hausman test

[Alex Mueller]

Hi al,

I am running a panel model on the effect of promotion on working performance.
I tried to implement a hausman test for deciding if i shall use fe or re:

 xtreg (variables), fe
ewstimates store fe
xtreg (variables), re
estimates store re

delivers the following output:

   Test:  Ho:  difference in coefficients not systematic

                 chi2(23) = (b-B)'[(V_b-V_B)^(-1)](b-B)
                          =    -4.65    chi2<0 ==> model fitted on these
                                        data fails to meet the asymptotic
                                        assumptions of the Hausman test;
                                        see suest for a generalized test

Does anyone have experience with this problem? I fail to understand why the test will not work.

Best, Alexander-Florian

[Carlo Lazzaro]

Alex:
it's a really frequent nuisance of the -hausman- test (with a limited sample, you cannot take for granted that VCE Matrix is positive definite).
A fix in point is to add the -sigmamore- option available with -hausman-.
If what above won't do the trick, you can consider the user-written programme -xtoverid-

[Alex Mueller]

Carlo,

hello again and welcome to my second post

Thanks for your hints,

the sigmamore option delivers

 Test:  Ho:  difference in coefficients not systematic

                 chi2(23) = (b-B)'[(V_b-V_B)^(-1)](b-B)
                          =        1.35
                Prob>chi2 =      1.0000

the p-value of 1.000 makes me a little suspicious (as it seems "too high")
Does this simply imply it stongly recommends the random effects model? Or points the 1.0000 out that there is a mistake somewhere?

Cheers, Alex

[Carlo Lazzaro]

Alex:
try -xtoverid- and see if it also recommends -re- specification

[Alex Mueller]

Carlo,

xtoverid doesnt work as it delivers:

 cAge#c:  operator invalid

I searched in earlier postst and saw that some facing the same problem concluded xtoverid cant handle factor variables. Is this still valid? And how would i deal with a assumed squared relationship of Age (c.Age##c.Age) and i.year

Cheers, Alex

[Alex Mueller]

Update:
carlo, I meanwhile found your solution.
So my Code is now like:

 xi: xtreg performance promoted_in_observation_period Age Age_squared (...) i.year, re robust

delivering:

Test of overidentifying restrictions: fixed vs random effects
Cross-section time-series model: xtreg re  robust cluster(ID)
Sargan-Hansen statistic  42.372  Chi-sq(23)   P-value = 0.0082

So am I fine with FE modell then? If i dont use the robust option in the 1. Code the p-value gets > 0.05. In this case RE would be better..

Cheers, Alex

[Carlo Lazzaro]

Alex:
- glad you have found the -xi:- prefix workaround yourself;
- robust/cluster standard errors can obviously impact on -xtoverid- oucome: if you detect evidence of heteroskedasticity and/or autocorrelation in your dataset invoking these options sounds wise;
- you asre correct in interpreting the results of -xtoverid- for robust/cluster and defaulst standard errors;
- by the way, with non-default standard errors, -hausman- is unfeasible, nor adopting cluster/robust standard errors after -hausman- test is correct.

[Alex Mueller]

Carlo, many thanks for your helpful comments!

To be honest I dont fully understand your last sidenote:

- by the way, with non-default standard errors, -hausman- is unfeasible, nor adopting cluster/robust standard errors after -hausman- test is correct.

Does that mean Hausman is not possible with heteroskedasticic data? Nevertheless, my -xtoverid- output stays unaffected of this, right?

Additionally, 2 issues arise from that conclusion:

1) Now I want to analyze whether my dataset suffers from heteroskedasticity.
Since I am working on a fe panel model the hettest command is not possible.
During my research how to test for heteroskedasticity I stumbled over user-written xttest2 and xttest3. Xttest2 doesnt work, but xttest3 does.
So after running my regression

 xtreg performane promoted_in_observation_period (…) c.Age##c.Age i.year, fe

xttest3 delivers

 H0: sigma(i)^2 = sigma^2 for all i
chi2 (195)  =   3.0e+34
Prob>chi2 =      0.0000

implying that null hypothesis of homoscedasticity is rejected strongly.

Do I have a massive heteroscedasticity problem now?
is

 xtreg (…) , fe robust

“enough” to deal with that strong heteroskedasticity?

