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st: RE: RE: RE: RE: Interpreting Kleibergen Paap weak instrument statistic
From
"Schaffer, Mark E" <[email protected]>
To
<[email protected]>
Subject
st: RE: RE: RE: RE: Interpreting Kleibergen Paap weak instrument statistic
Date
Mon, 25 Jun 2012 18:08:21 +0100
James,
> -----Original Message-----
> From: [email protected]
> [mailto:[email protected]] On Behalf Of
> Fitzgerald, James
> Sent: 25 June 2012 16:50
> To: [email protected]
> Subject: st: RE: RE: RE: Interpreting Kleibergen Paap weak
> instrument statistic
>
> Mark,
>
> ________________________________________
> From: [email protected]
> [[email protected]] on behalf of Schaffer,
> Mark E [[email protected]]
> Sent: 25 June 2012 15:54
> To: [email protected]
> Subject: st: RE: RE: Interpreting Kleibergen Paap weak
> instrument statistic
>
> James,
>
> > -----Original Message-----
> > From: [email protected]
> > [mailto:[email protected]] On Behalf Of
> Fitzgerald,
>
> > James
> > Sent: 25 June 2012 14:53
> > To: [email protected]
> > Subject: st: RE: RE: Interpreting Kleibergen Paap weak instrument
> > statistic
> >
> > Mark,
> >
> > Thank you very much for your reply.
> >
> > I have a few follow-up questions that you might be able to help me
> > with. First though I thought it might be helpful if I gave a quick
> > synopsis of my research question.
> >
> > I am investigating the determinants of capital structure in UK Plcs,
> > and my main hypothesis is that the theories espoused in the extant
> > literature are only applicable to certain types of firms.
> > As such, I divide my sample into sub-samples based on certain firm
> > characteristics i.e. size, tangibility of assets etc., and compare
> > regressor coefficients across the sub-samples.
>
> I'm not sure I understand. Do you estimate separately for
> the different subsamples, or do you interact your
> coefficients with indicator variables and estimate one big regression?
>
> I estimate separately for the different sub-samples.
I'm still confused. How then do you test that the coefficients are
different in the different estimations? This is possible with ivreg2
and xtivreg2 but it requires some gymnastics. See e.g. Austin Nichols'
post here:
http://www.stata.com/statalist/archive/2009-11/msg01485.html
> I
> decided to take this approach as I am interested in how the
> effects of a number of the independent variables vary across
> the sub-samples, and was advised that indicator variables can
> only be employed for one variable at a time in a model.
> Furthermore, it was pointed out to me that a binary indicator
> variable is no longer binary after a fixed effects
> transformation i.e. indicator variables coded as 1 or 0 can
> take the values -1, 0, 1 after a first differences
> transformation, and can take T values after an about-the-mean
> transformation.
I don't get this at all. The fixed effects transformation is equivalent
to including a full set of dummies and then partialling them out. There
is nothing special about partialling out dummies - the
Frisch-Waugh-Lovell theorem says that you can partial out anything in a
regression and get the same results. If your argument about indicator
variables and fixed effects were correct, it would apply equally to any
regressors used alongside an indicator variable - partial out the
regressor and the indicator variable is no longer 0/1. So what?
> > However, I was initially worried that such a categorisation
procedure
> > might introduce endogeneity issues that might vary across
sub-samples,
> > and thus I would not be able to reliably compare coefficients across
> > sub-samples. Hence I decided to employ instrumental variables
(lagged
> > independent variables) to over come such issues. Within each
> > sub-sample I test the orthogonality assumption of my included
> > regressors (on an individual basis) using the orthog option in
> > xtivreg2. Any variables I find to be potentially endogenous (C-stat
> > p-value
> > <0.100) are then instrumented where instruments are available.
> > I am currently unaware of any method to correctly test the i.i.d.
> > assumption using xtivreg2, and so I have decided to drop the
> > assumption, and hence my question with regards the KP stat.
> >
> > With regards to your earlier reply, the following are some
> follow up
> > questions I still have.
> >
> > 1. Is there an option in ivreg2 to test the i.i.d.
> > assumption, and if not, how would i go about testing same?
>
> This amounts to testing for heteroskedasticity or autocorrelation.
> -ivhettest- and -ivactest- will report such tests for IV
> models. But you are using a fixed effects model, which
> complicates things a bit.
