TSCS with Poisson Dependent Variable: choosing among ppml, nbreg, xinb, zip
Dear Statalisters,
I am currently working with a T dominant panel --time-series cross-section dataset-- that has N = 8 (the eight European countries) and T = 32 (32 quarters for each country). The dependent variable -bailout conditionality- follows a Poisson distribution, with excess of 0s (66.67%) and a lot of over-dispersion (with mean = 4.74 and standard deviation = 12.23). There are just 4 regressors, the one of interest and three controls. Overall, this means that, a priori, I can estimate my model using Negative Binomial Regression -nbreg-, Zero Inflated Negative Binomial -zinb-, Zero Inflated Poisson -zip- and Poisson Pseudo Maximum Likelihood, implemented as the user-written function -ppml-. The first thing I do is testing for potential problems of the data. Just let me point out that the data shows (a) contemporaneous autocorrelation, (b) panel heteroskedasticity, (c) endogeneity and (d) non-stationarity (luckily, though, errors seem to be serially independent). To meet the exogeneity assumption, all the covariates are conveniently lagged. To meet the stationarity assumption, those variables that show unit roots are first differentiated. Let me show you now an extract of my dataset (the initial missing values are due to lagging and differentiation).
I am currently running these models:
(1) Poisson Pseudo Maximum Likelihood with Fixed Effects (ppml)
(2) Poisson Pseudo Maximum Likelihood with Fixed Effects and Quadratic Time Trend (ppml)
(3) Negative Binomial with Fixed Effects (nbreg)
(4) Negative Binomial with Fixed Effects and Quadratic Time Trend (nbreg)
(5) Zero Inflated Negative Binomial with Fixed Effects (zinb)
(6) Zero Inflated Negative Binomial with Fixed Effects and Quadratic Time Trend (zinb)
(7) Zero Inflated Poisson with Fixed Effects (zip)
(8) Zero Inflated Poisson with Fixed Effects and Quadratic Time Trend (zip)
The output of each model can be seen here:
I am actually comparing the models as follows. I first run -zinb [...], zip vuong". Since both tests are positive and highly significant, -zinb- is preferred to -zip- or -nbreg-. Then, I test how ppml performs with -hpc-. Here, -ppml- seems to clearly beat -xtnbreg-, but it does not seem to perform better than the other three models that -xtnbreg- does beat. I think this is puzzling. The actual results of the tests are not provided. Let me now go with the actual questions.
(a) Do unit fixed effects dummies make sense or will any of the previous models suffer the incidental parameters bias?
(b) Is stationarity still an assumption to be met on these models? And strict exogeneity? In case they are, is my approach to solve them correct?
(c) Does robustness against general forms of heteroskedasticity protect against panel heteroskedasticity?
(d) Since there is no serial correlation but there is contemporaneous correlation, I am clustering on quarter_id (time) and not country_id (unit). Does it make any sense?
(e) In zero-inflated models, I am assuming that the process that leads an observation to be a "true" zero or not is the same that determines the count but dropping one regressor for theoretical reasons. To what extent is this valid? And, more importantly, what are the consequences of specifying the logit model as a single constant? And, finally, can inflate be a function of variables not included in the main model?
(f) Zero Inflated models assume two underlying processes generating the excess of zeroes. That is true in my case, but only 6% of the zeroes are not true zeroes. Then, knowing this and the results of the tests, should I still go for Zero Inflated models?
(g) Following the tests and all the information provided, which set of models should I consider the best?
(h) I understand when I may need to use nbreg, zip and zinb; but I do not see when ppml should be used. When is adequate to use -ppml-? Or, more sincerely, is it adequate to use ppml with my dataset? Why? This is actually a crucial question.
(i) In Zero Inflated models, how shall coefficients in the logit be interpreted? Do negative coefficients indicate higher or lower probability of being a true zero? Is it relevant if coefficients of the logit model are not significant at all?
(j) Am I missing any other potentially proper way of estimating the model?
