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Assignment 2 -- Solution
ECON6002 Panel Data and Spatial Econometrics, Term II 2020-21
Due date: Friday Week 6, February 19, 2021, 5:00pm. Submission: Email your completed assignments in one pdf file to: [email protected], with file name: YourName_A2.pdf; and email Subject: ECON6002 Assignment 2 1. Download the “Spanish Dairy Farm Production” data from the website: http://people.stern.nyu.edu/wgreene/Econometrics/PanelDataEconometrics.htm
(a) Write a paragraph to introduce the data, including the variables names, number of cross-sectional units and time periods, purpose of study, etc.
(b) Using the data, demonstrate the applications of the Stata command xtreg with options (be, fe, re, mle), in fitting the one-way effects panel model. Explain your results briefly.
(c) Extend your analysis by including the time-specific effects to fit (i) two-way fixed effects model and (ii) a mixed model with individual random effects and time-fixed effects. Explain your results briefly.
(d) Using the Stata command xtmixed, fit a two-way random effects model to the data. Explain your results.
Solution: (a) Spanish Dairy Farm Production, N = 247, T = 6. Variables in the file are:
FARM: Farm ID YEAR: year, 93, 94, ..., 98
Input variables: COWS: number of cows LAND: land size in hectares LABOR: number of works FEED: amount of food fed X1, X2, X3, X4: log of input variables, deviations from means (in logs) X11, X22, X33, X44: squares of X1, X2, X3, X4 X12, X13, X14, X23, X24, X34: cross product of X1, X2, X3, X4 YEAR93, . . . , YEAR98 = year dummy variables Output MILK = milk production each farm in each year YIT = log of MILK production Purpose of Study Identify factors determining the Spanish dairy farm production YIT, and specify a ‘good’ panel data model for predicting the milk production.
(b) The between estimator (be) of the one-way effects model, the Model (2.1) in Chapter 2
of lecture notes, is obtained by first averaging Model (2.1) over t to give 𝑦𝑦�𝑖𝑖∙ = 𝛼𝛼 + 𝑋𝑋�𝑖𝑖∙′ 𝛽𝛽 + (𝜇𝜇𝑖𝑖 + �̅�𝑣𝑖𝑖∙),
and the performing an OLS regression of the averaged model. Consistency of the be estimator requires that the error term (𝜇𝜇𝑖𝑖 + �̅�𝑣𝑖𝑖∙) to be uncorrelated with 𝑋𝑋�𝑖𝑖∙, which is the case when 𝜇𝜇𝑖𝑖 is a random effect but not when 𝜇𝜇𝑖𝑖 is a fixed effect. The other three estimators are well described in Chapter 2 of lecture notes. The outputs for xtreg with (be, fe, re, mle): by regressing YIT on X1, X2, X3, X4: Table 1.1a. The between estimation . xtreg yit x1 x2 x3 x4, be
Between regression (regression on group means) Number of obs = 1,482
Group variable: farm Number of groups = 247
R-sq: Obs per group:
within = 0.8309 min = 6
between = 0.9634 avg = 6.0
overall = 0.9524 max = 6
F(4,242) = 1593.64
sd(u_i + avg(e_i.))= .1191294 Prob > F = 0.0000
------------------------------------------------------------------------------
yit | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
x1 | .5625965 .0475769 11.82 0.000 .4688788 .6563142
x2 | .0254032 .0260339 0.98 0.330 -.0258787 .0766851
x3 | .0154496 .0292668 0.53 0.598 -.0422006 .0730998
x4 | .4779786 .0265409 18.01 0.000 .425698 .5302591
_cons | 11.57749 .00758 1527.37 0.000 11.56256 11.59242
------------------------------------------------------------------------------
Table 1.1b. The fixed effects estimation . xtreg yit x1 x2 x3 x4, fe
Fixed-effects (within) regression Number of obs = 1,482
Group variable: farm Number of groups = 247
R-sq: Obs per group:
within = 0.8359 min = 6
between = 0.9615 avg = 6.0
overall = 0.9513 max = 6
F(4,1231) = 1568.11
corr(u_i, Xb) = 0.1089 Prob > F = 0.0000
------------------------------------------------------------------------------
yit | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
x1 | .6620012 .0246784 26.83 0.000 .6135847 .7104177
x2 | .0373524 .0161331 2.32 0.021 .005701 .0690038
x3 | .0303996 .0232078 1.31 0.190 -.0151316 .0759307
x4 | .3825104 .0120169 31.83 0.000 .3589345 .4060862
_cons | 11.57749 .0021151 5473.85 0.000 11.57334 11.58164
-------------+----------------------------------------------------------------
sigma_u | .12198441
sigma_e | .08142265
rho | .69178541 (fraction of variance due to u_i)
------------------------------------------------------------------------------
F test that all u_i=0: F(246, 1231) = 12.84 Prob > F = 0.0000
Table 1.1c. The random effects estimation . xtreg yit x1 x2 x3 x4, re
Random-effects GLS regression Number of obs = 1,482
Group variable: farm Number of groups = 247
R-sq: Obs per group:
within = 0.8358 min = 6
between = 0.9621 avg = 6.0
overall = 0.9518 max = 6
Wald chi2(4) = 12563.20
corr(u_i, X) = 0 (assumed) Prob > chi2 = 0.0000
------------------------------------------------------------------------------
yit | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
x1 | .6502721 .0208835 31.14 0.000 .6093412 .691203
x2 | .0300488 .0133827 2.25 0.025 .0038193 .0562784
x3 | .03507 .0173829 2.02 0.044 .0010002 .0691398
x4 | .3995279 .0108786 36.73 0.000 .3782062 .4208497
_cons | 11.57749 .0076015 1523.04 0.000 11.56259 11.59239
-------------+----------------------------------------------------------------
sigma_u | .11439792
sigma_e | .08142265
rho | .66375185 (fraction of variance due to u_i)
------------------------------------------------------------------------------
Table 1.1d. The maximum likelihood estimation . xtreg yit x1 x2 x3 x4, mle
Fitting constant-only model:
Iteration 0: log likelihood = -221.37283
Iteration 1: log likelihood = -221.35168
Fitting full model:
Iteration 0: log likelihood = 1284.8672
Iteration 1: log likelihood = 1297.033
Iteration 2: log likelihood = 1297.1861
Iteration 3: log likelihood = 1297.1861
Random-effects ML regression Number of obs = 1,482
Group variable: farm Number of groups = 247
Random effects u_i ~ Gaussian Obs per group:
min = 6
avg = 6.0
max = 6
LR chi2(4) = 3037.08
Log likelihood = 1297.1861 Prob > chi2 = 0.0000
------------------------------------------------------------------------------
yit | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
x1 | .6505191 .0208955 31.13 0.000 .6095647 .6914734
x2 | .0301504 .0133864 2.25 0.024 .0039136 .0563873
x3 | .0350755 .0173955 2.02 0.044 .0009809 .06917
x4 | .3992413 .0109443 36.48 0.000 .3777909 .4206918
_cons | 11.57749 .0076555 1512.32 0.000 11.56248 11.59249
-------------+----------------------------------------------------------------
/sigma_u | .115638 .0056798 .1050248 .1273236
/sigma_e | .0813718 .0016398 .0782205 .08465
rho | .6688242 .0238522 .6208686 .7141585
------------------------------------------------------------------------------
LR test of sigma_u=0: chibar2(01) = 975.02 Prob >= chibar2 = 0.000
i) All four estimation methods show that X1 (COWS) and X4 (FEED) are highly
significant to the MILK production; ii) The X2 (LAND) and X3 (LABOR) are insignificant in be estimation, X2 (LAND) is
significant at 5% level in fe, re and mle estimation; and X3 (LABOR) is also significant in re and mle estimation but not in fe estimation at 5% level.
iii) The highly significance of X1 and X4 suggest that their squared terms and cross-product may be included in the model. Indeed, the re or fe estimation of such a model show that X11 and/or X44 should be added into the model.
iv) The re and mle estimation methods produce similar results.
(c) The outputs for xtreg (fe and re) on X1, X2, X3, X4, and time dummies: Table 1.2a. FE estimation with time dummies . xtreg yit x1 x2 x3 x4 i.year, fe
Fixed-effects (within) regression Number of obs = 1,482
Group variable: farm Number of groups = 247
R-sq: Obs per group:
within = 0.8517 min = 6
between = 0.9593 avg = 6.0
overall = 0.9493 max = 6
F(9,1226) = 782.05
corr(u_i, Xb) = 0.4929 Prob > F = 0.0000
------------------------------------------------------------------------------
yit | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
x1 | .6379655 .0237985 26.81 0.000 .5912751 .6846559
x2 | .0412755 .0154446 2.67 0.008 .0109747 .0715763
x3 | .0281924 .0221732 1.27 0.204 -.0153093 .071694
x4 | .3081603 .0132257 23.30 0.000 .2822127 .3341078
|
year |
94 | .0329188 .0071309 4.62 0.000 .0189286 .046909
95 | .0613667 .0074861 8.20 0.000 .0466797 .0760537
96 | .0719498 .0080094 8.98 0.000 .0562361 .0876635
97 | .0753031 .0084325 8.93 0.000 .0587594 .0918468
98 | .0940052 .0089244 10.53 0.000 .0764965 .111514
|
_cons | 11.52156 .0057982 1987.08 0.000 11.51019 11.53294
-------------+----------------------------------------------------------------
sigma_u | .14561471
sigma_e | .07758351
rho | .77889157 (fraction of variance due to u_i)
------------------------------------------------------------------------------
F test that all u_i=0: F(246, 1226) = 14.54 Prob > F = 0.0000
Adding the time dummies to the fe estimation seems improve the overall model fitting. The X1 and X4 remain highly significant, X2 becomes more significant with p-value 0.008, and the time dummies are all highly significant. The X3 remains insignificant. Adding the time dummies to the re estimation also improves the overall model fitting. The X1 and X4 remain highly significant, X2 and X3 become more significant with p-values 0.004 and 0.001, respectively, and the time dummies are all highly significant.