> How long is your T dimension? I see from the estimation
> below that you are using a kernel-robust VCE, which implies T
> is biggish. If so, you could apply the fixed effects
> transformation to your data by hand (e.g., using Ben Jann's
> -center- command) and then use these programs. But this is a
> bit tricky.
>
> The simplest way to test the i.i.d. assumption is to do an
> eyeball version of a White-type test. Estimate the model
> using kernel-robust VCEs, and then again without this option,
> i.e., using the classical VCE.
> Do the SEs look very different? If so, it's likely that the i.i.d.
> assumption would fail if you tested formally using a
> White-type test, since the same principle is involved - the
> test stat is based on a vector of contrasts between the
> robust and classical VCEs.
>
> I am using an unbalanced panel dataset, so my T dimension
> varies from 1 to 20. My understanding of the kernel-robust
> option is very limited, and I specify it so that my output is
> robust to autocorrelation.
I think you have a problem, but one that is perhaps easily addressed.
The kernel-robust covariance estimator requires T to go to infinity for
the asymptotics to work. 20 isn't very far on the way to infinity, and
1 is obviously not even getting started. Since you have a large
cross-section, you can instead use the cluster-robust covariance
estimator, i.e., cluster on the panel identifier.
> I think I will try your "eyeball" test suggestion, as I have
> about reached the limit of my econometric abilities! Thus, if
> I "see" major differences in the SEs the i.i.d assumption is invalid?
Yes, that's right.
> > 2. With regards to the Anderson-Rubin statistic and the Stock-Wright
> > LM S statistic, both of which are reported by xtivreg2, am I correct
> > in my interpretation that given that they both test the joint
> > hypotheses of weak instruments and orthogonality, the statistics are
> > only interpretable from a weak instruments perspective as long as
the
> > Hansen J test of all excluded instruments indicates orthogonality
> > conditions are valid?
>
> Sort of ... it's a litte more complicated than that. I
> recommend reading the Finlay-Magnusson paper on this.
>
> > 3.Included below is the first stage regression results from
> one of the
>
> > tests I run.
>
> Maybe I am misreading the output, but it looks like only the
> summary stats for the first stage are reported.
>
> Yes, I only included the summary first stage regression
> results. Below is the complete output produced by STATA.
>
> > As you can see the Cragg Donald and
> > Kleibergen Paap stats both suggest that the instruments are not
weak.
> > However, the AR and SW stats suggest that the instruments, given
that
> > the Hansen J-test does not reject the null, are potentially weak.
>
> No, that's a misintepretation of the AR and SW tests. See below.
>
> > From the output these stats
> > appear to me to be testing the explanatory power of the instrument
> > rather than whether or not it is weak
>
> Neither. These are not tests of the strength or explanatory
> power of the IV. They are just what the output says: tests
> of the significance of the endogenous regressor.
>
> Your endogenous regressor is liq. In the main output, the
> coeff on liq is -.0085538, with a z-stat of -1.73 and a
> p-value of 0.084. That is, the Wald test stat for the null
> that the coeff on liq=0 has a p-value of 0.084.
>
> The A-R test stat (F version) for the same hypothesis, i.e.,
> B1=0, augmented by the additional hypothesis that the IVs are
> exogenous, has a p-value of 0.0607. Very similar.
>
> The A-R-type approach can be extended to generate
> weak-instrument-robust confidence intervals. That's what
> Finlay & Magnusson's -rivtest- will do for you.
>
> I think I now understand what the AR tests are reporting; the
> AR stat p-value (0.067) is interpreted in the same manner as
> the p-value for liq in the main output (0.084), but with the
> added orthogonality condition.
Right.
> And given that both p-values
> are very similar, I can infer with some degree of reliability
> that the instrument is not weak (that degree of reliability
> being dependent on the confidence intervals I can generate
> using Finlay and Magnusson's -rivtest-). Is that correct?
Not really. "Weakness" or otherwise of the instruments is not being
tested in the A-R test of the null that the coeff=0. What you can infer
is just what the test stat allows you to infer about the null.
Finlay and Magnusson have a discussion of the relationship between the
A-R stat and the J stat (which tests the orthgonality conditions) -
worth having a look.
That said, the A-R test does have the feature that the weaker the
instruments, the wider the confidence interval for the parameter of
interest; this is its main appeal (in my view, anyway). But you can see
that you can reject the null of a zero coeff for two reasons: (a) the
coefficient is hugely different from zero; (b) the coefficient is very
close to zero, but your estimation determines its value so precisely
that you can reject it being zero. In other words, even if your
instruments are weak (case a) you can still reject the null of a zero
coeff. And anyway, if you want to know whether or not your instruments
are correlated with your endogenous regressor, you can just test this
directly - it's the same thing as a test of underidentification.
