Questions regarding Quantile Regression combined with fixed effect using xtqreg and mmqreg

[Jun He]

Hi, everyone.

I am recently curious about using quantile regression model to detect effects from extreme observations.

So far, I have reviewed many posts and come up with the idea to do quantile regression with a fixed effect.

However, it seems that the results do not present any information about the R^2 or Within R^2 while I find some packages used in R like rqpd may show the result as well but I am not sure whether it functions correctly because it shares totally different result with those I get using xtqreg and mmqreg in stata.

First question: Whether my Stata code presented below is incorrect or not? I find that the coefficients are similar in both Stata codes but totally different from that in R.

bootstrap, cluster(companyid) idcluster(_company) reps(100) seed(1234): xtqreg pb ///
interest_dev lev_fin size liq_curr roa tangibility i.year if floating_exposure == 1, i(_company) ls q(0.25)

while

mmqreg qt ///
interest_dev $xlist i.year if floating_exposure == 1, q(25) abs(_company)

The R code is (if you are curious about):

fit.rqpd <- rqpd(QT ~ Interest_DEV + Liq + Curr + Lev_fin + ROA + Size + Tangibility |
Company, data = pdata, panel(lambda = 5, taus=c(0.1, 0.25, 0.5, 0.75, 0.9), tauw=rep(1/5, 5)), method = "pfe")

Second question: Whether it is possible to get R^2 or Within R^2 if I use xtqreg and mmqreg in Stata as my professor is curious about R^2 to see how well it fits.

[Joao Santos Silva]

Dear Jun He

First of all, please note that both xrqreg and mmqreg implement the estimator proposed by

Machado, J.A.F. and Santos Silva, J.M.C. (2019), Quantiles via Moments, Journal of Econometrics, 213(1), pp. 145–173.

and therefore their results should be essentially the same. In contrast, rqpd implements a different FE estimator, which was suggested by

Koenker, R. (2004). Quantile Regression for Longitudinal Data, Journal of Multivariate Analysis. 91-1. p. 74--89.

Please see the original papers to understand the difference between the two approaches, but in a nutshell Koenker's estimator imposes that the fixed effects have the same impact in all quantiles, while the estimator based on moments does not impose this restriction, but assumes a location-scale model.

Finally, the R2 does not make much sense in the context of quantile regression that is why it is not reported in xrqreg and mmqreg. If you want to compute the R2, you can simply compute it as the square of the correlation between the fitted values and the dependent variable.

Best wishes,

Joao

[Jun He]

Dear Joao Santos Silva

Thanks for your explicit explanation of the differences between the two types of codes and for your good references.

I will read the papers thoroughly. However, it seems that the estimator proposed by Machado & Santos caters to my requirement more.

I hope you have a nice day

Regards.

[Rupali Vashisht]

Dear Joao Santos Silva,

I am new to quantile regression - I am using the following mmqreg command to estimate my quantile coefficients:

mmqreg lnSPREAD ROA ICR GROWTH LEV ESG lnMKTCAP MATURITY_IN_YEARS RATING lnISSUESIZE i.t, abs(id) cluster(id) q(10 20 30 40 50 60 70 80 90)

I have a few questions:

1. I have data on 64 bonds for 6 years i.e. my N is larger than T. Is the mmqreg command suitable for data with large N and relatively small T?
2. I have specified time dummies for time fixed effects - is that the correct way to control for time fixed effects? Alternatively, can specifying abs(id t) to absorb time FEs along with bond FEs control for year-specific effects? Which way is more appropriate?
3. What are the advantages of using mmqreg over Koenker's (2004) method?

Thank you in advance!