Thank you all very much for your time.
Best,
Héctor.
I am currently working with a T dominant panel --time-series cross-section dataset-- that has N = 8 (the eight European countries) and T = 32 (32 quarters for each country). The dependent variable -bailout conditionality- follows a Poisson distribution, with excess of 0s (66.67%) and a lot of over-dispersion (with mean = 4.74 and standard deviation = 12.23). There are just 4 regressors, the one of interest and three controls. Overall, this means that, a priori, I can estimate my model using Negative Binomial Regression -nbreg-, Zero Inflated Negative Binomial -zinb-, Zero Inflated Poisson -zip- and Poisson Pseudo Maximum Likelihood, implemented as the user-written function -ppml-. The first thing I do is testing for potential problems of the data. Just let me point out that the data shows (a) contemporaneous autocorrelation, (b) panel heteroskedasticity, (c) endogeneity and (d) non-stationarity (luckily, though, errors seem to be serially independent). To meet the exogeneity assumption, all the covariates are conveniently lagged. To meet the stationarity assumption, those variables that show unit roots are first differentiated. Let me show you now an extract of my dataset (the initial missing values are due to lagging and differentiation).
Code:
"IRELAND" 1 "2007Q1" 1 2007 0 . . . . "IRELAND" 1 "2007Q2" 2 2007 0 . . . . "IRELAND" 1 "2007Q3" 3 2007 0 6.3 . . . "IRELAND" 1 "2007Q4" 4 2007 0 6.3 . . . "IRELAND" 1 "2008Q1" 5 2008 0 5.9 . . . "IRELAND" 1 "2008Q2" 6 2008 0 5.9 -.19999886 0 .4000001 "IRELAND" 1 "2008Q3" 7 2008 0 5.9 4 0 0 "IRELAND" 1 "2008Q4" 8 2008 0 5.9 -3.700001 0 0 "IRELAND" 1 "2009Q1" 9 2009 0 5.9 2.5 0 -.2000003 "IRELAND" 1 "2009Q2" 10 2009 0 5.9 4.800001 0 .4000001 "IRELAND" 1 "2009Q3" 11 2009 0 5.9 5.899998 0 .0999999 "IRELAND" 1 "2009Q4" 12 2009 0 5.9 5.300003 0 -.0999999 "IRELAND" 1 "2010Q1" 13 2010 0 5.9 6.599998 0 .9000001 "IRELAND" 1 "2010Q2" 14 2010 0 5.9 7.599998 -.5 0 "IRELAND" 1 "2010Q3" 15 2010 0 5.9 2.600002 0 -.4000001 "IRELAND" 1 "2010Q4" 16 2010 0 5.9 2.5999985 -1 -.2999997 "IRELAND" 1 "2011Q1" 17 2011 27 5.9 12.399998 0 -.10000038 "IRELAND" 1 "2011Q2" 18 2011 10 5.9 .20000458 0 .3000002 "IRELAND" 1 "2011Q3" 19 2011 11 5.5 9 0 .5999999 "IRELAND" 1 "2011Q4" 20 2011 14 5.5 3.4000015 -2 2.1 "IRELAND" 1 "2012Q1" 21 2012 5 5.5 6.599998 0 1.5 "IRELAND" 1 "2012Q2" 22 2012 4 5.5 .5999985 0 1.4000006 "IRELAND" 1 "2012Q3" 23 2012 3 5.5 16.300003 0 -.4000006 "IRELAND" 1 "2012Q4" 24 2012 9 5.5 -1.2000046 0 -1.8000002 "IRELAND" 1 "2013Q1" 25 2013 5 5.5 2.4000015 0 -1.1999998 "IRELAND" 1 "2013Q2" 26 2013 4 5.5 2.300003 0 -.1999998 "IRELAND" 1 "2013Q3" 27 2013 3 5.5 6 0 -1.1999998 "IRELAND" 1 "2013Q4" 28 2013 2 5.5 .2999954 -.5 -1.1000004 "IRELAND" 1 "2014Q1" 29 2014 0 5.5 4.0999985 0 -.7999997 "IRELAND" 1 "2014Q2" 30 2014 0 5.5 1 0 -.10000014 "IRELAND" 1 "2014Q3" 31 2014 0 5.5 -2.399994 0 .10000014 "IRELAND" 1 "2014Q4" 32 2014 0 5.5 -2.800003 0 -.3000002 "IRELAND" 1 "2015Q1" 33 2015 0 5.5 -1.199997 0 -.3999999 "IRELAND" 1 "2015Q2" 34 2015 0 5.5 -4.4000015 0 -.5 "IRELAND" 1 "2015Q3" 35 2015 0 5.5 -1.7000046 1 -.7 "IRELAND" 1 "2015Q4" 36 2015 0 5.5 -5.199997 0 -.5 "GREECE" 2 "2007Q1" 1 2007 0 . . . . [...]