--Mark
> Thanks again for your help
>
> James
>
>
> > i.e.
> >
> > Weak-instrument-robust inference
> > Tests of joint significance of endogenous regressors B1 in main
> > equation
> > Ho: B1=0 and orthogonality conditions are valid
> >
> > The coefficient significance level of the instrumented
> variable (liq)
> > is relatively low (p-value = 0.084), but the instrument does not
> > appear to be weak (based on CD and KP stats). However, I would
> > conclude that it potentially is weak based on the AR and SW stats.
> > Is my interpretation incorrect, and if so could you
> indicate how these
>
> > stats ought to be interpreted?
> >
> > I greatly appreciate any help you can offer
> >
> > Best regards
> >
> > James
> >
> . xtivreg2 ltdbv lnsale tang itang itangdum tax prof mtb
> capexsa ndts yr* (liq=tang1 itang1 mtb1 liq1) if lnsalesubs<1 & ta
> > ngsubs<1, fe robust bw(2) gmm2s first
> Warning - singleton groups detected. 91 observation(s) not used.
> Warning - collinearities detected
> Vars dropped: yr08
> FIXED EFFECTS ESTIMATION
> Number of groups = 449 Obs per
> group: min = 2
> avg = 6.7
> max = 19
> First-stage regressions
>
> First-stage regression of liq:
> FIXED EFFECTS ESTIMATION
> Number of groups = 449 Obs per
> group: min = 2
> avg = 6.7
> max = 19
> OLS estimation
>
> Estimates efficient for homoskedasticity only Statistics
> robust to heteroskedasticity and autocorrelation
> kernel=Bartlett; bandwidth=2 time variable (t): year group
> variable (i): firm
> Number of obs = 3021
> F( 31, 2541) = 8.82
> Prob > F = 0.0000
> Total (centered) SS = 6087.457806
> Centered R2 = 0.2732
> Total (uncentered) SS = 6087.457806
> Uncentered R2 = 0.2732
> Residual SS = 4424.113333 Root
> MSE = 1.32
>
> Robust
> liq Coef. Std. Err. t P>t [95% Conf. Interval]
> lnsale -.3992946 .1006038 -3.97 0.000 -.5965684
> -.2020207
> tang -6.503772 1.007147 -6.46 0.000 -8.478685 -4.528859
> itang -2.818454 .3907103 -7.21 0.000 -3.584597 -2.052311
> itangdum .003545 .1125097 0.03 0.975 -.217075
> .2241649
> tax .0972279 .1132478 0.86 0.391 -.1248395 .3192952
> prof .0405595 .0546733 0.74 0.458 -.0666492 .1477683
> mtb -.0525982 .0277353 -1.90 0.058 -.1069843 .0017878
> capexsa .8377125 .3265792 2.57 0.010 .197324
> 1.478101
> ndts -.0143917 .0282565 -0.51 0.611 -.0697998 .0410164
> yr90 1.155508 3.618686 0.32 0.750 -5.940366 8.251382
> yr91 -.2388175 .2513692 -0.95 0.342 -.7317268 .2540919
> yr92 -.3008198 .2453313 -1.23 0.220 -.7818894 .1802499
> yr93 -.1499197 .2490001 -0.60 0.547 -.6381835 .338344
> yr94 -.2144308 .2420701 -0.89 0.376 -.6891055 .2602439
> yr95 -.2142347 .2435146 -0.88 0.379 -.691742 .2632725
> yr96 -.0750504 .2473898 -0.30 0.762 -.5601566 .4100559
> yr97 -.0568015 .2405942 -0.24 0.813 -.5285822 .4149792
> yr98 -.2275228 .2263855 -1.01 0.315 -.6714416 .216396
> yr99 .065933 .2331514 0.28 0.777 -.3912531 .5231191
> yr00 .3334675 .2521301 1.32 0.186 -.1609339 .8278688
> yr01 -.0156419 .2300491 -0.07 0.946 -.4667446 .4354608
> yr02 .1622597 .2160337 0.75 0.453 -.2613603 .5858797
> yr03 .0200205 .2144716 0.09 0.926 -.4005365 .4405775
> yr04 .2405879 .219952 1.09 0.274 -.1907155 .6718912
> yr05 .1176199 .2308627 0.51 0.610 -.3350784 .5703182
> yr06 -.1331952 .2180932 -0.61 0.541 -.5608537 .2944633
> yr07 -.370854 .2144122 -1.73 0.084 -.7912944 .0495865
> tang1 2.766925 .7109139 3.89 0.000 1.372896 4.160955