Table 1.2a. RE estimation with time dummies . xtreg yit x1 x2 x3 x4 i.year, re
Random-effects GLS regression Number of obs = 1,482
Group variable: farm Number of groups = 247
R-sq: Obs per group:
within = 0.8498 min = 6
between = 0.9605 avg = 6.0
overall = 0.9510 max = 6
Wald chi2(9) = 12872.02
corr(u_i, X) = 0 (assumed) Prob > chi2 = 0.0000
------------------------------------------------------------------------------
yit | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
x1 | .6622073 .0205154 32.28 0.000 .6219979 .7024166
x2 | .0376141 .0131975 2.85 0.004 .0117475 .0634808
x3 | .0551804 .0173129 3.19 0.001 .0212478 .089113
x4 | .353735 .0117757 30.04 0.000 .330655 .3768149
|
year |
94 | .0263511 .007207 3.66 0.000 .0122256 .0404765
95 | .0489399 .0074386 6.58 0.000 .0343606 .0635193
96 | .0528781 .0077166 6.85 0.000 .0377538 .0680024
97 | .0522242 .0079423 6.58 0.000 .0366575 .0677909
98 | .0664853 .0081929 8.11 0.000 .0504275 .0825432
|
_cons | 11.53634 .0092748 1243.83 0.000 11.51816 11.55452
-------------+----------------------------------------------------------------
sigma_u | .11484174
sigma_e | .07758351
rho | .68662771 (fraction of variance due to u_i)
------------------------------------------------------------------------------
(d) The two-way RE model is fitted using the general Stata command xtmixed with
options: || _all: R.year || farm:, mle. It produces results very similar to those by xtreg yit x1 x2 x3 x4 i.year, re
The difference between two-way FE and two-way RE estimation suggest more need to be done in choosing a panel model with FE or RE. Table 1.3. Two-way RE estimation . xtmixed yit x1 x2 x3 x4 || _all: R.year || farm:, mle
Performing EM optimization:
Performing gradient-based optimization:
Iteration 0: log likelihood = 1325.9818
Iteration 1: log likelihood = 1325.9818
Computing standard errors:
Mixed-effects ML regression Number of obs = 1,482
-------------------------------------------------------------
| No. of Observations per Group
Group Variable | Groups Minimum Average Maximum
----------------+--------------------------------------------
_all | 1 1,482 1,482.0 1,482
farm | 247 6 6.0 6
-------------------------------------------------------------
Wald chi2(4) = 8784.31
Log likelihood = 1325.9818 Prob > chi2 = 0.0000
------------------------------------------------------------------------------
yit | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
x1 | .6618469 .0205223 32.25 0.000 .621624 .7020698
x2 | .0376961 .0132336 2.85 0.004 .0117588 .0636334
x3 | .0537612 .0174256 3.09 0.002 .0196076 .0879148
x4 | .3543165 .0116974 30.29 0.000 .33139 .3772429
_cons | 11.57749 .0120504 960.76 0.000 11.55387 11.60111
------------------------------------------------------------------------------
------------------------------------------------------------------------------
Random-effects Parameters | Estimate Std. Err. [95% Conf. Interval]
-----------------------------+------------------------------------------------
_all: Identity |
sd(R.year) | .0220552 .0069488 .011894 .0408974
-----------------------------+------------------------------------------------
farm: Identity |
sd(_cons) | .1217421 .0061302 .110301 .1343699
-----------------------------+------------------------------------------------
sd(Residual) | .0782755 .0015957 .0752097 .0814664
------------------------------------------------------------------------------
LR test vs. linear model: chi2(2) = 1032.61 Prob > chi2 = 0.0000
Note: LR test is conservative and provided only for reference.
2. Consider the Cigarette Demand Data available from the course website as Cigar.txt. Label the nine columns as State, Year, Price, Pop, Pop16, CPI, NDI, C, and PIMIN, and Define
LnC = Ln(C), LnP = Ln(Price), LnNDI = Ln(NDI), and LnPmin = Ln(PIMIN). Consider a two-way fixed effects (FE) model for LnC, based on LnP, LnNDI, LnPmin, and Year. (a) Compute the within estimators of the coefficients of LnP, LnNDI, LnPmin, and Year,
and the corresponding estimated standard errors of the within estimators (you may need to write your own STATA code). Comment on the results.
(b) Assuming the idiosyncratic errors are iid normal, test the significance of individual FE given the non-existence of time FE, and the significance of time FE given the non-existence of individual FE.
(c) Assuming the idiosyncratic errors are iid normal, test the significance of individual FE allowing the existence of time FE, and the significance of time FE allowing the existence of individual FE.
(d) Consider fitting a one-way FE model for LnC, based on all possible regressors. Report a model you think the best in describing the relationship and discuss the results.
Solution: (a) We suppose to apply the Q-transformation to remove the two-way fixed effects, and then
run an OLE regression, compute the standard errors manually. A directly use of Stata fe option with i.Year dummies (as in Table 2.1 below) gives the same estimates, but slightly different standard errors. From Table 2.1, we see that all of the coefficients are statistically significant. LnP has a negative coefficient as expected, i.e., people buy less when the price is higher. LnNDI has a positive coefficient, with cigarette consumption showing some significant income effect. Surprisingly, LnPmin has a negative coefficient, showing that when the minimum price of cigarettes in neighboring states increase, cigarette consumption in the current state decreases. Year variable shows a positive price trend, but Year-dummies show negative effects, showing that with Year-dummies, there is probably no need for a Year-trend variable or with Year-trend variable there is no need for Year-dummies.
Table 2.1. Two-way FE estimation . xtreg LnC LnP LnNDI LnPmin Year i.Year, fe
note: 92.Year omitted because of collinearity
Fixed-effects (within) regression Number of obs = 1,380
Group variable: State Number of groups = 46
R-sq: Obs per group:
within = 0.6779 min = 30
between = 0.3178 avg = 30.0
overall = 0.4328 max = 30
F(32,1302) = 85.65
corr(u_i, Xb) = 0.0497 Prob > F = 0.0000
------------------------------------------------------------------------------
LnC | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
LnP | -1.023062 .0418181 -24.46 0.000 -1.1051 -.9410237
LnNDI | .5200042 .0466836 11.14 0.000 .4284208 .6115876
LnPmin | -.1172486 .0541294 -2.17 0.030 -.223439 -.0110582
Year | .0271425 .0056747 4.78 0.000 .01601 .038275
|
Year |
64 | -.0552901 .0154889 -3.57 0.000 -.0856761 -.0249042
65 | -.0963049 .0162297 -5.93 0.000 -.1281441 -.0644658
66 | -.1038394 .0165225 -6.28 0.000 -.1362531 -.0714257
67 | -.1250987 .0176757 -7.08 0.000 -.1597748 -.0904227
68 | -.1228057 .0172665 -7.11 0.000 -.1566789 -.0889324
69 | -.1584459 .018693 -8.48 0.000 -.1951175 -.1217742
70 | -.1497712 .0178417 -8.39 0.000 -.1847728 -.1147696
71 | -.1283003 .0183839 -6.98 0.000 -.1643656 -.092235
72 | -.1238581 .0196831 -6.29 0.000 -.1624723 -.085244
73 | -.1890391 .0217798 -8.68 0.000 -.2317664 -.1463117
74 | -.1943228 .0233824 -8.31 0.000 -.2401941 -.1484516
75 | -.1738359 .0230157 -7.55 0.000 -.2189878 -.128684
76 | -.1339885 .0221764 -6.04 0.000 -.1774938 -.0904831
77 | -.1767688 .0239316 -7.39 0.000 -.2237175 -.1298201
78 | -.1396288 .0217205 -6.43 0.000 -.1822399 -.0970178
79 | -.1828466 .0225567 -8.11 0.000 -.2270981 -.1385952
80 | -.1875899 .0229984 -8.16 0.000 -.2327079 -.1424718
81 | -.2052302 .0237856 -8.63 0.000 -.2518926 -.1585678
82 | -.1576212 .0217146 -7.26 0.000 -.2002207 -.1150218
83 | -.0804423 .0181705 -4.43 0.000 -.116089 -.0447956
84 | -.0476163 .0162976 -2.92 0.004 -.0795888 -.0156438
85 | -.0482089 .0163213 -2.95 0.003 -.0802279 -.0161899
