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

[Austin Nichols <austinnichols@gmail.com>]

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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