> itang1 1.893136 .3687716 5.13 0.000 1.170012
> 2.616259
> mtb1 .1395775 .0310299 4.50 0.000 .078731 .200424
> liq1 .3000688 .0442671 6.78 0.000 .2132655 .3868721
> Included instruments: lnsale tang itang itangdum tax prof mtb
> capexsa ndts yr90
> yr91 yr92 yr93 yr94 yr95 yr96 yr97 yr98 yr99 yr00 yr01
> yr02 yr03 yr04 yr05 yr06 yr07 tang1 itang1 mtb1 liq1 F test
> of excluded instruments:
> F( 4, 2541) = 20.20
> Prob > F = 0.0000
> Angrist-Pischke multivariate F test of excluded instruments:
> F( 4, 2541) = 20.20
> Prob > F = 0.0000
>
> Summary results for first-stage regressions
>
> (Underid) (Weak id)
> Variable F( 4, 2541) P-val AP Chi-sq( 4) P-val AP
> F( 4, 2541)
> liq 20.20 0.0000 81.78 0.0000 20.20
> NB: first-stage test statistics heteroskedasticity and
> autocorrelation-robust Stock-Yogo weak ID test critical
> values for single endogenous regressor:
> 5% maximal IV relative bias 16.85
> 10% maximal IV relative bias 10.27
> 20% maximal IV relative bias 6.71
> 30% maximal IV relative bias 5.34
> 10% maximal IV size 24.58
> 15% maximal IV size 13.96
> 20% maximal IV size 10.26
> 25% maximal IV size 8.31
> Source: Stock-Yogo (2005). Reproduced by permission.
> NB: Critical values are for Cragg-Donald F statistic and
> i.i.d. errors.
>
> Underidentification test
> Ho: matrix of reduced form coefficients has rank=K1-1
> (underidentified)
> Ha: matrix has rank=K1 (identified)
> Kleibergen-Paap rk LM statistic Chi-sq(4)=58.30
> P-val=0.0000
>
> Weak identification test
> Ho: equation is weakly identified
> Cragg-Donald Wald F statistic
> 78.65
> Kleibergen-Paap Wald rk F statistic
> 20.20
> Stock-Yogo weak ID test critical values for K1=1 and L1=4:
> 5% maximal IV relative bias 16.85
> 10% maximal IV relative bias 10.27
> 20% maximal IV relative bias 6.71
> 30% maximal IV relative bias 5.34
> 10% maximal IV size 24.58
> 15% maximal IV size 13.96
> 20% maximal IV size 10.26
> 25% maximal IV size 8.31
> Source: Stock-Yogo (2005). Reproduced by permission.
> NB: Critical values are for Cragg-Donald F statistic and
> i.i.d. errors.
>
> Weak-instrument-robust inference
> Tests of joint significance of endogenous regressors B1 in
> main equation
> Ho: B1=0 and orthogonality conditions are valid
> Anderson-Rubin Wald test F(4,2541)= 2.26
> P-val=0.0607
> Anderson-Rubin Wald test Chi-sq(4)= 9.14
> P-val=0.0577
> Stock-Wright LM S statistic Chi-sq(4)= 9.22
> P-val=0.0557
> NB: Underidentification, weak identification and
> weak-identification-robust test statistics heteroskedasticity
> and autocorrelation-robust
>
> Number of observations N = 3021
> Number of regressors K = 28
> Number of endogenous regressors K1 = 1
> Number of instruments L = 31
> Number of excluded instruments L1 = 4
> 2-Step GMM estimation
>
> Estimates efficient for arbitrary heteroskedasticity and
> autocorrelation Statistics robust to heteroskedasticity and
> autocorrelation kernel=Bartlett; bandwidth=2 time variable
> (t): year group variable (i): firm
> Number of obs = 3021
> F( 28, 2544) = 3.02
> Prob > F = 0.0000
> Total (centered) SS = 21.06783592
> Centered R2 = 0.0261
> Total (uncentered) SS = 21.06783592
> Uncentered R2 = 0.0261
> Residual SS = 20.51803233 Root
> MSE = .08932
>
> Robust
> ltdbv Coef. Std. Err. z P>z [95% Conf. Interval]
> liq -.0085538 .0049465 -1.73 0.084 -.0182487 .0011411
> lnsale .0053743 .0052578 1.02 0.307 -.0049307
> .0156794
