RE: st: RE: Testing for instrument relevance and overidentification when the endogeneous variable is used in interaction terms

["Schaffer, Mark E" <M.E.Schaffer@hw.ac.uk>]

Jason,

> -----Original Message-----
> From: owner-statalist@hsphsun2.harvard.edu [mailto:owner-
> statalist@hsphsun2.harvard.edu] On Behalf Of Jason Wichert
> Sent: 05 June 2013 21:49
> To: statalist@hsphsun2.harvard.edu
> Subject: Re: st: RE: Testing for instrument relevance and overidentification
> when the endogeneous variable is used in interaction terms
> 
> Alright, now here s some more issues I have encountered.
> 
> Using just one endogenous variable “en” in the model
> 
> [1] ivreg2 y ex1 ex2 controls (en = z1 z2),
> 
> the respective test statistics are just fine. However, when also incorporating
> interaction terms of the kind
> 
> [2] ivreg2 y ex1 ex2 control (en en_ex1 en_ex2 = z1 z2 z1_ex1 z1_ex2
> z2_ex1 z2_ex2)
> 
> as well as quadratic interaction terms, I’m having issues with the test results.
> In particular:
> 
> - Stock/Yogo (2005) have calculated critical values for the Cragg-Donald (1993)
> F-statistic only for up to three endogenous variables. While the critical values
> provided don’t differ too much among 1, 2, and 3 endogenous variables and
> such references might be eyeballed, does anyone know about exact critical
> values in the case of more than three endogenous regressors?

I don't think they've been compiled.  But no one should mind if you are a bit hand-wavey in your writeup at this point.  Rough magnitudes are still informative.

> - as further regards the Cragg-Donald (1993) F-statistic to test for weak
> identification, I notice an implosion of the F-statistic from model [1] to model
> [2].

Do you mean it gets very small, so the model becomes weakly identified?

> Since the null of C-D states that the instruments are *jointly* only weakly
> correlated with the endogenous regressors, I naively assume the small F-
> statistic results from the 2SLS procedure in my case, since many of the
> instruments are strongly correlated to the endogenous variables, i.e. the
> interaction terms, by construction (e.g. z1_ex1 is highly correlated to
> en_ex1). Could somebody confirm this?

Not sure what you mean, to be honest.  My gut feeling is that you are expecting a lot of your instruments for them to be correlated with the endogenous regressors not just in levels but also via interactions.

> - on a side note, the Kleibergen-Paap (2006) statistic of underidentification
> does just fine in each model.
> 
> - a similar concerns regards the Sargan/Hansen statistic of overidentification,
> which tests whether *any* of the instruments fail the orthogonality
> criterion. Since I know ex1 and ex2 are highly correlated to y, so should the
> constructed instruments z1_ex1, z1_ex2, etc., right? Therefore, I naively
> interpret the exploding Sargan-statistic (from failure to reject the null of
> overidentification with a p-value of around 0.7 to complete rejection at 0.00)
> as a mere by-product of my model specification, correct?

Again, not sure what you mean.  The only thing I can suggest here is that perhaps you can work out which of the orthogonality conditions you're violating.  One of the ways to think about a failure of the Sargan-Hansen test is that your instruments are identifying "different betas", in the same way that a Hausman test gives you a big test stat when the two estimated betas are very different.  It might be worth comparing the results using the full set of IVs based on z1 and z2 and their interactions, the results using just z1 and its interactions, and the results using just z2 and its interactions.

> If my naïve assumptions were true, would it allow for a stricter testing
> procedure to use a different approach to the model specification?
> 
> In particular, instead of the setup as indicated by [2], I might be tempted to
> try a different approach, such as regressing “en” on all instruments (included
> exogenous controls as well as excluded
> instruments) to get predictions enhat and then forming interactions
> enhat_ex1, enhat_ex2, enhat_(ex1)^2, enhat_(ex2)^2, taking into account
> the incorrect standard errors. Would that seem likely to help?