86 | -.0456368 .0159865 -2.85 0.004 -.076999 -.0142747
87 | -.0560178 .0161595 -3.47 0.001 -.0877194 -.0243162
88 | -.0677205 .0161209 -4.20 0.000 -.0993464 -.0360947
89 | -.0689491 .0158214 -4.36 0.000 -.0999873 -.037911
90 | -.0666174 .0155136 -4.29 0.000 -.0970517 -.0361831
91 | -.0701181 .0158837 -4.41 0.000 -.1012786 -.0389576
92 | 0 (omitted)
|
_cons | 2.890875 .1509644 19.15 0.000 2.594715 3.187035
-------------+----------------------------------------------------------------
sigma_u | .15466985
sigma_e | .07458789
rho | .81132297 (fraction of variance due to u_i)
------------------------------------------------------------------------------
F test that all u_i=0: F(45, 1302) = 125.73 Prob > F = 0.0000
(b) First, to test the hypothesis: 𝐻𝐻0: 𝜇𝜇1 = ⋯ = 𝜇𝜇𝑁𝑁, given 𝜆𝜆1 = ⋯ = 𝜆𝜆𝑇𝑇, the F test is used:
𝐹𝐹C1 =(RRSS − URSS)/(𝑁𝑁 − 1)
URSS/(𝑁𝑁𝑁𝑁 − 𝑁𝑁 − 𝐾𝐾) 𝐻𝐻0~ 𝐹𝐹𝑁𝑁−1, 𝑁𝑁(𝑇𝑇−1)−𝐾𝐾 = 𝐹𝐹45,1330
We have 𝐹𝐹C1 = 116.85, with p-value = 0.0000 from Table 2.2a below. Therefore, we reject 𝐻𝐻0 at any conventional level of significance, and conclude that the data provide significant evidence to show the existence of individual fixed effects. Table 2.2a Test for the existence of individual FE, given the non-existence of time FE . xtreg LnC LnP LnNDI LnPmin Year, fe
Fixed-effects (within) regression Number of obs = 1,380
Group variable: State Number of groups = 46
R-sq: Obs per group:
within = 0.6367 min = 30
between = 0.2938 avg = 30.0
overall = 0.4030 max = 30
F(4,1330) = 582.77
corr(u_i, Xb) = 0.0508 Prob > F = 0.0000
------------------------------------------------------------------------------
LnC | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
LnP | -.8514155 .0385828 -22.07 0.000 -.9271053 -.7757258
LnNDI | .6042318 .0352055 17.16 0.000 .5351674 .6732963
LnPmin | .2191902 .0371634 5.90 0.000 .146285 .2920955
Year | -.0100831 .0035202 -2.86 0.004 -.0169888 -.0031774
_cons | 2.8971 .095711 30.27 0.000 2.709339 3.084861
-------------+----------------------------------------------------------------
sigma_u | .1573968
sigma_e | .07837842
rho | .80130054 (fraction of variance due to u_i)
------------------------------------------------------------------------------
F test that all u_i=0: F(45, 1330) = 116.85 Prob > F = 0.0000
Second, switching the roles of State and Year using xtset, we perform the test for time FE, given non-existence of individual FE. However, the Year as an independent variable in the model is not allowed to run a time FE estimation, and thus is dropped.
To test 𝐻𝐻0: 𝜆𝜆1 = ⋯ = 𝜆𝜆𝑇𝑇, given 𝜇𝜇1 = ⋯ = 𝜇𝜇𝑁𝑁, we use the following F statistic:
𝐹𝐹C2 =(RRSS − URSS)/(𝑁𝑁 − 1)
URSS/(𝑁𝑁𝑁𝑁 − 𝑁𝑁 − 𝐾𝐾) 𝐻𝐻0~ 𝐹𝐹𝑇𝑇−1, (𝑁𝑁−1)𝑇𝑇−𝐾𝐾 = 𝐹𝐹29,1347
We have 𝐹𝐹C2 = 2.20 and p-value = 0.0003 from Table 2.2b below. Again 𝐻𝐻0 is strongly rejected, and data provide significant evidence to show the existence of time FE, given non-existence to State FE.
Table 2.2b. Test for the existence of time FE given the non-existence of individual FE . xtset Year
panel variable: Year (balanced)
. xtreg LnC LnP LnNDI LnPmin, fe
Fixed-effects (within) regression Number of obs = 1,380
Group variable: Year Number of groups = 30
R-sq: Obs per group:
within = 0.3446 min = 46
between = 0.6454 avg = 46.0
overall = 0.2531 max = 46
F(3,1347) = 236.07
corr(u_i, Xb) = -0.8881 Prob > F = 0.0000
------------------------------------------------------------------------------
LnC | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
LnP | -1.276568 .0598275 -21.34 0.000 -1.393933 -1.159203
LnNDI | .5687895 .0305808 18.60 0.000 .5087984 .6287806
LnPmin | .1602595 .0593774 2.70 0.007 .0437773 .2767416
_cons | 4.384547 .3359196 13.05 0.000 3.725565 5.04353
-------------+----------------------------------------------------------------
sigma_u | .21671531
sigma_e | .16954404
rho | .62032872 (fraction of variance due to u_i)
------------------------------------------------------------------------------
F test that all u_i=0: F(29, 1347) = 2.20 Prob > F = 0.0003
(c) First, to test 𝐻𝐻0: 𝜇𝜇1 = ⋯ = 𝜇𝜇𝑁𝑁 allowing 𝜆𝜆𝑡𝑡 ≠ 𝜆𝜆𝑠𝑠, 𝑡𝑡 ≠ 𝑠𝑠 = 1, . . . , 𝑁𝑁, we use the F- test:
𝐹𝐹M1 =(RRSS − URSS)/(𝑁𝑁 − 1)URSS/((N - 1)(T -1) − 𝐾𝐾)
𝐻𝐻0~ 𝐹𝐹𝑁𝑁−1, (𝑁𝑁−1)(𝑇𝑇−1)−𝐾𝐾 = 𝐹𝐹45, 1302
We have, 𝐹𝐹M1= 125.73, and p-value = 0.0000 from Table 2.3a below. Therefore, we reject 𝐻𝐻0 at any conventional level of significance, and conclude that the data provide sufficient evidence to show the existence of State FE, even the time FE is allowed in the model.
Table 2.3a. Test for the existence of individual FE allowing the existence of time FE
. xtset State
panel variable: State (balanced)
. xtreg LnC LnP LnNDI LnPmin Year i.Year, fe
note: 92.Year omitted because of collinearity
Fixed-effects (within) regression Number of obs = 1,380
Group variable: State Number of groups = 46
R-sq: Obs per group:
within = 0.6779 min = 30
between = 0.3178 avg = 30.0
overall = 0.4328 max = 30
F(32,1302) = 85.65
corr(u_i, Xb) = 0.0497 Prob > F = 0.0000
------------------------------------------------------------------------------
LnC | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
LnP | -1.023062 .0418181 -24.46 0.000 -1.1051 -.9410237
LnNDI | .5200042 .0466836 11.14 0.000 .4284208 .6115876
LnPmin | -.1172486 .0541294 -2.17 0.030 -.223439 -.0110582
Year | .0271425 .0056747 4.78 0.000 .01601 .038275
|
Year |
64 | -.0552901 .0154889 -3.57 0.000 -.0856761 -.0249042
65 | -.0963049 .0162297 -5.93 0.000 -.1281441 -.0644658
66 | -.1038394 .0165225 -6.28 0.000 -.1362531 -.0714257
67 | -.1250987 .0176757 -7.08 0.000 -.1597748 -.0904227
68 | -.1228057 .0172665 -7.11 0.000 -.1566789 -.0889324
69 | -.1584459 .018693 -8.48 0.000 -.1951175 -.1217742
70 | -.1497712 .0178417 -8.39 0.000 -.1847728 -.1147696
71 | -.1283003 .0183839 -6.98 0.000 -.1643656 -.092235
72 | -.1238581 .0196831 -6.29 0.000 -.1624723 -.085244
73 | -.1890391 .0217798 -8.68 0.000 -.2317664 -.1463117
74 | -.1943228 .0233824 -8.31 0.000 -.2401941 -.1484516
75 | -.1738359 .0230157 -7.55 0.000 -.2189878 -.128684
76 | -.1339885 .0221764 -6.04 0.000 -.1774938 -.0904831
77 | -.1767688 .0239316 -7.39 0.000 -.2237175 -.1298201
78 | -.1396288 .0217205 -6.43 0.000 -.1822399 -.0970178
79 | -.1828466 .0225567 -8.11 0.000 -.2270981 -.1385952
80 | -.1875899 .0229984 -8.16 0.000 -.2327079 -.1424718
81 | -.2052302 .0237856 -8.63 0.000 -.2518926 -.1585678
82 | -.1576212 .0217146 -7.26 0.000 -.2002207 -.1150218
83 | -.0804423 .0181705 -4.43 0.000 -.116089 -.0447956
84 | -.0476163 .0162976 -2.92 0.004 -.0795888 -.0156438
85 | -.0482089 .0163213 -2.95 0.003 -.0802279 -.0161899
86 | -.0456368 .0159865 -2.85 0.004 -.076999 -.0142747
87 | -.0560178 .0161595 -3.47 0.001 -.0877194 -.0243162
88 | -.0677205 .0161209 -4.20 0.000 -.0993464 -.0360947
89 | -.0689491 .0158214 -4.36 0.000 -.0999873 -.037911
90 | -.0666174 .0155136 -4.29 0.000 -.0970517 -.0361831
91 | -.0701181 .0158837 -4.41 0.000 -.1012786 -.0389576
92 | 0 (omitted)
|
_cons | 2.890875 .1509644 19.15 0.000 2.594715 3.187035
-------------+----------------------------------------------------------------
sigma_u | .15466985
sigma_e | .07458789
rho | .81132297 (fraction of variance due to u_i)
------------------------------------------------------------------------------
F test that all u_i=0: F(45, 1302) = 125.73 Prob > F = 0.0000
Second, to test 𝐻𝐻0: 𝜆𝜆1 = ⋯ = 𝜆𝜆𝑇𝑇 allowing 𝜇𝜇𝑖𝑖 ≠ 𝜇𝜇𝑗𝑗, 𝑖𝑖 ≠ 𝑗𝑗 = 1, . . . , 𝑁𝑁, we use the F-test:
𝐹𝐹M2 =(RRSS − URSS)/(𝑁𝑁 − 1)URSS/((N - 1)(T -1)− 𝐾𝐾)
𝐻𝐻0~ 𝐹𝐹𝑇𝑇−1, (𝑁𝑁−1)(𝑇𝑇−1)−𝐾𝐾 = 𝐹𝐹29, 1302
We have, 𝐹𝐹M2= 6.06, and p-value = 0.0000 from Table 2.3b below. Therefore, we reject 𝐻𝐻0 at any conventional level of significance, and conclude that the data provide sufficient evidence to show the existence of Year FE, even the State FE is allowed in the model.