> tang .1170177 .0610377 1.92 0.055 -.0026139 .2366493
> itang .0557467 .0239463 2.33 0.020 .0088127 .1026806
> itangdum .0123551 .0065003 1.90 0.057 -.0003853
> .0250955
> tax -.0193497 .00924 -2.09 0.036 -.0374598 -.0012396
> prof .0025405 .0027681 0.92 0.359 -.0028849 .0079659
> mtb -.0019451 .0019992 -0.97 0.331 -.0058635 .0019733
> capexsa .0108254 .0087886 1.23 0.218 -.0064
> .0280507
> ndts -.0022495 .0032416 -0.69 0.488 -.008603 .004104
> yr90 -.0860865 .1693451 -0.51 0.611 -.4179968 .2458238
> yr91 -.0057954 .0156291 -0.37 0.711 -.036428 .0248371
> yr92 .0060493 .0148008 0.41 0.683 -.0229596 .0350583
> yr93 -.0066494 .0154936 -0.43 0.668 -.0370163 .0237174
> yr94 -.0038801 .0137634 -0.28 0.778 -.0308559 .0230956
> yr95 -.0021814 .0139629 -0.16 0.876 -.0295482 .0251854
> yr96 .007044 .0137418 0.51 0.608 -.0198895 .0339775
> yr97 .0119441 .0134385 0.89 0.374 -.0143949 .0382831
> yr98 .0069794 .013185 0.53 0.597 -.0188627 .0328216
> yr99 .0132963 .0125952 1.06 0.291 -.0113898 .0379825
> yr00 .0080221 .0119826 0.67 0.503 -.0154633 .0315074
> yr01 -.0000815 .0107388 -0.01 0.994 -.0211291 .0209661
> yr02 .0001449 .0106504 0.01 0.989 -.0207295 .0210193
> yr03 .0106314 .0115621 0.92 0.358 -.0120299 .0332926
> yr04 .0097052 .0102908 0.94 0.346 -.0104643 .0298748
> yr05 .0156916 .0108831 1.44 0.149 -.0056388 .0370221
> yr06 .0093837 .0108831 0.86 0.389 -.0119467 .0307142
> yr07 .005672 .0086985 0.65 0.514 -.0113768 .0227207
>
> Underidentification test (Kleibergen-Paap rk LM statistic):
> 58.301
> Chi-sq(4) P-val = 0.0000
>
> Weak identification test (Cragg-Donald Wald F statistic):
> 78.647
> (Kleibergen-Paap rk Wald F statistic):
> 20.198
> Stock-Yogo weak ID test critical values:
> 5% maximal IV relative bias 16.85
> 10% maximal IV relative bias 10.27
> 20% maximal IV relative bias 6.71
> 30% maximal IV relative bias 5.34
> 10% maximal IV size 24.58
> 15% maximal IV size 13.96
> 20% maximal IV size 10.26
> 25% maximal IV size 8.31
> Source: Stock-Yogo (2005). Reproduced by permission.
> NB: Critical values are for Cragg-Donald F statistic and
> i.i.d. errors.
> Hansen J statistic (overidentification test of all
> instruments): 5.596
>
> Chi-sq(3) P-val = 0.1330
> Instrumented: liq
> Included instruments: lnsale tang itang itangdum tax prof mtb
> capexsa ndts yr90
> yr91 yr92 yr93 yr94 yr95 yr96 yr97 yr98 yr99 yr00 yr01
> yr02 yr03 yr04 yr05 yr06 yr07
> Excluded instruments: tang1 itang1 mtb1 liq1
> Dropped collinear: yr08
>
> .
>
> >
> > ________________________________________
> > From: [email protected]
> > [[email protected]] on behalf of
> Schaffer, Mark E
> > [[email protected]]
> > Sent: 25 June 2012 12:33
> > To: [email protected]
> > Subject: st: RE: Interpreting Kleibergen Paap weak instrument
> > statistic
> >
> > James,
> >
> > > -----Original Message-----
> > > From: [email protected]
> > > [mailto:[email protected]] On Behalf Of
> > > Fitzgerald, James
> > > Sent: 21 June 2012 14:02
> > > To: [email protected]
> > > Subject: st: Interpreting Kleibergen Paap weak instrument
> statistic
> > >
> > > Hi Statalist users
> > >
> > > I am using xtivreg2 to estimate a GMM-IV model (I specify the
> > > following options; fe robust bw(2) gmm2s). I am not
> assuming i.i.d
> > > errors, and thus when testing for weak instruments I am using the
> > > Kleibergen Paap rk wald F statistic rather than the Cragg Donald
> > > wald F statistic.
