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Re: st: Using ivregress when the endogenous variable is used in an interaction term in the main regression


From   Tirthankar Chakravarty <tirthankar.chakravarty@gmail.com>
To   statalist@hsphsun2.harvard.edu
Subject   Re: st: Using ivregress when the endogenous variable is used in an interaction term in the main regression
Date   Wed, 21 Dec 2011 10:30:56 -0800

Nick,

I don't have a specific reference in mind, but I suppose you should be
able to construct a workable explanation from Prof. Wooldridge's reply
(indeed, you can probably directly cite it) and some directed
googling.

T

On Wed, Dec 21, 2011 at 10:01 AM, Nick Kohn <coffeemug.nick@gmail.com> wrote:
> My apologies for spamming but I also wanted to mention that I'm trying
> out the specification that includes the endogenous variables as stand
> alone terms.
>
> I'm not sure whether I'll use it in my paper though because I'll need
> to provide a justification of why I deviate from the paper I cite, and
> going into long winded econometric arguments is beyond the scope of
> what I'm doing.
>
> Is there a paper or book I can cite that explains why adding the
> levels is appropriate?
>
> On Wed, Dec 21, 2011 at 6:58 PM, Nick Kohn <coffeemug.nick@gmail.com> wrote:
>> Sorry for the confusion - X1 is included as a stand alone term.
>>
>> To be more detailed, my model looks like this (X is exogenous, E is endogenous):
>>
>> dY = X1 + X2
>>     + X1*X3
>>     + X1*X3*E1
>>     + X1*X3*E2
>>     + X1*X3*E3
>>     + controls
>>
>> X3 is an indicator variable that is equal to 1 when X1 <= 0
>>
>> On Wed, Dec 21, 2011 at 6:44 PM, Austin Nichols <austinnichols@gmail.com> wrote:
>>> Tirthankar Chakravarty <tirthankar.chakravarty@gmail.com>:
>>> I don't see anywhere that the X1 is included as a main effect as
>>> opposed to just being included in the product X1*X2.  (Though it is
>>> not clear what is included in "+controls" in the post.) It seems that
>>> X1 is exogenous by assumption, i.e. X1 is uncorrelated with e while X2
>>> is correlated with e. There are no quadratic terms in Z in my
>>> suggestion. Note that you suggested instrumenting with X2hat*X1 and
>>> X2hat is linear in Z.
>>>
>>> On Wed, Dec 21, 2011 at 12:15 PM, Tirthankar Chakravarty
>>> <tirthankar.chakravarty@gmail.com> wrote:
>>>> " It does not seem too much of a stretch to assume Z*X1
>>>> uncorrelated with e as well (which implies X2hat*X1 uncorrelated with
>>>> e)"
>>>>
>>>> This part is the problem. When you form cross-products of the
>>>> instrument matrix, you will end up with quadratic terms in Z, coming
>>>> from terms like the one you mention, which will need to be
>>>> uncorrelated with the structural errors, hence the independence
>>>> requirement.
>>>>
>>>> Again, note that X1 is included so there is no overidentification (or,
>>>> at best, the same degree of overidentification as without the
>>>> interaction term).
>>>>
>>>> T
>>>>
>>>> On Wed, Dec 21, 2011 at 8:57 AM, Austin Nichols <austinnichols@gmail.com> wrote:
>>>>> Tirthankar Chakravarty <tirthankar.chakravarty@gmail.com>:
>>>>> No conditional independence assumed, though of course an independence
>>>>> assumption lets you form all kinds of transformations of Z to use as
>>>>> excluded instruments.
>>>>>
>>>>> We need Z, Z*X1, and X1 uncorrelated with e, but Z and e were already
>>>>> assumed uncorrelated and X1 is exogenous by assumption as well, in the