(1) Poisson Pseudo Maximum Likelihood with Fixed Effects (ppml)
(2) Poisson Pseudo Maximum Likelihood with Fixed Effects and Quadratic Time Trend (ppml)
(3) Negative Binomial with Fixed Effects (nbreg)
(4) Negative Binomial with Fixed Effects and Quadratic Time Trend (nbreg)
(5) Zero Inflated Negative Binomial with Fixed Effects (zinb)
(6) Zero Inflated Negative Binomial with Fixed Effects and Quadratic Time Trend (zinb)
(7) Zero Inflated Poisson with Fixed Effects (zip)
(8) Zero Inflated Poisson with Fixed Effects and Quadratic Time Trend (zip)
The output of each model can be seen here:
Code:
(1)
. ppml bailout_conditionality_a A_government_partisanship M_sovereign_debt M_fitch_rating M_sovereign_
> interest_rate d_country_id1 d_country_id2 d_country_id3 d_country_id4 d_country_id5 d_country_id6 d_
> country_id7 d_country_id8, cluster(quarter_id)
note: checking the existence of the estimates
WARNING: M_sovereign_debt has very large values, consider rescaling or recentering
note: starting ppml estimation
[...]
Number of parameters: 13
Number of observations: 248
Number of observations dropped: 0
Pseudo log-likelihood: -1434.555
R-squared: .33843253
(Std. Err. adjusted for 31 clusters in quarter_id)
-------------------------------------------------------------------------------------------
| Robust
bailout_conditionality_a | Coef. Std. Err. z P>|z| [95% Conf. Interval]
--------------------------+----------------------------------------------------------------
A_government_partisanship | -.2232219 .1128043 -1.98 0.048 -.4443143 -.0021295
M_sovereign_debt | .0825423 .0369093 2.24 0.025 .0102014 .1548831
M_fitch_rating | -.533758 .1892423 -2.82 0.005 -.904666 -.1628499
M_sovereign_interest_rate | .2165422 .1079706 2.01 0.045 .0049237 .4281608
d_country_id1 | -1.136513 .3853649 -2.95 0.003 -1.891814 -.3812115
d_country_id2 | .3066859 .3146976 0.97 0.330 -.3101101 .9234819
d_country_id3 | -2.179942 .6242554 -3.49 0.000 -3.40346 -.9564239
d_country_id4 | -1.690297 .5361264 -3.15 0.002 -2.741086 -.639509
d_country_id5 | -.234904 .4186861 -0.56 0.575 -1.055514 .5857057
d_country_id6 | -1.737111 .3672441 -4.73 0.000 -2.456896 -1.017326
d_country_id7 | -.4952458 .507592 -0.98 0.329 -1.490108 .4996163
d_country_id8 | 0 (omitted)
_cons | 3.110228 .5331654 5.83 0.000 2.065243 4.155213
-------------------------------------------------------------------------------------------
Number of regressors dropped to ensure that the estimates exist: 0
Option strict is off
(2)
. ppml bailout_conditionality_a A_government_partisanship M_sovereign_debt M_fitch_rating M_sovereign_interest_rate quarter_id quarter_id2 d_country_id1 d_country_id2 d_country_id3 d_country_id4 d_country_
> id5 d_country_id6 d_country_id7 d_country_id8, cluster(quarter_id)
note: checking the existence of the estimates
WARNING: M_sovereign_debt has very large values, consider rescaling or recentering
WARNING: quarter_id has very large values, consider rescaling or recentering
WARNING: quarter_id2 has very large values, consider rescaling or recentering
note: starting ppml estimation
[...]