> > >
> > > xtivreg2 produces Stock-Yogo critical values for the Cragg Donald
> > > statistic assuming i.i.d errors, so I'm not sure how to interpret
> > > the KP rk wald F stat.
> > >
> > > The help file for ivreg2 (Baum, Schaffer and Stillman, 2010) does
> > > however mention the following:
> > >
> > > When the i.i.d. assumption is dropped and ivreg2 is
> invoked with the
>
> > > robust, bw or cluster options, the Cragg-Donald-based weak
> > > instruments test is no longer valid.
> > > ivreg2 instead reports a correspondingly-robust
> Kleibergen-Paap Wald
>
> > > rk F statistic. The degrees of freedom adjustment for the rk
> > > statistic is (N-L)/L1, as with the Cragg-Donald F
> statistic, except
> > > in the cluster-robust case, when the adjustment is N/(N-1) *
> > > (N_clust-1)/N_clust, following the standard Stata small-sample
> > > adjustment for cluster-robust. In the case of two-way clustering,
> > > N_clust is the minimum of N_clust1 and N_clust2. The critical
> > > values reported by ivreg2 for the Kleibergen-Paap
> statistic are the
> > > Stock-Yogo critical values for the Cragg-Donald i.i.d. case.
> > > The critical values reported with 2-step GMM are the
> Stock-Yogo IV
> > > critical values, and the critical values reported with
> CUE are the
> > > LIML critical values.
> > >
> > >
> > > My understanding of the end of the paragraph is that the
> KP stat can
>
> > > still be compared to the Stock-Yogo values produced by STATA in
> > > determining whether or not instruments are weak.
> > >
> > > If someone could confirm or reject this I would be eternally
> > > grateful!!
> >
> > I wrote that paragraph, so the ambiguity is partly my
> fault. But the
> > problem is that there are no concrete results in the literature for
> > testing for weak IVs when the i.i.d. assumption fails. The
> only thing
>
> > one can do (that I'm aware of, anyway) is to point to stats
> that have
> > an asymptotic justification in a test of
> underidentification, which is
>
> > what the output of -ivreg2- does. That is, the K-P stat
> can be used
> > to test for underidentification without the i.i.d. assumption, and
> > under i.i.d.
> > it has the same distribution under the null as the
> Cragg-Donald stat.
> > This justification is different from that underlying the Stock-Yogo
> > critical values, so this is pretty hand-wavey.
> >
> > The alternative is weak-instrument-robust estimation, a la
> > Anderson-Rubin, Moreira, Kleibergen, etc. The Finlay-Magnusson
> > -rivtest- command, available via ssc ideas in the usual
> way, supports
> > this. Also see their accompanying SJ paper (vol. 9 no. 3).
> > The command
> > doesn't directly support panel data estimation, which is what you
> > have, but you could just demean your variables by hand.
> >
> > HTH,
> > Mark
> >
> >
> > > Best wishes
> > >
> > > James Fitzgerald
> > > *
> > > * For searches and help try:
> > > * http://www.stata.com/help.cgi?search
> > > * http://www.stata.com/support/statalist/faq
> > > * http://www.ats.ucla.edu/stat/stata/
>
>
> --
> Heriot-Watt University is the Sunday Times Scottish
> University of the Year 2011-2012
>
> Heriot-Watt University is a Scottish charity registered under
> charity number SC000278.
>
>
> *
> * For searches and help try:
> * http://www.stata.com/help.cgi?search
> * http://www.stata.com/support/statalist/faq
> * http://www.ats.ucla.edu/stat/stata/
> *
> * For searches and help try:
> * http://www.stata.com/help.cgi?search
> * http://www.stata.com/support/statalist/faq
> * http://www.ats.ucla.edu/stat/stata/
>
--
Heriot-Watt University is the Sunday Times
Scottish University of the Year 2011-2012
Heriot-Watt University is a Scottish charity
registered under charity number SC000278.
*
* For searches and help try:
* http://www.stata.com/help.cgi?search
* http://www.stata.com/support/statalist/faq
* http://www.ats.ucla.edu/stat/stata/