This sounds an awful lot like the "Forbidden Regression".  (And the name pretty much tells you how this is going to pan out.)  If you google that term you'll find it very quickly, or if you have Angrist and Pischke's "Mostly Harmless Econometrics" it's covered in there.

--Mark

> 
> Again, thanks much in advance for anyone (putting my hopes on Mark
> here) providing useful advice!
> 
> 
> On Tue, Jun 4, 2013 at 8:49 PM, Jason Wichert <jasonw8907@gmail.com>
> wrote:
> > Mark,
> >
> > Again, thank you so much for your feedback.
> >
> > As regards the endogeneity tests, I’m actually using the endog option
> > in ivreg2. On a side note, thank you guys for this excellent tool and
> > the detailed explanations in the articles/versions of 2003 and 2007.
> >
> > As regards your fourth point, concerning additional interactions: in
> > my case, ex1 and ex2 are distinct constructs, measures of good (ex1)
> > and poor (ex2) company performance in a certain sense, similar to
> > Herzberg’s two-factor theory of motivators and hygiene factors. They
> > both have (different) non-linear associations to my measure of
> > financial performance (y), which luckily has been largely established
> > in empirical research, as have the influences of both ex1 and ex2 on
> > the endogenous variable. Less established so far, however, is the
> > moderating effect of my endogenous variable on either link between
> > ex1/ex2 and y. This moderating effect (at least according to my humble
> > analyses), which my research focuses on, differs between the levels of
> > ex1 and ex2, as indicated by the significant interaction terms of
> > different sign between say ex1_en and (ex1)^2_en; hence my
> > interactions of the linear and quadratic terms of both ex1 and ex2
> > with en. Leaving ex2, en and controls aside, my results indicate
> >
> > y = 0.363 ex1 – 0.032 (ex1)^2 – 0.007 ex1_en + 0.001 (ex1)^2_en
> >
> > With all coefficients highly significant, I interpret these results as
> > decreasing marginal returns to ex1 or an inverted U-shaped
> > relationship between ex1 and y with the inflection point in the first
> > quadrant. While the moderating effect of en on ex1 is largely negative
> > (as indicated by the negative coefficient on ex1_en), this negative
> > effect is attenuated for high levels of ex1 (as indicated by the
> > positive coefficient on (ex1)^2_en). Unrelated to my initial
> > questions, does this interpretation seem to make sense?
> >
> > In various preliminary analyses, luckily(?!) I did not find any
> > non-linear associations between en and y, at the very least saving me
> > additional nightmares of the econometric and economic kinds.
> >
> > Kind regards,
> >
> > Jason
> >
> > On Tue, Jun 4, 2013 at 8:07 PM, Schaffer, Mark E <M.E.Schaffer@hw.ac.uk>
> wrote:
> >> Jason,
> >>
> >>> -----Original Message-----
> >>> From: owner-statalist@hsphsun2.harvard.edu [mailto:owner-
> >>> statalist@hsphsun2.harvard.edu] On Behalf Of Jason Wichert
> >>> Sent: 04 June 2013 12:47
> >>> To: statalist@hsphsun2.harvard.edu
> >>> Subject: Re: st: RE: Testing for instrument relevance and
> >>> overidentification when the endogeneous variable is used in
> >>> interaction terms
> >>>
> >>> Mark,
> >>>
> >>> Thank you very much for your feedback (and all the other excellent
> >>> comments on 2SLS you made on statalist). It's not the usual
> >>> regression 101, so it actually took me a couple of days to work
> >>> through all the respective IV statistics, hence my late reply.
> >>>
> >>> My analyses start with just the one endogenous regressor and are
> >>> subsequently extended to incorporate the endogenous interaction
> terms.
> >>>
> >>> In the base case of just that one endogenous variable, i.e.
> >>>
> >>>  ivreg2 y ex1 ex2 (en = z1 z2)
> >>>
> >>> I intend to present the first stage F-statistics (to reject weak
> >>> identification of my endogenous variable), results from the