Number of parameters: 15
Number of observations: 248
Number of observations dropped: 0
Pseudo log-likelihood: -1315.9455
R-squared: .37590262
(Std. Err. adjusted for 31 clusters in quarter_id)
-------------------------------------------------------------------------------------------
| Robust
bailout_conditionality_a | Coef. Std. Err. z P>|z| [95% Conf. Interval]
--------------------------+----------------------------------------------------------------
A_government_partisanship | -.2363886 .1331831 -1.77 0.076 -.4974226 .0246455
M_sovereign_debt | .0503378 .0229217 2.20 0.028 .0054121 .0952634
M_fitch_rating | -.3278799 .1733408 -1.89 0.059 -.6676216 .0118619
M_sovereign_interest_rate | .1765052 .0960089 1.84 0.066 -.0116688 .3646793
quarter_id | .3508606 .1065146 3.29 0.001 .1420959 .5596253
quarter_id2 | -.0079596 .0027167 -2.93 0.003 -.0132841 -.002635
d_country_id1 | -1.066567 .4007632 -2.66 0.008 -1.852049 -.2810858
d_country_id2 | .3761135 .340171 1.11 0.269 -.2906093 1.042836
d_country_id3 | -2.159306 .6578863 -3.28 0.001 -3.448739 -.8698722
d_country_id4 | -1.779881 .5969538 -2.98 0.003 -2.949889 -.6098734
d_country_id5 | -.1030793 .5012644 -0.21 0.837 -1.08554 .8793809
d_country_id6 | -1.62356 .3953631 -4.11 0.000 -2.398457 -.8486624
d_country_id7 | -.4166418 .5191953 -0.80 0.422 -1.434246 .6009622
d_country_id8 | 0 (omitted)
_cons | -.1420932 1.027489 -0.14 0.890 -2.155935 1.871748
-------------------------------------------------------------------------------------------
Number of regressors dropped to ensure that the estimates exist: 0
Option strict is off
(3)
. zinb bailout_conditionality_a A_government_partisanship M_sovereign_debt M_fitch_rating M_sovereign_interest_rate d_country_id1 d_country_id2 d_country_id3 d_country_id4 d_country_id5 d_country_id6 d_country_id7 d_country_id8, inflate(M_sovereign_deb
> t M_fitch_rating M_sovereign_interest_rate) cluster(quarter_id)
[...]