Table 2.3b. Test for the existence of time FE allowing the existence of individual FE . xtset Year
panel variable: Year (balanced)
. xtreg LnC LnP LnNDI LnPmin i.State, fe
Fixed-effects (within) regression Number of obs = 1,380
Group variable: Year Number of groups = 30
R-sq: Obs per group:
within = 0.8774 min = 46
between = 0.5945 avg = 46.0
overall = 0.4960 max = 46
F(48,1302) = 194.10
corr(u_i, Xb) = -0.8344 Prob > F = 0.0000
------------------------------------------------------------------------------
LnC | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
LnP | -1.023062 .0418181 -24.46 0.000 -1.1051 -.9410237
LnNDI | .5200042 .0466836 11.14 0.000 .4284208 .6115876
LnPmin | -.1172486 .0541294 -2.17 0.030 -.223439 -.0110582
|
State |
3 | -.0942399 .0209363 -4.50 0.000 -.1353124 -.0531674
4 | .0652249 .019404 3.36 0.001 .0271584 .1032914
5 | -.1828052 .0277137 -6.60 0.000 -.2371736 -.1284368
7 | -.0800961 .0320071 -2.50 0.012 -.1428871 -.017305
8 | .1944474 .0225727 8.61 0.000 .1501645 .2387302
9 | .0131377 .0301371 0.44 0.663 -.045985 .0722603
10 | .1138834 .0227796 5.00 0.000 .0691946 .1585722
11 | -.0158117 .0204961 -0.77 0.441 -.0560207 .0243973
13 | -.1734154 .0203114 -8.54 0.000 -.213262 -.1335688
14 | -.0826502 .0277457 -2.98 0.003 -.1370814 -.028219
15 | .0020156 .0235876 0.09 0.932 -.0442583 .0482894
16 | -.0830408 .0219061 -3.79 0.000 -.1260158 -.0400657
17 | -.1476527 .0229396 -6.44 0.000 -.1926552 -.1026501
18 | .1969145 .0228691 8.61 0.000 .1520502 .2417787
19 | .1011496 .0194426 5.20 0.000 .0630073 .1392919
20 | .1333838 .0204419 6.53 0.000 .0932812 .1734864
21 | -.1509335 .0258846 -5.83 0.000 -.2017137 -.1001534
22 | -.0365199 .0269326 -1.36 0.175 -.089356 .0163161
23 | .0146083 .0231935 0.63 0.529 -.0308924 .0601091
24 | -.0606817 .0223601 -2.71 0.007 -.1045475 -.016816
25 | .0454322 .020374 2.23 0.026 .0054627 .0854017
26 | -.0328212 .023741 -1.38 0.167 -.079396 .0137535
27 | -.0993292 .0205569 -4.83 0.000 -.1396575 -.0590008
28 | -.1517093 .0218189 -6.95 0.000 -.1945133 -.1089053
29 | .2711074 .0270335 10.03 0.000 .2180734 .3241413
30 | .4794221 .0254298 18.85 0.000 .4295343 .5293099
31 | -.0744017 .031457 -2.37 0.018 -.1361136 -.0126897
32 | -.1977903 .0194705 -10.16 0.000 -.2359874 -.1595933
33 | -.0465274 .0269565 -1.73 0.085 -.0994104 .0063555
35 | -.142697 .0203435 -7.01 0.000 -.1826067 -.1027874
36 | -.0520312 .0240028 -2.17 0.030 -.0991196 -.0049428
37 | .0291263 .0199561 1.46 0.145 -.0100234 .068276
39 | -.060841 .0228124 -2.67 0.008 -.1055942 -.0160879
40 | .1122522 .0238575 4.71 0.000 .0654488 .1590556
41 | -.0606001 .0204522 -2.96 0.003 -.1007229 -.0204772
42 | -.1397424 .0199737 -7.00 0.000 -.1789266 -.1005582
43 | -.0026864 .021861 -0.12 0.902 -.0455731 .0402002
44 | -.0196305 .0216586 -0.91 0.365 -.0621202 .0228591
45 | -.5592524 .0199262 -28.07 0.000 -.5983434 -.5201615
46 | .1273323 .0214664 5.93 0.000 .0852199 .1694448
47 | -.15095 .0249201 -6.06 0.000 -.1998381 -.102062
48 | -.194523 .0252826 -7.69 0.000 -.2441221 -.1449239
49 | .0282018 .0217202 1.30 0.194 -.0144086 .0708122
50 | -.0761815 .0222461 -3.42 0.001 -.1198236 -.0325394
51 | -.0240692 .0225262 -1.07 0.285 -.0682609 .0201225
|
_cons | 4.910075 .5067728 9.69 0.000 3.915894 5.904255
-------------+----------------------------------------------------------------
sigma_u | .26197707
sigma_e | .07458789
rho | .92501748 (fraction of variance due to u_i)
------------------------------------------------------------------------------
F test that all u_i=0: F(29, 1302) = 6.06 Prob > F = 0.0000
(d) A panel regression with State FE including all regressors, i.e., “xtreg LnC LnP LnNDI LnPmin Year CPI Pop Pop16, fe”shows that Pop and Pop16 are insignificant at all and thus are dropped. Further, CPI is insignificant in the presence of Year-trend and thus is dropped. As discussed in (a), with the inclusion of Year-trend variable, we may not consider year-dummies as it may be the reason for the wrong sign of LnPmin.
The results of the best fitted model are given in Table 2.4 below: Teble 2.4. One-way FE model . xtreg LnC LnP LnNDI LnPmin Year, fe
Fixed-effects (within) regression Number of obs = 1,380
Group variable: State Number of groups = 46
R-sq: Obs per group:
within = 0.6367 min = 30
between = 0.2938 avg = 30.0
overall = 0.4030 max = 30
F(4,1330) = 582.77
corr(u_i, Xb) = 0.0508 Prob > F = 0.0000
------------------------------------------------------------------------------
LnC | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
LnP | -.8514155 .0385828 -22.07 0.000 -.9271053 -.7757258
LnNDI | .6042318 .0352055 17.16 0.000 .5351674 .6732963
LnPmin | .2191902 .0371634 5.90 0.000 .146285 .2920955
Year | -.0100831 .0035202 -2.86 0.004 -.0169888 -.0031774
_cons | 2.8971 .095711 30.27 0.000 2.709339 3.084861
-------------+----------------------------------------------------------------
sigma_u | .1573968
sigma_e | .07837842
rho | .80130054 (fraction of variance due to u_i)
------------------------------------------------------------------------------
F test that all u_i=0: F(45, 1330) = 116.85 Prob > F = 0.0000
When only state FE is included, all the coefficients have the expected signs. Year has a negative coefficient, indicating that less people smoke over time, possibly indicating an increasing awareness of the unhealthy effects of smoking.
3. Consider the cigarette demand data used in Problem 2.
(a) Fit a two-way random effects (RE) model to the sale variable LnC, based on LnP, LnNDI, LnPmin, and Year. Discuss your results. Comment (based on relevant tests) on the significance of individual and time specific random effects.
(b) Fit a one-way individual RE model to the sale variable LnC, based on LnP, LnNDI, LnPmin, Year, and the time dummies. Discuss your results. Comment (based on relevant tests) on the significance of the individual RE, and the significance of time FE.
(c) Fit a one-way time RE model to the sale variable LnC, based on LnP, LnNDI, LnPmin, Year, and the individual dummies. Discuss your results. Comment (based on relevant tests) on the significance of the time RE, and the significance of individual FE.
