002 73928.81 30.02672 28.42257 0.5496323 0.0000 3.402088 3.347184 #R> 6 0.5222276 128639.62 110.67650 37.66002 0.1502981 0.0000 4.706612 3.628599 #R> lnsdroa scope ms_county #R> 1 -0.7914014 0.4801792 0.3466652 #R> 2 -1.4703018 0.5254947 0.8830606 #R> 3 -0.9482036 0.8897177 1.1430008 #R> 4 -0.9004720 0.3917692 0.1708316 #R> 5 -0.5985058 0.5523984 0.3415328 #R> 6 -1.8951346 0.4073012 0.5942832
Specification
Define the specification (formula) that will be used:
# Translog cost function specification spe.tl <- log(TC) ~ (log(Y1) + log(Y2) + log(W1) + log(W2))^2 + I(0.5 * log(Y1)^2) + I(0.5 * log(Y2)^2) + I(0.5 * log(W1)^2) + I(0.5 * log(W2)^2)
Linear Regression via MLE
To estimate simple OLS using MLE
# ------------------------------------------------------------- # Specification 1: homoskedastic noise (ln.var.v = NULL) # ------------------------------------------------------------- formSV <- NULL m1 <- lm.mle( formula = spe.tl, data = banks07, ln.var.v = formSV ) #R> theta0: #R> (Intercept) log(Y1) log(Y2) #R> -4.386991731 0.415874408 0.370050861 #R> log(W1) log(W2) I(0.5 * log(Y1)^2) #R> 0.524152369 1.342097991 0.047454756 #R> I(0.5 * log(Y2)^2) I(0.5 * log(W1)^2) I(0.5 * log(W2)^2) #R> 0.077484315 0.034996665 -0.226281101 #R> log(Y1):log(Y2) log(Y1):log(W1) log(Y1):log(W2) #R> -0.057609631 -0.020462972 -0.005676474 #R> log(Y2):log(W1) log(Y2):log(W2) log(W1):log(W2) #R> 0.013639077 0.018840745 -0.154925076 #R> lnVARv0i_(Intercept) #R> -3.687587268 #R> #R> -------------- Regression with normal errors: --------------- #R> #R> initial value -212.427550 #R> iter 1 value -212.427550 #R> final value -212.427550 #R> converged #R> Estimate Std.Err Z value Pr(>z) #R> (Intercept) -4.3869917 3.0631686 -1.4322 0.1520939 #R> log(Y1) 0.4158744 0.1494204 2.7832 0.0053817 ** #R> log(Y2) 0.3700509 0.2827874 1.3086 0.1906756 #R> log(W1) 0.5241524 0.2828127 1.8534 0.0638314 . #R> log(W2) 1.3420980 1.2504546 1.0733 0.2831419 #R> I(0.5 * log(Y1)^2) 0.0474548 0.0060002 7.9088 2.665e-15 *** #R> I(0.5 * log(Y2)^2) 0.0774843 0.0225416 3.4374 0.0005873 *** #R> I(0.5 * log(W1)^2) 0.0349967 0.0273035 1.2818 0.1999249 #R> I(0.5 * log(W2)^2) -0.2262811 0.3426730 -0.6603 0.5090349 #R> log(Y1):log(Y2) -0.0576096 0.0113060 -5.0955 3.479e-07 *** #R> log(Y1):log(W1) -0.0204630 0.0121495 -1.6843 0.0921316 . #R> log(Y1):log(W2) -0.0056765 0.0386938 -0.1467 0.8833669 #R> log(Y2):log(W1) 0.0136391 0.0161416 0.8450 0.3981301 #R> log(Y2):log(W2) 0.0188407 0.0596386 0.3159 0.7520666 #R> log(W1):log(W2) -0.1549251 0.0695098 -2.2288 0.0258257 * #R> lnVARv0i_(Intercept) -3.6875873 0.0632455 -58.3059 < 2.2e-16 *** #R> --- #R> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 #R> _____________________________________________________________ coef(m1) #R> (Intercept) log(Y1) log(Y2) #R> -4.386991731 0.415874408 0.370050861 #R> log(W1) log(W2) I(0.5 * log(Y1)^2) #R> 0.524152369 1.342097991 0.047454756 #R> I(0.5 * log(Y2)^2) I(0.5 * log(W1)^2) I(0.5 * log(W2)^2) #R> 0.077484315 0.034996665 -0.226281101 #R> log(Y1):log(Y2) log(Y1):log(W1) log(Y1):log(W2) #R> -0.057609631 -0.020462972 -0.005676474 #R> log(Y2):log(W1) log(Y2):log(W2) log(W1):log(W2) #R> 0.013639077 0.018840745 -0.154925076 #R> lnVARv0i_(Intercept) #R> -3.687587268 # ------------------------------------------------------------- # Specification 2: heteroskedastic noise (variance depends on TA) # ------------------------------------------------------------- formSV <- ~ log(TA) m2 <- lm.mle( formula = spe.tl, data = banks07, ln.var.v = formSV ) #R> theta0: #R> (Intercept) log(Y1) log(Y2) #R> -4.386991731 0.415874408 0.370050861 #R> log(W1) log(W2) I(0.5 * log(Y1)^2) #R> 0.524152369 1.342097991 0.047454756 #R> I(0.5 * log(Y2)^2) I(0.5 * log(W1)^2) I(0.5 * log(W2)^2) #R> 0.077484315 0.034996665 -0.226281101 #R> log(Y1):log(Y2) log(Y1):log(W1) log(Y1):log(W2) #R> -0.057609631 -0.020462972 -0.005676474 #R> log(Y2):log(W1) log(Y2):log(W2) log(W1):log(W2) #R> 0.013639077 0.018840745 -0.154925076 #R> lnVARv0i_(Intercept) lnVARv0i_log(TA) #R> -3.687587268 0.000000000 #R> #R> -------------- Regression with normal errors: --------------- #R> #R> initial value -212.427550 #R> iter 2 value -212.435637 #R> iter 2 value -212.435637 #R> iter 2 value -212.435638 #R> final value -212.435638 #R> converged #R> Estimate Std.Err Z value Pr(>z) #R> (Intercept) -4.38699173 3.07286157 -1.4277 0.1533907 #R> log(Y1) 0.41587442 0.14997873 2.7729 0.0055561 ** #R> log(Y2) 0.37005087 0.29216784 1.2666 0.2053093 #R> log(W1) 0.52415237 0.28241172 1.8560 0.0634555 . #R> log(W2) 1.34209799 1.25258014 1.0715 0.2839596 #R> I(0.5 * log(Y1)^2) 0.04745481 0.00604526 7.8499 4.219e-15 *** #R> I(0.5 * log(Y2)^2) 0.07748436 0.02254101 3.4375 0.0005871 *** #R> I(0.5 * log(W1)^2) 0.03499667 0.02724489 1.2845 0.1989592 #R> I(0.5 * log(W2)^2) -0.22628109 0.34575683 -0.6545 0.5128209 #R> log(Y1):log(Y2) -0.05760953 0.01151446 -5.0032 5.638e-07 *** #R> log(Y1):log(W1) -0.02046294 0.01213895 -1.6857 0.0918485 . #R> log(Y1):log(W2) -0.00567644 0.03861266 -0.1470 0.8831243 #R> log(Y2):log(W1) 0.01363911 0.01617084 0.8434 0.3989832 #R> log(Y2):log(W2) 0.01884078 0.06006921 0.3137 0.7537860 #R> log(W1):log(W2) -0.15492507 0.06936169 -2.2336 0.0255105 * #R> lnVARv0i_(Intercept) -3.68758724 1.00965557 -3.6523 0.0002599 *** #R> lnVARv0i_log(TA) -0.00036448 0.08630781 -0.0042 0.9966305 #R> --- #R> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 #R> _____________________________________________________________ coef(m2) #R> (Intercept) log(Y1) log(Y2) #R> -4.3869917305 0.4158744170 0.3700508703 #R> log(W1) log(W2) I(0.5 * log(Y1)^2) #R> 0.5241523719 1.3420979943 0.0474548059 #R> I(0.5 * log(Y2)^2) I(0.5 * log(W1)^2) I(0.5 * log(W2)^2) #R> 0.0774843635 0.0349966695 -0.2262810941 #R> log(Y1):log(Y2) log(Y1):log(W1) log(Y1):log(W2) #R> -0.0576095323 -0.0204629424 -0.0056764377 #R> log(Y2):log(W1) log(Y2):log(W2) log(W1):log(W2) #R> 0.0136391061 0.0188407818 -0.1549250651 #R> lnVARv0i_(Intercept) lnVARv0i_log(TA) #R> -3.6875872440 -0.0003644791
Linear Regression with Skew-Normal Errors
snregfits a linear regression model where the disturbance term follows a skew-normal distribution.