Zero-inflated negative binomial regression Number of obs = 248
Nonzero obs = 96
Zero obs = 152
Inflation model = logit Wald chi2(11) = 71.86
Log pseudolikelihood = -479.0765 Prob > chi2 = 0.0000
(Std. Err. adjusted for 31 clusters in quarter_id)
-------------------------------------------------------------------------------------------
| Robust
bailout_conditionality_a | Coef. Std. Err. z P>|z| [95% Conf. Interval]
--------------------------+----------------------------------------------------------------
bailout_conditionality_a |
A_government_partisanship | -.3144175 .1103015 -2.85 0.004 -.5306045 -.0982305
M_sovereign_debt | .030327 .0150157 2.02 0.043 .0008969 .0597572
M_fitch_rating | -.2300645 .1684376 -1.37 0.172 -.5601962 .1000672
M_sovereign_interest_rate | .1404795 .0656083 2.14 0.032 .0118896 .2690693
d_country_id1 | -.5401905 .3440433 -1.57 0.116 -1.214503 .1341219
d_country_id2 | .346963 .2893588 1.20 0.230 -.2201698 .9140958
d_country_id3 | -.0407952 .7554484 -0.05 0.957 -1.521447 1.439856
d_country_id4 | .0036681 .4887628 0.01 0.994 -.9542893 .9616255
d_country_id5 | .3373251 .3641198 0.93 0.354 -.3763366 1.050987
d_country_id6 | -1.420092 .3754115 -3.78 0.000 -2.155885 -.6842989
d_country_id7 | .1183418 .3919655 0.30 0.763 -.6498964 .88658
d_country_id8 | 0 (omitted)
_cons | 4.102541 .5358301 7.66 0.000 3.052333 5.152748
--------------------------+----------------------------------------------------------------
inflate |
M_sovereign_debt | -.118818 .0815607 -1.46 0.145 -.2786741 .0410381
M_fitch_rating | .9684598 .5973118 1.62 0.105 -.2022497 2.139169
M_sovereign_interest_rate | -.110943 .1953075 -0.57 0.570 -.4937386 .2718526
_cons | .6802068 .2684972 2.53 0.011 .1539619 1.206452
--------------------------+----------------------------------------------------------------
/lnalpha | -.6217049 .1756533 -3.54 0.000 -.965979 -.2774308
--------------------------+----------------------------------------------------------------
alpha | .5370281 .0943307 .3806104 .757728
-------------------------------------------------------------------------------------------
(4)
. zinb bailout_conditionality_a A_government_partisanship M_sovereign_debt M_fitch_rating M_sovereign_interest_rate quarter_id quarter_id2 d_country_id1 d_country_id2 d_country_id3 d_country_id4 d_country_id5 d_country_id6 d_country_id7 d_country_id8,
> inflate(M_sovereign_debt M_fitch_rating M_sovereign_interest_rate) cluster(quarter_id)
[...]
Zero-inflated negative binomial regression Number of obs = 248
Nonzero obs = 96
Zero obs = 152
Inflation model = logit Wald chi2(13) = 87.91
Log pseudolikelihood = -463.4255 Prob > chi2 = 0.0000
(Std. Err. adjusted for 31 clusters in quarter_id)
-------------------------------------------------------------------------------------------
| Robust
bailout_conditionality_a | Coef. Std. Err. z P>|z| [95% Conf. Interval]
--------------------------+----------------------------------------------------------------
bailout_conditionality_a |
A_government_partisanship | -.3946015 .1213576 -3.25 0.001 -.632458 -.1567451
M_sovereign_debt | .0119484 .0144604 0.83 0.409 -.0163934 .0402902
M_fitch_rating | -.0653695 .1739451 -0.38 0.707 -.4062957 .2755567
M_sovereign_interest_rate | .0989751 .0421587 2.35 0.019 .0163455 .1816048
quarter_id | -.1534731 .089372 -1.72 0.086 -.3286389 .0216928
quarter_id2 | .0012115 .0018029 0.67 0.502 -.0023222 .0047452
d_country_id1 | -.3765907 .2352637 -1.60 0.109 -.8376991 .0845176
d_country_id2 | .6977724 .2503369 2.79 0.005 .2071212 1.188424
d_country_id3 | .5658847 .6195751 0.91 0.361 -.6484602 1.78023
d_country_id4 | 1.143576 .571354 2.00 0.045 .0237426 2.263409
d_country_id5 | -.1220788 .4171782 -0.29 0.770 -.939733 .6955754
d_country_id6 | -2.535287 .4798754 -5.28 0.000 -3.475825 -1.594748
d_country_id7 | .1807495 .3298403 0.55 0.584 -.4657256 .8272246
d_country_id8 | 0 (omitted)
_cons | 7.248888 1.15053 6.30 0.000 4.99389 9.503885
--------------------------+----------------------------------------------------------------
inflate |
M_sovereign_debt | -.103604 .0830426 -1.25 0.212 -.2663645 .0591566
M_fitch_rating | .951205 .5824234 1.63 0.102 -.1903239 2.092734
M_sovereign_interest_rate | -.0922096 .1861618 -0.50 0.620 -.4570801 .2726609
_cons | .570622 .3036781 1.88 0.060 -.0245761 1.16582
--------------------------+----------------------------------------------------------------
/lnalpha | -.8812492 .1682219 -5.24 0.000 -1.210958 -.5515403
--------------------------+----------------------------------------------------------------
alpha | .4142651 .0696885 .2979117 .5760618
-------------------------------------------------------------------------------------------
(5)
. zip bailout_conditionality_a A_government_partisanship M_sovereign_debt M_fitch_rating M_sovereign_interest_rate i.country_id, inflate(M_sovereign_debt M_fitch_rating M_sovereign_interest_rate) cluster(quarter_id)
[...]