2) I used the “ ,fe robust” option for the -xtoverid- in order to check whether to use fe or re (see above).
Isn’t it somehow circular to conclude to use the fe model, from testing a heteroskedasticity-corrected model, and afterwards check whether heteroscedasticity is a problem at all?
But if I go the other way round I don’t know whether I can use the xttest3 command, as I didn’t check firstly whether I can use a fe model rather than a re. (xtoverid)
I hope it is clear, what I mean

Cheers, Alex

[Carlo Lazzaro]

Alex:
1) -hausman- test allows default standard errors only. Hence, if you suspect heteroskedasticity and/or autocorrelation with you -xtreg- suitable data you should invoke robust/cluster standard error, which points you directly to -xtoverid- to choose between the -fe-and -re- specification.
2) if you have heteroskeadsticity and/or autocorrelation and you wisely invoke robust/cluster standard errors, there's no scope in re-testing if your data suffers from heteroskeadsticity and/or autocorrelation.

[Alex Mueller]

Carlo,
thanks for the reply.

1) I see. But given hausman can deal only with default std.err. and xtoverid can deal with non-default std.err., is there any point in running xtoverid without the , fe robust option?
2) Can I simply assume i have heteroskedasticity in my data set and therefore use the , robust option by default for xtreg as well as for xtoverid? That sounds somehow not really academic unassailable to me?

Thx, Alex

[Carlo Lazzaro]

Alex:
1) as per -xtoverid- help file, you can run this test with or without the cluster/robust option. If you invoke default standard errors, the results are the same as the ones reported by -hausman-, as you can see from the following toy-example:

. use "http://www.stata-press.com/data/r15/nlswork.dta"
(National Longitudinal Survey.  Young Women 14-26 years of age in 1968)

. xtreg ln_wage age, re

Random-effects GLS regression                   Number of obs     =     28,510
Group variable: idcode                          Number of groups  =      4,710

R-sq:                                           Obs per group:
     within  = 0.1026                                         min =          1
     between = 0.0877                                         avg =        6.1
     overall = 0.0774                                         max =         15

                                                Wald chi2(1)      =    3140.35
corr(u_i, X)   = 0 (assumed)                    Prob > chi2       =     0.0000

------------------------------------------------------------------------------
     ln_wage |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         age |   .0185667   .0003313    56.04   0.000     .0179174    .0192161
       _cons |   1.120439   .0112038   100.01   0.000      1.09848    1.142398
-------------+----------------------------------------------------------------
     sigma_u |  .36972456
     sigma_e |  .30349389
         rho |  .59743613   (fraction of variance due to u_i)
------------------------------------------------------------------------------

. xtoverid

Test of overidentifying restrictions: fixed vs random effects
Cross-section time-series model: xtreg re  
Sargan-Hansen statistic  17.401  Chi-sq(1)    P-value = 0.0000

. xtreg ln_wage age, fe

Fixed-effects (within) regression               Number of obs     =     28,510
Group variable: idcode                          Number of groups  =      4,710

R-sq:                                           Obs per group:
     within  = 0.1026                                         min =          1
     between = 0.0877                                         avg =        6.1
     overall = 0.0774                                         max =         15

                                                F(1,23799)        =    2720.20
corr(u_i, Xb)  = 0.0314                         Prob > F          =     0.0000

------------------------------------------------------------------------------
     ln_wage |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         age |   .0181349   .0003477    52.16   0.000     .0174534    .0188164
       _cons |   1.148214   .0102579   111.93   0.000     1.128107     1.16832
-------------+----------------------------------------------------------------
     sigma_u |  .40635023
     sigma_e |  .30349389
         rho |  .64192015   (fraction of variance due to u_i)
------------------------------------------------------------------------------
F test that all u_i=0: F(4709, 23799) = 8.81                 Prob > F = 0.0000

. estimates store fe

. xtreg ln_wage age, re

Random-effects GLS regression                   Number of obs     =     28,510
Group variable: idcode                          Number of groups  =      4,710

R-sq:                                           Obs per group:
     within  = 0.1026                                         min =          1
     between = 0.0877                                         avg =        6.1
     overall = 0.0774                                         max =         15

                                                Wald chi2(1)      =    3140.35
corr(u_i, X)   = 0 (assumed)                    Prob > chi2       =     0.0000