# ------------------------------------------------------------- # Specification 1: homoskedastic & symmetric noise # ------------------------------------------------------------- formSV <- NULL # variance equation formSK <- NULL # skewness equation m1 <- snreg( formula = spe.tl, data = banks07, ln.var.v = formSV, skew.v = formSK ) #R> theta0: #R> (Intercept) log(Y1) log(Y2) #R> -4.386991731 0.415874408 0.370050861 #R> log(W1) log(W2) I(0.5 * log(Y1)^2) #R> 0.524152369 1.342097991 0.047454756 #R> I(0.5 * log(Y2)^2) I(0.5 * log(W1)^2) I(0.5 * log(W2)^2) #R> 0.077484315 0.034996665 -0.226281101 #R> log(Y1):log(Y2) log(Y1):log(W1) log(Y1):log(W2) #R> -0.057609631 -0.020462972 -0.005676474 #R> log(Y2):log(W1) log(Y2):log(W2) log(W1):log(W2) #R> 0.013639077 0.018840745 -0.154925076 #R> lnVARv0i_(Intercept) Skew_v0i_(Intercept) #R> -3.687587268 3.000000000 #R> #R> -------------- Regression with skewed errors: --------------- #R> #R> Iteration 0 (at starting values): log likelihood = 5.146960520337 #R> #R> Iteration 1 (hessian is provided, 1 in total): log likelihood = 201.5619797261 #R> Iteration 2 (hessian is provided, 2 in total): log likelihood = 218.5157385369 #R> Iteration 3 (hessian is provided, 3 in total): log likelihood = 219.8304053615 #R> Iteration 4 (hessian is provided, 4 in total): log likelihood = 221.1570395072 #R> Iteration 5 (hessian is provided, 5 in total): log likelihood = 221.4173977326 #R> Iteration 6 (hessian is provided, 6 in total): log likelihood = 221.45910305 #R> Iteration 7 (hessian is provided, 7 in total): log likelihood = 221.4742381631 #R> Iteration 8 (hessian is provided, 8 in total): log likelihood = 221.4742892007 #R> Iteration 9 (hessian is provided, 9 in total): log likelihood = 221.4772909678 #R> #R> Convergence given g inv(H) g' = 1.147956e-05 < lmtol #R> #R> Final log likelihood = 221.4772909678 #R> #R> Estimate Std.Err Z value Pr(>z) #R> (Intercept) -5.88439482 3.00884171 -1.9557 0.050500 . #R> log(Y1) 0.43986913 0.15724371 2.7974 0.005152 ** #R> log(Y2) 0.50497847 0.27574767 1.8313 0.067055 . #R> log(W1) 0.54680112 0.27757332 1.9699 0.048846 * #R> log(W2) 1.62221319 1.19336826 1.3594 0.174034 #R> I(0.5 * log(Y1)^2) 0.04152960 0.00579831 7.1624 7.929e-13 *** #R> I(0.5 * log(Y2)^2) 0.06775246 0.02251267 3.0095 0.002617 ** #R> I(0.5 * log(W1)^2) 0.03985513 0.02624261 1.5187 0.128833 #R> I(0.5 * log(W2)^2) -0.28421570 0.32037935 -0.8871 0.375013 #R> log(Y1):log(Y2) -0.05595449 0.01162258 -4.8143 1.477e-06 *** #R> log(Y1):log(W1) -0.02195339 0.01251279 -1.7545 0.079349 . #R> log(Y1):log(W2) 0.00032291 0.04106930 0.0079 0.993727 #R> log(Y2):log(W1) 0.01294569 0.01577160 0.8208 0.411747 #R> log(Y2):log(W2) 0.00975499 0.05798118 0.1682 0.866391 #R> log(W1):log(W2) -0.15803381 0.06620501 -2.3870 0.016985 * #R> lnVARv0i_(Intercept) -3.01076847 0.11366937 -26.4871 < 2.2e-16 *** #R> Skew_v0i_(Intercept) 1.86603587 0.32264117 5.7836 7.311e-09 *** #R> --- #R> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 #R> _____________________________________________________________ coef(m1) #R> (Intercept) log(Y1) log(Y2) #R> -5.884394817 0.439869125 0.504978469 #R> log(W1) log(W2) I(0.5 * log(Y1)^2) #R> 0.546801118 1.622213193 0.041529597 #R> I(0.5 * log(Y2)^2) I(0.5 * log(W1)^2) I(0.5 * log(W2)^2) #R> 0.067752458 0.039855132 -0.284215702 #R> log(Y1):log(Y2) log(Y1):log(W1) log(Y1):log(W2) #R> -0.055954492 -0.021953387 0.000322907 #R> log(Y2):log(W1) log(Y2):log(W2) log(W1):log(W2) #R> 0.012945689 0.009754988 -0.158033814 #R> lnVARv0i_(Intercept) Skew_v0i_(Intercept) #R> -3.010768472 1.866035868 # ------------------------------------------------------------- # Specification 2: heteroskedastic + skewed noise # ------------------------------------------------------------- formSV <- ~ log(TA) # heteroskedasticity in v formSK <- ~ ER # skewness driven by equity ratio m2 <- snreg( formula = spe.tl, data = banks07, ln.var.v = formSV, skew.v = formSK ) #R> theta0: #R> (Intercept) log(Y1) log(Y2) #R> -4.386991731 0.415874408 0.370050861 #R> log(W1) log(W2) I(0.5 * log(Y1)^2) #R> 0.524152369 1.342097991 0.047454756 #R> I(0.5 * log(Y2)^2) I(0.5 * log(W1)^2) I(0.5 * log(W2)^2) #R> 0.077484315 0.034996665 -0.226281101 #R> log(Y1):log(Y2) log(Y1):log(W1) log(Y1):log(W2) #R> -0.057609631 -0.020462972 -0.005676474 #R> log(Y2):log(W1) log(Y2):log(W2) log(W1):log(W2) #R> 0.013639077 0.018840745 -0.154925076 #R> lnVARv0i_(Intercept) lnVARv0i_lnVARv0i_log(TA) Skew_v0i_(Intercept) #R> -3.687587268 0.000000000 3.000000000 #R> Skew_v0i_Skew_v0i_ER #R> 0.000000000 #R> #R> ----------------- Regression with skewed errors: ----------------- #R> #R> Iteration 0 (at starting values): log likelihood = 5.146960520337 #R> #R> Iteration 1 (hessian is provided, 1 in total): log likelihood = 204.6988401214 #R> Iteration 2 (hessian is provided, 2 in total): log likelihood = 221.6390613234 #R> Iteration 3 (hessian is provided, 3 in total): log likelihood = 224.8771023643 #R> Iteration 4 (hessian is provided, 4 in total): log likelihood = 225.6867346145 #R> Iteration 5 (hessian is provided, 5 in total): log likelihood = 225.9380532273 #R> Iteration 6 (hessian is provided, 6 in total): log likelihood = 226.0484166119 #R> Iteration 7 (hessian is provided, 7 in total): log likelihood = 226.0721225638 #R> Iteration 8 (hessian is provided, 8 in total): log likelihood = 226.0820635056 #R> Iteration 9 (hessian is provided, 9 in total): log likelihood = 226.0835072457 #R> Iteration 10 (hessian is provided, 10 in total): log likelihood = 226.0909506421 #R> #R> Convergence given g inv(H) g' = 9.357213e-05 < lmtol #R> #R> Final log likelihood = 226.0909506421 #R> #R> Estimate Std.Err Z value Pr(>z) #R> (Intercept) -5.71501334 2.98991742 -1.9114 0.055950 . #R> log(Y1) 0.45216186 0.15708195 2.8785 0.003996 ** #R> log(Y2) 0.64671047 0.28153250 2.2971 0.021613 * #R> log(W1) 0.48917765 0.27664356 1.7683 0.077018 . #R> log(W2) 1.15530138 1.17400483 0.9841 0.325082 #R> I(0.5 * log(Y1)^2) 0.04070906 0.00590546 6.8935 5.445e-12 *** #R> I(0.5 * log(Y2)^2) 0.06170967 0.02265802 2.7235 0.006459 ** #R> I(0.5 * log(W1)^2) 0.04271441 0.02667730 1.6012 0.109343 #R> I(0.5 * log(W2)^2) -0.15999322 0.31994376 -0.5001 0.617028 #R> log(Y1):log(Y2) -0.05757860 0.01197968 -4.8064 1.537e-06 *** #R> log(Y1):log(W1) -0.01887894 0.01276125 -1.4794 0.139035 #R> log(Y1):log(W2) 0.00165155 0.04117676 0.0401 0.968006 #R> log(Y2):log(W1) 0.00516054 0.01583258 0.3259 0.744467 #R> log(Y2):log(W2) -0.00022261 0.05922730 -0.0038 0.997001 #R> log(W1):log(W2) -0.13107647 0.06626509 -1.9781 0.047922 * #R> lnVARv0i_(Intercept) -0.89452888 1.03829604 -0.8615 0.388943 #R> lnVARv0i_lnVARv0i_log(TA) -0.17730037 0.08736480 -2.0294 0.042415 * #R> Skew_v0i_(Intercept) 3.10567750 0.76059557 4.0832 4.442e-05 *** #R> Skew_v0i_Skew_v0i_ER -9.55208958 4.87607389 -1.9590 0.050116 . #R> --- #R> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 #R> __________________________________________________________________ coef(m2) #R> (Intercept) log(Y1) log(Y2) #R> -5.7150133374 0.4521618623 0.6467104718 #R> log(W1) log(W2) I(0.5 * log(Y1)^2) #R> 0.4891776489 1.1553013814 0.0407090640 #R> I(0.5 * log(Y2)^2) I(0.5 * log(W1)^2) I(0.5 * log(W2)^2) #R> 0.0617096653 0.0427144069 -0.1599932243 #R> log(Y1):log(Y2) log(Y1):log(W1) log(Y1):log(W2) #R> -0.0575785975 -0.0188789363 0.0016515541 #R> log(Y2):log(W1) log(Y2):log(W2) log(W1):log(W2) #R> 0.0051605363 -0.0002226092 -0.1310764688 #R> lnVARv0i_(Intercept) lnVARv0i_lnVARv0i_log(TA) Skew_v0i_(Intercept) #R> -0.8945288842 -0.1773003673 3.1056774984 #R> Skew_v0i_Skew_v0i_ER #R> -9.5520895773
Stochastic Frontier Model with a Skew-Normally Distributed Error Term
snsfperforms maximum likelihood estimation of the parameters and technical or cost efficiencies in a Stochastic Frontier Model with a skew-normally distributed error term.