Zero-inflated Poisson regression Number of obs = 248
Nonzero obs = 96
Zero obs = 152
Inflation model = logit Wald chi2(11) = 60.45
Log pseudolikelihood = -745.2113 Prob > chi2 = 0.0000
(Std. Err. adjusted for 31 clusters in quarter_id)
-------------------------------------------------------------------------------------------
| Robust
bailout_conditionality_a | Coef. Std. Err. z P>|z| [95% Conf. Interval]
--------------------------+----------------------------------------------------------------
bailout_conditionality_a |
A_government_partisanship | -.25843 .0597643 -4.32 0.000 -.3755659 -.141294
M_sovereign_debt | .0314059 .0208239 1.51 0.132 -.0094081 .07222
M_fitch_rating | -.2845093 .1639349 -1.74 0.083 -.6058158 .0367972
M_sovereign_interest_rate | .1366641 .0673081 2.03 0.042 .0047427 .2685854
|
country_id |
2 | .9922899 .3678644 2.70 0.007 .2712889 1.713291
3 | .3631831 .5408212 0.67 0.502 -.696807 1.423173
4 | .2537689 .3312795 0.77 0.444 -.3955269 .9030647
5 | .8156418 .3039924 2.68 0.007 .2198275 1.411456
6 | -.568403 .2822637 -2.01 0.044 -1.12163 -.0151764
7 | .8606325 .4337827 1.98 0.047 .0104341 1.710831
8 | .4228438 .3118131 1.36 0.175 -.1882987 1.033986
|
_cons | 3.261526 .3712896 8.78 0.000 2.533811 3.98924
--------------------------+----------------------------------------------------------------
inflate |
M_sovereign_debt | -.1220591 .0776563 -1.57 0.116 -.2742627 .0301445
M_fitch_rating | .9690261 .5705194 1.70 0.089 -.1491714 2.087224
M_sovereign_interest_rate | -.1190887 .1970787 -0.60 0.546 -.5053559 .2671784
_cons | .7676975 .2493916 3.08 0.002 .2788989 1.256496
-------------------------------------------------------------------------------------------
(6)
. zip bailout_conditionality_a A_government_partisanship M_sovereign_debt M_fitch_rating M_sovereign_interest_rate quarter_id quarter_id2 i.country_id, inflate(M_sovereign_debt M_fitch_rating M_sovereign_interest_rate) cluster(quarter_id)
[...]