>>>>> original post.  It does not seem too much of a stretch to assume Z*X1
>>>>> uncorrelated with e as well (which implies X2hat*X1 uncorrelated with
>>>>> e), but if we use all 3 as instruments we will see evidence of any
>>>>> violations of assumptions in the overid test (assuming no weak
>>>>> instruments problem).
>>>>>
>>>>> On Wed, Dec 21, 2011 at 11:44 AM, Tirthankar Chakravarty
>>>>> <tirthankar.chakravarty@gmail.com> wrote:
>>>>>> Austin,
>>>>>>
>>>>>> I agree re: well-cited papers.
>>>>>>
>>>>>> Note that the efficiency you mention comes at a cost. As I pointed out
>>>>>> in my previous Statalist reply:
>>>>>> http://www.stata.com/statalist/archive/2011-08/msg01496.html
>>>>>> the instrumenting strategy you suggest requires the instruments to be
>>>>>> conditionally independent rather than just uncorrelated with the
>>>>>> structural errors.
>>>>>>
>>>>>> T
>>>>>>
>>>>>> On Wed, Dec 21, 2011 at 7:57 AM, Austin Nichols <austinnichols@gmail.com> wrote:
>>>>>>> Nick Kohn <coffeemug.nick@gmail.com>:
>>>>>>> Or better, instrument for X1*X2 using Z, Z*X1, and X1.
>>>>>>> For maximal efficiency given your assumptions you may prefer
>>>>>>> to instrument for X1*X2 using Z*X1, or even
>>>>>>> to instrument for X1*X2 using X2hat*X1,
>>>>>>> but you should build in an overid test whenever feasible.
>>>>>>>
>>>>>>> Just because a well-cited paper does something wrong does not mean you
>>>>>>> have to, though.
>>>>>>>
>>>>>>> Including the main effects of X1 and X2 makes for harder interpretation, but
>>>>>>> will make you a lot more confident of your answers once you have worked out the
>>>>>>> interpretation.
>>>>>>>
>>>>>>> On Wed, Dec 21, 2011 at 9:20 AM, Tirthankar Chakravarty
>>>>>>> <tirthankar.chakravarty@gmail.com> wrote:
>>>>>>>> In that case, none of this is necessary. Just instrument for X1*X2
>>>>>>>> using Z. All standard results apply.
>>>>>>>>
>>>>>>>> T
>>>>>>>>
>>>>>>>> On Wed, Dec 21, 2011 at 6:03 AM, Nick Kohn <coffeemug.nick@gmail.com> wrote:
>>>>>>>>> Hmmm I see what you mean, but I'm following the methodology of a well
>>>>>>>>> cited paper that does the same thing.
>>>>>>>>>
>>>>>>>>> I'll be sure to discuss this limitation, but in terms of using this
>>>>>>>>> model, would the 3 steps in my last message be correct?
>>>>>>>>>
>>>>>>>>> On Wed, Dec 21, 2011 at 2:56 PM, Tirthankar Chakravarty
>>>>>>>>> <tirthankar.chakravarty@gmail.com> wrote:
>>>>>>>>>> I wanted to indirectly confirm that you did have the main effect in
>>>>>>>>>> the regression because even though I don't know the nature of your
>>>>>>>>>> study, a hard-to-defend methodological position arises when you
>>>>>>>>>> include interaction terms without including the main effect. You might
>>>>>>>>>> want to take that on the authority of someone who (literally) wrote
>>>>>>>>>> the book on the subject:
>>>>>>>>>>
>>>>>>>>>> http://www.stata.com/statalist/archive/2011-03/msg00188.html
>>>>>>>>>>
>>>>>>>>>> and reconsider your decision to not include the main effect.
>>>>>>>>>>
>>>>>>>>>> T
>>>>>>>>>>
>>>>>>>>>> On Wed, Dec 21, 2011 at 5:46 AM, Nick Kohn <coffeemug.nick@gmail.com> wrote:
>>>>>>>>>>> My model doesn't have X2 as a separate term, so in terms of the model
>>>>>>>>>>> you had it looks like:
>>>>>>>>>>>  Y = b*X1*X2 + controls
>>>>>>>>>>> So the only place the endogenous variable comes up is the interaction term
>>>>>>>>>>>