myprod <- FALSE # ------------------------------------------------------------- # Specification 1: homoskedastic & symmetric # ------------------------------------------------------------- formSV <- NULL # variance equation formSK <- NULL # skewness equation formSU <- NULL # inefficiency equation (unused here) m1 <- snsf( formula = spe.tl, data = banks07, prod = myprod, ln.var.v = formSV, skew.v = formSK ) #R> #R> -------------- Regression with skewed errors: --------------- #R> #R> initial value -105.392823 #R> iter 2 value -105.561746 #R> iter 3 value -105.568000 #R> iter 4 value -105.836407 #R> iter 5 value -106.255928 #R> iter 6 value -107.989401 #R> iter 7 value -129.390401 #R> iter 8 value -148.614164 #R> iter 8 value -148.614164 #R> iter 9 value -148.627674 #R> iter 10 value -148.628304 #R> iter 10 value -148.628304 #R> iter 10 value -148.628304 #R> final value -148.628304 #R> converged #R> Estimate Std.Err Z value Pr(>z) #R> X(Intercept) -4.40500089 3.80254827 -1.1584 0.246687 #R> Xlog(Y1) 0.45706967 0.18307828 2.4966 0.012540 * #R> Xlog(Y2) 0.31560584 0.33532057 0.9412 0.346599 #R> Xlog(W1) 0.48827077 0.38637994 1.2637 0.206335 #R> Xlog(W2) 1.29215252 1.70829901 0.7564 0.449411 #R> XI(0.5 * log(Y1)^2) 0.06252731 0.00718640 8.7008 < 2.2e-16 *** #R> XI(0.5 * log(Y2)^2) 0.19680294 0.03130033 6.2876 3.225e-10 *** #R> XI(0.5 * log(W1)^2) 0.05041646 0.03723206 1.3541 0.175700 #R> XI(0.5 * log(W2)^2) -0.28522159 0.46697561 -0.6108 0.541342 #R> Xlog(Y1):log(Y2) -0.14258698 0.00610987 -23.3372 < 2.2e-16 *** #R> Xlog(Y1):log(W1) 0.00029592 0.01705705 0.0173 0.986158 #R> Xlog(Y1):log(W2) 0.17610934 0.05558196 3.1685 0.001532 ** #R> Xlog(Y2):log(W1) 0.01454480 0.02162340 0.6726 0.501175 #R> Xlog(Y2):log(W2) -0.09354141 0.07710129 -1.2132 0.225043 #R> Xlog(W1):log(W2) -0.21197450 0.09475124 -2.2372 0.025275 * #R> lnVARv0i_(Intercept) -2.66540968 0.11013951 -24.2003 < 2.2e-16 *** #R> Skew_v0i_(Intercept) 0.98854923 0.03434255 28.7850 < 2.2e-16 *** #R> sv 0.26376286 0.00766250 34.4226 < 2.2e-16 *** #R> --- #R> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 #R> _____________________________________________________________ #R> theta0: #R> (Intercept) log(Y1) log(Y2) #R> -4.4050008900 0.4570696682 0.3156058426 #R> log(W1) log(W2) I(0.5 * log(Y1)^2) #R> 0.4882707672 1.2921525200 0.0625273146 #R> I(0.5 * log(Y2)^2) I(0.5 * log(W1)^2) I(0.5 * log(W2)^2) #R> 0.1968029394 0.0504164557 -0.2852215911 #R> log(Y1):log(Y2) log(Y1):log(W1) log(Y1):log(W2) #R> -0.1425869800 0.0002959179 0.1761093362 #R> log(Y2):log(W1) log(Y2):log(W2) log(W1):log(W2) #R> 0.0145448021 -0.0935414140 -0.2119744969 #R> lnVARv0i_(Intercept) Skew_v0i_(Intercept) lnVARu0i_(Intercept) #R> -4.2020745955 0.3982617494 -4.5984104618 #R> #R> ---------------------- The main model: ---------------------- #R> #R> Iteration 0 (at starting values): log likelihood = 21.64346032248 #R> #R> Iteration 1 (hessian is provided, 1 in total): log likelihood = 186.4880063145 #R> Iteration 2 (hessian is provided, 2 in total): log likelihood = 214.3430643968 #R> Iteration 3 (hessian is provided, 3 in total): log likelihood = 220.957921695 #R> Iteration 4 (hessian is provided, 4 in total): log likelihood = 223.1040419712 #R> Iteration 5 (hessian is provided, 5 in total): log likelihood = 224.3725358921 #R> Iteration 6 (hessian is provided, 6 in total): log likelihood = 224.6133195507 #R> Iteration 7 (hessian is provided, 7 in total): log likelihood = 224.7096271178 #R> Iteration 8 (hessian is provided, 8 in total): log likelihood = 224.7298349674 #R> Iteration 9 (hessian is provided, 9 in total): log likelihood = 224.8005650898 #R> Iteration 10 (hessian is provided, 10 in total): log likelihood = 224.8011109934 #R> Iteration 11 (hessian is provided, 11 in total): log likelihood = 224.8019248405 #R> Iteration 12 (hessian is provided, 12 in total): log likelihood = 224.8021079368 #R> Iteration 13 (hessian is provided, 13 in total): log likelihood = 224.8054321542 #R> Iteration 14 (hessian is provided, 14 in total): log likelihood = 224.8420044777 #R> Iteration 15 (hessian is provided, 15 in total): log likelihood = 225.4135588864 #R> Iteration 16 (hessian is provided, 16 in total): log likelihood = 225.5282232035 #R> Iteration 17 (hessian is provided, 17 in total): log likelihood = 225.533351752 #R> Iteration 18 (hessian is provided, 18 in total): log likelihood = 225.5368333384 #R> Iteration 19 (hessian is provided, 19 in total): log likelihood = 225.5371105355 #R> Iteration 20 (hessian is provided, 20 in total): log likelihood = 225.539300346 #R> Iteration 21 (hessian is provided, 21 in total): log likelihood = 225.539317247 #R> Iteration 22 (hessian is provided, 22 in total): log likelihood = 225.539324279 #R> #R> Convergence given g inv(H) g' = 1.027857e-06 < lmtol #R> #R> Final log likelihood = 225.539324279 #R> #R> #R> Log likelihood maximization completed in #R> 0.18 seconds #R> _____________________________________________________________ #R> #R> Cross-sectional stochastic (cost) frontier model #R> #R> Distributional assumptions #R> #R> Component Distribution Assumption #R> 1 Random noise: skew normal homoskedastic #R> 2 Inefficiency: exponential homoskedastic #R> #R> Number of observations = 500 #R> #R> -------------------- Estimation results: -------------------- #R> #R> Coef. SE z P>|z| #R> _____________________________________________________________ #R> Frontier #R> #R> (Intercept) -6.3606 3.0602 -2.08 0.0377 * #R> log(Y1) 0.4425 0.1767 2.50 0.0123 * #R> log(Y2) 0.5660 0.2753 2.06 0.0398 * #R> log(W1) 0.4983 0.2835 1.76 0.0788 . #R> log(W2) 1.6557 1.1889 1.39 0.1637 #R> I(0.5 * log(Y1)^2) 0.0449 0.0066 6.76 0.0000 *** #R> I(0.5 * log(Y2)^2) 0.0650 0.0222 2.93 0.0034 ** #R> I(0.5 * log(W1)^2) 0.0363 0.0258 1.41 0.1597 #R> I(0.5 * log(W2)^2) -0.3099 0.3170 -0.98 0.3283 #R> log(Y1):log(Y2) -0.0586 0.0117 -4.99 0.0000 *** #R> log(Y1):log(W1) -0.0227 0.0127 -1.78 0.0750 . #R> log(Y1):log(W2) -0.0006 0.0420 -0.01 0.9895 #R> log(Y2):log(W1) 0.0117 0.0152 0.77 0.4393 #R> log(Y2):log(W2) 0.0106 0.0563 0.19 0.8505 #R> log(W1):log(W2) -0.1375 0.0682 -2.02 0.0438 * #R> ------------------------------------------------------------- #R> Variance of the random noise component: log(sigma_v^2) #R> #R> lnVARv0i_(Intercept) -3.8207 0.2255 -16.94 0.0000 *** #R> ------------------------------------------------------------- #R> Skewness of the random noise component: `alpha` #R> #R> Skew_v0i_(Intercept) -1.5007 0.8755 -1.71 0.0865 . #R> ------------------------------------------------------------- #R> Inefficiency component: log(sigma_u^2) #R> #R> lnVARu0i_(Intercept) -4.3422 0.2398 -18.11 0.0000 *** #R> _____________________________________________________________ #R> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 #R> #R> --------------- Summary of cost efficiencies: --------------- #R> #R> JLMS:= exp(-E[ui|ei]) #R> #R> TE_JLMS #R> Obs 500 #R> NAs 0 #R> Mean 0.8955 #R> StDev 0.0715 #R> IQR 0.0645 #R> Min 0.4559 #R> 5% 0.7420 #R> 10% 0.7959 #R> 25% 0.8780 #R> 50% 0.9208 #R> 75% 0.9425 #R> 90% 0.9520 #R> Max 0.9691 coef(m1) #R> (Intercept) log(Y1) log(Y2) #R> -6.3606134654 0.4425312174 0.5660453675 #R> log(W1) log(W2) I(0.5 * log(Y1)^2) #R> 0.4983319531 1.6557355502 0.0448519992 #R> I(0.5 * log(Y2)^2) I(0.5 * log(W1)^2) I(0.5 * log(W2)^2) #R> 0.0649995029 0.0362747554 -0.3099280819 #R> log(Y1):log(Y2) log(Y1):log(W1) log(Y1):log(W2) #R> -0.0586056701 -0.0226917826 -0.0005509121 #R> log(Y2):log(W1) log(Y2):log(W2) log(W1):log(W2) #R> 0.0117430476 0.0106095399 -0.1375404433 #R> lnVARv0i_(Intercept) Skew_v0i_(Intercept) lnVARu0i_(Intercept) #R> -3.8206601915 -1.5006628040 -4.3422317628 # ------------------------------------------------------------- # Specification 2: heteroskedastic + skewed noise # ------------------------------------------------------------- formSV <- ~ log(TA) # heteroskedastic variance formSK <- ~ ER # skewness driver formSU <- ~ LA + ER # inefficiency m2 <- snsf( formula = spe.tl, data = banks07, prod = myprod, ln.var.v = formSV, skew.v = formSK ) #R> #R> -------------- Regression with skewed errors: --------------- #R> #R> initial value -105.392823 #R> iter 2 value -105.561746 #R> iter 3 value -105.568000 #R> iter 4 value -105.836407 #R> iter 5 value -106.255928 #R> iter 6 value -107.989401 #R> iter 7 value -129.390401 #R> iter 8 value -148.614164 #R> iter 8 value -148.614164 #R> iter 9 value -148.627674 #R> iter 10 value -148.628304 #R> iter 10 value -148.628304 #R> iter 10 value -148.628304 #R> final value -148.628304 #R> converged #R> Estimate Std.Err Z value Pr(>z) #R> X(Intercept) -4.40500089 3.80254827 -1.1584 0.246687 #R> Xlog(Y1) 0.45706967 0.18307828 2.4966 0.012540 * #R> Xlog(Y2) 0.31560584 0.33532057 0.9412 0.346599 #R> Xlog(W1) 0.48827077 0.38637994 1.2637 0.206335 #R> Xlog(W2) 1.29215252 1.70829901 0.7564 0.449411 #R> XI(0.5 * log(Y1)^2) 0.06252731 0.00718640 8.7008 < 2.2e-16 *** #R> XI(0.5 * log(Y2)^2) 0.19680294 0.03130033 6.2876 3.225e-10 *** #R> XI(0.5 * log(W1)^2) 0.05041646 0.03723206 1.3541 0.175700 #R> XI(0.5 * log(W2)^2) -0.28522159 0.46697561 -0.6108 0.541342 #R> Xlog(Y1):log(Y2) -0.14258698 0.00610987 -23.3372 < 2.2e-16 *** #R> Xlog(Y1):log(W1) 0.00029592 0.01705705 0.0173 0.986158 #R> Xlog(Y1):log(W2) 0.17610934 0.05558196 3.1685 0.001532 ** #R> Xlog(Y2):log(W1) 0.01454480 0.02162340 0.6726 0.501175 #R> Xlog(Y2):log(W2) -0.09354141 0.07710129 -1.2132 0.225043 #R> Xlog(W1):log(W2) -0.21197450 0.09475124 -2.2372 0.025275 * #R> lnVARv0i_(Intercept) -2.66540968 0.11013951 -24.2003 < 2.2e-16 *** #R> Skew_v0i_(Intercept) 0.98854923 0.03434255 28.7850 < 2.2e-16 *** #R> sv 0.26376286 0.00766250 34.4226 < 2.2e-16 *** #R> --- #R> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 #R> _____________________________________________________________ #R> theta0: #R> (Intercept) log(Y1) log(Y2) #R> -4.405001e+00 4.570697e-01 3.156058e-01 #R> log(W1) log(W2) I(0.5 * log(Y1)^2) #R> 4.882708e-01 1.292153e+00 6.252731e-02 #R> I(0.5 * log(Y2)^2) I(0.5 * log(W1)^2) I(0.5 * log(W2)^2) #R> 1.968029e-01 5.041646e-02 -2.852216e-01 #R> log(Y1):log(Y2) log(Y1):log(W1) log(Y1):log(W2) #R> -1.425870e-01 2.959179e-04 1.761093e-01 #R> log(Y2):log(W1) log(Y2):log(W2) log(W1):log(W2) #R> 1.454480e-02 -9.354141e-02 -2.119745e-01 #R> lnVARv0i_(Intercept) lnVARv0i_log(TA) Skew_v0i_(Intercept) #R> -4.202075e+00 2.212626e-16 3.982617e-01 #R> Skew_v0i_ER lnVARu0i_(Intercept) #R> 0.000000e+00 -4.598410e+00 #R> #R> ---------------------- The main model: ---------------------- #R> #R> Iteration 0 (at starting values): log likelihood = 21.64346032248 #R> #R> Iteration 1 (hessian is provided, 1 in total): log likelihood = 133.2018982456 #R> Iteration 2 (hessian is provided, 2 in total): log likelihood = 183.2824024606 #R> Iteration 3 (hessian is provided, 3 in total): log likelihood = 214.7412845581 #R> Iteration 4 (hessian is provided, 4 in total): log likelihood = 223.1763227676 #R> Iteration 5 (hessian is provided, 5 in total): log likelihood = 226.655477596 #R> Iteration 6 (hessian is provided, 6 in total): log likelihood = 226.7105002202 #R> Iteration 7 (hessian is provided, 7 in total): log likelihood = 226.8074129256 #R> Iteration 8 (hessian is provided, 8 in total): log likelihood = 226.8082101988 #R> Iteration 9 (hessian is provided, 9 in total): log likelihood = 226.8082151929 #R> Iteration 10 (hessian is provided, 10 in total): log likelihood = 226.808224564 #R> #R> Convergence given g inv(H) g' = 2.529443e-06 < lmtol #R> #R> Final log likelihood = 226.808224564 #R> #R> #R> Log likelihood maximization completed in #R> 0.103 seconds #R> _____________________________________________________________ #R> #R> Cross-sectional stochastic (cost) frontier model #R> #R> Distributional assumptions #R> #R> Component Distribution Assumption #R> 1 Random noise: skew normal heteroskedastic #R> 2 Inefficiency: exponential homoskedastic #R> #R> Number of observations = 500 #R> #R> -------------------- Estimation results: -------------------- #R> #R> Coef. SE z P>|z| #R> _____________________________________________________________ #R> Frontier #R> #R> (Intercept) -5.9441 3.0205 -1.97 0.0491 * #R> log(Y1) 0.4589 0.1647 2.79 0.0053 ** #R> log(Y2) 0.6135 0.2820 2.18 0.0296 * #R> log(W1) 0.5168 0.2747 1.88 0.0599 . #R> log(W2) 1.2707 1.1934 1.06 0.2870 #R> I(0.5 * log(Y1)^2) 0.0414 0.0061 6.81 0.0000 *** #R> I(0.5 * log(Y2)^2) 0.0614 0.0226 2.72 0.0065 ** #R> I(0.5 * log(W1)^2) 0.0439 0.0262 1.68 0.0938 . #R> I(0.5 * log(W2)^2) -0.2078 0.3247 -0.64 0.5223 #R> log(Y1):log(Y2) -0.0575 0.0120 -4.79 0.0000 *** #R> log(Y1):log(W1) -0.0215 0.0128 -1.68 0.0927 . #R> log(Y1):log(W2) 0.0000 0.0417 0.00 0.9997 #R> log(Y2):log(W1) 0.0068 0.0158 0.43 0.6682 #R> log(Y2):log(W2) 0.0086 0.0589 0.15 0.8836 #R> log(W1):log(W2) -0.1368 0.0661 -2.07 0.0384 * #R> ------------------------------------------------------------- #R> Variance of the random noise component: log(sigma_v^2) #R> #R> lnVARv0i_(Intercept) -1.8839 1.6595 -1.14 0.2563 #R> lnVARv0i_log(TA) -0.1558 0.1328 -1.17 0.2407 #R> ------------------------------------------------------------- #R> Skewness of the random noise component: `alpha` #R> #R> Skew_v0i_(Intercept) 2.0043 1.0468 1.91 0.0555 . #R> Skew_v0i_ER -7.2632 5.0581 -1.44 0.1510 #R> ------------------------------------------------------------- #R> Inefficiency component: log(sigma_u^2) #R> #R> lnVARu0i_(Intercept) -4.6407 0.3416 -13.58 0.0000 *** #R> _____________________________________________________________ #R> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 #R> #R> --------------- Summary of cost efficiencies: --------------- #R> #R> JLMS:= exp(-E[ui|ei]) #R> #R> TE_JLMS #R> Obs 500 #R> NAs 0 #R> Mean 0.9084 #R> StDev 0.0564 #R> IQR 0.0547 #R> Min 0.5206 #R> 5% 0.8096 #R> 10% 0.8379 #R> 25% 0.8903 #R> 50% 0.9241 #R> 75% 0.9450 #R> 90% 0.9568 #R> Max 0.9758 coef(m2) #R> (Intercept) log(Y1) log(Y2) #R> -5.944090e+00 4.589482e-01 6.134700e-01 #R> log(W1) log(W2) I(0.5 * log(Y1)^2) #R> 5.167646e-01 1.270694e+00 4.138294e-02 #R> I(0.5 * log(Y2)^2) I(0.5 * log(W1)^2) I(0.5 * log(W2)^2) #R> 6.141001e-02 4.385504e-02 -2.077538e-01 #R> log(Y1):log(Y2) log(Y1):log(W1) log(Y1):log(W2) #R> -5.745504e-02 -2.151307e-02 1.351571e-05 #R> log(Y2):log(W1) log(Y2):log(W2) log(W1):log(W2) #R> 6.752251e-03 8.627646e-03 -1.368348e-01 #R> lnVARv0i_(Intercept) lnVARv0i_log(TA) Skew_v0i_(Intercept) #R> -1.883867e+00 -1.558322e-01 2.004266e+00 #R> Skew_v0i_ER lnVARu0i_(Intercept) #R> -7.263169e+00 -4.640750e+00
Additional Resources
See
Oleg Badunenko and Daniel J. Henderson (2023). "Production analysis with asymmetric noise". Journal of Productivity Analysis, 61(1), 1–18. DOI
for more details
Try the snreg package in your browser
Any scripts or data that you put into this service are public.
R Package Documentation
Browse R Packages
We want your feedback!
Note that we can't provide technical support on individual packages. You should contact the package authors for that.
Embedding an R snippet on your website
Add the following code to your website.
For more information on customizing the embed code, read Embedding Snippets.