(d) Compare the three models fitted in (a)-(c). Solution: (a) From Table 3.1 below, we have the LR test statistic for the existence of both State and
Year RE has a value of 1991.31, with a p-value of 0.0000. Thus, we can reject the null hypothesis that the SD of both State and Year RE are zero, for the alternative hypothesis that at least one of the RE standard deviations are non-zero. Further, the 95% confidence interval (CI) for 𝜎𝜎𝜇𝜇 is (.1240788, .1873762), and hence we are at least 97.5% confident that 𝜎𝜎𝜇𝜇 > 0. Similarly, the 95% CI for 𝜎𝜎𝜆𝜆 is (.0209396, .0520159), and hence we are at least 97.5% confident that 𝜎𝜎𝜆𝜆 > 0.
Table 3.1. Two-way random effects estimation . xtmixed LnC LnP LnNDI LnPmin Year || _all: R.Year || State:, mle
Performing EM optimization:
Performing gradient-based optimization:
Iteration 0: log likelihood = 1472.667
Iteration 1: log likelihood = 1472.667
Computing standard errors:
Mixed-effects ML regression Number of obs = 1,380
-------------------------------------------------------------
| No. of Observations per Group
Group Variable | Groups Minimum Average Maximum
----------------+--------------------------------------------
_all | 1 1,380 1,380.0 1,380
State | 46 30 30.0 30
-------------------------------------------------------------
Wald chi2(4) = 1011.13
Log likelihood = 1472.667 Prob > chi2 = 0.0000
------------------------------------------------------------------------------
LnC | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
LnP | -.9438117 .0392337 -24.06 0.000 -1.020708 -.8669151
LnNDI | .5311831 .0429266 12.37 0.000 .4470484 .6153177
LnPmin | .0596247 .0453152 1.32 0.188 -.0291914 .1484408
Year | .0117017 .0049095 2.38 0.017 .0020792 .0213242
_cons | 2.85403 .1359743 20.99 0.000 2.587525 3.120535
------------------------------------------------------------------------------
------------------------------------------------------------------------------
Random-effects Parameters | Estimate Std. Err. [95% Conf. Interval]
-----------------------------+------------------------------------------------
_all: Identity |
sd(R.Year) | .0330029 .0076607 .0209396 .0520159
-----------------------------+------------------------------------------------
State: Identity |
sd(_cons) | .1524776 .0160338 .1240788 .1873762
-----------------------------+------------------------------------------------
sd(Residual) | .0749673 .0014913 .0721006 .077948
------------------------------------------------------------------------------
LR test vs. linear model: chi2(2) = 1991.31 Prob > chi2 = 0.0000
(b) From Table 3.2, we see that all the four main variables are significant. However, as in
the two-way FE case the inclusion of time-dummies changes the sign of LnPmin, which may be an undesirable feature. LR test for State RE has a value of 2042.81 and a p-value of 0.000, showing that the State RE is highly significant. Each of the estimated time dummies are highly significant, from which we can infer that the time FE is highly significant.
Table 3.2. One-way State RE with time-dummies . xtreg LnC LnP LnNDI LnPmin Year i.Year, mle
note: 92.Year omitted because of collinearity
Fitting constant-only model:
Iteration 0: log likelihood = 764.58324
Iteration 1: log likelihood = 764.58491
Fitting full model:
Iteration 0: log likelihood = 1496.3129
Iteration 1: log likelihood = 1523.7647
Iteration 2: log likelihood = 1528.905
Iteration 3: log likelihood = 1528.9753
Iteration 4: log likelihood = 1528.9753
Random-effects ML regression Number of obs = 1,380
Group variable: State Number of groups = 46
Random effects u_i ~ Gaussian Obs per group:
min = 30
avg = 30.0
max = 30
LR chi2(32) = 1528.78
Log likelihood = 1528.9753 Prob > chi2 = 0.0000
------------------------------------------------------------------------------
LnC | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
LnP | -1.0264 .0409447 -25.07 0.000 -1.106651 -.9461501
LnNDI | .522502 .0440898 11.85 0.000 .4360877 .6089164
LnPmin | -.1097341 .0526832 -2.08 0.037 -.2129914 -.0064769
Year | .0267034 .0053999 4.95 0.000 .0161197 .037287
|
Year |
64 | -.0550973 .0152925 -3.60 0.000 -.08507 -.0251245
65 | -.095866 .0159862 -6.00 0.000 -.1271985 -.0645336
66 | -.1032692 .016253 -6.35 0.000 -.1351244 -.071414
67 | -.1243342 .0173335 -7.17 0.000 -.1583073 -.0903612
68 | -.1221058 .0169345 -7.21 0.000 -.1552968 -.0889148
69 | -.1575054 .018286 -8.61 0.000 -.1933453 -.1216655
70 | -.1488741 .0174707 -8.52 0.000 -.183116 -.1146322
71 | -.1272949 .0179817 -7.08 0.000 -.1625385 -.0920513
72 | -.1226912 .019225 -6.38 0.000 -.1603714 -.0850109
73 | -.1877838 .0212737 -8.83 0.000 -.2294795 -.1460881
74 | -.192947 .0228136 -8.46 0.000 -.2376608 -.1482333
75 | -.1725179 .0224656 -7.68 0.000 -.2165496 -.1284862
76 | -.1327843 .0216585 -6.13 0.000 -.1752342 -.0903345
77 | -.1754749 .0233576 -7.51 0.000 -.221255 -.1296948
78 | -.1386318 .0212456 -6.53 0.000 -.1802724 -.0969913
79 | -.1819006 .0220619 -8.25 0.000 -.225141 -.1386601
80 | -.1866629 .0224923 -8.30 0.000 -.230747 -.1425788
81 | -.2043288 .0232583 -8.79 0.000 -.2499142 -.1587434
82 | -.1568713 .021257 -7.38 0.000 -.1985342 -.1152085
83 | -.0799712 .0178345 -4.48 0.000 -.1149262 -.0450163
84 | -.0474666 .0160192 -2.96 0.003 -.0788636 -.0160696
85 | -.0479929 .0160555 -2.99 0.003 -.0794611 -.0165246
86 | -.0454833 .0157439 -2.89 0.004 -.0763408 -.0146258
87 | -.0557555 .0159163 -3.50 0.000 -.0869508 -.0245601
88 | -.0674467 .0158837 -4.25 0.000 -.0985783 -.0363151
89 | -.068717 .0156017 -4.40 0.000 -.0992959 -.0381382
90 | -.0664853 .0153123 -4.34 0.000 -.0964968 -.0364738
91 | -.069805 .0156696 -4.45 0.000 -.1005168 -.0390932
92 | 0 (omitted)
|
_cons | 2.886191 .1457899 19.80 0.000 2.600448 3.171934
-------------+----------------------------------------------------------------
/sigma_u | .1522974 .016011 .1239384 .1871454
/sigma_e | .0736886 .0014266 .0709448 .0765385
rho | .8103021 .0328727 .7393529 .867932
------------------------------------------------------------------------------
LR test of sigma_u=0: chibar2(01) = 2042.81 Prob >= chibar2 = 0.000
(c) With the state dummies in the model, the Year-trend is not allowed, and thus is dropped.
All the three main variables are highly significant, and the sign of LnPmin is back to positive, showing the effect of time-dummies (or time FE). LR test for time RE has a value or 55.23, and a p-value of 0.000, showing the time RE is highly significant.