Kind regards,
Rupali

[FernandoRios]

Hi Rubali
1. I think your panel is too short to use mmqreg or xtqreg. The coefficients may be very biased. One could apply the Jacknife correction suggested in Joao's paper, but that is a manual implementation. (not too difficult tho)
2. you can "add" time fixed effects both ways. THey are equivalent. In fact, if for some reason they produce different estimates, let me know. It could be a sign of multicolinearity with your variables, or a bug on my end. (mmqreg)
3. Koenker implementation is more flexible for control heterogeneity, but imposes homogeneity on the fixed effects. mmqreg/xtqreg imposes restrictions on the coefficients of controls but allows fixed effects to have an heterogenous effect on conditional quantile of the outcome.


Best wishes
Fernando

[Rupali Vashisht]

Dear FernandoRios,

Thank you very much for your response!

I understand why the mmqreg method might yield biased estimates.

I have a general question (not related to the mmqreg command) but it would be very helpful for me if you could guide me on this - I wanted to understand if dummy variables can be used to control for fixed effects in quantile regression in general? I ask this because effects are nonadditive in quantile regression. Also, are there any specific conditions that need to be met for dummy variables to successfully control for fixed effects?

Kind regards,
Rupali

[FernandoRios]

Roughly...when using Quantile regression via moments (mmqreg or xtqreg) you need around 20+ observations within each fixed effect. For example 20+ period in panel.
This number is just a rule of thumb. Not a hard rule.

If you use standard QREG you may need even more observations per Fixed effect. It could actually be an interesting exercise (if it has not been done before) to check how many observations one would really need to reduce the impact of the bias.

Again, these are my observations, not hard rules.

[Rupali Vashisht]

Dear FernandoRios,

Thank you very much for your help!

I tried implementing the jackknife technique mentioned in Joao's paper. Following is my code and the results:

 .bs, cluster(id) rep(20): xtqreg lnSPREAD ESG MATURITY_IN_YEARS RATING lnISSUESIZE ROA    ICR    GROWTH    LEV    lnMKTCAP    i.t
(running xtqreg on estimation sample)
WARNING: 1.9900498% of the fitted values of the scale function are not positive

Bootstrap replications (20): .........10.........20 done

Bootstrap results                                          Number of obs = 402
Replications  =  20

(Replications based on 67 clusters in id)

  

Observed
Bootstrap


Normal
based

lnSPREAD
coefficient
std. err.
z
P>z
[95% conf.
interval]









ESG
0.1042956
0.0347165
3
0.003
0.0362525
0.1723386

MATURITY_IN_YEARS
0
(omitted)





RATING
0
(omitted)





lnISSUESIZE
0
(omitted)





ROA
-0.0280936
0.0052645
-5.34
0
-0.0384119
-0.0177754

ICR
0.0120381
0.0040686
2.96
0.003
0.0040639
0.0200123

GROWTH
-0.0037824
0.0029164
-1.3
0.195
-0.0094984
0.0019335

LEV
0.0127927
0.0046965
2.72
0.006
0.0035877
0.0219976

lnMKTCAP
-0.0762047
0.0286668
-2.66
0.008
-0.1323905
-0.0200188









t







2
0.5513821
0.0278471
19.8
0
0.4968027
0.6059614

3
0.0436829
0.055554
0.79
0.432
-0.0652009
0.1525667

4
-0.0958948
0.0707094
-1.36
0.175
-0.2344827
0.0426932

5
-0.1909287
0.075549
-2.53
0.011
-0.339002
-0.0428553

6
0.1460688
0.0678321
2.15
0.031
0.0131204
0.2790172


mat V=e(V)

. matrix b=e(b_location)

. matrix g=e(b_scale)

. matrix Q50=e(q)

. scalar N=e(N)

. 
. xtqreg lnSPREAD ESG MATURITY_IN_YEARS RATING lnISSUESIZE ROA    ICR    GROWTH LEV lnMKTCAP    i.t    if    s==0     



MM-QR regression results
Number of obs = 199
WARNING: 2.5125628% of the fitted values of the scale function    are    not positive
.5 Quantile regression
  