> >>> Sargan/Hansen overidentification test (to test whether the
> >>> instruments are jointly exogenous), as well as a partial R² (to
> >>> assess instrument relevance), and a Hausman test for endogeneity.
> >>
> >> That sounds fine.  Two minor suggestions:
> >>
> >> 1.  The first-stage F stat makes the partial R-sq redundant.  No need to
> report it or anything like it in the case of a single endogenous regressor.
> >>
> >> 2.  You can get ivreg2 to report an endogeneity test for you by using the
> endog option.
> >>
> >>>
> >>> In the extended case of (*gasp*)
> >>>
> >>>  ivreg2 y ex1 ex2 (ex1)^2 (ex2)^2 (en ex1_en ex2_en (ex1)^2_en
> >>> (ex2)^2_en = z1 z2 ex1_z1 ex1_z2 (ex1)^2_z1 (ex1)^2_z2 ex2_z1 ex2_z2
> >>> (ex2)^2_z1 (ex2)^2_z2)
> >>>
> >>> I intend to present results from the Sargan/Hansen
> >>> overidentification test, results from the Anderson/Rubin (1949) [or
> >>> potentially Stock/Wright (2000)] test to indicate that all the
> >>> endogenous regressors are jointly significant in the second stage,
> >>> the Kleibergen/Paap (2006) statistic of underidentification of the
> >>> model (i.e. the joint endogenous regressors) , the Cragg/Donald
> >>> (1993) statistic of weak identification of the model, the
> >>> Angrist/Pischke
> >>> (2009) statistics for identification of each of the endogenous
> >>> regressors, as well as a Hausman test for endogeneity.
> >>>
> >>> Is there something blatant obvious I’m missing or anything I could
> >>> well leave out, particularly in the extended case? In particular I’m
> >>> wondering about
> >>>
> >>> a) the necessity of the A/R-test, considering most all of my
> >>> endogenous variables are highly significant in the second stage, as
> >>> indicated by their respective t- and p-values,
> >>>
> >>> b) the necessity of presenting both K/P as well as C/D statistics,
> >>>
> >>> c) the necessity of the Hausman test in the extended case.
> >>>
> >>> Again, thank you very much in advance for your feedback!
> >>
> >> Let's see...
> >>
> >> 1.  The K-P test for underidentification is reported by ivreg2 mostly for
> completeness.  If you reject weak identification based on C-D, you are also
> rejecting underidentification.  So you could omit K-P.
> >>
> >> 2.  Most people probably wouldn't report the A-R test unless there were
> signs of weak identification (in which case they might consider using weak-
> instrument-robust methods, e.g., rivtest).
> >>
> >> 3.  On Hausman, sample point above applies - you can get ivreg2 to report
> the endog test by using the endog option.  Maybe you have priors about
> whether one or a subset of your endogenous regressors should be tested
> rather than the whole lot at once.
> >>
> >> 4.  You didn't ask about this but worth mentioning anyway - when people
> introduce quadratics in the way you are doing, they often include the
> interactions.  In your case that means the interaction of ex1 and ex2 and
> similarly for the other regressors and instruments (and if you were really
> serious about it, you'd probably interact the endogenous and exogenous
> regressors too).  The slightly hand-wavey justification would be a Taylor
> approximation.
> >>
> >> HTH,
> >> Mark
> >>
> >>>
> >>> On Fri, May 31, 2013 at 8:30 PM, Schaffer, Mark E
> >>> <M.E.Schaffer@hw.ac.uk>
> >>> wrote:
> >>> > Jason,
> >>> >
> >>> > I think the key point is that in your estimation
> >>> >
> >>> > ivreg2 y ex (en en_ex = z ex_z)
> >>> >
> >>> > just looking at the two standard first-stage F stats isn't enough.
> >>> > You can
> >>> easily get 2 large first-stage F stats, and yet the model is
> >>> underidentified because there isn't enough information in your
> >>> instruments to simultaneously identify the coeffs on both your
> endogenous regressors.
> >>> >
> >>> > To see if both coeffs are identified, you should use either the
> >>> > weak- or the
> >>> under-identification statistic reported by ivreg2.  You can also use