Table 3.3. One-way time RE model with individual dummies . xtset Year
panel variable: Year (balanced)
. xtreg LnC LnP LnNDI LnPmin i.State, mle
Fitting constant-only model:
Iteration 0: log likelihood = 169.71765
Iteration 1: log likelihood = 169.71789
Fitting full model:
Iteration 0: log likelihood = 1599.403
Iteration 1: log likelihood = 1604.1888
Iteration 2: log likelihood = 1604.4563
Iteration 3: log likelihood = 1604.4632
Iteration 4: log likelihood = 1604.4632
Random-effects ML regression Number of obs = 1,380
Group variable: Year Number of groups = 30
Random effects u_i ~ Gaussian Obs per group:
min = 46
avg = 46.0
max = 46
LR chi2(48) = 2869.49
Log likelihood = 1604.4632 Prob > chi2 = 0.0000
------------------------------------------------------------------------------
LnC | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
LnP | -.8975275 .0363402 -24.70 0.000 -.9687531 -.8263019
LnNDI | .5886631 .0290623 20.26 0.000 .531702 .6456241
LnPmin | .1257777 .0422316 2.98 0.003 .0430054 .2085501
|
State |
3 | -.0953094 .0201388 -4.73 0.000 -.1347808 -.0558379
4 | .0774022 .0191828 4.03 0.000 .0398047 .1149998
5 | -.2132145 .0227722 -9.36 0.000 -.2578472 -.1685817
7 | -.1426293 .0230455 -6.19 0.000 -.1877977 -.0974609
8 | .1848792 .0208157 8.88 0.000 .1440813 .2256772
9 | .0249555 .0266934 0.93 0.350 -.0273626 .0772736
10 | .0830323 .0200961 4.13 0.000 .0436447 .12242
11 | .0039473 .0201618 0.20 0.845 -.035569 .0434637
13 | -.1582158 .0199874 -7.92 0.000 -.1973903 -.1190412
14 | -.0625993 .0255646 -2.45 0.014 -.1127049 -.0124936
15 | .0506663 .0226457 2.24 0.025 .0062815 .0950511
16 | -.0890431 .0205654 -4.33 0.000 -.1293505 -.0487357
17 | -.1515802 .0212465 -7.13 0.000 -.1932225 -.1099379
18 | .2613197 .0210214 12.43 0.000 .2201186 .3025208
19 | .095641 .0192334 4.97 0.000 .0579442 .1333377
20 | .1346986 .0198959 6.77 0.000 .0957033 .1736939
21 | -.1251522 .0242253 -5.17 0.000 -.1726328 -.0776715
22 | -.0607483 .0227774 -2.67 0.008 -.1053912 -.0161054
23 | .018648 .0217115 0.86 0.390 -.0239058 .0612018
24 | -.0873185 .0200369 -4.36 0.000 -.1265901 -.0480469
25 | .0561629 .0195933 2.87 0.004 .0177608 .0945651
26 | .0090979 .0228495 0.40 0.691 -.0356863 .0538821
27 | -.0886813 .0201186 -4.41 0.000 -.128113 -.0492497
28 | -.1461788 .0208555 -7.01 0.000 -.1870549 -.1053027
29 | .2514699 .0230123 10.93 0.000 .2063667 .2965731
30 | .4632949 .0224709 20.62 0.000 .4192528 .507337
31 | -.1302755 .0232231 -5.61 0.000 -.175792 -.084759
32 | -.197185 .0192857 -10.22 0.000 -.2349843 -.1593856
33 | -.0841859 .0218829 -3.85 0.000 -.1270756 -.0412961
35 | -.1489439 .0196883 -7.57 0.000 -.1875323 -.1103555
36 | -.0156392 .0230855 -0.68 0.498 -.060886 .0296076
37 | .0322404 .0196162 1.64 0.100 -.0062067 .0706875
39 | -.0700525 .0209907 -3.34 0.001 -.1111934 -.0289115
40 | .0655532 .0201308 3.26 0.001 .0260976 .1050089
41 | -.0461755 .0201979 -2.29 0.022 -.0857625 -.0065884
42 | -.126918 .0197235 -6.43 0.000 -.1655753 -.0882607
43 | .0420017 .0211064 1.99 0.047 .0006339 .0833695
44 | -.0434668 .019814 -2.19 0.028 -.0823015 -.0046321
45 | -.5420722 .019692 -27.53 0.000 -.5806679 -.5034766
46 | .127251 .0204742 6.22 0.000 .0871222 .1673797
47 | -.095014 .023725 -4.00 0.000 -.1415142 -.0485138
48 | -.2279928 .0213031 -10.70 0.000 -.2697462 -.1862394
49 | .0762542 .0205311 3.71 0.000 .036014 .1164944
50 | -.1044423 .0199201 -5.24 0.000 -.1434851 -.0653996
51 | -.0189179 .0214157 -0.88 0.377 -.060892 .0230562
|
_cons | 2.835894 .1179413 24.04 0.000 2.604733 3.067055
-------------+----------------------------------------------------------------
/sigma_u | .0245583 .0046739 .0169121 .0356613
/sigma_e | .0741888 .0014358 .0714274 .077057
rho | .0987555 .0343306 .0469077 .1835962
------------------------------------------------------------------------------
LR test of sigma_u=0: chibar2(01) = 55.23 Prob >= chibar2 = 0.000
(d) In all the 3 models, the coefficients for LnP, LnNDI, and Year have remained statistically significant, and maintained their respective signs. This indicates that the coefficient estimates have some robustness against slightly different model specifications. The coefficient of LnPmin was only significant for the model with State RE and Time FE, but the p-value was only 0.037, indicating marginal significance, and will change depending on model specifications.
The effects were significant in all model specifications, regardless of FE or RE specification. But the Time RE and State FE specification had the highest log-likelihood value of 1605.9949, indicating the best fit with the data.
4. Consider the Bank Cost Data described in Chapter 3, Lecture Notes, also available at http://people.stern.nyu.edu/wgreene/Econometrics/PanelDataEconometrics.htm (a) Obtain the pooled OLS estimator with cluster-robust standard errors (CRSE) of panel
regression of C on W1-W4, and Q1-Q5, first using regress command, and then using xtreg, pa command. Comment on your results.
(b) For the panel regression of C on W1-W4, and Q1-Q5, obtain the pooled FGLS or PA estimator with CRSE and unstructured error correlation. Comment on your results.
(c) For the one-way individual FE panel regression of C on W1-W4, Q1-Q5 and time dummies, obtain the within estimator with CRSE. Comment on your results.
(d) For the model considered in (c), obtain the first-difference estimator with CRSE. Compare the results with those in (c).
(e) For the model considered in (c), find the LSDV estimator using the command areg. Compare the results with those in (c) and comment.
(f) For the one-way individual RE panel regression of C on W1-W4, Q1-Q5 and time dummies, obtain the re, mle and pa corr(exchangeble) estimators with CRSE. Comment on your results.
(g) Using STATA to compare all the estimators obtained in (a)-(f).
Solution
(a) The coefficient estimates are exactly the same in both cases, but the standard error estimation differs slightly. Standard errors in the population-averaged model are slightly smaller than those of the pooled OLS. All coefficients are statistically significant. . regress c w1 w2 w3 w4 q1 q2 q3 q4 q5, vce(cluster bank)
Linear regression Number of obs = 2,500
F(9, 499) = 3532.78
Prob > F = 0.0000
R-squared = 0.9547
Root MSE = .25046
(Std. Err. adjusted for 500 clusters in bank)
------------------------------------------------------------------------------
| Robust
c | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
w1 | .423211 .0201681 20.98 0.000 .3835861 .4628358
w2 | .0365596 .009008 4.06 0.000 .0188612 .0542579
w3 | .177698 .016407 10.83 0.000 .1454628 .2099333
w4 | .1060201 .0125327 8.46 0.000 .0813967 .1306434
q1 | .1033549 .0094003 10.99 0.000 .0848858 .1218241
q2 | .3749293 .0107371 34.92 0.000 .3538338 .3960249
q3 | .0965813 .015679 6.16 0.000 .0657762 .1273864
q4 | .0562386 .00525 10.71 0.000 .0459238 .0665533
q5 | .2860344 .0140998 20.29 0.000 .2583321 .3137367
_cons | .5636438 .1653867 3.41 0.001 .2387037 .8885839
------------------------------------------------------------------------------
. xtreg c w1 w2 w3 w4 q1 q2 q3 q4 q5, pa corr(independent) vce(robust)
Iteration 1: tolerance = 7.542e-12
GEE population-averaged model Number of obs = 2,500
Group variable: bank Number of groups = 500
Link: identity Obs per group:
Family: Gaussian min = 5
Correlation: independent avg = 5.0
max = 5
Wald chi2(9) = 31909.90
Scale parameter: .062477 Prob > chi2 = 0.0000
Pearson chi2(2500): 156.19 Deviance = 156.19
Dispersion (Pearson): .062477 Dispersion = .062477
(Std. Err. adjusted for clustering on bank)
------------------------------------------------------------------------------
| Robust
c | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
w1 | .423211 .0201317 21.02 0.000 .3837535 .4626684
w2 | .0365596 .0089918 4.07 0.000 .0189359 .0541832
w3 | .177698 .0163774 10.85 0.000 .1455989 .2097971
w4 | .1060201 .0125101 8.47 0.000 .0815007 .1305394
q1 | .1033549 .0093834 11.01 0.000 .0849638 .1217461
q2 | .3749293 .0107178 34.98 0.000 .3539229 .3959358
q3 | .0965813 .0156508 6.17 0.000 .0659063 .1272562
q4 | .0562386 .0052405 10.73 0.000 .0459674 .0665098
q5 | .2860344 .0140744 20.32 0.000 .2584491 .3136197
_cons | .5636438 .1650886 3.41 0.001 .2400761 .8872115
------------------------------------------------------------------------------
(b) When placing no restrictions on the structure of serial correlation between errors, the coefficient estimates and the standard errors changed very slightly. All coefficients still remain statistically significant. . xtset bank t
panel variable: bank (strongly balanced)
time variable: t, 1 to 5
delta: 1 unit
. xtreg c w1 w2 w3 w4 q1 q2 q3 q4 q5, pa corr(unstructured) vce(robust)
Iteration 1: tolerance = .01326228
Iteration 2: tolerance = .00021266
Iteration 3: tolerance = 8.629e-06
Iteration 4: tolerance = 5.371e-07
GEE population-averaged model Number of obs = 2,500
Group and time vars: bank t Number of groups = 500
Link: identity Obs per group:
Family: Gaussian min = 5
Correlation: unstructured avg = 5.0
max = 5
Wald chi2(9) = 33247.43
Scale parameter: .0624809 Prob > chi2 = 0.0000
(Std. Err. adjusted for clustering on bank)
------------------------------------------------------------------------------
| Robust
c | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
w1 | .4190001 .0201006 20.85 0.000 .3796037 .4583965
w2 | .0369044 .00895 4.12 0.000 .0193628 .054446
w3 | .180744 .0163357 11.06 0.000 .1487266 .2127613
w4 | .1056251 .0123766 8.53 0.000 .0813674 .1298827
q1 | .1037522 .0093492 11.10 0.000 .0854281 .1220764
q2 | .3748237 .0107855 34.75 0.000 .3536844 .3959629
q3 | .0969988 .0154105 6.29 0.000 .0667948 .1272028
q4 | .0559306 .0051648 10.83 0.000 .0458079 .0660533
q5 | .2863292 .0139839 20.48 0.000 .2589213 .313737
_cons | .5847327 .1647887 3.55 0.000 .2617529 .9077126
------------------------------------------------------------------------------
(c) The Within estimators of the coefficients are still largely close in value to the pooled OLS
and pooled FGLS. Standard errors are higher, but still close to those of the pooled OLS and pooled FGLS estimates. All coefficients are still statistically significant.