>>>>>>>>>>> At the risk of being repetitive, would these be the correct steps (so
>>>>>>>>>>> essentially only step 3 changes from what you said):
>>>>>>>>>>> 1) regress X2 on all instruments, exogenous variables and controls
>>>>>>>>>>> 2) Form interactions of X2hat with the exogenous variable X1, that is, X2hat*X1
>>>>>>>>>>> 3) ivregress instrumenting for X2*X1 using X2hat*X1.
>>>>>>>>>>>
>>>>>>>>>>> On Wed, Dec 21, 2011 at 1:44 PM, Tirthankar Chakravarty
>>>>>>>>>>> <tirthankar.chakravarty@gmail.com> wrote:
>>>>>>>>>>>> Not quite; here is the recommended procedure (I am assuming that you
>>>>>>>>>>>> have the main effect of the endogenous variable in there as in Y =
>>>>>>>>>>>> a*X2 + b*X1*X2 + controls):
>>>>>>>>>>>>
>>>>>>>>>>>> 1) -regress- X2 on _all_ instruments (included exogenous controls and
>>>>>>>>>>>> excluded instruments) and get predictions X2hat.
>>>>>>>>>>>>
>>>>>>>>>>>> 2) Form interactions of X2hat with the exogenous variable X1, that is, X2hat*X1.
>>>>>>>>>>>>
>>>>>>>>>>>> 3) -ivregress- instrumenting for X2 and X2*X1 using X2hat and X2hat*X1.
>>>>>>>>>>>>
>>>>>>>>>>>> Note that there is distinction between two calls to -regress- and
>>>>>>>>>>>> using -ivregress- for 3).
>>>>>>>>>>>>
>>>>>>>>>>>> T
>>>>>>>>>>>>
>>>>>>>>>>>> On Wed, Dec 21, 2011 at 3:43 AM, Nick Kohn <coffeemug.nick@gmail.com> wrote:
>>>>>>>>>>>>> Thanks for the reply.
>>>>>>>>>>>>>
>>>>>>>>>>>>> My simplified model is (X2 is endogenous):
>>>>>>>>>>>>> Y = b*X1*X2 + controls
>>>>>>>>>>>>>
>>>>>>>>>>>>> In regards to the third option you suggest, would I do the following?
>>>>>>>>>>>>>
>>>>>>>>>>>>>  1) First stage regression to get X2hat using the instrument Z
>>>>>>>>>>>>>  2) Run the first stage again but use X1*X2hat as the instrument for
>>>>>>>>>>>>> X1*X2 (so Z is no longer used)
>>>>>>>>>>>>>  3) Run the second stage using (X1*X2)hat (so the whole product is
>>>>>>>>>>>>> fitted from step 2))
>>>>>>>>>>>>>
>>>>>>>>>>>>> On Wed, Dec 21, 2011 at 12:24 PM, Tirthankar Chakravarty
>>>>>>>>>>>>> <tirthankar.chakravarty@gmail.com> wrote:
>>>>>>>>>>>>>> You can see my previous reply to a similar question here:
>>>>>>>>>>>>>> http://www.stata.com/statalist/archive/2011-08/msg01496.html
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> T
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> On Wed, Dec 21, 2011 at 2:24 AM, Nick Kohn <coffeemug.nick@gmail.com> wrote:
>>>>>>>>>>>>>>> Hi,
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>> I have a specification in which the endogenous variable is interacted
>>>>>>>>>>>>>>> with an exogenous variable. Since I cannot multiply the variables
>>>>>>>>>>>>>>> directly in the regression, I create a new variable. In ivregress it
>>>>>>>>>>>>>>> makes no sense to use the entire interaction term as the endogenous
>>>>>>>>>>>>>>> variable.
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>> I can do the first stage manually (and then use the fitted value in
>>>>>>>>>>>>>>> the main regression), however, from what I remember the standard
>>>>>>>>>>>>>>> errors will be wrong when doing it manually.
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>> Is there a way to overcome this?
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>> Thanks
>>>
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-- 
Tirthankar Chakravarty
tchakravarty@ucsd.edu
tirthankar.chakravarty@gmail.com

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