Zero-inflated Poisson regression Number of obs = 248
Nonzero obs = 96
Zero obs = 152
Inflation model = logit Wald chi2(13) = 93.75
Log pseudolikelihood = -708.4248 Prob > chi2 = 0.0000
(Std. Err. adjusted for 31 clusters in quarter_id)
-------------------------------------------------------------------------------------------
| Robust
bailout_conditionality_a | Coef. Std. Err. z P>|z| [95% Conf. Interval]
--------------------------+----------------------------------------------------------------
bailout_conditionality_a |
A_government_partisanship | -.2488389 .062078 -4.01 0.000 -.3705095 -.1271683
M_sovereign_debt | .0232429 .0198515 1.17 0.242 -.0156653 .0621511
M_fitch_rating | -.1942654 .1905968 -1.02 0.308 -.5678282 .1792974
M_sovereign_interest_rate | .1095538 .0703574 1.56 0.119 -.0283441 .2474517
quarter_id | -.0727754 .0880591 -0.83 0.409 -.2453681 .0998172
quarter_id2 | .0005671 .001943 0.29 0.770 -.0032411 .0043753
|
country_id |
2 | 1.00438 .3742975 2.68 0.007 .27077 1.737989
3 | .5049016 .5083995 0.99 0.321 -.491543 1.501346
4 | .5607791 .3410848 1.64 0.100 -.1077347 1.229293
5 | .3959234 .3427282 1.16 0.248 -.2758116 1.067658
6 | -1.133306 .3998859 -2.83 0.005 -1.917068 -.3495439
7 | .796642 .4162661 1.91 0.056 -.0192245 1.612508
8 | .4218333 .3104565 1.36 0.174 -.1866503 1.030317
|
_cons | 4.604716 1.164914 3.95 0.000 2.321526 6.887905
--------------------------+----------------------------------------------------------------
inflate |
M_sovereign_debt | -.1179454 .0782564 -1.51 0.132 -.2713252 .0354344
M_fitch_rating | .9599652 .5687244 1.69 0.091 -.1547141 2.074645
M_sovereign_interest_rate | -.1149118 .1946103 -0.59 0.555 -.496341 .2665173
_cons | .7311145 .2582689 2.83 0.005 .2249169 1.237312
-------------------------------------------------------------------------------------------
(7)
. nbreg bailout_conditionality_a A_government_partisanship M_sovereign_debt M_fitch_rating M_sovereign_interest_rate d_country_id1 d_country_id2 d_country_id3 d_country_id4 d_country_id5 d_country_id6 d_country_id7 d_country_id8, cluster(quarter_id)
[...]
Negative binomial regression Number of obs = 248
Wald chi2(11) = 98.29
Dispersion = mean Prob > chi2 = 0.0000
Log pseudolikelihood = -506.86994 Pseudo R2 = 0.0398
(Std. Err. adjusted for 31 clusters in quarter_id)
-------------------------------------------------------------------------------------------
| Robust
bailout_conditionality_a | Coef. Std. Err. z P>|z| [95% Conf. Interval]
--------------------------+----------------------------------------------------------------
A_government_partisanship | .0429828 .1050286 0.41 0.682 -.1628696 .2488351
M_sovereign_debt | .0841299 .021235 3.96 0.000 .0425102 .1257497
M_fitch_rating | -.778131 .2938107 -2.65 0.008 -1.353989 -.2022725
M_sovereign_interest_rate | .2651588 .0659782 4.02 0.000 .1358439 .3944737
d_country_id1 | -1.596139 .4241074 -3.76 0.000 -2.427374 -.7649036
d_country_id2 | -.0086451 .3706302 -0.02 0.981 -.735067 .7177768
d_country_id3 | -2.839683 .7393523 -3.84 0.000 -4.288787 -1.390579
d_country_id4 | -1.72478 .4915958 -3.51 0.000 -2.688291 -.7612704
d_country_id5 | -1.117178 .3758329 -2.97 0.003 -1.853797 -.3805592
d_country_id6 | -1.800344 .5819865 -3.09 0.002 -2.941016 -.6596713
d_country_id7 | -.9247609 .5242241 -1.76 0.078 -1.952221 .1026995
d_country_id8 | 0 (omitted)
_cons | 1.997088 .5753089 3.47 0.001 .8695037 3.124673
--------------------------+----------------------------------------------------------------
/lnalpha | 1.609591 .1554772 1.304862 1.914321
--------------------------+----------------------------------------------------------------
alpha | 5.000767 .7775052 3.687179 6.782332
-------------------------------------------------------------------------------------------
(8)
. nbreg bailout_conditionality_a A_government_partisanship M_sovereign_debt M_fitch_rating M_sovereign_interest_rate quarter_id quarter_id2 d_country_id1 d_country_id2 d_country_id3 d_country_id4 d_country_id5 d_country_id6 d_country_id7 d_country_id8,
> cluster(quarter_id)
[...]