. xtreg c w1 w2 w3 w4 q1 q2 q3 q4 q5 i.t, fe vce(robust)
Fixed-effects (within) regression Number of obs = 2,500
Group variable: bank Number of groups = 500
R-sq: Obs per group:
within = 0.9577 min = 5
between = 0.9473 avg = 5.0
overall = 0.9557 max = 5
F(13,499) = 2351.12
corr(u_i, Xb) = -0.0021 Prob > F = 0.0000
(Std. Err. adjusted for 500 clusters in bank)
------------------------------------------------------------------------------
| Robust
c | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
w1 | .4156845 .0219445 18.94 0.000 .3725694 .4587996
w2 | .0390743 .009717 4.02 0.000 .0199831 .0581655
w3 | .1876276 .0168927 11.11 0.000 .154438 .2208172
w4 | .0914774 .013212 6.92 0.000 .0655195 .1174354
q1 | .1052981 .0103676 10.16 0.000 .0849286 .1256676
q2 | .3768671 .0117705 32.02 0.000 .3537413 .3999929
q3 | .1022029 .0156095 6.55 0.000 .0715345 .1328713
q4 | .0529178 .0054736 9.67 0.000 .0421637 .063672
q5 | .2839846 .0148466 19.13 0.000 .2548151 .3131541
|
t |
2 | -.0005003 .0159987 -0.03 0.975 -.0319335 .0309329
3 | -.0324821 .0159782 -2.03 0.043 -.0638749 -.0010892
4 | -.0642705 .0160141 -4.01 0.000 -.0957339 -.0328071
5 | -.1015606 .0167178 -6.08 0.000 -.1344064 -.0687147
|
_cons | .5913814 .1745729 3.39 0.001 .248393 .9343699
-------------+----------------------------------------------------------------
sigma_u | .11882953
sigma_e | .2431313
rho | .19281473 (fraction of variance due to u_i)
------------------------------------------------------------------------------
(d) The coefficient estimates for the first-difference estimator are very close to the within estimators. The standard errors for the first-difference estimators are slightly larger than
those of the within estimator, but all coefficients are still statistically significant. The number of time-dummies are reduced by 1 due to the first-differencing. The time dummies were mostly statistically significant in the within estimation, but are now all statistically insignificant in the first-difference estimation. . regress D.(c w1 w2 w3 w4 q1 q2 q3 q4 q5) i.t, vce(cluster bank)
Linear regression Number of obs = 2,000
F(12, 499) = 1832.54
Prob > F = 0.0000
R-squared = 0.9542
Root MSE = .34386
(Std. Err. adjusted for 500 clusters in bank)
------------------------------------------------------------------------------
| Robust
D.c | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
w1 |
D1. | .3943297 .0260517 15.14 0.000 .3431452 .4455143
w2 |
D1. | .028143 .011224 2.51 0.012 .0060908 .0501952
w3 |
D1. | .2032593 .0194798 10.43 0.000 .1649867 .2415319
w4 |
D1. | .0893374 .0156151 5.72 0.000 .0586579 .1200169
q1 |
D1. | .1064259 .0118681 8.97 0.000 .0831083 .1297434
q2 |
D1. | .3768837 .0128699 29.28 0.000 .3515978 .4021697
q3 |
D1. | .0948604 .0175255 5.41 0.000 .0604276 .1292932
q4 |
D1. | .0542789 .0067099 8.09 0.000 .0410957 .067462
q5 |
D1. | .2867459 .0161605 17.74 0.000 .2549948 .318497
t |
3 | -.0207953 .0284985 -0.73 0.466 -.076787 .0351965
4 | -.0189306 .0227362 -0.83 0.405 -.063601 .0257399
5 | -.0299231 .0228045 -1.31 0.190 -.0747277 .0148815
|
_cons | -.0102601 .0169456 -0.61 0.545 -.0435536 .0230333
------------------------------------------------------------------------------
(e) The coefficient estimates and time dummy coefficients using areg are exactly the same as those in the Within estimation. The standard errors obtained from areg are higher than those of the Within estimation.
. areg c w1 w2 w3 w4 q1 q2 q3 q4 q5 i.t, absorb(bank) vce(cluster bank)
Linear regression, absorbing indicators Number of obs = 2,500
Absorbed variable: bank No. of categories = 500
F( 13, 499) = 1879.20
Prob > F = 0.0000
R-squared = 0.9659
Adj R-squared = 0.9571
Root MSE = 0.2431
(Std. Err. adjusted for 500 clusters in bank)
------------------------------------------------------------------------------
| Robust
c | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
w1 | .4156845 .0245459 16.94 0.000 .3674585 .4639105
w2 | .0390743 .0108688 3.60 0.000 .01772 .0604286
w3 | .1876276 .0188952 9.93 0.000 .1505037 .2247516
w4 | .0914774 .0147781 6.19 0.000 .0624424 .1205124
q1 | .1052981 .0115966 9.08 0.000 .082514 .1280822
q2 | .3768671 .0131658 28.62 0.000 .3509999 .4027343
q3 | .1022029 .0174598 5.85 0.000 .067899 .1365067
q4 | .0529178 .0061225 8.64 0.000 .0408889 .0649468
q5 | .2839846 .0166065 17.10 0.000 .2513574 .3166119
|
t |
2 | -.0005003 .0178952 -0.03 0.978 -.0356596 .034659
3 | -.0324821 .0178723 -1.82 0.070 -.0675962 .0026321
4 | -.0642705 .0179124 -3.59 0.000 -.0994636 -.0290774
5 | -.1015606 .0186995 -5.43 0.000 -.1383 -.0648211
|
_cons | .5913814 .1952668 3.03 0.003 .207735 .9750278
------------------------------------------------------------------------------
(f) The coefficient estimates are very close in all 3 estimation methods, and are exactly identical in MLE and population-averaged (PA) estimation. Standard errors are highest in the GLS estimation and lowest in the PA estimation. All coefficients are statistically significant in all methods, suggesting robustness to estimation methods. Time dummies of all 3 estimation methods also show the same pattern, with all except t2 coefficient being statistically significant.