------------------------------------------------------------------------------
     ln_wage |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         age |   .0185667   .0003313    56.04   0.000     .0179174    .0192161
       _cons |   1.120439   .0112038   100.01   0.000      1.09848    1.142398
-------------+----------------------------------------------------------------
     sigma_u |  .36972456
     sigma_e |  .30349389
         rho |  .59743613   (fraction of variance due to u_i)
------------------------------------------------------------------------------

. estimates store re

. hausman fe re

                 ---- Coefficients ----
             |      (b)          (B)            (b-B)     sqrt(diag(V_b-V_B))
             |       fe           re         Difference          S.E.
-------------+----------------------------------------------------------------
         age |    .0181349     .0185667       -.0004318        .0001055
------------------------------------------------------------------------------
                           b = consistent under Ho and Ha; obtained from xtreg
            B = inconsistent under Ha, efficient under Ho; obtained from xtreg

    Test:  Ho:  difference in coefficients not systematic

                  chi2(1) = (b-B)'[(V_b-V_B)^(-1)](b-B)
                          =       16.76
                Prob>chi2 =      0.0000

.

2) the issue with your suggested approach is that -xtreg- allows cluster/robust (put differently, you cannot choose between robust and cluster option, like you do under -regress-) standard errors, which may be biased (in fact, more biased that default standard errors) if the number of clusters is too small. Hence, researcher's judgement call is unavodable.

[Alex Mueller]

Carlo,
thanks for the clarification.
But how can I justify the use of xtreg ..., fe robust option the without testing for heteroskedasticity at all?
I am still a little confused why I would not use the xttest3.

Best, Alex

[Carlo Lazzaro]

Alex:
you can also visually investigate the residual distribution.
Hence, you are not forced to use -xttest3-.
However, if you decide to stick with -xttest3- you have two possible outcomes:
1) your residual distribution is homoskedastic; no need to re-run -xtreg,fe- with cluster/robust standard errors; run -xtreg,re- with default standard errors; test via -hausman- which specification fits your data better;
2 your residual distribution is heterooskedastic; you should re-run -xtreg,fe- with cluster/robust standard errors; run -xtreg,re- with cluster/robust standard errors; test via -xtoverid- which specification fits your data better.

[Alex Mueller]

Carlo,

thanks again.

1) For the visual inspection I stumbled over

1. predict s1, xb
2. predict s2, residual
3. scatter s2 s1

Is this correct for my setting with my fe panel data? It confuses me somehow, as i would expect the residuals on the y-axis and the years of my panel (i.e. 2000-2015) on the x axis.

2) Thanks a lot, Carlo. This is very helpful!
Linked with Heteroskedasticity there is the issue of Autocorrelation. As I understand from literature and prior posts the - , robust - option can control for both under -xtreg-.
Does this imply when I act as discussed above I dont need to worry about autocorrelation (given I have heteroskedasticity)? Because if AC occurs its corrected for and if no AC occurs it is no problem anyway.

Cheers, A.

[Carlo Lazzaro]

Alex:
1) visual inspection:

. use "http://www.stata-press.com/data/r15/nlswork.dta"
(National Longitudinal Survey.  Young Women 14-26 years of age in 1968)

. xtreg ln_wage age, fe

Fixed-effects (within) regression               Number of obs     =     28,510
Group variable: idcode                          Number of groups  =      4,710

R-sq:                                           Obs per group:
     within  = 0.1026                                         min =          1
     between = 0.0877                                         avg =        6.1
     overall = 0.0774                                         max =         15

                                                F(1,23799)        =    2720.20
corr(u_i, Xb)  = 0.0314                         Prob > F          =     0.0000

------------------------------------------------------------------------------
     ln_wage |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         age |   .0181349   .0003477    52.16   0.000     .0174534    .0188164
       _cons |   1.148214   .0102579   111.93   0.000     1.128107     1.16832
-------------+----------------------------------------------------------------
     sigma_u |  .40635023
     sigma_e |  .30349389
         rho |  .64192015   (fraction of variance due to u_i)
------------------------------------------------------------------------------
F test that all u_i=0: F(4709, 23799) = 8.81                 Prob > F = 0.0000

. predict res, e
(24 missing values generated)

. histogram res
(bin=44, start=-1.9891924, width=.11140382)

.

2) with the robust/cluster options you do not have to worry about heteroskedasticity and/or autocorrelation under -xtreg-.