Coefficient
Std. err.
z
P>z
[95% conf.
interval]









ESG
0.2515409
0.0613034
4.1
0
0.1313885
0.3716933

MATURITY_IN_YEARS
0
(omitted)





RATING
0
(omitted)





lnISSUESIZE
0
35103.32
0
1
-68801.24
68801.24

ROA
-0.0150663
0.0106696
-1.41
0.158
-0.0359782
0.0058457

ICR
0.0153956
0.0057658
2.67
0.008
0.0040948
0.0266963

GROWTH
-0.0096028
0.0037398
-2.57
0.01
-0.0169328
-0.0022729

LEV
0.0234749
0.0058527
4.01
0
0.0120038
0.0349461

lnMKTCAP
-0.1035034
0.0791106
-1.31
0.191
-0.2585574
0.0515505









t







2
0.5683364
0.0745813
7.62
0
0.4221596
0.7145131

3
-0.1054839
0.0995585
-1.06
0.289
-0.3006149
0.0896472

4
-0.2697122
0.120996
-2.23
0.026
-0.50686
-0.0325644

5
-0.3852656
0.1376914
-2.8
0.005
-0.6551357
-0.1153954

6
0.0487684
0.1160295
0.42
0.674
-0.1786452
0.2761819


scalar N0=e(N)

. matrix Q050=e(q)

. matrix g0=e(b_scale)

. 
. xtqreg lnSPREAD ESG MATURITY_IN_YEARS RATING lnISSUESIZE ROA    ICR    GROWTH LEV lnMKTCAP    i.t    if    s==1      



MM-QR regression results
Number of obs = 203
WARNING: 5.4187192% of the fitted values of the scale function    are    not positive
.5 Quantile regression

  

Coefficient
Std. err.
z
P>z
[95% conf.
interval]









ESG
0.1105789
0.0730302
1.51
0.13
-0.0325577
0.2537156

MATURITY_IN_YEARS
0
(omitted)





RATING
0
(omitted)





lnISSUESIZE
0
41616.34
0
1
-81566.53
81566.53

ROA
-0.0332894
0.012659
-2.63
0.009
-0.0581005
-0.0084782

ICR
0.0057916
0.0102622
0.56
0.573
-0.0143219
0.0259051

GROWTH
0.0003493
0.0021738
0.16
0.872
-0.0039112
0.0046098

LEV
0.0030016
0.0069255
0.43
0.665
-0.010572
0.0165752

lnMKTCAP
-0.0505127
0.0810601
-0.62
0.533
-0.2093876
0.1083621









t







2
0.4245002
0.1167092
3.64
0
0.1957545
0.653246

3
0.0396073
0.1040884
0.38
0.704
-0.1644022
0.2436168

4
-0.0852346
0.1371361
-0.62
0.534
-0.3540164
0.1835471

5
-0.21846
0.1430045
-1.53
0.127
-0.4987435
0.0618236

6
0.0738827
0.1550519
0.48
0.634
-0.2300134
0.3777788



scalar N1=e(N)

. matrix Q150=e(q)

. matrix g1=e(b_scale)

. mat bjk=b+(2*g-g1*(N1/N)-g0*(N0/N))*(2*Q50-Q150*(N1/N)-Q050*(N0/N)) 

. ereturn post bjk V

. ereturn display

  

Coefficient
Std. err.
z
P>z
[95% conf.
interval]









ESG
0.097211
0.0347165
2.8
0.005
0.0291679
0.1652541

MATURITY_IN_YEARS
0
(omitted)





RATING
0
(omitted)





lnISSUESIZE
0
(omitted)