. xtreg c w1 w2 w3 w4 q1 q2 q3 q4 q5 i.t, re vce(robust)
Random-effects GLS regression Number of obs = 2,500
Group variable: bank Number of groups = 500
R-sq: Obs per group:
within = 0.9577 min = 5
between = 0.9476 avg = 5.0
overall = 0.9557 max = 5
Wald chi2(13) = 33153.58
corr(u_i, X) = 0 (assumed) Prob > chi2 = 0.0000
(Std. Err. adjusted for 500 clusters in bank)
------------------------------------------------------------------------------
| Robust
c | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
w1 | .4269637 .0202635 21.07 0.000 .3872478 .4666795
w2 | .0332008 .0088759 3.74 0.000 .0158043 .0505973
w3 | .1836653 .0163038 11.27 0.000 .1517105 .2156201
w4 | .0885393 .012077 7.33 0.000 .0648688 .1122097
q1 | .1024541 .0093011 11.02 0.000 .0842242 .1206839
q2 | .3752844 .0107354 34.96 0.000 .3542435 .3963254
q3 | .098474 .0152924 6.44 0.000 .0685014 .1284466
q4 | .0540584 .0052547 10.29 0.000 .0437593 .0643575
q5 | .2900255 .0139809 20.74 0.000 .2626234 .3174277
|
t |
2 | -.0014721 .0162073 -0.09 0.928 -.0332379 .0302937
3 | -.0344234 .015634 -2.20 0.028 -.0650654 -.0037814
4 | -.0667354 .0159238 -4.19 0.000 -.0979455 -.0355253
5 | -.1036345 .0166382 -6.23 0.000 -.1362448 -.0710242
|
_cons | .5496466 .1664326 3.30 0.001 .2234447 .8758485
-------------+----------------------------------------------------------------
sigma_u | .04796794
sigma_e | .2431313
rho | .03746592 (fraction of variance due to u_i)
vce(robust) is not allowed for MLE
. xtreg c w1 w2 w3 w4 q1 q2 q3 q4 q5 i.t, mle
Fitting constant-only model:
Iteration 0: log likelihood = -3948.1225
Iteration 1: log likelihood = -3948.1224
Fitting full model:
Iteration 0: log likelihood = -48.543891
Iteration 1: log likelihood = -48.498592
Iteration 2: log likelihood = -48.498587
Random-effects ML regression Number of obs = 2,500
Group variable: bank Number of groups = 500
Random effects u_i ~ Gaussian Obs per group:
min = 5
avg = 5.0
max = 5
LR chi2(13) = 7799.25
Log likelihood = -48.498587 Prob > chi2 = 0.0000
------------------------------------------------------------------------------
c | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
w1 | .4269868 .0176047 24.25 0.000 .3924823 .4614913
w2 | .0331889 .0079966 4.15 0.000 .0175159 .0488619
w3 | .1836575 .0149601 12.28 0.000 .1543363 .2129787
w4 | .0885328 .0118133 7.49 0.000 .0653793 .1116864
q1 | .1024483 .0073384 13.96 0.000 .0880653 .1168313
q2 | .3752812 .0069884 53.70 0.000 .3615843 .3889781
q3 | .0984666 .0095068 10.36 0.000 .0798337 .1170996
q4 | .0540608 .0040136 13.47 0.000 .0461943 .0619273
q5 | .2900378 .0096068 30.19 0.000 .2712089 .3088667
|
t |
2 | -.001474 .0161605 -0.09 0.927 -.033148 .0302
3 | -.0344273 .0162771 -2.12 0.034 -.06633 -.0025247
4 | -.0667405 .0162657 -4.10 0.000 -.0986206 -.0348603
5 | -.1036388 .0162975 -6.36 0.000 -.1355813 -.0716963
|
_cons | .5495577 .123002 4.47 0.000 .3084782 .7906371
-------------+----------------------------------------------------------------
/sigma_u | .0474584 .010149 .0312092 .0721676
/sigma_e | .2424231 .0038352 .2350216 .2500577
rho | .03691 .0156746 .0150649 .0797028
------------------------------------------------------------------------------
LR test of sigma_u=0: chibar2(01) = 6.33 Prob >= chibar2 = 0.006
. xtreg c w1 w2 w3 w4 q1 q2 q3 q4 q5 i.t, pa corr(exchangeable) vce(robust)
Iteration 1: tolerance = .00424965
Iteration 2: tolerance = .00002336
Iteration 3: tolerance = 1.296e-07
GEE population-averaged model Number of obs = 2,500
Group variable: bank Number of groups = 500
Link: identity Obs per group:
Family: Gaussian min = 5
Correlation: exchangeable avg = 5.0
max = 5
Wald chi2(13) = 33321.75
Scale parameter: .0610213 Prob > chi2 = 0.0000
(Std. Err. adjusted for clustering on bank)
------------------------------------------------------------------------------
| Robust
c | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
w1 | .4269868 .0202116 21.13 0.000 .3873728 .4666007
w2 | .0331889 .0088526 3.75 0.000 .0158381 .0505398
w3 | .1836575 .0162633 11.29 0.000 .151782 .2155331
w4 | .0885328 .0120463 7.35 0.000 .0649226 .1121431
q1 | .1024483 .0092761 11.04 0.000 .0842674 .1206291
q2 | .3752812 .0107063 35.05 0.000 .3542973 .3962651
q3 | .0984666 .0152536 6.46 0.000 .0685701 .1283632
q4 | .0540608 .0052414 10.31 0.000 .0437879 .0643337
q5 | .2900378 .0139443 20.80 0.000 .2627074 .3173682
t |
2 | -.001474 .0161659 -0.09 0.927 -.0331587 .0302107
3 | -.0344273 .015593 -2.21 0.027 -.064989 -.0038657
4 | -.0667405 .0158825 -4.20 0.000 -.0978697 -.0356112
5 | -.1036388 .0165952 -6.25 0.000 -.1361648 -.0711129
|
_cons | .5495577 .166003 3.31 0.001 .2241978 .8749175
------------------------------------------------------------------------------
(g) The Stata commands for comparison:
quietly regress c w1 w2 w3 w4 q1 q2 q3 q4 q5, vce(cluster bank)
estimates store OLS_Est
quietly xtreg c w1 w2 w3 w4 q1 q2 q3 q4 q5, pa corr(independent) vce(robust)
estimates store PA_EstI
quietly xtreg c w1 w2 w3 w4 q1 q2 q3 q4 q5, pa corr(unstructured) vce(robust)
estimates store PA_EstU
quietly xtreg c w1 w2 w3 w4 q1 q2 q3 q4 q5 i.t, fe vce(robust)
estimates store FE_EstR
quietly regress D.(c w1 w2 w3 w4 q1 q2 q3 q4 q5) i.t, vce(cluster bank)
estimates store FD_EstR
quietly areg c w1 w2 w3 w4 q1 q2 q3 q4 q5 i.t, absorb(bank) vce(cluster bank)
estimates store LSDV_EstR
quietly xtreg c w1 w2 w3 w4 q1 q2 q3 q4 q5 i.t, re vce(robust)
estimates store RE_EstR
quietly xtreg c w1 w2 w3 w4 q1 q2 q3 q4 q5 i.t, mle
estimates store MLE_Est
quietly xtreg c w1 w2 w3 w4 q1 q2 q3 q4 q5 i.t, pa corr(exchangeable) vce(robust)
estimates store PA_EstE
The comparison results are given below. . estimates table OLS_Est PA_EstI PA_EstU FE_EstR LSDV_EstR RE_EstR PA_EstE, b(%7.3f) se ------------------------------------------------------------------------------------
Variable | OLS_Est PA_EstI PA_EstU FE_EstR LSDV_~R RE_EstR PA_EstE
-------------+----------------------------------------------------------------------
w1 | 0.423 0.423 0.419 0.416 0.416 0.427 0.427
| 0.020 0.020 0.020 0.022 0.025 0.020 0.020
w2 | 0.037 0.037 0.037 0.039 0.039 0.033 0.033
| 0.009 0.009 0.009 0.010 0.011 0.009 0.009
w3 | 0.178 0.178 0.181 0.188 0.188 0.184 0.184
| 0.016 0.016 0.016 0.017 0.019 0.016 0.016
w4 | 0.106 0.106 0.106 0.091 0.091 0.089 0.089
| 0.013 0.013 0.012 0.013 0.015 0.012 0.012
q1 | 0.103 0.103 0.104 0.105 0.105 0.102 0.102
| 0.009 0.009 0.009 0.010 0.012 0.009 0.009
q2 | 0.375 0.375 0.375 0.377 0.377 0.375 0.375
| 0.011 0.011 0.011 0.012 0.013 0.011 0.011
q3 | 0.097 0.097 0.097 0.102 0.102 0.098 0.098
| 0.016 0.016 0.015 0.016 0.017 0.015 0.015
q4 | 0.056 0.056 0.056 0.053 0.053 0.054 0.054
| 0.005 0.005 0.005 0.005 0.006 0.005 0.005
q5 | 0.286 0.286 0.286 0.284 0.284 0.290 0.290
| 0.014 0.014 0.014 0.015 0.017 0.014 0.014
t |
2 | -0.001 -0.001 -0.001 -0.001
| 0.016 0.018 0.016 0.016
3 | -0.032 -0.032 -0.034 -0.034
| 0.016 0.018 0.016 0.016
4 | -0.064 -0.064 -0.067 -0.067
| 0.016 0.018 0.016 0.016
5 | -0.102 -0.102 -0.104 -0.104
| 0.017 0.019 0.017 0.017
|
_cons | 0.564 0.564 0.585 0.591 0.591 0.550 0.550
| 0.165 0.165 0.165 0.175 0.195 0.166 0.166
------------------------------------------------------------------------------------
legend: b/se
. estimates table FD_EstR, b(%7.3f) se
------------------------
Variable | FD_EstR
-------------+----------
w1 |
D1. | 0.394
| 0.026
w2 |
D1. | 0.028
| 0.011
w3 |
D1. | 0.203
| 0.019
w4 |
D1. | 0.089
| 0.016
q1 |
D1. | 0.106
| 0.012
q2 |
D1. | 0.377
| 0.013
q3 |
D1. | 0.095
| 0.018
q4 |
D1. | 0.054
| 0.007
q5 |
D1. | 0.287
| 0.016
t |
3 | -0.021
| 0.028
4 | -0.019
| 0.023
5 | -0.030
| 0.023
|
_cons | -0.010
| 0.017
------------------------
legend: b/se
. estimates table MLE_Est, b(%7.3f) se
------------------------
Variable | MLE_Est
-------------+----------
c |
w1 | 0.427
| 0.018
w2 | 0.033
| 0.008
w3 | 0.184
| 0.015
w4 | 0.089
| 0.012
q1 | 0.102
| 0.007
q2 | 0.375
| 0.007
q3 | 0.098
| 0.010
q4 | 0.054
| 0.004
q5 | 0.290
| 0.010
t |
2 | -0.001
| 0.016
3 | -0.034
| 0.016
4 | -0.067
| 0.016
5 | -0.104
| 0.016
|
_cons | 0.550
| 0.123
-------------+----------
sigma_u |
_cons | 0.047
| 0.010
-------------+----------
sigma_e |
_cons | 0.242
| 0.004
------------------------
legend: b/se