> >>> the Angrist-Pischke (A-P) first-stage F stats to see whether one or
> >>> the other coeffs is identified.  More details about these in the
> >>> ivreg2 help file and the references therein.
> >>> >
> >>> > HTH,
> >>> > Mark
> >>> >
> >>> >> -----Original Message-----
> >>> >> From: owner-statalist@hsphsun2.harvard.edu [mailto:owner-
> >>> >> statalist@hsphsun2.harvard.edu] On Behalf Of Jason Wichert
> >>> >> Sent: 29 May 2013 21:18
> >>> >> To: statalist@hsphsun2.harvard.edu
> >>> >> Subject: st: Testing for instrument relevance and
> >>> >> overidentification when the endogeneous variable is used in
> >>> >> interaction terms
> >>> >>
> >>> >> Dear Statalisters,
> >>> >>
> >>> >> I have encountered some difficulties concerning 2SLS estimation
> >>> >> when the endogeneous variable is also used to construct interaction
> terms.
> >>> >>
> >>> >> After digging through the archives, I found a lot of helpful
> >>> >> comments concerning the procedure:
> >>> >>
> >>> >> http://www.stata.com/statalist/archive/2012-05/msg00970.html
> >>> >> http://www.stata.com/statalist/archive/2011-08/msg01485.html
> >>> >> http://www.stata.com/statalist/archive/2011-12/msg00705.html
> >>> >> http://www.stata.com/statalist/archive/2010-04/msg00759.html
> >>> >> http://www.stata.com/statalist/archive/2005-05/msg00150.html
> >>> >> http://www.stata.com/statalist/archive/2008-10/msg01009.html
> >>> >> http://www.stata.com/statalist/archive/2004-08/msg00779.html
> >>> >>
> >>> >> Following this advice, I am running an equation of the basic form
> >>> >>
> >>> >> ivreg2 y ex (en en_ex = z ex_z)
> >>> >>
> >>> >> In my case, there are two exogeneous variables interacted with
> >>> >> the endogeneous variable. Furthermore, I need interactions of
> >>> >> those squared exogeneous variables and the endogeneous variables.
> >>> >> Leaving additional control variables and further instruments
> >>> >> aside, this already leads to the following simplified regression:
> >>> >>
> >>> >> ivreg2 y ex1 ex2 (en ex1_en ex2_en (ex1)^2_en (ex2)^2_en = z
> >>> >> ex1_z ex2_z (ex1)^2_z (ex2)^2_z)
> >>> >>
> >>> >> So far, so good. However, I’m not sure as to how exactly examine
> >>> >> instrument relevance and exogeneity, and which statistics/tests to
> report.
> >>> >>
> >>> >> As regards instrument relevance, as to be assessed by the first
> >>> >> stage F statistic, the F-statistics on “en” clearly differ
> >>> >> depending on whether I instrument solely for “en”, or whether I
> >>> >> also instrument for the linear and non-linear interaction terms built
> around “en”.
> >>> >> Which F statistic is the correct one to refer to?
> >>> >>
> >>> >> Considering I have multiple instruments Z, I am also not sure
> >>> >> which overidentification tests and results I should rely on and
> >>> >> report.  As holds for the F statistics, the tests of
> >>> >> overidentifying restrictions (Sargan N*R-sq test as well as
> >>> >> Basmann test) provided by both ivreg2 and overid differ between
> >>> >> instrumenting solely “en” or also for the
> >>> interaction terms build around “en”.
> >>> >>
> >>> >> Any help is greatly appreciated!
> >>> >> Jason
> >>> >>
> >>> >> *
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> >>> >
> >>> >
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> >>
> >>
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----- 
Sunday Times Scottish University of the Year 2011-2013
Top in the UK for student experience
Fourth university in the UK and top in Scotland (National Student Survey 2012)


We invite research leaders and ambitious early career researchers to 
join us in leading and driving research in key inter-disciplinary themes. 
Please see www.hw.ac.uk/researchleaders for further information and how
to apply.

Heriot-Watt University is a Scottish charity
registered under charity number SC000278.


*
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