ROA
-0.0281969
0.0052645
-5.36
0
-0.0385151
-0.0178786

ICR
0.0127317
0.0040686
3.13
0.002
0.0047574
0.0207059

GROWTH
-0.004097
0.0029164
-1.4
0.16
-0.0098129
0.001619

LEV
0.0134528
0.0046965
2.86
0.004
0.0042478
0.0226577

lnMKTCAP
-0.0741985
0.0286668
-2.59
0.01
-0.1303844
-0.0180127









t







2
0.5482974
0.0278471
19.69
0
0.493718
0.6028767

3
0.0493426
0.055554
0.89
0.374
-0.0595412
0.1582265

4
-0.0916401
0.0707094
-1.3
0.195
-0.230228
0.0469478

5
-0.1892661
0.075549
-2.51
0.012
-0.3373394
-0.0411927

6
0.1589342
0.0678321
2.34
0.019
0.0259857
0.2918826


.
end    of    do-file

I have the following questions about the results:

1. Does the warning 'WARNING: 1.9900498% of the fitted values of the scale function are not positive' indicate some problem with my data/suitability of the code?

2. I have 3 time-invariant variables namely, MATURITY-IN-YEARS, RATING, and, lnISSUESIZE - the first 2 drop (as expected) in the first and the second table, but the 3rd one doesn't. However, all the 3 drop in the last table. Is there any explanation for this observation?

3. In the code you provide (which I have modified according to my data), you mention 'xtset idcode' to specify that you are dealing with panel data - would specifying panel data as 'xtset idcode t' make a difference?

4. Does using the Jackknife technique eliminate the bias caused by a small T completely?

5. Finally, if I estimate the model for multiple quantiles, do you have a code handy that combines the final result?

Looking forward to your response,

Kind regards,
Rupali

[FernandoRios]

1. The warning refers to the assumtions of nonegative predicted Sigma. Not an issue perse, just something to be noticed
2. Exclude the time invariant variables by hand to avoid this
3. No, xtset idcode is sufficient. the time variable is not used for xtqreg (nor mmqreg)
4. JK does not eliminate the bias completely. (and does it only in multiple regression settings. With one case alone, may be better or worse than not using it)
5. No. usually built code like that as needed

[Rupali Vashisht]

Dear FernandoRios,

Thanks a lot for clearing up all my doubts - your responses are very helpful!

Kind regards,
Rupali

[Rupali Vashisht]

Dear FernandoRios,

I have one final question - if I run the analysis for each quantile separately (using your code), the resulting matrices should be correct right? i.e. I do not need to do it all in one single code? I am asking because I am struggling to amend the final part of the command where you combine the results of the entire sample and the sub-samples.

Thank you again for taking the time to answer my queries!

Kind regards,
Rupali

[Rupali Vashisht]

Dear Joao Santos Silva,

I was reading your paper (Quantile via Moments, 2019) and I have the following query regarding the model:

I understand from the model specification in (3) that the heterogeneity in the quantile process is due to the existence of a scale function that depends on a vector of regressors (X or Z, according to the notation in the paper). In Stata, I just introduce our set of regressors X for the conditional quantile process so can I assert that these regressors also affect the scale process? I am saying this because I find heterogeneity in the slope coefficient beta across taus so according to your model, this heterogeneity is due to the presence of a scale function that depends on X. Otherwise, the beta(tau) function should be constant. Is our understanding correct?

Any insights or guidance you could offer on this matter would be very helpful!

Many thanks in advance.

Kind regards,
Rupali

[Joao Santos Silva]

Dear Rupali Vashisht,

Your understanding is correct. If you use the ls option, you will see the location and scale estimates.

Best wishes,

Joao

[Rupali Vashisht]

Dear Joao Santos Silva,

Thank you very much for your response.

I also wanted to clarify if the model is constructed in a way that it imposes monotonicity of betas? I am asking this because I am getting parameters that are stricty increasing/decreasing with successive quantiles.

Thank you again for taking the time to answer my queries!

Kind regards,
Rupali

[Joao Santos Silva]

Dear Rupali Vashisht,

Yes, that is the case and it ensures that the quantiles do not cross.

Best wishes,

Joao