Negative binomial regression Number of obs = 248
Wald chi2(13) = 218.90
Dispersion = mean Prob > chi2 = 0.0000
Log pseudolikelihood = -495.7899 Pseudo R2 = 0.0608
(Std. Err. adjusted for 31 clusters in quarter_id)
-------------------------------------------------------------------------------------------
| Robust
bailout_conditionality_a | Coef. Std. Err. z P>|z| [95% Conf. Interval]
--------------------------+----------------------------------------------------------------
A_government_partisanship | -.0057892 .0925691 -0.06 0.950 -.1872214 .1756429
M_sovereign_debt | .0781654 .0299451 2.61 0.009 .019474 .1368569
M_fitch_rating | -.4685302 .2934263 -1.60 0.110 -1.043635 .1065748
M_sovereign_interest_rate | .3515901 .0900972 3.90 0.000 .175003 .5281773
quarter_id | .5686942 .1370709 4.15 0.000 .3000402 .8373483
quarter_id2 | -.0124353 .0027232 -4.57 0.000 -.0177728 -.0070978
d_country_id1 | -1.883564 .4754332 -3.96 0.000 -2.815396 -.9517326
d_country_id2 | .2999586 .3884831 0.77 0.440 -.4614543 1.061372
d_country_id3 | -3.043286 .7101179 -4.29 0.000 -4.435092 -1.651481
d_country_id4 | -1.274951 .5938348 -2.15 0.032 -2.438846 -.1110561
d_country_id5 | -.7350396 .5716031 -1.29 0.198 -1.855361 .3852819
d_country_id6 | -1.087217 .7629414 -1.43 0.154 -2.582554 .4081208
d_country_id7 | -1.343735 .5463081 -2.46 0.014 -2.414479 -.272991
d_country_id8 | 0 (omitted)
_cons | -3.52121 1.617225 -2.18 0.029 -6.690913 -.351506
--------------------------+----------------------------------------------------------------
/lnalpha | 1.460274 .1250615 1.215158 1.70539
--------------------------+----------------------------------------------------------------
alpha | 4.307139 .5386572 3.370826 5.503531
-------------------------------------------------------------------------------------------
(a) Do unit fixed effects dummies make sense or will any of the previous models suffer the incidental parameters bias?
(b) Is stationarity still an assumption to be met on these models? And strict exogeneity? In case they are, is my approach to solve them correct?
(c) Does robustness against general forms of heteroskedasticity protect against panel heteroskedasticity?
(d) Since there is no serial correlation but there is contemporaneous correlation, I am clustering on quarter_id (time) and not country_id (unit). Does it make any sense?
(e) In zero-inflated models, I am assuming that the process that leads an observation to be a "true" zero or not is the same that determines the count but dropping one regressor for theoretical reasons. To what extent is this valid? And, more importantly, what are the consequences of specifying the logit model as a single constant? And, finally, can inflate be a function of variables not included in the main model?
(f) Zero Inflated models assume two underlying processes generating the excess of zeroes. That is true in my case, but only 6% of the zeroes are not true zeroes. Then, knowing this and the results of the tests, should I still go for Zero Inflated models?
(g) Following the tests and all the information provided, which set of models should I consider the best?
(h) I understand when I may need to use nbreg, zip and zinb; but I do not see when ppml should be used. When is adequate to use -ppml-? Or, more sincerely, is it adequate to use ppml with my dataset? Why? This is actually a crucial question.
(i) In Zero Inflated models, how shall coefficients in the logit be interpreted? Do negative coefficients indicate higher or lower probability of being a true zero? Is it relevant if coefficients of the logit model are not significant at all?
(j) Am I missing any other potentially proper way of estimating the model?
Thank you all very much for your time.
Best,
Héctor.
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